Characterizing Many-Body Localization by Out-of-Time-Ordered Correlation
aa r X i v : . [ c ond - m a t . d i s - nn ] F e b Characterizing Many-Body Localization by Out-of-Time-Ordered Correlation
Rong-Qiang He
1, 2, ∗ and Zhong-Yi Lu † Institute for Advanced Study, Tsinghua University, Beijing 100084, China Department of Physics, Renmin University of China, Beijing 100872, China (Dated: February 14, 2017)The out-of-time-ordered (OTO) correlation is a key quantity for quantifying quantum chaoticityand has been recently used in the investigation of quantum holography. Here we use it to study andcharacterize many-body localization (MBL). We find that a long-time logarithmic variation of theOTO correlation occurs in the MBL phase but is absent in the Anderson localized and ergodic phases.We extract a localization length in the MBL phase, which depends logarithmically on interactionand diverges at a critical interaction. Furthermore, the infinite-time ‘thermal’ fluctuation of theOTO correlation is zero (finite) in the ergodic (MBL) phase and thus can be considered as anorder parameter for the ergodic-MBL transition, through which the transition can be identified andcharacterized. Specifically, the critical point and the related critical exponents can be calculated.
PACS numbers: 71.23.An, 72.15.Rn, 71.30.+h, 05.70.Jk
Recently, the proposal and studies of a simplequantum-mechanical model, known as the Sachdev-Ye-Kitaev (SYK) model [1–3], show that it has some inter-esting features shared by a black hole, and it is proposedto be a model of quantum holography. A key quantityin this context is the out-of-time-ordered (OTO) correla-tion [4, 5], which is generalized from a quantity describ-ing classical chaos and can be used to diagnose quantumchaos as well as scrambling of quantum information inblack holes. These attract a lot of attention and there arealso experimental proposals to simulate the SYK model[6] and measure this quantity in cold atom systems [7] orvia a ‘quantum clock’ [8].The SYK model is a disordered model in zero dimen-sion. The disorder plays a key role in making the modelmost chaotic. However, when disorder is introduced infinite-dimensional models, one usually finds Anderson lo-calization [9] or many-body localization (MBL) [10–15]if the system is weakly interacting. An MBL systemis effectively integrable because of the emergence of acomplete set of local integrals of motion [13, 14, 16–18],thus any chaoticity is suppressed and ergodicity is brokendown as well. This makes MBL special. Consequently,it is difficult to describe intriguing properties of MBLby conventional correlation functions and convectionalmethods, especially the critical behavior of the ergodic-MBL transition and the subtle distinction between An-derson localization and MBL.In this paper, we employ the OTO correlation to studyand characterize MBL first via phenomenological analysisand then by numerical calculations for a one-dimensionalinteracting spinless fermionic model. We find that theOTO correlation initially decreases polynomially in shorttime, then it reaches zero in an ergodic phase but remainsfinite in the noninteracting Anderson localized phase.What is interesting is that in an MBL phase the OTOcorrelation decreases logarithmically to zero in long time,reminiscent of the behavior of the logarithmic increase in time of the entanglement entropy [19–22]. A localizationlength for the MBL can be further extracted, which de-pends logarithmically on the interaction, diverges at acritical interaction, and hence predicts a transition—theergodic-MBL transition. Furthermore, we find that the‘thermal’ fluctuation of the OTO correlation at infinitetime is zero for an ergodic phase but finite for an MBLphase. Thus this fluctuation can be used as an order pa-rameter to characterize the ergodic-MBL transition, ofwhich the critical point and related critical exponentscan be calculated, for example.
Model. —For the sake of concreteness, a one-dimensional spinless fermionic model with nearest-neighbor hoppings, nearest-neighbor density-density in-teractions, and disordered on-site potentials is studied.The Hamiltonian isˆ H = − X h ij i ( c † i c j + c † j c i )+ V X h ij i (ˆ n i −
12 )(ˆ n j −
12 ) − X i µ i ˆ n i , (1)where µ i is randomly chosen in [ − w, w ] with uniform dis-tribution and ˆ n i = c † i c i . L is the number of lattice sites.A periodic boundary condition is chosen. This model canbe transformed into a random-field XXZ spin chain bythe Jordan-Wigner transformation. For
V >
0, there isan ergodic to MBL phase transition as w increases, whilethe system is Anderson localized for V = 0 and w > OTO correlation. —The OTO correlation for two com-muting/anticommuting operators ˆ W and ˆ V is defined as F ( t ) = ±h ˆ W † ( t ) ˆ V † (0) ˆ W ( t ) ˆ V (0) i , (2)where + ( − ) is chosen when ˆ W and ˆ V commute (anticom-mute) and ˆ W ( t ) = exp( iHt ) ˆ W exp( − iHt ) with ~ = 1,and the average h· · · i is taken on some ensemble de-scribed by a density operator ˆ ρ , i.e., h ˆ O i = tr ˆ ρ ˆ O . TheOTO correlation arises from a commutator square C ( t ) = 12 h [ ˆ W ( t ) , ˆ V (0)] †± [ ˆ W ( t ) , ˆ V (0)] ± i , (3)which is non-negative and is zero at t = 0. Let us call it‘OTO commutativity’ for convenience. Usually, ˆ W andˆ V are chosen to be local operators in different locations.ˆ W ( t ) spreads in space as time. C ( t ) becomes significantwhen the spread of ˆ W ( t ) reaches the location of ˆ V . Whenˆ W and ˆ V are unitary, C ( t ) = 1 − ℜ F ( t ), and this is thecase on which we focus below. For clarity of later ref-erences, when choosing ˆ W = κ i and ˆ V = κ j with κ i being an operator at site i , we denote the OTO corre-lation/commutativity as F ij ( κ ; t )/ C ij ( κ ; t ), and furtherdenote as F ( n ) ij ( κ ; t )/ C ( n ) ij ( κ ; t ) when choosing ˆ ρ = | n ih n | with | n i being an eigenstate of the system. Phenomenological analysis. —In the MBL phase, theHamiltonian (1) can be transformed into a simple formin terms of a complete set of local integrals of motion { ˆ τ i } ([ˆ τ i , ˆ τ j ] = [ˆ τ i , ˆ H ] = 0) [16, 17]:ˆ H = X i ξ i ˆ τ i + X ij V ij ˆ τ i ˆ τ j + X ijk V ijk ˆ τ i ˆ τ j ˆ τ k + · · · , (4)where ξ i , V ij , V ijk , . . . are coupling coefficients and V ’sdecays exponentially as the distance between i, j, k, . . . increases. The locality of ˆ τ i is manifested by ˆ τ i = ˆ U ˆ n i ˆ U † ,where ˆ U is a local unitary operator and diagonalizes theHamiltonian, as shown in Ref. [18]. The OTO correlation F ij ( η ; t ) can be formally obtained for this model, where η i ≡ ˆ U γ i ˆ U † with γ i ≡ c i + c † i being a Majorana fermionoperator. Note that η i and γ i are unitary and Hermitian. F ij ( η ; t ) = X n ρ n F ( n ) ij ( η ; t ) (5)= X n ρ n exp[ it ∆ τ ( n ) i ∆ τ ( n ) j ˜ V ( n ) ij ] (6)= Z ∞−∞ f ( x ) exp( itx ) dx, (7)where ˆ ρ = P n ρ n | n ih n | describes an ensemble and | n i isan eigenstate of ˆ H . F ( n ) ij ( η ; t ) = exp[ it ∆ τ ( n ) i ∆ τ ( n ) j ˜ V ( n ) ij ]. τ ( n ) i ∈ { , } is the eigenvalue of ˆ τ i on | n i . | n \ i i ≡ η i | n i is also an eigenstate of ˆ H and ∆ τ ( n ) i ≡ τ ( n \ i ) i − τ ( n ) i with | ∆ τ ( n ) i | = 1. ˜ V ( n ) ij = V ij + P k V ijk τ ( n ) k + · · · is an effectivecoupling strength between ˆ τ i and ˆ τ j for state | n i , whichis bounded and decays exponentially as r ≡ | j − i | in-creases, i.e., ˜ V ( n ) ij ∼ V r ≡ V exp( − r/ξ ), where ξ defines alocalization length. f ( x ) = P n ρ n δ ( x − ∆ τ ( n ) i ∆ τ ( n ) j ˜ V ( n ) ij )can be considered as a probability density function witha standard deviation ∼ V r .In the thermodynamic limit, f ( x ) is a continuous func-tion so that F ij ( η ; t ) decreases from 1 to 0 as time goesfrom 0 to ∞ . For example, F ij ( η ; t ) = exp( − V r t / f ( x ) ∝ exp( − x / V r ) being a normal distributionand F ij ( η ; t ) = exp( − V r t ) for f ( x ) = V r /π ( x + V r ) be-ing a Lorentz distribution. Generally, F ij ( η ; t ) may bemodeled by F ij ( η ; t ) = exp[ − ( bV r t ) α ] with b ∼ α > C ij ( η ; t ) = 1 − exp[ − ( bV r t ) α ] . (8)For small t , C ij ( η ; t ) increases polynomially as time,which is essentially different from the exponential in-crease of C ij ( η ; t ) for a quantum chaotic system. Moreinterestingly, C ij ( η ; t ) ≈ − e − + e − α ln bV r t (9)for bV r t ∼
1, i.e., at intermediate time C ij ( η ; t ) in-creases logarithmically as time. This resembles the log-arithmic growth of entanglement entropy [19–22]. Ac-tually, they share the same origin of the interaction in-duced dephasing. For C ij ( η ; t ) becoming significant, say C ij ( η ; t ) = 1 − e − , one obtains r = ξ ln bV t , implyingthat ˆ W ( t ), carrying some information, spreads logarith-mically slowly in space.In the Anderson localization case ( V = 0), one findsimmediately that ˜ V ( n ) ij = 0 [16–18] so that F ij ( η ; t ) = 1and C ij ( η ; t ) = 0, which is in sharp contrast with theMBL case where C ij ( η ; t ) increases from 0 and saturatesfinally to 1 as time increases. So the OTO correlation canbe used to distinguish MBL from Anderson localization.A conventional time-ordered correlation function, such as h η i ( t ) η j ( t ) η i (0) η j (0) i or h η i (0) η i ( t ) η j ( t ) η j (0) i , lacks thisfeature because contributions from the one-body term P i ξ i ˆ τ i in the Hamiltonian (4) cannot be canceled out.Refer to Refs. [23–25] for related discussions.The imperfection of this phenomenological analysis isthe choice of the two operators ˆ W and ˆ V in the OTOcorrelation. They are chosen to be η i and η j above. η i isquasilocalized in the MBL phase, but becomes extendedwhen the system enters the ergodic phase. This quali-tative change of η i across the transition hides in someextent the singular behavior of the transition from theOTO correlation. It may thus be difficult to identify andcharacterize the transition with this choice of the OTOcorrelation. A resolution to this difficulty is a new choiceof the OTO correlation with ˆ W = γ i and ˆ V = γ j totallylocalized, namely, F ij ( γ ; t ), which is the same choice asthat in the studies of the Sachdev-Ye-Kitaev model. Thecost is that now it is difficult to find the properties of theOTO correlation analytically. So we resort to a numericalcalculation, as presented below. Numerical results. —We use the microcanonical ensem-ble in the calculation. A relative energy ǫ ≡ ( E − E min ) / ( E max − E min ) is introduced for convenience, where E is the energy of the system and E max and E min are themaximal and minimal energies of the many-body eigenen-ergy spectrum for a specific disorder realization. ǫ = 0 . -3 -1 -9 -7 -5 -3 -1 V = 0 C r=1 r=2 r=3 r=4 r=5 V = 0.2(a) r=1 r=2 r=3 r=4 r=5V = 0 V = 0.2(b) r=1 r=2 r=3 r=4 r=5 C t FIG. 1. (Color online) The disorder averaged OTO commu-tativity C r ( γ ; t ) for V = 0 and 0 . w = 8, ǫ = 0 .
5, and L = 14. For small t , C increases polynomially as time and has no apparent dif-ference for V = 0 and V > r = 1 case). Then, C increases logarithmically as time for V >
0, while it saturatesfor V = 0. Finally, C for V > F r ( γ ; t ). calculation, the OTO correlation is averaged over a num-ber of ( ∼ ) independent disorder realizations. Afteraveraging, ij -subscripted quantities [e.g., C ij ( γ ; t )] de-pend only on r = | j − i | and we replace the subscript ij with r [e.g., C r ( γ ; t )].A comparison for the OTO commutativity C r ( γ ; t ) be-tween the Anderson localized ( V = 0) and the MBL( V > C r ( γ ; t ) shares nearly the same values (exceptfor the most beginning stage for r = 1) and increasespolynomially for small t [see Fig. 1(b)]. In this stage,the interaction does not play a role, as we can see. Theaction of γ i or γ j on an eigenstate will generate a statecomposed of a number of eigenstates. The subsequentshort-time evolution, being a local relaxation, is deter-mined by the one-body energies ξ i ∼ w in short time t < w − . The interaction energies ∼ V r ≡ V exp( − r/ξ )are small and have no effect for small t . At t ∼ w − theperturbation of γ i or γ j is locally fully relaxed.For t > w − , there is not any essential change in C r ( γ ; t ) for the noninteracting case [see Fig. 1(b)]. In -6 -4 -2 0 2 4 60.00.51.01.5 exp[ (bV r t) ] C lnbV r t r=1 r=2 r=3 r=4 r=5 FIG. 2. (Color online) Data collapse for the disorder averagedOTO commutativity C r ( γ ; t ) for different r , which can beapproximated by 1 − exp[ − ( bV r t ) α ] from Eq. (8) with V r ≡ V exp( − r/ξ ). w = 8, V = 0 . ǫ = 0 .
5, and L = 14. Fitparameters: ξ = 0 .
406 defines a localization length, α < r increases, and b = 3 . contrast, for the MBL case, the effective two-body in-teraction energy between sites i and j , ∼ V r , will causedephasing in the time evolution of many-body eigenstatesas t increases and approaches V − r = V − exp( r/ξ ), i.e.,the exponentially small effective interaction will cause arelaxation in an exponentially long time. Thus we see alogarithmic increase as time in C r ( γ ; t ) for t ∼ V − r andthen a saturation when t > V − r , consistent with Eqs. (8)and (9). A localization length ξ can be extracted by adata collapse with C r ( γ ; t ) guided by Eq. (8) as we doin Fig. 2. The model for the OTO commutativity (8)matches excellently with the numerical data, as shown inFig. 2, except for a little deviation at and before the onsetof the logarithmic increase. This deviation may be ac-counted for by the additional short-time local relaxationin C r ( γ ; t ) rather than in C r ( η ; t ).To investigate the effect of the interaction more care-fully, we have calculated C r ( γ ; t ) for several different in-teractions in the MBL phase. A good data collapse isshown in Fig. 3 when the horizontal axis t is rescaled as tV a with a = 1 + u ( r −
2) (where u > r ≥
2, implying that for the long-range cases the effec-tive two-body interactions ˜ V ( n ) ij ∼ V a exp[ − ( r − /ξ ] = V exp[ − ( r − /ξ ] with ξ − = ξ − − u ln V (10)and ξ independent of V . Therefore, we find that thelocalization length ξ depends logarithmically on V . Asexpected, ξ increases as V increases. Remarkably, ξ willdiverge as V approaches V c ≡ exp( u − ξ − ), thus anergodic-MBL phase transition is predicted. As ξ goesto infinity, the time range for the logarithmic increase of -3 -1 C tV a a = 1, 1, 1.2, 1.4, 1.6 forr = 1, 2, 3, 4, 5, respectivelyr = 1 2 3 4 5 FIG. 3. (Color online) Data collapse for the disorder averagedOTO commutativity C r ( γ ; t ) for V = 0 .
01 (open), 0 .
05 (openwith cross), and 0 .
20 (solid) in the MBL phase. w = 8, ǫ =0 .
5, and L = 14. The horizontal axis is rescaled, where a = 1for r = 1 and a = 1 + 0 . r −
2) for r = 2 , . . . , C r ( γ ; t ) will shrink to zero, which can be readily inferredfrom Eq. (8). In the ergodic phase, C r ( γ ; t ) increasespolynomially fast to 1 in short time as expected (datanot shown). The ergodic-MBL transition. —A more careful inspec-tion on the OTO correlation allows us to identify andcharacterize the ergodic-MBL transition. We define a‘thermal’ fluctuation for F ij ( κ ; t ) as∆ F ij ( κ ; t ) = sX n ρ n | F ( n ) ij ( κ ; t ) − F ij ( κ ; t ) | . (11)∆ F ij ( η ; ∞ ) = 1 because | F ( n ) ij ( η ; t ) | = 1 and F ij ( η ; ∞ ) =0. But ∆ F ij ( γ ; t ) < γ i on aneigenstate | n i will generate a state composed of a num-ber of [denote this number as N ( n ) ( γ i )] eigenstates [de-noted as {| ψ ( n ) k ( γ i ) i} with k = 1 , , . . . , N ( n ) ( γ i )] andthen the time evolution will result in dephasing and make | F ( n ) ij ( γ ; t ) | <
1. Note that ˆ U † η i ˆ U ≡ γ i and ˆ U is a localunitary transformation in the MBL phase, so N ( n ) ( γ i )is finite and | ψ ( n ) k ( γ i ) i (for different k ) differ from eachother only locally within a localization length [18], re-sulting in a limited dephasing and hence a lower butfinite thermal fluctuation for F ij ( γ ; t ). In contrast, forthe ergodic phase ˆ U is a global transformation so that N ( n ) ( γ i ) = ∞ and the dephasing will result in a com-plete destructive interference and hence F ( n ) ij ( γ ; t ) = 0at a sufficiently long time and ∆ F ij ( γ ; ∞ ) = 0. Thisanalysis is supported by the numerical result shown inFig. 4.For a fixed V , the system is in the ergodic phase for w < w c and enters the MBL phase for w > w c . As shownin Fig. 4(a), the fluctuation of C ij ( γ ; ∞ ) approaches zero C w L=8 L=10 L=12 L=14(a) L=8 L=10 L=12 L=14 C L (w w c ) L(b)
FIG. 4. (Color online) (a) Disordered averaged thermal fluc-tuation ∆ C r ( γ ; t ) [ ∝ ∆ F r ( γ ; t )] at t = ∞ , serving as an orderparameter for the transition from the ergodic phase (small w )to the MBL phase (large w ), vanishes in the ergodic phase,while it remains finite in the MBL phase as L → ∞ , which ismore evidently shown in (b) by the data collapse. V = 1 and ǫ = 0 .
5. ∆ C r ( γ ; ∞ ) ∼ | w − w c | β for L → ∞ with w c = 3 . β = 0 . ν = 1 .
2. The r = 2 case is shown, and ∆ C r ( γ ; ∞ )hardly depends on r . as the system size L increases for small w and remainsfinite for large w . A careful scaling analysis in Fig. 4(b)yields w c = 3 . V = 1, being well consistent withother results in the literature. Effectively, this fluctua-tion can be taken as an order parameter for the ergodic-MBL transition and the related critical exponents can becalculated. Furthermore, one may find the mobility edgeof the system with this quantity simply by varying ǫ [26]. Conclusion. —In summary, we have used the out-of-time-ordered (OTO) correlation/commutativity, a keyquantity for the description of quantum chaoticity andquantum holography, to study and characterize many-body localization (MBL). We find a short-time polyno-mial increase and long-time logarithmic increase of theOTO commutativity in the MBL phase but an absenceof a long-time logarithmic increase in the Anderson lo-calized and ergodic phases. The saturate value of theOTO commutativity can be reached finally in the MBLand ergodic phases but not in the Anderson localizedphase. A localization length can be extracted in theMBL phase, which depends logarithmically on the in-teraction, diverges at a critical interaction, and predictsa transition—the ergodic-MBL transition. Moreover, the‘thermal’ fluctuation of the OTO correlation at infinitetime is zero (finite) for the ergodic (MBL) phase, andthus can be considered as an order parameter for theergodic-MBL transition and can be used to identify andcharacterize the transition, of which the critical point andrelated critical exponents can be calculated.
Note added.
Recently, we became aware of a few re-lated works [27–30].This work was supported by National Natural Sci-ence Foundation of China (Grants No. 11474356 andNo. 91421304). R.Q.H. was supported by China Post-doctoral Science Foundation (Grant No. 2015T80069).Computational resources were provided by National Su-percomputer Center in Guangzhou with Tianhe-2 Super-computer and Physical Laboratory of High PerformanceComputing in Renmin University of China. ∗ [email protected] † [email protected][1] A. Kitaev, “A Simple Model of Quantum Holography,”KITP Strings Seminar and Entanglement 2015 Program(Feb. 12, April 7, and May 27, 2015) (unpublished).[2] J. Polchinski and V. Rosenhaus, J. High Energy Phys. , 1 (2016).[3] J. Maldacena and D. Stanford, Phys. Rev. D , 106002(2016).[4] A. I. Larkin and Y. N. Ovchinnikov, Sov. Phys. JETP , 1200 (1969).[5] A. Kitaev, “Hidden Correlations in the Hawking Radi-ation and Thermal Noise,” Talk given at FundamentalPhysics Prize Symposium (Nov. 10, 2014) (unpublished).[6] I. Danshita, M. Hanada, and M. Tezuka,arXiv:1606.02454 (2016). [7] B. Swingle, G. Bentsen, M. Schleier-Smith, and P. Hay-den, Phys. Rev. A , 040302 (2016).[8] G. Zhu, M. Hafezi, and T. Grover, Phys. Rev. A ,062329 (2016).[9] P. W. Anderson, Phys. Rev. , 1492 (1958).[10] D. M. Basko, I. L. Aleiner, and B. L. Altshuler, Ann.Phys. (NY) , 1126 (2006).[11] I. V. Gornyi, A. D. Mirlin, and D. G. Polyakov, Phys.Rev. Lett. , 206603 (2005).[12] R. Nandkishore and D. A. Huse, Annu. Rev. Condens.Matter Phys. , 15 (2015).[13] J. Z. Imbrie, Phys. Rev. Lett. , 027201 (2016).[14] J. Z. Imbrie, J. Stat. Phys. , 998 (2016).[15] J.-y. Choi, S. Hild, J. Zeiher, P. Schauß, A. Rubio-Abadal, T. Yefsah, V. Khemani, D. A. Huse, I. Bloch,and C. Gross, Science , 1547 (2016).[16] M. Serbyn, Z. Papi´c, and D. A. Abanin, Phys. Rev. Lett. , 127201 (2013).[17] D. A. Huse, R. Nandkishore, and V. Oganesyan, Phys.Rev. B , 174202 (2014).[18] R.-Q. He and Z.-Y. Lu, arXiv:1606.09509 (2016).[19] M. ˇZnidariˇc, T. Prosen, and P. Prelovˇsek, Phys. Rev. B , 064426 (2008).[20] J. H. Bardarson, F. Pollmann, and J. E. Moore, Phys.Rev. Lett. , 017202 (2012).[21] M. Serbyn, Z. Papi´c, and D. A. Abanin, Phys. Rev. Lett. , 260601 (2013).[22] I. H. Kim, A. Chandran, and D. A. Abanin,arXiv:1412.3073 (2014).[23] M. Serbyn, Z. Papi´c, and D. A. Abanin, Phys. Rev. B , 174302 (2014).[24] M. Serbyn, M. Knap, S. Gopalakrishnan, Z. Papi´c, N. Y.Yao, C. R. Laumann, D. A. Abanin, M. D. Lukin, andE. A. Demler, Phys. Rev. Lett. , 147204 (2014).[25] R. Vasseur, S. A. Parameswaran, and J. E. Moore, Phys.Rev. B , 140202 (2015).[26] D. J. Luitz, N. Laflorencie, and F. Alet, Phys. Rev. B91