#  $Q$-Deformed Rainbows: a Universal Simulator of Free Entanglement Spectra

Lucy Byles<EMAIL_ADDRESS>School of Physics and Astronomy, University of
Leeds, Leeds, LS2 9JT, United Kingdom  Germán Sierra  Instituto de Física
Teórica UAM/CSIC, Universidad Autónoma de Madrid, Madrid, 28049, Spain
Jiannis K. Pachos  School of Physics and Astronomy, University of Leeds,
Leeds, LS2 9JT, United Kingdom

###### Abstract

The behaviour of correlations across a bipartition is an indispensable tool in
diagnosing quantum phases of matter. Here we present a spin chain with
position-dependent XX couplings and magnetic fields, that can reproduce
arbitrary structure of free fermion correlations across a bipartition. In
particular, by choosing appropriately the strength of the magnetic fields we
can obtain any single particle energies of the entanglement spectrum with high
fidelity. The resulting ground state can be elegantly formulated in terms of
$q$-deformed singlets. To demonstrate the versatility of our method we
consider certain examples, such as a system with homogeneous correlations and
a system with correlations that follow a prime number decomposition. Hence,
our entanglement simulator can be easily employed for the generation of
arbitrary entanglement spectra with possible applications in quantum
technologies and condensed matter physics.

##  I Introduction

Entanglement lies at the heart of the disparity between classical and quantum
mechanics. As such it has long been at the forefront of both theoretical [1,
2] and experimental [3, 4] investigations into the foundations of quantum
mechanics. More recently, entanglement has gained renewed interest with the
development of quantum information theory [5, 6]. In this framework, quantum
entanglement is viewed as a valuable resource [7], with several quantum
protocols, such as teleportation [8], able to be realised exclusively with the
use of entangled states. This new focus has stimulated intensive research into
how specific patterns of entanglement can be created and manipulated in
quantum many-body systems [9].

One such controllable entanglement property is the scaling of the entanglement
entropy, $S_{A}$, within a bipartite system. The ground states of local
quantum lattice Hamiltonians typically obey an ‘area law’ such that the
entanglement entropy is proportional to the size of the boundary of the chosen
subsystem, $A$ [10, 11]. In 2010, Vitagliano, Riera and Latorre showed how
tuning the coupling profile of the inhomogeneous XX model allows its ground
state to transition smoothly from obeying an area law of entanglement entropy
scaling to a volume law [12]. The ground state of this model is termed the
‘concentric singlet phase’ [12] or simply ‘rainbow state’ [13], due to its
distinctive structure of maximally entangled valence bonds connecting pairs of
sites distributed symmetrically across the centre of the chain. This simple
model hosts a rich variety of properties [14, 15] and has been the subject of
much interest in recent years [16, 17, 18, 19].

In this work we present a generalisation of the rainbow state model, whereby,
with the introduction of staggered transverse field terms to the inhomogeneous
XX model, the degree of entanglement between each concentric pair on the chain
can be independently varied. Using a Real-Space Renormalization Group approach
we derive recursive expressions for the induced effective coupling and
transverse field terms. These expressions have an elegant description in terms
of the formalism of $q$-deformed algebra [20, 21]. For a chain of $2N$ sites
the ground state is a tensor product of $N$ concentric $q$-deformed singlets,
each with an associated deformation parameter, $q_{i}$, dependent on the
transverse field and coupling parameters of our model. The variation of these
physical parameters allows for the generation of any arbitrary set of single-
particle entanglement energies. To verify the validity of our results we
perform a detailed numerical analysis. This analysis reveals that appropriate
choices of the values of the transverse field parameter, and ordering of the
degree of entanglement ensures a high fidelity between the exact ground state
and the $q$-deformed rainbow. Moreover, we consider two special cases to
demonstrate the applicability of our method. First, we consider the case
$q_{1}=q_{2}=\dots=q_{N}$ such that each concentric pair has the same degree
of entanglement. Second, we consider the case where the single-particle
entanglement energies follow the ‘prime number spectrum’. This prime number
decomposition employs the Moebius function to naturally mirror the
entanglement spectra of free fermionic systems.

While our quantum simulator gives rise to effective couplings between sites on
opposite ends of our chain, our model is completely local, given in terms of
XX interactions and local magnetic fields. Thus, it directly lends itself to
experimental verification and practical applications. Indeed, recent
developments in cold atom experiments [22, 23, 24, 25, 26, 27] have offered
unique opportunities to simulate such systems and access quantities related to
entanglement [28, 29, 30, 31, 32, 33]. We expect that our quantum simulator
can have direct applications in condensed matter or quantum technologies where
specific structures of correlation patterns are requested between two
subsystems.

##  II The $q$-Deformed Model

In order to introduce our model for a chain of $2N$ spin-${1\over 2}$
particles, we first present a two-site Hamiltonian that allows for direct
continuous variation of the degree of entanglement between its spins.

###  II.1 Two Spins Hamiltonian

Consider the two-spins Hamiltonian

$\mathcal{H}=J_{1}\left(\sigma_{-1}^{x}\sigma_{1}^{x}+\sigma_{-1}^{y}\sigma_{1}^{y}\right)+h_{1}\left(\sigma_{-1}^{z}-\sigma_{1}^{z}\right).$
(1)

The ground state of $\mathcal{H}$ is given by

$\ket{\psi_{1}}=\frac{1}{\sqrt{[2]_{q_{1}}}}\left(q_{1}^{-\frac{1}{2}}\ket{\uparrow\downarrow}_{-1,1}-q_{1}^{\frac{1}{2}}\ket{\downarrow\uparrow}_{-1,1}\right),$
(2)

and has ground state energy

$E_{1}=-[2]_{q_{1}}J_{1},$ (3)

where

$q_{1}=e^{\gamma_{1}},\hskip 5.69046pt\sinh{\gamma_{1}}=\frac{h_{1}}{J_{1}},$
(4)

and $[x]_{q}$ is the so-called quantum dimension

$[x]_{q}=\frac{q^{x}-q^{-x}}{q-q^{-1}}.$ (5)

This ground state is the singlet of the quantum group $SU(2)_{q_{1}}$ [34].
Such $q$-deformed valence bonds have been considered in relation to a range of
quantum many-body models [35, 36, 37, 38], including the anisotropic
$q$-deformed generalization of the spin-$1$ AKLT chain as considered in [39,
40]. In the limit $h_{1}\rightarrow 0$ such that $q_{1}\rightarrow 1$, we
recover the maximally entangled singlet state of the standard $SU(2)$ Lie
algebra. The degree of entanglement between this pair is directly related to
the value of the deformation parameter, $q_{1}$, which is in turn directly
related to our coupling and transverse field parameters via equation (4). To
investigate this, we bipartition the system down the centre of the chain into
region $A$, and its complement $B$. The reduced density matrix of (2) is then
determined for region $A$. The corresponding Renyi entropy of order $\alpha$
is given by

$S_{A,1}^{(\alpha)}=\frac{1}{1-\alpha}\ln{\frac{1+q_{1}^{2\alpha}}{(1+q_{1}^{2})^{\alpha}}},\hskip
14.22636pt\alpha>0,\alpha\neq 1$ (6)

and takes the maximum value $\ln{2}$ when $q_{1}=1$ for all $\alpha$ as shown
in Figure 1. This expression for the Renyi entropy reflects a symmetry of our
model as under the transformation $h_{1}\rightarrow-h_{1}$, such that
$q_{1}\rightarrow\frac{1}{q_{1}}$, the value of the Renyi entropy of order
$\alpha$ is unchanged.

By considering the limit of $S_{A,1}^{(\alpha)}$ as $\alpha\rightarrow 1$ we
obtain the expression for the von Neumann entanglement entropy of the pair

$S_{A,1}=\ln{(1+q_{1}^{2})}-\frac{q_{1}^{2}}{1+q_{1}^{2}}\ln{q_{1}^{2}}.$ (7)

This entropy can be varied continuously to achieve all values in the maximal
range $0\leq S_{A,1}\leq\ln{2}$, by varying $0\leq q_{1}\leq\infty$, or
equivalently $-\infty\leq\frac{h_{1}}{J_{1}}\leq\infty$. We see that by
varying the physical parameters of our model we can achieve all degrees of
pairwise entanglement between the two spins.

Figure 1: The variation of the Renyi entropy, $S_{A,1}^{(\alpha)}$ for states
given in (2) as a function of their deformation parameter $q_{1}$, for a range
of fixed values of $\alpha$. The Renyi entropy takes maximal value
$S_{A,1}^{(\alpha)}=\ln{2}$ when the deformation parameter $q_{1}=1,$ for all
$\alpha$, and is unchanged under the transformation, $h_{1}\rightarrow-h_{1}$,
such that $q_{1}\rightarrow\frac{1}{q_{1}}$, reflecting a symmetry of the
Hamiltonian (1).

###  II.2 $2N$ Spin Hamiltonian

The simple two-spin Hamiltonian presented above is the basis on which we
construct our general model for a chain of any even number of spins. We now
consider a chain of $2N$ spin-$\frac{1}{2}$ particles with the following
Hamiltonian

$\mathcal{H}=\sum_{i=1}^{N}h_{i}\left(\sigma_{-i}^{z}-\sigma_{i}^{z}\right)+J_{1}\left(\sigma_{-1}^{x}\sigma_{1}^{x}+\sigma_{-1}^{y}\sigma_{1}^{y}\right)\\\
+\sum_{i=2}^{N}J_{i}\left(\sigma_{-i}^{x}\sigma_{-(i-1)}^{x}+\sigma_{-i}^{y}\sigma_{-(i-1)}^{y}+\sigma_{i-1}^{x}\sigma_{i}^{x}+\sigma_{i-1}^{y}\sigma_{i}^{y}\right).$
(8)

We have introduced the site labelling
$\\{-N,-(N-1),\dots,-2,-1,1,2,\dots,N-1,N\\}$ such that sites $-i$ and $i$ are
equidistant from a central bipartition of the chain, as shown in Figure 2.

Figure 2: The $q$-deformed spin model for a chain of $2N$ sites. The blue
lines represent the XX coupling terms, $J_{i}$, and the red arrows represent
the magnitude and direction of the transverse magnetic field, $h_{i}$. The
magnitude of both the coupling and transverse field are symmetric about the
centre of the chain, with decreasing strength moving outwards.

In order to find the ground state of our model we have used the Real-Space
Renormalization Group approach as first introduced by Ma and Dasgupta in [41]
and later developed by Fisher with the application of the method to the Random
Transverse Field Ising Chain [42]. This approach allows us to consider the
ground state properties of random quantum chains by iteratively decimating the
degrees of freedom with highest energy in order to derive an overall effective
low-energy model. We start by first briefly reviewing this method with
reference to the known results of the $h_{i}\rightarrow 0$ limit of our model.

####  II.2.1 XX Model Renormalization Group

In the limit, $h_{i}=0$, our Hamiltonian (8) is equivalent to that of the
inhomogeneous XX model acting on a chain of $2N$ spins

$H_{XX}=\sum_{i=1}^{2N}J_{i}\left(\sigma_{i}^{x}\sigma_{i+1}^{x}+\sigma_{i}^{y}\sigma_{i+1}^{y}\right),$
(9)

where we have re-adopted the standard site labelling
$\\{1,2,\dots,2N-1,2N\\}$. Using the Renormalization Group (RG) approach for
some random coupling profile, the highest energy term such that $J_{i}\gg
J_{i-1},J_{i+1}$, is identified and diagonalised independently of the rest of
the chain. To zeroth-order in perturbation theory, the ground state of the
system is then

$\ket{\psi}=\ket{\psi_{j<i}}\otimes\ket{\psi^{-}_{i}}\otimes\ket{\psi_{j>i}},$
(10)

where
$\ket{\psi^{-}_{i}}=\frac{1}{\sqrt{2}}(\ket{\uparrow\downarrow}_{i,i+1}-\ket{\downarrow\uparrow}_{i,i+1})$
is the maximally entangled singlet ground state of the two-site XX model and
$\ket{\psi_{j<i}}$, $\ket{\psi_{j>i}}$ refer to the state of the spins to the
left and right of the singlet, respectively. To compute higher order
corrections to the ground state of our system we initial consider the spins
$i$ and $i+1$ to be ‘frozen’ into this singlet state. Then we employ
perturbation theory to find the effect induced by quantum fluctuations on the
neighbouring spins, as shown in [12]. It is found that an effective coupling
arises between sites $i-1$ and $i+2$ of strength

$\tilde{J}_{i-1,i+2}=\frac{J_{i-1}J_{i+1}}{J_{i}}.$ (11)

In this way the coupling between sites $i$ and $i+1$ is replaced by effective
longer range interaction that captures the low-energy properties of the model.
For a random coupling profile, successive iterations of this procedure yield a
‘random singlet phase’, as singlets form between the pairs of spins most
strongly coupled after each decimation. In [12] Vitagliano, Riera and Latorre
demonstrated how a coupling profile that decays exponentially away from the
centre of the chain produces a special form of ground state known as the
‘concentric singlet phase’. This ground state is also known as the ‘rainbow
state’, due to it’s distinctive structure of a series of singlets
symmetrically distributed around the centre of the chain. For any given
bipartition, the entanglement entropy is directly proportional to the number
of singlets ‘cut’ by the bipartition. Thus, for such a coupling profile, the
area law of entanglement entropy is maximally violated.

####  II.2.2 $q$-Deformed Model Renormalization Group

We now apply the Real-Space RG approach to the generalised model defined in
(8). In the limit $J_{1},h_{1}\gg J_{2},h_{2}$, this yields the ground state

$\ket{\psi}=\ket{\psi_{i<-1}}\otimes\ket{\psi_{1}}\otimes\ket{\psi_{i>1}},$
(12)

to zeroth-order in perturbation theory, where $\ket{\psi_{1}}$ is the
$q_{1}$-deformed singlet as defined in equation (4). To compute corrections to
the ground state at higher orders, second-order perturbation theory is used,
as illustrated in Figure 3 (see also Appendix A). We derive an effective
Hamiltonian of the form (1) acting between sites $-2$ and $2$ with a
renormalized coupling

$\tilde{J_{2}}=\frac{4J_{2}^{2}}{[2]_{q_{1}}^{2}J_{1}},$ (13)

and transverse field terms

$\tilde{h}_{2}=h_{2}-\frac{2\left(q_{1}-\frac{1}{q_{1}}\right)J_{2}^{2}}{[2]_{q_{1}}^{2}J_{1}}.$
(14)

In the case that $\tilde{J_{2}},\tilde{h}_{2}\gg J_{3},h_{3}$ this effective
Hamiltonian can be diagonalised to yield an additional $q_{2}$-deformed
singlet, $\ket{\psi_{2}}$, between sites $-2$ and $2$, where
$q_{2}=e^{\gamma_{2}},$
$\sinh{\gamma_{2}}=\frac{\tilde{h}_{2}}{\tilde{J_{2}}}$.

Figure 3: The Real-Space RG procedure. (a) Our model (8) acting on a chain of
four spins. For $J_{1},h_{1}\gg J_{2},h_{2}$ perturbation theory yields a
$q$-deformed singlet (4) between the central two spins. (b) These spins are
integrated out and an effective Hamiltonian of the form (1) is found to act
between sites $-2$ and $2$ with renormalized coupling $\tilde{J_{2}}$ and
transverse field $\tilde{h}_{2}$. (c) Diagonalization of this effective
Hamiltonian yields the ground state
$\ket{\psi}=\ket{\psi_{1}}\otimes\ket{\psi_{2}}$. The difference in colour of
the bonds between the two pairs indicates the difference in correlations that
can be achieved by appropriately tuning $J_{1},J_{2},h_{1}$ and $h_{2}$.

If the couplings throughout the chain are selected such that
$\tilde{J_{i}},\tilde{h}_{i}\gg J_{i+1},h_{i+1}$, then repeated iterations of
this renormalization process will eventually yield the overall ground state

$\ket{\psi}=\ket{\psi_{1}}\otimes\ket{\psi_{2}}\otimes\dots\otimes\ket{\psi_{N}},$
(15)

where

$\ket{\psi_{i}}=\frac{1}{\sqrt{[2]_{q_{i}}}}\left(q_{i}^{-\frac{1}{2}}\ket{\uparrow\downarrow}_{-i,i}-q_{i}^{\frac{1}{2}}\ket{\downarrow\uparrow}_{-i,i}\right),$
(16)

and for $i>1$

$q_{i}=e^{\gamma_{i}}\text{ with
}\sinh{\gamma_{i}}=\frac{\tilde{h}_{i}}{\tilde{J_{i}}}.$ (17)

The renormalized coupling and transverse field parameters for the effective
Hamiltonian between spins $-i$ and $i$ are given by the recursive expressions

$\tilde{J_{i}}=\frac{4J_{i}^{2}}{[2]_{q_{i-1}}^{2}\tilde{J}_{i-1}},$ (18)
$\tilde{h}_{i}=h_{i}-\frac{2\left(q_{i-1}-\frac{1}{q_{i-1}}\right)J_{i}^{2}}{[2]_{q_{i-1}}^{2}\tilde{J}_{i-1}}.$
(19)

From equations (18) and (19), we see that the expressions for $\tilde{J_{i}}$
and $\tilde{h}_{i}$ are dependent on all previous $\tilde{J}_{j<i}$,
$\tilde{h}_{j<i}$. By fixing all previous $i-1$ values, it is always possible
to vary the associated physical parameters $J_{i}$ and $h_{i}$ such as to
achieve any $1\leq q_{i}\leq\infty$. In this way, we will show that the
deformation of each $q$-singlet can be individually tuned to achieve any
degree of pairwise entanglement between a given pair of spins.

##  III Entanglement Properties of the $q$-Deformed Rainbow

The ground state (15) of our Hamiltonian (8) has a tensor product form.
Subsequently, the reduced density matrix across a central bipartition admits
the tensor product decomposition

$\rho_{A}=\rho_{1}\otimes\rho_{2}\otimes\dots\otimes\rho_{N},$ (20)

where each $\rho_{i}$ is diagonal, given by

$\rho_{i}=\begin{pmatrix}\frac{1}{1+q_{i}^{2}}&0\\\
0&\frac{q_{i}^{2}}{1+q_{i}^{2}}\end{pmatrix}.$ (21)

This decomposition yields simple expressions for many of the entanglement
properties of the $q$-deformed rainbow, as we will see in the following.

###  III.1 Renyi and von Neumann entropies

Using the reduced density matrix tensor product decomposition we derive the
form of the Renyi entropy of order $\alpha$ of the ground state (15) across a
central bipartition

$S_{A}^{(\alpha)}=\frac{1}{1-\alpha}\sum_{i=1}^{N}\ln{\frac{1+q_{i}^{2\alpha}}{(1+q_{i}^{2})^{\alpha}}},\hskip
8.5359pt\alpha>0,\alpha\neq 1.$ (22)

In the limit $\alpha\rightarrow 1$ we obtain an expression for the von Neumann
entropy of the ground state

$S_{A}=-\sum_{i=1}^{N}\left[\ln{(1+q_{i}^{2})}-\frac{q_{i}^{2}}{1+q_{i}^{2}}\ln{q_{i}^{2}}\right].$
(23)

In Section II.1 we found the von Neumann entropy, $S_{A,1}$, of a single pair
of spins as a function of the deformation parameter $q_{1}$, as given by (6).
By extending this definition to that of the von Neumann entropy of the state
$\ket{\psi_{i}}$ between spins $-i$ and $i$

$S_{A,i}=\ln{(1+q_{i}^{2})}-\frac{q_{i}^{2}}{1+q_{i}^{2}}\ln{q_{i}^{2}},$ (24)

it is clear that the total von Neumann entropy is a sum of the individual von
Neumann entropies of each concentric pair of spins on the chain. This is also
true for the Renyi entropy, and is a natural consequence of the tensor product
form of the reduced density matrix. By independently varying each $q_{i}$, we
can therefore achieve all degrees of entanglement in the allowed maximal range
$0\leq S_{A}\leq N\ln{2}$.

###  III.2 Entanglement Spectrum

The entanglement spectrum was introduced by Li and Haldane [43] as an
alternative entanglement measure that aimed to capture a complete
representation of the entanglement between two subsystems [44]. The values of
the spectrum, $E_{i}$, are related to the eigenvalues of the reduced density
matrix, $\lambda_{i}$, via

$\lambda_{i}=e^{-E_{i}}.$ (25)

The exponential relationship means that the dominant quantum correlations
depend predominantly on the ‘lowest’ part of the entanglement spectrum.

The entanglement spectrum reflects many of the physical properties of the
system [45, 46, 47, 48, 49] and serves as a fingerprint of topological order
[50, 51, 52]. For any non-interacting model, Wick’s theorem shows that the
spectrum can be constructed from a set of single-particle entanglement
energies as

$E^{f}_{j}(\epsilon)=E_{0}+\sum_{i=1}^{N}n_{i}(j)\epsilon_{i},$ (26)

where $E_{0}$ is a normalization constant and each $n_{j}=\\{0,1\\}$ [53].

For our $q$-deformed rainbow, we find that

$E_{0}=\sum_{i=1}^{N}\ln{(1+q_{i}^{2})},$ (27)

and

$\epsilon_{i}=-\ln{q_{i}^{2}}.$ (28)

Hence, the deformation parameters, $q_{i}$, of the $q$-deformed singlets
directly determine the single-particle entanglement energies. As each $q_{i}$
can take any value in the range $0\leq q_{i}\leq\infty$, each $\epsilon_{i}$
can be individually tuned to take any value
$-\infty\leq\epsilon_{i}\leq\infty$.

Figure 4: The values of $h_{2}$ required to generate any
$0\leq\epsilon_{2}\leq 10$ for fixed $J_{1}=h_{1}=1$, $J_{2}=0.01$
($\epsilon_{1}=-2\sinh^{-1}{(1)}$). Any desired value of $\epsilon_{2}$ in
this range can be obtained by selecting the corresponding value of $h_{2}$.

By combining (17) and (28) we derive the simple relationship

$\gamma_{i}=-\frac{\epsilon_{i}}{2}.$ (29)

This in turn yields an expression for the required ratio of the renormalized
parameters for a given pair in order to produce a specific desired single-
particle entanglement energy

$\frac{\tilde{h}_{i}}{\tilde{J_{i}}}=-\sinh{\left(\frac{\epsilon_{i}}{2}\right)}.$
(30)

As a result, each single particle energy of the entanglement spectrum can be
directly obtained by appropriately tuning a single effective magnetic field.
In Appendix B we expand these expressions to derive closed forms for the
required ratio of the physical coupling parameters. In Figure 4 the dependence
of $\epsilon_{2}$ on $h_{2}$ for fixed $J_{1},h_{1}$ and $J_{2}$ is
illustrated. For the shown range, any desired $\epsilon_{2}$ can be simulated
by simply reading off the corresponding value of $h_{2}$. In this way, by
fixing all previous $i-1$ single-particle entanglement energies,
$\epsilon_{i}$ can be tuned to any desired value by appropriately varying
$h_{i}$.

##  IV Fidelity Optimisation

In the previous Section, we have demonstrated how controlled variation of the
parameters of our model in the strong inhomogeneity limit
$\tilde{J_{i}},\tilde{h}_{i}\gg J_{i+1},h_{i+1}$ allows for the generation of
any arbitrary pattern of correlations given in terms of the entanglement
entropy (23) or the single-particle entanglement energies (30). In this
Section we present how the parameters of our model can be chosen such that the
fidelity is maximised for any desired entanglement profile. In quantum
information theory, fidelity is a measure of the ‘closeness’ of two quantum
states, $\ket{\psi_{A}}$ and $\ket{\psi_{B}}$, given by the squared overlap,
$\absolutevalue{\bra{\psi_{A}}\ket{\psi_{B}}}^{2}$ [54]. To optimise the
choice of parameters for any desired correlation profile, we consider the
variation of the fidelity between the exact ground state of our model and the
$q$-deformed rainbow in the case $N=4$.

###  IV.1 Optimising $h_{2}$

Figure 5: Variation of the ground state fidelity with the entanglement entropy
of the outer pair for different fixed values of $h_{1}$ with $J_{1}=1$ and
$J_{2}=0.1$. The grey line shows an example of a desired outer entanglement
entropy, $S_{A,2}=0.5$. The two intersections with each curve for $h_{1}>0$
indicate two possible values of $h_{2}$ to generate the desired $S_{A,2}$ with
a distinct difference in fidelity. This choice can be used to optimise the
accuracy of our model.

In Section II.1, we noted that a symmetry of our two-site Hamiltonian (1)
results in the preservation of the von Neumann entropy, $S_{A,1}$, under the
transformation $h_{1}\rightarrow-h_{1}$. Here, we will show that although
$S_{A,2}$ possesses a similar symmetry under the transformation
$\tilde{h}_{2}\rightarrow-\tilde{h}_{2}$, one of these values will yield a
significantly higher fidelity than the other corresponding to the choice of
sign of $\frac{h_{1}}{J_{1}}$.

In Figure 5 we plot the fidelity between the $q$-deformed rainbow and the
exact ground state of (8) as a function of the entanglement entropy between
sites $-2$ and $2$ for a range of constant values of $h_{1}$. For each curve
$J_{1},h_{1}$ and $J_{2}$ are fixed such that $S_{A,2}$ is a function of
$h_{2}$. We see that for any desired value of $S_{A,2}$, for example
$S_{A,2}=0.5$ as indicated by the vertical grey line, the two intersections
with each curve indicate two values of $h_{2}$ that correspond to the same
degree of entanglement, but with a distinct difference in fidelity. As
described, these two solutions arise due to the natural symmetry of the
entanglement entropy about the value of $h_{2}$ yielding maximal entanglement
between sites $-2$ and $2$. To find the value $h_{2}^{\text{max}}$ that
maximises $S_{A,2}$ we set $\tilde{h}_{2}=0$ in equation (14) and obtain

$h_{2}^{\text{max}}=\frac{2\left(q_{1}-\frac{1}{q_{1}}\right)J_{2}^{2}}{[2]_{q_{1}}^{2}J_{1}}.$
(31)

The symmetry of the entanglement entropy $S_{A,2}$ about $h_{2}^{\text{max}}$
is shown in Figure 6(a) for the case $J_{1}=h_{1}=1$, $J_{2}=0.1$. By mapping
these values onto the plot of fidelity with $S_{A,2}$ as shown in Figure 6(b),
we see that we have a ‘high fidelity branch’ corresponding to $h_{2}\geq
h_{2}^{\text{max}}$ and a ‘low fidelity branch’ for $h_{2}\leq
h_{2}^{\text{max}}$. For any desired value of $S_{A,2}$, the fidelity is
clearly maximised by choosing the appropriate value of $h_{2}\geq
h_{2}^{\text{max}}$. In contrast, if $\frac{h_{1}}{J_{1}}$ is negative as
shown in Figure 6(c) and (d), the opposite is true, and the fidelity is
maximised by selecting the value of $h_{2}$ from the branch $h_{2}\leq
h_{2}^{\text{max}}$. In this way, for fixed couplings $J_{1}$ and $J_{2}$, the
direction of the magnetic fields applied to sites $i=-1,1$ dictate the
magnitude and direction of the magnetic field that should be applied to sites
$i=-2,2$ in order to maximise the accuracy of our model.

Figure 6: (a) Variation of the entanglement entropy of the outer pair with
$h_{2}$ for $J_{1}=h_{1}=1$ and $J_{2}=0.1$. (b) The corresponding variation
of the ground state fidelity with the entanglement entropy for both the values
less than and greater than $h_{2}^{\text{max}}$. The higher fidelity branch
corresponds to the values $h_{2}\geq h_{2}^{\text{max}}$. (c) Variation of the
entanglement entropy of the outer pair with $h_{2}$, now for $J_{1}=1$,
$J_{2}=0.1$ and $h_{1}=-1$. (d) The corresponding variation of the ground
state fidelity with the entanglement entropy for both the values less than and
greater than $h_{2}^{\text{max}}$. The higher fidelity branch now corresponds
to the values $h_{2}\leq h_{2}^{\text{max}}$. On all four subfigures the solid
grey line serves to illustrate this fidelity optimisation for the specific
case $S_{A,2}=0.5$.

###  IV.2 Optimising Order of Pairs

Our system has a symmetry with respect to which pair $i,-i$ of spins is used
to tune a certain single particle entanglement energy $\epsilon_{k}$. We can
use this freedom, in conjunction with the optimisation procedure of the
previous Subsection to optimise the overall fidelity of our chain simulator.
To proceed, we adopt the appropriately restricted range of $h_{2}\geq
h_{2}^{\text{max}}$ that maximises the fidelity of two pairs of spins. We then
consider the variation of the fidelity with $h_{2}$ for a range of fixed
values of $h_{1}$ as shown in Figure 7(a). Here $\frac{h_{1}}{J_{1}}>0$ for
each curve such that we have selected the values $h_{2}\geq
h_{2}^{\text{max}}$. We first observe that for each fixed value of $h_{1}$, as
$S_{A,2}$ increases, the fidelity decreases. This result is more notable for
low values of $h_{1}$, such that the combination of parameters with lowest
fidelity corresponds to the reproduction of the rainbow state with
$h_{1}=h_{2}=0$. Hence, our model is best at accurately producing lower
degrees of entanglement between the concentric pairs of sites.

For fixed $J_{1}$, a larger magnitude of $h_{1}$ coresponds to a lower value
of $S_{A,1}$. Figure 7(a) therefore also shows that as the value of $h_{1}$
increases and $S_{A,1}$ decreases, the fidelity with which any desired
$S_{A,2}$ can be achieved increases. Consider the case in which we want to use
our model to generate a given pair of two-site von Neumann enanglement
entropies, for example, $S_{A,i}=0.2$ and $S_{A,j}=0.6$. Our simulator has the
freedom in the choice $i=1,j=2$ or $i=2,j=1$. We will employ this freedom to
choose the combination that maximises the fidelity. In Figure 7(b) we plot the
two curves corresponding to $S_{A,1}=0.2$, and $S_{A,1}=0.6$. It is clear from
this plot that the fidelity is always maximum by choosing the parameters such
that $S_{A,1}=0.2,S_{A,2}=0.6$. In general it is always true that the fidelity
is maximised by ordering the pairs such that $S_{A,i}\leq S_{A,i+1}$. Figure 8
further illustrates this with a direct comparison of the fidelity with the
degree of entanglement between one pair when the other is maximally entangled.
For all values of $S_{A,i}$, we observe that the fidelity is maximised when
the maximally entangled state lies between sites $-2$ and $2$.

Figure 7: (a) The variation of the fidelity with $S_{A,2}$ for a range of
values of $h_{1}$ and fixed $J_{1}=1$, $J_{2}=0.1$. We observe that always the
fidelity decreases as $S_{A,2}$ increases. For two curves with different value
of $h_{1}$, it is observed that all values of $S_{A,2}$ can be produced with a
higher fidelity by the curve with a higher value of $h_{1}$ or equivalently a
lower value of $S_{A,1}$. The implications of this on producing some desired
set of von Neumann entropies are examined further in (b). Here the dashed
lines serve to illustrate the higher value of fidelity achieved by choosing
$h_{1}$ and $h_{2}$ to obtain $S_{A,1}=0.2,S_{A,2}=0.6$ as opposed to
$S_{A,1}=0.6,S_{A,2}=0.2$. In general, the fidelity is optimised by selecting
the parameters of our model such that $S_{A,i}\leq S_{A,i+1}$.

By combining equations (24) and (28), we see that the condition $S_{A,i}\leq
S_{A,i+1}$ is equivalent to
$\absolutevalue{\epsilon_{i}}\geq\absolutevalue{\epsilon_{i+1}}$. For some
desired set of single-particle entanglement energies, $\\{\epsilon_{i}\\}$,
our model allows for complete freedom in assigning which pair of sites
corresponds to a given energy. In implementing our model we therefore choose
to tune the parameters such that the magnitude of the single-particle energy
generated by sites $-i$ and $i$ decreases with increasing $i$ in order to
optimise the fidelity.

Figure 8: The fidelity of the four-site $q$-deformed rainbow and the exact
ground state of our model when one pair is maximally entangled and the
entanglement entropy of the other is varied ($J_{1}=1,J_{2}=0.1$). For every
combination of entanglement entropies, the fidelity is maximised by choosing
our parameters such that the maximally entangled pair is between sites $-2$
and $2$.

##  V Special Cases

We have shown how the tuning of the parameters of our model allows for the
generation of any arbitrary set of single-particle entanglement energies. In
this section we highlight two interesting applications: the case in which all
deformation parameters are equal, and the reproduction of the single-particle
entanglement energies for the ‘prime number spectrum’ introduced below.

###  V.1 The $q_{1}=q_{2}=\dots=q_{N}=q$ Case

In the rainbow state model, the ground state is a tensor product of concentric
maximally entangled singlets, or in the language of our model, $q_{i}=1$, for
all $i$. Here, we show how the parameters of our model can be chosen such that
all deformation parameters take the same value, $q_{i}=q$, for some chosen $q$
in the allowed range $0<q<\infty$. In this way, each concentric pair on our
chain shares the same degree of pairwise entanglement, and all single-particle
entanglement energies are equal.

We have defined $q_{1}=e^{\sinh^{-1}\left(\frac{h_{1}}{J_{1}}\right)}$ and
$q_{i>1}=e^{\sinh^{-1}\left(\frac{\tilde{h}_{i}}{\tilde{J_{i}}}\right)}$, such
that the condition
$q_{1}=q_{i}\implies\frac{h_{1}}{J_{1}}=\frac{\tilde{h}_{i}}{\tilde{J_{i}}}$.
Re-arranging equation (4) and setting $q_{1}=q$ yields

$\frac{h_{1}}{J_{1}}=\frac{1}{2}\left(q-\frac{1}{q}\right),$ (32)

for some desired $q>0$.

For all other pairs of sites the relations for the renormalised couplings must
be used. For example. by dividing equation (14) by equation (13) and equating
with (32) we obtain

$h_{2}=\frac{2J_{2}^{2}}{h_{1}}\frac{\left(1-q^{2}\right)^{2}}{\left(1+q^{2}\right)^{2}}.$
(33)

For any fixed value of $J_{2}$ this relation can be easily implemented to find
the required transverse field parameter to produce some desired $q>0$.

In the same way, the ratio of equations (18) and (19) can be equated with (32)
in order to obtain the general formula

$h_{i}=\frac{4J_{i}^{2}}{h_{i-1}}\frac{\left(1-q^{2}\right)^{2}}{\left(1+q^{2}\right)^{2}},\hskip
5.69046pti>2$ (34)

By iterating through and systematically determining each successive value of
the required transverse field for some fixed coupling profile, these relations
allow us to produce a one-dimensional chain in which each concentric pair
shares the same degree of pairwise entanglement. In Figures 9(a) and (b) it is
shown how all values of $1\leq q\leq 10$ and the corresponding entanglement
entropies for the case $N=4$ can be produced with a very high level of
fidelity. For just $J_{1}=1,J_{2}=0.01$ the error is of the order of
$10^{-4}$.

Figure 9: (a) The variation of the ground state fidelity for $1\leq q\leq 10$
in the special case $q_{1}=q_{2}=q$ for $N=4$. Here $J_{1}=1,J_{2}=0.01$ (b)
The corresponding variation of the fidelity with the associated range of equal
von Neumann entanglement entropies across both pairs. All values in the range
shown can be produced with a very high level of fidelity.

###  V.2 Prime Number Spectrum

Prime numbers play an important role in number theory. The Fundamental Theorem
of Arithmetics [55] states that every natural number greater than one can be
uniquely factorised as a product of prime numbers

$N=2^{n_{2}}3^{n_{3}}\dots p^{n_{p}}\dots,$ (35)

where $p$ is a prime and $n_{p}$ counts the number of times that $p$ appears
in the factorisation of $N$. In this way, prime numbers can be thought of as
the building blocks of all natural numbers.

Let us introduce the Moebius function, $\mu(n)$:

$\mu(n)=\begin{cases}1,&n=1,\\\ (-1)^{r},&n=p_{1}\dots p_{r},\\\ 0,&\exists
p,p^{2}|n,\end{cases}$ (36)

where $p$ are prime numbers. The symbol $p^{2}|n$ means that $p^{2}$ divides
$n$. A square free integer is an integer whose factorization into products of
primes does not contain any square of a prime numbers. $\mu(n)$ is therefore
non-vanishing only on square free integers and its value is $+1$ if it
contains an even number of primes and $-1$ if it contains an odd number of
primes. In this way, the function $\mu$ is a sort of Fermi statistics if we
think of the primes as being fermions.

Let us now consider an entanglement spectrum of the form

$\lambda_{k}=\frac{A_{F}\absolutevalue{\mu(k)}}{k^{s}},\hskip
5.69046ptA_{F},s>0,\hskip 2.84544ptk=1,\dots,\infty.$ (37)

The normalization of the eigenvalues implies that

$1=\sum_{k=1}^{\infty}\lambda_{k}=A_{F}\prod_{p}(1+p^{-s})=A_{F}\frac{\zeta(s)}{\zeta(2s)},$
(38)

where we have used the Euler product formula

$\zeta(s)=\prod_{p}\frac{1}{1-p^{-s}},\hskip 5.69046pt\text{Re}(s)>1.$ (39)

Using equation (25), the entanglement energies for this spectrum are given by

$E_{k}=-\ln{\lambda_{k}}=-\ln{A_{F}}+s\ln{k}.$ (40)

Where $k$ is any square free integer. We equate this expression with that of
the spectrum of a free fermionic system

$-\ln{A_{F}}+s\ln{k}=E_{0}+\sum_{i=1}^{N}n_{i}(k)\epsilon_{i}.$ (41)

If $k$ is a square free integer then from the Fundamental Theorem of
Arithmetics one has

$k=2^{n_{2}}3^{n_{3}}\dots p^{n_{p}}\dots,\hskip 8.5359ptn_{i}=0,1,$ (42)

such that taking the logarithm of (42) yields

$\ln{k}=\sum_{p:prime}n_{p}\ln{p},\hskip 8.5359ptn_{p}=0,1.$ (43)

Hence, equation (41) is solved by

$E_{0}=-\ln{A_{F}}=\frac{\zeta(s)}{\zeta(2s)},$ (44) $\epsilon_{p}=s\ln{p}.$
(45)

The parameter $s$ can be thought of as an entanglement temperature since it is
common to all eigenenergies. The relation (45) has also been considered in
[56, 57] with $\ln{p}$ being the single-particle energies of the primon gas.
The partition function of this gas is related to the Riemann zeta function,
$\zeta(s)$. In recent work, a prime number eigenvalue spectrum has also been
experimentally realised by application of holographic optical traps [58], in
agreement with previous theoretical results [59].

In Section IV.2, we saw that the highest ground state fidelity is achieved by
fixing our parameters $\\{h_{i}\\}$ and $\\{J_{i}\\}$ such that
$\absolutevalue{\epsilon_{i}}\geq\absolutevalue{\epsilon_{i+1}}$. Thus, in
order for our model to most accurately reproduce this prime number spectrum,
we choose $\epsilon_{i}=s\ln{p_{i}}$ such that $p_{i}>p_{i+1}$. The required
values of the set $\\{h_{i}\\}$, for some fixed coupling profile
$\\{J_{i}\\}$, can then be simply read off from equations (67), (68) and (69)
in the Appendices.

##  VI Conclusions

In summary, we have introduced a spin chain that can produce arbitrary ground
state free-particle correlations across a given bipartition. Our scheme is a
generalisation of the rainbow states of concentric maximally entangled
singlets to the case of concentric pairs, each one with arbitrary
entanglement. The degree of entanglement is easily tuned by appropriately
choosing the magnitude of local magnetic fields. The entanglement across the
bipartition can be parametrised in terms of single particle energies of the
entanglement spectra. We find that for a fixed coupling profile these energies
are simple functions of the magnetic fields, thus providing direct
accessibility and tunability.

To test the validity and applicability of our method we compare the fidelity
of the predicted theoretical model with the exact diagonalisation of the spin
system. The employed perturbation method has a symmetry in terms of the
ordering of the concentric entangled states. By taking advantage of this
symmetry we find the optimal order of magnetic fields that gives the best
fidelities. Finally, we apply our method to two case scenarios. First, we
consider the homogeneous case of concentric pairs with the same entanglement.
Second, we consider the case of single particle energies of the entanglement
spectra that are parametrised by prime numbers. This model is inspired by the
similarity between the decomposition of free-system entanglement spectra in
terms of single particle energies and the decomposition of integers in terms
of prime numbers. In recent experimental work, holographic techniques have
been developed allowing for the tuning of the energy spectrum of the single-
particle Schrödinger equation [58]. Notably a ‘prime number quantum
potential’, $V_{N}(x)$, can be applied such that the single-particle
Schrödinger equation has the lowest $N$ prime numbers as eigenvalues. These
techniques demonstrate an interesting procedure for the reproduction of
arbitrary sequences of integers as energy spectra with possible applicability
to this work.

Our methodology can have a direct application in quantum technologies,
whenever a very specific pattern of quantum correlations is required [60, 61,
62]. It can also simulate quantum phases of matter that require specific
ground state correlations across a bipartition. Finally, our approach opens
the way to investigate inhomogeneous spin chains in the presence of disordered
magnetic fields, which is physically a common scenario, thus generalising
previous approaches [63, 17].

###### Acknowledgements.

We would like to thank Gabriel Matos and Andrew Hallam for inspiring
conversations. L.B. acknowledges support from EPSRC Grant No. EP/T517860/1.
This work was also in part supported by EPSRC Grant No. EP/R020612/1. G.S.
acknowledges financial support through the Spanish MINECO grant
PID2021-127726NB-I00, the Comunidad de Madrid Grant No. S2018/TCS-4342, the
Centro de Excelencia Severo Ochoa Program SEV-2016-0597 and the CSIC Research
Platform on Quantum Technologies PTI-001.

##  Appendix A Four Site Perturbation Theory

In order to illustrate the Real-Space RG approach, we apply perturbation
theory to our Hamiltonian (8) restricted to a chain of four sites

$H=H_{1}+\lambda V,$ (46)

where

$H_{1}=J_{1}(\sigma_{-1}^{x}\sigma_{1}^{x}+\sigma_{-1}^{y}\sigma_{1}^{y})+h_{1}(\sigma_{-1}^{z}-\sigma_{1}^{z}),$
(47)

and

$V=J_{2}^{\prime}(\sigma_{-2}^{x}\sigma_{-1}^{x}+\sigma_{-2}^{y}\sigma_{-1}^{y}+\sigma_{1}^{x}\sigma_{2}^{x}+\sigma_{1}^{y}\sigma_{2}^{y})+h_{2}^{\prime}(\sigma_{-2}^{z}-\sigma_{2}^{z}).$
(48)

Here the couplings $J_{2}=\lambda J_{2}^{\prime}$ and $h_{2}=\lambda
h_{2}^{\prime}$, such that for any $\lambda\ll 1$, the perturbative condition
$J_{2},h_{2}\ll J_{1},h_{1}$ is ensured.

On two sites, $H_{1}$ has the following eigenstates

$\ket{\psi^{-}}=\frac{1}{\sqrt{[2]_{q_{1}}}}\left(q_{1}^{-\frac{1}{2}}\ket{\uparrow\downarrow}-q_{1}^{\frac{1}{2}}\ket{\downarrow\uparrow}\right),$
(49) $\ket{\psi^{0}}=\ket{\uparrow\uparrow},$ (50)
$\ket{\psi^{1}}=\ket{\downarrow\downarrow},$ (51)
$\ket{\psi^{+}}=\frac{1}{\sqrt{[2]_{q_{1}}}}\left(q_{1}^{\frac{1}{2}}\ket{\uparrow\downarrow}+q_{1}^{-\frac{1}{2}}\ket{\downarrow\uparrow}\right),$
(52)

with eigenenergies $E_{1}=-[2]_{q_{1}}J_{1},E_{s}=0,E_{t}=0$ and
$E_{k}=+[2]_{q_{1}}J_{1}$ respectively, and $q_{1}$ as previously defined in
equation (4).

When extended to a chain of four spins, the ground state subspace of $H_{1}$
becomes four-fold degenerate. We represent this subspace with the basis
vectors: $\\{\ket{m}\\}=\\{\ket{m_{1}},\ket{m_{2}},\ket{m_{3}},\ket{m_{4}}\\}$
$=\\{\ket{\uparrow}_{-2}\ket{\psi^{-}}_{-1,1}\ket{\uparrow}_{2}$,
$\ket{\uparrow}_{-2}\ket{\psi^{-}}_{-1,1}\ket{\downarrow}_{2}$,
$\ket{\downarrow}_{-2}\ket{\psi^{-}}_{-1,1}\ket{\uparrow}_{2},$
$\ket{\downarrow}_{-2}\ket{\psi^{-}}_{-1,1}\ket{\downarrow}_{2}\\}$. The
first-order corrections arise due to the action of the perturbative term on
the ground state subspace. This is quantified via the computation of the
matrix elements of the effective Hamiltonian to first order

$H_{\alpha,\beta}^{eff\left(1\right)}=\bra{m_{\alpha}}V\ket{m_{\beta}},$ (53)

yielding

$H^{eff(1)}=2h_{2}^{\prime}\begin{pmatrix}0&0&0&0\\\ 0&1&0&0\\\ 0&0&-1&0\\\
0&0&0&0\end{pmatrix}$ (54)

in the basis $\\{\ket{m}\\}$. By inspection, it can be seen that the first-
order effective Hamiltonian term is therefore
$H^{eff(1)}=h_{2}^{\prime}(\sigma_{-2}^{z}-\sigma_{2}^{z})$. The first-order
ground state energy correction is found by diagonalizing the above matrix. It
is clear that the degeneracy is only partially lifted to first order. It is
therefore necessary to consider the second order corrections that arise due to
the overlap with states from each of the excited state subspaces. These
excited state subspaces are found in the same way as the set $\\{\ket{m}\\}$
by taking the tensor product of the two-qubit computational basis with the
excited eigenstates of $H_{0}$:
$\\{\ket{s}\\}=\\{\ket{s_{1}},\ket{s_{2}},\ket{s_{3}},\ket{s_{4}}\\}$
$=\\{\ket{\uparrow\uparrow\uparrow\uparrow}$,
$\ket{\uparrow\uparrow\uparrow\downarrow}$,
$\ket{\downarrow\uparrow\uparrow\uparrow}$,
$\ket{\downarrow\uparrow\uparrow\downarrow}\\}$,
$\\{\ket{t}\\}=\\{\ket{t_{1}},\ket{t_{2}},\ket{t_{3}},\ket{t_{4}}\\}$
$=\\{\ket{\uparrow\downarrow\downarrow\uparrow}$,
$\ket{\uparrow\downarrow\downarrow\downarrow}$,
$\ket{\downarrow\downarrow\downarrow\uparrow}$,
$\ket{\downarrow\downarrow\downarrow\downarrow}\\}$,
$\\{\ket{k}\\}=\\{\ket{k_{1}},\ket{k_{2}},\ket{k_{3}},\ket{k_{4}}\\}$
$=\\{\ket{\uparrow}_{-2}\ket{\psi^{+}}_{-1,1}\ket{\uparrow}_{2}$,
$\ket{\uparrow}_{-2}\ket{\psi^{+}}_{-1,1}\ket{\downarrow}_{2}$,
$\ket{\downarrow}_{-2}\ket{\psi^{+}}_{-1,1}\ket{\uparrow}_{2}$,
$\ket{\downarrow}_{-2}\ket{\psi^{+}}_{-1,1}\ket{\downarrow}_{2}\\}\\}$. Such
that the full set of excited states,
$\\{\ket{n}\\}=\\{\\{\ket{s}\\},\\{\ket{t}\\},\\{\ket{k}\\}\\}$.

The matrix elements of the effective Hamiltonian to second-order are found
from

$H_{\alpha,\beta}^{eff(2)}=\sum_{i=1}\frac{\bra{m_{\alpha}}V\ket{n_{i}}\bra{n_{i}}V\ket{m_{\beta}}}{E_{-}-E_{i}}.$
(55)

The computation of which yields

$H^{eff(2)}=\frac{(2J_{2}^{\prime})^{2}}{(1+q_{1}^{2})E_{1}}\begin{pmatrix}1+q_{1}^{2}&0&0&0\\\
0&2q_{1}^{2}&-2q_{1}&0\\\ 0&-2q_{1}&2&0\\\ 0&0&0&1+q_{1}^{2}\end{pmatrix}.$
(56)

Combining our first and second-order perturbative terms we derive an
expression for the effective Hamiltonian correct to $\mathcal{O}(\lambda^{2})$

$\displaystyle H^{eff}$ $\displaystyle\approx E_{1}\mathbf{1}_{4}+\lambda
H^{eff(1)}+\lambda^{2}H^{eff(2)}$ (57)
$\displaystyle=\left(E_{1}+\frac{(2\lambda
J_{2}^{\prime})^{2}}{E_{1}}\right)\mathbf{1}_{4}$ (58)
$\displaystyle\qquad-\frac{q_{1}(2\lambda
J_{2}^{\prime})^{2}}{(1+q_{1}^{2})E_{1}}(\sigma_{-2}^{x}\sigma_{2}^{x}+\sigma_{-2}^{y}\sigma_{2}^{y})$
$\displaystyle\qquad+\left(\lambda h_{2}^{\prime}-\frac{2(1-q_{1}^{2})(\lambda
J_{2}^{\prime})^{2}}{(1+q_{1}^{2})E_{1}}\right)(\sigma_{-2}^{z}-\sigma_{2}^{z})$
$\displaystyle=\mathcal{C}\mathbf{1}_{4}+\tilde{J_{2}}(\sigma_{-2}^{x}\sigma_{2}^{x}+\sigma_{-2}^{y}\sigma_{2}^{y})+\tilde{h}_{2}(\sigma_{-2}^{z}-\sigma_{2}^{z})$
(59)

In this way we obtain the following expressions for our renormalized
parameters

$\tilde{J_{2}}=\frac{4J_{2}^{2}}{[2]_{q_{1}}^{2}J_{1}},$ (60)
$\tilde{h}_{2}=h_{2}-\frac{2\left(q_{1}-\frac{1}{q_{1}}\right)J_{2}^{2}}{[2]_{q_{1}}^{2}J_{1}}.$
(61)

Note that, in the case $h_{1}=h_{2}=0$ such that $q_{1}=q_{2}=1$,
$\tilde{h}_{2}$ vanishes and $\tilde{J_{2}}$ returns to that of the
inhomogeneous XX model re-scaling as seen in equation (11) as expected.

Diagonalization of the effective Hamiltonian yields the ground state

$\displaystyle\ket{\psi}$
$\displaystyle=\frac{1}{\sqrt{[2]_{q_{2}}}}\left(q_{2}^{-\frac{1}{2}}\ket{m_{2}}-q_{2}^{\frac{1}{2}}\ket{m_{3}}\right)$
(62)
$\displaystyle=\left(\frac{1}{\sqrt{[2]_{q_{1}}}}(q_{1}^{-\frac{1}{2}}\ket{\uparrow\downarrow}_{-1,1}-q_{1}^{\frac{1}{2}}\ket{\downarrow\uparrow}_{-1,1})\right)$
(63)
$\displaystyle\qquad\otimes\left(\frac{1}{\sqrt{[2]_{q_{2}}}}(q_{2}^{-\frac{1}{2}}\ket{\uparrow\downarrow}_{-2,2}-q_{2}^{\frac{1}{2}}\ket{\downarrow\uparrow}_{-2,2})\right)$

with corresponding ground state energy

$E_{2}=E_{1}-\frac{4J_{2}^{2}}{[2]_{q_{1}}J_{1}}+\tilde{E_{2}},$ (64)

where

$\tilde{E_{2}}=-[2]_{q_{2}}\tilde{J_{2}}.$ (65)

##  Appendix B Relationship Between Real Model Parameters and Single-Particle
Entanglement Energies

In Section III.2 the following relationship was found between the renormalised
coupling parameters and single-particle entanglement energies

$\frac{\tilde{h}_{i}}{\tilde{J_{i}}}=-\sinh{\left(\frac{\epsilon_{i}}{2}\right)}.$
(66)

By combining this relation with equations (18) and (19), we can obtain
expressions that directly relate the desired single-particle entanglement
energies to the required ratio of real parameters in our model. The recursive
nature of the formulae for the re-scaled parameters means that in order to
engineer some single-particle energy $\epsilon_{i}$ it is necessary to have
previously established some value for all previous $i-1$ energies.

In the case $J_{1},h_{1}\gg J_{i\neq 1},h_{i\neq 1}$ that we consider, the
central terms $J_{1}$ and $h_{1}$ do not get re-scaled, therefore simply

$h_{1}=-J_{1}\sinh{\left(\frac{\epsilon_{1}}{2}\right)},$ (67)

for some desired single-particle energy, $\epsilon_{1}$, and fixed value of
the central coupling term. In Appendix A we have found exact expressions for
the renormalized parameters $\tilde{J_{2}}$ and $\tilde{h}_{2}$. By
substituting these into (30) and fixing all other parameters, we obtain an
expression for the required transverse field

$h_{2}=-\frac{J_{2}^{2}}{\cosh^{2}{\left(\frac{\epsilon_{1}}{2}\right)}J_{1}}\left[\sinh{\left(\frac{\epsilon_{2}}{2}\right)}+\sinh{\left(\frac{\epsilon_{1}}{2}\right)}\right].$
(68)

Repated iterations of this process yield a general form for the required
transverse field parameter for any pair of sites $-i$ and $i$, when $i>2$

$h_{i>2}=\frac{J_{i}^{2}}{\cosh^{2}\left(\frac{\epsilon_{i-1}}{2}\right)h_{i-1}}\left[\sinh{\left(\frac{\epsilon_{i}}{2}\right)}+\sinh{\left(\frac{\epsilon_{i-1}}{2}\right)}\right]\\\
\times\left[\sinh{\left(\frac{\epsilon_{i-1}}{2}\right)}+\sinh{\left(\frac{\epsilon_{i-2}}{2}\right)}\right].$
(69)

Thus, it is made explicit how the real parameters of our model can be selected
such as to produce any desired set of single-particle entanglement energies.

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