We investigate multipartite entanglement in quantum geometry states described by loop quantum gravity spin networks. We focus on states corresponding to bounded regions of 3D spacelike slices of spacetime with nonvanishing intrinsic curvature realized via spin networks defined on a graph with nontrivial S U 2  holonomies. The presence of intrinsic curvature in the region is encodedvia coarse graininginto tag spins attached to the vertices of the graph. The resulting states are generalized bulktoboundary maps defined in an extended boundary Hilbert space comprised by the spin representations carried by the tags and by the uncontracted edges at the boundary. We consider a tripartition of a random quantum geometry state in the extendedboundary space consisting of a bipartite boundary subsystem and a bulktags complement and we propose a measure of logarithmic negativity to study the change in the entanglement phases of the boundary marginal mixed state while varying the dimension of the bulk curvature environment. In the large spin regime we find that the typical entanglement negativity is well described by the generalized Page curve of a tripartite random state. In particular we find area scaling behavior of negativity for small curvature while for large curvature the negativity vanishes suggesting an effective thermalization of the boundary. Remarkably the positive partial transpose character of the mixed boundary state corresponds to a change in the effective topology of the network with the two boundary subregions becoming disconnected.
['High Energy Physics - Theory', 'General Relativity and Quantum Cosmology', 'Quantum Physics']
Curvature from multipartite entanglement in quantum gravity states
{'Curvature from multipartite entanglement in quantum gravity states': "Simone Cepollaro *  \nScuola Superiore Meridionale, Largo S. Marcellino 10, 80138 Napoli, Italy; INFN, Sezione di Napoli, Italy;  \nGoffredo Chirco, glyph[flat] Gianluca Cuffaro, § and Vittorio D'Esposito glyph[natural]  \nDipartimento di Fisica 'Ettore Pancini', Universit'a di Napoli Federico II, Napoli, Italy; INFN, Sezione di Napoli, Italy. (Dated: May 5, 2023)  \nWe investigate the multipartite entanglement of a uniformly curved quantum 3 D space region with boundary, realised in terms of spin networks defined on a graph with non trivial SU (2) holonomies, in the framework of loop quantum gravity. The presence of intrinsic curvature in the region is encoded in closure (topological) defects associated with tag -spins attached to the vertices of the graph. For such states, we generalise the bulk-to-boundary mapping as to include the space of tags in an extended boundary space: bulk information is shared among generically entangled boundary surfaces and intrinsic curvature degrees of freedom. We model the quantum region on a tripartite system composed by two (complementary) boundary subregions and the set of bulk tags. Via replica techniques, we can compute the typical value of the logarithmic negativity of the reduced boundary, described as an open quantum system, in a large spin regime.  \nWe find three entanglement regimes, depending on the ratio between the number of tags (curvature) and the area of the dual surface at the boundary. These are well described by the generalised Page curve of a tripartite random state. In particular, we find area scaling behaviour for negativity in case of small curvature, while for large curvature the negativity vanishes, suggesting an effective thermalization of the boundary. Remarkably, the PPT character of the mixed boundary state corresponds to a change in the effective topology of the network, with the two boundary subregions becoming disconnected.  \nKeywords: Quantum gravity, random tensor networks, multipartite entanglement, logarithmic negativity", 'I. INTRODUCTION': "In the last two decades, the idea that continuum spacetime geometry may be emerging from entanglement has radically changed all approaches to quantum gravity [19]. Beside the AdS/CFT framework [10], where the correspondence between geometry and entanglement is a necessary instance of the gauge/gravity duality[11-15], nonperturbative and background independent approaches to quantum gravity recently offered a different playground to investigate the roots of such interplay.  \nIn loop quantum gravity [16-18], and related generalizations [19, 20], like spin foam theory and group field theory, a beautiful quantization of space at one time, is realised in terms of quantum spin networks [21-23]. These are typically SU (2) gauge symmetric tensor networks, defined by graphs with edges labelled by SU (2) spin irreps and vertices dressed by intertwining operators [18].  \nFor given graph, the spin-networks Hilbert space essentially capture the kinematics of general relativity in its first order formulation [24], which is eventually described in analogy with the conventional Hilbert space of an SU (2) lattice Yang-Mills theory [25, 26].  \nIn this sense, spin networks provide a phenomenal quantitative tool to investigate the operational content of the gravitational field, reduced on 3 D spacetime slices, at the quantum scale [27-37]. Indeed, geometric and topological features of 3 D geometry get encoded into purely algebraic and combinatorial degrees of freedom, hence local, non-local and holographic features of the entanglement structure of quantum spin networks are expected to reflect the signatures of Einstein's equations and the invariance for diffeomorphisms at the quantum level.  \nIn this light, spin networks entanglement has become a central resource to characterize physical vacuum states of the theory [38-41], investigating local holographic properties of quantum spacetime geometry [42-44] and ultimately to study the emergence of classical spacetime geometry from its quantum description, with a renovate interplay of techniques and tools from quantum information theory, information geometry, quantum manybody theory and quantum computation theory more recently [45-56].  \nMost recent work in the field, in particular, has focused on quantum 3 D geometry states with 2 D boundaries, corresponding classically to space-time spatial slices with corners [57], thereby looking at quantum spin-networks with boundaries as boundary maps with corner states encoding the geometric and topological information stored in the bulk correlation structure. In this setting, different measures of multipartite entanglement have been proposed to investigate encoding and decoding of bulk information in the boundary/bulk mapping [58-60] and  \nits holographic behaviour.  \nIn [61-66], the holographic character of the spinnetworks boundary/bulk mapping has been investigated within a quantum typical regime starting from a random tensor-network description of 3 D quantum geometry states in the large spin limit [9, 67, 68]. Along this line, in [60], the authors first investigated the measure of entanglement negativity , along the lines of [69-72], to extend the study of the hierarchy of boundary correlations to mixed states, beyond the bipartite pure-state setting of entanglement entropy.  \nIn this work, we consider a synthetic description of a uniformly curved quantum 3 D space region with boundary, realised in terms of open tagged spin-networks states in loop quantum gravity [73, 74]. We show that the effective topology of this region is reflected in the multipartite entanglement of its tagged spin-networks state description. In presence of intrinsic curvature, we generalise the bulk-to-boundary mapping description as to include the space of bulk defects in a generalised boundary space. In the resulting extended boundary state, bulk information is shared among entangled boundary spins and tags: surface and intrinsic curvature of the quantum space region are entangled.  \nWe model the generalised boundary mapping on a tripartite random state and compute the logarithm negativity for the bipartite reduced boundary, focusing on the universal typical behaviour of the large spin regime.  \nWe find that the multipartite entanglement of the system depends not only on the bulk quantum correlations among the intertwiner states comprising the bulk of the spin network, but in an essential way, on the topology of the spin network state.  \nIn particular, we show that the degree of quantum correlations among boundary subregions depends on the dimension of the bulk tags system, which plays the role of a hidden environment from the viewpoint of a generic observer measuring correlations on the boundary. When the bulk curvature environment is smaller than the boundary system, the entanglement negativity of the boundary subregions displays the expected area law behaviour. However, when the environment is larger, the negativity vanishes suggesting an effective thermalization of the boundary. Remarkably, the PPT character of the mixed boundary state corresponds to change in the effective topology of the network in the large spin regime, with the two complementary boundary subregions becoming disconnected.  \nThe paper is organized as follows. Section II introduces loop quantum gravity quantum geometry states as superpositions of quantum spin networks. In Section III, this notion is generalised to include boundaries and non-trivial topology. In particular, here we define quantum states associated with regions of bounded 3 D space with non-vanishing intrinsic curvature. Section IV introduces the notion of tagged spin networks, which is the starting point of our characterisation of the bounded quantum geometry as a mixed state. In Section V, we  \ndefine the tripartition { A 1 , A 2 , B } of our system, with A 1 , A 2 corresponding to complementary subregions of the boundary and B referring to the set of tags produced by the integral (partial tracing) over the non-trivial loop holonomies in the bulk. For such a tripartite system, we quantify the entanglement of the boundary via a measure of typical entanglement negativity [75-81] by considering the boundary mapping on a bulk state prepared at random . The random measurement combined with the boundary map projection allows us to limit the partial tracing to the 'curvature' degrees of freedom, while (partially) retaining the information on the graph structure. We compute typical k -th order R'enyi log-negativity for even k via replicas by mapping momenta of the A 1 , A 2 reduced density matrix to partition functions of a classical generalised Ising model [69, 72], hence looking for the minimal free energy configurations of the model. In Section VI, we characterise the bulk states contributions to the free energy of the classical model. The analysis is thereby restricted to states with small intertwiner entanglement. Section VII contains an explicit example of open spin network state - a cluster of tags - with four vertices. For such a simplified setting, we can explicitly compute the typical logarithmic negativity of the mixed boundary state in the large spin regime and characterise the entanglement phase diagram of the boundary state. This section contains the main results of the paper. Finally, we close in Section VIII with a summary of the results and some closing remarks.", 'II. SPIN NETWORKS STATES': 'Loop quantum gravity (LQG) space of 3D geometry is realized as a space of square-integrable functions,  \nH γ = L 2 [ SU (2) × E /SU (2) × V ] , (1)  \nwith support on closed oriented graphs γ comprised by V vertices connected by E edges. States in H γ depend on one group element g e ∈ SU (2) for each edge e of the graph and are assumed to be invariant under the SU (2)action at each vertex v , that is, ∀ h ∈ SU (2),  \nψ γ : SU (2) × E → C (2) { g e } e ∈ γ ↦→ ψ γ ( { g e } ) = ψ γ ( { h t ( e ) g e h s ( e ) -1 } ) ,  \nwhere t ( e ) and s ( e ) respectively refer to the target and source vertices of the edge e [82].  \nFor SU (2) is compact, by the Peter-Weyl theorem, functions of d group elements g can be decomposed into irreducible representations (irreps) of the group,  \nf ( { g } ) = ∑ { j } ∑ { m }{ n } f { j } { m }{ n } d ∏ i =1 d j i D j i m i n i ( g i ) , (3)  \nwith { j i } ∈ N / 2 labelling irreps of SU (2); indices m i ( n i ) labelling a basis in the vector space V j i = span {| j i , m i 〉} ,  \ncarrying the representation j i , of dimension d j i = 2 j i + 1, and D j i m i n i ( g i ) = 〈 j i , n i | g i | j i , m i 〉 the Wigner matrix representing the group element g i .  \nThis allows to write SU (2)-invariant functions ψ γ as superpositions of gauge symmetric tensor networks with fixed spins { j } γ and the connectivity of the graph γ . Two vertices of the graph, say v and w , are connected by edge states labeled by a spin j irreducible representation (irrep) of SU (2) and dressed with a group element g e ∈ SU (2), that is  \n| e vw ( g e ) 〉 ≡ ∑ { m }{ n } ( -1) j -n √ d j D j mn ( g e ) | j, m 〉 v ⊗| j, n 〉 w (4)  \nin the Hilbert space H vw = [ V j v ⊗ V j w ] . At each vertex v of the graph, an intertwiner operator ι v enforces SU (2) gauge invariance via a projection  \nι v : ⊗ e ∈ v V j e → V 0 . (5)  \ncorresponding to a recoupling of the edges spins at the vertex in the SU (2)-symmetric (singlet) representation with j = 0. This associates to each vertex the degeneracy space of V 0 for fixed { j e } ,  \nH v = Inv SU (2) [ ⊗ e ∈ v V j e ] = span {| ι v 〉} , (6)  \nwith | ι v 〉 an orthonormal basis in H v .  \nA spin network state | γ, { j e } , { ι v }〉 is thereby defined as the assignment of representation labels { j e } to each edge of γ and the choice of a vector |{ ι v }〉 in H V = ⊗ V v H v for the vertices, corresponding to the contraction (see Fig 1)  \n| γ, { j e } , { i v }〉 = ( E ⊗ e =1 〈 e ( g e ) | ) |{ ι v }〉 . (7)  \nSuch spin-networks define an orthogonal basis in H γ , for the following isomorphism holds  \nH γ ≈ ⊕ { j } H V . (8)  \nAny quantum 3 D geometry state ψ γ can be constructed from the contraction of a generic state | ψ 〉 in H V with the set of E edge states, via a sum over spins  \n| ψ γ ( { g e } ) 〉 = ⊕ { j e } ( E ⊗ e =1 〈 e ( g e ) | ) | ψ 〉 , (9)  \nwhere the contraction amounts to a sum over bulk indices.  \n<latexit sha1\\_base64="/eseMWWAj/AUCjpMWZiw6LNN5Ww=">AAAB73icbVBNS8NAEJ3Ur1q/qh69BIvgqSQi6rGgB49V7Ae0oUy2m3bp7ibuboQS+ie8eFDEq3/Hm//GbZuDtj4YeLw3w8y8MOFMG8/7dgorq2vrG8XN0tb2zu5eef+gqeNUEdogMY9VO0RNOZO0YZjhtJ0oiiLktBWOrqd+64kqzWL5YMYJDQQOJIsYQWOldneAQmDvvleueFVvBneZ+DmpQI56r/zV7cckFVQawlHrju8lJshQGUY4nZS6qaYJkhEOaMdSiYLqIJvdO3FPrNJ3o1jZksadqb8nMhRaj0VoOwWaoV70puJ/Xic10VWQMZmkhkoyXxSl3DWxO33e7TNFieFjS5AoZm91yRAVEmMjKtkQ/MWXl0nzrOpfVP2780rtJo+jCEdwDKfgwyXU4Bbq0AACHJ7hFd6cR+fFeXc+5q0FJ585hD9wPn8A5MuP4g==</latexit>  \n<latexit sha1\\_base64="CMSAdXbUaIwktfz+mAbJSbclPW4=">AAAB7nicbVDLSgNBEOyNrxhfUY9eBoPgKeyKqMeAHjxGMA9IljA7mU2GzM4sM72BEPIRXjwo4tXv8ebfOEn2oNGChqKqm+6uKJXCou9/eYW19Y3NreJ2aWd3b/+gfHjUtDozjDeYltq0I2q5FIo3UKDk7dRwmkSSt6LR7dxvjbmxQqtHnKQ8TOhAiVgwik5qdYVG2hv3yhW/6i9A/pIgJxXIUe+VP7t9zbKEK2SSWtsJ/BTDKTUomOSzUjezPKVsRAe846iiCbfhdHHujJw5pU9ibVwpJAv158SUJtZOksh1JhSHdtWbi/95nQzjm3AqVJohV2y5KM4kQU3mv5O+MJyhnDhCmRHuVsKG1FCGLqGSCyFYffkvaV5Ug6tq8HBZqd3lcRThBE7hHAK4hhrcQx0awGAET/ACr17qPXtv3vuyteDlM8fwC97HN3GSj6Y=</latexit>  \n<latexit sha1\\_base64="JDoziWuZ/nsftHs6zypxoTpVVSI=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE1JMUvHisaG2hDWWznaRLN5uwuxFK6E/w4kERr/4ib/4bt20O2vpg4PHeDDPzglRwbVz32ymtrK6tb5Q3K1vbO7t71f2DR51kimGLJSJRnYBqFFxiy3AjsJMqpHEgsB2MbqZ++wmV5ol8MOMU/ZhGkoecUWOl+6iP/WrNrbszkGXiFaQGBZr96ldvkLAsRmmYoFp3PTc1fk6V4UzgpNLLNKaUjWiEXUsljVH7+ezUCTmxyoCEibIlDZmpvydyGms9jgPbGVMz1IveVPzP62YmvPJzLtPMoGTzRWEmiEnI9G8y4AqZEWNLKFPc3krYkCrKjE2nYkPwFl9eJo9nde+i7t2d1xrXRRxlOIJjOAUPLqEBt9CEFjCI4Ble4c0Rzovz7nzMW0tOMXMIf+B8/gBBXo3F</latexit>  \n<latexit sha1\\_base64="+V9PFpiRkEvyzb6ALDlOHOa+nO8=">AAAB8nicbVBNSwMxEM36WetX1aOXYBHqpeyKqMeiF48V7Ae0y5JNZ9vYbLIk2UpZ+jO8eFDEq7/Gm//GtN2Dtj4YeLw3w8y8MOFMG9f9dlZW19Y3Ngtbxe2d3b390sFhU8tUUWhQyaVqh0QDZwIahhkO7UQBiUMOrXB4O/VbI1CaSfFgxgn4MekLFjFKjJU6EGSjp0nlMYCzoFR2q+4MeJl4OSmjHPWg9NXtSZrGIAzlROuO5ybGz4gyjHKYFLuphoTQIelDx1JBYtB+Njt5gk+t0sORVLaEwTP190RGYq3HcWg7Y2IGetGbiv95ndRE137GRJIaEHS+KEo5NhJP/8c9poAaPraEUMXsrZgOiCLU2JSKNgRv8eVl0jyvepdV7/6iXLvJ4yigY3SCKshDV6iG7lAdNRBFEj2jV/TmGOfFeXc+5q0rTj5zhP7A+fwBB0KRFg==</latexit>  \n<latexit sha1\\_base64="R0oj1KadaMxFmWAU0EpToK+MK8k=">AAAB7nicbVDLSgNBEOz1GeMr6tHLYBA8hV0R9Rj04jGCeUCyhNnJbDJkdmaZ6VVCyEd48aCIV7/Hm3/jJNmDJhY0FFXddHdFqRQWff/bW1ldW9/YLGwVt3d29/ZLB4cNqzPDeJ1pqU0ropZLoXgdBUreSg2nSSR5MxreTv3mIzdWaPWAo5SHCe0rEQtG0UnNjtBIu0/dUtmv+DOQZRLkpAw5at3SV6enWZZwhUxSa9uBn2I4pgYFk3xS7GSWp5QNaZ+3HVU04TYcz86dkFOn9EisjSuFZKb+nhjTxNpRErnOhOLALnpT8T+vnWF8HY6FSjPkis0XxZkkqMn0d9IThjOUI0coM8LdStiAGsrQJVR0IQSLLy+TxnkluKwE9xfl6k0eRwGO4QTOIIArqMId1KAODIbwDK/w5qXei/fufcxbV7x85gj+wPv8AXJ8j6U=</latexit>  \nFIG. 1: Open spin network with support on the graph γ R . The boundary is comprised by the set of uncontracted edges with dangling magnetic indices pictured as empty circles.  \n<!-- image -->', 'III. SPIN-NETWORKS WITH BOUNDARY AND NON-TRIVIAL TOPOLOGY': "Imagine to cut a generic bounded region R out of quantum 3D space. The corresponding state will be defined with support on an open graph γ R ⊂ γ , which we can take to be comprised by the sets of vertices V R and edges E R , with |V R | = V R the number of vertices and |E R | = E R the number of edges respectively. The boundary of R , ∂R , is the set E ∂R of E ∂R edges which have only one end attached to a vertex laying in R . Such boundary edges are characterised by an open (uncontracted) dangling index m , while bulk edges connecting vertices within R are fully contracted. For states with support on open graphs, the contraction in (9) defines a mapping of the state | ψ 〉 ∈ H V R on the boundary space of uncontracted indices,  \nH ∂R = ⊕ { j e } ⊗ e ∈E ∂R V j e . (10)  \nIn particular, the gauge invariant property of the LQG wavefunctions defined on closed graphs reduces to SU (2)covariance of the bounded region wavefunction,  \nψ R ( { g e } e ∈E ) = ψ R ( { g e h s ( e ) -1 } e ∈E ∂R ) . (11)  \nIn the boundary mapping, the information on the bulk holonomies, as well as the correlation structure of ψ ∈ H V R gets encoded in the coefficients of the resulting boundary state | ψ R 〉 . We are interested in the information concerning the bulk curvature and, in particular, in the way this affects (gets reflected in) the correlation structure of the boundary state | ψ R 〉 .  \nBulk curvature is generally associated to the presence of loops of non-trivial holonomies in the graph. However, the topology of the graph is only partially relevant to characterise such a degree of freedom. Indeed, thanks to  \nthe local gauge invariance of ψ R , the structure of the bulk of γ R can be drastically simplified via a partial gaugefixing of the bulk holonomies [45, 73, 74]. In particular, for any open region R , the gauge-invariant Hilbert space on γ R is isomorphic to the gauge-invariant reduced space defined on a new graph Γ R consisting of a single vertex intertwining the external edges of the boundary ∂R together with a number of (independent) loops L fixed by the combinatorics of the region, that is L = E R -V R +1. If L = 0 there are no loops and γ R has trivial topology. In particular, this implies that R has vanishing intrinsic curvature.  \nMore generally, the Hilbert space of the reduced graph Γ R will consist of the intertwiner space of the single bulk vertex times the product of the boundary representations spaces, that is  \nH Γ R = L 2 [ SU (2) × ( E +2 L ) /SU (2)] = ⊕ { j glyph[lscript],e } H L v ⊗ ⊗ e ∈E ∂R V j e , (12)  \nwhere the degeneracy space at the vertex now also involves the irreps on the loop  \nH L v ≡ Inv SU (2) [ L ⊗ glyph[lscript] =1 ( V j glyph[lscript] ⊗ V j glyph[lscript] ) ⊗ ⊗ e ∈E ∂R V j e ] . (13)  \nThe space in (12) provides a very synthetic description of the region R in terms of a single 'loopy' vertex with both its boundary degrees of freedom and bulk nontrivial holonomies. The gauge reduction isomorphism does not produce any coarse graining on physical degrees of freedom. However, the information of the combinatorics of the original graph is lost. As a consequence, the correspondence between the original graph and its flower is many-to-one [74].  \nNow, imagine to operate a similar reduction-by-gaugefixing on a collection of subregions, so as to end up with a set of loopy vertices glued together via edges dressed with a trivial holonomy. In this way, we construct a new spin network state with support on an open graph ˜ γ R associated to an extended 3 D space region with distributed intrinsic curvature (see Fig. 2). Such an intermediate level of description allows to localize and keep track of the nontrivial holonomies { g glyph[lscript] } (classically closure defects) responsible for the curvature at the vertices of the graph, without trivializing the topology of the graph, which is going to play a role in the entanglement structure of the boundary.", 'IV. TAGS FROM SPIN-NETWORK KIRIGAMI': 'Along this line, let us consider for simplicity the case of a uniformly curved region of quantum 3 D space, given by an open-graph state | ψ R 〉 = | ψ R ( { g glyph[lscript] } ) 〉 with all V R vertices carrying one single loop holonomy g glyph[lscript] . 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This very simplified setting is what we need to neatly separate boundary from bulk degrees of freedom, as shown in Fig.(3). The first step in this sense consists in integrating out the loop holonomies. Following [83], for each spin k carried by the loop, one can separate the loop from the edges at each vertex by unfolding the intertwiner into two different intertwiners connected by a virtual link labelled by an intermediate spin J : one intertwiner recoupling the boundary edges spins j e at v into J and a second 3-valent intertwiner recoupling the two copies of the spin k into J at the virtual vertex ˜ v . Accordingly, the intertwiner space in (13) now decomposes as  \nH L v = ⊕ J, { k } { Inv SU (2) [ ⊗ e ∈ v V j e ⊗ V J ] ⊗ Inv SU (2) [ V J ⊗ ( V k ⊗ V k )] } . (14)  \nA basis in H L v is given by (see for instance [84])  \n| J, k, ι, { j e } [ g glyph[lscript] ] 〉 = D k ( g glyph[lscript] ) C k,J | k ˜ m,M | m 〈 J, M | ι 〉 (15)  \nwith  \nC k,J | k ˜ m,M | m = 〈 k, m | ( k, ˜ m ) ⊗ ( J, M ) 〉 (16)  \nthe Clebsh-Gordan coefficient associated to the 3-valent intertwiner at ˜ v .  \nAt each vertex of the graph, the integration over the loop holonomy only involves the Clebsh-Gordan coefficient and the Wigner matrix of the loop at the level of the basis in (15). One can then compute the vertex density matrix ρ k as  \nρ k = ∫ d g glyph[lscript] | J, k, ι, { j e } [ g glyph[lscript] ] 〉 〈 J, k, ι, { j e } [ g glyph[lscript] ] | (17)  \nand show that the sum of ρ k over the loop spin k gives exactly the identity matrix ρ = 〈 J, M | ι 〉 〈 ι | J, M 〉 , with | τ 〉 = 〈 J, M | ι 〉 a basis in the reduced vertex space [84]  \nH J v = V J ⊗ M { j e } J , (18)  \nAs a result of the integration, curvature is locally encoded into a virtual spin J , or tag , attached to the vertex. We can see the vector | J, M 〉 ∈ V J as the result of the recoupling of all the spins j e at the vertex into a non-vanishing overall spin J , a topological defect which breaks local gauge invariance. 1  \nwith M { j e } J ≡ Inv SU (2) [ V J ⊗ ⊗ e ∈ v V j e ] .  \nStarting from the tagged intertwiner description, we now define a homogeneously curved quantum 3 D bounded space geometry as the extended boundary map  \n| ψ τ 〉 = ( E ⊗ e =1 〈 e ( g e ) | ) | ψ b 〉 (20)  \nwhere the generic tagged state | ψ b 〉 is defined, for fixed spins { J, j e } v , as  \n| ψ b 〉 = ∑ { M v }{ ι v } C ( { J, j e } ) { M v } , { ι v } ⊗ v ∈V R | τ v 〉 (21)  \nin the tensor product space of V R tagged intertwiner spaces H V R ≡ ⊗ V R v H J v v . 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sha1\\_base64="DnlKK83Q9bteD3UKPxHVGfe7OwI=">AAAB6HicdZDLSgMxFIYz9VbrrerSTbAIroZMLW1dWdCFuGrBXqAdSibNtLGZzJBkhDL0Cdy4UKQufRu37nwbM62Civ4Q+Pn+c8g5x4s4UxqhdyuztLyyupZdz21sbm3v5Hf3WiqMJaFNEvJQdjysKGeCNjXTnHYiSXHgcdr2xudp3r6lUrFQXOtJRN0ADwXzGcHaoMZVP19A9mm1XCyVIbIRqjhFJzXFSumkBB1DUhXOXmepnuv9/FtvEJI4oEITjpXqOijSboKlZoTTaa4XKxphMsZD2jVW4IAqN5kPOoVHhgygH0rzhIZz+r0jwYFSk8AzlQHWI/U7S+FfWTfWftVNmIhiTQVZfOTHHOoQplvDAZOUaD4xBhPJzKyQjLDERJvb5MwRvjaF/5tW0XbKttNAhdoFWCgLDsAhOAYOqIAauAR10AQEUHAHHsCjdWPdW0/WbFGasT579sEPWS8fOXyRsw==</latexit>  \nFIG. 3: a) A single loopy vertex ; b) vertex unfolding: isolate the nontrivial loop holonomy; c) partial trace via integration over the loop holonomy; d) tagged vertex.  \n<!-- image -->  \nThe tagged open graph state is written as  \n| ψ τ 〉 = ∑ { M v ,m e } C ( { ι v , J v , j e } ) { M v ,m e } ⊗ v ∈V R | J v , M v 〉 ⊗ e ∈E ∂R | j e , m e 〉 , (23)  \nwith coefficients C ( { ι v , J v , j e } ) { M v ,m e } encoding the information on the quantum correlations among bulk intertwiners, the connectivity, and the topological defects of the graph. We expect states in (23) to be generically highly entangled, with information about bulk geometry encoded into the boundary state coefficients C in a complicated way. In the extended boundary system, such information is shared by boundary spins and bulk tags.  \nNow, consider an observer having access only to the actual boundary system. Such an observer is precluded from measuring the information on the bulk, hence they will typically describe the system via a mixed boundary state. Concretely, this amounts to a partial tracing over the bulk. In our analysis, we describe the coarsegrained viewpoint of the boundary observer by introducing a random measurement on the state | ψ τ 〉 . We then investigate the boundary-system/bulk-environment coupling via a combination of random projections and partial tracing over the tags.', 'V. NEGATIVITY OF A TRIPARTITE BOUNDARY-TAGS SYSTEM': 'Let us consider the extended boundary space (22). For convenience, let us assume curvature, in the form of closure defects, to be homogeneously distributed throughout the graph ˜ γ R . Accordingly, we set all the spins of the graph { j e } and { J v } to be equal to the single value j and J respectively, generally with j = J .  \nA random measurement on the bulk state | ψ b 〉 , prepared on ⊗ v H J v , is realized by a projection onto a set of independent and individually (Haar) random vertex states | f v 〉 in the extended vertex space  \nglyph[negationslash]  \nK J v = V J ⊗ ⊗ j e ∈ v V j e ⊗ M { j e } J , (24)  \nwhich comprises the intertwiner space at the vertex, the spin irrep spaces of the edges connecting the vertex with the rest of the graph, as well as the tag spin space. In particular, we can obtain the state | f v 〉 by acting on a reference state | 0 〉 in K J v at v with a Haar random unitary, that is | f v 〉 = U v | 0 〉 [67].  \nThe generic random vertex state is explicitly written as  \n| f v 〉 = ∑ ι v , { n e } f v ι v , { n e } 〈 J, M | ι v 〉 ⊗ ⊗ e ∈ v | j e , n e 〉 (25)  \nThe randomised extended boundary density matrix is then simply defined along with (20) by a trace over bulk indices, which contract ρ b = | ψ b 〉〈 ψ b | and ρ E = | E 〉〈 E | , for | E 〉 = ⊗ E e | e 〉 , with the further insertion of the random projector Π = ⊗ V R v Π v = ⊗ V R v | f v 〉〈 f v | :  \nρ τ = Tr[ ρ b · ρ E Π] . (26)  \nThe bulk state ρ b and the maximally entangled edges state ρ E enter as product states in the trace, while the information on the tags is enclosed in the bulk state.  \nWe shall define a tripartition of the extended boundary system into three subsystems A 1 , A 2 , B , with A 1 , A 2 complementary regions of the external boundary, and B the set of tags in the bulk. This corresponds to the following factorization of the extended boundary Hilbert space  \nH τ = ⊗ e ∈ A 1 V j e ⊗ ⊗ e ∈ A 2 V j e ⊗ ⊗ i ∈ B V J i . (27)  \nStarting from (27), we focus on the reduced state obtained tracing over the tag spins in B ,  \nρ A 1 A 2 = Tr B [ ρ τ ] . (28)  \nWe evaluate the entanglement of the mixed state ρ A 1 A 2 by computing the logarithmic negativity of the partial transpose of the state [28, 60, 69, 72, 85]. In the bipartite subsystem A 1 A 2 , with Hilbert space H A 1 ⊗ H A 2 , given  \nan orthonormal basis | i 〉 A 1 in H A 1 and similarly | j 〉 A 2 in H A 2 , the partial transpose of ρ A 1 A 2 with respect to the subsystem A 2 is defined as  \n( ρ i A 1 j A 2 ,k A 1 l A 2 ) T A 2 = ρ i A 1 l A 2 ,k A 1 j A 2 . (29)  \nThe eigenvalues of the partially transposed reduced density matrix ρ T A 2 A 1 A 2 are real since the partial transposition is an Hermitian and trace-preserving map. Yet ρ T A 2 A 1 A 2 is not completely positive, i.e. it may have negative eigenvalues. The presence of negative eigenvalues in the partial transpose witnesses quantum correlations in ρ A 1 A 2 . Concretely, the amount of entanglement of the state can be quantified by counting the number of negative eigenvalues of ρ T A 2 A 1 A 2 [80], as  \nN ( ρ AB ) ≡ ‖ ρ T B AB ‖ 1 -1 2 = ∑ i : λ i < 0 | λ i | , (30)  \nwhere ‖·‖ 1 is the trace norm, and the logarithmic negativity is defined as  \nE N ( ρ A 1 A 2 ) ≡ log ‖ ρ T A 2 A 1 A 2 ‖ 1 . (31)  \nBoth N and E N are entanglement monotone under general positive partial transpose (PPT) preserving operations [75]. In order to compute (31), the first key ingredient is to first look at the k -th R\'enyi negativity of ρ A 1 A 2 , 3  \nN k ( ρ A 1 A 2 ) = Tr [ ( ρ T A 2 A 1 A 2 ) k / (Tr[ ρ A 1 A 2 ]) k ] , (32)  \nhence, eventually recover the logarithmic negativity by taking the k → 1 limit of the logarithm of the analytic continuation of the momenta for even k [69]. In particular, since we took a random measurement on the bulk state, we will need to compute the R\'enyi negativity in expected value , with respect to the uniform Haar measure µ , namely  \nE µ [ N k ( ρ A 1 A 2 )] ≡ N k ( ρ A 1 A 2 ) . (33)  \nFollowing [60, 67, 69], we know that, due to concentration of the trace, a regime of large spins allows us to approximate the R\'enyi negativity by the ratio of expected values of the k -th moment and the k -th power of the partition function of ρ T A 2 A 1 A 2 , that is  \nN k ( ρ A 1 A 2 ) glyph[similarequal] Tr [ ( ρ T A 2 A 1 A 2 ) k ] (Tr [( ρ A 1 A 2 )]) k ≡ Z ( k ) 1 Z ( k ) 0 . (34)  \nof  \ne  \n2)  \ne-  \n3)  \na)  \nn-  \nm  \nr-  \nx,  \n,  \n4)  \nr-  \ne  \ny  \nh  \nly  \ni-  \nts  \ne-  \na)  \n)  \n-  \nlemma, see [  \n?  \n]) in a sum over unitary representations of  \nthe permutation group  \ng  \nZ  \n(  \nk  \n)  \nv  \n1  \n=  \nacting on the  \nk  \ncopies of the  \nsingle vertex degeneracy space  \nM  \n(  \nD  \n1)!  \nTr  \n(  \nk  \n)  \n(  \nk  \n)  \n1  \n1  \nj  \ne  \nJv  \n{  \n}  \nρ  \n⊗  \n,  \nρ  \n⊗  \nk  \nk  \nk  \ntags  \nE  \ntags  \n)  \n⊗  \n⊗  \nP  \nP  \n⊗  \ndefined by the action  \nP  \nv  \ng  \n⊗  \n∑  \nk  \nk  \ng  \n⊗  \nv  \nρ  \n⊗  \nρ  \n⊗  \nP  \nA  \ng  \n⊗  \nS  \nk  \nv  \nv  \n1  \nP  \nP  \nB  \n5  \nE  \nE  \nB  \nv  \n∈  \nX  \n-  \n(22)  \n⊗  \n(  \n)  \n∑  \n(  \ng  \nv  \n(  \nk  \n)  \n1  \n∈  \nS  \n∑  \nv  \nv  \n∈  \nS  \nA  \nX  \n-  \n⊗  \n(  \ng  \nv  \nv  \n(  \ng  \ng  \nv  \n]  \n]  \nA  \n⊗  \nB  \n]  \n+  \n) log  \nd  \nwhere the trace factorizes over the Hilbert spaces of a)  \ninternal edges, b) boundary spins and c) bulk intertwin-  \nwhere the trace factorizes over the Hilbert spaces of a)  \nwhere the trace factorizes over the Hilbert spaces of a)  \ninternal edges, b) boundary spins and c) bulk intertwin-  \ninternal edges, b) boundary spins and c) bulk intertwin-  \n1)!  \n-  \n|  \n〉〈  \ndefined by the action  \nwith dimension  \n(  \nk  \n)  \n1  \n+  \nZ  \n{  \n[  \n=  \nv  \nA  \ng  \n(  \nD  \n1)!  \n(  \nD  \n(  \nD  \nv  \n(  \nD  \nv  \n.  \nZ  \nv  \nv  \n+  \nk  \n-  \n(  \nk  \n)  \n1)!  \n1)!  \n-  \nreplaced by  \n.  \n[  \n0  \n1)!  \n-  \n+  \nk  \n1)!  \nBy performing the average, individually on each inde-  \npendent random vertex state, eventually we obtain  \n∑  \ne  \ni  \n(  \nk  \n)  \n1  \nv  \n¯  \nwith  \nξ  \n]  \ne  \ne  \nP  \n(  \ng  \n<!-- image -->  \n(  \nk  \n(  \nk  \n0  \n0  \nwith  \nP B ( ) d j i v + ∑ e i v ¯ v ∈ ¯ A ∆( g v ,X -) log d j i v + B ∆( g v , ) log D J i v + A tags + ξ (27) X -1 ¯ A 2 A 2 X -1 v w Now, to compute the action of the permutation operators on the different degrees of freedom at each vertex, we factorize P v ( g v ) into four different sets of operators, P v ( g v ) = P v, 0 ( g v ) ⊗ ⊗ e i vw ∈ E R P v,i ( g v ) ⊗ We consider a quasi-homogeneous spi all the edges carry the same spin j and same spin J , with j = J generally. Gi the dimension of the Hilbert spaces o and D J = 2 J +1 the dimension of the the tags spin, we define the strength of respectively as Now, to compute the action of the permutation operators on the different degrees of freedom at each vertex, we factorize P v ( g v ) into four different sets of operators, P v ( g v ) = P v, 0 ( g v ) ⊗ ⊗ i P v,i ( g v ) ⊗ We consider a quasi-homogene all the edges carry the same spin same spin J , with j = J genera the dimension of the Hilbert sp and D J = 2 J +1 the dimension the tags spin, we define the stre respectively as Now, to compute the action of the permutation operators on the different degrees of freedom at each vertex, we factorize P v ( g v ) into four different sets of operators, P v ( g v ) = P v, 0 ( g v ) ⊗ ⊗ i P v,i ( g v ) ⊗ We consider a quasi-homogene all the edges carry the same spin same spin J , with j = J genera the dimension of the Hilbert sp and D J = 2 J +1 the dimension the tags spin, we define the stre respectively as ρ ⊗ k E   ⊗ v ∑ g v ∈ S k P v ( g v )   · ( X ) ⊗ P B ( X -1 ) ⊗ P C ( ) ] , (23) A tags = -log { Tr bulk [ | ψ tags 〉〈 ψ tags | ⊗ k ( ⊗ v P v, 0 ( g v )) ] } (28) the bulk state contribution. In ( ?? ) and ( ?? ).The action A ( k ) 1 [ { g v } ] describes a two-body interaction between permutation elements, which therefore act as generalized spins , attached to the spin network vertices. FIG. 5: Permutation operator P v ( g v ) action factorizes on the indices of the extended vertex state. In the example in figure the operator acts as the identity connecting the set of internal indices of v to the same set in ¯ v , while P B ( ✶ ) realises the partial trace over the tags.  \nE  \nE  \nv  \n-  \nvw  \n∈  \nvw  \n∈  \nwhere the trace factorizes over the Hilbert spaces of a)  \ninternal edges, b) boundary spins and c) bulk intertwin-  \n]  \n]  \nR  \nR  \n)  \nP  \nP  \nv,i  \nv,i  \n⊗  \n⊗  \nNow, to compute the action of the permutation oper-  \nators on the different degrees of freedom at each vertex,  \nk  \nk  \nteractions are described by the Cayley distance ∆(  \ng,h  \n),  \nβ  \n,  \nlog  \nD  \n/negationslash  \n/negationslash  \nThese in-  \n+ 1  \nIn particular, maximal domains corres  \ntion  \nP  \nboundary conditions attached to the boundary and tags spins. 6 We consider a quasi-homogeneous spin network, where N k ( ρ A 1 A 2 ) glyph[similarequal] N k ( ρ A 1 A 2 ) , (35) that is, the function is well approximated by its typical value . Accordingly, the computation of the trace can be decomposed into four contributions, respectively for the a) internal edges, b) boundary spins and c) tags spins d) bulk intertwiners (for the technical aspects of the com5 and Accordingly, the computation of the trace can be decomposed into four contributions, respectively for the a) internal edges, b) boundary spins and c) tags spins d) bulk intertwiners (for the technical aspects of the com5 and Accordingly, the computation of the trace can be decomposed into four contributions, respectively for the a) internal edges, b) boundary spins and c) tags spins d) bulk intertwiners (for the technical aspects of the com5 and ⊗ ⊗ e i v ¯ v ∈ E ∂R P v,i ( g v ) ⊗ P v, tag ( g v ) , (24) where P v, 0 ( g v ) acts on k copies of the multiplicity interlog d j ≡ β , log D J ≡ β t (29) intended as inverse temperatures in analogy with standard Ising models. The action favours neighbouring By Schur\'s lemma, the average of the k copies of the vertex state in (37) results in a sum over unitary representations of the permutation group g v acting on the k copies of the extended vertex space K J v [60, 69, 71],  \nA tags = -log { Tr bulk [ | ψ tags 〉〈 ψ tags | ⊗ k ( ⊗ v P v, 0 ( g v )) ] } (28) the bulk state contribution. In ( ?? ) and ( ?? ).The acA ( k ) 1 [ { g v } ] describes a two-body interaction between permutation elements, which therefore act as generalized spins , attached to the spin network vertices. These infor g,h elements of the permutation group, while the pinning permutations fields X , X -1 and play the role of FIG. 4: a) Tripartition of the spin network extended boundary for a simple graph comprised by two 4-valent vertices glued via an edge; b) Partial tracing of the density matrix over the tags space. Moreover, in the large spin regime, we have e i v ¯ v ∈ E ∂R v,i v, tag v where P v, 0 ( g v ) acts on k copies of the multiplicity intertwiner space, ⊗ e i vw ∈ E R P v,i ( g v ) acts on k copies of the internal edges, ⊗ e i v ¯ v ∈ E ∂R P v,i ( g v ) acts on the boundary semi-edges, with ¯ v representing a virtual vertex which is connected to v by the boundary edge e i v ¯ v , and finally ⊗ J v i ∈ T R P v,i ( g v ) acting on k copies of the tags. We indicate with e i vw ∈ E R the i th internal edge which connects the vertex v to the vertex w . intended as inverse temperatures in an dard Ising models. The action favou \'spins\' to be parallel. Therefore, in t mension limit, corresponding to some st \'low temperature\' regime, the domina that minimize the action correspond to l domains separated by domain walls, w the energy cost of the configuration. We domains to be associated to equilibriu domain walls. ⊗ ⊗ e i v ¯ v ∈ E ∂R v,i ( g v ) ⊗ P v, tag ( g v ) , (24) where P v, 0 ( g v ) acts on k copies of the multiplicity intertwiner space, ⊗ e i vw ∈ E R P v,i ( g v ) acts on k copies of the internal edges, ⊗ e i v ¯ v ∈ E ∂R P ) acts on the boundary semi-edges, with ¯ v representing a virtual vertex which is connected to v by the boundary edge e i v ¯ v , and finally ⊗ J v i ∈ T R P v,i ( g v ) acting on copies of the tags. We indicate with e i vw ∈ E R the i th internal edge which connects the vertex v to the vertex w . log d j ≡ β , intended as inverse temperature dard Ising models. The actio \'spins\' to be parallel. Therefor mension limit, corresponding to s \'low temperature\' regime, the d that minimize the action correspo domains separated by domain w the energy cost of the configurati domains to be associated to equ In particular, maximal domains domain walls. ⊗ ⊗ e i v ¯ v ∈ E ∂R v,i ( g v ) ⊗ P v, tag ( g v ) , (24) where P v, 0 ( g v ) acts on k copies of the multiplicity intertwiner space, ⊗ e i vw ∈ E R P v,i ( g v ) acts on k copies of the internal edges, ⊗ e i v ¯ v ∈ E ∂R P ) acts on the boundary semi-edges, with ¯ v representing a virtual vertex which is connected to v by the boundary edge e i v ¯ v , and finally ⊗ J v i ∈ T R P v,i ( g v ) acting on copies of the tags. We indicate with e i vw ∈ E R the i th internal edge which connects the vertex v to the vertex w . log d j ≡ β , intended as inverse temperature dard Ising models. The actio \'spins\' to be parallel. Therefor mension limit, corresponding to s \'low temperature\' regime, the d that minimize the action correspo domains separated by domain w the energy cost of the configurati domains to be associated to equ In particular, maximal domains domain walls. ers, while C = ∏ v [ ( D v -1)! ( D v + k -1)! ] . Z ( k ) 0 has the same form with X and X -1 replaced by . we factorize P v ( g v ) into four different sets of operators, v ( g v ) = P v, 0 ( g v ) ⊗ ⊗ e i vw ∈ E R P v,i ( g v ) ⊗ teractions are described by the Cayley distance ∆( g,h ), for g,h elements of the permutation group, while the pinning permutations fields X , X -1 and play the role of boundary conditions attached to the boundary and tags spins. 6 We consider a quasi-homogeneous spin network, where all the edges carry the same spin j and all tags carry the same spin J , with j = J generally. Given d j = 2 j the dimension of the Hilbert spaces of the edges spin and D J = 2 J +1 the dimension of the Hilbert space of the tags spin, we define the strength of the interactions respectively as only the random tensors. From (26), we have Z ( k ) 1 = Tr [ ρ ⊗ k b ⊗ ρ ⊗ k E ( ⊗ v ( | f v 〉 〈 f v | ) ⊗ k ) · P A 1 ( X ) ⊗ P A 2 ( X -1 ) ⊗ P B ( ✶ ) ] , Z ( k ) 0 = Tr [ ρ ⊗ k b ⊗ ρ ⊗ k E ( ⊗ v ( | f v 〉 〈 f v | ) ⊗ k )] , (37) where the overall trace now runs over both bulk and boundary indices.  \n/negationslash  \nrespectively as log d j ≡ β , log D J ≡ β t Tr [ ( ρ T A 2 A 1 A 2 ) k ] = Tr [ ρ ⊗ k A 1 A 2 P A 1 ( X ) ⊗ P A 2 ( X -1 )] = Tr [ ρ ⊗ k R P A 1 ( X ) ⊗ P A 2 ( X -1 ) ⊗ P B ( ✶ ) ] . (36) Z ( k ) 1 / 0 = ∑ { g v } e -A ( k ) 1 / 0 [ Z ( k ) 1 / 0 = ∑ { g v } Z ( k ) 1 / 0 = ∑ { g v } J i ∈ T R v,i v cate with e i vw ∈ E R the i th internal edge which connects the vertex v to the vertex w .  \nall the edges carry the same spin j and all tags carry the same spin J , with j = J generally. Given d j = 2 j + 1 the dimension of the Hilbert spaces of the edges spin and D J = 2 J +1 the dimension of the Hilbert space of the tags spin, we define the strength of the interactions The expected values of the partition functions in (34) are computed via the standard replica technique. We first linearize the partial transpose matrix as follows, putation we refer the reader to []). As a result, the (normalised) k -th power of the partition function is mapped to the classical partition function of a generalized Ising-like model, A ( k ) 0 [ { g v } ] = ∑ e i vw ∈ L ∆( g v ,g w ) log d j i + ∑ J i v ¯ v ∈ T R ∪ E ∂R ∆( g v , ) log putation we refer the reader to []). As a result, the (normalised) k -th power of the partition function is mapped to the classical partition function A ( k ) 0 [ { g v } ] = ∑ e i vw ∈ L ∆( g v ,g w + ∑ J i v ¯ v ∈ T R ∪ E ∂R ∆( g v putation we refer the reader to []). As a result, the (normalised) k -th power of the partition function is mapped to the classical partition function A ( k ) 0 [ { g v } ] = ∑ e i vw ∈ L ∆( g v ,g w + ∑ J i v ¯ v ∈ T R ∪ E ∂R ∆( g v twiner space, ⊗ e i vw ∈ E R P v,i ( g v ) acts on k copies of the internal edges, ⊗ e i v ¯ v ∈ E ∂R P v,i ( g v ) acts on the boundary semi-edges, with ¯ v representing a virtual vertex which is connected to v by the boundary edge e i v ¯ v , and finally \'spins\' to be parallel. Therefore, in the large spin dimension limit, corresponding to some strong coupling or \'low temperature\' regime, the dominant configurations that minimize the action correspond to large uniform spin domains separated by domain walls, which in turn give ( | f v 〉 〈 f v | ) ⊗ k = ( D v -1)! ( D v + k -1)! ∑ g v ∈ S k P v ( g v ) , (38) with dimension D v = dim( K J v ).  \nof a  \ngeneralized  \nIsing-like model,  \nof a  \ngeneralized  \nIsing-like model,  \n{  \ng  \nv  \n(29)  \n5  \nthe energy cost of the configuration. We expect maximal  \nP  \n(  \ng  \n) acting on  \nk  \ncopies of the tags. We indi-  \ne  \n-  \nA  \ne  \n-  \nA  \nintended as inverse temperatures in analogy with standard Ising models. The action favours neighbouring \'spins\' to be parallel. Therefore, in the large spin dimension limit, corresponding to some strong coupling or \'low temperature\' regime, the dominant configurations that minimize the action correspond to large uniform spin domains separated by domain walls, which in turn give Here, we generally denote P I ( σ ) as a unitary representation of the permutation σ , I = A 1 , A 2 , B , with X , X -1 and ✶ the cyclic, anti-cyclic and identity permutations. On the spins and tags dangling indices, the operator P B ( ✶ ) realises the partial trace over the tags on ρ b , while P A 1 ( X ) ⊗ P A 2 ( X -1 ) act on the k copies of uncontracted spins in ρ ⊗ k E implementing the replica trick and the partial transposition respectively. Accordingly, the computation of the trace can be decomposed into four contributions, respectively for the a) internal edges, b) boundary spins and c) tags spins d) bulk intertwiners (for the technical aspects of the computation we refer the reader to []). As a result, the (normalised) k -th power of the partition function is mapped to the classical partition function  \nof a  \ngeneralized  \nIsing-like model,  \nIn particular, maximal domains correspond to minimal domain walls. For the linearity of the trace, the expectation value can be carried out before taking the trace and it will concern Z ( k ) 1 / 0 = ∑ { g v } e -A ( k ) 1 / 0 [ { g v } ] , (25) 6  \n5  \nand  \nvw  \n) log  \nd  \n) log  \nd  \nd  \n(26)  \n+  \nj  \ni  \nv  \n+  \n,  \n) lo  \n,  \n) lo  \n} ] , (25) respctively. 6 One can equivalently think of the pinning per X -1 and as being attached to virtual 1-va extended boundary. ( k ) 1 / 0 [ { g v } ] , (25) respctively. 6 One can equivalently think of the pin X -1 and as being attached to virt extended boundary. ( k ) 1 / 0 [ { g v } ] , (25) respctively. 6 One can equivalently think of the pin X -1 and as being attached to virt extended boundary. the energy cost of the configuration. We expect maximal domains to be associated to equilibrium configurations. In particular, maximal domains correspond to minimal domain walls. and A ( k ) 0 [ { g v } ] = ∑ e i vw ∈ L ∆( g v ,g w ) log d j i vw + By performing the average, individually on each independent random vertex, we obtain 4 Z ( k ) 1 = C Tr [ ρ ⊗ k b ⊗ ρ ⊗ k E   ⊗ v ∑ g v ∈ S k P v ( g v )   · P A 1 ( X ) ⊗ P A 2 ( X -1 ) ⊗ P B ( ✶ ) ] , (39) with C = ∏ v [ ( D v -1)! ( D v + k -1)! ] .  \ndomains to be associated to equilibrium configurations.  \n+ ∑ J i v ¯ v ∈ T R ∪ E ∂R ∆( g v , ) log d j i v + A ( ψ tags ) + ξ , On the bulk indices at each vertex, the permutation  \nrespctively.  \nOne can equivalently think of the pinning permutations fields  \nX  \n,  \nextended boundary.  \nreplaced by  \n.  \n[  \nreplaced by  \n.  \n1  \n⊗  \n(  \ng  \n(  \ng  \n.  \nZ  \n.  \nZ  \nP  \n(  \ng  \n)  \n,  \n(24)  \nv  \nv  \nj  \nJ  \nlo  \nlo  \n∈  \nA  \npermutation elements, which therefore a  \n-  \n-  \nj  \nP  \ni  \n=  \nA  \nA  \n,g  \n-  \nlog  \nTr  \ntags  \ntags  \nw  \n=  \n=  \n) log  \nd  \nbulk  \nlog  \nTr  \nlog  \nTr  \nv,  \n0  \nvw  \n(  \ng  \n+  \nthe bulk state contribution.  \nIn  \nv  \n)  \nthe bulk state contribution.  \nIn (  \n??  \n) a  \nR  \ntags  \nA  \n∆(  \ng  \nv  \n(  \nk  \n)  \ndescribes a two-body int  \nthe bulk state contribution.  \nIn  \ndescribes a two-b  \ndescribes a two-b  \nj  \ni  \n+  \nattached to the spin network ver  \n[  \n]  \npermutation elements, which the  \npermutation elements, which the  \nv  \nattached to the spin netw  \nattached to the spin netw  \nteractions are described by the Cayley  \nelements of the permutation gro  \nteractions are described by the  \nteractions are described by the  \n+  \nξ  \n(27)  \nelements of the permutati  \nelements of the permutati  \nning permutations fields  \nX  \n,  \nX  \n-  \nX  \n,  \nX  \n-  \n1  \nand  \n1  \n1  \nboundary conditions attached to the bo  \nning permutations fields  \nX  \n,  \nX  \n-  \nboundary conditions attached to  \nboundary conditions attached to  \nspins.  \nspins.  \n6  \n6  \nP  \nP  \n{  \n}  \ni  \n<!-- image -->  \n/negationslash  \n≡  \nP  \nX  \n-  \nk  \nk  \n(  \n(  \n(  \nX  \n)  \n(  \nX  \n)  \n)  \n[  \n·  \n)  \n)  \n]  \n·  \n·  \n∑  \nZ  \nZ  \nC  \n=  \n=  \nk  \ntags  \nTr  \nTr  \nC  \nC  \n⊗  \nρ  \n⊗  \nρ  \n⊗  \n⊗  \nv  \n(  \nf  \nv  \n<latexit 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sha1\\_base64="/vQMXDU5fuK1S5ridJxqAWttleY=">AAAB83icbVBNSwMxEJ3Ur1q/qh69BItQQcquiHos6MFjBfsB7bJk02wbms0uSbZQlv4NLx4U8eqf8ea/MW33oK0PBh7vzTAzL0gE18ZxvlFhbX1jc6u4XdrZ3ds/KB8etXScKsqaNBax6gREM8ElaxpuBOskipEoEKwdjO5mfnvMlOaxfDKThHkRGUgeckqMlXoNPxtfONPqwB+f++WKU3PmwKvEzUkFcjT88levH9M0YtJQQbTuuk5ivIwow6lg01Iv1SwhdEQGrGupJBHTXja/eYrPrNLHYaxsSYPn6u+JjERaT6LAdkbEDPWyNxP/87qpCW+9jMskNUzSxaIwFdjEeBYA7nPFqBETSwhV3N6K6ZAoQo2NqWRDcJdfXiWty5p7XXMfryr1+zyOIpzAKVTBhRuowwM0oAkUEniGV3hDKXpB7+hj0VpA+cwx/AH6/AH5nZEA</latexit>  \n<latexit sha1\\_base64="/vQMXDU5fuK1S5ridJxqAWttleY=">AAAB83icbVBNSwMxEJ3Ur1q/qh69BItQQcquiHos6MFjBfsB7bJk02wbms0uSbZQlv4NLx4U8eqf8ea/MW33oK0PBh7vzTAzL0gE18ZxvlFhbX1jc6u4XdrZ3ds/KB8etXScKsqaNBax6gREM8ElaxpuBOskipEoEKwdjO5mfnvMlOaxfDKThHkRGUgeckqMlXoNPxtfONPqwB+f++WKU3PmwKvEzUkFcjT88levH9M0YtJQQbTuuk5ivIwow6lg01Iv1SwhdEQGrGupJBHTXja/eYrPrNLHYaxsSYPn6u+JjERaT6LAdkbEDPWyNxP/87qpCW+9jMskNUzSxaIwFdjEeBYA7nPFqBETSwhV3N6K6ZAoQo2NqWRDcJdfXiWty5p7XXMfryr1+zyOIpzAKVTBhRuowwM0oAkUEniGV3hDKXpB7+hj0VpA+cwx/AH6/AH5nZEA</latexit>  \nlog  \nd  \n<latexit sha1\\_base64="6VHltoohaCGBOEOvOrMaR+aKcZ4=">AAAB83icbVBNSwMxEJ3Ur1q/qh69BItQQcquiHos6MFjBfsB7bJk02wbms0uSbZQlv4NLx4U8eqf8ea/MW33oK0PBh7vzTAzL0gE18ZxvlFhbX1jc6u4XdrZ3ds/KB8etXScKsqaNBax6gREM8ElaxpuBOskipEoEKwdjO5mfnvMlOaxfDKThHkRGUgeckqMlXoNPxtf8Gl14I/P/XLFqTlz4FXi5qQCORp++avXj2kaMWmoIFp3XScxXkaU4VSwaamXapYQOiID1rVUkohpL5vfPMVnVunjMFa2pMFz9fdERiKtJ1FgOyNihnrZm4n/ed3UhLdexmWSGibpYlGYCmxiPAsA97li1IiJJYQqbm/FdEgUocbGVLIhuMsvr5LWZc29rrmPV5X6fR5HEU7gFKrgwg3U4QEa0AQKCTzDK7yhFL2gd/SxaC2gfOYY/gB9/gBRZpE5</latexit>  \n|  \nψ  \nbulk  \ntags  \nbulk  \n〉〈  \n[  \n[  \nψ  \n|  \n|  \ntags  \nψ  \nψ  \ntags  \ntags  \n|  \n⊗  \n〉〈  \n〉〈  \nk  \nψ  \nψ  \nj  \nj  \noperator P v ( g v ) factorizes into four independent operators (see fig. 5),  \nP v ( g v ) = P v, 0 ( g v ) ⊗ ⊗ e i vw ∈E R P v,i ( g v ) ⊗ ⊗ e i v ¯ v ∈E ∂R P v,i ( g v ) ⊗ P v,τ ( g v ) , (40)  \nwhere P v, 0 ( g v ) acts on k copies of the multiplicity intertwiner space, ⊗ e i vw ∈E R P v,i ( g v ) acts on k copies of the internal edges, ⊗ e i v ¯ v ∈E ∂R P v,i ( g v ) acts on the boundary semi-edges, 5 and finally P v,τ ( g v ) acting on k copies of the recoupled tag spin at the vertex .  \nAccordingly, the trace in (39) factorizes over the Hilbert spaces of a) internal edges, b) boundary spins, c) tags spins and d) bulk intertwiners. As a result, we can eventually rewrite the k th power of the normalized partition function in (34) as  \nZ ( k ) 1 Z ( k ) 0 = ∑ { g v } e -A ( k ) 1 / 0 , (41)  \nwhere A ( k ) 1 / 0 ≡ A ( k ) 1 /A ( k ) 0 defines the (normalized) 6 action of a classical generalized Ising-like model:  \nA ( k ) 1 [ { g v } ] = A b + ξ + ∑ e i vw ∈E R ∆( g v , g w ) log d j i vw + ∑ e i v ¯ v ∈ A 1 ∆( g v , X ) log d j i v ¯ v + ∑ e i v ¯ v ∈ A 2 ∆( g v , X -1 ) log d j i v , (43)  \nwith ξ a constant term and  \nA b = -log { Tr [ ρ ⊗ k b ⊗ P B ( ✶ ) · · ( ⊗ v P v, 0 ( g v ) ⊗ P v,τ ( g v ) )]} (44)  \nthe bulk state contribution to the action, which accounts for the presence of curvature measured by the tags term in ρ b . In the analogy, the generalized spins correspond  \nA ( k ) 0 = ∑ e i vw ∈E R ∆( g v , g w ) log d j i vw + ∑ e i v ¯ v ∈ A 1 ∪ A 2 ∆( g v , ✶ ) log d j i v ¯ v + A b + ξ . (42)  \nto the permutation operators P v ( g v ) sitting on the vertices of the open spin network graph ˜ γ R acting on the k replicas of the vertex indices space. The permutations fields X , X -1 and ✶ play the role of boundary conditions set on the virtual one-valent vertices (¯ v ) attached to the dangling spins and tags of the extended boundary.  \nThe two-body interaction terms measure the length of the permutation-loops across the replicas, weighted by the (log of the) dimension of the representation carried by the edges of the graph. These lengths are conveniently described as Cayley distances ∆( g, h ) = k -χ ( g -1 h ) on the permutation group S k , where χ ( g ) indicates the number of cycles in a permutation g [60, 69]. 7  \nThe calculation of typical value of R\'enyi negativity is then mapped to the minimization of the action (43). In general, the three boundary conditions will propagate inside the spin network graph ˜ γ R . Depending on the couplings and the connectivity of the graph, different configurations of the model will be associated to different sets of domains for X,X -1 , 1 in the bulk. As usual, we expect maximal domains to be associated to equilibrium configurations. In particular, maximal domains correspond to minimal domain walls.  \nBy describing interactions via a Cayley distance we see that the R\'enyi negativity should depend directly on the area of such minimal domain walls in the network, given by the number of edges of ˜ γ R associated to a nonvanishing Cayley distance between permutations in the two domains. In particular, thanks to the existence of a well defined loop quantum gravity area operator acting on the edges of the spin network, we can map domain walls areas to actual quantum geometry surfaces in our 3 D space region R .', 'VI. GENERALIZED ISING MODEL: MODELLING BULK CONTRIBUTIONS': "The tensor product form of the edges and bulk density matrices in (39) corresponds to a factorization of the typical R'enyi negativity into a combinatorial contribution associated to the distribution of maximally entangled edge states e vw and a bulk contribution carrying physical correlations among intertwiners. The latter, in particular, contains the information on the intrinsic curvature of the 3 D geometry, in the form of tags, as apparent from (21). We would expect such information to be highly mixed in the bulk. However, after the partial tracing over the loop holonomies, the SU (2)-covariance of the tagged-bulk state induces SU (2)-invariance on the associated density matrix, which necessarily reads as a  \ntensor product for fixed tags spins { J v } , that is  \nρ b = V R ⊗ v ✶ 2 J v +1 ⊗ ρ { J v } (45)  \nwhich corresponds to a totally mixed state on the tags space and a generally non-trivial density matrix ρ { J v } ∈ End[ H V R ] [84, 86].  \nFrom the form of ρ b in (45), we have  \nρ ⊗ k b = V R ⊗ v =1 ( ✶ 2 J v +1 ) ⊗ k ⊗ ρ ⊗ k { J v } , (46)  \nthen the bulk term A b generally factorizes as follows  \nA b = -log { Tr [ ρ ⊗ k { J v } ⊗ ⊗ v ( ✶ 2 J v +1 ) ⊗ k (47) ⊗ P B ( ✶ ) ( ⊗ v P v, 0 ( g v ) ⊗ P v,τ ( g v ) )]} = A ι + A τ  \nwith a bulk intertwiner contribution A ι defined as  \nA ι ≡ -log { Tr [ ρ ⊗ k { J v } ( ⊗ v P v, 0 ( g v )) ]} , (48)  \nand a contribution of the tags, A τ , given by  \nA τ ≡ -log { Tr [ ⊗ v ( ✶ D J v ) ⊗ k ⊗ P B ( ✶ ) · · ( ⊗ v P v,τ ( g v ) ) ]} , (49)  \nwhere we used D J v = 2 J v +1, the dimension of the tag space. The result of the trace in (49) is  \n( ∏ v ∈V R D -k J v ) Tr   ⊗ 〈 v ¯ v 〉   ∑ g v ∈ S k P v ( g v )   ⊗ P B ( ✶ )   = ∑ { g v } ∏ v D -∆( g v , ✶ ) J v . (50)  \nTherefore, the tags contribution to the action reads  \nA τ = ∑ v ∆( g v , ✶ ) log D J v , (51)  \nwhich ends up being equivalent to a pairwise interaction term from an inner edge. The full action can be written  \nA ( k ) 1 = ξ + A ι + ∑ e i vw ∈E R ∆( g v , g w ) log d j i vw + ∑ e i v ¯ v ∈ A 1 ∆( g v , X ) log d j i v + ∑ e i v ¯ v ∈ A 2 ∆( g v , X -1 ) log d j i v + ∑ v ∆( g v , ✶ ) log D J v (52)  \nThe bulk contribution A ι is nontrivial as long as the bulk intertwiners are correlated. In fact, it vanishes only if we consider ρ J v to be a product state in End[ H V R ] as shown in [60]. On the other hand, both edges and tags contributions have to do with the topology of the graphs support ˜ γ R , connectivity and topological defects respectively.  \nAs a general result of our analysis, we can then formally divide the action in three terms as follows  \nA ( k ) 1 = A ( k ) topology + A ( k ) phys (53)  \nwith A ( k ) topology = A ( k ) edges + A ( k ) tags and A phys = A ι referring to the physical quantum correlations among bulk intertwiners.  \nIn the following analysis, we will consider the case where the intertwiners contribution is small, thereby looking at the correlations induced solely by the combinatorial structure of the graph, with a focus on the role of the tags. We leave the study of the intertwiners contribution in this setting for future work.", 'VII. LOGARITHMIC NEGATIVITY AND ENTANGLEMENT PHASES': 'By neglecting A ι , we are left with action A ( k ) 1 with two-body interactions only, which favours neighbouring \'spins\' to be parallel. Given d j = 2 j + 1 and D J = 2 J +1 the dimension of edge spins and tag Hilbert space respectively, we define  \nlog d j ≡ β , log D J ≡ β t (54)  \nthe interaction strengths intended as inverse temperatures, in analogy with the Ising model. Thereby, we can rewrite the action A ( k ) topology as  \nA ( k ) topology = β [ ∑ e i vw ∈E R ∆( g v , g w ) + ∑ e i v ¯ v ∈ A 1 ∆( g v , X ) + ∑ e i v ¯ v ∈ A 2 ∆( g v , X -1 ) ] + β t ∑ v ∈V R ∆( g v , ✶ ) = β H e + β t H tags (55)  \nwith generalised hamiltonians H e , H tags expressing the energy cost of the configuration. In general, we expect that the boundary conditions X , X -1 and 1 percolate  \ninside the network, flipping the generalized spins on the bulk vertices. This creates some internal domains filled with one of the three boundary conditions. Nonvanishing contributions to the action come from edges that connect different domains, which will be related to a non-zero Cayley distance. Given  \nN k ( ρ A 1 A 2 ) glyph[similarequal] ∑ { g v } e -A ( k ) topology / 0 , (56)  \nin the large spin (the strong coupling or \'low temperature\') regime, the leading contribution to N k corresponds to dominant { g v } configurations that minimize the action. Such configurations correspond to maximal uniform spin domains separated by extremal domain walls which minimize the energy cost (minimal surface areas) [60].  \nConcretely, we then proceed minimizing the action in (55) via a heuristic argument based on the property of the Cayley distance on the permutation group S k . Given two permutations g, h ∈ S k , the Cayley distance ∆( g, h ) defines a metric on the group, corresponding to the minimum number of swaps acting on g to give h . For some special permutations, the Cayley distance depends nicely on the order on the group k . For example, in the case of the (anti-) cyclic permutations we deal with, we have  \n∆( ✶ , X ) = ∆( ✶ , X -1 ) = k -1 (57)  \n∆( X,X -1 ) = { k -1 , k odd k -2 , k even (58)  \nWith this metric, we can define a geodesic on the group S k between g and k as the set of permutations { π } such that  \n∆( g, π ) + ∆( π, h ) = ∆( g, h ) (59)  \nIn particular, it is known that the set of permutations that are simultaneously geodesics between X , X -1 and 1 exist, and it is in a bijection with the set of non crossing partitions NC ( k ) of the set { 1 , . . . , k } , with cardinality given by the Catalan numbers C k [69]. Defining by τ the permutations on such geodesics, we have that  \n∆( 1 , τ ) = ⌊ k 2 ⌋ , ∆( X,τ ) = ∆( X -1 , τ ) = ⌈ k 2 ⌉ -1 (60)  \nGiven the homogeneity of our states, the minimization of the energy costs amounts to a minimization of the Cayley distances between any pairs of connected vertices.  \nLet us consider a single vertex v . To such a vertex the random averaging assigns a permutation element g v , which will eventually interact with the three permutations we choose as boundary conditions, namely X , X -1 and 1 . Recalling the symmetry between X and X -1 , we expect that only three possible minimal energy configurations can result from such interaction:  \n<latexit sha1\\_base64="DmdMzX+D+QW44o+0Ra4eMLVB8zs=">AAAB8XicbVBNSwMxEJ2tX7V+VT16CRbBU9kVUY9FLx6r2A9sl5JNp21oNrskWaEs/RdePCji1X/jzX9jtt2Dtj4IPN6bmcy8IBZcG9f9dgorq2vrG8XN0tb2zu5eef+gqaNEMWywSESqHVCNgktsGG4EtmOFNAwEtoLxTea3nlBpHskHM4nRD+lQ8gFn1FjpsRtTZTgV5L5XrrhVdwayTLycVCBHvVf+6vYjloQoDRNU647nxsZPs3lM4LTUTTTGlI3pEDuWShqi9tPZxlNyYpU+GUTKPmnITP3dkdJQ60kY2MqQmpFe9DLxP6+TmMGVn3IZJwYlm380SAQxEcnOJ32ukBkxsYQyxe2uhI2ooszYkEo2BG/x5GXSPKt6F1Xv7rxSu87jKMIRHMMpeHAJNbiFOjSAgYRneIU3RzsvzrvzMS8tOHnPIfyB8/kDOISQnw==</latexit>  \n<latexit sha1\\_base64="BG0hmqEYPi4+wZRv+Q+p9girVSw=">AAAB6HicbVBNS8NAEJ34WetX1aOXxSJ4KomIeix68diK/YA2lM120q7dbMLuRiihv8CLB0W8+pO8+W/ctjlo64OBx3szzMwLEsG1cd1vZ2V1bX1js7BV3N7Z3dsvHRw2dZwqhg0Wi1i1A6pRcIkNw43AdqKQRoHAVjC6nfqtJ1Sax/LBjBP0IzqQPOSMGivV73ulsltxZyDLxMtJGXLUeqWvbj9maYTSMEG17nhuYvyMKsOZwEmxm2pMKBvRAXYslTRC7WezQyfk1Cp9EsbKljRkpv6eyGik9TgKbGdEzVAvelPxP6+TmvDaz7hMUoOSzReFqSAmJtOvSZ8rZEaMLaFMcXsrYUOqKDM2m6INwVt8eZk0zyveZcWrX5SrN3kcBTiGEzgDD66gCndQgwYwQHiGV3hzHp0X5935mLeuOPnMEfyB8/kDrpmM3A==</latexit>  \nFIG. 6: Example: tagged open spin network graph corresponding to a cluster of four-valent tagged vertices, with T = 4 and E ∂R = 4.  \n<!-- image -->  \n- · g v = X,X -1 , in which case we minimize the distance between the vertex v and the neighboring vertices respectively contained in the X or X -1 domain;\n- · g v = 1 , in which case we minimize the distance between the vertex v and the neighboring vertices contained in the 1 domain;\n- · g v = τ , in which we minimize simultaneously the distance between the X , X -1 and the 1 domains.  \nNotice that the latter case is peculiar to the tripartite model only, since it can occur only when all three permutations interact in a single vertex [60, 67, 69].', 'A. Example: Cluster of four tags': 'Let us consider the example of an open spin network state with support on a graph ˜ γ R with E ∂R = 4 boundary edges, and V R = 4 tagged 4-valent vertices in the bulk (see Fig. 6). In the following we indicate with T the number of tags, which in the present case is equal to the number of vertices, namely T = V R . We refer to this setting as a cluster of tags , corresponding to a uniformly curved quantum 3 D space region R . Via random averaging, we get four permutations operators on the spin network, one for each tagged vertex. Following the discussion above, we have three possible configurations with minimal energy. The corresponding actions (for even k > 0) reads  \n- 1. cluster colored with X ( X -1 ):  \nA ( X ) k = β S ( k -2) + β t T ( k -1) (61)  \nHere S is the minimal number of edges cuts comprising the surface separating two domains with X and X -1 .  \n- 2. cluster colored with 1 :  \nA ( 1 ) k = βE ∂R ( k -1) (62)  \n<latexit sha1\\_base64="DmdMzX+D+QW44o+0Ra4eMLVB8zs=">AAAB8XicbVBNSwMxEJ2tX7V+VT16CRbBU9kVUY9FLx6r2A9sl5JNp21oNrskWaEs/RdePCji1X/jzX9jtt2Dtj4IPN6bmcy8IBZcG9f9dgorq2vrG8XN0tb2zu5eef+gqaNEMWywSESqHVCNgktsGG4EtmOFNAwEtoLxTea3nlBpHskHM4nRD+lQ8gFn1FjpsRtTZTgV5L5XrrhVdwayTLycVCBHvVf+6vYjloQoDRNU647nxsZPs3lM4LTUTTTGlI3pEDuWShqi9tPZxlNyYpU+GUTKPmnITP3dkdJQ60kY2MqQmpFe9DLxP6+TmMGVn3IZJwYlm380SAQxEcnOJ32ukBkxsYQyxe2uhI2ooszYkEo2BG/x5GXSPKt6F1Xv7rxSu87jKMIRHMMpeHAJNbiFOjSAgYRneIU3RzsvzrvzMS8tOHnPIfyB8/kDOISQnw==</latexit>  \n∂R  \nns of  \ntary representations of  \nf the  \non the  \nk  \ncopies of the  \nj  \ne  \nk  \n}  \n,  \nP  \nv  \n(  \ng  \nv  \n)  \n,  \n(22)  \n(22)  \n(  \nk  \n)  \n1  \nA  \n)  \n,  \n(22)  \n+  \nA  \n[  \n(  \nk  \n)  \ndefined by the action  \nlemma, see [  \n?  \n]) in a sum over unitary representations of  \n1  \nA  \n(  \nk  \n)  \ndefined by the action  \n1  \n5  \ng  \nv  \nlemma, see [  \n?  \n]) in a sum over unitary representations of  \n=  \n5  \n∆(  \ng  \ng  \nv  \n=  \n∆(  \ng  \nv  \n,g  \nw  \nthe permutation group  \ng  \ni  \n{  \nvw  \n(  \nk  \n)  \n1  \n[  \ne  \nA  \n) log  \nd  \nj  \ne  \nJv  \n) log  \nd  \n+  \nw  \n,g  \ni  \nsingle vertex degeneracy space  \nM  \nsingle vertex degeneracy space  \nM  \n]  \n[  \n{  \n}  \nv  \n∈  \n∑  \n{  \nrepresentations of  \ntations of  \nhe  \nk  \ncopies of the  \nies of the  \n(22)  \nv  \n∑  \ng  \n∈  \nS  \nk  \nions of  \nA  \n(  \nk  \n)  \nA  \n1  \nA  \n∆(  \ng  \n[  \n∑  \n(  \n]  \nv  \nthe permutation group  \ng  \nR  \nacting on the  \nk  \ncopies of the  \nR  \nE  \nsingle vertex degeneracy space  \nM  \n(  \ni  \nf  \nvw  \nv  \n∑  \n,X  \n) log  \nd  \ni  \n∑  \n+  \n]  \n∈  \nE  \n}  \n∆(  \ng  \nv  \ng  \nv  \nj  \ni  \nvw  \n+  \nj  \nvw  \n{  \n+  \n∆(  \ng  \n1)!  \nv  \nj  \ne  \nJv  \n{  \ni  \nacting on the  \nk  \ncopies of the  \nsingle vertex degeneracy space  \nM  \n|  \n∆(  \ng  \nf  \nR  \nv  \ni  \nvw  \nw  \n∑  \n,g  \nv  \n⊗  \nk  \nv  \n=  \n,X  \n) log  \nd  \nf  \nj  \n}  \n=  \n∆(  \ng  \ni  \n}  \ne  \n=  \n(  \nD  \nv  \nv  \n) log  \nd  \n+  \ne  \nE  \n∈  \ne  \ni  \nv  \n)  \nvw  \n(  \nD  \n〉〈  \n|  \n|  \n〉〈  \ndefined by the action  \ndefined by the action  \nv  \n∑  \n,X  \n) log  \nd  \n[  \n⊗  \n{  \nk  \nj  \n)  \n5  \ni  \nv  \n+  \n∈  \nv  \n¯  \nv  \nv  \nf  \n∈  \nj  \ne  \nJv  \nv  \n+  \nj  \ni  \nvw  \nk  \n⊗  \n(  \nD  \nv  \nv  \nE  \nj  \n+  \nk  \n¯  \n∑  \n5  \ni  \nv  \nv  \n¯  \nv  \n∑  \nv  \n{  \n,  \n,g  \n(  \nD  \n}  \n,X  \n-  \n(  \nD  \n∆(  \ng  \ni  \n-  \n=  \nR  \n)  \nv  \n1)!  \ni  \nv  \n¯  \n+  \n∈  \n∑  \n∈  \n|  \nv  \n∈  \n∆(  \ng  \nv  \n1)!  \nv  \nv  \n-  \ne  \nk  \ni  \nv  \n,X  \n-  \ng  \nw  \n∑  \n) log  \nd  \nv  \n1  \n∆(  \ng  \n+  \nk  \n¯  \n(  \nD  \nA  \nv  \n}  \n,  \n-  \n(  \nD  \n) log  \nd  \n1)!  \n) log  \nd  \n∈  \n∑  \nP  \nv  \n-  \nv  \nk  \nS  \nv  \nv  \nj  \ni  \n(  \ng  \nv  \nj  \n,X  \n-  \nvw  \ni  \n(  \ng  \nj  \n1)!  \nP  \nv  \n-  \n+  \n+  \n]  \n∆(  \ng  \n|  \n〉〈  \n|  \nwith dimension  \nv  \ng  \n+  \n∑  \n∑  \n}  \n+  \n1  \n+  \nv  \ng  \n)  \n,  \n(22)  \nv  \n,  \n{  \n5  \n}  \n,  \nP  \nj  \ne  \nJv  \n) log  \nd  \n∈  \n1)!  \nv  \nv  \nS  \n,X  \n-  \n)  \n,  \n(22)  \n5  \n1)!  \n(27)  \ntags  \n∆(  \ng  \n).  \n-  \nj  \ne  \nJv  \n+  \nA  \n∑  \nk  \n1  \n∑  \ng  \n+  \nξ  \n(27)  \nv  \n(  \ng  \nj  \ni  \nv  \n+  \nv  \nP  \n5  \n)  \n,  \n(22)  \nv  \n(  \ng  \ni  \nv  \nj  \n+  \nA  \n[  \nv  \n(  \nk  \n)  \n1  \nA  \n)  \n,  \n(22)  \nv  \n5  \n{  \n}  \n[  \nv  \n∈  \ng  \n]  \n∆(  \ng  \n]  \nA  \n(  \nk  \n)  \ndefined by the action  \n(  \nk  \n)  \n1  \n{  \ng  \ne  \n∈  \n+  \ni  \nv  \n¯  \n∑  \n∑  \nv  \ni  \nv  \n¯  \nA  \nwith  \nξ  \n∈  \n∑  \nwith dimension  \nry representations of  \nn the  \nk  \ncopies of the  \nof the  \n,  \n(22)  \ne  \ndefined by the action  \nally on each inde-  \nach inde-  \nlly we obtain  \ntain  \nk  \n}  \nP  \n,  \n)  \n,  \n(22)  \n(  \nk  \n)  \n1  \nA  \nv  \n(  \ng  \ninde-  \nv  \n+  \nwith  \nξ  \ng  \nX By performing the average, individually on each inde-  \ni  \nv  \n¯  \n{  \nwith  \nξ  \na constant term and  \nvidually on each inde-  \ntually we obtain  \n-  \n}  \n1  \n) log  \nd  \n∈  \ntags  \nS  \nk  \n+  \nξ  \n(27)  \nBy performing the average, individually on each inde-  \nBy performing the average, individually on each inde-  \ni  \nv  \n).  \n) log  \nd  \n+  \n+  \nv  \n=  \nBy performing the average, individually on each inde-  \npendent random vertex state, eventually we obtain  \ntags  \n}  \nv  \npendent random vertex state, eventually we obtain  \n∈  \n=  \n-  \ni  \n)  \n,  \n(22)  \nA  \nv  \n(  \ng  \nv  \nP  \nv  \nP  \n∑  \n(22)  \nP  \nv  \nv  \n∈  \n(  \ng  \nS  \nk  \n)  \nv  \n·  \nv  \nA  \n+  \ntags  \n[  \n=  \nv  \n¯  \nv  \nA  \nwith  \nξ  \ne  \n∑  \nally on each inde-  \nach inde-  \nS  \ntain  \n⊗  \n,  \n(23)  \nk  \n1  \n,  \n(23)  \n(  \ng  \n)  \nlly we obtain  \nC  \nP  \n∈  \n(23)  \n{  \n∑  \n]  \n∆(  \ng  \nA  \n-  \nlog  \nZ  \nthe bulk state contribution.  \n).The ac-  \ntion  \nA  \n{  \n(  \nk  \n)  \n1  \ntion  \nj  \ni  \nv  \npendent random vertex state, eventually we obtain  \npendent random vertex state, eventually we obtain  \nv  \n(27)  \nP  \n)  \nv,  \n0  \n,X  \n-  \n]  \ntags  \nv  \nv  \n∑  \n}  \n(  \ng  \n1  \nv  \n·  \nk  \n+  \nξ  \n(27)  \n(28)  \n·  \n∈  \nv  \n??  \n).The ac-  \n⊗  \nS  \nA  \n(27)  \nC  \nv  \n,X  \n) log  \nd  \n⊗  \nj  \nf  \nv  \nv  \n)  \n(  \n=  \n-  \nv  \n(  \nf  \ne  \nf  \nv  \nf  \n=  \n<!-- image -->  \n]  \n+  \n]  \ni  \n}  \nv  \n))  \n(  \ng  \n) log  \nd  \nP  \nv  \nv,  \n0  \n(28)  \n(  \ng  \nP  \nj  \n)  \nv  \nv  \nv  \n))  \n(  \ng  \n+  \n·  \nv  \ndescribes a two-body interaction between  \n}  \ng  \npermutation elements, which therefore act as  \ngeneralized  \n{  \n(  \nk  \n)  \n1  \npermutation elements, which therefore act as  \ngeneralized  \n·  \nthe bulk state contribution.  \n(  \n)  \n⊗  \n]  \nC  \nA  \ntion  \nA  \n⊗  \nk  \ntags  \nA  \nv  \n¯  \nlog  \n∈  \nv  \n=  \nTr  \n[  \n]  \n[  \n]  \n(  \nk  \n)  \n1  \ng  \nv  \n(  \n)  \n) and (  \n??  \n).The ac-  \nX  \n-  \n⊗  \ndescribes a two-body interaction between  \ntion  \nA  \n(  \nk  \n)  \n1  \ng  \nv  \n1  \nP  \n+  \nξ  \n(27)  \nk  \n)  \ntags  \nlog  \nTr  \nv  \n¯  \nv  \n∈  \nbulk  \nψ  \nψ  \ntags  \n⊗  \nk  \nC  \n(  \nv  \nP  \nv,  \n0  \n∑  \n⊗  \n,  \n(23)  \n1  \n(  \ng  \nv  \nC  \n(  \n)  \nwith  \nξ  \na constant term and  \n}  \nA  \n)  \nA  \ntags  \n(28)  \n,  \n(23)  \ntags  \n=  \n=  \n-  \n-  \nlog  \nthe bulk state contribution.  \n).The ac-  \ntion  \nA  \n{  \ntion  \n) log  \nd  \n+  \nw  \ni  \nvw  \nv  \n,g  \n5  \n∆(  \ng  \n) log  \nd  \ndefined by the action  \n1  \n=  \ni  \nvw  \nv  \n[  \n∑  \n5  \ng  \nv  \n∈  \n{  \ni  \nvw  \n(  \nk  \n)  \n1  \nA  \n,X  \n) log  \nd  \ni  \n∑  \n+  \nv  \n}  \n=  \n∆(  \ng  \nv  \ne  \n,X  \n) log  \nd  \n+  \n∑  \nE  \ne  \n∈  \nv  \n¯  \nv  \n+  \nR  \n∈  \n∑  \nj  \n+  \n∆(  \ng  \n+  \nj  \n,g  \n∈  \n∑  \n∆(  \ng  \n=  \nv  \ni  \nvw  \nw  \n,g  \nv  \nv  \n]  \nE  \nR  \n}  \n∆(  \ng  \ng  \n∆(  \ng  \n[  \n{  \nj  \nv  \n,g  \n,X  \n-  \nv  \ni  \nv  \n¯  \n+  \n∈  \nw  \nE  \n) log  \nd  \n=  \n,X  \n) log  \nd  \nj  \n+  \n∑  \ni  \nvw  \ne  \ne  \ni  \nv  \n¯  \n∑  \nv  \nR  \n∈  \n∈  \ni  \nv  \ni  \nvw  \ni  \nv  \n∆(  \ng  \nR  \nv  \nE  \nj  \n¯  \nA  \n∆(  \ng  \n,X  \n-  \nw  \nj  \ni  \nvw  \n) log  \nd  \n∆(  \ng  \n1  \n) log  \nd  \n) log  \nd  \nj  \nv  \ni  \nv  \n+  \nj  \ni  \nvw  \ni  \nv  \n,X  \n-  \n+  \ne  \ni  \n}  \nv  \n]  \n∆(  \ng  \n+  \nv  \n,X  \n) log  \nd  \nj  \ne  \ni  \nv  \n∆(  \ng  \nv  \n+  \n1  \n) log  \nd  \n+  \n,X  \n-  \n(27)  \ntags  \n+  \nA  \nv  \nP  \n(  \n1  \n+  \nξ  \n(27)  \ntags  \nv,  \n0  \nv  \n}  \n(28)  \n??  \n).The ac-  \n⊗  \n+  \nξ  \n(27)  \nv  \n(  \ng  \nP  \nv,  \n0  \n))  \n(  \ng  \n}  \nv  \n(28)  \n))  \n}  \n(28)  \n]  \n]  \ndescribes a two-body interaction between  \ng  \npermutation elements, which therefore act as  \ngeneralized  \n<!-- image -->  \n(  \nk  \n)  \n1  \npermutation elements, which therefore act as  \ngeneralized  \n·  \nthe bulk state contribution.  \n∈  \ndescribes a two-body interaction between  \npermutation elements, which therefore act as  \ngeneralized  \n]  \nidually on each inde-  \ninde-  \n-  \ntags  \n)  \nspins  \n,  \nattached to the spin network vertices.  \nThese in-  \nP  \n(  \n)  \n,  \n(23)  \nwith  \nξ  \na constant term and  \nually we obtain  \nP  \nv  \n(  \ng  \nv  \n)  \n·  \n·  \nilbert spaces of a)  \nces of a)  \n)  \nme form  \n)  \n,  \n(23)  \nP  \ntwin-  \n(  \nk  \n)  \n0  \n⊗  \nS  \nform  \nk  \naces of a)  \nion oper-  \nC  \n-  \n1  \nh vertex,  \nP  \nv  \n(  \ng  \nC  \n⊗  \n(  \n)  \n,  \nP  \n(23)  \n=  \nbulk  \n))  \nψ  \ntags  \nψ  \ntags  \n(  \nv  \nP  \nv,  \n0  \n(  \ng  \nv  \n))  \nspins  \n,  \nattached to the spin network vertices.  \nThese in-  \n(  \n)  \n(  \n)  \nP  \nA  \n(  \nX  \n)  \nP  \nB  \nX  \n-  \n1  \npermutation elements, which therefore act as  \ngeneralized  \nwith  \nξ  \na constant term and  \n[  \n|  \n⊗  \nk  \n1  \n]  \n⊗  \nP  \n(  \n)  \n,  \n(23)  \nP  \nA  \n(  \nX  \n)  \nP  \nB  \nX  \n-  \nteractions are described by the Cayley distance ∆(  \ng,h  \n),  \nteractions are described by the Cayley distance ∆(  \ng,h  \n),  \nA  \n)  \nfor  \n(  \nintertwin-  \noper-  \nrtex,  \n)  \name form  \nerators,  \ns of a)  \ntors,  \nv  \nge  \nnally  \n)  \nP  \nv,  \ntag  \n/negationslash  \n/negationslash  \nthe tags spin, we define the strength of the interactions  \nmension limit, corresponding to some strong coupling or  \nv  \n(  \ng  \nv  \n)  \n,  \nβ  \n,  \nlog  \nD  \nJ  \nβ  \nt  \n}  \n{  \n(  \nk  \n)  \n1  \n[  \n]  \n[  \n]  \ntion  \nA  \ng  \nv  \n) and (  \n??  \n).The ac-  \ndescribes a two-body interaction between  \ntion  \nA  \n(  \nk  \n)  \n1  \ng  \nv  \ndescribes a two-body interaction between  \npermutation elements, which therefore act as  \ngeneralized  \n]  \nspins  \n,  \nattached to the spin network vertices.  \nThese in-  \nP  \n]  \nC  \n(  \n)  \n,  \n(23)  \nwhere the trace factorizes over the Hilbert spaces of a)  \n}  \nwhere the trace factorizes over the Hilbert spaces of a)  \n|  \n|  \n(  \n(  \nψ  \n(  \nD  \n〉〈  \n·  \nP  \ntags  \n∏  \n[  \n⊗  \nv  \n}  \n⊗  \n]  \n]  \n⊗  \n))  \n|  \n⊗  \n)  \nv,  \n0  \nψ  \nv  \n+  \nk  \nboundary conditions attached to the boundary and tags  \nv  \n|  \n〉〈  \n⊗  \n.  \nZ  \n(  \nk  \n)  \nv  \nv  \n1)!  \ntags  \n(  \ng  \ntags  \ng  \n(  \nhas the same form  \n1  \nv  \n.  \nZ  \n|  \n1)!  \n⊗  \n-  \n-  \n1)!  \n.  \nZ  \n=  \n/negationslash  \nP  \n0  \n0  \n(  \nD  \nplay the role of  \nreplaced by  \n.  \n(  \ng  \n))  \nv  \nP  \nv,  \n0  \n{  \nspins  \n,  \nattached to the spin network vertices.  \nThese in-  \nteractions are described by the Cayley distance ∆(  \ng,h  \n),  \nteractions are described by the Cayley distance ∆(  \ng,h  \n),  \n[  \n]  \nhas the same form  \n(  \nk  \n)  \n(  \ng  \ngeneralized  \n}  \n{  \n}  \n]  \n[  \n{  \n}  \n]  \npermutation elements, which therefore act as  \ngeneralized  \nboundary conditions attached to the boundary and tags  \nboundary conditions attached to the boundary and tags  \nhas the same form  \n(28)  \nv  \nand  \nplay the role of  \n6  \n6  \nning permutations fields  \nX  \n,  \nX  \n-  \n1  \nand  \nplay the role of  \nning permutations fields  \nX  \n,  \nX  \n-  \n1  \nand  \nplay the role of  \nboundary conditions attached to the boundary and tags  \nspins.  \nWe consider a quasi-homogeneous spin network, where  \nspins.  \nWe consider a quasi-homogeneous spin network, where  \nboundary conditions attached to the boundary and tags  \n/negationslash  \n/negationslash  \nthe dimension of the Hilbert spaces of the edges spin  \nthe dimension of the Hilbert spaces of the edges spin  \nnd c) bulk intertwin-) has the same form e permutation operedom at each vertex, nt sets of operators, ) ⊗ ⊗ (24) he multiplicity interts on k copies of the virtual vertex which edge e i v ¯ v , and finally of the tags. We indiedge which connects the trace can be deespectively for the a) and c) tags spins d) aspects of the comh power of the partical partition function g v } ] , (25) for g,h elements of the permutation group, while the pinning permutations fields X , X -1 and play the role of boundary conditions attached to the boundary and tags spins. 6 We consider a quasi-homogeneous spin network, where all the edges carry the same spin j and all tags carry the same spin J , with j = J generally. Given d j = 2 j + 1 the dimension of the Hilbert spaces of the edges spin and D J = 2 J +1 the dimension of the Hilbert space of the tags spin, we define the strength of the interactions respectively as log d j ≡ ≡ intended as inverse temperatures in analogy with standard Ising models. The action favours neighbouring \'spins\' to be parallel. Therefore, in the large spin di\'low temperature\' regime, the dominant configurations that minimize the action correspond to large uniform spin domains separated by domain walls, which in turn give the energy cost of the configuration. We expect maximal domains to be associated to equilibrium configurations. In particular, maximal domains correspond to minimal domain walls. 5 and A ( k ) 0 [ { g v } ] = ∑ e i vw ∈ L ∆( g v ,g w ) log d j i vw + + ∑ J i v ¯ v ∈ T R ∪ E ∂R ∆( g v , ) log d j i v + A ( ψ tags ) + ξ , (26) respctively. 6 One can equivalently think of the pinning permutations fields X , X -1 and as being attached to virtual 1-valent vertices on the extended boundary. rtwinform operertex, ators, interof the which finally e indinnects be dethe a) ins d) compartinction (25) for g,h elements of the permutation group, while the pinning permutations fields X , X -1 and play the role of boundary conditions attached to the boundary and tags spins. 6 We consider a quasi-homogeneous spin network, where all the edges carry the same spin j and all tags carry the same spin J , with j = J generally. Given d j = 2 j + 1 the dimension of the Hilbert spaces of the edges spin and D J = 2 J +1 the dimension of the Hilbert space of respectively as log d j ≡ β , log D ≡ intended as inverse temperatures in analogy with standard Ising models. The action favours neighbouring \'spins\' to be parallel. Therefore, in the large spin di\'low temperature\' regime, the dominant configurations that minimize the action correspond to large uniform spin domains separated by domain walls, which in turn give the energy cost of the configuration. We expect maximal domains to be associated to equilibrium configurations. In particular, maximal domains correspond to minimal domain walls. 5 and A ( k ) 0 [ { g v } ] = ∑ e i vw ∈ L ∆( g v ,g w ) log d j i vw + + ∑ J i v ¯ v ∈ T R ∪ E ∂R ∆( g v , ) log d j i v + A ( ψ tags ) + ξ , (26) respctively. 6 One can equivalently think of the pinning permutations fields X , X -1 and as being attached to virtual 1-valent vertices on the extended boundary. permutation operm at each vertex, sets of operators, ⊗ P v, tag ( g v ) , (24) multiplicity intern k copies of the on the boundary tual vertex which ge e i v ¯ v , and finally ge which connects e trace can be deectively for the a) c) tags spins d) pects of the comower of the partipartition function ] , (25) spins. 6 We consider a quasi-homogeneous spin network, where all the edges carry the same spin j and all tags carry the same spin J , with j = J generally. Given d j = 2 j + 1 the dimension of the Hilbert spaces of the edges spin and D J = 2 J +1 the dimension of the Hilbert space of the tags spin, we define the strength of the interactions respectively as log d j ≡ β , log D J ≡ β t (29) intended as inverse temperatures in analogy with standard Ising models. The action favours neighbouring \'spins\' to be parallel. Therefore, in the large spin dimension limit, corresponding to some strong coupling or \'low temperature\' regime, the dominant configurations that minimize the action correspond to large uniform spin domains separated by domain walls, which in turn give the energy cost of the configuration. We expect maximal domains to be associated to equilibrium configurations. In particular, maximal domains correspond to minimal domain walls. 5 and A ( k ) 0 [ { g v } ] = ∑ e i vw ∈ L ∆( g v ,g w ) log d j i vw + + ∑ J i v ¯ v ∈ T R ∪ E ∂R ∆( g v , ) log d j i v + A ( ψ tags ) + ξ , (26) respctively. 6 One can equivalently think of the pinning permutations fields X , X -1 and as being attached to virtual 1-valent vertices on the extended boundary. ion operch vertex, perators, v ) , (24) city interies of the boundary ex which nd finally connects an be defor the a) spins d) the comthe partifunction (25) spins. 6 We consider a quasi-homogeneous spin network, where all the edges carry the same spin j and all tags carry the same spin J , with j = J generally. Given d j = 2 j + 1 the dimension of the Hilbert spaces of the edges spin and D J = 2 J +1 the dimension of the Hilbert space of the tags spin, we define the strength of the interactions respectively as log d j ≡ β , log D J ≡ β t (29) intended as inverse temperatures in analogy with standard Ising models. The action favours neighbouring \'spins\' to be parallel. Therefore, in the large spin dimension limit, corresponding to some strong coupling or \'low temperature\' regime, the dominant configurations that minimize the action correspond to large uniform spin domains separated by domain walls, which in turn give the energy cost of the configuration. We expect maximal domains to be associated to equilibrium configurations. In particular, maximal domains correspond to minimal domain walls. 5 and A ( k ) 0 [ { g v } ] = ∑ e i vw ∈ L ∆( g v ,g w ) log d j i vw + + ∑ J i v ¯ v ∈ T R ∪ E ∂R ∆( g v , ) log d j i v + A ( ψ tags ) + ξ , (26) respctively. 6 One can equivalently think of the pinning permutations fields X , X -1 and as being attached to virtual 1-valent vertices on the extended boundary. ⊗ P v, tag ( g v ) , (24) multiplicity interon k copies of the s on the boundary tual vertex which e i v ¯ v , and finally the tags. We indige which connects e trace can be deectively for the a) d c) tags spins d) spects of the comower of the partil partition function } ] , (25) the tags spin, we define the strength of the interactions respectively as log d j ≡ β , log D J ≡ β t (29) intended as inverse temperatures in analogy with standard Ising models. The action favours neighbouring \'spins\' to be parallel. Therefore, in the large spin dimension limit, corresponding to some strong coupling or \'low temperature\' regime, the dominant configurations that minimize the action correspond to large uniform spin domains separated by domain walls, which in turn give the energy cost of the configuration. We expect maximal domains to be associated to equilibrium configurations. In particular, maximal domains correspond to minimal domain walls. 5 and A ( k ) 0 [ { g v } ] = ∑ e i vw ∈ L ∆( g v ,g w ) log d j i vw + + ∑ J i v ¯ v ∈ T R ∪ E ∂R ∆( g v , ) log d j i v + A ( ψ tags ) + ξ , (26) respctively. 6 One can equivalently think of the pinning permutations fields X , X -1 and as being attached to virtual 1-valent vertices on the extended boundary. ) , (24) ity interes of the oundary ex which d finally We indiconnects n be deor the a) spins d) the comhe partifunction (25) respectively as log d j ≡ β , log D J ≡ β t (29) intended as inverse temperatures in analogy with standard Ising models. The action favours neighbouring \'spins\' to be parallel. Therefore, in the large spin dimension limit, corresponding to some strong coupling or \'low temperature\' regime, the dominant configurations that minimize the action correspond to large uniform spin domains separated by domain walls, which in turn give the energy cost of the configuration. We expect maximal domains to be associated to equilibrium configurations. In particular, maximal domains correspond to minimal domain walls. 5 and A ( k ) 0 [ { g v } ] = ∑ e i vw ∈ L ∆( g v ,g w ) log d j i vw + + ∑ J i v ¯ v ∈ T R ∪ E ∂R ∆( g v , ) log d j i v + A ( ψ tags ) + ξ , (26) respctively. 6 One can equivalently think of the pinning permutations fields X , X -1 and as being attached to virtual 1-valent vertices on the extended boundary. (24) interf the dary hich indinects he a) ns d) comartiction (25) and D J = 2 J +1 the dimension of the Hilbert space of the tags spin, we define the strength of the interactions respectively as log d j ≡ β , log D J ≡ β t (29) intended as inverse temperatures in analogy with standard Ising models. The action favours neighbouring \'spins\' to be parallel. Therefore, in the large spin di\'low temperature\' regime, the dominant configurations that minimize the action correspond to large uniform spin the energy cost of the configuration. We expect maximal domains to be associated to equilibrium configurations. In particular, maximal domains correspond to minimal domain walls. 5 and A ( k ) 0 [ { g v } ] = ∑ e i vw ∈ L ∆( g v ,g w ) log d j i vw + + ∑ J i v ¯ v ∈ T R ∪ E ∂R ∆( g v , ) log d j i v + A ( ψ tags ) + ξ , (26) respctively. 6 One can equivalently think of the pinning permutations fields X , X -1 and as being attached to virtual 1-valent vertices on the extended boundary. rent sets of operators, ( g v ) ⊗ v ) ⊗ P v, tag ( g v ) , (24) the multiplicity intercts on k copies of the acts on the boundary virtual vertex which edge e i , and finally s of the tags. We indil edge which connects respectively for the a) and c) tags spins d) l aspects of the com-. th power of the partisical partition function { g v } ] , (25) the dimension of the Hilbert spaces of the edges spin and D J = 2 J +1 the dimension of the Hilbert space of the tags spin, we define the strength of the interactions respectively as log d j ≡ β , log D J ≡ β t (29) intended as inverse temperatures in analogy with standard Ising models. The action favours neighbouring \'spins\' to be parallel. Therefore, in the large spin dimension limit, corresponding to some strong coupling or \'low temperature\' regime, the dominant configurations that minimize the action correspond to large uniform spin domains separated by domain walls, which in turn give the energy cost of the configuration. We expect maximal domains to be associated to equilibrium configurations. In particular, maximal domains correspond to minimal domain walls. 5 and A ( k ) 0 [ { g v } ] = ∑ e i vw ∈ L ∆( g v ,g w ) log d j i vw + + ∑ J i v ¯ v ∈ T R ∪ E ∂R ∆( g v , ) log d j i v + A ( ψ tags ) + ξ , (26) respctively. 6 One can equivalently think of the pinning permutations fields X , X -1 and as being attached to virtual 1-valent vertices on the extended boundary. Such configurations define the three equilibrium phases of our simple model, corresponding to three different entanglement regimes for the boundary state ρ A 1 A 2 . The parameter space of such regimes is completely specified by the following three inequalities: A ( 1 ) k < A ( τ ) k ⇐⇒ βE ∂R < β t T (64) A ( τ ) k < A ( X ) k ⇐⇒ βE ∂R < β t T +2 βS (65) A ( X ) k < A ( 1 ) k ⇐⇒ βE ∂R > β t T + βS ( 1 -1 k -1 ) (66) where the last condition (66) is relevant to establish a hierarchy between the three configurations, while it does not play a role in the search for the minimal energy cost. Notably, this rules out from the parameter space the only condition which depends explicitly on the order of the permutation group k . Consider the (positive definite) quantity βE ∂R , namely the set of edges comprising the external surface of our 3 D region weighted with the factor β . If βE ∂R < β t T , as 2 βS > 0, also (65) is satisfied, then A ( 1 ) k < A ( X ) k , A ( τ ) k . The configuration with a bulk colored with g v = 1 ∀ v dominates. In particular, this corresponds to the formation of a domain wall around the cluster of tags. The domain wall sharply divides the spin network boundary from the bulk of the region R . We refer to such a regime as the hole regime (see Fig. 7). Notice that, due to the bound on the maximum dimension of the tag spin [74], in the present example we can have at most P v ( g v ) = P v, 0 ( g v ) ⊗ ⊗ e i vw ∈ E R P v,i ( g v ) ⊗ ⊗ ⊗ e i v ¯ v ∈ E ∂R P v,i ( g v ) ⊗ P v, tag ( g v ) , (24) where P v, 0 ( g v ) acts on k copies of the multiplicity intertwiner space, ⊗ e i vw ∈ E R P v,i ( g v ) acts on k copies of the internal edges, ⊗ e i v ¯ v ∈ E ∂R P v,i ( g v ) acts on the boundary semi-edges, with ¯ representing a virtual vertex which is connected to v by the boundary edge e i v ¯ v , and finally ⊗ J v i ∈ T R P v,i ( g v ) acting on k copies of the tags. We indicate with e i vw ∈ E R the i th internal edge which connects the vertex v to the vertex w . Accordingly, the computation of the trace can be decomposed into four contributions, respectively for the a) internal edges, b) boundary spins and c) tags spins d) bulk intertwiners (for the technical aspects of the computation we refer the reader to []). As a result, the (normalised) k -th power of the partition function is mapped to the classical partition function of a generalized Ising-like model, Z ( k ) 1 / 0 = ∑ { g v } e -A ( k ) 1 / 0 [ { g v } ] , (25) respectively as log d j ≡ β , log D J ≡ β t (29) intended as inverse temperatures in analogy with standard Ising models. The action favours neighbouring \'spins\' to be parallel. Therefore, in the large spin dimension limit, corresponding to some strong coupling or \'low temperature\' regime, the dominant configurations that minimize the action correspond to large uniform spin domains separated by domain walls, which in turn give the energy cost of the configuration. We expect maximal domains to be associated to equilibrium configurations. In particular, maximal domains correspond to minimal domain walls. 5 and A ( k ) 0 [ { g v } ] = ∑ e i vw ∈ L ∆( g v ,g w ) log d j i vw + + ∑ J i v ¯ v ∈ T R ∪ E ∂R ∆( g v , ) log d j i v + A ( ψ tags ) + ξ , (26) respctively. 6 One can equivalently think of the pinning permutations fields X , X -1 and as being attached to virtual 1-valent vertices on the extended boundary. P v ( g v ) = P v, 0 ( g v ) ⊗ ⊗ e i vw ∈ E R P v,i ( g v ) ⊗ ⊗ ⊗ e i v ¯ v ∈ E ∂R P v,i ( g v ) ⊗ P v, tag ( g v ) , (24) where P v, 0 ( g v ) acts on k copies of the multiplicity intertwiner space, ⊗ e i vw ∈ E R P v,i ( g v ) acts on k copies of the internal edges, ⊗ e i v ¯ v ∈ E ∂R P v,i ( g v ) acts on the boundary semi-edges, with ¯ v representing a virtual vertex which is connected to v by the boundary edge e i v ¯ v , and finally ⊗ J v i ∈ T R P v,i ( g v ) acting on k copies of the tags. We indicate with e i vw ∈ E R the i th internal edge which connects the vertex v to the vertex w . Accordingly, the computation of the trace can be deinternal edges, b) boundary spins and c) tags spins d) bulk intertwiners (for the technical aspects of the computation we refer the reader to []). As a result, the (normalised) k -th power of the partition function is mapped to the classical partition function of a generalized Ising-like model, Z ( k ) 1 / 0 = ∑ { g v } e -A ( k ) 1 / 0 [ { g v } ] , (25) respectively as log d j ≡ β , log D J ≡ β t (29) intended as inverse temperatures in analogy with standard Ising models. The action favours neighbouring \'spins\' to be parallel. Therefore, in the large spin dimension limit, corresponding to some strong coupling or \'low temperature\' regime, the dominant configurations that minimize the action correspond to large uniform spin domains separated by domain walls, which in turn give the energy cost of the configuration. We expect maximal domains to be associated to equilibrium configurations. In particular, maximal domains correspond to minimal domain walls. 5 and A ( k ) 0 [ { g v } ] = ∑ e i vw ∈ L ∆( g v ,g w ) log d j i vw + + ∑ J i v ¯ v ∈ T R ∪ E ∂R ∆( g v , ) log d j i v + A ( ψ tags ) + ξ , (26) respctively. 6 One can equivalently think of the pinning permutations fields X , X -1 and as being attached to virtual 1-valent vertices on the extended boundary. we factorize P v ( g v ) into four P v ( g v ) = P v, 0 ( g v ) ⊗ ⊗ e i vw ∈ E R P v,i ( g v ) ⊗ ⊗ ⊗ e i v ¯ v ∈ E ∂R P v,i ( g v ) ⊗ P v, tag ( g v ) , (24) where P v, 0 ( g v ) acts on k copies of the multiplicity intertwiner space, ⊗ e i vw ∈ E R P v,i ( g v ) acts on k copies of the internal edges, ⊗ e i v ¯ v ∈ E ∂R P v,i ( g v ) acts on the boundary semi-edges, with ¯ v representing a virtual vertex which is connected to v by the boundary edge ⊗ J v i ∈ T R P v,i ( g v ) acting on k copies of the tags. We indicate with e i vw ∈ E R the i th internal edge which connects the vertex v to the vertex w . Accordingly, the computation of the trace can be decomposed into four contributions, respectively for the a) internal edges, b) boundary spins and c) tags spins d) bulk intertwiners (for the technical aspects of the computation we refer the reader to []). As a result, the (normalised) k -th power of the partition function is mapped to the classical partition function of a generalized Ising-like model, Z ( k ) 1 / 0 = ∑ { g v } e -A ( k ) 1 / 0 [ { g v } ] , (25) and D J = 2 J +1 the dimension of the Hilbert space of the tags spin, we define the strength of the interactions respectively as log d j ≡ β , log D J ≡ β t (29) intended as inverse temperatures in analogy with standard Ising models. The action favours neighbouring \'spins\' to be parallel. Therefore, in the large spin dimension limit, corresponding to some strong coupling or \'low temperature\' regime, the dominant configurations that minimize the action correspond to large uniform spin domains separated by domain walls, which in turn give the energy cost of the configuration. We expect maximal domains to be associated to equilibrium configurations. In particular, maximal domains correspond to minimal domain walls. 5 and A ( k ) 0 [ { g v } ] = ∑ e i vw ∈ L ∆( g v ,g w ) log d j i vw + + ∑ J i v ¯ v ∈ T R ∪ E ∂R ∆( g v , ) log d j i v + A ( ψ tags ) + ξ , (26) respctively. 6 One can equivalently think of the pinning permutations fields X , X -1 and as being attached to virtual 1-valent vertices on the extended boundary. we factorize different sets of operators, P v ( g v ) = P v, 0 ( g v ) ⊗ ⊗ e i vw ∈ E R P v,i ( g v ) ⊗ ⊗ ⊗ e i v ¯ v ∈ E ∂R P v,i ( g v ) ⊗ P v, tag ( g v ) , (24) where P v, 0 ( g v ) acts on k copies of the multiplicity intertwiner space, ⊗ P v,i ( g v ) acts on k copies of the internal edges, ⊗ e i v ¯ v ∈ E ∂R P v,i ( g v ) acts on the boundary semi-edges, with ¯ v representing a virtual vertex which is connected to , and finally ⊗ J v i ∈ T R P v,i ( g v ) acting on k copies of the tags. We indicate with e i vw ∈ E R the i th internal edge which connects the vertex v to the vertex w . Accordingly, the computation of the trace can be decomposed into four contributions, respectively for the a) internal edges, b) boundary spins and c) tags spins d) bulk intertwiners (for the technical aspects of the computation we refer the reader to []). As a result, the (normalised) k -th power of the partition function is mapped to the classical partition function of a generalized Ising-like model, Z ( k ) 1 / 0 = ∑ { g v } e -A ( k ) 1 / 0 [ { g v } ] , (25) and D J = 2 J +1 the dimension of the Hilbert space of the tags spin, we define the strength of the interactions respectively as log d j ≡ β , log D J ≡ β t (29) intended as inverse temperatures in analogy with standard Ising models. The action favours neighbouring \'spins\' to be parallel. Therefore, in the large spin dimension limit, corresponding to some strong coupling or \'low temperature\' regime, the dominant configurations that minimize the action correspond to large uniform spin domains separated by domain walls, which in turn give the energy cost of the configuration. We expect maximal domains to be associated to equilibrium configurations. In particular, maximal domains correspond to minimal domain walls. 5 and A ( k ) 0 [ { g v } ] = ∑ e i vw ∈ L ∆( g v ,g w ) log d j i vw + + ∑ J i v ¯ v ∈ T R ∪ E ∂R ∆( g v , ) log d j i v + A ( ψ tags ) + ξ , (26) respctively. 6 One can equivalently think of the pinning permutations fields X , X -1 and as being attached to virtual 1-valent vertices on the extended boundary. More generally, the hole regime will be favoured by tags recoupling a large number of edge spins, so for coarsegrained vertices with high valence. When the number of tagged vertices in the bulk and boundary edges is balanced we exit the hole regime and we need to consider βE ∂R ∈ [ β t T, β t T + 2 βS ]. From eq. (64), we see that in this range of values A ( τ ) k < A ( 1 ) k . At the same time, eq. (65) implies that A ( τ ) k < A ( X ) k hence the minimal configuration is the one in which the geodesic permutation τ takes all the bulk, with the action A ( τ ) k dominating. We call this setting the island regime. The island domain wall stands between the three boundary conditions domains preventing them to access the bulk of R (see Fig. 8). Differently from the hole regime, the connectivity of the graph is not canceled in this case. Tags and boundary spins are connected through the island. Finally, for βE ∂R > β t T + 2 βS , namely when the tags environment is small compared with the boundary system, from eq. (64) and eq. (65) it follows that A ( X ) k < A ( 1 ) k , A ( τ ) k . In this regime, the minimal energy configuration is associated to an effective bipartition of R into two domains, one colored with the X permutation and the other with the X -1 permutation (see Fig. 9). 8 We refer to this setting as the bipartite regime.  \n(24)  \nhas the same form  \n).The ac-  \n(  \n(28)  \n]  \ndifferent sets of operators,  \npermutation elements, which therefore act as  \ngeneralized  \n/negationslash  \n/negationslash  \nteractions are described by the Cayley distance ∆(  \ng,h  \n),  \nv  \nv  \nv  \ne  \nβ  \ng,h  \ni  \nv  \n¯  \n(29)  \nJ  \nX  \n,  \nX  \n-  \n.  \n) into  \nfour  \ni  \nvw  \nt  \n(29)  \n∈  \nE  \nR  \ndomains separated by domain walls, which in turn give  \nv  \n¯  \nthe tags. We indi-  \ncts on the boundary  \ne  \ni  \nv  \n¯  \nv  \n, and finally  \nv  \nby the boundary edge  \ne  \nv  \nmension limit, corresponding to some strong coupling or  \nmension limit, corresponding to some strong coupling or  \nf the trace can be de-  \ncomposed into four contributions, respectively for the a)  \nJ ( max ) = 4 j = ⇒ β ( max ) t = β +log4 . (67)  \nThen, in a large spins regime, we can generally consider  \nWe indi-  \nthe dimension of the Hilbert spaces of the edges spin  \n-  \n{  \n-  \n{  \n|  \n〉〈  \n{  \n]  \n·  \n ) ] , (23) Hilbert spaces of a) A tags = -log { Tr bulk [ ⊗ k ] the bulk state contribution. In ( ?? ) and ( ?? tion A ( k ) 1 [ { g v } ] describes a two-body interaction between permutation elements, which therefore act as generalized spins , attached to the spin network vertices. These interactions are described by the Cayley distance ∆( ), tags = -log { Tr bulk [ | ψ tags 〉〈 ψ tags ⊗ k ] } the bulk state contribution. In ( ?? ) and ( ?? ).The action A ( k ) 1 [ { g v } ] describes a two-body interaction between permutation elements, which therefore act as generalized spins , attached to the spin network vertices. These in-] (23) ilbert spaces of a) c) bulk intertwinas the same form (28) the bulk state contribution. In ( ?? ) and ( ?? ).The action A ( k ) 1 [ { g v } ] describes a two-body interaction between permutation elements, which therefore act as spins , attached to the spin network vertices. These interactions are described by the Cayley distance ∆( g,h ), for g,h elements of the permutation group, while the pinning permutations fields X , X -1 and play the role of boundary conditions attached to the boundary and tags ] (28) the bulk state contribution. In ( ?? ) and ( ?? ).The action A ( k ) 1 [ { g v } ] describes a two-body interaction between spins , attached to the spin network vertices. These interactions are described by the Cayley distance ∆( g,h ), for g,h elements of the permutation group, while the pinning permutations fields X , X -1 and play the role of boundary conditions attached to the boundary and tags c) bulk intertwinhas the same form permutation operom at each vertex, sets of operators, for g,h elements of the permutation group, while the pinning permutations fields X , X -1 and play the role of boundary conditions attached to the boundary and tags spins. 6 We consider a quasi-homogeneous spin network, where all the edges carry the same spin j and all tags carry the same spin J , with j = J generally. Given d j = 2 j + 1 the dimension of the Hilbert spaces of the edges spin and D J = 2 J +1 the dimension of the Hilbert space of ntertwinfor g,h elements of the permutation group, while the pinning permutations fields X , X -1 and play the role of spins. 6 We consider a quasi-homogeneous spin network, where all the edges carry the same spin j and all tags carry the same spin J , with j = J generally. Given d j = 2 j + 1 the dimension of the Hilbert spaces of the edges spin and D J = 2 J +1 the dimension of the Hilbert space of of a) spins , attached to the spin network vertices. These interactions are described by the Cayley distance ∆( g,h ), g,h elements of the permutation group, while the pinning permutations fields X , X -1 and boundary conditions attached to the boundary and tags spins. 6 We consider a quasi-homogeneous spin network, where all the edges carry the same spin j and all tags carry the same spin J , with j = J generally. Given d j = 2 j + 1 e Hilbert spaces of a) and c) bulk intertwinhas the same form the permutation opereedom at each vertex, spins , attached to the spin network vertices. These interactions are described by the Cayley distance ∆( g,h ), for g,h elements of the permutation group, while the pinning permutations fields boundary conditions attached to the boundary and tags spins. 6 We consider a quasi-homogeneous spin network, where all the edges carry the same spin j and all tags carry the same spin J , with j = J generally. Given d j = 2 j + 1 FIG. 7: Domain walls configuration in the hole regime. 3. cluster colored with τ : A ( τ ) k = βE ∂R ( k 2 -1 ) + β t T k 2 (63) internal edges, b) boundary spins and c) bulk intertwiners, while C = ∏ [ ( D -1)! ] ( k ) with X and X -Now, to compute the action of the permutation operators on the different degrees of freedom at each vertex, we factorize P v ( g v ) into four different sets of operators, for g,h elements of the permutation group, while the pinning permutations fields X , X -1 and play the role of all the edges carry the same spin j and all tags carry the same spin J , with j = J generally. Given d j = 2 j + 1 and D J = 2 J +1 the dimension of the Hilbert space of the tags spin, we define the strength of the interactions internal edges, b) boundary spins and c) bulk intertwiners, while C = ∏ v [ ( D v -1)! ( D v + ] ( k ) with X and X -1 replaced by Now, to compute the action of the permutation operators on the different degrees of freedom at each vertex, we factorize P v ( g v ) into four different sets of operators, for g,h elements of the permutation group, while the pinning permutations fields X , X -1 and play the role of all the edges carry the same spin j and all tags carry the same spin J , with j = J generally. Given d j = 2 j + 1 and D J = 2 J +1 the dimension of the Hilbert space of the tags spin, we define the strength of the interactions where the trace factorizes over the Hilbert spaces of a) internal edges, b) boundary spins and c) bulk intertwiners, while C = ∏ v [ ( D v -1)! ( D v + k -1)! 0 with X and X -1 replaced by . Now, to compute the action of the permutation operators on the different degrees of freedom at each vertex, spins , attached to the spin network vertices. These interactions are described by the Cayley distance ∆( g,h ), for g,h elements of the permutation group, while the pinspins. 6 We consider a quasi-homogeneous spin network, where all the edges carry the same spin j and all tags carry the same spin J , with j = J generally. Given d j = 2 j + 1 the dimension of the Hilbert spaces of the edges spin where the trace factorizes over the Hilbert spaces of a) internal edges, b) boundary spins and c) bulk intertwiners, while C v ( D v + k -1)! 0 with X and X -1 replaced by . Now, to compute the action of the permutation operators on the different degrees of freedom at each vertex, spins , attached to the spin network vertices. These interactions are described by the Cayley distance ∆( g,h ), for g,h elements of the permutation group, while the pinspins. 6 We consider a quasi-homogeneous spin network, where all the edges carry the same spin j and all tags carry the same spin J , with j = J generally. Given d j = 2 j + 1 the dimension of the Hilbert spaces of the edges spin FIG. 8: Domain walls configuration in the island regime. β ( max ) t glyph[revsimilar] β in facts. Thereby, in order for the hole regime condition to be verified we must have T > E ∂R . (68)  \nthe tags spin, we define the strength of the interactions  \n/negationslash  \n/negationslash  \ne de-  \nndary  \n/negationslash  \n/negationslash  \n-  \n.  \nZ  \n·  \nv  \n]  \nj  \ni  \nv  \n+  \n) log  \nd  \n∑  \nj  \n∑  \n∑  \n1  \n¯  \nA  \n∑  \n∑  \nA  \ne  \nv  \nA  \n]  \n]  \nv  \n∑  \nv  \nS  \nP  \n]  \n(  \n[  \n∈  \nA  \nZ  \n+  \nv  \n(  \nD  \n+  \nk  \n∑  \n1)!  \n-  \n|  \n〉〈  \nA  \n1  \n+  \nk  \nA  \ne  \n|  \ne  \ni  \nv  \n¯  \ng  \nv  \ni  \nv  \n¯  \ne  \n+  \nP  \nv  \n(  \ng  \nv  \nv  \ne  \n<latexit 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sha1\\_base64="NiZ/UMWOLqmTj0A+Z+Nzi9MY8pE=">AAACCnicbVC7TsNAEDzzDOFloKQxRJGoIhshoIygoQyIPKTYWOvLJTnl7mzdnZEik5qGX6GhACFavoCOv+GSuICEkVYazexqdydKGFXadb+thcWl5ZXVwlpxfWNza9ve2W2oOJWY1HHMYtmKQBFGBalrqhlpJZIAjxhpRoPLsd+8J1LRWNzqYUICDj1BuxSDNlJoH5Qf/ETRMPN7wDmEN6O7zNeQjnwJosdIsRXaJbfiTuDMEy8nJZSjFtpffifGKSdCYwZKtT030UEGUlPMyKjop4okgAfQI21DBXCigmzyysgpG6XjdGNpSmhnov6eyIArNeSR6eSg+2rWG4v/ee1Ud8+DjIok1UTg6aJuyhwdO+NcnA6VBGs2NASwpOZWB/dBAtYmvaIJwZt9eZ40jiveacW7PilVL/I4CmgfHaIj5KEzVEVXqIbqCKNH9Ixe0Zv1ZL1Y79bHtHXBymf20B9Ynz+m6pre</latexit>  \n<latexit 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sha1\\_base64="sXr1R3W7vz5ZgzsLbC9hidV1dds=">AAAB63icbVBNS8NAEJ3Ur1q/qh69LBbBU0lE1GNBDx4r2A9oQ9lsN+3SzSbsToQS+he8eFDEq3/Im//GTZuDtj4YeLw3w8y8IJHCoOt+O6W19Y3NrfJ2ZWd3b/+genjUNnGqGW+xWMa6G1DDpVC8hQIl7yaa0yiQvBNMbnO/88S1EbF6xGnC/YiOlAgFo5hLfaTpoFpz6+4cZJV4BalBgeag+tUfxiyNuEImqTE9z03Qz6hGwSSfVfqp4QllEzriPUsVjbjxs/mtM3JmlSEJY21LIZmrvycyGhkzjQLbGVEcm2UvF//zeimGN34mVJIiV2yxKEwlwZjkj5Oh0JyhnFpCmRb2VsLGVFOGNp6KDcFbfnmVtC/q3lXde7isNe6KOMpwAqdwDh5cQwPuoQktYDCGZ3iFNydyXpx352PRWnKKmWP4A+fzByO/jlA=</latexit>  \n<latexit 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sha1\\_base64="ScGYA68kzMdsJa5n37hZOEO15qw=">AAACD3icbVC7TsNAEDzzDOFloKSxiIJoiGyEgDKChjIg8pDixFpfLskpd2fr7owUGf8BDb9CQwFCtLR0/A2XRwEJI600mtnV7k4YM6q0635bC4tLyyurubX8+sbm1ra9s1tTUSIxqeKIRbIRgiKMClLVVDPSiCUBHjJSDwdXI79+T6SikbjTw5i0OPQE7VIM2kiBfVh88GNFg9TvAecQ3Gbt1NeQZL4E0WMk32inx14W2AW35I7hzBNvSgpoikpgf/mdCCecCI0ZKNX03Fi3UpCaYkayvJ8oEgMeQI80DRXAiWql438yp2iUjtONpCmhnbH6eyIFrtSQh6aTg+6rWW8k/uc1E929aKVUxIkmAk8WdRPm6MgZheN0qCRYs6EhgCU1tzq4DxKwNhHmTQje7MvzpHZS8s5K3s1poXw5jSOH9tEBOkIeOkdldI0qqIowekTP6BW9WU/Wi/VufUxaF6zpzB76A+vzBzo8nMQ=</latexit>  \nj  \ni  \nv  \n+  \nhe  \nk  \ncopies of the  \nes of the  \nns of  \ntary representations of  \nf the  \non the  \nk  \ncopies of the  \nj  \nS  \ne  \nk  \ng  \n.  \n)  \n,  \n(22)  \nv  \n(  \ng  \n}  \n,  \nP  \nv  \n(22)  \n(22)  \nv  \n∑  \n(  \nk  \n)  \n1  \nA  \n)  \n,  \n(22)  \n{  \n}  \n[  \nv  \n∈  \ng  \n]  \n∆(  \ng  \n]  \nA  \n(  \nk  \n)  \ndefined by the action  \n(  \nk  \n)  \n1  \nA  \n[  \nv  \n+  \ni  \nv  \n¯  \n{  \ng  \nv  \ni  \nv  \n¯  \n∑  \ne  \n∈  \n∑  \ne  \nA  \n∈  \nally on each inde-  \nS  \nk  \nach inde-  \nlly we obtain  \ntain  \ninde-  \nwith  \nξ  \nvidually on each inde-  \ntually we obtain  \nwith  \nξ  \na constant term and  \nA  \ntags  \nA  \n)  \nv  \n(  \ng  \nv  \nP  \n∈  \n⊗  \n,  \n(23)  \nS  \n-  \n)  \nthe bulk state contribution.  \n).The ac-  \ntion  \nA  \n{  \ntion  \n) log  \nd  \n+  \nw  \ni  \nvw  \nv  \n,g  \n5  \n∆(  \ng  \n) log  \nd  \ndefined by the action  \n1  \n=  \ni  \nvw  \nv  \n[  \n∑  \n5  \ng  \nv  \n∈  \n{  \ni  \nvw  \n(  \nk  \n)  \n1  \nA  \n,X  \n) log  \nd  \ni  \n∑  \n+  \nv  \n}  \n=  \n∆(  \ng  \nv  \ne  \n,X  \n) log  \nd  \n+  \n∑  \nE  \ne  \n∈  \nv  \n¯  \nv  \n+  \nR  \n∈  \n∑  \nj  \n+  \n∆(  \ng  \n+  \nj  \n,g  \n∈  \n∑  \n∆(  \ng  \n=  \nv  \ni  \nvw  \nw  \n,g  \nv  \nv  \n]  \nE  \nR  \n}  \n∆(  \ng  \ng  \n∆(  \ng  \n[  \n{  \nj  \nv  \n,g  \n,X  \n-  \nv  \ni  \nvw  \ni  \nv  \n∆(  \ng  \nR  \nv  \n∈  \nw  \nE  \n) log  \nd  \n=  \n,X  \n) log  \nd  \nj  \n+  \n∑  \ni  \nvw  \ne  \ni  \nv  \n¯  \n∑  \nv  \nR  \n∈  \n∈  \ni  \nv  \nE  \nj  \n¯  \nA  \n∆(  \ng  \n,X  \n-  \nw  \nj  \ni  \nvw  \n) log  \nd  \n∆(  \ng  \n1  \n) log  \nd  \n) log  \nd  \nj  \nv  \ni  \nv  \n+  \nj  \ni  \nvw  \ni  \nv  \n,X  \n-  \n+  \ne  \ni  \n}  \nv  \n]  \n∆(  \ng  \n+  \nv  \n,X  \n) log  \nd  \nj  \ne  \ni  \nv  \ni  \nv  \n¯  \n+  \n∆(  \ng  \nv  \n+  \n1  \n) log  \nd  \n+  \n,X  \n-  \n(27)  \ntags  \n+  \nA  \nv  \nP  \n(  \n]  \n1  \n+  \nξ  \n(27)  \ntags  \nv,  \n0  \nv  \n}  \n(28)  \n) and (  \n??  \n).The ac-  \n⊗  \n+  \nξ  \n(27)  \nv  \n(  \ng  \nP  \nv,  \n0  \n))  \n(  \ng  \n}  \nv  \n(28)  \n))  \n}  \n(28)  \n]  \n]  \ndescribes a two-body interaction between  \ng  \npermutation elements, which therefore act as  \ngeneralized  \n<!-- image -->  \ntags  \n=  \n=  \n-  \n(  \nk  \n)  \n1  \npermutation elements, which therefore act as  \ngeneralized  \n,  \n(23)  \n)  \nv  \nP  \nP  \nC  \n(23)  \n·  \nthe bulk state contribution.  \n(  \ng  \n(  \n)  \n⊗  \n(  \nk  \n)  \n1  \n[  \n]  \n-  \nlog  \n[  \n]  \ntion  \nA  \ng  \nv  \n) and (  \n??  \n).The ac-  \ndescribes a two-body interaction between  \ntion  \nA  \n(  \nk  \n)  \n1  \ng  \nv  \ndescribes a two-body interaction between  \npermutation elements, which therefore act as  \ngeneralized  \nspins  \n,  \nattached to the spin network vertices.  \nThese in-  \nk  \n1  \nP  \n]  \nC  \n(  \n)  \n,  \n(23)  \nilbert spaces of a)  \nces of a)  \n{  \nspins  \n,  \nattached to the spin network vertices.  \nThese in-  \nteractions are described by the Cayley distance ∆(  \ng,h  \n),  \nteractions are described by the Cayley distance ∆(  \ng,h  \n),  \n]  \nspins  \n,  \nattached to the spin network vertices.  \nThese in-  \n[  \n]  \n)  \n[  \n]  \npermutation elements, which therefore act as  \ngeneralized  \nspins  \n,  \nattached to the spin network vertices.  \nThese in-  \nhas the same form ning permutations fields X , X -1 and play the role of boundary conditions attached to the boundary and tags 6 me form ning permutations fields X , X -1 and play the role of boundary conditions attached to the boundary and tags 6 of a) twinteractions are described by the Cayley distance ∆( g,h ), for g,h elements of the permutation group, while the pinning permutations fields X , X -1 and play the role of e Hilbert spaces of a) and c) bulk intertwin-( k ) teractions are described by the Cayley distance ∆( g,h ), for g,h elements of the permutation group, while the pinning permutations fields X , X -1 and play the role of FIG. 9: Domain walls configuration in the bipartite regime.  \nc) bulk intertwin-  \nntertwin-  \nform  \n0  \nion oper-  \n}  \nfor  \ng,h  \nelements of the permutation group, while the pin-  \nfor  \ng,h  \nelements of the permutation group, while the pin-  \nhas the same form  \nspins.  \nboundary conditions attached to the boundary and tags  \nspins.  \nspins.  \nWe consider a quasi-homogeneous spin network, where  \nWe consider a quasi-homogeneous spin network, where  \nboundary conditions attached to the boundary and tags  \nspins.  \n6  \npermutation oper-  \nall the edges carry the same spin  \nj  \nand all tags carry the  \nom at each vertex,  \nh vertex,  \nsets of operators,  \n6', 'same spin J , with j = J generally. Given d j = 2 j + 1 same spin J , with j = J generally. Given d j = 2 j + 1 We consider a quasi-homogeneous spin network, where all the edges carry the same spin j and all tags carry the We consider a quasi-homogeneous spin network, where all the edges carry the same spin j and all tags carry the B. Entanglement Phases': 'the permutation oper-  \noper-  \nrtex,  \nerators,  \nall the edges carry the same spin  \nj  \nand all tags carry the  \n/negationslash  \nsame spin  \nJ  \n,  \nwith  \nj  \nsame spin  \nJ  \n,  \nwith  \nj  \needom at each vertex,  \nP  \n/negationslash  \nthe dimension of the Hilbert spaces of the edges spin  \nj  \n/negationslash  \n=  \nJ  \ngenerally.  \nGiven  \nd  \n= 2  \nj  \n+ 1  \n/negationslash  \n=  \nJ  \ngenerally.  \nGiven  \nd  \n= 2  \nj  \n+ 1  \n(24) log d j ≡ β , log D J ≡ β t log d j ≡ β , log D J ≡ β t (29) respectively as In the hole regime, from the action in (62), we get  \nthe dimension of the Hilbert spaces of the edges spin  \n⊗ and D J = 2 J +1 the dimension of the Hilbert space of the tags spin, we define the strength of the interactions respectively as and D J = 2 J +1 the dimension of the Hilbert space of the tags spin, we define the strength of the interactions respectively as tors, j the dimension of the Hilbert spaces of the edges spin and D J = 2 J +1 the dimension of the Hilbert space of the tags spin, we define the strength of the interactions respectively as rent sets of operators, ( g v ) ⊗ the dimension of the Hilbert spaces of the edges spin and D J = 2 J +1 the dimension of the Hilbert space of the tags spin, we define the strength of the interactions Wecan now compute the typical logarithmic negativity of the random mixed state ρ A 1 A 2 in the three regimes considered in the given example.  \n)  \n,  \n(24)  \nv  \nv,  \ntag  \n(24)  \n)  \nP  \ndary  \n(  \ng  \nv,  \ntag  \n)  \n,  \n(  \ng  \nv  \n)  \n,  \nJ  \nβ  \nt  \n(29)  \n(29)  \nthat minimize the action correspond to large uniform spin  \nv  \nmultiplicity interon k copies of the s on the boundary tual vertex which intended as inverse temperatures in analogy with standard Ising models. The action favours neighbouring \'spins\' to be parallel. Therefore, in the large spin dimension limit, corresponding to some strong coupling or \'low temperature\' regime, the dominant configurations ity interes of the oundary ex which intended as inverse temperatures in analogy with standard Ising models. The action favours neighbouring \'spins\' to be parallel. Therefore, in the large spin dimension limit, corresponding to some strong coupling or \'low temperature\' regime, the dominant configurations interf the log d j ≡ β , log D J ≡ β t (29) intended as inverse temperatures in analogy with standard Ising models. The action favours neighbouring \'spins\' to be parallel. Therefore, in the large spin dimension limit, corresponding to some strong coupling or ⊗ (24) the multiplicity intercts on k copies of the log d j ≡ β , log D ≡ intended as inverse temperatures in analogy with standard Ising models. The action favours neighbouring \'spins\' to be parallel. Therefore, in the large spin dimension limit, corresponding to some strong coupling or E (hole) N ( ρ A 1 A 2 ) glyph[similarequal] -lim k → 1 A ( 1 ) k = -lim k → 1 βE ∂R ( k -1) = 0 . (69)  \nthat minimize the action correspond to large uniform spin  \nge e i v ¯ v , and finally the tags. We indige which connects e trace can be deectively for the a) d c) tags spins d) spects of the comower of the partil partition function domains separated by domain walls, which in turn give the energy cost of the configuration. We expect maximal domains to be associated to equilibrium configurations. In particular, maximal domains correspond to minimal domain walls. 5 and A ( k ) 0 [ { g v } ] = ∑ e i vw ∈ L ∆( g v ,g w ) log d j i vw + + ∑ J i v ¯ v ∈ T R ∪ E ∂R ∆( g v , ) log d j i v + A ( ψ tags ) + ξ , (26) d finally We indiconnects n be deor the a) spins d) the comhe partifunction domains separated by domain walls, which in turn give the energy cost of the configuration. We expect maximal domains to be associated to equilibrium configurations. In particular, maximal domains correspond to minimal domain walls. 5 and A ( k ) 0 [ { g v } ] = ∑ e i vw ∈ L ∆( g v ,g w ) log d j i vw + + ∑ J i v ¯ v ∈ T R ∪ E ∂R ∆( g v , ) log d j i v + A ( ψ tags ) + ξ , (26) hich nally indinects e dehe a) ns d) comparti\'low temperature\' regime, the dominant configurations that minimize the action correspond to large uniform spin domains separated by domain walls, which in turn give the energy cost of the configuration. We expect maximal domains to be associated to equilibrium configurations. In particular, maximal domains correspond to minimal domain walls. 5 and A ( k ) 0 [ { g v } ] = ∑ e i vw ∈ L ∆( g v ,g w ) log d j i vw + virtual vertex which edge e i v ¯ v , and finally s of the tags. We indil edge which connects f the trace can be derespectively for the a) and c) tags spins d) al aspects of the com-. th power of the parti\'low temperature\' regime, the dominant configurations that minimize the action correspond to large uniform spin domains separated by domain walls, which in turn give the energy cost of the configuration. We expect maximal domains to be associated to equilibrium configurations. In particular, maximal domains correspond to minimal domain walls. 5 and A ( k ) 0 [ { g v } ] = ∑ e i vw ∈ L ∆( g v ,g w ) log d j i vw + A vanishing negativity tells us that the boundary state has Positive Partial Transpose (PPT), hence, according to the Peres criterion [75], that boundary state could be separable. This suggests that ρ A 1 A 2 is well described by a maximally mixed state. The boundary system is almost fully entangled with the large number of degrees of freedom environment with little quantum correlation between A 1 and A 2 . This effectively corresponds to a thermalization of the boundary state, in strong analogy with the results in [72, 87], where the typicality of the random state, in the large spin regime, similarly played a crucial role.  \nacts on the boundary  \nction  \nsical partition function  \n}  \n{  \nOne can equivalently think of the pinning permutations fields  \nX  \n,  \nrespctively. 6 respctively. + ∑ J i v ¯ v ∈ T R ∪ E ∂R ∆( g v , ) log d j i v + A ( ψ tags ) + ξ , (26) + ∑ J i v ¯ v ∈ T R ∪ E ∂R ∆( g v , ) log d j i v + A ( ψ tags ) + ξ , Moving to the island regime, the minimal action reads  \nOne can equivalently think of the pinning permutations fields  \nX  \n,  \n6  \nextended boundary.  \n(25) X -1 and as being attached to virtual 1-valent vertices on the extended boundary. X -1 and as being attached to virtual 1-valent vertices on the extended boundary. respctively. One can equivalently think of the pinning permutations fields X , X -1 and as being attached to virtual 1-valent vertices on the (25) respctively. 6 One can equivalently think of the pinning permutations fields X , X -1 and as being attached to virtual 1-valent vertices on the extended boundary. A ( τ ) k = βE ∂R ( k 2 -1 ) + β t T k 2 (70)  \nand the typical value of the logarithmic negativity in the k → 1 limit is given by  \nE (island) N ( ρ A 1 A 2 ) glyph[similarequal] -lim k → 1 [ A ( i ) k +log C k ] = = -lim k → 1 [ βE ∂R ( k 2 -1 ) + β t T k 2 +log C k ] = = 1 2 ( βE ∂R -β t T ) + log 8 3 π , (71)  \nwith the Catalan numbers C k and C 1 = 8 3 π here taking into account the degeneracy in the choice of the geodesic  \n[  \n,  \ng  \nv  \n(25)  \n(25)  \n,  \n]  \n]  \n(26)  \n·  \nv  \n{  \n{  \n}  \nj  \ni  \nv  \n+  \n) log  \nd  \n∑  \nj  \n∑  \n∑  \ne  \n1  \n¯  \nA  \n∑  \n∑  \nA  \ne  \nv  \n]  \nA  \n+  \nP  \nv  \n(  \ng  \n}  \nv  \n6  \nA  \n<latexit 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sha1\\_base64="ScGYA68kzMdsJa5n37hZOEO15qw=">AAACD3icbVC7TsNAEDzzDOFloKSxiIJoiGyEgDKChjIg8pDixFpfLskpd2fr7owUGf8BDb9CQwFCtLR0/A2XRwEJI600mtnV7k4YM6q0635bC4tLyyurubX8+sbm1ra9s1tTUSIxqeKIRbIRgiKMClLVVDPSiCUBHjJSDwdXI79+T6SikbjTw5i0OPQE7VIM2kiBfVh88GNFg9TvAecQ3Gbt1NeQZL4E0WMk32inx14W2AW35I7hzBNvSgpoikpgf/mdCCecCI0ZKNX03Fi3UpCaYkayvJ8oEgMeQI80DRXAiWql438yp2iUjtONpCmhnbH6eyIFrtSQh6aTg+6rWW8k/uc1E929aKVUxIkmAk8WdRPm6MgZheN0qCRYs6EhgCU1tzq4DxKwNhHmTQje7MvzpHZS8s5K3s1poXw5jSOH9tEBOkIeOkdldI0qqIowekTP6BW9WU/Wi/VufUxaF6zpzB76A+vzBzo8nMQ=</latexit>  \nj  \ni  \nv  \n+  \ncoloring τ for the bulk.  \nWe have then a transition from an unentangled phase to an area law behavior, as expected by the topological character of the entanglement contributions. In fact, it is possible to write E (island) N ( ρ A 1 A 2 ) as a generalisation of the Ryu-Takayanagi formula for entanglement entropy [69]. By defining the minimal surfaces that separate the entanglement wedges relative to each subsystem as | γ A 1 | , | γ A 2 | and | γ B | , it is straightforward to see that  \n| γ A 1 | = E A 1 , | γ A 2 | = E A 2 , | γ B | = T (72)  \nwhere E A 1 , E A 2 are the number of edges in A 1 and A 2 respectively.  \nHence, since E A 1 + E A 2 = E ∂R we can write eq. (71) as  \nE (island) N ( ρ A 1 A 2 ) = 1 2 [ β ( | γ A 1 | + | γ A 2 | ) + β t | γ B | ] +log 8 3 π (73)  \nThis is a generalisation of the RT formula found in [60] to the case of non-homogeneous mixed spin network states (two temperatures). As for the previous works, it can be expressed as a mutual information between the subsystem A 1 and A 2 :  \nE (island) N = 1 2 [ S ( ρ A 1 ) + S ( ρ A 2 ) -S ( ρ A 1 A 2 ) ] +log 8 3 π = 1 2 I ( A 1 : A 2 ) + log 8 3 π (74)  \nIn the island regime, then, the logarithmic negativity is always non-zero, and scales with the total area of the boundary of R , namely βE ∂R , minus a negative correction that depends linearly on the number of tags. This implies that the boundary state is necessarily entangled. In particular, the presence of the environment reflects in a saturation of the entanglement of the two boundary subsystems. The logarithmic negativity only depends on the size of the system and of the environment but it is independent of the specific bipartition of the boundary.  \nFinally, in the bipartite regime, the action with minimal energy is given by  \nA ( X ) k = βS ( k -2) + β t T ( k -1) (75)  \nand the typical logarithmic negativity reads  \nE (bipartite) N ( ρ A 1 A 2 ) glyph[similarequal] -lim k → 1 A ( X ) k = (76) = -lim k → 1 [ βS ( k -2) + β t T ( k -1)] = = βS  \nIn this regime, negativity scales with the area of the internal domain wall S . In particular, we see from (76) that in this regime we could achieve a PPT state for the boundary only by setting S to 0, where a PPT state  \nwould correspond to a disconnected graph as expected.  \nSuch a behaviour is consistent with a limit toward a bipartite pure boundary state setting corresponding to vanishing tags (trivial topology) [63].  \nAltogether, the effective topology of the bounded region is reflected in the multipartite entanglement of its tagged spin-networks state description. Beside the toy model example, the proposed analysis can be easily generalised for graphs with generic number of vertices, with more than one open edge attached to each boundary vertex and with clusters of tags restricted to a subregion of the bulk graph. The universal character of the result is indeed associated to the typical behaviour of the randomised system in the large spin regime.', 'VIII. DISCUSSION': 'In the framework of loop quantum gravity, quantum geometry states corresponding to bounded regions of 3 D space with uniform curvature can be synthetically described via superpositions of tagged spin-networks with boundary. Curvature naturally builds up at the spinnetwork vertices in the form of topological defects described by spin tags, as the result of the partial tracing over nontrivial SU (2) holonomies of the Ashtekar connection along the network.  \nIn presence of a boundary, a quantum geometry wavefunction generally behaves as a map encoding the bulk information into the boundary Hilbert space. In the proposed analysis, we generalize such a bulk-to-boundary mapping as to include the space of bulk topological defects, effectively described as extra (inner) boundary degrees of freedom. 9 In the resulting extended boundary state, bulk information is shared among generically entangled boundary spins and tags: the surface and the intrinsic curvature of the quantum space region are entangled. Remarkably, the degree of quantum correlations among boundary subregions is affected by the presence of the bulk curvature, which plays the role of a hidden environment for the mixed outer boundary state.  \nThe effect of the curvature on the boundary entanglement is the focus of our analysis. We model the generalised boundary mapping on a tripartite ( A 1 , A 2 , B ) system and consider the reduced boundary state ρ A 1 A 2 obtained via a trace over the space of tags (system B ). We further characterise the coarse viewpoint of a boundary observer by considering a random measurement on the bulk state, thereby ultimately dealing with a random mixed state for the boundary.  \nThe entanglement of the bipartite mixed boundary is quantified via a measure of logarithmic negativity, along  \nthe lines of the recent results in [60, 69, 72]. Concretely, we restrict the analysis to a class of states characterised by graphs coloured by the same spin j on each edge and dressed with a tag of corresponding recoupled spin J at each vertex. We call this setting a cluster of tags . The full computation is carried on for a simple cluster of four tags is given in Section VII A. By using standard replica techniques for random tensor network, this computation is ultimately mapped to the evaluation of the minimal action of a classical generalised Ising-like statistical model on the spin network graph, in the limit of vanishing temperature (see [60] and references therein).  \nAs a first general result, we find that the Ising-like model action dual to the random spin network generally splits into two main contributions:  \nA ( k ) 1 = A ( k ) topology + A ( k ) phys (77)  \nthat is a contribution A ( k ) topology which accounts for the entanglement induced by the topology of the spin network graph and a second one, A ( k ) phys , encoding the bulk intertwiner entanglement. The former, in particular, further splits into two well distinct contributions,  \nA ( k ) topology = A ( k ) edges + A ( k ) tags ,  \nrespectively associated to the connectivity of the bulk network (trivial topology), induced by maximally entangled edge states, and to the presence of the tags. The last term is therefore specific to the case of spin network states with topological defects.  \nIn a topologically trivial setting, as showed in [60], the first term in (77) is associated to an area law scaling of the entanglement of the boundary, while bulk intertwiners entanglement can be interpreted in analogy with [67]. We leave the characterisation of the random bulk contribution for future work, while focusing on the nontrivial topological contribution to the action.  \nWithin the typical regime, the entanglement phases of the random boundary state can be described solely in terms of the relative dimensions of the three subsystems A 1 , A 2 and B , 10 via the parameter  \nq = β t T/βE ∂R = log[dim( B )] log[dim( A 1 )dim( A 2 )] , (78)  \nwhich expresses the ratio of bulk curvature over boundary surface, with E ∂R = E A 1 + E A 2 . Two main entanglement phases are separated by a critical point at q = 1.  \nFor 1 -2 S/E ∂R < q < 1, the curvature environment mediates the entanglement of the boundary. The logarithmic negativity, modulo degeneracy, scales with the area of the cluster boundary with a negative correction  \nFIG. 10: In red the generalized Page curve of the logarithmic negativity for the random mixed state ρ A 1 A 2 . In red, how the Page curve would appear for random pure states.  \n<!-- image -->  \nwhich depends linearly on the number of tags. We have  \nE (island) N ( ρ A 1 A 2 ) ∝ 1 2 βE ∂R (1 -q ) (79) = 1 2 [ β ( | γ A 1 | + | γ A 2 | ) + β t | γ B | ]  \nwith | γ A 1 | = E A 1 , | γ A 2 | = E A 2 and | γ B | = T , the minimal surfaces separating the entanglement wedges relative to each subsystem. This is a generalisation of the RyuTakayanagi formula for entanglement entropy for the tripartite setting, in agreement with [60, 69] and it can be expressed as a mutual information between the subsystem A 1 and A 2 .  \nFinally, for q < 1 -2 S/E ∂R , when the curvature environment is much smaller than the boundary system, we tend to a bipartite setting and the typical logarithmic negativity scales with the area of the surface S separating A 1 and A 2 through the boundary. In this case, we find  \nE (bipartite) N ( ρ A 1 A 2 ) ∝ βS . (80)  \nNotice that the value of S cannot be tuned in our analysis, as the surface is fixed by the spin network connectivity for the given graph. However, assuming some regularity of the graph, one can geometrically relate this value to the bipartition of the boundary. For instance, assuming a generic tagged graph with a large density of vertices, dual to a 3-ball, we can imagine the minimal surface S to be approximated by the intersection of the sphere and a generic plane, which divides A 1 from A 2 on the boundary of R . In this case, we can approximately  \nS ≈ j 2 E A 1 E A 2 E ∂R (81)  \nTherefore, we see that the island condition is favoured when the dimension of the two subsystems A 1 and A 2 is well balanced, since S is maximized by high values of both E A 1 and E A 2 = E ∂R -E A 1 . Conversely, from the condition q < 1 -2 S/E ∂R , the bipartite regime is favoured when the two boundary subregions A 1 and A 2 are extremely asymmetric in extension (smaller S ).  \nIn particular, we can describe the transition from the island to the bipartite regime by using (81) in the condition A ( bipartite ) k < A ( island ) k . We find  \n( E A 1 E ∂R ) 2 -E A 1 E ∂R +(1 -q ) > 0 (82)  \nThereby, for 0 < q < 1, we can identify two critical points between the island and the bipartition phases,  \nc ∗ 1 , 2 ( T ) = 1 2 ± 1 2 √ q . (83)  \nglyph[negationslash]  \nThe behaviour of the negativity is well described by the generalized Page curve in fig. 10. As a general feature of a tripartite random state, we find that the degree of correlation of the two boundary subregions A 1 , A 2 depends on the degree of quantum correlations the two boundary systems have with the curvature environment. By varying the relative size of the two subsystems, the logarithmic negativity shows an initial increase and a final decrease in analogy with the Page curve. When T = 0 (trivial topology), c ∗ 1 (0) = c ∗ 2 (0) = 1 / 2 and we get back the standard Page curve [88] (no tripartition). As soon as T = 0, the intermediate island regime forms and the negativity has a plateau. In this regime the logarithmic negativity only depends on the size of the system and of the environment but not on how the system is partitioned.  \nOn the other hand, for q > 1, corresponding to a large tags environment with respect to the boundary system, the logarithmic negativity vanishes. The boundary system is almost fully entangled with the environment with little quantum correlation between A 1 and A 2 . In particular, the area law behaviour is lost. This suggests that large curvature is associated to a maximally mixed boundary state ρ A 1 A 2 . Such a thermalization of the boundary due to curvature is in strong analogy to the quantum gravity formulation of the hoop conjecture proposed in [87], where again the typicality of the random state, in the large spin regime, plays a crucial role.  \nIf we consider that the large spin regime we have been working in is necessarily associated with a semiclassical limit for our quantum geometry states, then we can tentatively interpret the formulas found for the entanglement negativity in terms of areas of a bounded 3 D region of a Riemannian manifold and look for a direct relation with  \nits Ricci curvature. Indeed, we know that the value of the scalar curvature R of a Riemannian n -manifold M at a point p can be quantified by the ratio of the area of the ( n -1)-dimensional boundary of a ball of radius ε in M to that of a corresponding ball in the Euclidean space. For small ε , the ratio is given by [89, 90]  \nArea( ∂B ε ( p ) ⊂ M ) Area( ∂B ε (0) ⊂ R n ) = 1 -R 6 n ε 2 + O ( ε 3 ) . (84)  \nNow, in the semiclassical limit, for n = 3, we can easily associate the area of the boundary of a flat three ball region with the sum of the areas of the triangles dual to the open edges spins comprising the boundary of R (no tags), that is Area( ∂B ε (0)) glyph[similarequal] βE ∂R ( ε ), where E ∂R ( ε ) counts the number of boundary edges at a certain radius ε , now measured via some distance on the spin network graph. In case of the curved three ball in M , however, we expect this value to be modified by the presence of tags. In particular, if we assume  \nArea( ∂B ε ( p )) glyph[similarequal] βE ∂R ( ε ) + κβ t T ( ε ) (85)  \nnamely a linear correction in the number of tags (positive or negative depending on the sign of κ ), where T ( ε ) indicates the number of tags within a certain radius ε . Thereby, the ratio in (84) gives us a direct relation  \nq ( ε ) glyph[similarequal] -R 18 κ ε 2 (86)  \nbetween the entanglement phase parameter and the Ricci curvature. For instance, in this light, we find a nice relation between the area scaling negativity (or mutual information) in (79) and the geometric relation in (84). Such a relation is consistent with the interpretation of our results and hints toward a characterisation of the curvature in purely information theoretic terms [91].  \nLet us then conclude with a few further remarks. A first remark is in order concerning the non-gauge invariant character of the degrees of freedom considered in our analysis. Spin network\'s tags are not different from boundary dangling spins, whose presence indeed breaks gauge invariance at the boundary. The entanglement we describe among boundary spins and tags cannot be measured in terms of gauge invariant observables, hence it is considered as non physical in loop quantum gravity [92]. Nevertheless, we expect such a topological layer of entanglement to become essential in modelling the emergence of spacetime connectivity as soon as we allow both edge and tags spins to be physical at some higher energy scale, where we expect gauge invariance to break. This perspective would require an extension of the kinematical description of quantum geometry in loop quantum gravity, which is in part provided by the second quantized formalism of group field theory [20]. Differently, one can take the present analysis as a toy model for an ultimate description of the entanglement structure of edge modes degrees of freedom at the interface of subsystems [93-95].  \n<latexit sha1\\_base64="qL73LpXLpFerIQVGSKN2yhCMepI=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE1JMUvHhswX5AG8pmO2nXbjZhdyOU0F/gxYMiXv1J3vw3btsctPXBwOO9GWbmBYng2rjut1NYW9/Y3Cpul3Z29/YPyodHLR2nimGTxSJWnYBqFFxi03AjsJMopFEgsB2M72Z++wmV5rF8MJME/YgOJQ85o8ZKDdMvV9yqOwdZJV5OKpCj3i9/9QYxSyOUhgmqdddzE+NnVBnOBE5LvVRjQtmYDrFrqaQRaj+bHzolZ1YZkDBWtqQhc/X3REYjrSdRYDsjakZ62ZuJ/3nd1IQ3fsZlkhqUbLEoTAUxMZl9TQZcITNiYgllittbCRtRRZmx2ZRsCN7yy6ukdVH1rqpe47JSu83jKMIJnMI5eHANNbiHOjSBAcIzvMKb8+i8OO/Ox6K14OQzx/AHzucP4O2M+g==</latexit>  \n<latexit sha1\\_base64="BG0hmqEYPi4+wZRv+Q+p9girVSw=">AAAB6HicbVBNS8NAEJ34WetX1aOXxSJ4KomIeix68diK/YA2lM120q7dbMLuRiihv8CLB0W8+pO8+W/ctjlo64OBx3szzMwLEsG1cd1vZ2V1bX1js7BV3N7Z3dsvHRw2dZwqhg0Wi1i1A6pRcIkNw43AdqKQRoHAVjC6nfqtJ1Sax/LBjBP0IzqQPOSMGivV73ulsltxZyDLxMtJGXLUeqWvbj9maYTSMEG17nhuYvyMKsOZwEmxm2pMKBvRAXYslTRC7WezQyfk1Cp9EsbKljRkpv6eyGik9TgKbGdEzVAvelPxP6+TmvDaz7hMUoOSzReFqSAmJtOvSZ8rZEaMLaFMcXsrYUOqKDM2m6INwVt8eZk0zyveZcWrX5SrN3kcBTiGEzgDD66gCndQgwYwQHiGV3hzHp0X5935mLeuOPnMEfyB8/kDrpmM3A==</latexit>  \n<latexit sha1\\_base64="8yaoKCskarkczxuKjVJDmOoaVJQ=">AAAB6HicbVBNS8NAEJ34WetX1aOXxSJ4KomIeix68diC/YA2lM120q7dbMLuRiyhv8CLB0W8+pO8+W/ctjlo64OBx3szzMwLEsG1cd1vZ2V1bX1js7BV3N7Z3dsvHRw2dZwqhg0Wi1i1A6pRcIkNw43AdqKQRoHAVjC6nfqtR1Sax/LejBP0IzqQPOSMGivVn3qlsltxZyDLxMtJGXLUeqWvbj9maYTSMEG17nhuYvyMKsOZwEmxm2pMKBvRAXYslTRC7WezQyfk1Cp9EsbKljRkpv6eyGik9TgKbGdEzVAvelPxP6+TmvDaz7hMUoOSzReFqSAmJtOvSZ8rZEaMLaFMcXsrYUOqKDM2m6INwVt8eZk0zyveZcWrX5SrN3kcBTiGEzgDD66gCndQgwYwQHiGV3hzHpwX5935mLeuOPnMEfyB8/kD6DGNAg==</latexit>  \n<latexit sha1\\_base64="Xo4o1CRk3duVZ6f7kO5zwIA1aaU=">AAACEXicbVDLSgMxFM3UV62vUZdugkWomzJTRF22iuBKKtoHdIYhk6ZtaCYZkoxQhv6CG3/FjQtF3Lpz59+YtrPQ1gOBwzn3cnNOGDOqtON8W7ml5ZXVtfx6YWNza3vH3t1rKpFITBpYMCHbIVKEUU4ammpG2rEkKAoZaYXDy4nfeiBSUcHv9SgmfoT6nPYoRtpIgV2CV8FNyZMDEaS1wK0FlfEx9LSAd3NiYBedsjMFXCRuRoogQz2wv7yuwElEuMYMKdVxnVj7KZKaYkbGBS9RJEZ4iPqkYyhHEVF+Ok00hkdG6cKekOZxDafq740URUqNotBMRkgP1Lw3Ef/zOonunfsp5XGiCcezQ72EQZN4Ug/sUkmwZiNDEJbU/BXiAZIIa1NiwZTgzkdeJM1K2T0tu7cnxepFVkceHIBDUAIuOANVcA3qoAEweATP4BW8WU/Wi/VufcxGc1a2sw/+wPr8AaO0mwU=</latexit>  \n<latexit sha1\\_base64="JWF40CSeyS3xJALLLU2IOoMg3i4=">AAACAnicbVDLSgMxFM3UV62vUVfiJliEuikzRdRlqwiupIJ9QGcYMmnahmaSIckIZShu/BU3LhRx61e4829M21lo64HA4Zx7uTknjBlV2nG+rdzS8srqWn69sLG5tb1j7+41lUgkJg0smJDtECnCKCcNTTUj7VgSFIWMtMLh1cRvPRCpqOD3ehQTP0J9TnsUI22kwD6A18FtyZMDEaS1wK0FlfEJ9LSATmAXnbIzBVwkbkaKIEM9sL+8rsBJRLjGDCnVcZ1Y+ymSmmJGxgUvUSRGeIj6pGMoRxFRfjqNMIbHRunCnpDmcQ2n6u+NFEVKjaLQTEZID9S8NxH/8zqJ7l34KeVxognHs0O9hEETcdIH7FJJsGYjQxCW1PwV4gGSCGvTWsGU4M5HXiTNStk9K7t3p8XqZVZHHhyCI1ACLjgHVXAD6qABMHgEz+AVvFlP1ov1bn3MRnNWtrMP/sD6/AEp7pVc</latexit>  \n<latexit sha1\\_base64="DmdMzX+D+QW44o+0Ra4eMLVB8zs=">AAAB8XicbVBNSwMxEJ2tX7V+VT16CRbBU9kVUY9FLx6r2A9sl5JNp21oNrskWaEs/RdePCji1X/jzX9jtt2Dtj4IPN6bmcy8IBZcG9f9dgorq2vrG8XN0tb2zu5eef+gqaNEMWywSESqHVCNgktsGG4EtmOFNAwEtoLxTea3nlBpHskHM4nRD+lQ8gFn1FjpsRtTZTgV5L5XrrhVdwayTLycVCBHvVf+6vYjloQoDRNU647nxsZPs3lM4LTUTTTGlI3pEDuWShqi9tPZxlNyYpU+GUTKPmnITP3dkdJQ60kY2MqQmpFe9DLxP6+TmMGVn3IZJwYlm380SAQxEcnOJ32ukBkxsYQyxe2uhI2ooszYkEo2BG/x5GXSPKt6F1Xv7rxSu87jKMIRHMMpeHAJNbiFOjSAgYRneIU3RzsvzrvzMS8tOHnPIfyB8/kDOISQnw==</latexit>  \nFIG. 11: Milne spacetime sliced by 3 D surfaces with hyperbolic embedding. Close to Milne\'s big bang, the quantum realisation of such surfaces have vanishing negativity, E N ( ρ A 1 A 2 ) → 0. As time increases, the negativity of the 2 D cosmological horizon grows and tends to bipartite entanglement entropy, E N ( ρ A 1 A 2 ) → S ( ρ A 1 A 2 ) at large times.  \n<!-- image -->  \nIn this sense, we expect the described universal features of the interface entanglement in the large spin regime to be general.  \nA final remark has to do with the interplay of intrinsic and extrinsic geometry in loop quantum gravity. As showed in [83, 96], in loop quantum geometry states, nontrivial Ashtekar-Barbero holonomies can be thought alternatively as intrinsic geometry or extrinsic curvature . In particular, it is possible to compensate the defects of the intrinsic geometry appearing while coarse-graining spin networks by Lorentz boosts which change the local embedding of the spin network description of quantum 3 D space in the (3 + 1)-D geometry. This suggests a natural relation between embedding and entanglement to which we shall dedicate further investigation in future work. For instance, one could figure the phase transition previously described, between a maximally entangled ( q glyph[lessmuch] 1) to a maximally mixed ( q glyph[greatermuch] 1) boundary state, as being described by a sequence of quantum 3 D spacelike slices with a gradient in curvature. If we embed such slices, via boost, in the hyperbolic space, we would get a transition from almost flat bounded regions with large corner surface corresponding to a maximally entangled boundary state to highly curved regions with small corner surface corresponding to maximally mixed boundary state. A similar setting naively recalls the geometry of the Milne spacetime cosmology [97] (see Fig. 11). 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