Many-body localization characterized from a one-particle perspective
Soumya Bera, Henning Schomerus, Fabian Heidrich-Meisner, Jens H. Bardarson
MMany-body localization characterized from a one-particle perspective
Soumya Bera, Henning Schomerus,
2, 1
Fabian Heidrich-Meisner, and Jens H. Bardarson Max-Planck-Institut f¨ur Physik komplexer Systeme, 01187 Dresden, Germany Department of Physics, Lancaster University, LA1 4YB Lancaster, United Kingdom Department of Physics and Arnold Sommerfeld Center for Theoretical Physics,Ludwig-Maximilians-Universit¨at M¨unchen, 80333 M¨unchen, Germany (Dated: July 6, 2015)We show that the one-particle density matrix ρ can be used to characterize the interaction-drivenmany-body localization transition in closed fermionic systems. The natural orbitals (the eigenstatesof ρ ) are localized in the many-body localized phase and spread out when one enters the delocalizedphase, while the occupation spectrum (the set of eigenvalues of ρ ) reveals the distinctive Fock-space structure of the many-body eigenstates, exhibiting a step-like discontinuity in the localizedphase. The associated one-particle occupation entropy is small in the localized phase and large in thedelocalized phase, with diverging fluctuations at the transition. We analyze the inverse participationratio of the natural orbitals and find that it is independent of system size in the localized phase. PACS numbers: 72.15.Rn 05.30.Rt 05.30.Fk
Introduction.
While the theory of noninteracting dis-ordered systems is well developed [1, 2], the possibility ofa localization transition in closed interacting systems hasonly recently been firmly established [3–23]. This many-body localization (MBL) transition occurs at finite en-ergy densities and is not a conventional thermodynamictransition [24, 25]. Instead, it can be understood as adynamical phase transition, associated with the emer-gence of a complete set of local conserved quantities inthe localized phase, which thus behaves as an integrablesystem [26–30]. This restricts the entanglement entropyof the eigenstates to an area law [31], in contrast to thevolume law predicted by the eigenstate thermalizationhypothesis for the ergodic delocalized phase [32–34]. Atthe localization transition, the fluctuations of the entan-glement entropy diverge [16, 35]. The effects of MBLare also observed in the dynamics following, for example,a global quench from a product state, wherein dephas-ing between the effective degrees of freedom leads to acharacteristic logarithmic growth of the entanglement en-tropy [6, 10, 12]. These features comprise a much richerset of signatures than in the context of noninteractingsystems, for which, in the spirit of one-parameter scaling,the notion of a localization length based on single-particlewave functions generally suffices [1, 2].In view of the rich phenomenology of many-body local-ization it is natural to ask, both from a fundamental pointof view as well as for the interpretation of experimentaldata [36, 37], to which extent (if at all) the MBL tran-sition can be detected and characterized from a single-particle perspective. Here we show that a rather com-plete characterization of many-body localization is in-deed possible based on the eigenvalues (occupations) andeigenstates (natural orbitals) of the one-particle densitymatrix. The one-particle density matrix was originallyintroduced by Onsager and Penrose to extend the notionof a Bose-Einstein condensate to interacting systems [38]. Importantly, the natural orbitals take a Bloch form intranslationally invariant systems, providing a true many-body generalization of the Bloch theorem [39]. This nat-urally suggests studying the effect of disorder, as in recentstudies of localization of thermalized hard-core bosons ina quasi-periodic potential [40, 41]. However, so far noconnection to many-body localization has been made.We are further motivated to consider the one-particledensity matrix because it naturally focusses on the dy-namics of one particle in the presence of all the others,without the need to resort to a mean-field theory or tosacrifice particle indistinguishability. As we will see, thisperspective retains sufficient information to capture thegenuine many-body aspects that set many-body local-ization apart from Anderson localization transitions innoninteracting systems. In particular, the occupationssharply reorganize themselves from being close to eitherzero or one in the localized phase to being in betweenthese extremal values in the delocalized phase, thus re-flecting a delocalization transition in Fock space that cor-responds to a mixing of product states. It follows thatin the localized phase the occupation spectrum devel-ops a step-like discontinuity, similar to a Fermi-liquid.The associated one-particle occupation entropy is largeand proportional to the system size in the delocalizedphase, corresponding to the volume law of thermal states,while in the localized phase it is small. The dynamics ofone particle in the effective bath of the others thus pro-vides complementary information to the dynamics of aspatially confined region in the effective bath of its sur-rounding. In addition, we show that the transition leavesdirect signatures in the natural orbitals, which are local-ized in the many-body localized phase and spread outover the system when one enters the delocalized phase.We show that the inverse participation ratio (IPR) of thenatural orbitals depends on the system size in the delo-calized phase, while it is independent of system size in a r X i v : . [ c ond - m a t . s t r- e l ] J u l the localized phase. Model and method.
We consider spinless fermions inone dimension with a nearest-neighbor repulsion and di-agonal disorder, described by the Hamiltonian H = t L (cid:88) i =1 (cid:20) −
12 ( c † i c i +1 + h . c . ) + (cid:15) i (cid:18) n i − (cid:19) + V (cid:18) n i − (cid:19) (cid:18) n i +1 − (cid:19)(cid:21) . (1)Here c † i creates a fermion on site i = 1 , , . . . , L and n i = c † i c i is the associated number operator. Energiesare expressed in units of the nearest-neighbor hoppingconstant t , so that V is a dimensionless measure of thestrength of the nearest-neighbor repulsive interactions.The diagonal disorder is introduced via a box distribu-tion of the onsite potentials (cid:15) i ∈ [ − W, W ]. We studythis system using exact diagonalization at finite sizes L = 10 , ,
14 (10 disorder realizations), L = 16 (10 realizations) and L = 18 (500 realizations), imposing pe-riodic boundary conditions and fixing the overall occu-pation at half filling (number of particles N = L/ ε = 1 where ε = 2( E − E min ) / ( E max − E min ) with E max and E min the maximum and minimum energy foreach disorder realization, and take the 6 eigenstates clos-est to this energy. This energy corresponds to infinitetemperature in the thermodynamic limit. At the fixedinteraction strength V = 1, the critical disorder strength W c is found to take values in the range between threeand four [7, 8, 42–44].Given a many-body eigenstate | ψ n (cid:105) of the Hamilto-nian (1), the one-particle density matrix is defined as ρ ij = (cid:104) ψ n | c † i c j | ψ n (cid:105) . (2)The natural orbitals | φ α (cid:105) with α = 1 , , . . . , L , are ob-tained by diagonalizing ρ , ρ | φ α (cid:105) = n α | φ α (cid:105) , (3)which delivers a basis of single-particle states. Theeigenvalues n α are interpreted as occupations, with (cid:80) Lα =1 n α = tr ρ = N equal to the total number of par-ticles in the system. We order the natural orbitals bydescending occupation, n ≥ n ≥ . . . ≥ n L . Occupation spectrum.
In a noninteracting fermionicsystem, barring degeneracies, each many-body eigenstate | ψ n (cid:105) can be written as a Slater determinant of N single-particle states. The occupations from the one-particledensity matrix are then fixed to n α = 0 or 1, with thenatural orbitals with n α = 1 spanning the space of thesingle-particle states used in the Slater determinant. Oc-cupations departing from n α = 0 or 1 can therefore beinterpreted as a signature of the true many-body natureof the eigenstates in the interacting system and, thus, FIG. 1. (Color online) (a) The main panel shows the disorder-averaged occupation of the natural orbitals with index α fordifferent values of disorder strength (system size L = 16, in-teraction strength V = 1). The dotted line shows the oc-cupation in a noninteracting system, which is independentof the disorder strength. The vertical line indicates the ex-pected discontinuous behavior of the occupations in the in-finite system-size limit of the MBL phase. The emergenceof this discontinuity is further illustrated in the inset, whichshows the average occupations (cid:104) n N (cid:105) and (cid:104) n N +1 (cid:105) as a functionof disorder strength, for L = 16 and three values of the inter-action ( V = 0 . , , . n asa function of disorder strength and energy density (left panel)or interaction strength (right panel). (c) Distribution of oc-cupations n α in the delocalized phase ( W = 0 . , . W = 3 . W = 6 . , . L = 16 and interactionstrength V = 1. as a proxy of the delocalization of such states in Fockspace. We expect this Fock-space delocalization to bepronounced in the metallic phase, while it should be sup-pressed in the MBL phase [3, 4].In Fig. 1(a) we show the disorder-averaged occupationsfor different values of disorder for L = 16, with (cid:104) . (cid:105) denot-ing the disorder average. The horizontal dashed lines rep-resent the occupations (cid:104) n α (cid:105) = 1 for α ≤ N , (cid:104) n α (cid:105) = 0 for α > N in the noninteracting limit V = 0, where the sys-tem is Anderson localized for any finite disorder strength.The quasi-discontinuous jump ∆ n = n N − n N +1 = 1between these values is indicated by a vertical line.In the interacting system, deep in the localized phase( W = 6 , (cid:104) n α (cid:105) ≈
1, with the other half being al-most unoccupied, (cid:104) n α (cid:105) ≈
0. As one decreases the disor-der and approaches the transition ( W = 3), more orbitalsacquire a finite occupation, while for even smaller disor-der, in the delocalized phase ( W = 0 . , . (cid:104) n α (cid:105) ≈ N/L = 1 / n . A more detailed view of thisaspect is provided by the inset of Fig. 1(a), showing thedisorder dependence of the occupations (cid:104) n N (cid:105) , (cid:104) n N +1 (cid:105) forthree values of interaction strength V = 0 . , . , .
5. Inthe delocalized phase, both occupations are close to themean filling fraction, (cid:104) n α (cid:105) ≈ N/L = 1 /
2, while deep inthe localized phase they tend to their asymptotic val-ues (cid:104) n N (cid:105) = 1, (cid:104) n N +1 (cid:105) = 0 [45]. The dependence ofthe discontinuity ∆ n on energy density, shown in theleft panel of Fig. 1(b), recovers the many-body mobil-ity edge [3, 16, 44, 46], while at small and large interac-tion strengths (right panel) the delocalized phase shrinks,consistent with observations from dynamics in the samemodel [43]. According to these results, the occupationspectrum serves as a reliable indicator of many-body lo-calization. One-particle occupation entropy.
A well documentedaspect of MBL is the appearance of strong fluctuationsaround the localization-delocalization transition [16, 23,47–49]. In terms of the occupations, this is addressed inFig. 1(c), which shows the probability distribution func-tions P ( n α ) for different disorder strengths in a semi-log plot. In the large disorder limit the distributionis bimodal with peaks at n α = 0 ,
1, with very littleweight in the central region between these extremal val-ues. This bimodal distribution is characteristic of the lo-calized state, in analogy to the noninteracting scenario.As expected, close to the transition ( W = 3) the distri-bution is wide, with significant weight across the wholerange of occupations. Finally, in the delocalized phasewith low enough disorder the distribution becomes againnarrower, but now is concentrated around the filling frac-tion N/L = 1 / S = − tr ρ ln ρ = − (cid:88) α n α ln( n α ) . (4)As this entropy is determined by the occupations of the W › S fi L =10 L =12 L =14 L =16 L =18 L ln 2 V =1 . W v a r S FIG. 2. (Color online) Dependence of the disorder averagedone-particle occupation entropy (cid:104) S (cid:105) , defined in Eq. (4), onthe disorder strength, for different system sizes at interactionstrength V = 1. The dashed lines indicate the maximal value L ln 2, corresponding to the volume law for the entropy in afully delocalized system. In contrast, in the localized phasethe entropy becomes small. Inset: Variance var S of the en-tropy due to sample-to-sample fluctuations in the disorderensemble as a function of disorder strength, for different sys-tem sizes at interaction strength V = 1. The peak in thevariance indicates the location of the MBL transition. natural orbitals we call this the one-particle occupationentropy, to distinguish it from the entanglement entropyof the many-body eigenstates. The disorder-averaged en-tropy (cid:104) S (cid:105) is shown in the main panel of Fig. 2, as a func-tion of disorder strength for different system sizes. Inthe delocalized phase the entropy approaches the max-imal value L ln 2, indicated by the dashed lines. Thiscorresponds to a volume law as displayed, in general,by extensive thermodynamic properties and many-bodyeigenstates in ergodic systems. In contrast, the entropyin the localized phase is much smaller [45].In the inset of Fig. 2 we show the variance var S = (cid:104) S (cid:105) − (cid:104) S (cid:105) of the entropy as a function of disorderstrength. For the corresponding case of the entanglemententropy, it is known [16, 31] that the variance vanishesin the thermodynamic limit ( L → ∞ ) both in the lo-calized and in the delocalized phase, where in the latterphase this is consistent with the eigenstate thermaliza-tion hypothesis. Furthermore, in finite systems, the vari-ance of the entanglement entropy is sharply peaked inthe crossover regime, which is associated with the mixingand coexistence of localized and delocalized regions nearthe transition, becoming sharper with increasing systemsize [16]. This universal behavior of the entanglement en-tropy is mirrored by the one-particle occupation entropy.The occupation spectrum therefore recovers a reliable sig-nature of the MBL transition, giving quantitative accessto the locus of the transition. Delocalization of natural orbitals.
Because of theemerging degeneracy of the occupations deep in the local-ized and delocalized phase, one may wonder whether the
FIG. 3. (Color online) Evolution of the probability distribu-tion of the IPR for increasing system size (a) in the delocalizedphase ( W = 0 . W = 3 . W = 8 . (cid:104) / IPR (cid:105) = ξ of the natural orbitalsas a function of disorder strength. In the localized phase ξ is independent of the system size, while for small disorderit saturates at ξ ≈ L/
2. Inset: Average participation ratioas a function of interaction strength V for several values ofdisorder strength ( L = 16). natural orbitals themselves display any signatures of theMBL transition. As we show in the insets of Figs. 3(a)and (c), the orbitals indeed turn out to be well localizedin the MBL phase, while they are far more extended inthe delocalized phase. The multiply-peaked structure ofthe natural orbitals for weak disorder suggests that thedelocalization transition involves the formation of a chainthroughout the system via which the particle can hopresonantly. Given the complete set of natural orbitals,a measure of the localization of the occupied states canthen be derived from the inverse participation ratioIPR = 1 N L (cid:88) α =1 n α L (cid:88) i =1 | φ α ( i ) | . (5)The IPR is normalized to take the maximal value 1 fora system in which all occupied states are fully localized,while it takes the minimal value 1 /L when all occupiedstates are fully extended. In between these two extremes, the resonant-hopping picture for the delocalized phasesuggests that the IPR scales inversely with the systemsize, while in the localized phase it should be indepen-dent of system size. These tendencies are confirmed inthe main panels of Figs. 3(a-c), which show, for threedisorder strengths in the delocalized, transitional, andlocalized regime, how the probability distribution of theIPR depends on the system size. In the delocalized phase(a), the flow with system size is indicative of a 1 /L be-havior, while in the localized phase (c) the distributionis almost independent of system size, with a peak closeto the maximal value IPR = 1. Close to the transi-tion (b), the IPR distribution is wide, with no discernibletrend with system size. It is therefore suggestive to intro-duce the characteristic length ξ = (cid:104) / IPR (cid:105) . Figure 3(d)shows the disorder-strength-dependence of ξ for differentsystem sizes. In the localized regime this characteristiclength is independent of system size. With decreasingdisorder strength ξ increases, whereas at very small dis-order it approaches the value ξ ≈ L/
2. While ξ is stillsmall at the transition in the accessible system sizes, theorbitals spread out significantly once one enters into thedelocalized phase. Moreover, as shown in the inset ofFig. 3(d), ξ depends non-monotonically on V : it firstincreases as V increases, then takes a maximum at a W -dependent value and finally decreases again in the large V -limit. A similar behavior was observed in spectral fluc-tuations in this model [43]. Our quantity ξ thus capturesthe delocalizing effect of interactions, both in the delo-calized and in the MBL phase. Summary and outlook.
In conclusion, the one-particledensity matrix uncovers essential many-body aspects ofinteracting disordered fermions. Our results suggest thatin the thermodynamic limit the one-particle occupationspectrum is continuous in the delocalized phase but de-velops a finite discontinuity in the localized phase. Thecorresponding occupation entropy shares features withthe many-body entanglement entropy, one of the prin-cipal vehicles for the theoretical characterization of themany-body localization transition. The delocalization isalso observed in the structure of the natural orbitals,which is reflected in a system-size dependent inverse par-ticipation ratio. These findings support the conceptualpicture that the many-body localization transition in-volves delocalization both in Fock space and in real space.Our approach should therefore apply to a broad range ofsystems that follow this scenario, which can be furtherenriched when the particle number is not conserved. Aninteresting and timely application of our work would con-sist in analyzing the one-particle density matrix for thesystem that was experimentally realized in Ref. 37.
Acknowledgment.
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In the main text, we showed that the occupation spec-trum (cid:104) n α (cid:105) derived from the one-particle density matrixallows to distinguish the many-body localized phase fromthe delocalized phase. In the localized phase, the occu-pations tend to be close to 1 and 0, which suggests adiscontinuous dependence (finite jump ∆ n ) to persist inthe thermodynamic limit. Likewise, in the delocalizedphase, one expects that the eigenstate thermalization hy-pothesis holds, which corresponds to a smooth behaviorof the occupation spectrum in the thermodynamic limit.While the system sizes accessible in exact numerics arestill relatively small, Fig. 4 shows that these expecta-tions are consistent with our numerical results. In thedelocalized phase [panels (a)-(c)], increasing the systemsize smoothes out the occupation spectrum (closer to thetransition this happens more slowly), while in the local-ized phase [panels (d)-(f)] we maintain two branches atoccupations close to 1 or 0. In a more detailed scenariostill consistent with this data, the jump ∆ n would con-tinuously increase from 0 at the MBL transition to 1 deepin the MBL phase. (a) (b)(d) (c)(e) (f) FIG. 4. Occupation spectrum (cid:104) n α (cid:105) at fixed disorder strengths for (a) W = 0 .
4, (b) W = 0 .
8, (c) W = 2 .
0, (d) W = 3 .
0, (e) W = 6 .
0, and (f) W = 8 .
0, for different system sizes L [cf. main panel of Fig. 1(a) in the main text]. The index α is scaled sothat in the thermodynamic limit the horizontal axis runs from 0 to 1. The interaction strength is V = 1.
10 11 12 13 14 15 16 17 18 L › S fi W =0 . W =0 . W =2 . W =3 . W =6 . W =8 . FIG. 5. System-size dependence of the occupation entropy (cid:104) S (cid:105) versus L for disorder strength W = 0 . , . , . , . , . , . V = 1. Appendix B: System-size dependence of theoccupation entropy
Figure 5 shows the L -dependence of the occupation en-tropy for several values of L . The data is consistent with the overall trend discussed in the main text: S is large inthe delocalized phase and then becomes much smaller inthe many-body localized phase, approaching zero for verystrong disorder. In the delocalized phase, the entropy isproportional to L and becomes independent of disorderat large system sizes, consistent with thermalization. Forthe accessible system sizes, we observe that the entropystill increases in the localized phase, which is related tothe existence of occupations n α (cid:54) = 0 ,