Many-body mobility edges in a one-dimensional model of interacting fermions
aa r X i v : . [ c ond - m a t . d i s - nn ] A ug Many-body mobility edges in a one-dimensional system of interacting fermions
Sabyasachi Nag and Arti Garg
Condensed Matter Physics Division, Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Kolkata 700 064, India
We study many-body localization (MBL) in an interacting one-dimensional system with a de-terministic aperiodic potential. Below the threshold potential h < h c , the non-interacting systemhas single particle mobility edges (MEs) at ± E c while for h > h c all the single particle states arelocalized. We demonstrate that even in the presence of single particle MEs, interactions do not al-ways delocalise the system and the interacting system can have MBL. Our numerical calculation ofenergy level spacing statistics, participation ratio in the Fock space and Shannon entropy shows thatfor some regime of particle densities, even for h < h c many-body states at the top (with E > E )and the bottom of the spectrum (with E < E ) remain localized though states in the middle of thespectrum are delocalized. Variance of entanglement entropy (EE) also diverges at E , indicatinga transition from MBL to delocalized regime though transition from volume to area law scaling forEE and from thermal to non-thermal behavior occur inside the MBL regime much below E andabove E . PACS numbers: 72.15.Rn, 71.10.Fd, 72.20.Ee, 05.30.-d, 05.30.Fk, 05.30.Rt
Interplay of disorder and interactions in quantum sys-tems is a topic of great interest in condensed matterphysics. In a non-interacting system with random disor-der, any small amount of disorder is sufficient to localizeall the single particle states in one and two dimensions [1–3], except in systems where back scattering is suppressede.g. in graphene [4, 5], while in three dimensions (3-d)there occurs a single particle mobility edge (ME) leadingto a metal-Anderson Insulator transition. The questionof immense interest, that has remained unanswered fordecades, is what happens to Anderson localization whenboth disorder and interactions are present in a system.Recently based on perturbative treatment of interactionsfor the case where all the single particle states are lo-calised, it has been established that Anderson localiza-tion can survive interactions and disordered many-bodyeigenstates can be localized resulting in a many-body lo-calized (MBL) phase, provided that interactions are suf-ficiently weak [6]. The question we want to answer inthis work is what happens in the presence of interac-tions when the non-interacting system has single particleMEs? Conventional wisdom says that in the presence ofinteractions, localised states will get hybridised with theextended states resulting in delocalization. In this workbased on exact diagonalisation (ED) study of an interact-ing model of spin-less fermions in the presence of a de-terministic aperiodic potential, where the non-interactingsystem has MEs, we demonstrate that for some parame-ter regimes, many-body states at the top and the bottomof the spectrum remain localised even in the presence ofinteractions.The MBL phase and the MBL transition are uniquefor several reasons and challenge the basic foundations ofquantum statistical physics [7, 8]. In the MBL phase thesystem explores only an exponentially small fraction ofthe configuration space and local observables do not ther- malize leading to violation of eigenstate thermalisationhypothesis (ETH) [9–11]. MBL phase has been shownto have similarity with integrable systems [12, 13] withan extensive number of local integrals of motion [14, 15].Recently a lot of progress has been made in the fieldbased on numerical analysis of interacting one dimen-sional models of spin-less fermions or spins with com-pletely random disorder [16, 17, 22–24] as well as mod-els where there is no randomness but have a determin-istic (quasi-periodic) potential [18–21, 25–28] for exam-ple Aubry-Andre (AA) model [29] and models with Fi-bonacci potentials [30] which show a localization to de-localization transition even in 1-d. (a) h=1.75, ρ =0.25 ’phase_all_V_1_h=1.75’ u 1:2:3 V -3-2-1 0 1 2 3 E -3-2-1 0 1 2 3 -3-2.5-2-1.5-1-0.5 0 0.5 1 1.5 2 2.5 3-3-2.5-2-1.5-1-0.5 0 0.5 1 1.5 2 2.5 3-3-2.5-2-1.5-1-0.5 0 0.5 1 1.5 2 2.5 3-3-2.5-2-1.5-1-0.5 0 0.5 1 1.5 2 2.5 3 0.1-3-2.5-2-1.5-1-0.5 0 0.5 1 1.5 2 2.5 3 0.1 (b) h=2.10, ρ =0.5 ’phase_all_V_1_h=2.1’ u 1:2:3 V -2-1 0 1 2 FIG. 1: Phase diagram for model in Eq.1. Density plot showsthe value of η ( E ) in the thermodynamic limit. Black curvesrepresent MEs E , obtained from the level spacing statisticsand scaling of NPR and Shannon entropy, while pink, greenand sky-blue curves represent transition points from area tovolume scaling law of EE, non-thermal to thermal transitionpoints from ETH and points where variance of EE is peakedrespectively. In this work we consider a 1-d deterministic modelwhich has been explored in context of Anderson local-ization [31] and has been shown to have tunable singleparticle MEs at ± E c [32] for the strength of aperiodicpotential h < h c . Recently this model was studied withinteractions [28] and it was concluded that the modeldoes not show MBL for h < h c . We demonstrate, basedon a careful energy resolved analysis, that for a broadrange of parameters, interactions do not delocalize theentire spectrum. Main findings of our work, presented inthe phase diagram of Fig. 1, are following. For h < h c with system less than half-filled, such that in the non in-teracting system the Fermi energy E f is sufficiently be-low − E c , the interacting system has localized many-bodystates with energy density E < E and E > E while theintermediate energy states with E < E < E are delo-calized resulting in two MEs. The characteristic energyscales E and E are obtained from analysis of normal-ized participation ratio (NPR) in the Fock space, energylevel spacing statistics and Shannon entropy. To the bestof our knowledge, it is for the first time that Shannon en-tropy has been used for characterization of MBL phase.These MEs are also consistent with the transition pointsat which variance of entanglement entropy (EE) divergesin the thermodynamic limit. However, crossover fromarea to volume law scaling of EE happens at ˜ E , insidethe localised regime with a broad regime of MBL statesobeying ETH and the volume law scaling of EE speciallyfor weak interactions.The model we study has the Hamiltonian of the form H = − t X i [ c † i c i +1 + h.c. ] + X i h i n i + V X i n i n i +1 (1)Here t is the nearest neighbor hopping amplitude forspin-less fermions on a 1 − d chain, V is the nearestneighbor repulsion and h i is the on site potential of form h i = h cos(2 παi n + φ ) where α is an irrational numberand φ is an offset. Note that n = 1 for V = 0 corre-sponds to AA model which has all single particle statesdelocalized for h < t but in this work we study thismodel for n < E c = ±| t − h | for h < t [32]. For h > t , allthe single particle states are localized for any value of n .The phase diagram shown in Fig. 1 has been obtainedby solving this model using ED on finite size chains withopen boundary conditions. Below we present analysis ofvarious quantities used to obtain the phase diagram ofFig. 1, mainly for two parameters; h = 1 . t , which isless than h c , for a quarter-filled system( ρ = 0 .
25) and h = 2 . t , which is just above h c , at half-filling ( ρ = 0 . α = √ − , n = 0 . φ . Normalized Participation ratio in Fock space(NPR) : Wecalculate the NPR η ( E ) which represents the fraction ofconfiguration space participating in a many-body stateand is defined as η ( E ) = 1 h P i,n | Ψ n ( i ) | δ ( E − E n ) i C V H (2) η H (a) h=1.75, V =0.1E=-2.00E=-1.06E=0.00 η H (b) h=1.75, V =0.8E=1.06E=2.00 η H (c) h=2.1, V =0.1E=-2.00E=-0.59E=0.00 η H (d) h=2.1, V =0.8E=1.06E=2.00 FIG. 2: η ( E ) vs 1 /V H for various values of E . Here for V =0 . t , η ( E ) ∼ b ∗ V − cH for all values of E except for a few stateswith E ∼ η ( E ) ∼ a + b ∗ V − cH . On increasing V moremany-body states get delocalized as indicated in panel (b). where Ψ n ( i ) is an eigenfunction with eigenvalue E n of theHamiltonian in Eq. 1, V H is the volume of the Fock space,and hi C indicates the configuration averaging. Fig. 2shows scaling of η ( E ) w.r.t 1 /V H for h = 1 . t for al-most quarter filled system ( ρ ∼ . η ( E ) ∼ bV − cH and goes to zero inthe thermodynamic limit indicating localized nature ofthese many-body states. For states in the middle of thespectrum η ( E ) ∼ a + b (1 /V H ) c with finite value of a inthe thermodynamic limit indicating the ergodic natureof these states. From the extrapolated values of η ( E ) inthe thermodynamic limit, shown in the density plot ofFig. 1, we obtained two transitions from MBL states todelocalized states at energy densities E and E (shownin black curves in Fig. 1) such that states with E < E and E > E are localized while states with E < E < E are extended. For h = 2 . t and ρ = 0 .
5, we get a similarpicture from the analysis of NPR, shown in the bottompanel of Fig. 2 except that here at V = 0 all the many-body states are localised. r( E ) E (a) V =0.3E E L=16L=20 E (b) V =0.5E E FIG. 3: Ratio of successive gaps r ( E ) vs E for h = 1 . t and ρ = 0 .
25 for different system sizes. For
E < E and E > E , r ( E ) is close to its value for PS and does not increase with L . But for intermediate energy values r ( E ) increases with L and approaches the value for WDS. Level spacing statistics : The distribution of energylevel spacings is expected to follow Poisson statistics(PS) for localized phase while it follows Wigner-Dysonstatistics (WDS) for the ergodic phase [33]. We calcu-late the ratio of successive gaps in energy levels r n = min ( δ n ,δ n +1 ) max ( δ n ,δ n +1 ) [25] with δ n = E n +1 − E n at a given Eigenenergy E n of the Hamiltonian. For PS, the disorder aver-aged value of h r i is 2 ln − ≈ . h r i ≈ . r ( E ) is close to the PSvalue and does not change significantly with the systemsize [37] while in the middle of the spectrum, r ( E ) in-creases with the system size approaching the WDS valueindicating delocalized nature of these states. Character-istic energy scales obtained are shown in Fig. 3 which arevery close to the MEs E , obtained from NPR. Plots for r , averaged over the entire spectrum [35] show that for h ≫ h c there exists an infinite temperature MBL phasewhere all the many-body states are localised. H (a) h=1.75, ρ ~0.25,V =0.3 H (b) h=1.75, ρ ~0.25,V =1.0 f f f H (c) h=2.1, ρ =0.5,V =0.3 H (d) h=2.1, ρ =0.5,V =1.0 FIG. 4: Scaling of f , f and f vs 1 /V H . Both for V =0 . t and V = 1 . t , f and f vanish in the thermodynamiclimit indicating localised nature of states while f stays finiteindicating the delocalized nature of states in the intermediateenergy range. Shannon Entropy : To further analyze the ergodicity ofmany-body states, we calculate Shannon entropy for ev-ery eigenstate S ( E n ) = − P V H i =1 | Ψ n ( i ) | ln | Ψ n ( i ) | . Fora many-body state which gets contribution from all thebasis states in the Fock space S ( E n ) ∼ ln( V H ) and thus f ( E n ) = exp( S ( E n )) /V H ∼ N l of the basis states, f ( E n ) ∼ N l /V H and vanishesto zero in the thermodynamic limit. Fig. 4 shows thescaling of f , , which are obtained by averaging f ( E n )over three regions of the spectrum, namely, E n < E , E < E n < E and E n > E obtained from NPR anal-ysis. In the thermodynamic limti, f is finite while f and f vanish [35] indicating the transition from MBL todelocalized states across the spectrum. Comparison of h < h c quarter-filled and half-filledcase : So far we presented results for h < h c quarter-filledcase. Now consider the half-filled system ( ρ = 0 .
5) for p V (a) E FIG. 5: Left panel shows p vs V for various values of h and ρ . Note that for h = 1 . t, ρ = 0 . p is finite while for h = 1 . t and ρ = 0 . p is vanishingly small. For h > h c ,even at half-filling there is a finite fraction of states in thelocalized sector. Right panel shows p , fraction of states inthe delocalized part of the spectrum. h < h c where for the non-interacting case − E c < E f 25. This indicates that there is no MBL phasefor h < h c half-filled system which is consistent with ear-lier results [28] while for ρ = 0 . 25 case presented above,a finite fraction of many body spectrum is localized. Return Probability : Next we calculate the probabilityof return of particles to their initial position i as a func-tion of time C i ( τ ) = V H P m h Ψ m | ( n i ( τ ) − )( n i (0) − ) | Ψ m i , which is averaged over all the sites to obtain C ( τ ). For h = 6 t ≫ h c , where all the many body statesare localised, C ( τ ) remains constant with time in thelarge time limit as shown in Fig. 6. For h < h c and ρ = 0 . 5, where the system remains delocalised even inthe presence of interactions, C ( τ ) decays with time inthe large time limit. On the contrary for h < h c quarter-filled case where there are two many-body MEs, C ( τ )decreases very slowly (remaining almost constant withtime) and has value much larger than that for h < h c half-filled system which indicates that interactions havenot delocalised the entire many-body spectrum. Entanglement Entropy and ETH : Entanglement en-tropy (EE) is a useful tool to distinguish between theergodic and many-body localized phases. We divide thelattice into two subsystems A and B of sites L/ R ( E n ) = − log [ T r A ρ A ( E n ) ]where ρ A is the reduced density matrix obtained by in-tegrating the total density matrix ρ total ( E n ) = | Ψ n ih Ψ n | over the degree of freedom of subsystem B. EE is ex-pected to obey the volume law of scaling R ∼ L d in theergodic phase while it is suppressed for the MBL phaseshowing an area law scaling R ∼ L d − [18, 27, 34] with d = 1 for the model under study. As shown in Fig. 7,for low and high energy states average R ( E ) is same for C ( τ ) time τ h=6t, ρ=0.5 ,L=16h=1.75t, ρ=0.25, L=20h=1.5t, ρ=0.5, L=16 FIG. 6: Return probability C ( τ ) for V = 1 . t . In the largetime limit, C ( τ ) stays finite and constant for h = 6 t indicat-ing localised nature of the system while for h = 1 . t , C ( τ ) de-creases with τ indicating its delocalised nature. For h = 1 . t and ρ = 0 . C ( τ ) decreases very slowly indicating localisednature of most of the spectrum. various L values indicating that the states in this en-ergy regime are localized while for states in the middleof the spectrum R ( E ) increases with L indicating theirdelocalised nature though the transition points ˜ E , ob-tained from EE are little off from E , as shown in Fig. 1.This indicates that there is a regime where many-bodystates are localized in the Fock space but still EE in-creases with L . Though NPR overestimates the MBLregime specially for weak interactions [18] having zerothermodynamic value for partially extended states, EEmight give an over estimate of extended regime. The EEdata for non-interacting case shows [36] that even formany-body states, which are slater determinants of alllocalised single particle states and hence are definitelylocalised, R increases with the system size. Hence theactual many-body ME lies somewhat little below E butdefinitely above ˜ E . R ( E ) E(a) V =0.1 L=12L=16L=20 =0.7 2 2.2 2.4 2.6 2.8 3 -2 -1 0 1 2 O ( E ) E(c) V =0.1, L=20 2 2.2 2.4 2.6 2.8 3 -2 -1 0 1 2E(d) V =0.7, L=20 FIG. 7: Top panel shows R ( E ) vs E . For E < ˜ E and E > ˜ E , R ( E ) is same for all L but for the intermediate states R ( E )increases with L indicating their ergodic nature. The bottompanel shows O ( E ) vs E which shows large fluctuations in itsvalue for near by eigenstates for E < ˜˜ E ∼ ˜ E and E > ˜˜ E ∼ ˜ E . This data is for h = 1 . t with ρ = 0 . 25. Correspondingfigure for h = 2 . t is shown in supplement [35]. To check for the ETH in various parameter regimeswe calculated expectation value of the number operator on subsystem A which has L/ O = P L/ i =1 ˆ n i where ˆ n i is the numberoperator for spin-less fermions at site i . As shown inbottom panel of Fig. 7, for E < ˜˜ E and E > ˜˜ E , many-body system is not thermal showing large fluctuations in O ( E ) for nearby energy states while for ˜˜ E < E < ˜˜ E ,system is ergodic and obeys ETH. As shown in Fig. 7,˜˜ E , ∼ ˜ E , within numerical errors.We also calculated the variance of EE δ R ( E ) = h R ( E ) i − h R ( E ) i shown in Fig. 8. In the thermody-namic limit δ R should be zero deep inside the localizedand delocalised phases but at the transition point it di-verges due to contribution from both the extended andthe localized states [16] which is reflected as a peak infinite size calculations. Our data shows two clear peaksin δ R ( E ) vs E curve for intermediate values of V indi-cating two transition points which are very close to E , as shown in Fig. 1 where sky-blue curves represent peakpositions in δ R ( E ) vs E curve. Note that for small valuesof V the two peaks in δ R ( E ) vs E curve are very close toeach other and it is difficult to identify the peak positionsprecisely. δ R ( E ) E(a) h=1.75, ρ =0.25, L=20V =0.3V =0.6V =0.8V =1.0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 -1 0 1 2E(b) h=2.1, ρ =0.5, L=16 V =0.3V =0.5V =0.8V =1.0 FIG. 8: Variance δ R ( E ) of EE as a function of E for var-ious values of V . For V > . t , δ R ( E ) shows two clearpeaks, indicating localisation to delocalization transition inthe many-body spectrum. In summary we have analysed MBL in an interact-ing 1-d model of spin-less fermions in the presence of anaperiodic potential where the non-interacting system hasmobility edges at ± E c for h < h c . We demonstrated thatfor the system less than half-filled such that E f < − E c ,the interacting system has two mobility edges with lo-calised states living on the low and very high energy partof the spectrum while the middle of the spectrum has de-localized states. 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