Many-body localization in a fragmented Hilbert space
MMany-body localization in a fragmented Hilbert space
Lo¨ıc Herviou,
1, 2
Jens H. Bardarson, and Nicolas Regnault
3, 4 Department of Physics, KTH Royal Institute of Technology, Stockholm, 106 91 Sweden Institute of Physics, Ecole Polytechnique F´ed´erale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland Joseph Henry Laboratories and Department of Physics,Princeton University, Princeton, New Jersey 08544, USA Laboratoire de Physique de l’ ´Ecole normale sup´erieure,ENS, Universit´e PSL, CNRS, Sorbonne Universit´e,Universit´e Paris-Diderot, Sorbonne Paris Cit´e, Paris, France
We study many-body localization (MBL) in a pair-hopping model exhibiting strong fragmentationof the Hilbert space. We show that several Krylov subspaces have both ergodic statistics in thethermodynamic limit and a dimension that scales much slower than the full Hilbert space, but stillexponentially. Such a property allows us to study the MBL phase transition in systems includingmore than 50 spins. The different Krylov spaces that we consider show clear signatures of a many-body localization transition, both in the Kullback-Leibler divergence of the distribution of theirlevel spacing ratio and their entanglement properties. But they also present distinct scalings withsystem size. Depending on the subspace, the critical disorder strength can be nearly independentof the system size or conversely show an approximately linear increase with the number of spins.
I. INTRODUCTION
Many-body localization (MBL) and its transition have been the subjects of numerous studies over the re-cent decades. They are directly related to core physicalconcepts and properties of the physics of closed quan-tum systems, namely thermalization, transport, and theeffects of disorder. Interacting systems at weak disor-der thermalize and present ergodic features seeminglyfollowing the so-called strong eigenstate thermalizationhypothesis (ETH) . It states that, at high energy, ageneric closed quantum system has all its eigenstatesdisplay thermal values for all local observables. Atstrong disorder, on the other hand, theoretical argu-ments and numerical studies show a breakdown of theETH in one dimensional systems, arising from emergentintegrability and approximate integrals of motions .The high-energy eigenstates are then characterized bylow entanglement, following an area law instead of avolume law as in the ergodic phase . While theexistence of the MBL phase is now generally well ac-cepted, despite a recent debate , describing the tran-sition from the ergodic phase to the localized phase re-mains a numerical and theoretical challenge. Indeed, thephysics of MBL arises from the rich interplay of variousphenomena: many-body interactions allowing for non-integrability, (strong) disorder leading to localization,and high-energy physics. Many theoretical studies relyon phenomenological renormalization group arguments,based on various physical arguments on thermalizationand predictions of random matrix theory . Numer-ical studies are also limited by the complexity of theproblems: matrix-product-states based approaches perform well deep inside the MBL phase but become un-reliable close to the transition due to rapidly increasingentanglement, leaving exact diagonalization and variantsthereof, with its generally limited system sizes, as themain source of exact numerical resources . The existence of MBL, as a means to break the strong ETHbeyond integrability, spurred the growth of interest inother phenomenas leading to such a breakdown.
Twomajors archetypes have emerged: many-body quantumscars and Krylov fragmentation . Systems withquantum scars present a set of measure zero of highlyexcited non-thermal eigenstates, typically characterizedby a sub-volume law entropy. The other eigenstates re-main thermal. The presence of these states has especiallystrong consequences on nonequilibrium dynamics in suchsystems, with partially suppressed thermalization .Depending on the initial state, time-evolution under aHamiltonian presenting these scar states can typicallypresent much slowler relaxation of observables towardsthe thermal equilibrium states, with slowly suppressedrevivals at long times. Generic methods to embed suchstates into a thermal spectrum have been proposed and scars have been proved to be resilient to the effect ofdisorder . More relevant to this work is the concept ofKrylov subspaces or Hilbert space fragmentation .Due to the interplay between different U (1) symmetriessuch as charge and dipole conservation, each symme-try sector of the Hilbert space shatters into an expo-nential number of sectors or Krylov subspaces that arenot connected by the Hamiltonian dynamics. Impor-tantly, these subspaces are not fully labeled by quan-tum numbers. The exponential number of small dis-connected sectors leads to anomalous and effectively lo-calized dynamics . Conversely, exponentially largeKrylov subspaces have recently been the subject of sev-eral studies.
Remarkably, in the same model and inthe absence of disorder, some of these Krylov subspacesfollow the ETH, while other subspaces have completelyintegrable statistics.A natural question for these systems is the effect ofdisorder on these Krylov subspaces, and in particularwhether it can preserve the fragmented nature of theHilbert space, and lead to a localization of the different a r X i v : . [ c ond - m a t . d i s - nn ] N ov ergodic subspaces. More importantly, we identify sets ofergodic subspaces whose dimension grows much slowerthan the total Hilbert space dimension, albeit still inan exponentional fashion. This gives us the possibilityto investigate through exact (and full) diagonalizationone-dimensional systems of unprecedented physicalsizes. A similar approach was proposed in Ref. 77: inthe conventional XXZ model, strong disorder permitsto approximately separate the Hilbert space into quasi-independent subspaces. There, the separation is only astrong-disorder induced approximation. In our model,it is exact at all disorder strengths. Our approach alsoshares some similarities with the studies of frustratedmodels such as quantum dimers whose Hilbert spaceshows slow exponential scalings and signs of MBL evenin two dimensions .The outline of our paper is as follows. In Section II,we introduce the pair-hopping model and its main prop-erties and symmetries. Section III is dedicated to Krylovsubspaces. After a formal definition and a discussionof the Krylov subspaces studied in Ref. 68, we intro-duce our slowly-growing ergodic subspaces. We study thelevel spacing ratio statistics in the pair-hopping modelin Section IV. We discuss the distribution of the levelspacing ratio of the full Hilbert space, showcasing theneed to consider the individual Krylov subspaces. Toprobe the ETH-MBL transition within the Krylov sub-spaces, we rely on the Kullback-Leibler divergence of thelevel ratio distribution and the reference GOE and Pois-son distributions . We identify the critical disor-der strength as the point of maximum confusion. Weshow that our Krylov subspaces present all signs of theMBL phase transitions. The different critical disordersnonetheless have radically different scalings with systemsize, ranging from quasi absent finite-size effects to an ap-proximately linear shift (within the size we have accessto). Our results underline the importance of the struc-ture of the Hilbert space in the behavior of the MBLphase transition. We also briefly discuss the existence ofa mobility edge . We then turn to the von Neumannentanglement entropy (vNEE) of highly excited statesin Section V. The critical disorder strengths, identifiedby the maximum of the standard deviation of the mid-chain entanglement entropy , are in qualitative agree-ment with our previous results. We also carefully dis-cuss the scaling of the vNEE with subsystem size whichpresent unusual plateaus due to the strongly constrainednature of our model. II. PAIR HOPPING MODEL IN ATRANSVERSE FIELD
The pair hopping model is an interacting modelof spinless fermions on a one-dimensional lattice whose Hamiltonian is given by:˜ H PP = (cid:88) j J j ( c † j c j +1 c j +2 c † j +3 + h.c. ) , (1)where c j ( c † j ) is the fermionic annihilation (creation) op-erator on site j , J j are site-dependent pair hopping termsand we fix the number of sites to L . We take J j uniformlysampled in [0 . , .
1] in order to break inversion symmetryand translation symmetry. For convenience, we performa Jordan-Wigner transform, and work with the spin- Hamiltonian H PP = (cid:88) j J j ( σ + j σ − j +1 σ − j +2 σ + j +3 + h.c. ) , (2)and consider either open boundary conditions (OBC) orperiodic boundary conditions (PBC) in the spin basis.PBC in the spin basis are not equivalent to fermionicPBC due to the presence of the fermionic string, but theobservables we consider in the remainder of this articleare largely unaffected. This model preserves the totalpolarization (or charge in the fermionic language) P tot = L (cid:88) j =1 σ zj (3)and the dipole moment (or center-of-mass position) de-fined as C = (cid:80) j jσ zj if OBCexp iπL (cid:80) j jσ zj if PBC. (4)Additionally, for L even with PBC and all L ’s withOBC, the pair hopping terms preserve the sublatticesymmetry, i.e., P o − P e where P e ( P o ) is the total chargeof the even (odd) sites, and therefore these two chargesare also conserved quantities. We denote with p e , p o , p tot and c the quantum numbers respectively associatedto P e , P o , P tot and C . Similar pair-hopping termsappear naturally in different experimentally relevantset-ups, such as electrons in a Landau level andin the Wannier-Stark problem .We introduce disorder in the form of a transverse field(a random on-site chemical potential in the fermionic lan-guage), resulting in the total hamiltonian H = H PP + L (cid:88) j =1 W j σ zj , (5)where W j is taken uniformly in [ − W, W ]. This disorderdoes not break any of the identified symmetries.To better understand the physics of the hopping terms,it is convenient to represent the system in terms of pairof spins, using the notations introduced in Ref. 68. Forconvenience, we assume L even in the rest of the paper.Let the local Hilbert space be defined by σ z | (cid:105) = | (cid:105) , σ z | (cid:105) = − | (cid:105) . (6)The system is composed of N = L/ |↑(cid:105) = | (cid:105) , |↓(cid:105) = | (cid:105) , |−(cid:105) = | (cid:105) and | + (cid:105) = | (cid:105) . (7)We denote |↑(cid:105) and |↓(cid:105) as pseudo-spins, | + (cid:105) and |−(cid:105) asfractons and the combination | + −(cid:105) and |− + (cid:105) as dipoles.While σ z ’s are still diagonal in this basis, the hoppingterms of Eq. (1) lead to some complex algebra. Thetransformation rules are the following: |↑↓(cid:105) ↔ |↓↑(cid:105) (8) |↑ + −(cid:105) ↔ | + − ↑(cid:105) , |↓ − + (cid:105) ↔ |− + ↓(cid:105) (9) | + − + (cid:105) ↔ |↑ + ↓(cid:105) , |− + −(cid:105) ↔ |↓ − ↑(cid:105) (10)Pseudo-spins exchange with each other (Eq. (8)), anddipoles can move each in one flavor of pseudo-spins(Eq. (9)). Conversely, well-chosen trio of fractons cantransform into a fracton and a pair of pseudo-spins, andreversely the presence of a fracton may lead a pair ofpseudo-spins to transform into a pair of fractons. III. KRYLOV SUBSPACES
In this Section, we introduce the notion of Krylovsubspaces, vector subspaces of the symmetry-resolvedHilbert space which are stable under the application ofthe Hamiltonian. In particular, we present several fami-lies of Krylov subspaces, ergodic at zero disorder despitetheir simple structure, whose dimension grows slowlywith system size.
A. Definition
A natural approach to find the stable subspaces in-duced by the fragmentation of the Hilbert space is towork with Krylov subspaces . Generally, theKrylov space K ( | Ψ (cid:105) , H ) generated by the state | Ψ (cid:105) andthe Hamiltonian H is defined as follows: K ( | Ψ (cid:105) , H ) = Span( | Ψ (cid:105) , H | Ψ (cid:105) , H | Ψ (cid:105) ... ) . (11)This definition has the drawbacks of being numericallyunstable for generic Hamiltonians and requiring the or-thogonalization of a set of large vectors. Working in theconfiguration basis B = { N (cid:79) j =1 | α j (cid:105) with α j ∈ {↑ , ↓ , + , −}} (12)= { L (cid:79) j =1 | α j (cid:105) with α j ∈ { , }} , (13) it is more convenient to use the modified definition: K ( | Ψ (cid:105) , H ) = Span( {| w (cid:105) ∈ B|∃ n ∈ N , (cid:104) w | H n | Ψ (cid:105) (cid:54) = 0 } ) . (14)For simplicity, we consider in the following | Ψ (cid:105) ∈ B , i.e., aproduct state in the σ z ’s basis. Then, seeing the Hamilto-nian as a graph in the configuration space B , where eachnode is an element of the basis and each link a non-zerocoefficient of the Hamiltonian, K ( | Ψ (cid:105) , H ) is simply theconnected subgraph including | Ψ (cid:105) . This definition is nu-merically stable and straightforward to compute. Notethat contrary to K , K strongly depends on the basis B , which should be specifically crafted for the studiedmodel. Our choice for B for the pair-hopping model en-sures the following property: K ( | Ψ (cid:105) , H ) = (cid:91) ∀ J j ,W j K ( | Ψ (cid:105) , H ) . (15)In fact, K and K here nearly always coincide . Wetherefore only work with K ( | Ψ (cid:105) , H ) and drop the sub-script and explicit dependence on the Hamiltonian inwhat follows. B. Ergodic Krylov subspace with a single pair ofdipoles
In Refs. 68 and 73, the authors observed that exponen-tially large subspaces with either Poissonian or ergodicstatistics might coexist in the absence of disorder. An es-pecially convenient family of exponentially large ergodicsubspaces were generated by the pair of dipoles |− + + −(cid:105) or | + − − + (cid:105) in a sea of pseudo spins |↑(cid:105) and |↓(cid:105) . In theabsence of dipoles, the pair-hopping Hamiltonian acts inthe sea of pseudo spins, with conservation of each flavourof pseudo-spins. Introducing a pair of dipoles breaksdown integrability, as each can only move through a sin-gle flavour of pseudo-spins. The pair-hopping Hamilto-nian acting on this subspace conserves the number ofeach flavor of pseudo-spin and fracton, leading to a re-markably simple structure of the corresponding Krylovsubspace.The first Krylov subspace we consider is therefore gen-erated by the state (cid:12)(cid:12) Ψ n (cid:11) = | ( ↑↓ ) n − + + − ( ↑↓ ) n (cid:105) . (16)where ( w ) n marks that we repeat n times the sequence w . The system is comprised of N = 4 n + 4 pairs ofspins and − + + − is placed exactly at the center of thechain. This Krylov space was shown to be ergodic in theabsence of a transverse field and its dimension can bereadily computed. For simplicity, we first consider peri-odic boundary conditions. The dipoles |− + (cid:105) and | + −(cid:105) can be seen as separating the pseudo-spins into two se-quences: either in between |− + (cid:105) and | + −(cid:105) or outside ofthem. We denote by N ↑ = 2 n the conserved total num-ber of pseudo-spins ↑ , and by n ↑ (resp. n ↓ ) the numberof ↑ (resp. of ↓ ) between |− + (cid:105) and | + −(cid:105) . As a concreteexample, the state |↑↑↓↑↑ − + n ↑ + n ↓ =5 (cid:122) (cid:125)(cid:124) (cid:123) ↑ ↓↓↓ (cid:124)(cid:123)(cid:122)(cid:125) n ↓ =3 ↑ + − ↓↓↑↓↑↓(cid:105) (17)belongs to K ( (cid:12)(cid:12) Ψ n (cid:11) , H PBC ) with n = 4, N ↑ = 8, n ↑ = 2and n ↓ = 3. The dimension of the Krylov subspace issimply given by:dim K ( (cid:12)(cid:12) Ψ n (cid:11) ) PBC = N ↑ (cid:88) n ↑ ,n ↓ =0 (cid:18) N ↑ − n ↑ − n ↓ N ↑ − n ↑ (cid:19)(cid:18) n ↑ + n ↓ n ↓ (cid:19) = (2 N ↑ + 1)! N ↑ ! ≈ N √ N e √ π , (18) using the Chu-Vandermonde identity for simplification,and the Stirling’s approximation for the factorial. Thedimension of this Krylov subspace therefore scales as2 N = √ L , i.e., much slower than the full Hilbert space’sdimension, which scales as 2 L .For OBC, the two dipoles separate the chain in three.There is a fixed number N L ↑ = n of pseudo-spins ↑ ( ↓ )to the left (right) of the dipoles. The n ↓ ↓ -pseudo-spinsin between |− + (cid:105) and | + −(cid:105) originally came from the leftof |− + + −(cid:105) via acting by H OBC ; similarly the n ↑ ↑ inbetween pseudo-spins came from the right. Hence thestate represented in Eq. (17) also belongs to K ( (cid:12)(cid:12) Ψ n (cid:11) ) OBC with N L ↑ = 4, n ↑ = 2 and n ↓ = 3. The dimension of K ( (cid:12)(cid:12) Ψ n (cid:11) ) OBC is thus given bydim K ( (cid:12)(cid:12) Ψ n (cid:11) ) OBC = N L ↑ (cid:88) n ↑ ,n ↓ =0 (cid:18) N L ↑ − n ↑ N L ↑ (cid:19)(cid:18) N L ↑ − n ↓ N L ↑ (cid:19)(cid:18) n ↑ + n ↓ n ↓ (cid:19) . (19)Using twice the Chu-Vandermonde equality (seeApp. A 1), we obtaindim K ( (cid:12)(cid:12) Ψ n (cid:11) ) OBC = (cid:18) N L ↑ + 12 N L ↑ (cid:19) ≈ N √ πN . (20)The exponential scaling is similar, albeit with a morefavorable prefactor.For both boundary conditions, the reduced Hilbertspace scaling is not due to an extremal choice of quantumnumbers (such as the linear scaling of the one particlesector of a particle number conserving U (1) model). Itis a simple consequence of the presence of an extensivenumber ( ∝ N ) of freely exchanging pseudo-spins (com-posed of two real spins) that make most of the degreesof freedom. A table summarizing the properties of theKrylov subspace for numerically relevant values of N canbe found in App. A 1. Due to the significantly smallerKrylov space’s dimension, we focus on OBC for this fam-ily. C. Slowly-growing ergodic Krylov subspaces
It is possible to construct a series of ergodic sub-spaces with even more favorable scaling with system size,which cannot be mapped to any simple quasi-particlepicture. Working with sets of |↑↓(cid:105) and pairs of dipoles |− + + −(cid:105) allows us to keep a simple analytical struc-ture of the Krylov subspace while working in the sector( p o , p e , c ) = (0 , , N . Anatural way to go beyond this limit is to alternate be-tween a finite number of pairs of pseudo-spins and theset of two dipoles. The less pseudo-spins, the slower thegrowth of the Hilbert space. The pair-hopping Hamilto-nian in Eq. (1) cancels the state |− + + − − + + − ... (cid:105) ,which therefore forms a Krylov subspace of dimension 1.We define the state | Φ mn (cid:105) for periodic boundary condi-tions as: | Φ mn (cid:105) = | (( ↑↓ ) n − + + − ) m (cid:105) , (21)where ( w ) m again marks that we repeat m times thesequence w . Hence, (cid:12)(cid:12) Φ (cid:11) = |↑↓↑↓↑↓ − + + − ↑↓↑↓↑↓ − + + −(cid:105) (22) | Φ mn (cid:105) is therefore a state of size N = (2 n + 4) m , andcan be seen as several dipoles oscillating in a sea ofpseudo-spins. Note that (cid:12)(cid:12) Ψ n (cid:11) also belongs to this familywhen considering periodic boundary conditions.The dimensions and scaling of the Krylov spaces withtotal system sizes at fixed n can also be computed analyt-ically using a transfer matrix approach, for both open andperiodic boundary conditions. The dimension asymptot-ically scales as t m + where t + is the largest eigenvalue ofthe transfer matrix T ( n ) whose entries are given by T ( n ) x,y = (cid:18) n + x − y + 1 n (cid:19) (23)We summarize in Table I the asymptotical scaling of theKrylov subspaces dimension for different values of n . De-tails of the computation of T ( n ) are kept in App. A 2. In n 1 2 3 4 5 ≈ dim K ( | Φ mn (cid:105) , H ) 1 . N . N . N . N . N TABLE I. We summarize the scaling of the Krylov subspacesgenerated by the repeating sequences | (( ↑↓ ) n − + + − ) m (cid:105) .The scaling is irrational both in L and N for all values of n . particular, for n = 1, we show that t + = 3 + 2 √ K ( | Φ m (cid:105) ) ≈ ( t + ) m ∝ . N ∝ . L . (24)In this example, the slow growth of the Krylov space can-not be understood from a simple quasi-particle picture.Indeed we prove in App. A 2 that there exists no p ∈ N ∗ such that t p + is rational. The subspaces also always scaleslower than 2 N as can be seen from Table I (see alsoApp. A 2). Actually, 2 N matches the Krylov spaces scal-ing when n → + ∞ , keeping m fixed. In the rest of thepaper, we focus on the n = 1 and n = 2 families, i.e.,the families with the two slowest scalings. We will showin Secs. IV B and V that these two families have indeedergodic statistics at zero and low disorders. IV. LEVEL SPACING RATIO STATISTICS
Level spacing statistics are a convenient tool to de-termine whether a system is integrable or ergodic .Random matrices without conservation laws, i.e., de-scribing non-integrable models, have level-repulsion: theprobability of having two eigenstates with the same en-ergy is vanishing. Integrable models, on the other hand,are characterized by the presence of an extensive num-ber of conserved quantities. Each sector then behavesas an independent random matrix and therefore there isno level repulsion between different sectors. Additionally,given a symmetry sector, directly studying the level spac-ing statistics requires unfolding the spectrum. Indeed, inorder to obtain universal signatures, we are required towork with a uniform density of states. Several unfold-ing procedures exist, but finite-size effects may lead todifferent physical interpretations depending on the exactchoice of method.
Instead, an efficient way to char-acterize quantitatively the level repulsion is to look atthe level spacing ratio defined as follow . The studyof this quantity does not require flattening the densityof states. Let { e n } be the ordered eigenspectrum of theHamiltonian. We denote by r n the level spacing ratio r n = e n +2 − e n +1 e n +1 − e n . (25)Its probability distribution P ( r ) distinguishes betweenergodic and integrable models. For an integrable model, P ( r ) is the Poisson distribution P Poi ( r ) = r ) , whilefor non-integrable systems, it depends on the symmetries of the Hamiltonian and is well-approximated by function-als of the form : P β ( r ) = 1 Z β ( r + r ) β (1 + r + r ) β . (26)The real Hermitian Hamiltonians we consider fall into theGaussian Orthogonal Ensemble (GOE) with Z β = and β = 1. In practice, it is more convenient to study˜ r n = min( r n , r n ) (27)which is bounded between 0 and 1 and therefore hasno heavy tails. For the classes we are interested in, P (˜ r ) = 2 P ( r ) θ (1 − r ). In the following, references tolevel ratio are references to ˜ r .Finally, we remind the reader of the definition of theKullback-Leibler (KL-)divergence: D KL ( P, Q ) = (cid:90) dxp ( x ) log( p ( x ) /q ( x )) , (28)where p and q are the probability densities associatedto the distributions P and Q . It trivially satisfies D KL ( P, P ) = 0. The KL − divergence is asymmetric in( P , Q ). It corresponds to the relative entropy from Q to P , that is to say the amount of additional informationrequired to model P starting from the prior Q . Hence,when D KL ( P num (˜ r ) , P Poi (˜ r )) < D KL ( P num (˜ r ) , P GOE (˜ r )),the numerical distribution P num is better modelled by thePoisson distribution. A. Level spacing ratio statistics of the full Hilbertspace
Before we turn to the study of the individual Krylovsubspaces themselves in Section IV B, we point out thatit is crucial to decompose the symmetry resolved Hilbertspace into its fractured components, in order to studyany thermalization properties and transition.We study a system of length N = 16 ( L = 32 spins)with OBC in the symmetry sector p e = p o = c = 0 (seeEqs. (3) and (4)). The dimension of this symmetry sectoris 4 . × , beyond the reach of full diagonalization. Itfractures into approximately 2 . × Krylov subspaces,whose dimension varies from 1 to 12870. To emphasizethe role of the dipole conservation, note that a system of32 spins with only the two U (1) sublattice symmetrieshas already a symmetry sector ( p e = 0 , p o = 0) thatincludes 165M states.We compute the exact full spectrum taking advantageof the decomposition into Krylov subspaces and iden-tify whether each Krylov subspace has GOE or Poisso-nian statistics by computing the KL-divergences definedin Eq. (28), using the theoretical distributions as prior.We fix W = 0 .
01 in order to avoid accidental degen-eracies and limit finite-size effects, and average over 100disorder realizations. We consider the Krylov subspaceto have Poissonian statistics if D KL ( P num (˜ r ) , P Poi (˜ r )) < . , (29)and to have GOE statistics if D KL ( P num (˜ r ) , P GOE (˜ r )) < . . (30)Otherwise, we do not assign a label as the subspace iseither afflicted by finite-size effects or presents signs ofcriticality. For comparison, the KL-divergences betweenthe Poisson and GOE distributions are given by D KL ( P GOE (˜ r ) , P Poi (˜ r )) = 13 + log √ ≈ . , (31) D KL ( P Poi (˜ r ) , P GOE (˜ r )) = 5 π √ − − log 278 ≈ . . (32)In practice, as we do not compare the distributions di-rectly but histograms with 50 bins between 0 and 1, theeffective divergence D KL ( P Poi (˜ r ) , P GOE (˜ r )) is slightly re-duced to 0 .
305 [ D KL ( P GOE (˜ r ) , P Poi (˜ r )) is almost unaf-fected]. Our choice of cut-off comes from the followingobservation: distributions that are maximally confusingwith the KL-divergence verify D KL ( P num (˜ r ) , P Poi (˜ r )) = D KL ( P num (˜ r ) , P GOE (˜ r )) ≈ . , (33)as will be discussed in Sec. IV B. Choosing a thresholdlower than 0 .
05 allows us to only select distributionsthat are convincingly Poissonian or GOE. For the sakeof simplicity, we also focus only on Krylov spaces ofdimension larger than 50 to minimize the number ofsamples to average over. This removes around 2 . × Krylov spaces associated to 1 . × states (41% ofthe symmetry sector), including 5 × dark states,i.e., Krylov spaces consisting of a single state. Thefractions of dark states decreases with system size.Fig. 1 summarizes our results and the nature of theKrylov spaces. Of the approximately 1 . × remainingspaces, a significant fraction present intermediate statis-tics (around 1 . × spaces, comprising 1 . × states,i.e., 23% of the symmetry sector). 4 . × Krylovspaces present clear Poissonian statistics and the re-maining 2 . × have GOE statistics. They nonethelessrepresent a significant proportion of the total symmetrysector, approximately 7 . × states (16% of the totalsymmetry sector) and 9 . × states (20%) respectively.We now turn towards the study of the level spacing ra-tio statistics in this symmetry sector, without resolvingthe Krylov spaces. As shown in Fig. 1d, the statisticsare essentially undistiguishable from Poisson. In a givensymmetry sector, the occupancies of each Krylov sub-space act as an exponential number of additional good Dimension of the Krylov subspace10 K r y l ov s ub s p ace s c) Unassigned . × Total dim: 1 . × K r y l ov s ub s p ace s b) GOE . × Total dim: 9 . × K r y l ov s ub s p ace s a) Poisson . × Total dim: 7 . × . . . . . . r . . . . . P ( ˜ r ) d) PoissonGOE NumericsNumerics ≥ FIG. 1. a), b) and c) : histograms of the dimensions ofthe Krylov subspaces for N = 16 in the symmetry sector p e = p o = c = 0 depending on their level spacing ratio dis-tribution with W = 0 .
01. We only represent subspaces withdimension larger than 50. In a), we represent the Krylov sub-spaces that present ergodic level spacing ratio statistics, inb) Poissonian. c) provides the Krylov spaces that cannot beclassified by our KL-divergence criteria defined in Eqs. (29)and (30). In each panel, we also give the number of Krylovspaces in that category, and the total dimension of theseKrylov spaces. d): distribution of the level ratio for the samesystem if we do not resolve the Krylov spaces. Even if wediscard the smallest Krylov subspaces, the level spacing ra-tio distribution is virtually undistiguishable from the Poissondistribution. quantum numbers. Therefore there is no apparent levelrepulsion. Theoretically, analytical formulas have beenrecently derived to predict the distribution of thelevel ratios for matrices decomposing in several indepen-dent blocks. These studies computed the level spacingratio distribution obtained from considering a small num-ber (up to 12, but easily generalizable) of independent er-godic blocks as a single matrix. In Ref. 96, it was numer-ically shown that the mean level spacing ratio obtainedfrom M ergodic blocks converges toward the Poissonianstatistics approximately as M − . This means that, al-ready for M = 12, the two average values differ only by10 − . With the exponentially large number of blocks,and the additional scrambling induced by our Poisso-nian blocks, the precision required to differentiate our nu-merically obtained distribution from the true Poissoniandistribution goes well-beyond any numerically achievablesampling. Indeed, we numerically obtain that the fullnumerical distribution P full , including all the Krylov sub-spaces, has a KL-divergence with respect to P Poi of D KL ( P full (˜ r ) , P Poi (˜ r )) ≈ . × − . (34) B. Level spacing ratio statistics in a single Krylovsubspace
To study the effects of disorder, we focus on the fami-lies of Krylov subspaces defined in Sec. III. In particular,we specifically do not consider the largest Krylov sub-space. Indeed, for OBC, the Hamiltonian restricted tothis largest Krylov space is equivalent to a random XXmodel in a transverse field for all system sizes we con-sidered. It is integrable and localizes at arbitrarily lowdisorder. Let the reduced energy of an eigenstate of en-ergy E be ε = E − E min E max − E min , (35)with E min ( E max ) the lowest (highest) energy of thereduced Hamiltonian in the Krylov subspace. In therest of this article, we focus on states in the bulk of thespectrum with ε ∈ [0 . , . N = 8) down to 500 for thelarger systems (for N = 30). We represent in Fig. 2a-cthe KL divergence of the distribution of the level spacingratio in the three families of Krylov subspaces defined inSecs III B and III C, using GOE and Poisson distributionsas prior. For all families, at low disorder, we observea quick convergence towards the GOE distribution ofthe level spacing ratio distribution, when increasing thesystem size N . The three families of Krylov subspacesappear indeed ergodic in the thermodynamic limit.Crossing of the KL-divergence universally occurs for D KL ( P num (˜ r ) , P Poi (˜ r )) ≈ .
05. This implies that we are maximally confused about which theoretical distributionbetter approximates the numerical one at this value ofthe KL-divergence. Thus, we take this crossing as amarker of the phase transition.We first turn to the Krylov spaces generated by | Φ m (cid:105) defined in Eq. (21), working with periodic boundaryconditions due to the favorable scaling. As shown inFig. 2a, for m >
2, the crossing point of the KL diver-gences shows very small finite-size effects at W c ≈ . | Φ m (cid:105) , also with PBC, we observesimilar results in Fig. 2b. Due to the faster growth of theKrylov subspaces, we are effectively limited to smallersystems plagued by stronger finite-size effects. For each m , we observe a transition from an ergodic phase to alocalized phase, albeit at a significantly larger disorderstrength, despite the similar Hilbert space dimensionsand structures. The effective critical disorder strengthsdo not display any simple convergence behavior whenincreasing m , at least within the accessible system sizes.Finally, the family (cid:12)(cid:12) Ψ n (cid:11) — studied with OBC due to theslower scaling — also exhibits signs of an MBL phasetransition, as shown in Fig. 2c. Interestingly enough,the crossing point admits an approximately linear driftwith increasing system sizes (see inset in Fig. 2c).Additionally, we observe in all Krylov subspaces thatthe critical disorder strength strongly depends on therelative energies of the eigenstates. Mobility edges are therefore also present in these constrained systems.More details can be found in App. C 2.Note that the three Krylov spaces considered here orig-inate from the same initial Hamiltonian (up to boundaryconditions) and therefore for a given W and N have thesame disorder and hopping amplitudes in configurationspace. Still, the corresponding critical values, as pre-dicted by the level spacing ratio distributions, and scal-ing behavior are radically different. K ( (cid:12)(cid:12) Φ (cid:11) ) and K ( (cid:12)(cid:12) Φ (cid:11) )both correspond to N = 24, K ( (cid:12)(cid:12) Ψ (cid:11) ) and K ( (cid:12)(cid:12) Φ (cid:11) ) to N = 16, and K ( (cid:12)(cid:12) Ψ (cid:11) ) and K ( (cid:12)(cid:12) Φ (cid:11) ) to N = 12 and yetadmit different transition points. Conversely, the Krylovspace dimension alone is also not a good indicator of thecritical disorder: both K ( (cid:12)(cid:12) Φ (cid:11) ) ( N = 16) and K ( (cid:12)(cid:12) Ψ (cid:11) )( N = 20) have a dimension close to 2 . × . K ( (cid:12)(cid:12) Φ (cid:11) )appears to localize at a larger disorder than K ( (cid:12)(cid:12) Ψ (cid:11) ) eventhough the disorder in the Fock basis is averaged over lesssites.The family K ( (cid:12)(cid:12) Ψ n (cid:11) ) shows a strong drift of the criti-cal disorder towards larger values with increasing systemsizes. This could be a sign of an absence of a transitionfor these subspaces in the thermodynamic limit. Para-doxically, this family also has a structure very close to anintegrable one. Indeed, the Hamiltonian acting on the seaof pseudo-spins |↑(cid:105) and |↓(cid:105) reduces to a non-interactingXX Hamiltonian. The pair of dipoles breaks integrability . . . . . . . . . D K L t o t h e o r e t i c a l a) D KL ( ., P Poi ) D KL ( ., P GOE )m = 2 m = 3m = 4m = 5 . . . . . . . . . D K L t o t h e o r e t i c a l b) m = 1m = 2m = 3 . . . . W . . . . D K L t o t h e o r e t i c a l c) 12 16 20N0 . . W c n = 2n = 3n = 4 FIG. 2. Level spacing ratio statistics for the Krylov sub-spaces generated by the state | Φ m (cid:105) (a), | Φ m (cid:105) (b) and (cid:12)(cid:12) Ψ n (cid:11) (c) for ε ∈ [0 . .
6] and different values of m and n . Wecompute D KL ( P num , P Poi ) (full line) and D KL ( P num , P GOE )(dotted line). The red dotted line represent D KL ( P Poi , P
GOE )(see Eq. (32)) and the full red line D KL ( P GOE , P
Poi ) (seeEq. (31)). For very weak disorder and small Krylov space di-mensions, the level spacing ratio distributions present strongfinite-size effects due to the sparsity of the model and quasi-degeneracies. Nonetheless, when increasing system sizes, allthree families convincingly have ergodic statistics. For all sys-tems, we observe an effective transition from GOE to Poissonstatistics. The critical disorder is identified with the crossingpoints of the two divergences. Note the different effective crit-ical disorder ( W c ≈ .
75 for | Φ m (cid:105) , W c ≈ .
15 for | Φ m (cid:105) ) for thefirst two families with PBC. For the subspaces generated by (cid:12)(cid:12) Ψ n (cid:11) (OBC), on the other hand, we observe an approximatelylinear shift of the effective critical disorder with increasing sys-tem sizes, as is shown in inset. Error bars (generally too smallto be seen) are obtained through subsampling of our data. by stitching together a set of triplets—the sea of pseudo-spins to the left, in-between, and to the right of the pair of dipoles—of integrable spaces. Yet, while the XX Hamil-tonian is localized at arbitrarily low-disorder, with an ef-fective critical disorder strength decreasing with systemsize, we observe the exact opposite for (cid:12)(cid:12) Ψ n (cid:11) .The different behavior observed in our Krylov sub-spaces reinforces the need to distinguish between theKrylov subspaces if we want to study the MBL phasetransition and the effect of disorder. In App. C 3, weshow some additional numerical results showing the levelspacing ratio statistics obtained when mixing the sub-spaces generated by (cid:12)(cid:12) Φ (cid:11) and (cid:12)(cid:12) Φ (cid:11) . We observe a signif-icant difference between the distribution at low-disorderand P GOE , and a smoother crossover when studying theKL-divergences.
V. ENTANGLEMENT ENTROPY IN ACONSTRAINED MODEL
In the previous Section, we have seen that the differ-ent Krylov subspaces appear to undergo an MBL phasetransition at different critical disorder strengths, accord-ing to their level spacing ratio distributions. We nowturn to the study of the von-Neumann entanglement en-tropy (vNEE) of the many-body eigenstates as anothercomplementary probe of this transition. For a subsystem A , the vNEE of the pure state | Ψ (cid:105) is given by: S ( A ) = − Tr( ρ A log ρ A ) with ρ A = Tr A | Ψ (cid:105) (cid:104) Ψ | , (36)where Tr A marks the trace on the degrees of freedom notin A . We denote by H A (resp. H A ) the Hilbert spaceof ρ A (resp. ρ A ). In terms of the entanglement entropy,the MBL phase transition can be seen as a transitionfrom thermal volume-law to an area-law . In onedimension, the volume law is to be understood as S ( A ) ∝ s th l A for l A (cid:28) L (37)with l A the number of sites (degrees of freedom) in A . s th takes the value log 2 in the thermal phase for conven-tional spin- systems. The strongly constrained modelwe study sees very irregular growth of the Hilbert space H A with subsystem size. Instead, we consider the entan-glement entropy to be ergodic if it verifies: S ( A ) ≈ S Page ( A ) , (38)where the Page entropy S Page is the average entangle-ment entropy of uniformly distributed random states. Inthe absence of symmetries or of Hilbert space fragmen-tation, the Page entropy S Page ( A ) is given by S Page ( A ) ≈ log m − m M for 1 ≤ m < M, (39)with m = min(dim H A , dim H A ) , (40) M = max(dim H A , dim H A ) . (41)The Page entropy trivially satisfies the volume law aslog m is roughly proportional to the number of degreesof freedom in A . Due to the presence of the multiple U (1)symmetries, we have to take into account the splitting ofthe wave functions down to submatrices in different sym-metry subsectors. Correspondingly, the Hilbert space (orKrylov subspace) can be split into: H = (cid:77) j H A ,j ⊗ H A ,j (42)where the subspaces H A ,j and H A ,j have dimension m j = min(dim H A ,j , dim H A ,j ) , (43) M j = max(dim H A ,j , dim H A ,j ) , (44)such that dim H = (cid:80) j m j M j . The Page entropy is thengiven by S Page ( A ) = (cid:88) j (cid:20) m j M j dim H (cid:18) log m j − m j M j (cid:19) − m j M j dim H log m j M j dim H (cid:21) (45)The area-law remains here defined as S ( A ) = O (1) when l A , L → + ∞ (46)We compute the entanglement entropy in the differentKrylov spaces introduced in Secs. III B and III C, andaverage over all states with ε ∈ [0 . , .
6] and over alarge number of disorder realizations (see Sec. IV B).We work in the original spin basis ( | (cid:105) , | (cid:105) ). We assumePBC for (cid:12)(cid:12) Φ m (cid:11) and (cid:12)(cid:12) Φ m (cid:11) , and OBC for (cid:12)(cid:12) Ψ n (cid:11) due tothe favorable scalings. In Fig. 3, we show the scalingof the vNEE S ( l A ) as a function of the subsystem size,where A is the segment made of the l A consecutivespins [[1 , l A ]] for different disorder values for (cid:12)(cid:12) Φ (cid:11) , (cid:12)(cid:12) Φ (cid:11) and (cid:12)(cid:12) Ψ (cid:11) . The entropy we obtain therefore matches theone we would obtain in the pseudo-spins basis when l A is even. At low disorder values, the vNEE remainsroughly proportional to the Page entropy, following theaforementioned volume law. The entropy varies onlyweakly with the disorder strength. At stronger disorder,we observe a crossover towards an area law where theentanglement remains (nearly) constant over severaldecades. This area law is typical of the predicted MBLphase and shows no signs of increasing again at largerscales.The exact pattern followed by the vNEE depends onthe Krylov subspaces, and can be very irregular. Inparticular, the family | Φ m (cid:105) , for the cut we chose, has S (3 l + 1) = S (3 l + 2) = S (3 l + 3) for l ≥
1. It is nota consequence of any effective three-spin quasi-particlesbut a non-trivial interplay between the pair-hoppingterms and the chosen starting state (see App. B). Additionally, as can be seen in Fig. 3a, in the ergodicphase, the growth of the entanglement entropy fromone plateau to the next alternates between large andsmall jumps. This irregular growth pattern comes fromthe lack of translation invariance at the single spinlevel in the starting generating state (while it remainsinvariant by translation of 12 spins). The dimensionof the reduced density matrix grows faster when l A goes through a higher entropy jump. This irregulargrowth also affects the MBL phase. A larger growth ofthe reduced Hilbert space translates into more statesconnected by a pair-hopping term going through theentanglement cut. As the entanglement entropy at largedisorder mainly arises from local resonant pairs, thisstructure leads to the observed alternating high and lowplateaus. More details on the growth of the dimensionof the reduced density matrix and the pairing structurecan be found in App. B).To pinpoint the transition, it is convenient to studythe standard deviation of the entanglement entropy (typ-ically at the midchain point) . The transition point istaken to be at its maxima: the system can there be eitherin a thermal state with high volume-law entanglement orin a localized states with low entanglement. In Fig. 4,we show the standard deviation of the entanglement en-tropies obtained for all states with ε ∈ [0 . , .
6] and fordifferent disorder realisations for the Krylov subspace weconsidered. For all families, the larger the system, themore peaked the standard deviation is. For the familygenerated by | Φ m (cid:105) , the peaks clearly concentrate aroundthe critical disorder value W c ≈ .
75. For | Φ m (cid:105) there isno clear tendency emerging. Finally, for (cid:12)(cid:12) Ψ n (cid:11) , the effec-tive critical disorder values increase quasi-linearly withsystem size, preventing pinpointing any phase transition.The obtained values are in qualitative agreement withthose obtained considering the level spacing ratio. Dueto the limited number of sizes available in each family,we cannot perform a reliable scaling analysis. VI. DISCUSSIONS AND CONCLUSIONS
In this paper, we have provided numerical evidenceof a many-body localisation type transition within theergodic Krylov subspaces of constrained models present-ing a strong fragmentation of the Hilbert space. Dueto the slow scaling of the Hilbert space dimensions, wehave been able to study systems comprised of up to 60spins using exact diagonalization. We observe the transi-tion from an approximate linear scaling of the entangle-ment entropy at low-disorder to a clear area law oversignificantly larger scales than conventionally studied.We see no signs of a general breakdown of the many-body localization phenomenon in these large systems,though the Krylov spaces’ dimensions remain compara-ble to other models which have been studied. Withinthe same constrained model, the different Krylov sub-0 S ( l A ) a) l A log mW . . . . . S ( l A ) / S P ag e ( l A ) b) W S ( l A ) c) W . . . . . S ( l A ) / S P ag e ( l A ) d) W l A S ( l A ) e) W l A . . . . . S ( l A ) / S P ag e ( l A ) f) W FIG. 3. Entanglement entropy as function of the subsystem size for several disorder strengths W for (cid:12)(cid:12) Φ (cid:11) (a-b), (cid:12)(cid:12) Φ (cid:11) (c-d)and (cid:12)(cid:12) Ψ (cid:11) (e-f). a), c) and e): vNEE. b), d) and f): vNEE normalized by the Page entropy given in Eq. (45). For all Krylovsubspaces, we observe a clear transition between a volume-law at low-disorder, where the entropy varies little with disorderstrength. The slight dip in the middle of the chain for the normalized vNEE is characteristic of finite-size systems as m j and M j in Eqs. (43) and (44) become of the same order. It is unrelated to any breakdown of the volume law. At stronger disorder,the vNEE transitions towards an area law. In the inset in a), we display log m as defined in Eq. (40) for l A in [[1 , spaces see a transition occuring at wildly different disor-der strengths. This reinforces the importance of consid-ering separately each Krylov space to study the localiza-tion properties of systems that see such a fragmentationof the full Hilbert space: without doing so, any sign ofthe transition will be blurred towards Poissonian statis-tics. More importantly, we see no significant correlationsbetween effective critical disorder strengths and Krylovspace dimensions or system sizes, as illustrated in Fig. 5.The role of the structure of the Krylov space is there-fore key to explaining the MBL phase transition in thesemodels, and a detailed study is left for future works.The subspaces generated by the families | Φ m (cid:105) appear topresent a stable MBL phase transition in the thermody-namic limit, whether we consider the level spacing ratiodistributions or the entropy properties. On the otherhand, the subspaces generated by (cid:12)(cid:12) Ψ n (cid:11) show an approx- imately linear scaling of the critical disorder strengthswith system size, within the sizes and the numerical pre-cision we have access to. It raises the question whetherthis subspace actually always thermalizes in the thermo-dynamic limit. This is especially remarkable given thatthis subspace and the action of the constrained Hamil-tonian on it appear the closest to an effective integrableXX model given the presence of a single pair of dipolesin a sea of integrable spins. ACKNOWLEDGMENTS
We thank David Aceituno, Fabien Alet, Jeremy Ben-sadon, Marta Brzezi´nska, Vardan Kaladzhyan, NicolasLaflorencie and Nicolas Mac´e for useful discussions. N.R.is also grateful to B.A. Bernevig and S. Moudgalya for1 . . . . . . . . . ∆ S ( L / ) a) m = 2m = 3 m = 4m = 5 . . . . . . . . . . ∆ S ( L / ) b) m = 1m = 2 m = 3 .
00 0 .
25 0 .
50 0 .
75 1 .
00 1 .
25 1 . W . . . . ∆ S ( L / ) c) n = 2n = 3 n = 4 FIG. 4. Standard deviation of the mid-chain entanglemententropy for | Φ m (cid:105) (a), | Φ m (cid:105) (b) and (cid:12)(cid:12) Ψ n (cid:11) for different m and n as a function of the disorder strength W . The maximumof the standard deviation is supposed to capture the phasetransition. For | Φ m (cid:105) (a), we observe a clear and more andmore marked peak around W = 0 .
75. For | Φ m (cid:105) (b), it isharder to extract a tendency within the system size available,due to a strong finite size effect. Finally, for (cid:12)(cid:12) Ψ n (cid:11) (c), we ob-serve an approximately linear increase of the effective criticaldisorder with system sizes. In all cases, the predictions qual-itatively agree with the ones obtained from the level spacingratio statistics. Error bars are too small to be seen. collaboration on previous related works. L.H and J.B.were supported by the ERC Starting Grant No. 679722,the Roland Gustafsson’s Foundation for TheoreticalPhysics and the Karl Engvers foundation. N.R. was sup-ported by NSF through the Princeton University’s Ma-terials Research Science and Engineering Center DMR-2011750B, the DOE Grant No. DE-SC0016239, theSchmidt Fund for Innovative Research, Simons Investi-gator Grant No. 404513, the Packard Foundation, theNSF-EAGER No. DMR 1643312, NSF-MRSEC No.DMR-1420541 and DMR-2011750, ONR No. N00014-20-1-2303, Gordon and Betty Moore Foundation throughGrant GBMF8685 towards the Princeton theory pro-gram, BSF Israel US foundation No. 2018226, and thePrinceton Global Network Funds. N = L/ . . . . . W c | Φ m i| Φ m i| Ψ n i dim K . . . . . Level statisticsEntropy
FIG. 5. Critical disorder strengths obtained from the study ofthe level spacing ratio distributions and of the entanglemententropy for the different Krylov spaces we investigated in thispaper. No distinct pattern emerges relating system sizes (a)or dimensions of the Krylov subspaces (b) to the critical dis-order strengths. Error bars for entanglement are obtained byfitting randomly generated sequences with similar propertiesof our data and measuring the variation of the interpolatedmaximum.
Appendix A: Computation of the dimension of theKrylov subspaces1. Krylov subspaces generated by (cid:12)(cid:12) Ψ n (cid:11) We discuss in this Appendix the simplification of theformulas given in Eqs. (18) and (19). We start with pe-riodic boundary conditions, where the dimension of theKrylov subspace is given by:dim K ( (cid:12)(cid:12) Ψ n (cid:11) ) PBC = N ↑ (cid:88) n ↑ ,n ↓ =0 (cid:18) N ↑ − n ↑ − n ↓ N ↑ − n ↑ (cid:19)(cid:18) n ↑ + n ↓ n ↓ (cid:19) . (A1)We reorganize the double summation introducing s = n ↑ + n ↓ ,dim K ( (cid:12)(cid:12) Ψ n (cid:11) ) PBC = N ↑ (cid:88) s =0 min( N ↑ ,s ) (cid:88) n ↑ =max(0 ,s − N ↑ ) (cid:18) N ↑ − sN ↑ − n ↑ (cid:19)(cid:18) sn ↑ (cid:19) . (A2)The bounds of the sum on n ↑ can be simplified as eitherone of the binomial coefficient is 0 for n ↑ < max(0 , s − N ↑ )or n ↑ > min( N ↑ , s ). Namely, we getdim K ( (cid:12)(cid:12) Ψ n (cid:11) ) PBC = N ↑ (cid:88) s =0 N ↑ (cid:88) n ↑ =0 (cid:18) N ↑ − sN ↑ − n ↑ (cid:19)(cid:18) sn ↑ (cid:19) . (A3)From here, application of the Chu-Vandermonde identity k (cid:88) j =0 (cid:18) mj (cid:19)(cid:18) nk − j (cid:19) = (cid:18) n + mk (cid:19) (A4)leads todim K ( (cid:12)(cid:12) Ψ n (cid:11) ) PBC = N ↑ (cid:88) s =0 (cid:18) N ↑ N ↑ (cid:19) = (2 N ↑ + 1)! N ↑ ! . (A5)2 n N dim K ( (cid:12)(cid:12) Ψ n (cid:11) , H OBC ) C ( H OBC ) dim ρ OBC dim K ( (cid:12)(cid:12) Ψ n (cid:11) , H PBC ) C ( H PBC ) dim ρ PBC (cid:12)(cid:12) Ψ n (cid:11) . The columns 3 to 5 (6 to 8) arefor the OBC (PBC) case. The third and sixth columns list the dimensions of the Krylov subspaces according to Eqs. (18) and(19). The dimension of the Krylov spaces approximately grows as 2 N . The fourth and seventh columns show the connectivityof the Hamiltonian C (here defined as the ratio of the number of non-zero non-diagonal terms in the Hamiltonian over theHilbert space dimension). Finally, the fifth and eighth columns list the dimensions dim ρ OBC of the reduced density matrix fora cut exactly in the middle of the chain, i.e., separating the two +s in the generating state for different system sizes. We now turn to open boundary conditions. The di- mension of the Krylov space (denoted here d OBC for con-venience) is given by d OBC = N L ↑ (cid:88) n ↑ ,n ↓ =0 (cid:18) N L ↑ − n ↑ N L ↑ (cid:19)(cid:18) N L ↑ − n ↓ N L ↑ (cid:19)(cid:18) n ↑ + n ↓ n ↓ (cid:19) (A6)= N L ↑ (cid:88) s =0 min( N L ↑ ,s ) (cid:88) n ↑ =max(0 ,s − N L ↑ ) (cid:18) N L ↑ − n ↑ N L ↑ (cid:19)(cid:18) N L ↑ + n ↑ − sN L ↑ (cid:19)(cid:18) sn ↑ (cid:19) (A7)= N L ↑ (cid:88) s =0 N L ↑ (cid:88) n ↑ =0 (cid:18) N L ↑ − n ↑ N L ↑ (cid:19)(cid:18) N L ↑ + n ↑ − sN L ↑ (cid:19)(cid:18) sn ↑ (cid:19) (A8)= N L ↑ (cid:88) n ↑ =0 (cid:18) N L ↑ − n ↑ N L ↑ (cid:19) N ↑ (cid:88) s =0 (cid:18) N L ↑ + n ↑ − sN L ↑ (cid:19)(cid:18) sn ↑ (cid:19) (A9)Using the Chu-Vandermonde identity n (cid:88) m =0 (cid:18) mj (cid:19)(cid:18) n − mk − j (cid:19) = (cid:18) n + 1 k + 1 (cid:19) , (A10)and the fact that the term in the second sum is zero for s ≥ N L ↑ + 1, we obtain d OBC = N L ↑ (cid:88) n ↑ =0 (cid:18) N L ↑ − n ↑ N L ↑ (cid:19)(cid:18) N L ↑ + n ↑ + 1 N L ↑ (cid:19) . (A11)Now we introduce t = 2 N L ↑ − n ↑ such that d OBC = N L ↑ (cid:88) t = N ↑ (cid:18) tN L ↑ (cid:19)(cid:18) N L ↑ + 1 − tN L ↑ (cid:19) (A12)= N L ↑ (cid:88) t =0 (cid:18) tN L ↑ (cid:19)(cid:18) N L ↑ + 1 − tN L ↑ (cid:19) . (A13)Similarly, we can instead introduce ˜ t = 2 N L ↑ + n ↑ + 1 to obtain d OBC = N L ↑ +1 (cid:88) ˜ t =2 N ↑ +1 (cid:18) N L ↑ + 1 − ˜ tN L ↑ (cid:19)(cid:18) ˜ tN L ↑ (cid:19) (A14)= N L ↑ +1 (cid:88) ˜ t =2 N ↑ +1 (cid:18) ˜ tN L ↑ (cid:19)(cid:18) N L ↑ + 1 − ˜ tN L ↑ (cid:19) . (A15)This leaves us with d OBC = 12 N L ↑ +1 (cid:88) t =0 (cid:18) N L ↑ + 1 − tN L ↑ (cid:19)(cid:18) tN L ↑ (cid:19) (A16)= 12 (cid:18) N L ↑ + 22 N L ↑ + 1 (cid:19) = (cid:18) N L ↑ + 12 N L ↑ (cid:19) , (A17)where we used Eq. (A10) a second time and obtainEq. (20) in the main text. Note that as far as we know,this special identity for the triple sum of binomials is notregistered in conventional tables. It can be generalized3to d ( X, Y ) =
X,Y (cid:88) x,y =0 (cid:18) X − xX (cid:19)(cid:18) Y − yY (cid:19)(cid:18) x + yx (cid:19) = (cid:18) X + 2 Y + 1 X + Y (cid:19) , (A18)where we used, following Eq. (A11), d ( X, Y ) = X (cid:88) x =0 (cid:18) X − xX (cid:19)(cid:18) Y + x + 1 Y (cid:19) , (A19) d ( X, Y ) = Y (cid:88) y =0 (cid:18) Y − yY (cid:19)(cid:18) X + y + 1 X (cid:19) , (A20)the two changes of variables t x = 2 X − x and t y = 2 Y + y + 1, and applied Eq. (A10).
2. Krylov subspaces generated by | Φ mn (cid:105) We now turn to the asymptotic dimension scaling ofthe Krylov spaces generated by | Φ mn (cid:105) . Let us first con- sider periodic boundary conditions for simplicity. ↑ ’s canmove in between the two dipoles to their left (but notbeyond), and similarly for ↓ ’s to their right. We denote x j = 0 , ..., n and y j = 0 , ..., n the number of pseudo-spinsof the j th sequence of pseudo spins that have moved totheir left and to their right. For example, we considerthe Krylov subspace generated by (cid:12)(cid:12) Φ (cid:11) , i.e., | ↑↓↑↓ − + + − ↑↓↑↓ − + + − ↑↓↑↓ − + + −(cid:105) (A21)A typical configuration connected to this initial statelooks like |↑↑↓ − + ↑↓↑ (cid:124)(cid:123)(cid:122)(cid:125) y = 1 x = 2 + − ↓ − + ↓↑ (cid:124)(cid:123)(cid:122)(cid:125) y = 1 x = 1 + − ↑ − + ↓↓ (cid:124)(cid:123)(cid:122)(cid:125) y = 2 x = 0 + −(cid:105) . (A22)It satisfies ( x , y ) = (0 ,
1) as only one of the ↓ pseudo-spins of the first subsequence of ↑↓↑↓ has moved the rightof the first (leftmost) dipole − +, and none to the left ofthe last (rightmost) dipole + − . As can be straightfor-wardly observed, it also satisfies ( x , y ) = (2 ,
1) and( x , y ) = (1 , K ( | Φ mn (cid:105) ) PBC = n (cid:88) x ,y ,... =0 (cid:18) n − x − y n − y (cid:19)(cid:18) y + x y (cid:19)(cid:18) n − x − y n − y (cid:19) ... (cid:18) y m + x y m (cid:19) . (A23)This can be rewritten asdim K ( | Φ mn (cid:105) ) PBC = n (cid:88) x ,y =0 f ( x , y ) g m ( y , x ) , (A24)with f ( x, y ) = (cid:18) n − x − yn − y (cid:19) (A25)and g m ( y , x ) = (cid:88) x ,y ,... (cid:18) y + x y (cid:19)(cid:18) n − x − y n − y (cid:19) ... (cid:18) y m + x y m (cid:19) = n (cid:88) z =0 T ( n ) y ,z g m − ( z, x ) . (A26)where the ( n +1) × ( n +1) transfer matrix T ( n ) has entriesgiven by T ( n ) x,y = n (cid:88) z =0 (cid:18) n − y − zn − z (cid:19)(cid:18) x + zz (cid:19) = (cid:18) n + 1 + x − yn (cid:19) , (A27) and x, y = 0 , ..., n . Defining the matrices F x,y = f ( x, y )and G x,y = (cid:0) x + yx (cid:1) , we obtain the simple expression:dim K ( | Φ mn (cid:105) ) PBC = Tr (cid:18) F (cid:104) T ( n ) (cid:105) m − G (cid:19) . (A28)Thus, the dimension of the Krylov subspace scalesas ( t ( n ) ) m with t ( n ) the largest eigenvalue of T ( n ) ,as it corresponds to the translation by a single motif( ↑↓ ) n − + + − . Using the relation N = (2 n + 4) m , wereadily obtain that the dimension of K ( | Φ mn (cid:105) ) PBC scalesas ( t ( n ) ) N/ (4+2 n ) = ( t ( n ) ) L/ (8+4 n ) .Let us consider n = 1 as a concrete example. Thematrix T (1) is given by T (1) = (cid:18) (cid:19) . (A29)Its eigenvalues are t (1) ± = 3 ± √
2, and thereforedim K ( | Φ m (cid:105) ) PBC ≈ ( t (1)+ ) m = ( t (1)+ ) N = ( t (1)+ ) L (A30) ≈ . N ≈ . L . (A31)The dimension of the Hilbert space cannot be understoodfrom a simple quasi-particle picture. Indeed, there exists4no p ∈ N ∗ such that ( t (1)+ ) p is rational. The proof goesas follows: if such a p exists, then there exists ˜ p ∈ N ∗ such that ( T (1) ) ˜ p has rational eigenvalues. Eigenvaluesof ( T (1) ) ˜ p are roots of the polynomial det(( T (1) ) ˜ p − λI ),which has integer coefficients, and a leading coefficient of1. The eigenvalues are therefore real irrational integers,whose intersection with Q are integers only. They are given by12 (cid:16) (3 − √ ˜ p + (3 + 2 √ ˜ p (cid:17) ± (cid:114) (cid:16) (3 − √ ˜ p + (3 + 2 √ ˜ p (cid:17) − . (A32)The first parenthesis is trivially an integer, while thesecond term is of the form √ A −
1, with A ∈ N and A >
1. This second term is therefore never aninteger, and ( T (1) ) ˜ p can have neither integer nor rationaleigenvalues.For open boundary conditions, the Krylov subspacedimension can be obtained from a similar formula:dim K ( | Φ mn (cid:105) ) OBC = (cid:88) y ,... (cid:18) n − y n (cid:19)(cid:18) y + x y (cid:19)(cid:18) n − x − y n − y (cid:19) ... (cid:18) n − x m n (cid:19) (A33)= (cid:88) y ,x m f (0 , y ) g m ( y , x m ) f ( x m , , (A34)The dimension of the OBC Krylov subspace thereforescales as in the periodic case.In Tables III and IV, we summarize the dimensions ofthe Krylov subspace and of the reduced density matrixfor a half-chain cut. Appendix B: Scaling of the reduced density matrixand entanglement properties of | Φ m (cid:105) In the Krylov subspace built from | Φ m (cid:105) , the reduceddensity matrix follows an interesting simple pattern. Ascan be seen in Fig. 3a, each cut l = 3 j + 1, l = 3 j + 2 and l = 3 j + 3 has the same entropy for j ≥ S ( l A = 3 j + 1) = S ( l A = 3 j + 2) = S ( l A = 3 j + 3) . (B1)For OBC, this property is true already at j = 0. In thisAppendix, we show that the reduced density matricesobtained for these cuts are actually identical. It can beproven by rewriting | Φ m (cid:105) in terms of states combiningthree consecutive spins. We use the compact notation | ν ν ν (cid:105) = | ν + 2 ν + ν (cid:105) where ν j = 0 or 1. In thisnotation, | Φ m (cid:105) can be written as | (4474) m (cid:105) . The relevanttransformation rules induced by the pair-hopping termsnow read | (cid:105) ↔ | (cid:105) and | (cid:105) ↔ | (cid:105) (B2)All other configurations of | (cid:105) , | (cid:105) , | (cid:105) and | (cid:105) are can-celled by the four fermions hopping terms and preservedby the transverse field. No state containing | (cid:105) , | (cid:105) , | (cid:105) , | (cid:105) and | (cid:105) is therefore connected to | Φ m (cid:105) . The Krylovsubspace only contains (a subset of the) states builtfrom the triplets | (cid:105) , | (cid:105) , | (cid:105) and | (cid:105) .For clarity, we start with the OBC case, and discussthe PBC case later. We will consider separately the case j = 0, j = 1 and j = 2 by direct inspection. Then wewill address the generic j value. For any given startingstate, we can write the single-site density matrix of theleft most state explicitely as: ρ ( l A = 1) = α | (cid:105) (cid:104) | + β | (cid:105) (cid:104) | + ( γ | (cid:105) (cid:104) | + h.c. ) , (B3)where α, β are positive real coefficients and γ is a complexconstant. Using the transformation rules in Eq. (B2),the configurations | (cid:105) can only be mapped to | (cid:105) or | (cid:105) . This means the configuration of the first triplet iseither | (cid:105) or | (cid:105) . Once we fix the first spin, the next twoare determined. Using Eq. (B3), we obtain the followingexpressions for the reduced density matrices at l A = 2and l A = 3. ρ ( l A = 2) = α | (cid:105) (cid:104) | + β | (cid:105) (cid:104) | + ( γ | (cid:105) (cid:104) | + h.c. ) , (B4) ρ ( l A = 3) = α | (cid:105) (cid:104) | + β | (cid:105) (cid:104) | + ( γ | (cid:105) (cid:104) | + h.c. ) . (B5)The coefficients of the density matrices, and therefore thevNEE, remain the same whether we cut after the first,second or third spin. Now, we turn to a cut through thesecond triplet (corresponding to j = 1 in Eq. (B1)). The5 m N PBC dim H PBC C ( H PBC ) dim ρ PBC N OBC dim H OBC C ( H OBC ) dim ρ OBC ↑↓ − + + − ) m . The column3 to 5 (6 to 8 are for the OBC (PBC) case. Note that for OBC, we add a sequence ↑↓ at the right end of the generating statefor symmetry. The third and sixth columns list the dimensions of the Krylov subspaces. The dimension of the Krylov spacesapproximately grows as 1 . N . The fourth and seventh columns show the connectivity of the Hamiltonian C (here defined asthe ratio of the number of non-zero non-diagonal terms in the Hamiltonian over the Hilbert space dimension). Finally, the fifthand eighth columns list the dimensions dim ρ OBC of the reduced density matrix for a cut exactly in the middle of the chain.m N PBC dim H PBC C ( H PBC ) dim ρ PBC N OBC dim H OBC C ( H OBC ) dim ρ OBC ↑↓↑↓ − ++ − ) m . The column3 to 5 (6 to 8 are for the PBC (OBC) case. Note that for OBC, we add a sequence ↑↓ at the right end of the generating statefor symmetry. The third and sixth columns list the dimensions of the Krylov subspaces. The dimension of the Krylov spacesapproximately grows as 1 . N . The fourth and seventh columns show the connectivity of the Hamiltonian C (here defined asthe ratio of the number of non-zero non-diagonal terms in the Hamiltonian over the Hilbert space dimension). Finally, the fifthand eighth columns list the dimensions dim ρ OBC of the reduced density matrix for a cut exactly in the middle of the chain. triplet itself can be either | (cid:105) , | (cid:105) or | (cid:105) . On the otherhand, also taking into account the first triplet, lead tothe following three combinations | (cid:105) = | (cid:105) , | (cid:105) = | (cid:105) and | (cid:105) = | (cid:105) . (B6)Therefore, fixing the first 4 spins, i.e., the first triplet andthe first spin of the second triplet again entirely deter-mines the states of the second and third spin. Eq. (B1) istherefore also valid for j = 1. A similar reasoning can beapplied to the third triplet and j = 2, with the sequences: | (cid:105) , | (cid:105) , | (cid:105) and | (cid:105) . (B7)To straightforwardly extend the results to the rest ofthe system, it is enough to consider all possible fourtriplets sequences in the Krylov subspace. There areonly 44 such sequences (out of 2 = 4096 possible spinconfiguration and 4 = 256 combination of triplets),given in Tab. V. They can be obtained by brute force for m = 3 ( m = 2 for PBC), and the limited propagationof the dipoles through the pseudo-spins ensures thatno other configurations arise for larger systems. For allthese sequences, fixing the first three triplets and thefirst spin of the fourth triplet is enough to determine thewhole sequence. Eq. (B1) is therefore valid for any j .For PBC, the same analysis can be performed, leadingto the same property and pattern observed in Fig. 3a.The only difference is that the reduced density matrixdoes change going from the first spin to the secondspin as the left-most triplet can be | (cid:105) , | (cid:105) and | (cid:105) . Forfurther spins and triplets, the proof is similar to the one | Φ m (cid:105) we derived for OBC.Additionally, the growth of the reduced Hilbertspace from one three-site plateau to the next is veryirregular, as shown in Fig. 3a. The alternating small andlarge jumps in entropy observed in the ergodic phasetranslates into alternating low and high entanglementplateaus in the MBL phase at strong disorder. Bothphenomena can be explained by simple perturbativeexpansion arguments in the triplet language. Atstrong disorder, the dominant energy terms are thedisorder terms, and we can assume that the eigen-states are generally close to product states in the z spin-basis, and therefore in the triplet basis. Non-zerocontributions to the entropy mainly come from localresonances, with two nearest neighbours triplet forminga pair due to the corresponding hopping term. InTable VI, we summarize how many pairs of statesthe hopping terms can generate depending on wherethey are applied on the Krylov spaces generated by | Φ m (cid:105) .6 Hopping term j ↔ j + 1 0 − − − − (cid:12)(cid:12) Φ (cid:11) (cid:12)(cid:12) Φ (cid:11) (cid:12)(cid:12) Φ (cid:11)
29 70 29 70Number of pairs of states in (cid:12)(cid:12) Φ (cid:11)
169 408 169 408Number of pairs of states in (cid:12)(cid:12) Φ (cid:11)
985 2378 985 2378TABLE VI. Number of pairs of states created by the hop-ping terms connecting the triplet j (that is to say the sites3 j + 1, 3 j + 2 and 3 j + 3) and the triplet j + 1 for the Krylovspaces generated from the state | Φ m (cid:105) for different values of m , taking PBC. The remaining terms can be obtained due tothe invariance by translation of 12 sites. Due to the patternof the generating state, we alternate between large and smallnumbers of connected states. This pattern explains the al-ternating large and small increase of entropies in the ergodicphase, and the alternating high and low entropy links in theMBL phase seen in Fig. 3a. We treat as an example the case m = 1. With peri-odic boundary conditions, the Krylov subspace consistsof only 6 states: | i (cid:105) ≡ | (cid:105) , | ii (cid:105) ≡ | (cid:105) , | iii (cid:105) ≡ | (cid:105) , | iv (cid:105) ≡ | (cid:105) , | v (cid:105) ≡ | (cid:105) and | vi (cid:105) ≡ | (cid:105) . The pair-hopping terms linking the first two triplets only trans-form the state | i (cid:105) into | ii (cid:105) (and vice versa). Similarly,those connecting the 3rd and 4th triplets only transform | v (cid:105) into | vi (cid:105) . On the other hand, the pair-hopping termsconnecting the 2nd and the 3rd triplet map | i (cid:105) into | iii (cid:105) and | iv (cid:105) into | vi (cid:105) . The alternating low and high numberof pairs perfectly explain the observed entropy patterns.If this number is small, the average entropy in the MBLphase is lower as a limited number of local resonant pairscan exist. Conversely, at low disorder, the number ofpairs reflect the number of connections in the configura-tion basis, that is to say the growth of the reduced Hilbertspace when increasing system size. A lower number ofpairs implies that the subspace grows less and thereforethat the entropy increases less in the ergodic phase. Appendix C: Additional numerical data
In this Appendix, we present briefly some additionalnumerical results mentioned in the main text.
1. Mean level spacing
We first turn towards the computation of the meanvalue of the energy level ratio. The mean level ratio issimply defined as the average of the distribution intro-duced in Eq. (27). It is a good indicator of the MBLphase transition as it crosses from 0 . . . On theother hand, it generally shows a significant shift with sys-tem size and captures only partially the behavior of thedistributions through the phase transition. In Fig. 6, we represent the mean level ratio for the three Krylov sub-spaces we consider. We observe in all cases, a crossoverfrom GOE statistics to Poisson statistics. The behaviorin each family is qualitatively different, with a sharpertransition for | Φ m (cid:105) , significant changes with system sizefor | Φ m (cid:105) (compared to the other two Krylov subspaces),and a slower transition for (cid:12)(cid:12) Ψ n (cid:11) . We also observe a sig-nificant shift of the transition point with system size forthe family generated by (cid:12)(cid:12) Ψ n (cid:11) . These results are consis-tent with those obtained by studying the KL-divergencein the main text. . . . . . . . . < ˜ r > a) PoissonGOEm = 2 m = 3m = 4m = 5 . . . . . . . . < ˜ r > b) PoissonGOEm = 1 m = 2m = 3 .
00 0 .
25 0 .
50 0 .
75 1 .
00 1 .
25 1 . W . . . < ˜ r > c) PoissonGOEn = 2 n = 3n = 4
FIG. 6. Mean value of the level spacing ratio for the threefamilies | Φ m (cid:105) , | Φ m (cid:105) and (cid:12)(cid:12) Ψ n (cid:11) for different values of m and n . We generically observe a crossover from the GOE valuetowards the Poisson value. The observed behavior is compat-ible with the results based on the KL-divergence in the maintext.
2. Mobility edge
As was observed in similar models, a MBL tran-sition typically presents a mobility edge, that is to saythat the critical disorder strength depends on the en-ergy of the eigenstates. The same mobility edge can alsobe observed in our constrained model, as illustrated inFig. 7. We compute the level spacing ratio distributionof states of normalized energies ε ∈ [ ε t − . , ε t + 0 . ε t varying from 0 .
075 to 0 . . . . . . W c . . . . ε t | Φ i| Φ i| Ψ i FIG. 7. Estimated critical disorder strengths W c for the threefamilies | Φ m (cid:105) (a), | Φ m (cid:105) (b) and (cid:12)(cid:12) Ψ n (cid:11) (c) for different valuesof m and n , as a function of the normalized energy. We com-pute the distribution of level ratio of states with normalizedenergies ε ∈ [ ε t − . , ε t + 0 . ε t . We es-timate the critical disorder strength using the KL-divergenceas shown in Sec. IV B. For all three families, we observe asignificant variation of the transition point with energy level.Error bars are obtained through subsampling of our data.
3. Level spacing ratio for two mixed Krylovsubspaces
Finally we study the level spacing ratio statistics ob-tained from mixing two Krylov subspaces seeing a transi-tion at different disorder strengths. More precisely here,we consider the two Krylov subspaces generated by (cid:12)(cid:12) Φ (cid:11) (of dimension 1154) and (cid:12)(cid:12) Φ (cid:11) (of dimension 21873), takenwith periodic boundary conditions with N = 24. Wecompute the level spacing ratio distributions by mixingthe spectra obtained in the two spaces for the same dis-order realisations. In Fig. 8, we show our results foreigenstates with ε ∈ [0 . , . D KL ( P num , P Poi ), with a sat- .
00 0 .
25 0 .
50 0 .
75 1 .
00 1 .
25 1 . W . . . . D K L t o t h e o r e t i c a l b) PoissonGOE . . . . . . r . . . . . P ( ˜ r ) a) PoissonGOEW = 0.0W = 0.3 W = 0.6W = 0.9W = 1.2W = 1.5
FIG. 8. a) Distribution of the level spacing ratio when mix-ing the Krylov subspaces generated by (cid:12)(cid:12) Φ (cid:11) and (cid:12)(cid:12) Φ (cid:11) , corre-sponding to N = 24. We observe a significant divergence fromthe ergodic GOE distribution at zero disoder, which convergesslowly towards the Poisson distribution at higher disorder. b)KL-divergence of that distribution with the two reference dis-tributions. The red lines mark the divergences between P num and P Poi ). The discrepancy at low disorder is better seen in D KL ( P num , P Poi ) than in D KL ( P num , P GOE ). The phase tran-sition is not as sharp than in Sec. IV B (see Fig. 2). Errorbars are too small to be seen. uration value significantly lower than D KL ( P GOE , P
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Quantum Signatures of Chaos (Springer BerlinHeidelberg, 2010). V. Oganesyan and D. A. Huse, “Localization of interactingfermions at high temperature,” Phys. Rev. B , 155111(2007). Y. Y. Atas, E. Bogomolny, O. Giraud, and G. Roux,“Distribution of the ratio of consecutive level spacings inrandom matrix ensembles,” Phys. Rev. Lett. , 084101(2013). Y. Y. Atas, E. Bogomolny, O. Giraud, P. Vivo, andE. Vivo, “Joint probability densities of level spacing ratiosin random matrices,” Journal of Physics A: Mathematicaland Theoretical , 355204 (2013). F. Sun, Y. Yi-Xiang, J. Ye, and W.-M. Liu, “Classifi-cation of the quantum chaos in colored sachdev-ye-kitaevmodels,” Phys. Rev. D , 026009 (2020). O. Giraud, N. Mac´e, E. Vernier, and F. Alet, “Probingsymmetries of quantum many-body systems through gapratio statistics,” arXiv:2008.11173. D. N. Page, “Average entropy of a subsystem,” Phys. Rev.Lett. , 1291–1294 (1993). In principle, due to constraints not directly taken intoaccount by the symmetries, it is possible that dim H < (cid:80) jj m jj
Quantum Signatures of Chaos (Springer BerlinHeidelberg, 2010). V. Oganesyan and D. A. Huse, “Localization of interactingfermions at high temperature,” Phys. Rev. B , 155111(2007). Y. Y. Atas, E. Bogomolny, O. Giraud, and G. Roux,“Distribution of the ratio of consecutive level spacings inrandom matrix ensembles,” Phys. Rev. Lett. , 084101(2013). Y. Y. Atas, E. Bogomolny, O. Giraud, P. Vivo, andE. Vivo, “Joint probability densities of level spacing ratiosin random matrices,” Journal of Physics A: Mathematicaland Theoretical , 355204 (2013). F. Sun, Y. Yi-Xiang, J. Ye, and W.-M. Liu, “Classifi-cation of the quantum chaos in colored sachdev-ye-kitaevmodels,” Phys. Rev. D , 026009 (2020). O. Giraud, N. Mac´e, E. Vernier, and F. Alet, “Probingsymmetries of quantum many-body systems through gapratio statistics,” arXiv:2008.11173. D. N. Page, “Average entropy of a subsystem,” Phys. Rev.Lett. , 1291–1294 (1993). In principle, due to constraints not directly taken intoaccount by the symmetries, it is possible that dim H < (cid:80) jj m jj M jj