Abelian Tensor Models on the Lattice

We consider a chain of Abelian Klebanov-Tarnopolsky fermionic tensor models coupled through quartic nearest-neighbor interactions. We characterize the gauge-singlet spectrum for small chains ($L=2,3,4,5$) and observe that the spectral statistics exhibits strong evidences in favor of quasi-many body localization.


Introduction
Tensor models [1][2][3][4][5] of fermions provide a novel class of quantum mechanical models where a qualitatively new kind of large-N limit could be studied. In their large N limits, they are intermediate between the familiar class of vector models and matrix models. On one hand, they share with vector models the ease of solvability: the dominating diagrams at large-N are of a simple type and can easily be resummed to give tractable Schwinger-Dyson equations. On the other hand, they share with matrix models the feature of non-trivial large-N dynamics.
The recent interest in tensor models is triggered by Witten's observation [6] that they are large-N equivalent to the maximally chaotic, disordered, vector-like fermion models of Sachdev-Ye-Kitaev (SYK) [7][8][9]. The interest on SYK from a quantum gravity viewpoint stems in turn from their similarity to black holes [10] in being maximally chaotic [8,9,11,12].
At finite N , unlike SYK models, tensor models have the merit of being completely unitary quantum mechanical models using which, for example, one might hope to understand more about finite-N restoration of unitarity in black holes. The disadvantage of tensor models lies in their relative unfamiliarity compared to matrix models and the rapid growth of the degrees of freedom with N , making numerical computations at finite N using existing techniques very expensive. There are also many fundamental questions about tensor models including the general structure of gauge invariants, in particular the set of them that dominate large-N limit (viz. the analogues of 'single trace' operators) which remain unsolved. There is a clear necessity for coming up with effective and efficient ways of tackling tensor models, which would allow us to work out the finite-N physics of these class of models.
In this work, we will embark on the analysis of what is perhaps the simplest of tensor models: those whose gauge groups are abelian. Our main focus would be lattice versions of minimally colored tensor models a la Klebanov-Tarnopolsky [5] 1 (Henceforth, we call it KT model.). These lattice tensor models (which we will refer to as KT chains here on) are unitary counterparts of lattice SYK model studied by Gu-Qi-Stanford [15]. 2 The large N versions of these KT chains will be studied in detail in an adjoining paper [21] by a different set of authors. We will refer the reader there for a description of how large N KT chains largely reproduce the phenomenology of Gu-Qi-Stanford model with its maximal chaos and characteristic Schwarzian diffusion. Our aim here would be to study the abelian counterparts of the models in [21] with a special focus on the singlet spectrum.
The abelian KT chains do not exhibit maximal chaos. In fact, as we will argue in the following sections, a variety of spectral diagnostics of singlet states show it to be closer to being integrable -may be even many-body-localized, though the lattice sizes we study here are unfortunately too small to resolve that question. In this, the abelian KT tensor chains are qualitatively different from their non-abelian large N cousins whose large time dynamics and spectral statistics of singlet states are governed by random-matrix like behavior [22,23]. Given this qualitative difference, the abelian KT chains are far from the classical black hole-like behavior that sparked the recent interest in tensor models. We will thus begin by explaining our motivations for studying these models.
First of all, it is logical to begin a finite N study of tensor models with the study of abelian models. Given the intricate and unfamiliar structure of singlet observables in tensor models, the abelian model provides a toy model to train our intuition. These abelian tensor chains are simple enough for us to exactly solve for their gauge-invariant spectrum (at least for small chain lengths). We hope that the results presented in this work would serve as a stepping stone for a similar analysis in non-abelian tensor models.
A second broader (if more vague) motivation is to have a toy model to see whether tensor models can be embedded within string theory. This is an outstanding and a crucial question on which hinges the utility of tensor models for quantum gravity and black holes: can we find exact holographic duals of tensor models with black hole solutions? 3 This may well require an embedding into string theory, however it is unclear at present how this might come about. Our hope is that the study of abelian tensor models can give us some intuition on the analogue of the 'coulomb branch' for tensor models. Like the abelian gauge theories which describe D-branes, abelian tensor models may give us intuitions about string theory particles on which the tensor models live on.
A third motivation is from the viewpoint of many body localization (MBL) [27][28][29] 4 . Many body localization is a phenomenon by which a quantum system (often with a quenched disorder) fails to thermalize in the sense of Eigenstate thermalization hypothesis (ETH) [30][31][32].
ETH posits that in an energy eigenstate of an isolated quantum many body system, any smooth local observable will eventually evolve to its corresponding microcanonical ensemble average. The idea of ETH is to hypothesize that, in this sense, every energy eigenstate behaves like a thermal bath for its subsystems and the subsystem is effectively in a thermal state. By now, many low dimensional disordered systems are known where ETH has been known to fail, thus leading to a many body localized (MBL) phase where even interactions fail to thermalize the system. MBL behavior signals a breakdown of ergodicity in the system and is often associated with integrability or near-integrability. Its name derives from the fact that it is the many-body and Hilbert space analogue of Anderson localization [33] whereby in low dimensions, a single particle moving in a disordered potential gets spatially localized.
MBL phase is a novel non-ergodic state of matter where standard statistical mechanical intuitions fail. Thus, the failure of thermalization and the emergence of MBL behavior has drawn a great amount of interest recently. Given the variety of disordered models which have been studied in the context of MBL, it is a natural question to enquire whether a quenched disorder is strictly necessary for MBL like behavior. An interesting question is to enquire whether one can achieve MBL behavior in a translation invariant unitary model [34][35][36][37][38][39]. This question has been vigorously debated in the recent literature with many authors [40][41][42] concluding that MBL like behavior in translation invariant systems is likely to be not as robust as the localizing behavior observed in disordered systems. When one tries to construct a unitary model which can naively exhibit MBL like behavior, one ends up instead with a quasi-many body localized state (qMBL) [41] where a many body localization like behavior persists for long but finite times, but thermalization does happen eventually. For example, in the systems studied by [41], the time scales involved for thermalization of modes with a small wavenumber k are non-perturbatively long (i.e., τ ∼ exp[1/(kξ)]) but finite even as system size is taken to be infinite. Such slowly thermalizing, almost MBL like systems are interesting on their own right since they show a transition from MBL-like behavior to ergodic behavior as they evolve in time. The abelian KT chains that we study in this work share many similarities with the model described in [41,42] and we expect a similar low temperature phase with anomalous diffusion in the thermodynamic limit.
In this work, we will present preliminary evidence that abelian KT chains at small L indeed seem to exhibit the necessary features to exhibit a quasi many body localized behavior. Since our main concern here would be the singlet spectrum, the main evidences we will present here will be the characteristically large degeneracies in the middle part of the singlet spectrum 5 and a Poisson-like spectral statistics (showing near integrability and consequently MBL like behavior). We will leave a more detailed analysis of possible quasi-localization in these set of models (like transport, entanglement etc.) at thermodynamic limit to future work. Our main concern in this work would be to study the spectrum of the fermionic tensor chain built out of tensor models by Klebanov-Tarnopolsky [5]. KT model is a quantum mechanical theory of a real fermionic field ψ ijk which transforms in the tri-fundamental representation of an SO(N ) 3 gauge group. These fermions interact via a Hamiltonian where we will be interested in the simplest case of N = 2, which we refer to as abelian KT model. The Hilbert space of this model is exceedingly simple with just 16 states. Of these 14 are degenerate and lie in the middle of the spectrum whereas the rest two states are split off from these mid-spectral states by an energy gap ±2g. We will choose the zero of our energy to lie in the middle of the spectrum so that whenever the spectrum is symmetric about the middle, the corresponding spectral reflection symmetry is manifest. We will use this simple model as the 'atom' to build an abelian KT chain. The abelian KT chain is made of L copies of abelian KT models arranged on a circle and with a Gu-Qi-Stanford type hopping term connecting the nearest neighbors: where we impose periodic boundary conditions for fermions: ψ (L+1) = ψ (1) . Here, λ r,g,b are the three Gu-Qi-Stanford couplings which differ from each other in which of the three SO(2) 3 index is contracted across the sites. The outline of this paper is as follows: after a brief review of SYK and tensor models in sec 2 and abelian KT models in sec 3.1, in sec 3.2 we will present a complete singlet spectrum of the simplest KT chain: the 2 site abelian KT chain. This is followed by a detailed analysis of the gauge singlet spectrum of the 3 site and 4 site abelian KT chains in sec 4 and sec 5 respectively. In these sections, we will focus on the various structural features of the spectrum which are generic to the abelian tensor chains. After a brief description of how these features generalize to the 5 site case in sec 6, we will analyze the spectra of these Each vertex of the tetrahedron corresponds to the fermion ψ ijk , and the each edge represents the gauge contraction of the corresponding color between two fermions. models and the associated thermodynamics in sec 7 and argue for the near-integrability of abelian KT chains. We will conclude in sec 8 with discussions on further directions. Some of the technical details about the spectrum of 4 site KT chain are relegated to appendix A.

Klebanov-Tarnopolsky Model (KT Tensor Model)
We begin with a review of the Klebanov-Tarnopolsky (KT) model [5] which is the simplest tensor model exhibiting maximal chaos in large N . The KT model is a unitary quantum mechanical model of a real fermion field ψ ijk (i, j, k = 1, 2, · · · N ) which transforms in the trifundamental representation of SO(N ) 3 gauge group. We will find it convenient to distinguish three SO(N ) gauge groups by RGB color. i.e., r, g and b denotes the color of the first, second and the third SO(N ) gauge groups. We will also correspondingly take the first, second and third gauge indices i, j and k of ψ ijk to also be of colors red, green and blue respectively.
The Hamiltonian of the KT model is given by Note that the four fermion gauge index contractions in tensor models with SYK-like behavior have a tetrahedron-like structure. Henceforth, we will call it tetrahedron interaction. Figure 1 represents the gauge contraction of four fermions in the Hamiltonian (2.1). Each vertex of the tetrahedron represents a fermion field ψ ijk whereas the edges denote their gauge indices. The tetrahedron is then a geometric representation of how the color indices contract.
The tetrahedral structure of the gauge contraction is crucial to the dominance of melonic diagrams in large N , which in turn enables us to solve the model in strong coupling limit. In the strong coupling limit, like SYK model, KT model also exhibits an emergent reparametrization symmetry. This reparametrization symmetry is (explicitly and spontaneously) broken, and the associated Goldstone boson leads to the characteristic Schwarzian (and inter alia maximally chaotic) behavior of the model. This maximal chaos is not restricted to the KT model, but has been found in a wide class of tensor models with tetrahedron interaction. For example, Gurau-Witten model [4,6] is also maximally chaotic. Fermions of the Gurau-Witten model have an additional flavor index, and similar features are observed in large N : the emergence and the breaking of reparametrization symmetry and maximal chaos. We refer the reader to [21] for a more detailed description of these models and a general technique which can be used to show maximal chaos not only in large N KT model but also in large N Gurau-Witten model and lattice generalizations thereof.

SYK Model and its Extension to a 1d Model
We will now very briefly review the SYK model [7][8][9]43] and its lattice generalization by Gu-Qi-Stanford [44] which inspired the lattice models of this work. SYK model is a vector-like quantum mechanical model of N SYK real fermions ψ i (i = 1, 2, · · · , N SYK ), but with disorder in form of a random four-fermion interaction. The Hamiltonian is given by where j klmn is a Gaussian-random coupling with variance j 2 klmn = 3!J 2 N 3 SYK . After disorder average, the melonic diagrams dominate two point functions in large N SYK , and reparametrization symmetry emerges in the strong coupling limit [9,[45][46][47][48]. Like KT model, the reparametrization symmetry is broken explicitly and spontaneously, which, in turn, leads to maximal chaos due to the corresponding Schwarzian pseudo-Goldstone boson [47].
A lattice generalization of the SYK model was studied by Gu-Qi-Stanford in [44] where L copies of SYK models on a L-site lattice are interacting via the nearest neighbor interaction: Here j klmn and j klmn are two Gaussian-random couplings with variances j 2 klmn = 3!J 2 0 N 3 and j 2 klmn = J 2 1 N 3 , respectively. We have also imposed periodic boundary conditions for fermions: ψ (L+1) = ψ (1) . Note that this model has a Z L 2 global symmetry which acts by flipping the sign of all fermions in a given site ψ (a) → −ψ (a) for each lattice index a (= 1, 2, · · · , L). This model also exhibits maximal chaos in large N . Furthermore, in this lattice generalization of the SYK model, one can evaluate the speed with which chaos propagates in space (the butterfly velocity).
In this work, we construct a similar lattice model as above where the SYK model is replaced with a KT tensor model instead. Further, our focus will be on the opposite limit to the large N limit studied in the works referenced above. Consequently, the phenomenology of our lattice chains would be very different from their large N cousins described above.

Hamiltonian
As mentioned before, the Hamiltonian of the KT model is given by Here we consider the abelian case that is, when N = 2. Since SO(2) 3 ∼ = U (1) 3 , we will find it convenient to think of the gauge group as U (1) 3 instead. We can then use these charges to define components of ψ with definite U (1) 3 charges. In this work, we will mostly be interested in U (1) 3 gauge-singlet states and the spectrum in the singlet sector.
We will find it convenient to introduce the following creation and annihilation operators which we will use from hereon: These operators satisfy the relations The creation and annihilation operators that we have formed out of the fermionic fields have definite charges under the U (1) 3 gauge symmetry as given in table 1.
In terms of these creation and annihilation operators, the Hamiltonian has the form In writing this expression, we have removed an irrelevant constant energy shift by g from the Hamiltonian given in (3.1). Written in this form, this Hamiltonian exhibits a spectral reflection symmetry and the energy eigenstates are symmetrically distributed on either side of E = 0 . It is interesting to see that this Hamiltonian also has two other symmetries : • Z 3 symmetry generated byΩ: where (234) · i is the cyclic permutation of i by (234) ∈ S 3 (i = 1, 2, 3, 4). e.g. • We note that the Z 2 action which maps ψ ijk to −ψ ijk is generated by the elementΣ 4 .
We find it useful to define the following notations for the products of the creation operators: We define a vacuum | which is annihilated by all the a i 's. (i.e., a i | = 0.) The Hilbert space for this theory is 16 dimensional. We find it useful to work with the following basis: (3.10) Here we have chosen a convenient notation for the basis states, which will be useful for later sections. More explicitly, we have a mapping as given in table 2.   zero energy), and they are given by 4 one-fermion states of the form |a i , 6 two-fermion states of the form |a 2 ij and 4 three fermion states of the form |a 3 i . The other states |A + and |A − have energies +2g and −2g, respectively.
In table 4, we provide these 16 states, the actions of operatorsΣ andΩ on each state, the corresponding energies of the states, and the U (1) 3 charges. We see that out of the 16 states, only 2 states, |A + and |A − , are invariant under U (1) 3 symmetry. Thus they span the singlet-sector of the theory.

An
Extension of the Abelian KT Model to a 1d Lattice: 2 Sites

Hamiltonian
We can extend the KT model to a 1d lattice in a manner similar to what was done in [15] for the SYK model. State Action ofΣ Action ofΩ Energy U (1) 3 charges  Let us consider copies of the KT model on each site of a one-dimensional lattice with L sites and introduce interactions between the nearest neighbors. The Hamiltonian is given by where we impose periodic boundary conditions for fermions: ψ (L+1) = ψ (1) . As in the (onesite) KT model, the fermion ψ (a) (a = 1, 2, · · · , L) in the KT chain model transform in the tri-fundamental representation of U (1) 3 . In addition to U (1) 3 symmetry, it also has (Z 2 ) L symmetry under ψ (a) → −ψ (a) for any a ∈ {1, 2, ..., L} where the corresponding (Z 2 ) L charge is denoted by (η 1 , η 2 , ..., η L ) (η i = ±). The simplest case is when there are two lattice sites. Let us first consider the L = 2 case. The Hamiltonian for L = 2 case is where ψ (i) is the fermionic field at the i th site and λ r , λ g and λ b are the couplings of the three different types of interaction terms shown above. As mentioned, there is Z 2 2 symmetry corresponding to ψ (i) → −ψ (a) for each value of a ∈ {1, 2}.
As before, we can define annihilation and creation operators a i 's and a † i 's in terms of the components of ψ (1) .
In a similar way, we can define annihilation and creation operators b i 's and b † i 's by the same linear combinations of the corresponding components of ψ (2) . As before, let us define operatorsΩ,Σ whose actions on the annihilation operators is as followŝ where (234) · i is the cyclic permutation of i by (234) ∈ S 3 (i = 1, 2, 3, 4) (e.g. See (3.7) and (3.8).) Furthermore, we define a lattice translation operatorT : The Hamiltonian expressed in terms of these creation and annihilation operators has the form where

Singlet Sector
The singlet sector of the theory is the subspace that is invariant under U (1) 3 transformations. For the generic L-site chain we can consider a basis of the total Hilbert space where each element has k 1 , k 2 , k 3 and k 4 creation operators with charges (+, +, +), (+, −, −), (−, +, −) and (−, −, +) respectively acting on the vacuum. Such a basis element has charges of U (1) 3 and hence can belong to the singlet sector if and only if For any basis vector that does belong to the singlet sector the k creation operators of any particular charge belong to k of the L sites. Thus for any particular charge we can choose k out of the L sites to place the corresponding creation operators to construct a basis vector that belongs to the singlet sector. Therefore, the total number of such basis vectors is  constructed from the vacuum by the action of the appropriate operators in the following way.
where σ 1 , σ 2 = ± and i, j = 1, 2, 3, 4. These states are related to each other by the operator Σ andT : It is also convenient to define the following states in order to diagonalize the Hamiltonian. , where σ is again +1 or −1, and (σ 1 , σ 2 , σ 3 ) can take values only from the following set

Spectrum
There are 4 middle states (states with zero energy) in the 2-site KT chain model given by . There are 12 states which become middle states only at the symmetric point of the couplings λ r = λ g = λ b = λ. Nine of them are found to be The Hamiltonian for the rest of the states can be written as

Large g Limit
In the large g limit (i.e., g λ r , λ g , λ b ), we can treat the hopping interaction λ r H (r) ab as perturbations over the on-site interaction In this limit, the states with absolute energies of order O(g) are located near the tail of the spectrum (spectral tail states) whereas the states with absolute energies of order O(λ) and less populate the central region of the spectrum (mid-spectral states).
When g is very large, we get spectral tail states proportional to with energies ±4g , respectively. In this limit, the energies of the other three mid-spectral states (up to order O(λ)) are given by We see that up to first order in perturbation, the energies of the 2 spectral tail states are unaffected by the perturbation.

Symmetric Coupling for All the Hopping Terms
When λ r = λ g = λ b = λ, we have 13 middle states as mentioned earlier, and 3 more middle states: Furthermore, there are 2 other states given by with energies ±2 4g 2 + 3λ 2 , respectively. We summarize the spectrum of the symmetric hopping coupling case in table 6.

Comments
• In the 2-site case, we see that there are far more middle states when the 3 hopping couplings λ r , λ g and λ b are symmetric. We see a similar behavior for the 3-site and 4-site cases as well. We expect that this would be true generally for the L-sites KT chain model.
• We see that 4 states, i.e., |A + B − , |A − B + , |ab 3 +++ and |a 3 b +++ , are middle states irrespective of the values of the couplings. i.e., their energies are protected under change of the couplings. In the 4-site case, we see similar protected middle states. Although in the 2-site case we do not find any other protected state with non-zero energy, we observe some such protected states in both the 3-site and 4-site cases with non-zero energies.
• As expected, the Hamiltonian does not mix states with different (Z 2 ) 2 charges. We will use this fact while studying the 3-site and the 4-site cases and look at the eigenstates and eigenvalues in each subsector with particular (Z 2 ) L charges.

Hamiltonian
The Hamiltonian of the 4-site KT chain model is defined by where on-site interaction is given by and the hopping interaction is and similar for H (r) bc etc.

Singlet Sector
In the 3-site KT chain model, the singlet sector has 164 states, and we define a basis for each subsector with definite (Z 2 ) 3 charges below.

Form of basis
Degeneracy

The (+, +, +) Subsector
The basis in this subsector are given in table 7. where we define Since σ, σ 1 , σ 2 , σ 3 can take the values +1 or −1, and (i, j) is a pair of distinct elements chosen from the set {1, 2, 3, 4}, the total number of states in (+, +, +) subsector is 44. Due to the lattice translational symmetry, it is useful to introduce a projection operator onto Z 3 charge eigenspace: In addition, we find it convenient to define the following states: where σ 1 and σ 2 can take the values in {+, −}. We call the states of the form |(a 2 b 2 −b 2 a 2 )C σ biquadratic difference state, and the states of the form |(a 2 b 2 +b 2 a 2 )C σ biquadratic sum state.
Biquadratic Difference States: There are 18 biquadratic difference eigenstates of the form where σ = ±1 and p = 0, 1, 2. Therefore, we have 18 states in this subsector given by Ψ p BiQdiff,σ , and the Hamiltonian is found to be where ω = e i 2πp Biquadratic Sum States and Their Partners: The remaining 26 states can be decomposed into blocks of states as described below such that the action of the Hamiltonian is closed within each block. Each of these blocks contain Bloch states obtained out of biquadratic sum states and their partners which are of the form There are 2 blocks of 5 states with zero Bloch momentum: where σ = ±. The Hamiltonian in these blocks is where we define In addition, we found 4 blocks of 4 states with Bloch momentum p = 1, 2: where σ = ± and p = 1, 2. The Hamiltonian in such blocks is given by where ω = e i 2πp 3 and we define In total this accounts for 26 biquadratic sum states and their partners.
Large g Spectrum: At large g, we have only spectral tail states that is, states with energies of O(g) in this sector. First of all, there are 2 states with energies ±6g: In addition, we have six states where the upper signs give an energy +2g at large g whereas lower signs give states with an energy −2g at large g, respectively. Finally, let us consider 36 states of the form (4.23) Among them 18 states (with plus signs) have energy +2g and the other 18 states (with minus signs) have energy −2g at large g. In total, we have 21 states with energy +2g and 21 states with energy −2g in this sector.
Symmetric Couplings: We summarize the spectrum of symmetric hopping couplings (i.e., are the 3 roots of the polynomial The general solution of the equation P (α) = 0 is found to be where ω = e 2πi 3 n (n = 0, 1, 2) and   where σ = ± and i, j, k and l are distinct elements chosen from the set {1, 2, 3, 4}.
There are 40 states in the (+, −, −) subsector which can be divided into 8 blocks of 5 states: and so on. The Hamiltonian in such blocks is given by Large g: At large g, we have the 16 spectral tail states (i.e., states with energy of O(g)) in this sector: where i = 1, 2, 3, 4. Moreover, there are 24 mid-spectral states in this sector of the following form.
and we summarize their energies in table 10.

Other Subsectors
The

A Comparison of the Spectra in Different Sectors
Large g: The energy eigenvalues (up to first order in perturbation) and their corresponding degeneracies in different sectors are shown in table 12.
ij kl      Eigenvalue Degeneracy 1 0 0 0 1 The Spectrum in the different subsectors for the case λ r = λ g = λ b = λ of the 3-site abelian KT chain model.

Band Diagrams for Eigenvalues
In figure 2, we show the band diagrams of rescaled eigenvalues E/ 1 2 (λ 2 r + λ 2 g + λ 2 b ) + g 2 of the 3-site abelian KT chain model with symmetric and asymmetric hopping couplings against the coupling ratio σ ≡ g 2 1 2 (λ 2 r +λ 2 g +λ 2 b )+g 2 . We see that when the couplings of the hopping terms vanish (i.e., in the limit λ r = λ g = λ b = 0 for the asymmetric case, and λ = 0 in the symmetric case) the bands collapse to give 1 level with energy 0 and degeneracy 72; 2 levels with energies 2g and −2g, each having a degeneracy 45; and 2 more non-degenerate levels with energies 6g and −6g.

Cumulative Spectral Function, Level Spacing Distribution and r-parameter Statistics
In the preceding sections, we have given a detailed description of the singlet spectrum of the L = 2, 3 site KT chain models. We will now turn to an analysis of the general characteristics of the spectrum. We will begin by reviewing some general notions about the nature of the spectrum and what they reveal about the Hamiltonian. It is an essential insight due to Wigner and Dyson [49][50][51] that the spectrum of any sufficiently generic (i.e., non-integrable) Hamiltonian can be modeled by a spectrum of a random matrix. Thus in a system which is ergodic, i.e., a system where eigenstate thermalization hypothesis (ETH) holds, the Hamiltonian effectively behaves like a random matrix. This in turn entails that the spectrum shows a characteristic level repulsion, i.e., the adjacent energies in an ergodic system tend not to cluster together but rather feel an effective repulsion resulting in a specific structure in the energy spectrum. Such a level repulsion is familiar from, say, the perturbation theory of two level systems where the off-diagonal entries of the interaction Hamiltonian mixes the levels resulting in a level repulsion. Thus, if we denote the level spacing between two adjacent energy levels in an ergodic system as δ, the probability of δ taking a value near zero is vanishingly small.
In contrast, in a many body localized state, the off-diagonal entries are very much suppressed thus resulting in a breakdown of ergodicity. Thus, the eigenvalues corresponding to localized states are essentially uncorrelated random numbers without any spectral rigidity and hence they fall into a Poisson distribution. Thus, an examination of the statistics of the spectrum gives us crucial clues as to the nature of the Hamiltonian and its ergodicity.
We can apply a statistical measure known as nearest neighbor spacing distribution in order to extract this information. The first step is to perform an unfolding procedure on the eigenvalues of the model. We need to define the spectral staircase function, also known as the cumulative spectral function, [52,53] for the unfolding procedure.
The spectral staircase function N (E) is defined as where Θ is the Heaviside step function and E n represents the n-th energy level from the ordered set of energy levels {E 1 , E 2 , · · · , E n } of the model. It is easy to see that N (E) is a counting function; it jumps by one unit each time an energy level E n is encountered. Thus N (E) gives the number of energy levels E n with energy less than E.
In figure 3, we show the spectral staircase function N (E) of the 3-site abelian KT chain model for asymmetric case of the hopping couplings. The plot is for the coupling ratio σ = 0.5, which corresponds to the hopping couplings λ r /g = 0.8255, λ g /g = 0.3005 and λ b /g = 1.1083. Note that in the plot energy E is given in units of the on-site coupling g.
We are now in a position to calculate the nearest neighbor spacing distribution. Let us define a quantity s ≡ ξ k+1 − ξ k , (4.36) which is the spacing between two neighboring energy levels. The nearest neighbor spacing distribution P (s) gives the probability that the spacing between two neighboring energy levels is s. The unfolding procedure mentioned above ensures that both P (s) and its mean are normalized to unity. We can use the nearest neighbor spacing distribution P (s) to study the short-range fluctuations in the spectrum. We also note that there is another statistical measure of the energy level spacings known as the spectral rigidity. It measures the long-range correlations in the model. We do not diagnose the spectral rigidity properties of our models in this paper.
We have a strong indication that the 3-site abelian KT chain model we consider here is integrable. The probability distribution P (s) behaves like which is the characteristic of a Poisson process. This in turn indicates that the energy levels are uncorrelated, that is, they are distributed at random. We also see that the maximum value of the distribution occurs at s = 0, indicating a level clustering in the model. In figure 4a we give the nearest neighbor spacing distribution P (s) against s for the 3-site abelian KT chain model with asymmetric hopping couplings. Level clustering is evident in the figure. It becomes more and more apparent as we go to 4-and 5-site models.
We also note that chaotic systems generally exhibit level repulsion. That is, the difference between neighboring eigenvalues is statistically unlikely to be small compared to the mean eigenvalue spacing.
We can also diagnose the ergodicity of the system using the statistics of a dimensionless quantity called the r-parameter [54]. This parameter characterizes the correlations between adjacent gaps in the energy spectrum. It is defined as the ratio For comparison, we also note that for large Gaussian Orthogonal Ensemble (GOE) random matrices the mean value is r GOE ≈ 0.5295. (4.42) In figure 4b, we give the r-parameter distribution P (r) against r for the 3-site abelian KT chain model with asymmetric hopping couplings. The fit is to a Poisson distribution.

Comments
• The r-parameter statistics (shown in figure 4) does not show a clear fit for Poisson (or for that matter random matrix) behavior. Due to the small number of states going into the fit, r-parameter statistics is inconclusive in this case. But as we will see in the following, with an increase in the number of sites, a better fit to Poisson like behavior and level statistics can be obtained.
• Here, we see that there are no middle states when the 3 couplings of the hopping terms are all different. This has also been seen in the 5-site case. We expect this to be a generic behavior whenever the number of sites is odd.
• As in the 2 site case we see that there are many middle states when λ r = λ g = λ b = λ.
In particular, we have 18 middle states in this case.
• There are 6 protected states 3 of which have energy 2g and the other 3 have energy −2g. These states are (4.43) The states with the (+) have energy 2g and those with the (−) have energy −2g.

Hamiltonian
The Hamiltonian is  14). We find that there are a large number of middle states in the spectrum. Thus we perform a detailed analysis of these middle states in appendix A.1 for the case of asymmetric hopping couplings and appendix A.2 for the case of symmetric hopping couplings. We enumerate the dimensions of these subsectors and the number of middle states in each of them in table 14.

The Spectrum at Symmetric Hopping
We summarize the spectrum in each sector in the following table.  The (+, +, +, +) Sector

A Comparison of the Spectra in Different Sectors at Symmetric Hopping
Eigenvalue Degeneracy  10 (1) 18

Band Diagrams for Eigenvalues
In figure 5, we show the band diagrams of rescaled eigenvalues E/ 1 2 (λ 2 r + λ 2 g + λ 2 b ) + g 2 of the 4-site abelian KT chain model hamiltonian with symmetric and asymmetric hopping couplings against the coupling ratio σ ≡ g 2 1 2 (λ 2 r +λ 2 g +λ 2 b )+g 2 . We see that when the couplings of the hopping terms vanish (that is, in the limit λ r = λ g = λ b = 0 for the asymmetric case, and λ = 0 in the symmetric case) the bands collapse to give 1 level with energy 0 and degeneracy 1056; 2 levels with energies 2g and −2g, each with degeneracy 288; 2 levels with energies 4g and −4g, each with degeneracy 88; and 2 more non-degenerate levels with energies 8g and −8g.

Cumulative Spectral Function, Level Spacing Distribution and r-parameter Statistics
In figure 6, we show the spectral staircase function N (E) of the 4-site abelian KT chain model for asymmetric case of the hopping couplings. The plot is for σ = 0.5, which corresponds to λ r /g = 0.8255, λ g /g = 0.3005 and λ b /g = 1.1083. Note that here E is measured in units of the on-site coupling g.
In figure 7a we show the histogram of nearest neighbor spacing distribution P (s) against s for the 4-site abelian KT chain model with asymmetric hopping couplings. In figure 7b we show the histogram of r-parameter distribution P (r) against r for the 4-site abelian KT chain model with asymmetric hopping couplings. In both cases the plots are for the coupling ratio σ = 0.5, which corresponds to λ r /g = 0.8255, λ g /g = 0.3005 and λ b /g = 1.1083; and the fits are to Poisson distribution. From figure 7 it is evident that the model exhibits the characteristics of an integrable system.  In both cases the plots are for the coupling ratio σ = 0.5, which corresponds to λ r /g = 0.8255, λ g /g = 0.3005 and λ b /g = 1.1083; and the fits are to Poisson distribution.

Comments
• We see from the graphs of the unfolded level spacing distribution and the r-parameter statistics that the 4-site KT chain has a spectrum that shows a more clear quasi many body localized behavior than the 3-site chain.
• The spectrum in this case has a much larger degeneracy in the middle when the hopping couplings are symmetric. In fact, more than half of the states (938 out of 1810 singlet states) are exactly degenerate with zero energy in the symmetric hopping case. This should be contrasted with the generic asymmetric hopping where only 107 states are degenerate. Both these degeneracies are a dramatic demonstration of the lack of level repulsion in these models.
• The above behavior is broadly similar to the 2-site and the 3-site cases, except for the huge degeneracies. We expect a very fast growing degeneracy in the middle part of the spectrum to persist in the case of even number of sites with symmetric hopping.

Spectral Properties of 5-site Abelian KT Chain
In the 5-site chain the Hamiltonian is (6.1) There are 21, 252 states in the singlet sector. We note that when the hopping couplings are unequal, there are no states at zero energy as expected from an odd site KT chain. In figure 8, we show the spectral staircase function N (E) of the 5-site abelian KT chain model for asymmetric case of the hopping couplings. The plot is for σ = 0.5, which corresponds to λ r /g = 0.8255, λ g /g = 0.3005 and λ b /g = 1.1083. Note that here E is measured in units of the on-site coupling g. It is clear from the r-parameter statistics (shown in figure 9) that this model is close to being quasi many body localized.
We see from the graphs of the unfolded level spacing distribution and the r-parameter statistics that the the 5-site chain has a spectrum that shows a more clear quasi many body localized behavior than the 3-site and the 4-site chain. We expect this qMBL behavior to become even clearer in larger number of sites implying that as L → ∞ this model is indeed quasi many body localized.  In both cases the plots are for the coupling ratio σ = 0.5, which corresponds to λ r /g = 0.8255, λ g /g = 0.3005 and λ b /g = 1.1083; and the fits are to Poisson distribution.

Spectral Form Factors
The spectral form factor was proposed in [55] to study the black hole information paradox, and it has been extensively used in measuring late-time discrete spectrum and in capturing the random matrix behavior of systems. (e.g. SYK model [22], tensor model [13,14], 2D CFT [56] and D1-D5 system [57]). This simple quantity could reveal the random matrix behavior of the SYK model at late times. The spectral form factors in that case can come with a dip, ramp and plateau [22]. At earlier times, the spectral form factor decreases until it reaches the minimum value at the dip, at a time t d , which is followed by a linear growth, the so-called ramp, until it reaches a plateau at a time t p . In large N SYK , the ramp and plateau can be easily distinguishable since the ratio of the plateau and dip time ). However, in N = 2 KT model, the ratio of those two times is not large enough (i.e., tp t d ∼ exp[ 2 3 2 ] ∼ O(10 2 )), and it would be difficult to capture the clear linear growth between t d and t p .
The spectral form factor is defined by where Z(β) = Tr (e −βH ) is the partition function (Here, β = 1 k B T is the inverse temperature, and we consider the trace only over states in the singlet sector). It can be understood as analytic continuation of the partition function. Note that the long time average of the spectral form factor is bounded below, and it saturates the bound when there is no degeneracy in each energy level [56] f (β) ≡ lim 3-site KT Chain Model: In figure 10, we show the spectral form factors f (β, t) against time tg for the 3-site abelian KT chain model at fixed coupling ratio σ = 0.5. Here σ is the effective coupling which appears in large N case for both the asymmetric and symmetric cases of the hopping couplings. The spectral from factor clearly exhibits a ballistic regime identified as the early-time plateau, a diffusive regime where the curve approaches a dip, an ergodic regime where it tries to climb back up and finally a quantum regime where it fluctuates around a mean value [58].

4-site KT Chain Model:
In figure 11, we show the spectral form factors f (β, t) against time tg for the 4-site abelian KT chain model at fixed coupling ratio σ.

5-site KT Chain Model:
In figure 12, we show the spectral form factors f (β, t) against time tg for the 5-site abelian KT chain model at fixed coupling ratio σ. )+g 2 . We take λ r /g = 0.8255, λ g /g = 0.3005 and λ b /g = 1.1083 for the asymmetric hopping coupling case and λ/g = 0.8165 for the symmetric hopping coupling case, respectively. In both cases gβ runs from 0 to 10. )+g 2 . We take λ r /g = 0.8255, λ g /g = 0.3005 and λ b /g = 1.1083 for the asymmetric hopping coupling case and λ/g = 0.8165 for the symmetric hopping coupling case, respectively. In both cases gβ runs from 0 to 10. )+g 2 . We take λ r /g = 0.8255, λ g /g = 0.3005 and λ b /g = 1.1083 for the asymmetric hopping coupling case and λ/g = 0.8165 for the symmetric hopping coupling case, respectively. In both cases gβ runs from 0 to 10.

Thermodynamic Properties
We also compute the thermodynamic quantities for the abelian KT chain model: the mean energy, the mean entropy and the specific heat. In Figs. 13, 14, 15 and 16, we show the mean energy and mean entropy of the 3-site and 4-site KT chain models against temperature T /g for asymmetric and symmetric cases, respectively.
In figure 18 we provide the specific heat of the 3, 4 and 5-site KT chain models against temperature T /g. We see that the specific heat falls off to zero exponentially quickly as T /g → 0. This fall off is expected since it indicates that the system possesses an energy gap, which we have already seen earlier. As T /g → ∞ the specific heat falls off at a slower (power law) rate indicating that the states are being occupied as temperature increases. There is a critical temperature T c at which the specific heat attains its maximum. Note that the critical temperature T c /g systematically shifts towards the low temperature region as the lattice size is increased. The peak value of the specific heat also increases as the lattice volume is increased, suggesting a possible phase transition in the infinite volume limit.
We attempt to fit the specific heat data to the following functional form [59] C ∼ Note that these expressions hold only in the vicinity of T c . There are four fit parameters on each side and performing a reliable fit to all four parameters is a highly non-trivial issue. The critical temperature region is more readily attainable on the high-temperature side and so we proceed to perform the fit to the high-temperature side, T > T c , of the specific heat data.
In figure 19, we fit the T > T c region of the specific heat data of the 3-, 4-and 5-site abelian KT chain model with asymmetric hopping couplings to the functional form given in 7.4. The critical parameters A + , α + and B + , extracted in each case are provided in  The critical parameters extracted from the specific heat data for L = 3, 4, 5 cases of the abelian KT chain model. The data are for σ = 0.5 and asymmetric case for the hopping couplings with λ r /g = 0.8255, λ g /g = 0.3005 and λ b /g = 1.1083. The fit is performed for a fixed value of the critical temperature T c for each L case and in the region T > T c .   Figure 18: Comparing the specific heat of the 3, 4 and 5-site KT chain models against temperature T /g. The plot is for σ = 0.5, asymmetric case. The hopping couplings are λ r /g = 0.8255, λ g /g = 0.3005 and λ b /g = 1.1083. Note that the critical temperature T c /g systematically shifts towards left as the lattice size is increased. The peak value of the specific heat also increases as the lattice size is increased, suggesting a phase transition in the infinite volume limit.  The specific heat C of L = 3, 4, 5 abelian KT chain models against temperature T /g. The plot is for σ = 0.5 and asymmetric case for the hopping couplings with λ r /g = 0.8255, λ g /g = 0.3005 and λ b /g = 1.1083. We fit the data in the region T > T c to the functional form described in the text.

Conclusions and Discussions
In this work, we have studied the spectrum of abelian KT chains made of L copies of abelian KT tensor models, connected by Gu-Qi-Stanford type hopping terms. Unlike their large N cousins [21], they do not exhibit fast scrambling or maximality of chaos. In contrast, they seem to fall into the class of quasi many body localized (qMBL) system as evinced by the lack of level repulsion in the spectrum. We give a detailed characterization of the energy eigenstates, which we hope will lead to a more deeper understanding of tensor models.
As we have discussed in the body of the paper, the spectral statistics of abelian KT chains seem to show evidences of quasi many-body localization. It would be good to confirm this by using other diagnostics of MBL phase available in the literature. Some of the proposed diagnostics are based on entanglement. Consider a system living in one spatial dimension which exhibits an ergodic phase, i.e., a phase where ETH holds. In this ergodic phase, if we follow the evolutions of the isolated system from an initial product state, one often sees a ballistic spread in the entanglement, i.e., a linear growth of entanglement entropy with time. In contrast, MBL systems are expected to exhibit a slower growth of entanglement, with the entanglement growing logarithmically in time [60][61][62][63]. Another diagnostic is the area law for entanglement entropy instead of volume law as is usual for excited states in an ergodic system [64]. It would be interesting to check whether the qMBL behavior of abelian KT chains also extend to their entanglement entropy. 6 We expect that the methods we describe in this work can be extended straightforwardly to abelian Gurau-Witten model [4,6] and more general tensor models on the lattice [21]. A numerical study of the abelian Gurau-Witten model has already appeared in the literature [13] which however seems to exhibit level repulsion unlike the abelian models studied in this work. This opens up the possibility that by interpolating between abelian Gurau-Witten model and abelian Klebanov-Tarnopolsky 7 model one can set up a tensor model with a quasimany body localization transition. Since the non-abelian tensor chains are known to exhibit random matrix like behavior, interspersing them with abelian tensor models would be another way to set up a system with qMBL to chaotic transition. We hope to return to these questions in the future. thank the International Centre for Theoretical Physics (ICTP) for the hospitality and Asia Pacific Center for Theoretical Physics (APCTP) for partial support during the completion of this work, within the program "Spring School on Superstring Theory and Related Topics". AJ thanks hospitality at Harish-Chandra Research Institute, where part of this work was completed. We thank Simons Foundation for partial support. We gratefully acknowledge support from International Centre for Theoretical Sciences (ICTS), Tata Institute of Fundamental Research, Bangalore. We would also like to acknowledge our debt to the people of India for their steady and generous support to research in the basic sciences.

A Energy Eigenstates in the Singlet Sector of the 4-site KT Model
In this appendix we describe with more detail the energy spectrum of the 4-site KT chain model, first in the case where we have a generic asymmetric hopping couplings, the classification is based on the symmetries, or more concretely based on charges associated to those symmetries. For the generic asymmetric coupling case we give a description of the middle states (states with zero energy). When all hopping couplings are the same we have additional symmetries which enlarge the subspace of middle states. We give a description of this sector. In the last part of the appendix we describe some special states which we dubbed protected, they are independent of the hopping coupling constants, and they are energy eigenstates for both the generic asymmetric hopping coupling case and the symmetric hopping coupling case.

A.1 Middle States for Asymmetric Couplings of the 3 Hopping Terms
There is a Z 4 2 symmetry in the model corresponding to a i → −a i and a † i → −a † i for all i's at a particular site. Hence, we can group the states with particular charges under this symmetry and the action of the Hamiltonian will be closed within each such sector.
The There are 2 middle states of the form |A ± B ∓ C ± D ∓ . These 2 states are related in the following way.
There are 24 other middle states of the form |a 2 ij b 2 jk c 2 kl d 2 li where (i, j, k, l) is some permutation of (1, 2, 3, 4). Thus, in total there are 26 middle states in the (+, +, +, +) sector.

A.1.2 The (−, +, −, +) Subsector
There is a state of the form These 2 states are related to each other by Then there are 24 states as given below: where (i, j) is an ordered pair chosen from the set {1, 2, 3, 4} with i = j. In total, we have 26 middle states in the (−, +, −, +) sector. from the set {1, 2, 3, 4}. Also, we define two states by The 2 middle states that can be constructed out of linear combinations of these 3 states are found to be |v 2 + |v 3 , |v 1 + |v 2 − |v 3 . (A.10) The first one is a Bloch state with Bloch momentum 0 and the second one has Bloch momentum π. To enumerate the other 27 middle states, it would be convenient to define |mn p ≡ P T p |(a 3 b 3 cd)mn mn (p = 0, 1, 2, 3 , m, n = 1, 2, 3, 4) , (A.11) where π 2 p is the Bloch momentum of the state. The remaining 27 middle states are given in

A.2 Middle States for Symmetric Couplings of the 3 Hopping Terms
In this section we will look at the middle states for the case when λ r = λ g = λ b = λ. We will try in most cases to write the states using the Z 3 charges and Z 4 charges (when translation symmetry is also present in the respective sector).

A.2.1 The (+, +, +, +) Subsector
There are 132 states in this sector with zero energy. Generically these states can depend on the coupling constants, namely, they will be linear combinations of the basis with coefficients which are dimensionless functions of g and λ. However what happens is that almost all of them are independent of the coupling, the subsector of these 132, which is independent of the coupling, has 104 members. For the case of symmetric hopping, it is useful to utilize Z 4 × Z 3 symmetry where Z 4 and Z 3 is generated byT andΩ defined in (3.13) and (3.15), respectively. Hence, we define a projection operator onto Z 4 × Z 3 eigenstates: where q = 0, 1, 2. We now describe other states that appear. They are linear combinations of the following states: and T 3 |ijkl ∓ , (A. 18) where T 3 is the translation by 3 sites on the lattice. The states are partial symmetric points like λ r = λ g,b , which means that in the general asymmetric coupling case they have eigenvalue λ r − λ g,b .
The count leaves 28 states out of the 134 which do depend on the coupling. On dimensional grounds the coefficients of the linear combinations are dimensionless functions of g and λ. They are given by at most quadratic functions in g λ and λ g . All other states we have not described explicitly are given by linear combinations of the following states: and translations, where (kl) = ( ij), (pq) = ( mn) and σ 1 , σ 2 , σ 3 , σ 4 = ±.
A. The first line corresponds to all choices for three different labels of a's and c's that is, 4 2 . The choice of label for b and d is uniquely determined by the previous choice, except that we can interchange b and d labels. This gives 4 2 × 2. In multiplying by 2 we are overcounting the choices with labels (1, 1), (2, 2), (3,3), (4,4). This gives 4 2 × 2 − 4 = 28. The same reorganization of the fields is true for the combination a i b 3 c j d 3 , changing the role of a, c with b, d.
The other states that come in the pack of 4, correspond to picking the same choice of labels for the bi-cubics, e.g. a 1 a 2 a 3 b 1 b 2 b 3 c 4 d 4 the cubic terms in a's are the same as for b's and this defines uniquely c, d labels. This gives exactly the count 4.
The states above are already quite simple, but the writing does not refer at all about the Z 4 × Z 3 charges. Let us summarize how the states in (A.21) decomposes under these charges, the first line can be replaced by P p,0 |a 3 1 b 1 c 3 1 d 1 , p = 0, 1, P p,q |a 3 i b i c 3 i d i , p = 0, 1, q = 0, 1, 2, one i = 1, P p,q |a 3 1 b 1 c 3 i d i , p = 0, 1, 2, 3, q = 0, 1, 2, one i = 1, P p,q |a 3 i b i c 3 j d j , p = 0, 1, 2, 3, q = 0, 1, 2, (i, j) = (2, 3), (2,4), (3,4) . In the (+, −, +, −) subsector we find 1 protected state with energy E = 4g and 1 protected state with energy E = −4g. These states are where the state with the (+) sign has energy 4g and the one with the (−) sign has energy −4g. This couple of states are both charge zero states of the Z 3 , symmetry, actually they are sum of two zero charge states, .
(A. 31) In the first state in the sum, we can actually remove the projection operator since, the state (1 + T 2 )|A ± b 1 C ± d 3 1 it is by itself an invariant.

A.3.3 Protected States in (−, +, −, +) Subsector
In the (−, +, −, +) subsector we find 1 protected state with energy E = 4g and 1 protected state with energy E = −4g. These states are obtained by translating the protected states in the (+, −, +, −) subsector by 1 step. Therefore these states are where, as before, the state with the (+) sign has energy 4g and the one with the (−) sign has energy −4g. We can also used the projector operator as in the previous section, where in the second line we used the fact that translations and the Z 3 transformations commute.