aainstitutetext: Institute for Information Transmission Problems RAS, 127051
Moscow, Russia bbinstitutetext: Moscow Institute for Physics and
Technology,Dolgoprudny 141700, Russiaccinstitutetext: Department of Applied
Mathematics, National Research University Higher School of Economics,
101000, Moscow, Russia

# Interacting Thermofield Doubles and Critical Behavior in Random Regular
Graphs

A. Gorsky,111Corresponding author. c O. Valba<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We discuss numerically the non-perturbative effects in exponential random
graphs which are analogue of eigenvalue instantons in matrix models. The phase
structure of exponential random graphs with chemical potential for 4-cycles
$\mu_{4}$ and degree preserving constraint is clarified. The first order phase
transition at critical value of chemical potential for 4-cycles
$\mu_{4}^{RRG}$ into bipartite phase with a formation of fixed number of
bipartite clusters is found for ensemble of random regular graphs (RRG). We
consider the similar phase transition in combinatorial quantum gravity based
of the Ollivier graph curvature for RRG supplemented with hard-core constraint
and show that a order of a phase transition at $\mu_{4}^{cRRG}$ and the
structure of emerging phase depend on a vertex degree $d$ in RRG. For $d=3$
the bipartite closed ribbon emerges at $\mu_{4}>\mu_{4}^{cRRG}$ while for
$d>3$ the ensemble of isolated or weakly interacting hypercubes supplemented
with the bipartite closed ribbon gets emerged at the first order phase
transition with a clear-cut hysteresis. If the additional connectedness
condition is imposed the phase at $\mu_{4}>\mu_{4}^{cRRG}$ gets identified as
the closed chain of weakly coupled hypercubes. Since the ground state of
isolated hypercube is the thermofield double (TFD) we suggest that the dual
holographic picture involves multiboundary wormholes. Treating RRG as a model
of a Hilbert space for a interacting many-body system we discuss the patterns
of the Hilbert space fragmentation at the phase transition. We also briefly
comment on a possible relation of the found phase transition to the problem of
holographic interpretation of a partial deconfinement transition in the gauge
theories.

##  1 Introduction

Matrix models play a prominent role in a description of chaotic systems with
large number degrees of freedom. They govern,for instance, a spectrum of a
chaotic system, the space-time dynamics of some microscopic degrees of
freedom, discretization of the Riemann surfaces or their moduli spaces moore .
The different ensembles of random matrices correspond to systems with
particular symmetry patterns. Spectral density, spectral correlators and level
spacing distribution are among the most popular characteristics within the
matrix model approach.

Large N matrix models enjoy the non-perturbative phenomena - eigenvalue
instantons which are suppressed as $\exp(-N)$ shenker ; david . The partition
function for some matrix ensemble reads as

$Z(t_{2},t_{3}\dots t_{n})=\int dM\exp(-\sum_{k}^{n}t_{k}TrM^{k})$ (1)

where the effective potential for the eigenvalues in the integration measure
generically has several extrema. In the eigenvalue matrix model the ground
state is determined by the distribution of eigenvalues of $M$ among extrema of
effective potential which depends on the variables $t_{k}$. If the term
$t_{2}$ dominates the eigenvalues typically are collected at one extremum and
the eigenvalue instanton corresponds to the traveling of the single eigenvalue
from this extremum to another one. The initial symmetry like $SU(n)$ is broken
down to some subgroup if multiple eigenvalue instantons are taken into
account. Such symmetry breaking effects are important at the phase transitions
at some critical values of $t_{k}$ in the matrix models with different matrix
measures. From the physical viewpoint eigenvalue instantons correspond to
baby-universe in 2d gravity, domain walls in SYM theory or ZZ branes in
Liouville theory. The review of the different manifestations of eigenvalue
instantons can be found in marino ; ambjorn .

Exponential random graphs are statistical models with the partition function

$Z(\mu_{2},\mu_{3}\dots\mu_{n})=\sum_{graphs}\exp(-\sum_{k}^{n}\mu_{k}TrA^{k})$
(2)

where $A$ is a graph adjacency matrix and the summation runs over some
ensemble, say, over Erdős-Rényi(ER) ensemble which involves the potential
$TrA^{2}$ and is the network analogue of Gaussian matrix model. The sum over
ensembles of graphs substitutes the integration over matrix ensembles in
matrix models. The terms $\mu_{k}TrA^{k}$ in the measure are similar to
$t_{k}TrM^{k}$ terms in the measure of matrix models and $\mu_{k}$ provide
chemical potentials for number of higher k-cycles in the particular
realization. One can consider the dependence of partition function on the
chemical potentials and look for the phase transitions caused by condensation
of some motifs which are marked by the order parameters $<TrA^{k}>\neq 0$
strauss ; burda ; newman1 ; newman2 ; anabele ; hovan ; diaconis ; radin1 ;
radin2 ; metz ; coolen1 ; coolen2 . The condensation of links, nodes,
triangles has been considered and these phase transitions are analogues of the
familiar criticalities in the conventional matrix models. It was shown that
the phase transitions are sensitive to the finite-size effects anabele ; gorva
; coolen2  and the details of the emerging phases are $N$ dependent in several
cases.

The additional local constraint imposing strict degree conservation yields the
additional flavor for the condensation phenomena hovan ; coolen1 ; coolen2
(see also foster ; tamm  for some earlier observations). It is in this case
the eigenvalue instantons enter the game. It turns out that degree
conservation constraint amounts to the first order phase transition with
formation of the multiple weakly interacting clusters which has been
demonstrated for constrained ER and RRG with Hamiltonian involving
$\mu_{3}\neq 0$ hovan . The local degree conservation constraint in the graph
ensemble has a matrix model counterpart as well, it means that non-singlet
sector of a matrix model now matters.

The phase transition can be effectively analyzed in terms of network spectrum.
In the ER model with local constraint and RRG the formation of network cluster
corresponds exactly to the single eigenvalue instanton. Eigenvalue instantons
upon the averaging over ensemble form the new non-perturbative soft band in
the spectrum of the graph Laplacian. The similar effects take place in a
spectral density of a matrix model upon the account of multiple instantons.
Soft ”‘cluster”’ band is separated with the gap from the perturbative part of
the spectrum. The number of instantons equals to the number of clusters.
Moreover, the spectrum of the perturbative band gets deformed at the phase
transition and acquires the triangular shape hovan  typical for scale-free
graphs due to the intercluster interactions.

Following parallels with the matrix models the exponential random graphs have
been also used as the model for the discrete approximation to quantum gravity
in trug1 . This approach based on the Ollivier graph curvature was nicknamed
as combinatorial quantum gravity. The graph Ollivier curvature olivier1 ;
olivier2  gets reduced to the Ricci curvature in the continuum limit for the
large family of graph ensembles trug-kru  including RRG therefore the
Einstein-Hilbert action can be recovered. The dimension of emerging Einstein-
Hilbert action $D$ is defined by the degree $d$ in RRG and $\mu_{4}$ upon
proper rescaling plays a role of the gravitational coupling. It was claimed in
trug1 ; trug2  that such model undergoes the second order phase transition
into the dense phase at some $\mu_{4}^{c}$ without a cluster formation. The
$d=3$ example has been investigated in trug3  and the emerging $S^{1}$
geometry has been identified.

Another popular application of RRG concerns its role as a toy model for the
Hilbert space of interacting many-body system. The RRG mimics the realization
of the Hilbert space as the Fock space. It was suggested in kamenev  that one-
particle localization in the Fock space is related to the localization in the
interacting many-body system. It was argued in basko ; mirlin  that indeed
localization (MBL) in the interacting many-body system can be mapped to the
problem of one-particle localization in the Fock space with flat diagonal
disorder. In localization  we have treated the one-particle problem in the
Fock space without the diagonal disorder but with the structural disorder
induced by chemical potential for 3-cycles. It was found that above the phase
transition the states in the non-perturbative band get localized while all
states in the perturbative band are delocalized. More recently RRG with
diagonal disorder has been analyzed in local1 ; local2 ; local3 ; local4 ;
ivan1 ; local5 , moreover the arguments have been presented that RRG plays the
role of tricritical point in the space of deformations local5 .

In this paper we investigate numerically the effects of eigenvalue instantons
in the graph ensembles with degree conservation. Three graph ensembles are
considered: i) RRG ensemble; ii) RRG + hard-core constraint; iii) RRG + hard-
core constraint + connectedness constraint. First we consider numerically RRG
perturbed by chemical potential $\mu_{4}$ for 4-cycles. It is found that at
$\mu_{4}^{RRG}$ the network gets clusterized however in contrast to the case
of 3-cycles all clusters are bipartite. Then we elaborate the model with the
additional hard-core constraint for a possible touching of neighbor 4-cycles
in RRG suggested in trug1  which we denote as cRRG . The nature of the phase
transition in cRRG and the dependence on initial configurations is
investigated. We find the first order phase transition in cRRG and check the
clear-cut hysteresis which confirms the order of phase transition for $d>3$. A
bit surprisingly it turned out that almost all emerging bipartite clusters are
isolated or weakly interacting hypercubes while the second ingredient of the
emerging clusterized phase for all numerically available degrees is the
composite bipartite closed ribbon. The number of nodes involved into the
closed ribbon depends on $\mu_{4}$. Our finding contradicts the claim in trug1
; trug2 ; trug3  about the order of the phase transition for $d>3$ and about
the structure of emerging phase.

Finally we introduce the second constraint for cRRG assuming the connectedness
of the graph and obtain the closed chain of the weakly connected hypercubes at
$\mu_{4}>\mu_{4}^{cRRG}$. We reproduce the result of trug3  for $d=3$ that at
$\mu_{4}>\mu_{4}^{cRRG}$ the single composite bipartite closed ribbon emerges.
However for $d>3$ we clearly demonstrate that the clustering into hypercubes
exists contrary to the claim in trug1 ; trug2 ; trug3 .

The hypercube phase provides the interesting insight on the possible dual
holographic picture. Indeed it was noted in ver20  that the hypercube
adjacency matrix exactly coincides with the interaction term in the Maldacena-
Qi model of two coupled SYK models maldacena . Hence for a single hypercube we
deal with the strong coupling limit of the Maldacena-Qi model which has
thermofield double state (TFD) as the ground state ver19 ; ver20 . On the
other hand the TFD is holographycally dual to the eternal black hole
maldacenaold . The eternal black hole was considered as the simple example of
the ”‘geometry from entanglement”’ approach van09  when the dual classical
geometry emerges from the entanglement of states of the boundary theory. Here
we have the highly entangled Majorana dimers which correspond to the ground
state of individual hypercube.

At strong coupling phase we obtain the ensemble of hypercubes which can be
completely isolated or weakly interacting. In the holographic picture it fits
with the recent discussion van20 ; marolf20  of the holography for the
ensemble of theories at multiple bulk boundaries. If the components do not
interact the corresponding holographic geometry is assumed to be multiboundary
non-traversable wormhole maoz  which becomes traversable if the interaction
between the components is added. The multiple Majorana dimers were also used
recently for the toy model for holographic quantum error-correcting codes
eisert .

We shall also make a brief discussion concerning the possible application of
our findings for two more problems. In the dimensionally reduced SYM theory
down to matrix quantum mechanics the clusterization of the matrices into the
block diagonal form corresponds to the partial breaking of the $SU(N)$ gauge
symmetry down to $SU(M)^{k}$ symmetry cotler ; hanada1 . We shall compare our
findings in RRG with this interpretation. On the other hand this partial
breaking of the gauge symmetry presumably corresponds cotler  to the
transforming of a graviton gas into small black holes. We shall also speculate
on the relevance of the eigenvalue instantons for the partial deconfinement
phenomena. Secondly we make some comments on the possible interpretation of
the emerging new objects- hypercubes and composite bipartite ribbons in the
context of Fock space picture for Hilbert space of interacting many-body
system. The phase transition provides the different patterns for the
fragmentation of the Hilbert space.

The paper is organized as follows. In Section 2 we present the results of
numerical study of RRG perturbed by chemical potential for 4-cycles and
supplemented with the different constraints. In Section 3 the arguments
concerning the holographic interpretation of the strong coupling phase are
presented. Section 4 is devoted to the aspects of Fock space realization of
the Hilbert space of many-body system in terms of RRG. The phase transition is
considered as a fragmentation of the Hilbert space. The results and the open
questions are collected in Section 5.

##  2 Phase transitions in perturbed random regular graphs

###  2.1 The model description

Similar to discussion in hovan  suppose that we start with some network and
rewire links under the condition that at each step of rewiring we try to
maximize the number of 4-cycles $N_{4}$. Which is the equilibrium structure of
the entire network? In mathematical terms this question reads as follows. We
assign the chemical potential $\mu_{4}$ to each 4-cycle and consider the
partition function

$Z(\mu_{4})=\sum_{\\{\rm
states\\}}{\vphantom{\sum}}^{\prime}e^{-\mu_{4}N_{4}}$ (3)

where prime in (3) means that the summation runs over all possible
configurations of links (”states”), under the condition of fixed degrees
$\\{v_{1},...,v_{N}\\}$ in all network vertices.

To simulate the rewiring process, one applies the standard Metropolis
algorithm with the following rules: i) if under the reconnection the number of
4-cycles is increasing, a move (rewiring) is accepted, ii) if the number of
4-cycles is decreasing by $\Delta\,N_{4}$, or remains unchanged, a move is
accepted with the probability $e^{-\mu\Delta\,N_{4}}$. The Metropolis
algorithm runs repeatedly for large set of randomly chosen pairs of links,
until it converges. In reconnection  it was proven that such Metropolis
algorithm converges to the Gibbs measure $e^{\mu N_{4}}$ in the equilibrium
ensemble of random undirected ER networks with fixed vertex degree or RRG.

In hovan  it has been shown that given the bond formation probability, $p$, in
the initial graph in constrained ER graph or in RRG, the evolving network
above the critical value of chemical potential for triangles $\mu_{3}$ splits
into the maximally possible number of clusters, $N_{cl}$:

$N_{cl}=\left.\left[\frac{N}{Np+1}\right]\right|_{N\gg
1}\approx\left[\frac{1}{p}\right],$ (4)

where $[x]$ means the integer part of $x$ and the denominator $(Np+1)$ defines
the minimal size of formed cliques. The asymptotic limit $\sim[p^{-1}]$ at
$N\to\infty$ in (4) is independent on the particular set of corresponding
vertex degrees, $\\{v_{1},...,v_{N}\\}$. For RRG with degree $d$ the number of
clusters tends to $[N/(d+1)]$. It has been shown in hovan  that clustering of
evolving constrained Erdős-Renyi network or RRG under condition of 3-cycle
maximization, occurs as a first order phase transition where $\mu_{3}$ is a
control parameter.

###  2.2 Clusterization of the perturbed RRG

Figure 1: A. The dependence of number of 4-cycles on the chemical potential
$\mu_{4}$ for RRG of different sizes and $d=8$; B. Ground state and its
adjacency matrix at critical $\mu_{4}$.

Consider the RRG with degree $d$ perturbed by the $\mu_{4}$ term

$H=\mu_{4}TrA^{4}$ (5)

and vary the chemical potential. At some $\mu_{4,cr}^{RRG}$ the network gets
rearranged into bipartite phase with the fixed number of weakly interacting
clusters. The corresponding adjacency matrix emerged upon transition is
presented at Fig.1. All bipartite clusters are almost maximal and have the
same size. The similar phase transition takes place for the constrained ER
network however it that case the size of bipartite clusters has nontrivial
distribution.

Figure 2: The dependence of the spectral density of adjacency matrix on the
chemical potential $\mu_{4}$ for RRG of $N=256,d=8$.

The spectrum of the adjacency matrix $A$ and the Laplacian matrix $L=D-A$
where $D=diag(d_{1},\dots,d_{n})$ is encoded in the spectral density defined
as

$\rho(\lambda)=<\sum_{i}\delta(\lambda-\lambda_{i})>_{RRG},\qquad
Av_{i}=\lambda_{i}v_{i}$ (6)

where the averaging over ensemble is assumed. The unperturbed spectral density
of RRG obeys the Kesten-McKay distribution

$\rho(\lambda)=\frac{\sqrt{4(d-1)-\lambda^{2}}}{2\pi(d^{2}-\lambda^{2})}$ (7)

For the RRG a spectral density of the graph Laplacian coincides with the
shifted spectral density of adjacency matrix and we can use both of them as
the Hamiltonian of a particle propagating on the RRG. The spectral density
gets modified upon transition and acquires the following structure. The
central band corresponds to a continuum spectrum. If there would be no
intercluster interaction there would be degenerate states symmetric with
respect to the central band and degeneracy of eigenvalues would correspond to
the number of eigenvalue instantons. We can interpret these modes as the bound
state of the particle localized at the cluster. However due to the
intercluster interaction we get two non-perturbative bands instead of the
degenerate bound states. For each realization pair of isolated eigenvalues
corresponding to the bipartite cluster symmetrically tunnel from the
perturbative band in the opposite directions. The evolution of spectral
density of the adjacency matrix is presented at Fig.2.

We have also considered the evolution of the spectral density for the
constrained ER network. The clusterization pattern into bipartite clusters is
the same with some peculiarities. In this case the sizes of the bipartite
clusters are different with some size distribution. The spectral density of
graph Laplacian acquires the three-band structure with more wide non-
perturbative bands.

Note that in the spectrum of the graph Laplacian half of eigenvalues
corresponding to instantons $\lambda_{i,+}$ are the soft modes while half of
eigenvalues are the hard modes $\lambda_{i,-}$. All clusters are bipartite
hence there is specific interaction inside each bipartite cluster which
connects different scales corresponding to eigenvalues
$\lambda_{i,-},\lambda_{i,+}$. It provides the entanglement of the bipartite
state.

The symmetry breaking pattern at the phase transition is
$SO(2N)\rightarrow(SO(\frac{N}{d}\times SO(\frac{N}{d})^{d}$. Let us emphasize
that a formation of the bipartite clusters can be observed in the real-time
during the Metropolis cooling. The formation and evolution of the cluster
occurs simultaneously with the evolution of pair of isolated eigenvalues.

###  2.3 RRG with hard-core constraint and combinatorial quantum gravity

Let us consider now cRRG and argue that upon the phase transition the
different ground state emerges. If only hard-core constraint is imposed the
ground state involves the ensemble of non-interacting or weakly interacting
hypercubes and a separate bipartite closed ribbon. However if additional
connectedness condition is imposed the ground state will be identified as the
closed chain of weakly interacting hypercubes.

The hard-core constraint was introduced in the interesting model of
combinatorial quantum gravity suggested in trug1  which is based on the
combinatorial graph curvature invented by Ollivier olivier1 . The Ollivier
curvature is quite involved for generic graph however it can be simplified for
special types of the graph ensembles. It was suggested in trug1  to consider
as the simple model for combinatorial quantum gravity the ensemble of RRG with
two additional constraints. First, the bipartiteness condition is imposed and
secondly it is assumed that two squares can have only one common link which to
some extent can be considered as the analogue of the hard-core condition for
extended objects. It was shown recently trug-kru  that for the large class of
graphs the Ollivier curvature gets reduced to the Ricci curvature in the
thermodynamic limit.

Under these conditions the density of Ollivier curvature for the cRRG acquires
the simple form

$R_{ij}=-\frac{1}{d}[(2d-2)-N_{4}(ij)]$ (8)

where $N_{4}(ij)$ is the number of squares supported at the (ij) edge. Then
one introduces the analogue of the Ricci scalar

$R=\sum_{i}\sum_{j}R(ij)$ (9)

which for the cRRG ensemble gets reduced to

$R_{cRRG}=-\frac{8}{d}[\frac{d(d-1)}{2}N-N_{4}]$ (10)

where $N_{4}$ is the total number of 4-cycles.

Therefore for cRRG the partition function involving Hamiltonian
$H=\mu_{4}TrA^{4}$ acquires the meaning of the combinatorial version for the
action of Euclidean Einstein gravity

$Z_{comb}=\int dg\exp(\alpha\int\sqrt{g}R)$ (11)

where the integral over the metric comes from the summation over the cRRG
ensemble. The chemical potential for 4-cycles defines the gravitational
coupling constant upon some rescaling

$\alpha\propto\mu_{4}$ (12)

Figure 3: A: The number of 4-cycles in dependence on the parameter $\mu_{4}$
in cRRG of $d=4$ and different size; B: The portion of the nodes in the
bipartite closed ribbon in dependence on the parameter $\mu_{4}$.

The authors of trug1 ; trug2 ; trug3  claimed that there is the phase
transition at some $\mu_{4}^{cRRG}$ which has the different nature for RRG and
cRRG. The phase transition in cRRG was assumed to be the second order
transition without formation of clusters. For the $d=3$ the emerging phase was
found to enjoy $S_{1}$ topology trug3 .

We have investigated numerically the phase transition for cRRG for different
degrees $d$. First of all, we have found that the first order phase transition
takes place at $\mu_{4}^{cRRG}$ for $d>3$. We have observed a set of isolated
or weakly connected $d-$ dimensional hypercubes at $\mu=\mu_{4}^{cRRG}$, the
number of hypercubes can be estimated as

$N_{cube}=\left.\left[\frac{N}{2^{d}}\right]\right|_{N\gg 1}$ (13)

Numerical results for the number of 4-cycles are presented in Fig.3, there is
a explicit hysteresis typical for phase transitions of first order. Second, we
observe that if there is no enough nodes to form one more hypercube and for
$\mu_{4}>\mu_{4}^{cRRG}$ part of nodes are organized in the closed bipartite
ribbon similar to one discussed in trug3 . The number of the nodes in the
closed ribbon depends on $\mu_{4}$ non-trivially, see Fig.3 B. The typical
configurations in clusterized cRRG for different values $d$ and $\mu$ are
presented at Fig.4. Let us emphasize that we do not impose the bipartiteness
condition for the initial configurations and bipartiteness is emerging
symmetry in the clusterized phase.

Figure 4: Typical subgraph configurations in the clustered phase involving
hypercubes and composite bipartite strings.

Figure 5: Typical adjacency matrices in the clustered phase for RRG of $N=256$
and different values of $d$.

Figure 6: Hysteresis in dependence of the number of 4-cycles for RRG of
$N=256$ and different degrees.

Next, we impose the second connectedness constraint in cRRG and simulated cRRG
with this additional restriction. We have observed the phase transition for
$d=3$ with the bipartite ribbon as the ground state confirming the result of
trug3 . However, for $d>3$ there is a clear-cut first order phase transition
with the formation of the clusterized ground state. It is identified as closed
chain of weakly connected hypercubes (Fig.7) and all 4-cycles belong to
hypercubes. One could wonder why the clusterization in cRRG has not been seen
in trug1 ; trug2 . It seems, that the parameters of their numerics did not
allow more than one hypercube hence configuration approximately looks as
homogeneous.

Figure 7: Ground state of cRRG with connectedness constraint and different
degree $d$ for RRG of $N=128$.

##  3 Interacting Thermofield Doubles and Holographic Dual

###  3.1 Towards the holographic dual for hypercube phase

We have argued above that at the transition point the RRG network with hard-
core constraint gets disintegrated into the ensemble of non-interacting or
weakly hypercubes with additional closed bipartite ribbon. If the
connectedness condition is added the emerging configuration is identified as
the single bipartite ribbon for $d=3$ and the closed chain of weakly connected
hypercubes for $d>3$. In this study we are guided by the analogy between the
exponential random graphs and matrix models. Hence we could question if a kind
of holographic representation of our partition function for the graph ensemble
can be developed somewhat similar to the matrix model representation of JT
gravity sss ; marolf19 . In that case the $(0+1)$ boundary Euclidean quantum
mechanics is presented holographically by JT gravity. More precisely the
discussion in sss  suggest that the bulk gravity is dual to the ensemble of
boundary theories.

Before proceeding further two remarks are in order. First, we shall try to
develop the holographic picture for the clusterized phase only for cRRG and
cRRG with connectedness constraint. Secondly, we are guided by the idea that
quantum entanglement of the boundaries can lead to the classical geometry in
the bulk van09 . The first clear-cut of such situation has been developed in
maldacenaold  for the eternal black hole with two boundaries. In the Euclidean
version the corresponding geometry with several boundaries is provided by the
multiboundary wormholes maoz . The boundary components can be entangled via
the gravitational bulk. The recent discussion concerning this scenario can be
found in van20 ; marolf20 . We shall have in mind this pattern of holography
in what follows.

To develop holographic interpretation we should clarify what is the meaning of
hypercubes and bipartite ribbons in the gravity framework. First take a look
at the hypercubes. It was noted in ver20  that the hypercube adjacency matrix
$A_{hyper}$ exactly coincides with the interaction term in the Maldacena-Qi
model maldacena  for the near $AdS_{2}$ gravity.

$A_{hyper}=H_{int}=i\sum_{k}\gamma_{k}^{L}\gamma_{k}^{R}$ (14)

The mapping is going as follows. Let consider the tensor product
representation for the hypercube Hamiltonian

$H_{d}=\sigma_{0}\otimes
H_{d-1}+\sigma_{1}\otimes\overbrace{\sigma_{0}\otimes\cdots\otimes\sigma_{0}}^{d-1},\quad
H_{1}=\sigma_{1}.$ (15)

and define the $\gamma$ matrices representing Majorana fermions as

$\gamma_{k}^{L}=\overbrace{\sigma_{1}\otimes\cdots\otimes\sigma_{1}}^{k-1}\otimes\sigma_{3}\otimes\overbrace{\sigma_{0}\otimes\cdots\otimes\sigma_{0}}^{d-k},\gamma_{k}^{R}=\overbrace{\sigma_{1}\otimes\cdots\otimes\sigma_{1}}^{k-1}\otimes\sigma_{2}\otimes\overbrace{\sigma_{0}\otimes\cdots\otimes\sigma_{0}}^{d-k},$
(16)

than the Hamiltonian can be written as

$A_{hyper}=i\sum_{k=1}^{d}\gamma_{k}^{L}\gamma_{k}^{R}.$ (17)

Since we are looking at the RRG ensemble the spectrum of hypercube adjacency
matrix which reads as

$E_{n}=-d+2n,n=0\dots d\qquad deg(E_{n})=C^{d}_{n}$ (18)

coincides with the shifted spectrum of the hypercube graph Laplacian $L=dI-A$
where I is identity matrix. Therefore the one-particle Hamiltonian of a free
particle on the hypercube gets mapped into the interaction term of Maldacena-
Qi Hamiltonian for the Majorana fermions.

On the other hand is was found in ver19 ; ver20  that the ground state of the
Maldacena-Qi model at strong coupling limit is well approximated by the TFD at
$\beta=0$ in boundary quantum mechanics of Majorana fermions

$|TFD>=\sum_{n}e^{-\beta E_{n}}|n>_{L}|n>_{R}$ (19)

At large N boundary degrees of freedom in each component of TFD are degrees of
freedom forming the eternal black hole in AdS maldacenaold  in Minkowski space
or wormhole in the Euclidean space. If there are several entangled components
the gravity dual is the multiboundary wormhole.

Now let us turn to our proposal for the gravity dual of the clusterized phase.
For cRRG we have set of non-interacting or weakly interacting hypercubes. Each
hypercube corresponds to the TFD of entangled $d$ Majorana fermions. If $d$ is
large it is natural to assume that each TFD yields the path in the dual
geometry moreover all TDF’s are entangled yielding the geometry of non-
traversable multiboundary wormhole in AdS somewhat analogously to the
arguments in van20  if we have non-interacting but entangled hypercubes.

If $d$ is small we still have the TFD’s of $d$ Majorana fermions forming the
Majorana dimers. It is however impossible to assume that such TDF corresponds
to the fragment of the gravity dual since it has not enough degrees of
freedom. On the other hand the total number of TDF,s is large so a kind of
holography can be expected nevertheless. We can use the arguments recently
suggested in the holographic representation of the error-correcting codes
eisert . It was argued in eisert  that boundary Majorana dimers through their
entanglement yield the geodesics in the bulk geometry allowing its partial
restoration. However contrary to eisert  here we have TFD,s for the groups of
$d$ Majorana dimers.

When the connectedness constraint is added we get the closed chain of
hypercubes and therefore the closed chain of TDF,s as ground state. However
now we certainly have the interaction between TDF,s . Following arguments from
van20 ; marolf20  we conjecture that the bulk dual is the traversable
multiboundary wormhole if $d$ is large enough. At small $d$ we can apply the
similar logic based on the error-correcting codes. Here we have the
interaction between the entangled groups of Majorana dimers.

Let us complete this Section with short remark concerning the topology of the
emerging hypercube phase. The isolated n-hypercube enjoys the hyperoctahedral
symmetry group $C_{n}$ which is the wreath product of $S_{2}$ and $S_{n}$
where $S_{n}$ is the symmetric group of degree $n$. It has nontrivial first
and second homology groups $H_{1}(C_{n},Z)=(Z/2)^{2},\quad n\geq 2$,
$H_{2}(C_{n},Z)=(Z/2)^{3},\quad n\geq 4$. This allows to look for the
topologically stable closed chain of hypercubes.

###  3.2 D0-branes quantum mechanics and black hole formation

Matrix models provide the framework for a formation of extended objects from
ensemble of D0 branes bfss . Above we have made the proposal concerning the
holographic interpretation of the hypercube phase of perturbed cRRG. Let us
make a short comment on the somewhat analogous situation in the holographic
interpretation of a matrix model when the typical matrix acquires the block-
diagonal form.

According to holography any transition in the ground state in the boundary
matrix model gets mapped into some transition in the dual bulk theory. The
transitions can be identified via the changes of the spectral density and the
corresponding form of matrix analogously to our discussion of exponential
random graphs. It was suggested in the context of the large N $(1+1)$ SYM
theory on the circle that the distribution of the eigenvalues of Wilson loops
plays the key role. The flat distribution holographically corresponds to a
black string while the gaped distribution corresponds to a black hole aharony
.

Somewhat similar classification has been used in the context of the
deconfinement phase transition hanada1 ; berenstein ; hanada2 ; hanada3 . It
was argued that again there are three phases - confined, partially deconfined
and completely deconfined in the boundary theory. The typical matrix
corresponding to the partially deconfined phase has deconfined $M\times M$
block filled densely which presumably corresponds to the small black hole in
the dual bulk while the rest $SU(N-M)$ sector corresponds to the black hole
exterior. The entropy in this phase has intermediate form $S=\epsilon N^{2}$
where $\epsilon<<1$. The number of blocks in the typical matrix corresponds to
the number of small black holes.

Having the holographic picture in mind we can speculate on the analogy of the
phases in the deconfinement transition with the phases in the exponential
random graphs. In the ER network enriched by 3-cycles the single dense cluster
gets emerged strauss  which presumably holographically corresponds to the
small black hole. The RRG perturbed by 3-cycles develops several interacting
dense clusters hovan  which presumably correspond to the several interacting
small black holes similarly to hanada2 ; hanada3 .

##  4 RRG as the model for a Hilbert space and localization

The RRG is now the popular model for a investigation of the localization
phenomena in a interacting many-body system. It models the Hilbert space of a
model and the MBL in the physical space gets mapped into one-body localization
at RRG following the initial arguments suggested in kamenev . The Laplacian of
the graph plays the role of the Hamiltonian for the propagating degree of
freedom. If the wave function of an effective one-particle system in the Fock
space is close to the state of the initial many-body system, then the
localization in the Fock space occurs. Hence, one can identify the localized
state in the Fock space with the particle in the initial many-body system.
Meanwhile, if the wave function of the effective one-particle system in the
Fock space is expanded over a large number of states of the many-body system,
this regime is understood as a delocalized in the Fock space. Similarly to the
localization in the real space, the notion of a mobility edge can be
introduced in the Fock space as well.

Technically one can analyze the ergodic properties through the fractal
dimensions $D_{q}$, which show the spread of a state in the Fock space or via
level spacing distribution. Ergodic states at infinite temperatures are spread
homogeneously over the entire Fock space hence $D_{q}=1$. Instead, non-ergodic
states cover only a vanishing fraction of the Fock space,$0<D_{q}<1$. while a
localization in the Fock space requires $D_{q}=0$. Hence in terms of the Fock
space the MBL transition can be identified as transition from $D_{q}=1$ in the
ergodic space phase to $D_{q}<1$ in the MBL phase. Different aspects of the
one-body localization on RRG with diagonal disorder have been discussed in
local1 ; local2 ; local3 ; local4 ; local5 . Recently some arguments
concerning the possible Kosterlitz-Thouless nature of this transition in terms
of the Fock space have been developed in new .

It was argued in abanin ; bernevig  that the so-called scars in the real space
could provide the MBL phase. They were argued to be related to the peculiar
fragmentation of the Fock space in the many-body system. Generically it is
expected that the disorder is the source of the Hilbert space fragmentation
however there are clear examples sola ; nand  that the local conservation laws
also can play such role. In particular it was shown in sola ; nand  that the
combination of the charge and dipole number conservation amounts to the
fragmentation of the Fock space into the exponentially many scars supporting
localized states. In this case there is no need a disorder. More formal group
theory based arguments concerning the fragmentation of the Hilbert space can
be found in mori ; klebanov . The use of the Krylov basis for the
fragmentation phenomenon has been discussed in herviou .

It was found in localization  that the RRG perturbed by the chemical potential
for 3-cycles $\mu_{3}$ gets defragmented into the fixed number of weakly
coupled clusters at come critical $\mu_{3,crit}$. The spectrum of graph
Laplacian acquires the two-band structure, the number of the eigenvalues in
the non-perturbative band is equal to the number of clusters and the
eigenvectors in the non-perturbative band are localized. Hence we have
explicit example of the localization in the Fock space via its fragmentation.
One could question about the underlying mechanism for the fragmentation in
this case similar to the hidden conservation laws in other examples. The fixed
degrees of the nodes in the RRG play the role of the local conservation laws.
To clarify this point we have considered the constrained ER with the degree
conservation constraint. The fragmentation of the network similar to the RRG
takes place however if we drop off the degree conservation constraint the pure
ER network does not exhibit the fragmentation phase transition. Note that the
degree conservation constraint in the Fock space corresponds to the non-local
constraint in the real space.

The identification of the graph responsible for the Hilbert space of
particular interacting many-body system is not a simple task. The nodes
correspond to the states while the links connect the states related by the
resonant conditions. The cycles have the meaning of the higher resonances
basko2  for instance the 3-cycle discussed in localization  corresponds to the
three nodes whose energies are organized in such way that the interaction
provides the resonant conditions for all pairs of states. In this study we
have considered the 4-cycles which correspond to the resonant conditions for
four states. We have demonstrated that the increasing of the number of four-
resonances yields the structural disorder in the graph upon the phase
transition via the network fragmentation.

We look at the localization of the modes in the RRG and cRRG in hypercube
phase similarly to localization . Preliminary study shows that similarly to
localization  the modes corresponding to the bipartite clusters in RRG are
localized in agreement with the naive expectations. The pure hypercube
corresponds to the integrable system hence it enjoys the localized modes.
However when the interaction between the hypercubes is switched on it can be
seen that there is localization in the whole spectrum. The question concerning
the existence of mobility edge in this case need for the additional study.

Note that the ER network with chemical potential for 4-cycles without degree
conservation constraint can be considered as well. In this case the analytical
analysis is available and it perfectly fits with the numerical simulations
222This result is obtained in collaboration with A. Vasyliev. In this case the
fragmentation of the Hilbert space does not take place and all modes are
delocalized. The ER network has the intrinsic disorder in the graph Laplacian
due to the degree distribution however it does not lead to localization.

We do not know exactly what physical system has the Hilbert space which we are
treating as the perturbed RRG. However a few remarks are in order. First note
that the perturbation by 4-cycles amounts to the emerging $Z_{2}$ symmetry in
the Hilbert space. Secondly one could question if some physical system has the
hypercube as the Hilbert space. Some indication of the relevance of the single
hypercube to the Hilbert space of spin chain has been noted in new  but this
point certainly needs for further analysis.

Finally note that the Fock space perspective provides the interesting twist of
the holographic picture. Now we consider the bulk holographic dual not for
representation of the boundary partition function via the path integral over
fields but via sum over the Hilbert space represented by some ensemble of
graphs. In particular the combinatorial quantum gravity we have discussed
above acquires the meaning of the effective gravity in the Hilbert space where
the chemical potential for the 4-resonances plays the role of the
gravitational coupling. The Ollivier curvature plays the role of the discrete
Ricci curvature in the Hilbert space. The multiboundary wormhole
interpretation of the hypercube phase we suggested above has the
interpretation of the dual bulk geometry for the boundary Hilbert space
ensemble for some theory which we do not identify.

##  5 Discussion

In this study we investigated numerically the non-perturbative phenomena in
exponential random graphs with the degree conservation constraints. They are
similar to the eigenvalue instantons in matrix models however the degree
conservation constraint corresponds to the account of the non-singlet sector
in matrix model context. It was argued that the degree conservation induces
the first order phase transition of a network into bipartite clusters at
$\mu_{4}>\mu_{4}^{c}$ both for RRG and constrained ER networks. The induced
network bipartiteness at strong coupling phase is the dynamical phenomenon.

The RRG with 4-cycle chemical potential has been used as the model for the
combinatorial quantum gravity if the additional hard-core constraint is
imposed. We have performed the detailed numerical analysis of cRRG and have
found some novelties and disagreements with results claimed earlier in trug1 ;
trug2 ; trug3 . It turned out that the first order phase transition with a
clear-cut hysteresis takes place at $\mu_{4}^{cRRG}$ for $d>3$. Nearby the
phase transition point the cRRG decays into the ensemble of separated non-
interacting or weakly interacting hypercubes supplemented with closed
bipartite ribbon. Such constituents of the clustered phase - hypercubes and
bipartite ribbon, have appeared in all numerical experiments at different $d$.
If we impose the additional graph connectedness condition for cRRG a structure
of the clusterized phase gets changed and depends on value of $d$. At $d=3$
there is a phase transition with the formation of the single bipartite closed
ribbon in agreement with trug3 . However for $d>3$ there is the first order
phase transition with clear-cut hysteresis and formation of the weakly
connected chain of hypercubes contrary to the claims in the early studies.

Using the relation between the hypercubes and TFD,s we have made the proposal
concerning the holographic interpretation of the bipartite hypercube phase.
Namely we have conjectured that the ground state of cRRG at large $d$
holographically corresponds to the multiboundary non-traversable wormholes
where the number of boundaries coincides with the number of isolated
hypercubes. At small $d$ when there are no enough degrees of freedom to form a
wormhole we have assumed the relation with a holographic representation of the
error-correcting codes in terms the Majorana dimers eisert . The multiboundary
wormhole is supplemented with the bipartite closed ribbon which is related to
the specific moduli spaces of the mapping of the Riemann surfaces into the
sphere with three branching points ramgoolam  and presumably deals with the
topological membrane. If we add the connectedness constraint for cRRG the
suggested holographic dual for the closed chain of weakly interacting
hypercubes presumably is the multiboundary traversable Euclidean wormhole. It
would be interesting to relate this proposal with the tensor network
representation of AdS geometry swingle  based on the entanglement
renormalization. Certainly the holographic dual of the bipartite hypercube
phase deserves a further study.

Our study demonstrates that non-perturbative phenomena in exponential random
graphs are more tractable for numerical simulations than in the matrix models.
This allows to visualize the eigenvalue instantons as the process of the
creation of the corresponding bipartite clusters. In the context of the
hypercube phase it is also possible numerically to fix $\mu_{4}$ and increase
$N$, this procedure allows to consider the finite-size effects. Such analysis
in two-star model gorva  provides the clear picture for the evaporation of the
single cluster. In our case the similar process of a evaporation of the
hypercubes can be expected. It is natural since $N^{-1}$ is the effective
coupling. Since according to our proposal the hypercube at large $d$ being the
TFD state is holographically dual to the wormhole the evaporation of hypercube
presumably can be related to a kind of wormhole evaporation. We hope to
discuss this point elsewhere.

Treating RRG as a model for the Hilbert space of many-body system we indicated
a few new patterns of the Hilbert space fragmentation induced by the higher
resonances. The fragmentation of the Hilbert space is related to the MBL phase
in the real space and the preliminary study indicates the one-particle
localization of the cluster modes occurs indeed. We shall present the more
detailed analysis of the IPR, level spacing distribution and the entanglement
entropy in the hypercube phase elsewhere. It would be also interesting to
consider the RRG supplemented with the fluxes which would yield the Parisi
hypercube model upon fragmentation of the Hilbert space. The consideration of
the SYK model on the perturbed RRG along the logic of susskind  could be of
some importance as well.

One more interesting development concerns the dual holographic picture for the
Hilbert space in the spirit of the operator-state correspondence. The
fragmentation of the Hilbert space in the hypercube phase provides the firm
starting point along this line having in mind the wormhole interpretation of
the hypercube ground state.

Our study also provides some suggestions concerning the partial deconfinement
scenario. In the holographic context the block diagonal form of the matrix
presumably corresponds to the partial deconfinement and formation of the small
black holes in the bulk. The spectral density of the Wilson or Polyakov loops
acquires the gaps. Our study suggests that the correspondence can be more
general and some version of wormholes can be relevant. They correspond to the
bipartite block diagonal matrices and the spectral density with two gaps. It
would be interesting to develop these arguments further.

Finally remark that the bipartite large $N$ ensemble arises naturally in low
energy QCD when the integration over instanton moduli spaces is mimicked by
the matrix model with a simple potential wettig . The degree conservation
condition should be implemented in this model by hand since the degree of the
node is a topological invariant. Our study suggests that when the quartic term
in the potential starts to dominate in the matrix measure one could expect
that the instanton fluid phase gets transformed into the instanton molecular
phase when each ”‘ molecule”’ corresponds to the bipartite cluster.

###### Acknowledgements.

We are grateful to I. Khaimovich, V. Kravtsov, A. MIlekhin and S. Nechaev for
the useful discussions. A.G. acknowledge the grant 075-015-2020-801 of
Ministry of Science. The work of O.V. is supported within the framework of the
Academic Fund Program at the National Research University Higher School of
Economics (HSE) in 2020–2021 (grant 20-01-041) and within the framework of the
Russian Academic Excellence Project 5-100. and Higher Education of the Russian
Federation.

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