# Entanglement of mid-spectrum eigenstates of chaotic many-body systems —
deviation from random ensembles

Masudul Haque  Department of Theoretical Physics, Maynooth University, Co.
Kildare, Ireland  Max-Planck-Institut für Physik komplexer Systeme, D-01187
Dresden, Germany  Paul A. McClarty  Max-Planck-Institut für Physik komplexer
Systeme, D-01187 Dresden, Germany  Ivan M. Khaymovich  Max-Planck-Institut für
Physik komplexer Systeme, D-01187 Dresden, Germany

###### Abstract

Eigenstates of local many-body interacting systems that are far from spectral
edges are thought to be ergodic and close to being random states. This is
consistent with the eigenstate thermalization hypothesis and volume-law
scaling of entanglement. We point out that systematic departures from complete
randomness are generically present in mid-spectrum eigenstates, and focus on
the departure of the entanglement entropy from the random-state prediction. We
show that the departure is (partly) due to spatial correlations and due to
orthogonality to the eigenstates at the spectral edge, which imposes structure
on the mid-spectrum eigenstates.

Ergodicity and equilibration in the quantum realm remain imperfectly
understood, and the characterization of quantum ergodicity is now an active
research front. One view is that quantum ergodicity corresponds to eigenstates
of many-body systems being effectively random. This idea is closely connected
to the eigenstate thermalization hypothesis (ETH) Deutsch (1991); Srednicki
(1994); Rigol  _et al._ (2008); Reimann (2015); D’Alessio  _et al._ (2016);
Deutsch (2018); Mori  _et al._ (2018), and to the ideas loosely known as
(canonical) typicality Tasaki (1998); Gemmer  _et al._ (2001); Goldstein  _et
al._ (2006); Popescu  _et al._ (2006); Sugita (2007); Reimann (2007); Gemmer
_et al._ (2009); Goldstein  _et al._ (2010); von Neumann (2010); Sugiura and
Shimizu (2013); Tasaki (2016); Mori  _et al._ (2018). For a non-integrable
(chaotic) many-body Hamiltonian $H$, it is expected that a state
$\ket{\psi_{R}}$ with the random coefficients independently Gaussian
distributed in a certain typical basis should be a good model for infinite-
temperature eigenstates, while eigenstates at energy corresponding to
temperature $1/\beta$ should be well-described by
$e^{-\frac{1}{2}\beta{H}}\ket{\psi_{R}}$ White (2009); Sugiura and Shimizu
(2012, 2013); Elsayed and Fine (2013); Nakagawa  _et al._ (2018). This
expectation is mirrored by the behavior of the entanglement entropy (EE) in
eigenstates of many-body systems with finite Hilbert spaces: at the spectral
edges, EE is low (“area law”) Eisert  _et al._ (2010); Amico  _et al._ (2008),
while in the infinite-temperature (mid-spectrum) regime, the eigenstates have
EE close to the value expected for random states. As a result, for chaotic
many-body systems, the scatter plot of EE versus eigenenergy takes the shape
of an arch (rainbow), Fig. 1, by now familiar from many numerical examples
Beugeling  _et al._ (2015); Garrison and Grover (2018); Turner  _et al._
(2018); Moudgalya  _et al._ (2018); Ok  _et al._ (2019); Mukherjee  _et al._
(2020a, b); Iadecola and Schecter (2020); Mark  _et al._ (2020); Shibata  _et
al._ (2020); Chattopadhyay  _et al._ (2020); Mark and Motrunich (2020);
McClarty  _et al._ (2020).

In this work, we consider the bipartite entanglement entropy of mid-spectrum
eigenstates. For definiteness, we will focus on spin-$1/2$ chains with $L$
sites with all symmetries broken, so that the Hilbert space is $D=2^{L}$, and
consider the entanglement between two subsystems ($A$, $B$) of equal size,
$L/2$ sites each EE- . In this case, the random states have an average EE
well-approximated by the Page formula Page (1993), $S_{\rm Page}=\log
D_{A}-\frac{1}{2}$, where $D_{A}=2^{L/2}$ is the size of the reduced Hilbert
space of the $A$ subsystem 111When the subsystems are unequal, the Page value
is $\log D_{\rm A}-D^{2}_{\rm A}/(2D)$, with $D=D_{A}D_{B}=2^{L}$. If the
Hilbert space is restricted by additional symmetries (e.g., number
conservation) then the expressions for entanglement entropy are not as simple.
Although the mid-spectrum eigenstates are expected to be random, numerically
calculated mid-spectrum entanglement in finite-size many-body systems – both
in the existing literature Beugeling  _et al._ (2015); Liu  _et al._ (2018);
LeBlond  _et al._ (2019) and in this work – systematically fall below the Page
value. The deviation decreases more slowly with system size than the width of
the state-to-state fluctuations of EEs, which means that the deviation is
significant at any finite size. To the best of our knowledge, the origin of
this subtle, systematic and seemingly universal effect has not been addressed
so far. In this work, we present a study of this discrepancy, uncovering the
ways in which mid-spectrum eigenstates deviate from random states.

Figure 1: Schematic, summarizing main ideas. Lower (red) rainbow shows block
EE against eigenenergies. Upper (black) rainbow is EE between “comb”
partitions. Partitions are shown in the left and right insets, respectively.
The difference between the top of the two rainbows is attributed to residual
spatial correlation in mid-spectrum eigenstates. The difference between the
Page value (dotted horizontal line) and the mid-spectrum comb entanglement is
attributed to the “orthogonality blockade” effect — orthogonality to the
special spectral edge states. 3D plots are cartoons of distributions of
eigenstate intensities, using a 2D space to visualize the Hilbert space. The
three cases correspond to low-entanglement states typical of spectral edges
(middle), fully random or ‘ergodic’ states (right), and many-body mid-spectrum
states (left). The latter demonstrates a depletion of weight in parts of the
Hilbert space where the spectral edges have large weight.

We find that the locality of the Hamiltonian leads to spatial correlations
persisting in mid-spectrum eigenstates of any finite system; we demonstrate
this through the mutual information between sites. The mid-spectrum
eigenstates thus differ in an important manner from random states. The
presence of spatial correlations manifests itself strongly in the entanglement
between spatially connected blocks (block bipartition) – a partitioning which
is naturally sensitive to spatial correlations in eigenstates. We show that
the departure of mid-spectrum entanglement from the Page value is smaller for
comb partitions that are, of all bipartitions, the least sensitive to spatial
variations of correlations Com . Nevertheless, even the comb entanglements
depart from the Page value. We argue that the reason for the departure from
Page value of the mid-spectrum eigenstates is their orthogonality to the
eigenstates at spectral edges. Orthogonality forces the mid-spectrum
eigenstates to live in an effectively lower-dimensional Hilbert space: part of
the physical Hilbert space is blocked off. This Hilbert space blockade
phenomenon manifests itself in the eigenstate coefficient distribution as an
enhanced weight around zero Beugeling  _et al._ (2018); Luitz and Bar Lev
(2016); Bäcker  _et al._ (2019); Luitz  _et al._ (2020). The orthogonality
blockade effect exists for any Hamiltonian, local or not. However, for local
Hamiltonians, eigenstates at the spectral edges have area law entanglement and
strong spatial correlations. The blockade effect then forces mid-spectrum
states to have the observed spatial correlations. The departure from the Page
value for comb partitions is thus due to a correlated Hilbert space blockade,
such that certain types of configurations in the Hilbert space are blocked
from appearing in the mid-spectrum eigenstates. This scenario is illustrated
in Fig. 1 and elaborated in the rest of this paper.

_Model_ – We focus on the spin-$\frac{1}{2}$ chain, with couplings between
sites $i$, $j$ having the XYZ form
$h_{i,j}[\eta,\Delta]=(1-\eta)S^{x}_{i}S^{x}_{j}+(1+\eta)S^{y}_{i}S^{y}_{j}+\Delta
S^{z}_{i}S^{z}_{j}$. The nearest-neighbor version of this model is integrable
through the algebraic Bethe ansatz McCoy (2009). Since our focus is on non-
integrable models, we add next-nearest neighbor couplings $h_{j,j+2}$ and/or
magnetic fields:

$H=J_{1}\sum_{j=1}^{L-1}h_{j,j+1}[\eta_{1},\Delta_{1}]+J_{2}\sum_{j=2}^{L-2}h_{j,j+2}[\eta_{2},\Delta_{2}]\\\
+h_{z}\sum_{j=1}^{L}(1-\tfrac{1}{2}\delta_{jL})S^{z}_{j}+h_{x}\sum_{j=1}^{L}(1-\tfrac{1}{2}\delta_{j1})S^{x}_{j}.$
(1)

The $h_{x}$ term breaks the parity of total-$S^{z}$. In addition, the $J_{2}$,
$h_{z}$ and $h_{x}$ terms are each tweaked at one edge of the chain so that
reflection symmetry is broken. Unless otherwise specified, we present data for
$J_{\alpha}=1$, $\eta_{\alpha}=0.5$, $\Delta_{\alpha}=0.9$, $h_{z}=0.8$,
$h_{x}=0.2$. For parameters that we used, the level spacing statistics of the
model is consistent with that of the Gaussian orthogonal ensemble (GOE),
indicating chaotic behavior.

Figure 2: EE for block bipartition. XYZ chain; parameters listed in text.
(a-c) EE versus energy for different system sizes. Horizontal line: Page
value. (d) Statistics of mid-spectrum eigenstates ($D/16$ states nearest to
peak of rainbow), for system sizes $L=$8,10,12,14,16. The standard deviation
(black circles) has similar scaling $\sim D^{-1/2}$ to the GOE case (yellow
stars). The departure from the Page value (red squares) falls off much more
slowly, possibly even saturating.

Entanglement for block partitions – In Fig. 2 we consider “block”
bipartitioning, i.e., the $A$ ($B$) bipartition is the left (right) half of
the chain. Fig. 2(a-c) shows how the spectral edges and mid-spectrum states
scale differently ($\sim{L^{0}}$ vs $\sim{L^{1}}$), resulting in the rainbow
or arch shape. The largest EE values are close to the EE values of random
states of the same Hilbert space size, whose average is here the Page value,
$S_{\rm Page}=\frac{L}{2}\ln 2-\frac{1}{2}$, because we have chosen a
spin-$\frac{1}{2}$ system with no symmetries. The EE being close to the Page
value indicates that the mid-spectrum eigenstates are close to being “random”
or “infinite-temperature”. Accordingly, the width of the distribution of mid-
spectrum EE values is expected to decrease as $\sim{D}^{-1/2}$, like
eigenstates of GOE/GUE matrices Vivo  _et al._ (2016). Fig. 2 (d) shows that
the mid-spectrum EE widths are larger than corresponding GOE values, but are
consistent with $\sim{D}^{-1/2}$ behavior.

The feature we focus on in this paper is the systematic departure from the
Page value even in the middle of the spectrum (Fig. 2 (c) inset). Fig. 2 (d)
shows how the departure of the mean mid-spectrum EE from the Page value scales
with system size. There is some ambiguity in how to choose the “mid-spectrum”
states. Fig. 2 (d) uses the 1/16th eigenstates closest to the top of the
rainbow, but our basic qualitative observations are insensitive to the exact
procedure (see Appendix B). The departure decreases with the system size
_remarkably slowly_. In fact, the data does not rule out saturation, i.e.,
nonzero departure in the thermodynamic limit. The departure is certainly much
larger than the width, which decreases much faster, $\sim{D}^{-1/2}$. This
means that, at _any_ system size, the Page value lies outside the distribution
of EE values. In this sense, the departure is not “merely a finite size
effect.”

_Spatial Correlations and comb entanglement_ – To uncover the reason for the
departure from random-matrix behavior, we first appeal to the best-known case
of such departures, namely the spectral edges, for which the EE is known to
violate the volume law Eisert  _et al._ (2010); Amico  _et al._ (2008). The
origin of the area law EE is the spatial locality of the Hamiltonian. This
ensures that the block entanglement receives its largest contribution from the
boundary region. (It also leads to local correlations that fall off
exponentially or as a power law with distance.) We ask whether some degree of
locality, in the above sense, also exists in the mid-spectrum eigenstates. We
quantify correlations via the quantum mutual information $I(i,j)\equiv S_{\rm
i}+S_{\rm j}-S_{\rm i\cup j}$ for pairs of spins $i$ and $j$, as in De Tomasi
_et al._ (2017); De Tomasi (2019). Fig. 3(a) shows $I(i,j)$ against the
distance $|i-j|$, for mid-spectrum eigenstates, low-energy eigenstates and
random states. The distance-dependence in mid-spectrum eigenstates is much
less pronounced than in low-energy eigenstates, as expected. However, there
exists an unmistakable dominance of small-distance correlations. Thus, we
identify distance-dependence as one reason for the departure of mid-spectrum
EE from the Page value.

Figure 3: Spatial correlations and comb EE. (a) Mutual information vs.
distance. Low energy states (black circles), mid-spectrum states (red squares)
of XYZ Hamiltonians, compared with sparse matrices (green stars) and GOE
matrices (blue diamonds). Statistics gathered from 32 eigenstates of each of
40 XYZ Hamiltonians, with parameters drawn from ranges shown. (b) EE for block
and comb bipartitions, versus energy. In (a,b), data are included for sparse
random matrices with sparsity close to the corresponding XYZ Hamiltonian. (c)
Scaling of mid-spectrum comb EE (of both types, filled/open symbols), similar
to Fig. 2(d). The departure from the Page value (red squares) falls much
slower than the width (black circles).

To ‘correct’ for this effect, we consider ‘comb’ bipartitions Keating  _et
al._ (2006); Chen  _et al._ (2006); Iglói and Peschel (2010); Rossignoli  _et
al._ (2011); He  _et al._ (2017) of two types: ‘comb1’ partitions spins
according to the pattern $ABABAB...$ and ‘comb2’ as $AABBAABB...$ Com . For
such bipartitions, EE should be insensitive to large-scale distance-dependence
of correlations, as two nearby points are as likely to be in different
partitions as two faraway points. (The boundary and bulk of partitions are not
spatially separated.) Fig. 3(b) shows that the mid-spectrum comb EE is indeed
closer to the Page value than the block EE. There is little difference between
comb1 and comb2, which supports the idea that this reduction of the departure
is due to removal of the effect of distance-dependence. There is no notion of
‘area law’ in the case of comb bipartitioning, so that the EE for spectral
edges scale as $\sim{L}$ instead of $\sim{L^{0}}$. Nevertheless, the comb EE’s
at the spectral edges are significantly smaller than the mid-spectrum ones,
despite having the same scaling. Thus the comb EE’s are arranged in an
arch/rainbow shape (see Appendix C), like the block EE.

Remarkably, even for comb partitions, the departure from the Page value
remains much larger than the width of mid-spectrum EE distributions. As in the
block case, the comb EE departure decreases far slower than the width, Fig.
3(c). (The available data would even be consistent with a _saturation_ of the
departure in the $L\to\infty$, as opposed to a slow decrease.) This means,
again, that the departure is a visible effect at any finite size, also for
partitions which (unlike block partitions) do not select for locality effects.
We are thus forced to look for additional mechanisms – beyond ‘locality’ as
discussed above – for the departure from the random-state behavior.

_Sparsity is not responsible_ – One possible source of the difference between
random states and the eigenstates of local many-body Hamiltonians is that such
Hamiltonians are generally sparse matrices in common basis choices. To examine
the consequence of sparsity, in Fig. 3(a,b) we include results (mutual
information, EE) for the eigenstates of matrices with sparsity close to the
physical (XYZ) Hamiltonian, and nonzero elements drawn from a Gaussian
distribution. We find only very slight differences from eigenstates of the
usual (full) GOE ensemble. Thus sparsity is not a significant factor in the
departure from the Page value.

_Orthogonality and blockade_ – We now introduce a framework for discussing the
deviation of mid-spectrum states from randomness (full ergodicity).
Eigenstates at the spectral edges are well-known to be special — they have
area-law instead of volume-law entanglement, and this is reflected in the
local structure of correlations, both aspects discussed above. These
eigenstates may be seen as occupying a specific tiny part of the Hilbert space
which promotes the special features. Because mid-spectrum eigenstates need to
be orthogonal to these special states, they are forced to _exclude_ that part
of the Hilbert space. Thus mid-spectrum eigenstates are distributed in a large
fraction of, but not the complete, Hilbert space: part of the Hilbert space is
blocked off.

This blockade phenomenon is illustrated in the cartoons of Fig. 1. If we use
real-space configurations as the basis, then these cartoons over-simplify in
showing the low-energy states as having exactly zero coefficients for most
basis states. In reality, the low-energy eigenstates are not completely
localized in configuration space — they have low but nonzero entanglement. In
other words, their participation ratios PR_  are much smaller than the random
state value of $D/3$, but are still much larger than $1$ Beugeling  _et al._
(2015).

Figure 4: Orthogonality blockade, illustrated. (a) Eigenstate coefficient
distribution, showing an excess of small values compared to Gaussian. Excess
seems not to scale with system size. (b) Scatter-plot between eigenstate
coefficients. Each point in the scatter-plot represents one real-space
configuration. Coefficients of a mid-spectrum state against those of a low-
energy (b1) and a high-energy (b2) state. Eigenstates (shown in brackets)
labeled from $1$ to $2^{10}=1024$. (c) EE of states obtained by
orthogonalizing a random state to the $k$ lowest-energy and $k$ highest-energy
eigenstates. $k=0$ is a random state with Page EE; $k=\frac{D}{2}-1$ is
essentially a mid-spectrum eigenstate.

Aspects of the blockade phenomenon are illustrated in Fig. 4. The coefficients
of mid-spectrum eigenstates, in the basis of real-space configurations, are
not entirely Gaussian (as would be the case for GOE eigenstates) but have
excess weight at small values Beugeling  _et al._ (2018); Luitz and Bar Lev
(2016); Bäcker  _et al._ (2019); Luitz  _et al._ (2020), Fig. 4(a). Some
configurations are “over-represented” in eigenstates at the spectral edges;
hence by orthogonality they have to be under-represented in mid-spectrum
eigenstates, leading to an excess of small values of coefficients. This effect
is seen more explicitly when coefficient magnitudes of different eigenstates
are plotted against each other – Fig. 4(b) shows that low/high-energy
eigenstates have anomalously large weights in a few basis states, which have
small weights in mid-spectrum eigenstates (see Appendix D).

In Fig. 4(c) we show how orthogonality affects entanglement. Starting from a
random state, we successively Gram-Schmidt-orthogonalize this state against
pairs of eigenstates at the outermost edges of the spectrum, i.e., first with
the lowest and highest eigenstates, then with the second-lowest and second-
highest eigenstates, and so on. The EE (both block and comb) of the resulting
states, Fig. 4(c), starts with zero departure and gradually achieves the
previously observed departure after all but the mid-spectrum eigenstates have
been orthogonalized away. The curves saturate at roughly the values observed
for the block and comb EE’s of mid-spectrum eigenstates. The curve has
fluctuations depending on the initial random state used, but the fluctuations
decrease with increasing system size and the overall picture is seen to be
valid for a variety of random states tried (see Appendix E).

Interestingly, the distribution of coefficients by itself does not predict the
correct mid-spectrum EE. Using a random state with coefficients drawn from the
observed mid-spectrum distribution, Fig. 4(a), we find the resulting EE to
have a departure one order of magnitude smaller than that observed in the comb
cases. Thus, a random state with an effective Hilbert space of reduced
dimension $D_{\rm eff}<D$ (as in De Tomasi and Khaymovich (2020)) is not
sufficient to model the departure. The orthogonality causes very particular
combinations of configurations to be missing from the mid-spectrum
eigenstates; this “correlated blockade” is necessary for the observed
departure (see Appendix F).

_Rainbow shape implies departure_ – The orthogonality mechanism has the
following implication. If the EE versus eigenenergy plot is rainbow- or arch-
shaped, then the correlations in spectral-edge eigenstates which cause those
to have low entanglement will affect the mid-spectrum eigenstates by
orthogonality, causing the mid-spectrum eigenstates to depart from Gaussian
randomness. In other words, a rainbow shape is necessarily accompanied by a
departure in the mid-spectrum states.

We can thus trace back the departure for comb EE to the fact that the EE plot
is rainbow-shaped for comb bipartitioning. Unlike block partitioning, there is
now no parametric argument ($\sim{L^{0}}$ vs $\sim{L^{1}}$) for the rainbow
shape, as the spectral edge EE now scales as $\sim{L}$ (the scaling of the
boundary between partitions), the same as the mid-spectrum EE. A general
argument for the rainbow shape is that, because the low-/high-energy states
are more constrained (less like random states) compared to mid-spectrum
eigenstates, the EE at spectral edges has to be farther from the Page value
compared to mid-spectrum EE. However, this is not obviously related to the
spatial structure of correlations. In fact, in a model without spatial
locality, where all eigenstates have volume-law scaling, the EE is found to
also have a rainbow structure Liu  _et al._ (2018). According to the picture
presented above, orthogonality should then force a departure of the mid-
spectrum EE; indeed this is observed Liu  _et al._ (2018).

Figure 5:  (a) Power-law random banded matrix of size $2^{10}$, treated as a
10-spin Hamiltonian. EE versus eigenvalues, for exponent $\alpha$ in the
ergodic/delocalized (a1), weakly ergodic (a2) and localized (a3) regimes. Case
(a2) is analogous to the many-body situation. (b) Lines are block EE
calculated from states $e^{-\frac{1}{2}\beta{H}}\ket{\psi_{R}}$, for different
realizations of random $\ket{\psi_{R}}$. On average, these lie above actual
eigenstate EE’s at all temperatures/energies.

Our picture is applicable also beyond the context of many-body physics. In
power-law random banded matrices (PLRBMs) Mirlin  _et al._ (1996); Kravtsov
and Muttalib (1997); Varga and Braun (2000); Mirlin and Evers (2000); Evers
and Mirlin (2008); Bogomolny and Giraud (2011); Rushkin  _et al._ (2011);
Mendez-Bermudez  _et al._ (2012); Bogomolny and Sieber (2018); Vega-Oliveros
_et al._ (2019); Nosov  _et al._ (2019); De Tomasi (2019) and ultrametric
matrices Fyodorov  _et al._ (2009); Bogomolny and Giraud (2011); Rushkin  _et
al._ (2011); Mendez-Bermudez  _et al._ (2012); von Soosten and Warzel (2017);
Bogomolny and Sieber (2018), there is a regime of parameters where the mid-
spectrum eigenstates are “weakly ergodic” in the sense that, even though the
scaling with matrix size is as in the ergodic case, there is deviation from
GOE/GUE ensembles at any finite size Bogomolny and Sieber (2018); Nosov  _et
al._ (2019); Bäcker  _et al._ (2019). To connect to the present topic: by
interpreting the indices of such matrices as spatial configurations (as
discussed in, e.g., Khaymovich  _et al._ (2019); Łydżba  _et al._ (2020) and
Appendix G), one can evaluate entanglements. For PLRBMs in the weakly ergodic
regime, we have found a rainbow-shaped dependence of EE versus eigenenergy,
with mid-spectrum falling short of the Page value, (Fig. 5(a) and Appendix G),
just as in the many-body case. The spectral edges are likely power-law-
localized (see Appendix G), so that the Hilbert space blockade effect is more
direct than in the many-body situation.

_“Weak ergodicity”_ – The departure is a signature of deviation from GOE/GUE
behavior at all finite sizes. Signatures of this departure are also seen in
eigenstate coefficient distributions (Fig. 4(a); see also Beugeling  _et al._
(2018); Luitz and Bar Lev (2016); Bäcker  _et al._ (2019); Luitz  _et al._
(2020)). Mid-spectrum eigenstates have the same scaling behavior as GOE-GUE,
but approach the thermodynamic limit differently, as also seen in
multifractality analysis Bäcker  _et al._ (2019). This behavior could
justifiably be called “weakly ergodic”, although the phrase does not yet have
a widely accepted definition Bogomolny and Sieber (2018); Nosov  _et al._
(2019); Nosov and Khaymovich (2019); Bäcker  _et al._ (2019); De Tomasi  _et
al._ (2020); Khaymovich  _et al._ (2020). In Ref. Khaymovich  _et al._ (2020),
weak ergodicity is associated with breaking of “basis rotation invariance.”
Our finding, that mid-spectrum eigenstates have different block EE and comb
EE, is another type of non-invariance under basis rotation.

_Context and consequences –_ The EE of non-extremal eigenstates has been the
focus of much attention recently, both for chaotic systems Mejía-Monasterio
_et al._ (2005); Deutsch (2010); Santos  _et al._ (2012); Deutsch  _et al._
(2013); Beugeling  _et al._ (2015); Zhang  _et al._ (2015); Vidmar and Rigol
(2017); Dymarsky  _et al._ (2018); Garrison and Grover (2018); Nakagawa  _et
al._ (2018); Fujita  _et al._ (2018); Dymarsky  _et al._ (2018); Fremling  _et
al._ (2018); Liu  _et al._ (2018); Murthy and Srednicki (2019); Wilming  _et
al._ (2019); Huang (2019); Huang and Gu (2019); Lu and Grover (2019); Miao and
Barthel (2019); Kaneko  _et al._ (2020) as in this work, and also for
integrable systems Alba  _et al._ (2009); Alcaraz  _et al._ (2011); Mölter
_et al._ (2014); Ares  _et al._ (2014); Storms and Singh (2014); Lai and Yang
(2015); Alba (2015); Beugeling  _et al._ (2015); Nandy  _et al._ (2016);
Vidmar  _et al._ (2017, 2018); Zhang  _et al._ (2018); Liu  _et al._ (2018);
Castro-Alvaredo  _et al._ (2018); Hackl  _et al._ (2019); LeBlond  _et al._
(2019); Jafarizadeh and Rajabpour (2019); Barthel and Miao (2019); Łydżba  _et
al._ (2020). The eigenstate EE plays a role in connecting quantum properties
to the thermodynamic entropy Gemmer  _et al._ (2001); Deutsch (2010); Santos
_et al._ (2012); Deutsch  _et al._ (2013); Storms and Singh (2014); Iyoda  _et
al._ (2017); Garrison and Grover (2018); Murthy and Srednicki (2019). Current
theory suggests that the mid-spectrum states are effectively random. We have
shown that a subleading deviation is present for any finite size, and have
developed concepts (orthogonality blockade, residual spatial correlations)
pertinent to understanding this deviation. We expect that our results to be
equally valid for systems with symmetries (for which the average random-state
EE is not given by the Page formula Page (1993)), and that the presented
concepts will have further applications, e.g., effects of orthogonality to the
spectral edge eigenstates has been exploited in recent literature Nosov  _et
al._ (2019); Nosov and Khaymovich (2019); Kutlin and Khaymovich (2020); Deng
_et al._ (2020). An open question is whether the deviation _saturates_ or
vanishes in the large-size limit.

The idea that the state $e^{-\frac{1}{2}\beta{H}}\ket{\psi_{R}}$ (with
$\ket{\psi_{R}}$ a random state) is a good model for finite-temperature
eigenstates Sugiura and Shimizu (2012, 2013); Nakagawa  _et al._ (2018) has
been fruitful for numerical computations of thermodynamic properties
Steinigeweg  _et al._ (2014); Monnai and Sugita (2014); Steinigeweg  _et al._
(2015); Jin  _et al._ (2015); Steinigeweg  _et al._ (2016); Endo  _et al._
(2018); Inoue  _et al._ (2019); Richter and Steinigeweg (2019); Rousochatzakis
_et al._ (2019); Schnack  _et al._ (2020a); Nishida  _et al._ (2020); Schnack
_et al._ (2020b). Our observation, that mid-spectrum (highest-EE or infinite-
temperature) eigenstates show departures from random state properties, implies
that the state $e^{-\frac{1}{2}\beta{H}}\ket{\psi_{R}}$ is an imperfect model
for lower-entanglement (finite-temperature) eigenstates as well. In fact, one
can attempt to reproduce the entanglement rainbow by plotting the EE of the
state $e^{-\frac{1}{2}\beta{H}}\ket{\psi_{R}}$ against the corresponding
energy. We find that this curve falls systematically above the EE scatter-plot
of actual eigenstates (Fig. 5(b) and Appendix H).

Acknowledgments – We thank A. Bäcker and G. De Tomasi for useful discussions.

## References

  * Deutsch (1991) J. M. Deutsch, “Quantum statistical mechanics in a closed system,” Phys. Rev. A 43, 2046–2049 (1991).
  * Srednicki (1994) Mark Srednicki, “Chaos and quantum thermalization,” Phys. Rev. E 50, 888–901 (1994).
  * Rigol  _et al._ (2008) M Rigol, V Dunjko, and M Olshanii, “Thermalization and its mechanism for generic isolated quantum systems.” Nature 452, 854 (2008).
  * Reimann (2015) Peter Reimann, “Eigenstate thermalization: Deutsch’s approach and beyond,” New Journal of Physics 17, 055025 (2015).
  * D’Alessio  _et al._ (2016) Luca D’Alessio, Yariv Kafri, Anatoli Polkovnikov, and Marcos Rigol, “From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics,” Advances in Physics 65, 239–362 (2016).
  * Deutsch (2018) Josh Deutsch, “Eigenstate thermalization hypothesis,” Reports on Progress in Physics (2018).
  * Mori  _et al._ (2018) Takashi Mori, Tatsuhiko N Ikeda, Eriko Kaminishi, and Masahito Ueda, “Thermalization and prethermalization in isolated quantum systems: a theoretical overview,” Journal of Physics B: Atomic, Molecular and Optical Physics 51, 112001 (2018).
  * Tasaki (1998) Hal Tasaki, “From quantum dynamics to the canonical distribution: General picture and a rigorous example,” Phys. Rev. Lett. 80, 1373–1376 (1998).
  * Gemmer  _et al._ (2001) Jochen Gemmer, Alexander Otte, and Günter Mahler, “Quantum approach to a derivation of the second law of thermodynamics,” Phys. Rev. Lett. 86, 1927–1930 (2001).
  * Goldstein  _et al._ (2006) Sheldon Goldstein, Joel L. Lebowitz, Roderich Tumulka, and Nino Zanghì, “Canonical typicality,” Phys. Rev. Lett. 96, 050403 (2006).
  * Popescu  _et al._ (2006) Sandu Popescu, Anthony J Short, and Andreas Winter, “Entanglement and the foundations of statistical mechanics,” Nature Physics 2, 754–758 (2006).
  * Sugita (2007) A Sugita, “On the basis of quantum statistical mechanics,” Nonlinear. Phenom. Complex Syst. 10, 192 (2007), arXiv:0602625.
  * Reimann (2007) Peter Reimann, “Typicality for generalized microcanonical ensembles,” Phys. Rev. Lett. 99, 160404 (2007).
  * Gemmer  _et al._ (2009) J. Gemmer, M. Michel, and G. Mahler,  _Quantum Thermodynamics: Emergence of Thermodynamic Behavior Within Composite Quantum Systems_ , Lecture Notes in Physics, Vol. 784 (Springer Berlin Heidelberg, 2009).
  * Goldstein  _et al._ (2010) Sheldon Goldstein, Joel L Lebowitz, Roderich Tumulka, and Nino Zanghi, “Long-time behavior of macroscopic quantum systems,” The European Physical Journal H 35, 173–200 (2010).
  * von Neumann (2010) John von Neumann, “Proof of the ergodic theorem and the h-theorem in quantum mechanics,” The European Physical Journal H 35, 201–237 (2010).
  * Sugiura and Shimizu (2013) Sho Sugiura and Akira Shimizu, “Canonical thermal pure quantum state,” Phys. Rev. Lett. 111, 010401 (2013).
  * Tasaki (2016) Hal Tasaki, “Typicality of thermal equilibrium and thermalization in isolated macroscopic quantum systems,” Journal of Statistical Physics 163, 937–997 (2016).
  * White (2009) Steven R. White, “Minimally entangled typical quantum states at finite temperature,” Phys. Rev. Lett. 102, 190601 (2009).
  * Sugiura and Shimizu (2012) Sho Sugiura and Akira Shimizu, “Thermal pure quantum states at finite temperature,” Phys. Rev. Lett. 108, 240401 (2012).
  * Elsayed and Fine (2013) Tarek A. Elsayed and Boris V. Fine, “Regression relation for pure quantum states and its implications for efficient computing,” Phys. Rev. Lett. 110, 070404 (2013).
  * Nakagawa  _et al._ (2018) Yuya O Nakagawa, Masataka Watanabe, Hiroyuki Fujita, and Sho Sugiura, “Universality in volume-law entanglement of scrambled pure quantum states,” Nature communications 9, 1–8 (2018).
  * Eisert  _et al._ (2010) J. Eisert, M. Cramer, and M. B. Plenio, “Colloquium: Area laws for the entanglement entropy,” Rev. Mod. Phys. 82, 277–306 (2010).
  * Amico  _et al._ (2008) Luigi Amico, Rosario Fazio, Andreas Osterloh, and Vlatko Vedral, “Entanglement in many-body systems,” Rev. Mod. Phys. 80, 517–576 (2008).
  * Beugeling  _et al._ (2015) W Beugeling, A Andreanov, and Masudul Haque, “Global characteristics of all eigenstates of local many-body hamiltonians: participation ratio and entanglement entropy,” Journal of Statistical Mechanics: Theory and Experiment 2015, P02002 (2015).
  * Garrison and Grover (2018) James R. Garrison and Tarun Grover, “Does a single eigenstate encode the full hamiltonian?” Phys. Rev. X 8, 021026 (2018).
  * Turner  _et al._ (2018) C. J. Turner, A. A. Michailidis, D. A. Abanin, M. Serbyn, and Z. Papić, “Quantum scarred eigenstates in a rydberg atom chain: Entanglement, breakdown of thermalization, and stability to perturbations,” Phys. Rev. B 98, 155134 (2018).
  * Moudgalya  _et al._ (2018) Sanjay Moudgalya, Nicolas Regnault, and B. Andrei Bernevig, “Entanglement of exact excited states of affleck-kennedy-lieb-tasaki models: Exact results, many-body scars, and violation of the strong eigenstate thermalization hypothesis,” Phys. Rev. B 98, 235156 (2018).
  * Ok  _et al._ (2019) Seulgi Ok, Kenny Choo, Christopher Mudry, Claudio Castelnovo, Claudio Chamon, and Titus Neupert, “Topological many-body scar states in dimensions one, two, and three,” Phys. Rev. Research 1, 033144 (2019).
  * Mukherjee  _et al._ (2020a) Bhaskar Mukherjee, Sourav Nandy, Arnab Sen, Diptiman Sen, and K. Sengupta, “Collapse and revival of quantum many-body scars via floquet engineering,” Phys. Rev. B 101, 245107 (2020a).
  * Mukherjee  _et al._ (2020b) Bhaskar Mukherjee, Arnab Sen, Diptiman Sen, and K Sengupta, “Dynamics of the vacuum state in a periodically driven rydberg chain,” arXiv preprint arXiv:2005.07715 (2020b).
  * Iadecola and Schecter (2020) Thomas Iadecola and Michael Schecter, “Quantum many-body scar states with emergent kinetic constraints and finite-entanglement revivals,” Phys. Rev. B 101, 024306 (2020).
  * Mark  _et al._ (2020) Daniel K. Mark, Cheng-Ju Lin, and Olexei I. Motrunich, “Unified structure for exact towers of scar states in the affleck-kennedy-lieb-tasaki and other models,” Phys. Rev. B 101, 195131 (2020).
  * Shibata  _et al._ (2020) Naoyuki Shibata, Nobuyuki Yoshioka, and Hosho Katsura, “Onsager’s scars in disordered spin chains,” Phys. Rev. Lett. 124, 180604 (2020).
  * Chattopadhyay  _et al._ (2020) Sambuddha Chattopadhyay, Hannes Pichler, Mikhail D. Lukin, and Wen Wei Ho, “Quantum many-body scars from virtual entangled pairs,” Phys. Rev. B 101, 174308 (2020).
  * Mark and Motrunich (2020) Daniel K Mark and Olexei I Motrunich, “Eta-pairing states as true scars in an extended hubbard model,” arXiv preprint arXiv:2004.13800 (2020).
  * McClarty  _et al._ (2020) Paul A McClarty, Masudul Haque, Arnab Sen, and Johannes Richter, “Disorder-free localization and many-body quantum scars from magnetic frustration,” arXiv preprint arXiv:2007.01311 (2020).
  * (38) Given a state $|\Psi\rangle$, the EE between $A$ and $B$ partitions is $S_{\rm A}={\rm Tr}\rho_{A}\log\rho_{A}$, where $\rho_{A}={\rm Tr}_{\rm B}\left(|\Psi\rangle\langle\Psi|\right)$ is the reduced density matrix of partition $A$.
  * Page (1993) Don N. Page, “Average entropy of a subsystem,” Phys. Rev. Lett. 71, 1291–1294 (1993).
  * Note (1) When the subsystems are unequal, the Page value is $\mathop{log}\nolimits D_{\rm A}-D^{2}_{\rm A}/(2D)$, with $D=D_{A}D_{B}=2^{L}$. If the Hilbert space is restricted by additional symmetries (e.g., number conservation) then the expressions for entanglement entropy are not as simple.
  * Liu  _et al._ (2018) Chunxiao Liu, Xiao Chen, and Leon Balents, “Quantum entanglement of the sachdev-ye-kitaev models,” Phys. Rev. B 97, 245126 (2018).
  * LeBlond  _et al._ (2019) Tyler LeBlond, Krishnanand Mallayya, Lev Vidmar, and Marcos Rigol, “Entanglement and matrix elements of observables in interacting integrable systems,” Phys. Rev. E 100, 062134 (2019).
  * (43) In ‘comb’ bipartitioning, the sites of the chain are assigned to partitions $A$ and $B$ according to either the pattern $ABABAB...$ (sublattice or ‘comb1’ bipartitioning), or the pattern $AABBAABB...$ (‘comb2’ bipartitioning) For comb2, when the number of sites $L$ is not divisible by $4$, the last two sites are set to be in partitions $A$ and $B$. E.g., for $L=10$, the comb2 partitioning is $AABBAABBAB$.
  * Beugeling  _et al._ (2018) Wouter Beugeling, Arnd Bäcker, Roderich Moessner, and Masudul Haque, “Statistical properties of eigenstate amplitudes in complex quantum systems,” Phys. Rev. E 98, 022204 (2018).
  * Luitz and Bar Lev (2016) David J. Luitz and Yevgeny Bar Lev, “Anomalous thermalization in ergodic systems,” Phys. Rev. Lett. 117, 170404 (2016).
  * Bäcker  _et al._ (2019) Arnd Bäcker, Masudul Haque, and Ivan M Khaymovich, “Multifractal dimensions for random matrices, chaotic quantum maps, and many-body systems,” Phys. Rev. E 100, 032117 (2019).
  * Luitz  _et al._ (2020) David J. Luitz, Ivan M. Khaymovich, and Yevgeny Bar Lev, “Multifractality and its role in anomalous transport in the disordered XXZ spin-chain,” SciPost Phys. Core 2, 6 (2020).
  * McCoy (2009) B.M. McCoy,  _Advanced Statistical Mechanics_ , International series of monographs on physics (Oxford University Press, 2009).
  * Vivo  _et al._ (2016) Pierpaolo Vivo, Mauricio P. Pato, and Gleb Oshanin, “Random pure states: Quantifying bipartite entanglement beyond the linear statistics,” Phys. Rev. E 93, 052106 (2016).
  * De Tomasi  _et al._ (2017) Giuseppe De Tomasi, Soumya Bera, Jens H. Bardarson, and Frank Pollmann, “Quantum mutual information as a probe for many-body localization,” Phys. Rev. Lett. 118, 016804 (2017).
  * De Tomasi (2019) Giuseppe De Tomasi, “Algebraic many-body localization and its implications on information propagation,” Phys. Rev. B 99, 054204 (2019).
  * Keating  _et al._ (2006) J. P. Keating, F. Mezzadri, and M. Novaes, “Comb entanglement in quantum spin chains,” Phys. Rev. A 74, 012311 (2006).
  * Chen  _et al._ (2006) Yan Chen, Paolo Zanardi, ZD Wang, and FC Zhang, “Sublattice entanglement and quantum phase transitions in antiferromagnetic spin chains,” New Journal of Physics 8, 97 (2006).
  * Iglói and Peschel (2010) Ferenc Iglói and Ingo Peschel, “On reduced density matrices for disjoint subsystems,” EPL (Europhysics Letters) 89, 40001 (2010).
  * Rossignoli  _et al._ (2011) R. Rossignoli, N. Canosa, and J. M. Matera, “Even-odd entanglement in boson and spin systems,” Phys. Rev. A 83, 042328 (2011).
  * He  _et al._ (2017) Temple He, Javier M. Magán, and Stefan Vandoren, “Entanglement entropy of periodic sublattices,” Phys. Rev. B 95, 035130 (2017).
  * (57) The participation ratio (PR) is the inverse of the IPR, which is here defined as $\sum_{i}|c_{i}|^{4}$, where $c_{i}$ are the normalized coefficients of a wavefunction Evers and Mirlin (2008). The GOE expectation value of the PR is $D/3$ and the minimum (maximum) possible value is $1$ ($D$). In Beugeling  _et al._ (2015), the IPR is defined as $\frac{1}{D}\sum_{i}|c_{i}|^{4}$, so that the GOE expectation value is $1/3$ and the bounds are $1/D$ ($1$).
  * De Tomasi and Khaymovich (2020) Giuseppe De Tomasi and Ivan M. Khaymovich, “Multifractality meets entanglement: Relation for nonergodic extended states,” Phys. Rev. Lett. 124, 200602 (2020).
  * Mirlin  _et al._ (1996) Alexander D. Mirlin, Yan V. Fyodorov, Frank-Michael Dittes, Javier Quezada, and Thomas H. Seligman, “Transition from localized to extended eigenstates in the ensemble of power-law random banded matrices,” Phys. Rev. E 54, 3221–3230 (1996).
  * Kravtsov and Muttalib (1997) V. E. Kravtsov and K. A. Muttalib, “New class of random matrix ensembles with multifractal eigenvectors,” Phys. Rev. Lett. 79, 1913–1916 (1997).
  * Varga and Braun (2000) Imre Varga and Daniel Braun, “Critical statistics in a power-law random-banded matrix ensemble,” Phys. Rev. B 61, R11859–R11862 (2000).
  * Mirlin and Evers (2000) A. D. Mirlin and F. Evers, “Multifractality and critical fluctuations at the anderson transition,” Phys. Rev. B 62, 7920–7933 (2000).
  * Evers and Mirlin (2008) Ferdinand Evers and Alexander D. Mirlin, “Anderson transitions,” Rev. Mod. Phys. 80, 1355–1417 (2008).
  * Bogomolny and Giraud (2011) E. Bogomolny and O. Giraud, “Eigenfunction entropy and spectral compressibility for critical random matrix ensembles,” Phys. Rev. Lett. 106, 044101 (2011).
  * Rushkin  _et al._ (2011) I Rushkin, A Ossipov, and YV Fyodorov, “Universal and non-universal features of the multifractality exponents of critical wavefunctions,” Journal of Statistical Mechanics: Theory and Experiment 2011, L03001 (2011).
  * Mendez-Bermudez  _et al._ (2012) JA Mendez-Bermudez, A Alcazar-Lopez, and Imre Varga, “Multifractal dimensions for critical random matrix ensembles,” EPL (Europhysics Letters) 98, 37006 (2012).
  * Bogomolny and Sieber (2018) E. Bogomolny and M. Sieber, “Power-law random banded matrices and ultrametric matrices: Eigenvector distribution in the intermediate regime,” Phys. Rev. E 98, 042116 (2018).
  * Vega-Oliveros  _et al._ (2019) Didier A. Vega-Oliveros, J. A. Méndez-Bermúdez, and Francisco A. Rodrigues, “Multifractality in random networks with power-law decaying bond strengths,” Phys. Rev. E 99, 042303 (2019).
  * Nosov  _et al._ (2019) P. A. Nosov, I. M. Khaymovich, and V. E. Kravtsov, “Correlation-induced localization,” Physical Review B 99, 104203 (2019).
  * Fyodorov  _et al._ (2009) Yan V Fyodorov, Alexander Ossipov, and Alberto Rodriguez, “The anderson localization transition and eigenfunction multifractality in an ensemble of ultrametric random matrices,” Journal of Statistical Mechanics: Theory and Experiment 2009, L12001 (2009).
  * von Soosten and Warzel (2017) Per von Soosten and Simone Warzel, “The localization transition in the ultrametric ensemble,” arXiv preprint arXiv:1705.04165 (2017).
  * Khaymovich  _et al._ (2019) Ivan M. Khaymovich, Masudul Haque, and Paul A. McClarty, “Eigenstate thermalization, random matrix theory, and behemoths,” Phys. Rev. Lett. 122, 070601 (2019).
  * Łydżba  _et al._ (2020) Patrycja Łydżba, Marcos Rigol, and Lev Vidmar, “Eigenstate entanglement entropy in random quadratic hamiltonians,” arXiv preprint arXiv:2006.11302 (2020).
  * Nosov and Khaymovich (2019) P. A. Nosov and I. M. Khaymovich, “Robustness of delocalization to the inclusion of soft constraints in long-range random models,” Phys. Rev. B 99, 224208 (2019).
  * De Tomasi  _et al._ (2020) Giuseppe De Tomasi, Soumya Bera, Antonello Scardicchio, and Ivan M. Khaymovich, “Subdiffusion in the anderson model on the random regular graph,” Phys. Rev. B 101, 100201 (2020).
  * Khaymovich  _et al._ (2020) IM Khaymovich, VE Kravtsov, BL Altshuler, and LB Ioffe, “Fragile ergodic phases in logarithmically-normal rosenzweig-porter model,” arXiv preprint arXiv:2006.04827 (2020).
  * Mejía-Monasterio  _et al._ (2005) Carlos Mejía-Monasterio, Guliano Benenti, Gabriel G. Carlo, and Giulio Casati, “Entanglement across a transition to quantum chaos,” Phys. Rev. A 71, 062324 (2005).
  * Deutsch (2010) JM Deutsch, “Thermodynamic entropy of a many-body energy eigenstate,” New Journal of Physics 12, 075021 (2010).
  * Santos  _et al._ (2012) Lea F. Santos, Anatoli Polkovnikov, and Marcos Rigol, “Weak and strong typicality in quantum systems,” Phys. Rev. E 86, 010102 (2012).
  * Deutsch  _et al._ (2013) J. M. Deutsch, Haibin Li, and Auditya Sharma, “Microscopic origin of thermodynamic entropy in isolated systems,” Phys. Rev. E 87, 042135 (2013).
  * Zhang  _et al._ (2015) Liangsheng Zhang, Hyungwon Kim, and David A. Huse, “Thermalization of entanglement,” Phys. Rev. E 91, 062128 (2015).
  * Vidmar and Rigol (2017) Lev Vidmar and Marcos Rigol, “Entanglement entropy of eigenstates of quantum chaotic hamiltonians,” Phys. Rev. Lett. 119, 220603 (2017).
  * Dymarsky  _et al._ (2018) Anatoly Dymarsky, Nima Lashkari, and Hong Liu, “Subsystem eigenstate thermalization hypothesis,” Phys. Rev. E 97, 012140 (2018).
  * Fujita  _et al._ (2018) Hiroyuki Fujita, Yuya O Nakagawa, Sho Sugiura, and Masataka Watanabe, “Page curves for general interacting systems,” Journal of High Energy Physics 2018, 112 (2018).
  * Fremling  _et al._ (2018) M Fremling, Cécile Repellin, Jean-Marie Stéphan, N Moran, JK Slingerland, and Masudul Haque, “Dynamics and level statistics of interacting fermions in the lowest Landau level,” New Journal of Physics 20, 103036 (2018).
  * Murthy and Srednicki (2019) Chaitanya Murthy and Mark Srednicki, “Structure of chaotic eigenstates and their entanglement entropy,” Phys. Rev. E 100, 022131 (2019).
  * Wilming  _et al._ (2019) H. Wilming, M. Goihl, I. Roth, and J. Eisert, “Entanglement-ergodic quantum systems equilibrate exponentially well,” Phys. Rev. Lett. 123, 200604 (2019).
  * Huang (2019) Yichen Huang, “Universal eigenstate entanglement of chaotic local hamiltonians,” Nuclear Physics B 938, 594–604 (2019).
  * Huang and Gu (2019) Yichen Huang and Yingfei Gu, “Eigenstate entanglement in the sachdev-ye-kitaev model,” Phys. Rev. D 100, 041901 (2019).
  * Lu and Grover (2019) Tsung-Cheng Lu and Tarun Grover, “Renyi entropy of chaotic eigenstates,” Phys. Rev. E 99, 032111 (2019).
  * Miao and Barthel (2019) Qiang Miao and Thomas Barthel, “Eigenstate entanglement: Crossover from the ground state to volume laws,” arXiv preprint arXiv:1905.07760 (2019).
  * Kaneko  _et al._ (2020) Kazuya Kaneko, Eiki Iyoda, and Takahiro Sagawa, “Characterizing complexity of many-body quantum dynamics by higher-order eigenstate thermalization,” Phys. Rev. A 101, 042126 (2020).
  * Alba  _et al._ (2009) Vincenzo Alba, Maurizio Fagotti, and Pasquale Calabrese, “Entanglement entropy of excited states,” J. Stat. Mech. 2009, P10020 (2009).
  * Alcaraz  _et al._ (2011) Francisco Castilho Alcaraz, Miguel Ibáñez Berganza, and Germán Sierra, “Entanglement of low-energy excitations in conformal field theory,” Phys. Rev. Lett. 106, 201601 (2011).
  * Mölter  _et al._ (2014) Jan Mölter, Thomas Barthel, Ulrich Schollwöck, and Vincenzo Alba, “Bound states and entanglement in the excited states of quantum spin chains,” Journal of Statistical Mechanics: Theory and Experiment 2014, P10029 (2014).
  * Ares  _et al._ (2014) F. Ares, J. G. Esteve, F. Falceto, and E. Sánchez-Burillo, “Excited state entanglement in homogeneous fermionic chains,” J. Phys. A: Math. Theor. 47, 245301 (2014).
  * Storms and Singh (2014) Michelle Storms and Rajiv R. P. Singh, “Entanglement in ground and excited states of gapped free-fermion systems and their relationship with fermi surface and thermodynamic equilibrium properties,” Phys. Rev. E 89, 012125 (2014).
  * Lai and Yang (2015) Hsin-Hua Lai and Kun Yang, “Entanglement entropy scaling laws and eigenstate typicality in free fermion systems,” Phys. Rev. B 91, 081110 (2015).
  * Alba (2015) Vincenzo Alba, “Eigenstate thermalization hypothesis and integrability in quantum spin chains,” Phys. Rev. B 91, 155123 (2015).
  * Nandy  _et al._ (2016) Sourav Nandy, Arnab Sen, Arnab Das, and Abhishek Dhar, “Eigenstate gibbs ensemble in integrable quantum systems,” Phys. Rev. B 94, 245131 (2016).
  * Vidmar  _et al._ (2017) Lev Vidmar, Lucas Hackl, Eugenio Bianchi, and Marcos Rigol, “Entanglement entropy of eigenstates of quadratic fermionic hamiltonians,” Phys. Rev. Lett. 119, 020601 (2017).
  * Vidmar  _et al._ (2018) Lev Vidmar, Lucas Hackl, Eugenio Bianchi, and Marcos Rigol, “Volume law and quantum criticality in the entanglement entropy of excited eigenstates of the quantum ising model,” Phys. Rev. Lett. 121, 220602 (2018).
  * Zhang  _et al._ (2018) Yicheng Zhang, Lev Vidmar, and Marcos Rigol, “Information measures for a local quantum phase transition: Lattice fermions in a one-dimensional harmonic trap,” Phys. Rev. A 97, 023605 (2018).
  * Castro-Alvaredo  _et al._ (2018) Olalla A. Castro-Alvaredo, Cecilia De Fazio, Benjamin Doyon, and István M. Szécsényi, “Entanglement content of quasiparticle excitations,” Phys. Rev. Lett. 121, 170602 (2018).
  * Hackl  _et al._ (2019) Lucas Hackl, Lev Vidmar, Marcos Rigol, and Eugenio Bianchi, “Average eigenstate entanglement entropy of the xy chain in a transverse field and its universality for translationally invariant quadratic fermionic models,” Phys. Rev. B 99, 075123 (2019).
  * Jafarizadeh and Rajabpour (2019) Arash Jafarizadeh and M. A. Rajabpour, “Bipartite entanglement entropy of the excited states of free fermions and harmonic oscillators,” Phys. Rev. B 100, 165135 (2019).
  * Barthel and Miao (2019) Thomas Barthel and Qiang Miao, “Scaling functions for eigenstate entanglement crossovers in harmonic lattices,” arXiv preprint arXiv:1912.10045 (2019).
  * Iyoda  _et al._ (2017) Eiki Iyoda, Kazuya Kaneko, and Takahiro Sagawa, “Fluctuation theorem for many-body pure quantum states,” Phys. Rev. Lett. 119, 100601 (2017).
  * Kutlin and Khaymovich (2020) A. G. Kutlin and I. M. Khaymovich, “Renormalization to localization without a small parameter,” SciPost Phys. 8, 49 (2020).
  * Deng  _et al._ (2020) X. Deng, A. L. Burin, and I. M. Khaymovich, “Anisotropy-mediated reentrant localization,”  (2020), arXiv:2002.00013 .
  * Steinigeweg  _et al._ (2014) Robin Steinigeweg, Jochen Gemmer, and Wolfram Brenig, “Spin-current autocorrelations from single pure-state propagation,” Phys. Rev. Lett. 112, 120601 (2014).
  * Monnai and Sugita (2014) Takaaki Monnai and Ayumu Sugita, “Typical pure states and nonequilibrium processes in quantum many-body systems,” Journal of the Physical Society of Japan 83, 094001 (2014).
  * Steinigeweg  _et al._ (2015) Robin Steinigeweg, Jochen Gemmer, and Wolfram Brenig, “Spin and energy currents in integrable and nonintegrable spin-$\frac{1}{2}$ chains: A typicality approach to real-time autocorrelations,” Phys. Rev. B 91, 104404 (2015).
  * Jin  _et al._ (2015) F. Jin, R. Steinigeweg, F. Heidrich-Meisner, K. Michielsen, and H. De Raedt, “Finite-temperature charge transport in the one-dimensional hubbard model,” Phys. Rev. B 92, 205103 (2015).
  * Steinigeweg  _et al._ (2016) Robin Steinigeweg, Jacek Herbrych, Frank Pollmann, and Wolfram Brenig, “Typicality approach to the optical conductivity in thermal and many-body localized phases,” Phys. Rev. B 94, 180401 (2016).
  * Endo  _et al._ (2018) Hiroyuki Endo, Chisa Hotta, and Akira Shimizu, “From linear to nonlinear responses of thermal pure quantum states,” Phys. Rev. Lett. 121, 220601 (2018).
  * Inoue  _et al._ (2019) Kouki Inoue, Yuto Maeda, Hiroki Nakano, and Yoshiyuki Fukumoto, “Canonical-ensemble calculations of the magnetic susceptibility for a spin-1/2 spherical kagome cluster with dzyaloshinskii–moriya interactions by using microcanonical thermal pure quantum states,” IEEE Transactions on Magnetics 55, 1–3 (2019).
  * Richter and Steinigeweg (2019) Jonas Richter and Robin Steinigeweg, “Combining dynamical quantum typicality and numerical linked cluster expansions,” Phys. Rev. B 99, 094419 (2019).
  * Rousochatzakis  _et al._ (2019) I. Rousochatzakis, S. Kourtis, J. Knolle, R. Moessner, and N. B. Perkins, “Quantum spin liquid at finite temperature: Proximate dynamics and persistent typicality,” Phys. Rev. B 100, 045117 (2019).
  * Schnack  _et al._ (2020a) Jürgen Schnack, Johannes Richter, and Robin Steinigeweg, “Accuracy of the finite-temperature lanczos method compared to simple typicality-based estimates,” Phys. Rev. Research 2, 013186 (2020a).
  * Nishida  _et al._ (2020) Hisao Nishida, Ryo Fujiuchi, Koudai Sugimoto, and Yukinori Ohta, “Typicality-based variational cluster approach to thermodynamic properties of the hubbard model,” Journal of the Physical Society of Japan 89, 023702 (2020).
  * Schnack  _et al._ (2020b) Jürgen Schnack, Johannes Richter, Tjark Heitmann, Jonas Richter, and Robin Steinigeweg, “Finite-size scaling of typicality-based estimates,” Zeitschrift für Naturforschung A 75, 465–473 (2020b).

##  Appendix A Overview

In these Supplemental Materials, we present supporting information and data:

  * •

In Section B, we explain and discuss methods to extract the statistics of mid-
spectrum entanglement entropies (EE).

  * •

In Section C, we show and discuss the rainbow shape of the EE for comb-
partitioning in more detail than was possible in the main text.

  * •

In Section D we show and discuss additional data on the correlations between
mid-spectrum and extremal eigenstates.

  * •

In Section E we present additional data on orthogonalizing random states
successively to the non-central states.

  * •

In Section F, we expand on the idea that the orthogonality blockade is not
just a reduction of the effective Hilbert space — rather, a _correlated_
blockade is required to cause the observed magnitude of the departure. To
illustrate this point, we show through a toy calculation the difference
between a naive blockade (random reduction of the Hilbert space) and a
correlated blockade.

  * •

In Section G, we discuss the analogy to the power-law random banded matrix
model, especially the entanglement rainbow and corresponding departure from
Page value displayed by this random matrix ensemble.

  * •

In Section H we comment on the typicality argument and the corresponding
finite-temperature entanglement curve.

Data shown in the Supplements for the XYZ chain all use the parameters
announced in the main text, sometimes referred to as the “usual parameters.”

##  Appendix B Statistics of mid-spectrum entanglement

In this section we consider the method used to extract the departure of the
entanglement entropy (EE) from the Page value in the middle of the spectrum
for a many-body non-integrable system.

The EE data versus energies of eigenstates, e.g., shown in Fig. 2(a)-(c) of
the main text, is arranged in a rainbow shape with some fluctuations around
this overall shape. We can model this as $S(E)=\overline{S}(E)+\delta S(E)$,
where $\overline{S}(E)$ is a smooth function of energy $E$, and $\delta S(E)$
represents fluctuations on top of the smooth curve.

We are interested in the highest-temperature states which are located around
the maximum of this smooth curve and form at least a finite fraction of all
states.

Figure 6:  Smoothed entanglement entropy and the location of its maximum for
different system sizes. Symbols show the raw EE data. Black lines are obtained
through smoothening by the $3$rd-order polynomial Savitzky-Golay filter with
the sliding window size $51$. The positions of maxima of the above smoothed
curves are given by the positions of circles, the energy interval containing
the chosen fraction $f=1/8$ of states is given by the vertical lines for each
system size. Inset: the position of the maxima of the smoothed curves for the
sliding window sizes $41$ (dash-dotted blue), $51$ (solid green), and $61$
(dashed red).

In order to extract the EE and the fluctuations of these states we follow the
following steps:

  1. 1.

First, we determine the smooth curve $\overline{S}(E)$. For this we average
our “noisy” data with the sliding-window moving average filter and select the
parameter range where the data is insensitive to the window size.

  2. 2.

Once we have the smooth curve $\overline{S}(E)$, we determine the location of
its maximum, i.e., the energy value $E_{\max}=E_{i_{\max}}$ where the peak of
the arch-shaped curve is located.

  3. 3.

Next, we select a finite fraction $f$ of all $D$ states around the maximum,
$E_{i}$, $|i-i_{\max}|\leq fD/2$ and use only these states for the statistics
of the EE of the mid-spectrum eigenstates. The fraction $f$ is selected in
such a way that the results obtained for the EE and its fluctuations are
insensitive to $f$.

  4. 4.

Finally, using the above range of states, $E_{i}$, $|i-i_{\max}|\leq fD$, we
can calculate the statistical properties of mid-spectrum EE, e.g., the mean EE
$S_{av}=\langle{S}\rangle_{f}$, the typical EE $S_{typ}=e^{\langle{\ln
S}\rangle_{f}}$, or the standard deviation $\Delta
S=\sqrt{\langle{S^{2}}\rangle_{f}-\langle{S}\rangle_{f}^{2}}$.

Figure 7:  Departure of EE from Page value (top) versus the system size $L$
and (bottom) versus Hilbert space dimension $D$. Solid (dashed) lines show the
difference between the Page value and the mean (typical) EE averaged over the
energy intervals containing $f=1/16$ (red), $1/8$ (light blue), and $1/4$
(dark blue) fraction of states around the peak of the rainbow. Dash-dotted
lines give power-law fits for $f=1/8$ with the exponents given in the legend.
Dotted lines in the bottom panel show the standard deviation of EE in the same
energy intervals.

Results of the above procedure are displayed in Figs. 6 and 7. One can see
that the smoothed curve in Fig. 6 provides reasonable locations of the
maximum, which has a very weak dependence on the sliding window size of the
filter (inset). We choose the window size of the filter to be $51$.

Figure 7 demonstrates the deviation of the mean and typical EE from the Page
value as well as the standard deviation for several fractions $f$. One can see
that for all considered fractions $f$ the mean and typical results coincide
and barely depend on $f$. Top (bottom) panel of Fig. 7 shows the above EE
measures versus system size (Hilbert space dimension) and the corresponding
power-law fitting.

Figure 8:  EE for all eigenstates, for 3 types of partitions. (a,b,c) Full EE
rainbows shown for block, comb1 and comb2 partitions, for three different
system sizes. (d,e) Scaling of the EE for the lowest-energy and highest-energy
eigenstates with system size $L$.

##  Appendix C Entanglement rainbows for comb partitions

In the main text, we summarized results for the mid-spectrum EE for comb
partitions, highlighting that they have a smaller (but nevertheless nonzero)
departure than block-partitioning EE. Here, in Figure 8, we show the full EE
rainbows in some detail.

For the comb EE’s, there is no parametric reason for the EE scatter-plot to
have a rainbow shape, as the spectral edges should have the same EE scaling
(volume law, $\sim L$) as the mid-spectrum EE scaling. This is in contrast to
block EE, for which a difference in scaling behavior gives an obvious
explanation for the shape.

In the bottom row of Figure 8, we show the scaling for the two extremal
eigenstates. The comb EE’s are larger than the block EE in each case. The
difference in parametric dependence on $L$ (i.e., $\sim L^{0}$ behavior for
block EE vs $\sim L^{1}$ behavior for comb EE) is clear for the highest-energy
eigenstate, Figure 8(e). The parametric difference is less obvious for the
ground state data, Figure 8(d); presumably the distinction becomes clearer at
larger sizes.

Figure 9:  Scatter plots of coefficient magnitudes in pairs of eigenstates.
$L=10$, XYZ chain, usual parameters. Each panel plots the magnitudes of the
coefficients ($|c|$) of one eigenstate against those of another. The
coefficients are in the basis of real-space configurations. In each case, the
two eigenstates are marked as $(\alpha,\beta)$; the horizontal and vertical
axes are the amplitude magnitudes of eigenstate $\ket{\phi_{\alpha}}$ and
eigenstate $\ket{\phi_{\beta}}$ respectively. Here the eigenstates are labeled
in order of increasing energy, from $\alpha=1$ to $\alpha=D=2^{10}$.

##  Appendix D Correlations between coefficients of different eigenstates

In the main text, we showed that the orthogonality blockade can be visualized
by comparing the amplitudes of eigenstates in real-space configurations — some
classes of configurations are favored in the extremal eigenstates and
therefore have low weight in the mid-spectrum eigenstates. Figure 4(b) of the
main text shows one example of a low-mid correlation and one example of a mid-
mid correlation, for the $L=10$ system.

Since the effect is inherently statistical, here in Figure 9 we display an
array of panels, each showing a scatter-plot for a different pair of
eigenstates. There are five examples each of low-mid, low-low, high-mid and
mid-mid pairs. The ‘low’ eigenstates are taken from the 10 lowest-energy ones,
$\alpha\in[1,10]$. Similarly, the ‘mid’ and ‘high’ correspond to indices
$\alpha\in[508,517]$ and $\alpha\in[1015,1024]$ respectively. Here $\alpha$
labels the eigenstates in order of increasing energy, from $1$ to
$D=2^{L}=1024$.

The orthogonality blockade effect is seen in the panels of the first and third
rows. There are some configurations in which extremal eigenstates have
atypically large weight, but mid-spectrum eigenstates have atypically small
weight. This is visually seen as a few points in the scatter-plot near the
horizontal axis, far to the right of the panel. The effect is seen in every
panel in the low-mid and high-mid scatter plots (first and third rows).

Comparing the mid-mid scatter-plots (4th-row) and the low-low scatter-plots
(2nd row), we see a visual difference. Because the extremal eigenstates have
large weight in a few configurations, there are a lot of points spread out at
larger values (well away from the origin), in the second row. This reflects
the fact that extremal eigenstates deviate more markedly from a Gaussian
distribution of coefficients (and from ergodicity) than mid-spectrum
eigenstates. As described in the main text (Figure 4(a) of the main text),
mid-spectrum eigenstates also have a small departure from Gaussianity
(stronger weight near zero values) but this is not visible in the scatter-
plots.

Figure 10:  Correlation (Pearson coefficient) between eigenstate amplitudes.
$L=10$, XYZ chain, usual parameters. The black symbols are the correlation
coefficients between the ground state and every other eigenstate. The blue
symbols are correlation coefficients between the $\alpha=512$ (mid-spectrum)
eigstate and every other eigenstate. Red symbols: same, with the highest-
energy eigenstate.

There are various questions one can ask about the observations we have made
above. For example, exactly which configurations are implicated in the
orthogonality effect, i.e., what are the configurations showing anomalous
behavior in the first and third row? This presumably depends on the details of
the particular system. (In contrast, in the present study we have focused on
generic statistical features.) Also, since each extremal eigenstate
contributes to the orthogonality blockade, one could ask whether the same
configurations are important in the blockade due to different extremal
eigenstates. This can be partially addressed by examining the correlation
between the eigenstate amplitudes seen through the Pearson correlation
coefficient, Figure 10. It appears that all the low-energy eigenstates are
strongly positively correlated to the ground state (and hence to each other),
and a similar picture holds for the highest-energy eigenstates. This suggests
that there is a significant intersection between configurations which play a
dominant role in the blockade phenomenon due to lowest-energy eigenstates, and
a similar effect for the highest-energy eigenstates. A thorough quantification
of this type of inter-eigenstate correlations remains an interesting topic for
future research.

##  Appendix E Orthogonalization to non-central eigenstates

A demonstration of the orthogonalization blockade and its effect on the mid-
spectrum EE is provided in the main text via a random state successively
orthogonalized to eigenstates, starting from the extremal eigenstates. As more
and more extremal eigenstates are orthogonalized away, the nature of the
resulting state increasingly resembles the mid-spectrum eigenstates.

Figure 11:  Orthogonalizing successively to eigenstates. In each panel, the
block, comb1 and comb2 entanglement entropies of orthogonalized states are
plotted against $k$, the number of eigenstate pairs that have been
orthogonalized away. (a) Three different realizations, i.e., three different
choices of the random state $\ket{\psi_{R}}$, for $L=12$. (b) Same, for
$L=14$. (c,d) EE values of orthogonalized states for many choices of
$\ket{\psi_{R}}$, averaged for each $k$. Dashed straight lines are guides to
the eye.

Let $\ket{\psi_{R}}$ denote the initial state, with amplitudes drawn from a
gaussian random distribution. The normalized eigenstates are denoted by
$\ket{\phi_{\alpha}}$, with $\alpha$ running from $1$ to $D$, ordered
according to increasing eigenenergy. The process of successive Gram-Schmidt
orthogonalization produces the sequence $\ket{\psi_{k}}$ as follows:

$k=0:\qquad\qquad\ket{\psi_{k=0}}=\frac{\ket{\psi_{R}}}{\langle\psi_{R}\ket{\psi_{R}}}$
(2) $k=1:\\\
\ket{\bar{\psi}_{k=1}}=\ket{\psi_{k=0}}-\langle\phi_{1}\ket{\psi_{k=0}}\ket{\phi_{1}}-\langle\phi_{D}\ket{\psi_{k=0}}\ket{\phi_{D}}\\\
\ket{\psi_{k=1}}=\frac{\ket{\bar{\psi}_{k=1}}}{\langle\bar{\psi}_{k=1}\ket{\bar{\psi}_{k=1}}}\
$ (3) $k=2:\\\
\ket{\bar{\psi}_{k=2}}=\ket{\psi_{k=1}}-\langle\phi_{2}\ket{\psi_{k=1}}\ket{\phi_{2}}-\langle\phi_{D-1}\ket{\psi_{k=1}}\ket{\phi_{D-1}}\\\
\ket{\psi_{k=2}}=\frac{\ket{\bar{\psi}_{k=2}}}{\langle\bar{\psi}_{k=2}\ket{\bar{\psi}_{k=2}}}$
(4)

and so on, until $k=(D/2)-1$. The entanglement entropies of the state
$\ket{\psi_{k}}$ are plotted against $k$, in the main text and in more detail
in Figure 11.

The last state $\ket{\psi_{k=(D/2)-1}}$ is orthogonal to all but the two
central eigenstates, and hence is a linear combination of the two central
eigenstates. The central eigenstates are not far from the infinite-temperature
(highest-EE) eigenstates — hence we expect $\ket{\psi_{k=(D/2)-1}}$ to have a
departure from the Page value typical of mid-spectrum eigenstates. Indeed the
EE versus $k$ plots decrease (on average) from the Page value to this
‘departure’ value.

In Figure 11 we show the effect of the choice of initial random state
$\ket{\psi_{R}}$. The top two rows show that, for accessible sizes, there is
considerable fluctuation depending on the initial state chosen, and that these
fluctuations are larger for smaller sizes. However, the general trend is clear
in each case — the EE values decrease from Page value at $k=0$ (randomly
chosen state $\ket{\psi_{R}}$) to the mid-spectrum value as $k\to D/2$. Here
$k$ is the number of pairs orthogonalized to.

In the bottom row, we have averaged over many realizations, i.e., many choices
of the $k=0$ state. For each value of $k$, the EE values are averaged. The
resulting dependence on $k$ appears to be close to linear for smaller sizes
(not shown). For $L=12$ and $L=14$, we notice on average a larger slope at
smaller $k$ and a smaller slope at larger $k$. (The dashed lines are guides to
the eye to assist in observing this feature.) This is consistent with the
expectation that the eigenstates nearer to the spectral edges should
contribute more to reducing the entanglement compared to eigenstates farther
from the edges. This is also consistent with the Pearson coefficient results,
Fig. 10, where the states more correlated to the ground and anti-ground states
contribute more to reducing the entanglement compared to the ones more
correlated to the mid-spectrum eigenstate.

Another characteristic of smaller sizes visible in panel (c) and barely
visible in panel (d) is that the average EE of random states deviates slightly
from the Page value — the Page value is exact only in the large-size limit.
However, this tiny deviation is much smaller than the departure under
discussion in this work and consistent with the standard deviation of GOE
ensemble.

##  Appendix F The ‘correlated’ blockade — a toy model

Figure 12: Effects of reducing Hilbert space dimension. Blue squares show the
EE obtained by setting a randomly chosen number of basis states
(configurations) to have zero amplitude, case (i). Orange circles show the EE
due to the highly correlated manner of blockading the Hilbert space, case
(ii).

We have observed in the main text that the orthogonality effect causes the
mid-spectrum eigenstates to have a slightly non-Gaussian coefficient
distribution — in the basis of real-space configurations, the distribution of
eigenstate coefficients has an excess around zero in comparison to the
Gaussian. We have also reported that this reduction of effective Hilbert space
dimension does not by itself explain the EE departure — a state with
amplitudes randomly chosen from the observed mid-spectrum distribution does
not have the observed large EE departure from Page value. The orthogonality
blockade is ‘correlated’ in some sense.

Here, we demonstrate the difference between correlated and uncorrelated
Hilbert space blockades, through a toy calculation.

We generate $10^{5}$ random states with $2^{12}=4096$ real gaussian amplitudes
of which $n_{0}$ are set equal to zero. The averaged von Neumann entanglement
entropy is computed from the resulting states. We point out that the choice of
which $n_{0}$ configurations are excluded is important in determining the
effect on the entanglement.

We consider:

  * (i)

choosing randomly the $n_{0}$ configurations whose coefficients are set to
zero (Figure 12, squares) and

  * (ii)

choosing the first $n_{0}$ coefficients to be zero (Figure 12, circles).

In the first case, when the zeros are chosen randomly, the entanglement falls
slowly as $n_{0}$ is increased, so that for $n_{0}=2048$, the drop in
entanglement is $\Delta S_{A}\approx 0.025$.

In the second case, we have referred implicitly to an ordering of the
configurations. We arranged configurations according to increasing values of
their binary interpretation, thus in the order

$\begin{split}000000000001\\\ 000000000010\\\ 000000000011\\\ 000000000100\\\
000000000101\\\ \vdots\end{split}$ (5)

When the first $2^{11}$ amplitudes are zero, the first qubit (spin)
effectively factors out of the wavefunction and the entanglement is reduced
from Page value (the case with $n_{0}=0$) by $\log 2\approx 0.69$. This is a
highly non-random way of blocking off half the Hilbert space.

The first case may be regarded as a cartoon for an “uncorrelated blockade”
while the second case is a highly correlated blockade.

Of course, this model is too simple to directly compare to our mid-spectrum
states, for which the amplitude distribution does not have a delta function,
i.e., the coefficients of suppressed configurations are not set exactly to
zero. We could regard the physical situation for mid-spectrum states to be
somewhat intermediate between the two extremes discussed above. In the main
text in Figure 2(d), the departure is about $\Delta S_{A}\approx 0.12$.

##  Appendix G Power-law random banded matrices

As presented in the main text, our proposed mechanism of Hilbert space
blockade — orthogonality to special states at the spectral edges influencing
mid-spectrum eigenstates — is more generally applicable, even beyond many-body
Hamiltonians. We have found similar structures appearing in power-law random
banded matrices (PLRBMs), as reported in the main text. We describe some of
these here in more detail.

Figure 13:  Entanglement entropy (EE) for the power-law random banded matrix
model, interpreted as a spin chain as described in text. The upper row of five
panels shows the EE for each eigenstate plotted against the ‘energy’ of the
eigenstate, each panel corresponding to a different value of the parameter
$\alpha$. The lower row contains the same information but the EE is plotted
instead against eigenstate index. The EE lies on top of the Page value
(horizontal line in all panels) in the ergodic phase, $\alpha<0.5$, and
deviates from it in the weakly ergodic, $0.5\leq\alpha\leq 1$, and localized,
$\alpha>1$, phases. For $\alpha=0.9$ the comparison with the EE of the non-
integrable many-body model (green symbols) is given.

PLRBMs have elements which are Gaussian random variables (like GOE matrices),
but the widths of the random variates fall off with diagonal distance from the
main diagonal. The elements have the form

$H_{mn}=G_{mn}\;A(|m-n|)\ .$

The amplitudes $G_{mn}$ are independent Gaussian random variables with zero
mean. Viewed as a matrix itself, $G$ is drawn from Gaussian orthogonal
ensemble (GOE), i.e., the diagonal elements $G_{mm}$ have variance twice that
of the off-diagonal elements $G_{m\neq n}$. The function $A(r)$ is a
decreasing function, and decreases as a power law for large $|r|$.

There are several variants in use in the literature for the function $A(r)$.
We use

$A(r)=\frac{1}{1+(r/b)^{\alpha}}$ (6)

which decays as a power law $A(r)\propto(b/r)^{\alpha}$ with the distance $r$
from the main diagonal when $r\gtrsim b$ and saturates to $A(r)\simeq 1$ at
$r\lesssim b$. For the purposes of this work we restrict to $b\approx 1$.

The usual interpretation of PLRBM is as a single-particle quantum system on a
1D lattice with long-range hoppings, with hopping strengths decreasing as a
power law. This model undergoes an Anderson localization transition at
$\alpha=1$ as a function of the parameter $\alpha$; at larger $\alpha$ the
eigenstates are power-law localized. For $\alpha<1/2$, the eigenstates are
ergodic in the sense of having Porter-Thomas distribution of weights (Gaussian
distributions of amplitudes). The interesting regime for our purposes is
$\alpha\in(0.5,1)$, which is thought to be neither fully ergodic nor fully
localized; we loosely use the term “weakly ergodic” for this parameter regime.

We are interested in interpreting the PLRBM in the many-body sense. Thus, we
will consider matrices of size $2^{L}$ only, and interpret each row/column not
as a single-particle site, but rather as a many-body configuration, ordered as
described in the previous section, Eq. (5). (The exact choice of ordering
presumably affects the results and these effects may be explored in future
work.) With this interpretation, we can regard each eigenstate as a many-body
(spin chain) state, and hence calculate the entanglement entropy for each
eigenstate.

Some results are shown in Figure 13. The fully ergodic phase (left panels,
$\alpha<0.5$) is characterized by a flat EE profile fluctuating around the
Page value and by the absence of special eigenstates at spectral edges. This
is in agreement with the eigenstates being essentially ergodic and, in the
large size limit, indistinguishable from GOE eigenstates.

The localized phase (right panels, $\alpha>1$) has EE values falling far below
the Page value. Presumably, this non-ergodic behavior (and large discrepancy
from Page value) becomes even more pronounced at large sizes.

In the intermediate phase, $\alpha\in(0.5,1)$, we obtain rainbows as in
chaotic many-body systems. To highlight the analogy, in the lower central
panel ($\alpha=0.9$) we plot the EE for the XYZ model on top of the PLRBM
results. A preliminary numerical survey has suggested to us that, in the
single-particle language, the extremal eigenstates are power-law-localized
while the mid-spectrum eigenstates are not localized — this is a plausible
explanation for the rainbow shape. As we have argued in the main text, the
rainbow shape necessarily induces a departure from Page value in mid-spectrum
eigenstates, and this departure is clear in the plots shown.

Whether power-law-localization in the single-particle picture corresponds in
the many-body interpretation to area-law entanglement or to volume-law
entanglement with smaller prefactor, needs further investigation. In other
words, it is unclear at this stage whether the EE rainbows for PLRBM has a
closer analogy to block EE or to comb EE of the many-body system.

The panels for $\alpha=0.5$ (at the border between fully and weakly ergodic
phases) shows the beginnings of a rainbow as some of the extremal-state EE
values drops from the Page value. The large-size behavior at this point of
parameter space is an interesting question for future studies.

Figure 14: Comparison of entanglement rainbow of XYZ eigenstates to that
obtained from ‘typicality’. The state $e^{-\beta H/2}\ket{\psi_{R}}$ is
supposedly a good approximation to system properties at temperature $1/\beta$.
This idea is tested by plotting the EE of this state against $E(\beta)={\rm
tr}\left[He^{-\beta H}\right]/{\rm tr}\left[e^{-\beta H}\right]$, and
comparing against the entanglement rainbow of actual eigenstates. In the main
text, we showed this for a number of different $\ket{\psi_{R}}$. Here, the
EE’s are averaged over 200 realizations of $\ket{\psi_{R}}$, for $L=12$. Left
panel shows the block EE (as in main text) and right panel shows the comb1 EE.
Insets zoom into some part of the positive-temperature region in each case.

##  Appendix H Finite-temperature entanglement entropy from random infinite-
temperature states

In the main text we have shown a consequence of the departure that manifests
itself at all temperatures, not just at mid-spectrum infinite temperatures.
Based on ideas of (canonical) typicality, the state

$\ket{\psi_{\beta}}=e^{-\beta H/2}\ket{\psi_{R}}$ (7)

is commonly used as a wavefunction description for the properties of the
finite-temperature ensemble at temperature $1/\beta$. Using the equivalence of
ensembles, this means that the state $\ket{\psi_{\beta}}$ is expected to be a
good representative of the eigenstates in the microcanonical window around
energy

$E(\beta)=\frac{{\rm tr}\left[He^{-\beta H}\right]}{{\rm tr}\left[e^{-\beta
H}\right]}~{}=~{}\frac{\sum_{\alpha}E_{\alpha}e^{-\beta
E_{\alpha}}}{\sum_{\alpha}e^{-\beta E_{\alpha}}}\ .$ (8)

The sums in the last expression runs over all eigenvalues.

Of course, this idea presupposes that $\ket{\psi_{R}}$ is a faithful
representative of mid-spectrum (infinite-temperature) eigenstates. In this
work, we have shown that there is a subleading departure from this
correspondence. In Figure 5 of the main text, we demonstrated that this
departure manifests itself at all temperatures.

We do this by plotting the entanglement of $\ket{\psi_{\beta}}$ against
$E(\beta)$, and comparing the resulting rainbow against the EE rainbow of the
actual eigenstates. The typicality curve is found to be consistently above the
eigenstate rainbow. This is shown again in Figure 14, this time averaged over
many different realizations of the starting random state $\ket{\psi_{R}}$. The
actual eigenstates have lower entanglement than the typicality prediction.
This is more prominent for negative temperatures, which is the upper half of
the spectrum. Of course, thermodynamic calculations are usually done for
positive temperature, so in the insets we have zoomed into part of the
positive-temperature region. The departure at positive finite temperatures is
not very obvious in the comb entanglement; however, it is very clear in the
inset for block entanglement.