AA many body localization proximity effect
Rahul Nandkishore Princeton Center for Theoretical Science, Princeton University, Princeton, New Jersey 08544, USA
We examine what happens when a strongly many body localized system is coupled to a weakheat bath, with both system and bath containing similar numbers of degrees of freedom. Previousinvestigations of localized systems coupled to baths operated in regimes where the back action of thesystem on the bath is negligible, and concluded that the bath generically thermalizes the system. Inthis work we show that when the system is strongly localized and the bath is only weakly ergodic,the system can instead localize the bath. We demonstrate this both in the limit of weak couplingbetween system and bath, and in the limit of strong coupling, and for two different types of ‘weak’bath - baths which are close to an atomic limit, and baths which are close to a non-interactinglimit. The existence of this ‘many body localization proximity effect’ indicates that many bodylocalization is more robust than previously appreciated, and can not only survive coupling to a(weak) heat bath, but can even destroy the bath.
Quantum localized systems violate many of the foun-dational assumptions of quantum statistical physics(such as the ergodic hypothesis), and present an excitingnew frontier for research [1]. While localization was longbelieved to occur mainly in systems of non-interactingparticles, the recent discovery of many body localiza-tion (MBL) [2–5] has ignited a blaze of interest in thisfield. It has been realized that quantum localized systemscan display a cornucopia of exotic properties, includingan emergent integrability [6–8], exotic quantum statesof matter [9–18], and unexpected behavior in linear [19]and non-linear [20] response. These properties not onlydramatically revise our understanding of quantum statis-tical physics, but also offer a new route to dissipationlessquantum technologies. A summary of progress in thisfield can be found in the review article [21].Most works on MBL have focused on perfectly isolatedquantum systems. Experimental systems, however, arealways coupled (however weakly) to a thermalizing envi-ronment. The behavior of many body localized systemscoupled to a thermalizing environment was first exam-ined in [22–24], in the limit where back action on the bathwas negligible and the bath could be treated as Marko-vian. In this limit, it was argued that an arbitrarily weakcoupling to a thermodynamically large bath should re-store ergodicity, thermalizing the system. These resultssuggested that perfect MBL would be unobservable inexperiments, which would see instead only signatures ofproximity to a localized phase. However, these worksleft open the question of whether different physics couldresult if the localization in the system were strong, andthe bath were weak. Could many body localization thensurvive even after coupling to a heat bath?In this Letter, we show that when the system of interestis strongly localized, and the ‘heat bath’ is only weaklyergodic, then MBL in the system can not only survivecoupling to the bath, but can even localize the heat bath.We call this phenomenon a ‘many body localization prox-imity effect,’ and it establishes that MBL is much morerobust to coupling to an environment than was previ- ously appreciated. It also suggests a possible explanationfor the numerical results recently presented in [26, 27],which counter-intuitively observed many body localiza-tion in an interacting model, when the non-interactinglimit contained a single particle mobility edge.The system we consider consists of a D dimensionallattice which hosts two species of spinless fermions - c and d . The c fermions are present with density n c andhave Hamiltonian H c = (cid:88) (cid:104) ij (cid:105) t c c † i c j + U c † i c i c † j c j + (cid:88) i ε i c † i c i (1)where t c is the hopping, U is a nearest neighbor inter-action, and ε is a random potential, drawn from a dis-tribution of width W . The width of the distribution issufficiently large that the c particles in isolation are inan MBL phase, with a localization length ξ c . We do as-sume that W is the largest scale in this Hamiltonian, andsets the characteristic energy scale of the MBL system.Meanwhile, the d particles are present with density n d ,and have Hamiltonian H d = (cid:88) (cid:104) ij (cid:105) t d d † i d j + λd † i d i d † j d j (2)This Hamiltonian in isolation describes a system in anergodic phase (the ground state will be a Fermi liquid).The coupling between c and d systems is taken to havethe form H int = (cid:88) i g i c † i c i d † i d i (3)For simplicity in this work we assume n c = n d , althoughvarying the ratio of the two densities would also be an in-teresting parameter to tune in future work. We considertwo models. First, we consider a model (model I) wherethe coefficients g i are taken from some distribution (e.g.box distribution) of width G . Next, we consider a model(model II) where the g i are uniform , with strength G .Many of the results are easier to establish with a ran-dom coupling (model I), and we thus discuss this model a r X i v : . [ c ond - m a t . s t a t - m ec h ] J un first, but we will subsequently show that a many bodylocalization proximity effect can also arise with uniformcouplings (model II). Weak coupling (small G ): We first discuss the behaviorwith a weak random coupling g i taken from a distribu-tion of width G (cid:28) W . In this regime the disorder in the c sector is the primary source of randomness. A manybody localization proximity effect can be shown to arisein two limits: the atomic limit t d →
0, and the non-interacting d limit λ →
0. We establish by means of per-turbative expansions that this ‘many body localizationproximity effect’ survives for small (but non-zero) t d and λ respectively, such that there exists a finite sized regionof parameter space where coupling a localized system to abath results in localization of the bath. We subsequentlyshow that this ‘many body localization proximity effect’also arises if the coupling is uniform rather than random. Bath close to an atomic limit: t d < G < W . We startby taking the limit t d →
0. In this atomic limit for the d fermions, the eigenstates of the combined c and d systemtake the form | Ψ (cid:105) = | φ ( { i } ) (cid:105) ⊗ (cid:89) { i } d † i | (cid:105) (4)i.e. d fermions are present on a certain set of sites { i } , andthe c fermions are in a localized state, the precise wavefunction of which depends on the set of occupied sites { i } in the d sector. The wave function in the c sector dependson { i } because the distribution of d fermions affects therandom potential seen by the c fermions. We note thata random distribution of atomic-limit d fermions will in-crease the effective random potential disorder in the c sector, from W to √ W + G , and thus if the c fermionsare localized at G = 0 (uncoupled c and d sectors), theyshould also be localized at non-zero G .We now turn on a non-zero hopping in the d fermionsector t d (cid:54) = 0, and ask whether the resulting system isstill localized. The hopping of a single d fermion changesthe interaction energy between c and d fermion sectors byan amount of order G (we are working here with randomcouplings g i ). It may also change the interaction energyin the d fermion sector by an amount of order λ . Thus,naively a single hop by a d fermion takes the system ‘offshell’ by an random number drawn from a distributionof width at least G . However, the hopping of a single d fermion changes the potential seen by the c fermions, andthus in the c fermion sector acts like a ‘quantum quench.’We now discuss to what extent the c fermion system maybe able to relax to accommodate this change in energy.The c sector is in a localized phase, thus it will not beable to bring the system back precisely on shell. Rather,the c fermion sector should act like a ‘finite sized bath’,with the localization length setting the size, and s ( T )being the entropy density. The relevant level spacing willthus be ∼ W exp( − s ( T ) ξ Dc ). The hopping of a d fermion,combined with an appropriate relaxation in the c sector on length scales short compared to ξ c , must thus take thesystem off shell by at least ∆ E ≈ W exp( − s ( T ) ξ D ).To evaluate the convergence (or not) of the locatorexpansion [1], the change in energy must be comparedto the associated matrix element. The matrix elementbetween the two states is( (cid:104) | d i ⊗ (cid:104) φ ( i ) | ) t d d † i d j (cid:16) | φ ( j ) (cid:105) ⊗ d † j | (cid:105) (cid:17) ≈ t d (cid:104) φ ( i ) | φ ( j ) (cid:105) (5)Where | φ i (cid:105) and | φ ( j ) (cid:105) are the initial and final states inthe c sector. We assume that the two states differ onlyon length scales less than ξ c (ignoring rare long rangeresonances). We make the standard assumption thatthe eigenstates are ‘ergodic’ on length scales less than ξ c i.e. that the initial state has similar overlap with allexp( s ( T ) ξ Dc ) possible final states that differ from the ini-tial state only on length scales shorter than ξ c , and alsothat the exp( s ( T ) ξ Dc ) possible overlap matrix elementshave random phases. Demanding normalization of thefinal wave function, we then conclude that a typical over-lap matrix element (cid:104) φ ( i ) | φ ( j ) (cid:105) ≈ exp( − s ( T ) ξ D )The locator expansion will converge if the typical hop-ping matrix element is smaller than the amount by whichthe hop takes the system off shell. From the estimatesabove, we conclude that a locator expansion in small t d will converge iff t d < min( G, W exp( − s ( T ) ξ Dc )) (6)As long as this condition is satisfied, the combined c and d systems will both be localized, with eigenstates that are‘close’ to the form (4). Thus in this there regime is a clearMBL proximity effect where coupling a localized systemto a bath ends up localizing the bath instead. When thiscondition is violated, the locator expansion breaks down,which may indicate delocalization.The above analysis may be readily generalized to uni-form couplings of strength G (model II). As long as thedensity pattern in the c sector is spatially inhomogenous(which should be the case in the localized regime), the effective coupling between c and d sectors will be ran-dom, drawn from a distribution of width Gδn c , where δn c is the width of the density distribution in the c sec-tor. An analogous argument may then be constructed fora many body localization proximity effect at weak t d . Wecaution however that in the limit of site localization of c fermions t c /W →
0, with uniform couplings, the effectivedisorder introduced in the d fermion sector is binary, andhas percolating equipotential surfaces which break thelocator expansion. This problem is absent with model Icouplings. A weakly interacting low dimensional bath: λ (cid:28) G (cid:28) W, t d and D = 1 , λ →
0. In this limit, the d fermions aredescribed by a quadratic Hamiltonian, which takes theform H d = (cid:88) (cid:104) ij (cid:105) t d d † i d j + (cid:88) i V i d † i d i (7)where V i = g i c † i c i is a static random variable (since the c fermions are in a localized phase), and can be interpretedas a disorder potential, with the precise realization of thedisorder depending on the state in which the c fermionsare prepared. Thus, the d-fermion Hamiltonian describesfermions hopping in a random potential. It does not par-ticularly matter whether we work with model I couplings(random coupling constants) or model II (uniform cou-pling constants), since as long as the density pattern inthe c sector is inhomogenous the d fermions will see aneffective random potential anyway. We choose to workwith model I for convenience and specificity. As is wellknown, free fermions hopping in a random potential willinevitably localize in D = 1 , ξ d that is power law large in t d /G for D = 1 andexponentially large in t d /G for D = 2.The argument above is cleanest in the site localiza-tion limit ξ c →
0, when the c − d coupling is diag-onal in the c eigenbasis and there is no appreciableback action on the c system. At non-zero ξ c , therewill be an effective four d fermion interaction mediatedby the c fermions. This four fermion interaction maybe estimated in the manner of [16] and will be of or-der ( G /W ) ξ − D/ d exp( − /ξ c ) for ξ c < G /W ) ξ − D/ d ξ Dc for ξ c > c sector, we can also turn on the intrinsicinteraction λ (cid:54) = 0. The matrix elements of this interac-tion in the basis of localized wave functions | ϕ (cid:105) will be λ αβγδ = λ (cid:80) (cid:104) ij (cid:105) ϕ ∗ α ( i ) ϕ ∗ β ( j ) ϕ γ ( j ) ϕ δ ( i ) ≈ λξ − Dd , wherewe have made use of normalization of the wavefunctions.The locator expansion should converge (such that thedressed eigenstates of the interacting problem are closeto the eigenstates of the non-interacting problem), if thematrix elements of the interaction (both intrinsic andinduced) are less than the accessible level spacing t d ξ − Dd i.e.max (cid:32) G W t d ξ D/ d exp( − /ξ c ) , λt d (cid:33) ξ Dd < ξ c < (cid:32) G ξ Dc W t d ξ D/ d , λt d (cid:33) ξ Dd < ξ c > ξ d is power-law (exponentially) large in t d /G . For there tobe a well controlled weak localization regime, we there-fore require not only that λ is small, but also that either ξ c (cid:28) W (cid:29) t d . We note also that the con-vergence criterion (8) is simply an estimate obtained by considering the leading order terms in perturbation the-ory. A calculation to all orders along the lines of [3] isbeyond the scope of the present work.As long as the interactions (both intrinsic and induced)are sufficiently weak, the wavefunction will take the form | Ψ (cid:105) = | φ ( { α } ) (cid:105) ⊗ (cid:89) { α } d † α ( φ ) | (cid:105) (9)i.e. the d fermions are in a product state where a set { α } of the non-interacting wave functions are occupied,the precise shape of the non-interacting wave functionsdepends on the state | φ (cid:105) in which the c fermions are pre-pared, and the c -fermions are in a state | φ (cid:105) which is manybody localized, but depends in its detailed structure onthe state of the d -fermions. Strong coupling (large G ) We now point out that amany body localization proximity effect also arises for arbitrary λ and t in the limit G → ∞ . We demonstratethis for model II (uniform) couplings, but it is obviouslytrue also for model I.In the large G limit, we have a description in terms ofthree species: ‘bound states’ whereby a c and d fermionsit on the same site, unbound c s, and unbound d s. Abound state cannot break apart or form, because thiswould change the energy by an amount of order G , whichis the largest energy scale in the problem.The unbound c fermions are governed by a Hamilto-nian of the form (1), but now with certain lattice sitesforbidden (the sites occupied by the c − d pairs and theunpaired d s). Since the c fermions localized with all lat-tice sites allowed, and the forbidding of certain latticesites only presents an obstruction to transport, the un-paired c fermions should still be localized. Meanwhile,the bound c − d pairs are extremely heavy, with an effec-tive hopping matrix element of order t c t d /G (cid:28) t c , andthey see the same disorder potential as the c fermions.Thus, if the c fermions localized in a disorder potentialof magnitude W , then the c − d composites should lo-calize also. Finally, that leaves the unpaired d fermions.These live on a random lattice, obtained from the origi-nal lattice by deleting all sites on which unpaired c s and c − d pairs are present. (This is because hopping ontoan ‘occupied’ site changes the energy by an amount oforder G , and in the large G limit this effectively makesoccupied sites inaccessible). In one dimension, any finitedensity of deleted sites causes the ‘lattice’ on which theunpaired d fermions move to break up into finite sizedsegments, on which the d fermions are localized. Thus,in one dimension, in the G → ∞ limit, we clearly have asystem of three distinct simultaneously localized species,regardless of the values of t d and λ .In higher dimensions too, when a sufficiently largenumber of sites are deleted (large G limit), the latticeon which the unpaired d fermions live will break into dis-connected clusters, and at this point the d fermions willmost certainly be localized. Thus, there must be a regimein which the coupling of system and bath leads to the for-mation of localized species: localized c fermions, localized c − d composites, and d fermions which live on isolatedislands that are separated by regions containing either c fermions or c − d composites. In fact, work on quan-tum percolation [30] suggests that for non-interacting d fermions (or, presumably, weakly interacting d-fermions),the localization transition precedes the percolation tran-sition, and d fermions will quantum localize even beforethe lattice on which they live breaks up into disconnectedclusters. The density of deleted sites may be tuned byvarying either the density of particles or the energy ofthe state (and hence the number of c − d pairs), and thequantum percolation (and hence localization) transitionmay also be tuned in this way.We note that in principle there is an ‘exchange’ processby which an unbound c and a c − d composite on adjacentsites could exchange position by moving the d fermion,without taking the system off shell. This is really onlypossible if the c fermions are in the site-localization limit W (cid:29) t c (since the c − d composites are always site local-ized in the large G limit). However, in this limit of sitelocalization for c fermions, localization requires the addi-tional condition that the set of possible energy conserving d exchanges should not percolate.If G is large but not infinite, then d fermions will inprinciple be able to ‘tunnel through’ the barriers pre-sented by c fermions, but virtual processes involving c − d composites will then also give rise to a random onsitepotential for the lattice on which the d fermions live (in-herited from the inhomogenous density pattern for the c fermions). In low dimensions and in the non-interactinglimit λ →
0, and for sufficiently large G , this will be suffi-cient to localize the d fermions, and arguments along thelines of [3] suggest the d fermions will continue to be lo-calized for small but non-zero λ . Of course, the existenceof a ‘many body localization proximity effect in the limitof weakly interacting d fermions is not particularly sur-prising, since something similar occurs for weak G . Thetruly novel feature is the existence of an MBL proximityeffect at arbitrary t d , λ in the limit G → ∞ . Conclusion:
Thus, we have demonstrated that whena many body localized system is coupled to a bath, theend result can be the survival of MBL in the system, andthe localization of the bath. We have demonstrated thatthis happens in three distinct limits: a bath close to theatomic limit that is weakly coupled to a strongly local-ized system, a weakly interacting bath weakly coupled toa strongly localized system, and an bath (with arbitrarybath parameters) that is strongly coupled to a localizedsystem. The problem of localized systems coupled tobaths is thus much richer when the back action on thebath is taken into account, and indeed the coupling of alocalized system and a delocalized system can result inthe localization of both. This may explain recent numer- ical results [26, 27]. The development of a general theoryof localized systems coupled to baths is an interestingchallenge for future work.
Acknowledgements:
I thank David Huse for severalilluminating discussions regarding this work. I also thankSarang Gopalakrishnan for a useful conversation, and forfeedback on an early draft. I am supported by a PCTSfellowship. [1] P. W. Anderson, Phys. Rev. , 1492 (1958).[2] I. V. Gornyi, A. D. Mirlin and D. G. Polyakov, Phys.Rev. Lett. , 206603 (2005).[3] D. M. Basko, I. L. Aleiner and B. L. Altshuler, Annalsof Physics , 1126 (2006).[4] J. Z. Imbrie, arXiv:1403.7837.[5] A. Pal and D. A. Huse, Phys. Rev. B , 174411 (2010).[6] D.A. Huse and V. Oganesyan, arXiv: 1305.4915, D.A.Huse, R. Nandkishore and V. Oganesyan, Phys. Rev. B , 174202 (2014)[7] M. Serbyn, Z. Papic and D. A. Abanin, Phys. Rev. Lett. , 127201 (2013).[8] V.Ros, M.Mueller, A.Scardicchio, arXiv:1406.2175, Phys.Rev. B [in press].[9] D. A. Huse, R. Nandkishore, V. Oganesyan, A. Pal andS. L. Sondhi, Phys. Rev. B , 014206 (2013).[10] D. Pekker, G. Refael, E. Altman, E. Demler and V.Oganesyan, Phys. Rev. X , 011052 (2014).[11] R. Vosk and E. Altman, Phys. Rev. Lett. , 217204(2014).[12] J. A. Kjall, J. H. Bardarson and F. Pollmann,arXiv:1403.1568.[13] B. Bauer and C. Nayak, J. Stat. Mech. P09005 (2013).[14] Y. Bahri, R. Vosk, E. Altman and A. Vishwanath,arXiv:1307.4192.[15] A. Chandran, V. Khemani, C. R. Laumann and S. L.Sondhi, Phys. Rev. B , 144201 (2014).[16] R. Nandkishore and A. C. Potter, Phys. Rev. B ,195115 (2014)[17] . R. Vasseur, A.C. Potter and S.A. Parameswaran, Phys.Rev. Lett.
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Annual Reviews of Con-densed Matter Physics , 15-38 (2015)[22] R. Nandkishore, S. Gopalakrishnan and D.A. Huse, Phys.Rev. B , 064203 (2014)[23] S. Johri, R. Nandkishore and R.N. Bhatt, Phys. Rev.Lett. , , 117401 (2015)[24] S. Gopalakrishnan and R. Nandkishore, arXiv: 1405.1036[25] D.A. Huse, R. Nandkishore, S. Pietracaprina, V. Ros andA. Scardicchio, http://arxiv.org/abs/1412.7861[26] R. Modak and S. Mukerjee, arXiv: 1503.07620[27] X. Li, S. Ganeshan, J.H. Pixley and S. Das Sarma, arXiv:1504.00016 β" β"α" α"γ" γ"δ" δ"a" a"b" b"c"(i)" (ii)" FIG. 1. Processes contributing to the mediation of an effective four d fermion interaction through the c fermions, at leadingorder in perturbation theory in weak G . Solid lines denote d fermions and dashed lines denote c fermions. The state of the d fermions is labelled by Greek characters, while the state of the c fermions is labelled by English characters. We assume thelocalization centers of the d fermion wavefunctions φ α,β,γ,δ are all within ξ d of each other.[28] E. Abrahams, P.W. Anderson, D.C. Licciardello andT.V. Ramakrishnan, Phys. Rev. Lett. , 673 (1979)[29] R. Abou-Chacra, D.J. Thouless and P.W. Anderson, J.Phys. C: Solid State Phys. , 1734 (1973) [30] G. Schubert, A. Wei β e and H. Fehske, Phys. Rev. B ,045126 (2005) and references contained therein SUPPLEMENT: EFFECTIVE FOUR d FERMION INTERACTION MEDIATED BY c FERMIONS
When ξ c (cid:54) = 0, the coupling of c and d fermion sectors leads to an effective four d fermion interaction. In thissupplement we estimate this interaction, first for ξ c < ξ c >
1. The calculation follows [16]. The twoprocesses contributing to the effective interaction at lowest order in perturbation theory in weak G are shown in Fig.1.The process (i) dominates and will be discussed here, but the contribution of process (ii) can be calculated similarly.The process Fig.1(i) represents an effective T matrix element of the form T abcαβγδ = G (cid:90) d D r d D r φ ∗ α ( r ) φ β ( r ) φ γ ( r ) φ ∗ δ ( r ) ψ ∗ a ( r ) ψ b ( r ) ψ ∗ b ( r ) ψ c ( r )∆ E (10)where φ represents a c fermion wavefunction, ψ represents a d fermion wavefunction, and ∆ E represents the amountby which the intermediate state is off shell.If ξ c < | r − r | and the integral is effectively restrictedto neighboring sites. Meanwhile, the scattering in the c fermion sector takes us off shell by an amount of order W . Finally, since each site has in effect a single associated state ψ a,b,c , the matrix element is of order T abcαβγδ ≈ ( G /W ) ξ − Dd exp( − /ξ c ) exp( iφ abc ), where we have assumed the d wavefunctions φ are localized to a volume of radius ξ d , and where φ abc is a random phase. The effective induced interaction may be obtained by summing over allintermediate c fermion states i.e. λ effαβγδ = (cid:88) abc T abcαβγδ (11)There are ξ Dd terms contributing to the above sum (since the four d fermions involved can interact through c fermionsanywhere in the volume where they all overlap). Each term has similar magnitude but random phase. Summing overall the terms in rms fashion gives an additional factor of ξ D/ d , and as a result we conclude that the effective fourfermion interaction mediated by the c fermions is of order ( G /W ) ξ − D/ d exp( − /ξ c ), as quoted in the main text.The calculation is similar when ξ c >