Extremal statistics of entanglement eigenvalues can track the many-body localized to ergodic transition
EExtremal statistics of entanglement eigenvalues can track the many-body localized to ergodictransition
Abhisek Samanta, ∗ Kedar Damle, † and Rajdeep Sensarma ‡ Department of Theoretical Physics, Tata Institute of Fundamental Research, Mumbai 400005, India. (Dated: January 29, 2020)Some interacting disordered many-body systems are unable to thermalize when the quenched disorder be-comes larger than a threshold value. Although several properties of nonzero energy density eigenstates (in themiddle of the many-body spectrum) exhibit a qualitative change across this many-body localization (MBL) tran-sition, many of the commonly-used diagnostics only do so over a broad transition regime. Here, we provideevidence that the transition can be located precisely even at modest system sizes by sharply-defined changes inthe distribution of extremal eigenvalues of the reduced density matrix of subsystems. In particular, our resultssuggest that p ∗ = lim λ → ln(2) + P ( λ ) , where P ( λ ) is the probability distribution of the second lowestentanglement eigenvalue λ , behaves as an “order-parameter” for the MBL phase: p ∗ > in the MBL phase,while p ∗ = 0 in the ergodic phase with thermalization. Thus, in the MBL phase, there is a nonzero probabilitythat a subsystem is entangled with the rest of the system only via the entanglement of one subsystem qubit withdegrees of freedom outside the region. In contrast, this probability vanishes in the thermal phase. Introduction:
Strongly disordered interacting many body sys-tems in one dimension have been predicted to exhibit theabsence of transport at finite temperatures, a phenomenonknown as many-body localization (MBL) [1, 2]. Contrary tothermal or ergodic systems, where basic tenets of equilibriumstatistical mechanics hold, these systems cannot act as theirown heat bath when initialized to arbitrary initial states outof thermal equilibrium [3, 4]. Several candidate experimentalsystems [5, 6] and theoretical models [4, 7, 8] have been ar-gued to exhibit such a transition from ergodic behaviour to theMBL phase upon increasing the disorder strength [9, 10].MBL and ergodic phases can be distinguished by the prop-erties of many-body eigenstates in the middle of the spec-trum, i.e. with nonzero energy density relative to the groundstate [9, 10]. In the MBL phase, this eigenspectrum is charac-terized by energy gaps with a Poisson distribution, violationof the eigenvalue thermalization hypothesis (ETH) [11, 12],and short range (area law) entanglement entropy of subsys-tems even in eigenstates with nonzero energy density. In con-trast, ergodic systems show Wigner-Dyson gap statistics, fol-low the eigenvalue thermalization hypothesis and have longrange (volume law) entanglement entropy in the middle ofthe spectrum [13, 14]. MBL phases are also characterizedby the bimodal nature of the density of entanglement eigen-values [15], and by the logarithmic growth of entanglemententropy [16] at intermediate time scales upon evolving froman initial Fock state. While the MBL and ergodic phases areclearly distinguished by these contrasting properties, they donot provide a sharp distinction that can be used to preciselylocate the transition between the two phases at accessible sys-tem sizes. The level statistics, for example, changes behaviourover a broad transition regime, and has strong finite size ef-fects [4].The MBL-ergodic transition has been interpreted earlier [1,2] as an Anderson localization transition in Fock space: on theMBL side, the many-body eigenfunctions have support over afew basis states. Heuristically, this can be thought of as afragmentation of Fock space into small “clusters” with weak
Gap statistics (a) (b) L = L = . . . . . ˜ h ¯ r h ¯ r . . . . . FIG. 1. The average ratio of adjacent many body gaps ¯ r for therandom field Heisenberg model as a function of the disorder strengthin the magnetic field for (a) a uniform distribution of fields with width ˜ h and (b) a Gaussian distribution of standard deviation h . ¯ r goesfrom a Wigner-Dyson value of . in the weak disorder limit to aPoissonian value of . in the strong disorder limit. Note that theuniform distribution has a critical ˜ h about about twice the critical h for the Gaussian distribution. For each disorder strength, we haveused the number of samples ranging from × for L = 8 to for L = 16 . connectivity between them. In contrast, in the thermal phase,many-body eigenfunctions have support over a more exten-sive set of Fock states. In this Letter, we explore how thisdistinction is encoded in the largest few eigenvalues of the re-duced density matrix of a subsystem. We focus on the smallest few entangletment eigenvalues λ α ≡ − ln( ρ α ) , where ρ α arethe largest few eigenvalues of the subsystem’s reduced den-sity matrix corresponding to an eigenstate in the middle of thespectrum. Our key result is that the distribution of these ex-tremal entanglement eigenvalues provides us new informationabout the nature of the MBL phase, as well as a way of lo-cating the MBL-ergodic transition precisely, with weak finite-size corrections.Specifically, we propose that p ∗ = lim λ → ln(2) + P ( λ ) ,where P ( λ ) is the probability distribution of the secondlowest entanglement eigenvalue λ , behaves as an “order- a r X i v : . [ c ond - m a t . d i s - nn ] J a n parameter” for the MBL phase: p ∗ > in the MBL phase,while p ∗ = 0 in the ergodic phase with thermalization. Thus,in the MBL phase, there is a nonzero probability that a sub-system is entangled with the rest of the system only via theentanglement of one subsystem qubit with degrees of free-dom outside the region, while this probability is essentiallyzero even in the ergodic phase. Indeed, we find that P → exponentially quickly in the ergodic phase as λ approachesits lowest possible value of ln(2) . Importantly, this discrimi-nator is already sharp at relatively modest system sizes, andpinpoints the transition point within the somewhat broadertransition regime identified by more well-known diagnosticsof the transition.Additionally, we study the functional form of P ( λ ) , theprobability distribution of the smallest entanglement eigen-value λ . Although this is, by its very definition, identicalin the range λ ∈ [0 , ln(2)) to the density of entanglementeigenvalues studied earlier [15] as a probe of many-body lo-calization, we show that in this range P has a characteris-tic power law, whose exponent can also be used to track theMBL-ergodic transition. We note that this is different fromthe power law dependence of (cid:104) λ α (cid:105) with α seen in Ref [17].Finally, we also study the distributions P and P of the nexttwo lowest entanglement eigenvalues λ and λ , finding thatthey too bear a discernible signature of the same MBL to er-godic transition, although lacking precision and clarity of theorder parameter p ∗ obtained from P ( λ ) . Model and methods:
We work with spin S = 1 / degrees offreedom on a one dimensional lattice of L sites. Our Hamilto-nian is the Heisenberg model with random fields, which is thecanonical model for studying many body localization, H = L (cid:88) i =1 J (cid:126)S i .(cid:126)S i +1 + h i S zi . (1)Here (cid:126)S i = (cid:126)σ i and (cid:126)σ i are Pauli matrices at each site i .The Heisenberg coupling J is set to throughout this pa-per. h i is a random field, drawn independently for eachsite i from a Gaussian distribution of zero mean and stan-dard deviation h , which sets the strength of the disorder; i.e. P ( h i ) = (1 / √ πh ) e − h i / h . The z component of the to-tal spin S ztot is a conserved quantity in this model. We nu-merically diagonalize the Hamiltonian in the S ztot = 0 sectorto obtain the eigenvalues E n and the eigenstates | ψ n (cid:105) of theHamiltonian.This random field Heisenberg model has been studied be-fore with a uniform distribution of the fields; i.e. P ( h i ) =Θ(˜ h − h i )Θ( h i ) / ˜ h , where it undergoes an ergodic to MBLtransition at ˜ h ≈ . . To make a connection with ourmodel, we define the gap between successive eigenstates δ n = E n +1 − E n . The ratio of successive gaps, r n =Min( δ n , δ n +1 ) / Max( δ n , δ n +1 ) is then averaged over eigen-states and disorder realizations to obtain ¯ r , which interpolatesbetween its GOE value of . in the ergodic phase, to itsPoissonian value of . in the MBL phase. In Fig. 1(a) and (b) we plot ¯ r as a function of ˜ h and h for the uniform andthe Gaussian distribution respectively. From the crossing ofthe curves at largest system sizes, the transition in the uni-form distribution occurs at ˜ h ≈ . in the uniform case andat h ≈ . for the Gaussian distribution, although it showssignificant variations with the system size. The factor of be-tween the uniform and Gaussian distribution is explained bythe fact that the Gaussian distribution gives a finite probabil-ity for very large values of h i , whereas the uniform distribu-tion cuts off the possible values of h i . From Fig. 1(b), onecan clearly say that h = 0 . is deep in the ergodic phase and h = 6 . is deep in the MBL phase. These are the canonicalvalues we will use to describe the two phases. The ¯ r curveshows a region of h = 1 to h = 3 as the transition region forthe Gaussian distribution.We consider a subsystem of size L A and construct the den-sity matrix ˆ ρ ( n ) for this subsystem from each of the eigen-states | ψ n (cid:105) by tracing out degrees of freedom in the rest ofthe system. The eigenvalues of the density matrix ρ ( n ) α are re-lated to the entanglement eigenvalues λ ( n ) α = − ln ρ ( n ) α , where α = 1 , .. L A , and the eigenvalues ρ ( n ) α are arranged in de-scending order of values, i.e. ρ ( n )1 is the largest eigenvalueand so on, with the sum-rule (cid:80) α ρ ( n ) α = 1 . Consequentlythe entanglement eigenvalues are arranged in ascending or-der, i.e. λ ( n )1 is the lowest entanglement eigenvalue and soon. We tabulate the lowest four entanglement eigenvalues ob-tained from each of the eigenstates (in the middle one third ofthe spectrum) for different disorder realizations to constructthe distributions P α ( λ α ) for α = 1 , , , . As noted earlier, P in the range λ ∈ [0 , ln(2)) is identical to the density ofentanglement eigenvalues, but contains new information for λ > ln(2) .Indeed, the overall density of entanglement eigenvaluesconstructed by averaging over states in the middle of the spec-trum is affected by the trace constraint on ρ ( n ) α in a mannerthat is hard to disentangle: This density has contributionsfrom all the entanglement eigenvalues obtained from a singlestate n , in addition to contributions from other states. Con-tributions from a single state are constrained by a sum rulesince (cid:80) i ρ ( n ) α = 1 , whereas ρ ( n ) α for different n are not sim-ilarly constrained by each other’s values. This is one of ourmotivations for focusing on the distributions of the smallestfew entanglement eigenvalues, which are therefore expectedto have sharper signatures of the underlying localization phe-nomenon. The Ergodic phase:
We first consider the distribution of low-lying entanglement eigenvalues at a weak disorder of h = 0 . ,where, as seen from Fig. 1(b), the system is deep in the ergodicphase. For the delocalized eigenstates with volume law entan-glement entropy, one would expect the typical low lying en-tanglement eigenvalues to strongly depend on the system andsubsystem size. In Fig. 2(a), we plot the distribution of thelowest four entanglement eigenvalues λ ..λ for system size L = 16 and subsystem size L A = 8 . The distribution P α ( λ α ) . .
01 0 . MBL phase ln(2) -3 L = (c) . MBL phase L = ln(2) (d) Ergodic phase L = (b) h = MBL phase (e) ln(3) . MBL phase (f) ln(4) h = . P ( ) P ( ) P ( ) P ( ) P ( ) Ergodic phase (a) P ↵ ( ↵ ) ↵ FIG. 2. (a): Distribution of lowest four entanglement eigenvalues λ , ..λ in the ergodic phase ( h = 0 . ) for system size L = 16 and subsystem size L A = 8 . They all have similar skew symmet-ric structures, while the distribution becomes sharper and the peakposition increases from λ to λ . (b)-(c): Distribution of lowestentanglement eigenvalue λ for fixed subsystem size L A = 5 andsystem size L = 10 , , and 16. (b) Distribution in ergodic phase( h = 0 . ) showing the peak position increases and the distributionsharpens with increasing L for fixed L A . (c) Distribution in MBLphase ( h = 6 . ) showing power law upto ln(2) . Inset shows expo-nential distribution beyond ln(2) . The distribution is independent of L . (d): Distribution of second lowest entanglement eigenvalue λ in the MBL phase ( h = 6 . ) showing a finite value of the distri-bution at ln(2) for L A = 5 . The distribution is independent of L .(e)-(f): Distribution of (e) third lowest and (f) fourth lowest entan-glement eigenvalues for L = 16 and L A = 8 in the MBL phase( h = 4 . , . and 8.0). Note that the distributions go to zero at ln(3) and ln(4) respectively. are characterized by a sharp peak at λ pα with a skew symmet-ric tail, which is broader on the left than on the right. Empiri-cally, the large deviation function on the right (i.e. λ α > λ pα )is given by P α ( λ α ) ∼ e − (cid:104) λα − λpαlR (cid:105) , where as the tail on theleft (i.e. λ α < λ pα ) is given by P α ( λ α ) ∼ e − (cid:104) | λα − λpα | lL (cid:105) / .Fig. 2(b) shows the distribution of the lowest entanglement eigenvalue P α ( λ ) for a fixed L A = 5 for different systemsizes L = 10 , , , . The peak position scales linearlywith the size of the system, while the distribution narrowswith increasing system size in the ergodic phase of the sys-tem. Similar finite size effects are also seen in the distributionof other low lying entanglement eigenvalues [18]. The MBL phase:
We now consider the distributions of low-lying entanglement eigenvalues at a strong disorder of h =6 . , where the system is deep in the many body localizedphase. The distributions show several distinctive features inthis case.We first focus on the distribution P of the lowest entan-glement eigenvalue λ . The distribution function P ( λ ) isplotted in Fig. 2(c) on a log-log plot for a fixed subsystem size L A = 5 and different system sizes L = 10 , , , . Asexpected, deep in the localized phase, the distribution func-tion is insensitive to system size. There is a large weight inthe limit λ → , corresponding to the occurrence of productstates in the MBL phase. The distribution has a power-lawform for small λ (cid:28) ln(2) . Close to λ = 0 , this power-lawdivergence is cutoff in a characteristic manner [18]. Beyond λ = ln(2) , the distribution decays exponentially, as seen inthe semi-log plot in the inset of Fig. 2(c). This leads to a dis-tinctive kink in the distribution function at λ = ln(2) ; i.e. P [ λ ] ∼ [ λ ] − b for < λ < ln(2) ∼ e − λ /λ λ > ln(2) (2)or equivalently for the eigenvalues of the density matrix, P [ ρ ] ∼ ρ [ − ln( ρ )] b for ≤ ρ < ∼ [ ρ ] λ − > ρ > (3)Naturally, the power law behaviour we find for λ → matches the known behaviour [15] of the density of entan-glement eigenvalues in this range. As noted earlier, it canbe understood in terms of presence of local integrals of mo-tion (LIOM), whose weight decay exponentially with distancefrom its central location [17].We now turn our attention to the distribution of λ . Bydefinition, λ ∈ (ln(2) , L A ln(2)) . The distribution P ( λ ) for h = 6 . is plotted in Fig. 2(d) for a fixed L A = 5 for L = 10 , , , . At large λ (not shown in figure), it de-cays exponentially. λ = ln(2) is a special value which cor-responds to ρ = ρ = 1 / , and ρ α = 0 for α > . The manybody eigenstate which gives rise to this can be decomposed as | ψ (cid:105) = √ [ | ψ A (cid:105)| ψ B (cid:105) + | ψ A (cid:105)| ψ B (cid:105) ] , where | ψ iA ( B ) (cid:105) are respec-tively states in the Hilbert space of subsystems A and the restof the system B . In other words, exactly one subsystem qubitis maximally entangled with one degree of freedom from theenvironment.We finally consider the distribution of the rd and th low-est entanglement eigenvalues. We note that ln(3) < λ