Quantum ergodicity in the many-body localization problem
Felipe Monteiro, Masaki Tezuka, Alexander Altland, David A. Huse, Tobias Micklitz
QQuantum ergodicity in the many-body localization problem
Felipe Monteiro, Masaki Tezuka, Alexander Altland, David A. Huse, and T. Micklitz Centro Brasileiro de Pesquisas F´ısicas, Rua Xavier Sigaud 150, 22290-180, Rio de Janeiro, Brazil Department of Physics, Kyoto University, Kyoto 606-8502, Japan Institut f¨ur Theoretische Physik, Universit¨at zu K¨oln, Z¨ulpicher Str. 77, 50937 Cologne, Germany Department of Physics, Princeton University, Princeton, NJ 08544, USA (Dated: December 16, 2020)
Nonergodic extended states are a class of quantum states purported to form in between the phasesof weak disorder ergodicity and strong disorder localization in random many-body systems. Theirdefining feature is a combination of extendedness and incomplete coverage (‘nonergodicity’) of Fockspace. We here suggest an alternative description where non-localized many-body wave functionsremain ergodic in an orthodox sense: uniform, or thermal coverage of a shell of constant energy inFock space. We apply analytic methods of localization and information theory to describe this typeof quantum ergodicity under a mild set of assumptions. Comparison to numerical results for a SYKmodel corroborates the picture.
PACS numbers: 05.45.Mt, 72.15.Rn, 71.30.+h
Introduction:—
Complex quantum systems exposed toexternal disorder may enter a phase of strong localization.About two decades after the prediction of many-body lo-calization (MBL) [1–3], now there is still no strong con-sensus about the stability of the MBL phase and aboutpossibilities of an intermediate phase between the MBLphase and the thermal phase. One class of models wherethese questions can be explored with more analytic con-trol is confined many-body systems with long-range in-teractions. Under these conditions, the interaction op-erator couples all single-particle states, which facilitatesthe analysis. At the same time, the many-body Hilbertspace dimension is still exponentially large in the particlenumber. The interest in this system class is also moti-vated by pragmatism: Classical computers may neverbe strong enough to probe the stability and critical phe-nomena of short-range interacting MBL in the thermody-namic limit. A second motivation lies in the fascinatinglyrich physics of intermediate-scale MBL systems such ascorrelated chaotic quantum dots [4–8], small sized opticallattices [9–11], or qubit arrays [12, 13].At this stage the structure of the many-body quantumstates governing the physics even of long-range interact-ing MBL systems remains somewhat controversial. Themain question that we address in this Letter is whetherwave functions in regimes prior to the onset of strong lo-calization satisfy an ergodicity principle — in the sensethat they are uniformly (‘thermally’) distributed over ashell of constant energy in Fock space — or whether thereexists an intermediate phase of ‘nonergodically’ extended(NEE) states. This question is disputed including forsynthetic problems defined on artificial high-dimensionallattices [14–19] meant to mimic the complexity of Fockspaces.Our main conclusions obtained for the above class oflong range interacting systems are summarized as fol-lows: (1) throughout the entire delocalized phase eigen- states are ergodic and thermally extended over their en-ergy shell. However, (2) the structure of this shell isnon-trivial. It is interlaced into Fock space as a corre-lated structure whose volume diminishes with increasingdisorder. We describe this structure, and that of theeigenstates residing on it, by methods inspired by infor-mation theory.The structure of many-body wave functions | ψ (cid:105) is com-monly described in terms of their local moments |(cid:104) n | ψ (cid:105)| q in a Fock space basis {| n (cid:105)} . These quantities indicate thediminishing support of wave functions upon approach-ing a localization threshold. However, they do not re-veal the structure of the energy shells in Fock space, northe distributions of states over them. In this paper, weconsider pure state entanglement entropies as a comple-mentary diagnostic, tailored to monitor the ergodicityproperties of quantum states. For a given pure state, ρ = | ψ (cid:105)(cid:104) ψ | , these are defined as the von Neumann en-tropies, S A = tr A ( ρ A ln ρ A ) of the reduced density matrix ρ A = tr B ( | ψ (cid:105)(cid:104) ψ | ) relative to a partitioning of the sitesand the corresponding partitioning F = F A ⊗F B of Fockspace. Specifically, the entanglement entropies of puremaximally random states were calculated in the classicRef. [20]. More recent work [21] emphasizes the utility ofthe concept in the context of random matrix models serv-ing as proxies of high-dimensional localizing systems [19].In these systems, quantum interference shows in a contri-bution to the entanglement entropy proportional to theratio of subsystem Fock-space dimensions. A main find-ing of the present work is that the additional channel ofenergy-shell correlations present in microscopic systemsopens a second channel of quantum information and ex-ponentially enhances the suppression of the entanglementbelow its thermal value. In this way, the entanglementsharply distinguishes between genuine many-body wavefunctions and wave functions on generic high-dimensionalrandom lattices. a r X i v : . [ c ond - m a t . d i s - nn ] D ec In the rest of this Letter, we will compute the entangle-ment entropy of pure states prior to the onset of stronglocalization under a minimal set of assumptions. We willcompare our results to the entropies obtained for phe-nomenological models and to numerical data obtainedfor MBL in the SYK model of 2 N Majorana fermions.
Energy shell: –
We begin with a qualitative discussionof the energy shell supporting wave functions prior tolocalization. To this end, consider a prototypical many-body Hamiltonian ˆ H = ˆ H + ˆ H , where ˆ H is a one-bodyterm defined by a single-particle operator with eigenval-ues { v i } , i = 1 , . . . , N distributed over a range δ . Work-ing in the eigenbasis of ˆ H , Fock space is spanned bythe D ≡ N occupation number states n = ( n , . . . , n N ), n i = 0 , v n = (cid:80) n i v i with r.m.s. value ∆ ≡ N / δ . Individ-ual Fock states n are connected to a polynomially largenumber N α of ‘nearest neighbors’ m by the system’s in-teraction operator, ˆ H . For interaction matrix elements t nm ∼ gN − β/ , the r.m.s. eigenvalue of ˆ H scales as∆ ∼ gN ( α − β ) / , with g an N -independent coupling en-ergy for the interaction. These interactions change onlyan order-one number of occupation numbers, so | v n − v m | is of order δ and thus for large N much smaller than the‘bandwidth’ ∆ of ˆ H .In the competition of the operators ˆ H and ˆ H , Fockstate n may hybridize with nearest neighbor m due to thecoupling t nm . When the eigenstates of ˆ H are delocalizedin Fock space, this hybridization gives the local spectraldensity ν n ( E ) ≡ − π Im (cid:104) n | ( E + − ˆ H ) − | n (cid:105) , (1)a linewidth κ = κ ( v n , δ, g ) which must be self-consistentlydetermined [23]. As we are going to demonstrate in therest of the paper, the statistical properties of the ensem-ble { ν n } define both the structure of the energy shell andthe entanglement properties of quantum states defined onit.To prepare this discussion, we note that for genericvalues of the energy E (we set E = 0 for concreteness),four regimes of different disorder strength, δ , exist:(I) δ (cid:28) N − / ∆ : the characteristic disorder bandwidth δN / = ∆ (cid:28) ∆ is perturbatively small.In this regime, the spectral density, ν n ≡ ν is ap-proximately constant over energy scales ∼ ∆ .(II) N − / ∆ (cid:28) δ (cid:28) ∆ : the r.m.s. bandwidth ofˆ H exceeds that of the interaction ˆ H , but nearestneighbors remain energetically close | v n − v m | ∼ δ (cid:28) ∆ . In this regime, κ = ∆ , indicatingthat the full interaction Hamiltonian enters the hy-bridization of neighboring sites.(III) ∆ (cid:28) δ (cid:28) δ c : only a fraction ∼ (∆ /δ ) of nearest neighbors remain in resonance, and the broadeningis reduced to κ ∼ ∆ /δ .(IV) The threshold to localization, δ c , is reached whenless than one of the ∼ N α neighbors of character-istic energy separation δ falls into the broadenedenergy window. Up to corrections logarithmic in N (and neglecting potential modifications due toFock space loop amplitudes) this leads to the esti-mate δ c ∼ N α/ ∆ for the boundary to the stronglocalization regime.The energy shell in the delocalized regimes II and III isan extended cluster of resonant sites embedded in Fockspace. It owes its structure to the competition betweenthe large number O ( N α ) of nearest neighbor matrix el-ements and the detuning of statistically correlated near-est neighbor energies, v n , v m . In regime II, only a poly-nomially (in N ) small fraction κ/ ∆ II ∼ ∆ / ( δN / ) ofFock space sites lie in the resonant window defining theenergy shell, and in III this fraction is further reducedto III ∼ ∆ / ( δ N / ), before the shell fragments at theboundary to regime IV. In regimes II and III the eigen-states of H are approximately equilibrium microcanon-ical distributions over the eigenstates of H (the Fockstates) with a width κ in energy.We note that if a site, n , lies on the shell, the probabil-ity that its neighboring sites of energy v m = v n ±O ( δ ) arelikewise on-shell is parametrically enhanced compared tothat of generic sites with energy v n ± O (∆ ). It is thisprinciple which gives the energy shell of many-body sys-tems a high degree of internal correlations (missed byphenomenological approaches in terms of Fock space lat-tices with statistically independent on-site randomness).What physical quantities are sensitive to these correla-tions? And how do quantum states spread over the shellstructure? As we are going to demonstrate in the follow-ing, the pure state entanglement entropy, S A , containsthe answer to these questions. Entanglement entropy: —
Consider a Fock space(outer product) partitioning defined by n = ( l, m ) wherethe N A -bit vector l labels the states of subsystem A and the N B = N − N A (cid:29) N A vector m those of B . We are interested in the disorder averaged mo-ments M r ≡ (cid:104) tr A ( ρ rA ) (cid:105) , and the entanglement entropy S A = − ∂ r M r | r =1 of the reduced density matrix, ρ A =tr B ( | ψ (cid:105)(cid:104) ψ | ), defined by a realization-specific zero-energyeigenstate ˆ H | ψ (cid:105) = E | ψ (cid:105) with E ∼ = 0 exponentially smallin N . The bookkeeping of index configurations enteringthe moments tr A ( ρ rA ) = ψ l m ¯ ψ l m ψ l m . . . ψ l r m r ¯ ψ l m r is conveniently done in a tensor network representationas in Fig. 1. Introducing a multi-index N ≡ ( n , . . . , n r ),and analogously for N A,B , the figure indicates how theindex-data N and M carried by ψ and ¯ ψ is constrainedby the summation as M iB = N iB and M iA = N τiA , where τ i = ( i + 1)mod( r ). A further constraint, indicated by FIG. 1: Top left: graphic representation of the tensor ampli-tude ψ lm ¯ ψ m A m B . Top right: contraction of indices definingtr( ρ A ). Bottom: averaging enforces pairwise equality of in-dices n, n (cid:48) in tensor products (cid:104) . . . ψ n . . . ¯ ψ n (cid:48) . . . (cid:105) , as indicatedby red lines. Left: identity pairing of indices within the fivefactors (cid:104) tr A ( ρ A ρ A ρ A ρ A ρ A ) (cid:105) . Right: pairing of indices of thesecond and fourth factor. red lines in the bottom part of the figure, arises fromthe random phase cancellation under averaging, whichin the present notation requires N i ≡ M σi , for some permutation σ . (The figure illustrates this for the iden-tity, σ = id . and the transposition σ = (2 , M r = (cid:80) σ (cid:80) N (cid:81) i (cid:104)| ψ n i | (cid:105) δ N A ,σ ◦ τ N A δ N B ,σ N B . This ex-pression is universal in that it does not require assump-tions other than the random phase cancellation. In aless innocent final step we establish contact to the pre-viously discussed local density of states, ν n , and com-pare the two representations Dν ≡ (cid:80) α δ ( E − E α ) = (cid:80) n,α | ψ α,n | δ ( E − E α ) = (cid:80) n ν n to identify | ψ n | = ν n Dν .In other words, we identify the moduli | ψ n | of a fixedeigenstate ψ = ψ α with the realization specific local den-sity of states, ν n , at E = E α . The legitimacy of thisreplacement is backed by first principle analyses of wavefunctions in random systems [24] (see Ref. [25] for theparticular case of the SYK model). With this substitu-tion, we obtain the representation M r = (cid:88) σ (cid:88) N r (cid:89) i =1 λ n i δ N A , ( σ ◦ τ ) N A δ N B ,σ N B , λ n ≡ ν n Dν . (2)This expression describes two complementary perspec-tives of quantum states in Fock space: their support ona random energy shell defined by the coefficients λ n ∼ ν n ,and random phase cancellations implicit in the combina-torial structure. In the following, we discuss the mani-festations of these principles in the above regimes I-IV. Regime I, maximally random states: —
In this regime,the distribution of wave functions in Fock space is uni-form, ν n = ν , and the evaluation of Eq. (2) reduces toa combinatorial problem. The latter has has been ex-tensively discussed in the string theory literature [26, 27](where high-dimensional pure random states are consid-ered as proxies for black hole micro states.) For simplic- ity, we consider the case of small subsystems, where theratio D A /D B = 2 N A − N B is tiny. Inspection of the for-mula, or of the examples shown in Fig. (1), shows thatincreasing permutation complexity needs to be paid forin summation factors D B . Retaining only the leadingterm, σ = id . , and the next leading single transpositions σ = ( ij ), we obtain M r ≈ D − rA + (cid:0) r (cid:1) D − rA D − B . Differ-entiation in r then yields Page’s result [20] S A − S th = − D A D B , S th = ln D A . (3)While obtained under the above simplifying assumption,this equation actually is exact [20, 28, 29] for N A ≤ N B .(The opposite case is covered by an exchange A ↔ B .)It states that up to exponentially small corrections theentropy of the subsystem is that of a maximally random(‘thermal’) state, S th . The residual term results fromquantum interference of the wave function across systemboundaries. Reflecting a common signature of ‘interfer-ence contributions’ to physical observables, this term issuppressed relative to the leading contribution by a factorproportional to the Hilbert space dimension. Regime II & III, energy shell entanglement: —
In theseregimes, the energy shell is structured and correlationsin the local densities, { ν n } , lead to a much strongercorrection to the thermal entropy. Since these contri-butions come from the identity permutation (do not in-volve wave function interference), we ignore for the mo-ment σ (cid:54) = id . , reducing Eq. (2) to M r (cid:39) (cid:80) l λ rA,l with λ A ≡ tr B ( λ ). This expression suggests an interpretationof the unit normalized density { λ n } as a spectral measure , (cid:80) n λ n = 1, λ n ≥
0, and of λ A as the reduced density ofsystem A . With this identification, the entropy, S A ≈ S ρ ≡ tr A ( λ A ln( λ A )) (4)becomes the information entropy of that measure.This is as far as the model independent analysis goes.Further progress is contingent on two assumptions, whichwe believe should be satisfied for a wide class of systemsin their regimes II and III: First, the exponentially largenumber of sites entering the computation of the spectralmeasure justifies a self averaging assumption, (cid:88) X F ( v n X ) ≈ (cid:104) F ( v ) (cid:105) X ≡ D X √ π ∆ X (cid:90) dv X e − v X X F ( v X ) , where X = A, B, AB stands for the two subsystems, orthe full space, respectively, D X are the respective Hilbertspace dimensions, and ∆ X = δ √ N X . In other words, weapply the central limit theorem to replace the sum oversite energies by the average over the approximately Gaus-sian distributed contributions of the subsystem energiesto the total site energy. Second, when integrated againstthe distribution of subsystem energies v B , the local DoSat zero energy E (cid:39) δ -function, set-ting v = v A + v B (cid:39)
0. Since κ (cid:28) ∆ ∼ ∆ B , the detailedvalue of the width of the shell, κ , is of no significance inthis construction.Under these assumptions, we obtain the global den-sity of states as Dν = (cid:80) AB ν n ≈ D (cid:104) δ κ ( v ) (cid:105) AB = D √ πNδ ,and from there the spectral measure as ρ n (cid:39) D (cid:104) δ κ ( v l + v B ) (cid:105) B = D A ∆∆ B exp( − v A / B ). Subjecting this ex-pression to the central limit average over the A -spaceand including leading interference corrections resultingfrom single transpositions, a straightforward computa-tion yields [32] S A − S th = −
12 ln (cid:18) NN B (cid:19) + 12 N A N − (cid:114) N N A D A D B , (5)which again is exact. A number of comments on this re-sult: First, in the small subsystem limit, N A (cid:28) N , theinformation entropy S A − S th ≈ − (cid:0) N A N (cid:1) is exponen-tially enhanced compared to the correction in Eq. (3).Second, the result is independent of disorder, as long asthe above two conditions hold. This makes the entan-glement entropy (5) a universal signature of correlations (but not the volume) of the energy shell. Conversely,the wave function moments, | ψ n | q , computed under thetwo conditions describe the shrinking of the shell vol-ume (but not its correlations). To see that these areindependent features, consider the random energy model(REM) [22], a model differing from microscopic ones inthat the one-body randomness is replaced by a set ofstatistically independent Fock state potentials { v n } . In-creasing their variance leads to a diminishing of wavefunction moments similar to that in microscopic mod-els [30]. However, it is straightforward to check that theentanglement entropy of that model coincides with Page’sresult S A − S th = − D A D B . This is in agreement with recentnumerical work on the pure state entanglement entropyof sparse random states [31].A physically proper many-body model does have many“bodies”, which are the microscopic degrees of freedom.The Fock space is an outer product over the spaces ofthe degrees of freedom, and the Hamiltonian only con-tains operators that each involve only of order one differ-ent degrees of freedom. In this sense the REM is not amany-body model, since the energies of the Fock statesare given by fully nonlocal operators that act as productsover all (or most) degrees of freedom. Thus the entangle-ment entropy becomes a sensitive indicator of whether ornot the system is indeed a proper many-body system.We finally comment on how S A changes at the bound-aries I/II to the random matrix regime and III/IV to thestrongly localized regime. Approaching the I/II interface,the second condition gets compromised, i.e. the width κ of individual states ceases to be small compared to thestatistical fluctuations ∼ ∆ B . In a manner which we arenot going to discuss in analytic terms (but see below fornumerics) this collapses S A to Eq. (3). In the oppositeregime IV of the localized phase, eigenstates are concen- N A ( S A S t h ) = 0.01= 1.0 N A ( S A S t h ) FIG. 2: Entanglement entropies for a system of size N = 14in regime I, δ = 0 .
01 and III, δ = 1, respectively vs. analyticalprediction (solid). Inset: linear scale representation of thesame data. trated on a small number O (1) of Fock space states. Theentanglement entropy then scales as S A ∼ s ( δ/δ c ) N A /N ,where s is related to the entropy of the distribution of thelocalized eigenstate in Fock space. For 1 (cid:28) N A (cid:28) N , S A (cid:28) δ ∼ δ c , where it jumps to S A ∼ N A at the localization transition to regime III. Numerical analysis: —
Fig. (2) shows a comparison ofthe analytical predictions of Eqs. (3) and (5) with numer-ical results obtained for the SYK Hamiltonian [32]. Inthat case, ˆ H = (cid:80) Ni,j,k,l =1 J ijkl ˆ χ i ˆ χ j ˆ χ k ˆ χ l , where { ˆ χ l } are Majorana operators [33, 34]. The competing one-body operator reads ˆ H = (cid:80) Ni =1 v i (2 c † i c i − c i = ( ˆ χ i − + i ˆ χ i ) are complex fermion operators de-fined by the Majoranas [35, 36]. The agreement is ex-cellent and provides evidence for the robustness of theassumptions underlying the analytical result. Discussion: —
In this paper, we have taken a criti-cal view on the concept of nonergodic extendedness inrandom many-particle systems. We propose an alterna-tive scenario where the transition into a phase of stronglocalization is out of a phase of quantum states whichare almost perfectly ergodic on a shell of constant energyin Fock space. We corroborated this view for a class offermionic model Hamiltonians, ˆ H + ˆ H , defined by thefeature that the interaction, ˆ H , efficiently couples allone-body eigenstates of ˆ H .We reasoned that the description of ergodicity inmany-body systems requires two observables describingcomplementary aspects of the phenomenon, wave func-tion moments and entanglement entropies. The formerdescribe the reduction of the wave function support to anenergy shell that is a subset of states with its width in en-ergy continuously diminishing with the disorder concen-tration. The latter show the thermality of eigenstates onthese shells. We also showed how universal entanglementsignatures distinguish the energy shells of realistic many-body systems from those of synthetic high-dimensionalrandom systems.What is the scope of the above findings? Within theabove class of strong interaction coupled models, there issome freedom in the specific realization of the ˆ H eigen-states. Broadly speaking, this setup is realized in 3 typesof settings: i ) The system may not have any other geom-etry beyond that specified by the matrix elements of ˆ H ,as in a SYK model. ii ) The single-particle eigenstates ofˆ H may be localized in a d -dimensional real-space, andthe couplings in ˆ H are such that the long-range cou-plings dominate (e.g., a power-law in space that decaysslowly enough) [37]. In these first two cases, as we takethe limit of large N , the sparsity of the interactions canbe adjusted with N to set α , but we require α > iii )The single-particle eigenstates of ˆ H may be all delocal-ized in real space. In this case, even local interactionscouple all-to-all and for density-density interactions, forexample, we will have α = 4. In all three cases, as wetake the limit of large N , the strength of the interactionscan be adjusted with N to set β . Our analysis does notapply to models of MBL with only short-range interac-tions. At the same time, it does not specifically excludethis case, and it seems natural that the ergodicity pic-ture extends to it. However the corroboration of thatbelief requires further study. (For very recent work onthe entanglement entropy of extended random systems,see Ref. [21].) Acknowledgments: —
D.A.H. thanks SarangGopalakrishnan for helpful discussions. F. M andT. M. acknowledge financial support by Brazilianagencies CNPq and FAPERJ. A. A. acknowledgespartial support from the Deutsche Forschungsge-meinschaft (DFG) within the CRC network TR 183(project grant 277101999) as part of projects A03.The work of M. T. was supported in part by JSPSKAKENHI Grant Numbers JP17K17822, JP20K03787,and JP20H05270. D.A.H. is supported in part by DOEgrant DE-SC0016244. [1] B. Altshuler, Y. Gefen, A. Kamenev, and L.S. Levitov,
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Gapless spin-fluid ground state ina random quantum Heisenberg magnet , Phys. Rev. Lett. , 3339 (1993).[34] A. Kitaev, http://online.kitp.ucsb.edu/online/ entan-gled15/kitaev/ .... /kitaev2/ (Talks at KITP on April 7thand May 27th 2015).[35] A. M. Garc´ıa-Garc´ıa, B. Loureiro, A. Romero-Berm´udez,and M. Tezuka, Chaotic-Integrable Transition in theSachdev-Ye-Kitaev Model , Phys. Rev. Lett. , 241603(2018).[36] A. R. Kolovsky and D. L. Shepelyansky,
Dynamical ther-malization in isolated quantum dots and black holes , Eur.Phys. Lett. , 10003 (2017).[37] K.S. Tikhonov and A. D. Mirlin,
Many-body localizationtransition with power-law interactions: Statistics of eigen-states , Phys. Rev. B , 214405 (2018). upplementary Material to: “Quantum ergodicity in the many-body localizationproblem” Felipe Monteiro, Masaki Tezuka, Alexander Altland, David A. Huse, and T. Micklitz Centro Brasileiro de Pesquisas F´ısicas, Rua Xavier Sigaud 150, 22290-180, Rio de Janeiro, Brazil Department of Physics, Kyoto University, Kyoto 606-8502, Japan Institut f¨ur Theoretische Physik, Universit¨at zu K¨oln, Z¨ulpicher Str. 77, 50937 Cologne, Germany Department of Physics, Princeton University, Princeton, NJ 08544, USA (Dated: December 16, 2020)In this Supplemental Material, we provide details on the analytical calculation of the entanglemententropy in regimes II & III and on the numerical entanglement entropy for the SYK model.
PACS numbers: 05.45.Mt, 72.15.Rn, 71.30.+h
ENTANGLEMENT ENTROPY
Leading contribution:—
As discussed in the main text,the leading contribution to the entanglement entropycomes from the identity permutation M id r ( A ) = D A D rB (cid:104) ν rB,n A (cid:105) A ( Dν ) r , (6)where we introduced ν B,n A ≡ D B (cid:80) D B n B =1 ν n , and (cid:104) ... (cid:105) A ≡ D A (cid:80) D A n A =1 ( ... ). Building on the averaging assumption,we then find ν B,n A = (cid:90) dv e − v NBδ √ πN B δ δ κ ( v + v A ) = e − v A NBδ √ πN B δ , (7)where v X = (cid:80) N X n Xi =1 v i n Xi and v B = v for notationalsimplicity. Similarly, (cid:104) ν rB (cid:105) A = 1 √ N A (cid:18) √ πN B δ (cid:19) r (cid:114) N A N B rN A + N B . (8)and ν = (cid:90) dv e − v Nδ √ πN δ δ κ ( v ) = 1 √ πN δ . (9)Combining all contributions we arrive at M id r ( A ) = D − rA ( N/N B ) r/ (cid:112) rN A /N B . (10) Subleading contribution:—
Single transpositions σ =( ij ), give the subleading contribution to the entangle-ment entropy. Now observe that in the product of the r density of states coefficients Λ n i ∝ ν n i in Eq. (2), themajority of products is of the form ∝ ν n B n A ν n B n A withidentical n B but different n A coordinate. The summationover n A and n A in expressions containing such productsis dominated by the term n A = n A . To see this, note that for general functions F D A (cid:88) n A (cid:88) n A F ( v A , v A ) ν n B n A ν n B n A = 12 πN A δ (cid:90) dv A (cid:90) dv A e − v A δ NA e − v A δ NA ×× F ( v A , v A ) δ κ ( v A + v B ) δ κ ( v A + v B )= 12 πN A δ (cid:90) dv e − v δ NA F ( v, v ) δ κ ( v + v B ) (11)to find M σr ( A ) = (cid:18) r (cid:19) (2 πδ ) r − D − rA D B N r N − r B N A ×× (cid:90) dv e − v δ (cid:16) NA + r − NB (cid:17) . (12)Completing then the final integral, we arrive at M σr ( A ) = (cid:18) r (cid:19) D − rA D B N r/ √ N A N (2 − r ) / B (cid:112) N B + ( r − N A . (13) Remaining contributions:—
In regime I, the leading andsubleading contributions discussed above give the Pageentropy Eq. (2) in the main text [3]. That is, contri-butions from permutations that are not the identity orsingle transpositions vanish. This has been discussed indetail in the string theory literature [1, 2], and the argu-ments presented there also apply to regimes II & III. (Ba-sically, the combinatorial factor for contributions with agiven number of transpositions are the Narayana num-bers and vanish for more than one transposition in thereplica limit.) We thus conclude that, upon analyticalcontinuation and replica limit, the leading and sublead-ing terms discussed above also give the exact entangle-ment entropy in regimes II & III, Eq. (5) in the maintext.
EXACT DIAGONALIZATION
We numerically calculated the reduced density matrixand the average entanglement entropy for generic eigen-states (in the center of the band) of the SYK Hamilto-nian. Since the latter conserves fermion parity, the purestate density matrix is block diagonal, with non vanish-ing entries only in one given parity sector. Tracing out N B of the occupation states generates a block diagonalreduced density matrix ρ A , with finite entries in both par-ity blocks. Our analysis (presented in the main text) as-sumes generic systems without symmetries, which couldbe realized by adding a fermion parity breaking pertur-bation to the SYK Hamiltonian. Instead, we simply addthe two parity blocks of ρ A . One can convince oneselfthat the resulting density matrix describes a system of N − D = 2 N − , as shown (by the dashed line) in Fig. 2 ofthe main text. For our calculations, we average over 100 realizations of the Hamiltonian H (keeping H fixed).We obtain eigenstates corresponding to energies withinthe middle 1 / (cid:0) NN A (cid:1) Fock space bi-partitions. [1] Geoff Penington, Stephen H. Shenker, Douglas Stanford,Zhenbin Yang,
Replica wormholes and the black hole inte-rior , arXiv:1911.11977.[2] Hong Liu and Shreya Vardhan,
Entanglement entropiesof equilibrated pure states in quantum many-body systemsand gravity , arXiv:2008.01089.[3] D. N. Page,
Average entropy of a subsystem , Phys. Rev.Lett.71