Extracting many body localization lengths with an imaginary vector potential
EExtracting many body localization lengths with an imaginary vector potential
Sascha Heußen, Christopher David White, and Gil Refael Institute for Quantum Information and Matter and Department of Physics,California Institute of Technology, Pasadena, CA 91125, USA Condensed Matter Theory Center, University of Maryland, College Park, Md, 20742, USA
One challenge of studying the many-body localization transition is defining the length scale thatdiverges upon the transition to the ergodic phase. In this manuscript we explore the localizationproperties of a ring with onsite disorder subject to an imaginary magnetic flux. We connect theimaginary flux which delocalizes single-particle orbitals of an Anderson-localized ring with the lo-calization length of an open chain. We thus identify the delocalizing imaginary flux per site withan inverse localization length characterizing the transport properties of the open chain. We putthis intuition to use by exploring the phase diagram of a disordered interacting chain, and we findthat the inverse imaginary flux per bond provides an accessible description of the transition and itsdiverging localization length.
I. INTRODUCTION
A large body of evidence shows that one-dimensionalfermionic quantum systems with both (local) interactionsand sufficiently strong disorder will exhibit a cluster oftraits known as many-body localization . These includelong-time memory of an initial state, conductance expo-nentially small in length, slow entanglement growth, andnear-Poisson level statistics.The localization length of single particle orbitals is astandard measure of localization. This notion appliesto interacting localized systems as well as noninteract-ing systems. In fact, the localized nature of the many-body localized phase implies that it can be described by aso-called (cid:96) -bit
Hamiltonian . The Hamiltonian can bewritten in terms of mutually commuting single-particleoccupation operators ˜ n j with local support as H = (cid:88) E j ˜ n j + (cid:88) jk J (2) ˜ n j ˜ n k + (cid:88) j j j J (3) j j j ˜ n j ˜ n j ˜ n j . + · · · where the interactions J ( n ) are presumptively short-ranged. Explicitly constructing the (cid:96) -bits would fullysolve the quantum dynamics of a many-body localizedchain; the problem of doing so has attracted muchattention. The first phenomenological and numerical treatmentsof the MBL transition tended to concentrate on entangle-ment and transport times, and computed the gap ra-tio, the entanglement entropy of eigenstates, or decay oflocal observables.
But the microscopic avalanchepicture and recent renormalization analysis hingeon the correlation lengths of perturbatively-constructed (cid:96) -bits.The most direct way to probe many-body localization,however, should be through finding the appropriate local-ization length of single particle creation operators. Sucha localization length, which is analogous to the Andersonlocalization scale, would be most relevant to transport-related questions. A localization length could be defined from the support of the (cid:96) -bits. But constructing (cid:96) -bits isdifficult and relies on variants of exact diagonalization,Wegner flow or matrix product state methods; moreover,the same physical Hamiltonian can be described in termsof many different sets of (cid:96) -bits. Extracting localizationlength from (cid:96) -bits of limited-size systems is therefore, dif-ficult and ambiguous. Nonetheless, Refs. 23–25 succeedin extracting localization length by constructing (cid:96) -bit op-erators and exploring their decay.In this manuscript we show that exploring the responseof a system to non-Hermitian hoppings (namely, to animaginary flux) provides a direct way to address the def-inition of a localization length. Furthermore, it does notrequire knowledge of any (cid:96) -bits properties, and addressesdirectly the localization length most relevant for trans-port. We introduce an imaginary vector potential, whichmaps to a “tilt”—an asymmetry in the tunneling ratesbetween neighboring lattice sites. While at small tiltsall eigenvalues of the system’s Hamiltonian remain real,they develop imaginary parts at some critical tilt. Weargue that the critical tilt of a non-Hermitian systemprobes the (cid:96) -bit localization length ξ l-bit of its zero-tiltHermitian limit, while the distribution of points at whichsuccessive eigenvalues develop imaginary parts (“excep-tional points”) probes the (cid:96) -bit interaction scale J int . Weshow explicitly that the critical tilt in a ring (a chain withperiodic boundary conditions) is the inverse localizationlength of the open chain with the same disorder realiza-tion. We use this to describe the many-body localizationtransition and extract its phase diagram.We first introduce our model in Sec. II. We then ar-ticulate the connection between critical tilt in a ring andthe localization length of an open chain in the single-particle case (in Sec. III); in doing so, we extend thework of Ref. 26 to connect the critical tilt on a ring toa Green’s function on the open chain. We then considergeneralizations to the many-particle, non-interacting case(Sec. IV A) and to the interacting case (Sec. IV B), wherewe show a connection between the distribution of excep-tional points and the l-bit interaction strength. Finally,in Sec. V we use the critical tilt to map the MBL transi-tion of the isotropic random-field Heisenberg model. We a r X i v : . [ c ond - m a t . d i s - nn ] F e b find a phase diagram and critical exponents broadly con-sistent with . In addition, we see that the MBL regimeis reentrant as a function of interaction strength. II. INTRODUCING NON- HERMITIANHOPPING TO THE MANY-BODYLOCALIZATION MODELA. Background
The non-Hermitian hopping problem in a tilted dis-ordered lattice was proposed as an effective model forvortex pinning in non-parallel columnar defects . In-deed, the anhermiticity of the hopping operator repre-sented the tilt of the columnar pinning defects relativeto the external field. The critical tilt in this single par-ticle problem was shown to be intimately related to thelocalization properties of the zero-tilt system. The re-lationship between critical tilt and correlation length hasreceived much interest, as have various properties of thesingle-particle spectrum at fixed tilt. . Very recently,non-Hermitian tilt was also introduced to interactingsystems . B. Model
We study spinless fermions hopping on a one-dimensional lattice with a random onsite chemical po-tential and an imaginary vector potential. The system’sHamiltonian is H = t (cid:88) j [ e g c † j c j +1 + e − g c † j +1 c j ]+ U (cid:88) j n j n j +1 + (cid:88) j h j n j . (1)with the random onsite potential h j uniformly distributedon [ − W, W ]. We set the bare hopping to t = 1. When U = 2, this model is Jordan-Wigner equivalent to therandom-field Heisenberg model. In Sec. III we work inthe single-particle sector; in the subsequent sections wework at half-filling.The Hamiltonian (1) is non-Hermitian, with anher-miticity parametrized by the imaginary vector potential,or “tilt”, g . In an open chain the tilt g could be removedby a similarity transformation S = e (cid:80) j gjn j . (2)In a ring, the imaginary flux cannot be removed by sucha similarity transformation, and imaginary parts can ap-pear in the energy eigenvalues of the system.The imaginary eigenvalues are directly related to delo-calization on the ring. For g >
0, the system prefers left-wards hopping, but if g is small one expects the systemto remain localized. Localized orbitals cannot explore FIG. 1.
Eigenenergies for one particle on a chain of N =100 sites with disorder width W = 1 at three tilts g . Forsufficiently strong gauge field the single-particle spectra formellipses emerging from the band center. the entire ring, and therefore their energy eigenvalues re-main real. At large g the preferential leftward hoppingdominates, and one expects the system to be delocalized.Orbitals that wrap around the ring necessarily develop animaginary part to their energy. In the next section we ex-plain how to precisely relate the tilt g at which imaginaryparts appear to the localization length of orbitals. III. CRITICAL TILT AND LOCALIZATIONLENGTH: THE SINGLE-PARTICLE CASE
As the tilt in a ring increases, the Hamiltonian’s eigen-values develop imaginary parts. The point at which aneigenvalue develops an imaginary part is called an excep-tional point . We seek a precise relationship between thelocalization length of the open chain and the location ofthe exceptional point.We can get some intuition for the processes involvedand see what ingredients are required by a heuristic argu-ment, in which we gauge the imaginary flux to one linkand add that link perturbatively. Consider the Hamil-tonian of Eq. (1) on N sites with periodic boundaryconditions, and take the single-particle case. At g = 0its single-particle eigenstates are exponentially localized,and, therefore, cannot explore the flux penetrating thering. An eigenstate | n (cid:105) centered on some site j , for in-stance, can be asymptotically written as | n (cid:105) ∼ (cid:88) j (cid:48) e −| j − j (cid:48) | /ξ | j (cid:48) (cid:105) . (3)Through a similarity transformation as in Eq. (2), we canshift all of the imaginary vector potential to the far side ofthe system, away from | n (cid:105) ’s center site j . The imaginaryflux would then be shifted to ¯ j = j + N/ N , andthe Hamiltonian would be H = (cid:88) j (cid:48) (cid:54) =¯ j [ c † j (cid:48) c j (cid:48) +1 + h.c. ] + (cid:88) j (cid:48) h zj (cid:48) n j (cid:48) + e Ng c † ¯ j c ¯ j +1 + e − Ng c † ¯ j +1 c ¯ j . (4)Imagine now adding the anhermitian hopping on the link¯ j, ¯ j +1 perturbatively. The perturbation becomes impor-tant when the resulting change in energy is comparableto some energy difference ∆ E in the closed chain: thatis, when1 ∼ (cid:104) n | te Ng c † ¯ j +1 c ¯ j | n (cid:105) ∼ e N ( g − ξ − ) ∆ E − ; , (5)where ∆ E is some energy difference in the closed chain.Immediately we see that the anhermiticity is importantwhen g ∼ ξ − . This crucial insight—that the tilt com-petes directly with the localization properties of individ-ual eigenstates—goes back to the work of Hatano andNelson (e.g. Ref. 28). But we also see the three in-gredients that will be important in our detailed calcula-tion: the tilt g , the end-to-end hopping matrix elementin eigenstates of the open chain, and energy differencesin the open chain. Although our detailed calculationin Secs. III A-III D applies only to single-particle (non-interacting) systems, we hope that this perturbative ap-proach will in the future yield more precise insight intointeracting (many-body localized) systems; we will returnto it in interpreting our results for those systems.For g > g c , the eigenstates with complex energy eigen-values resemble a plane wave. (Recall that we work inthe single-particle sector.) Therefore these eigenstateshave (complex) energy (cid:15) k ≈ cos( k − ig ) (recall t = 1) andare distributed on an ellipse (cid:18) (cid:60) ( (cid:15) )cosh g (cid:19) + (cid:18) (cid:61) ( (cid:15) )sinh g (cid:19) = 1 (6)(cf Fig. 1).For the Hamiltonian (1) in the single-particle sector, aheuristic relationship between the critical tilt g c and theend-to-end Green’s function of an open chain was estab-lished in Ref. 26. There it was shown that the criticaltilt of a ring is (cid:0) e Ng c + e − Ng c − (cid:1) − = N (cid:81) i =1 t iN (cid:81) i =1 ( E − (cid:15) i ) (7)with (cid:15) i the spectrum of the ring (in the absence of atilt), E the energy at which the first eigenvalue devel-ops an imaginary part, and t i are the hoppings betweensite i and i + 1 modulo N . The right-hand side of thisrelationship is suggestive: were the (cid:15) i eigenvalues of the open chain at g = 0 and g = g c respectively, it wouldbe closely related to the end-to-end Green’s function ofthat open chain at energy E . Since the eigenvalues (cid:15) i will, in fact, approach the eigenvalues of the open chainin the long-system, strong-disorder limit, this providesgood intuition—but the connection is definitely not ex-act. ttt t t t t t μ ϵ ϵ ϵ N ϵ N −1 FIG. 2.
Lead setup for Sec. III : open chain (sites 1 through N with onsite chemical onsite potentials (cid:15) . . . (cid:15) N and uniformhopping amplitude t ) together with lead site (chemical poten-tial µ ) connected to chain by a tunneling t . A precise relationship between the critical tilt of a ringand the inverse localization length of an open chain doesexist; we work it out in this section. To expose the rela-tionship we start with an open chain, and add a “lead”site with some local potential µ (cf. Fig. 2). The leadsite is then connected weakly (with hopping t (cid:28)
1) toboth the first and the last sites of the open chain. Next,we calculate the determinant of the resulting closed chainin terms of the open-chain eigenvalues. We then con-nect this determinant on the one hand to the end-to-endGreen’s function of the open chain, and hence to end-to-end eigenstate correlations; and on the other to thecritical tilt. We ultimately find that for an appropriatechoice of chemical potential µ and tunneling strength t , e − ( N +1) g c ∼ (cid:104) n | c † N c | n (cid:105) (8)where | n (cid:105) is an eigenstate selected by µ of the open-chain Hamiltonian, while 1 and N are basis states on thefirst and last sites of the open chain.Having established this relationship, we go on to gen-eralize to generic lattice rings and to the many-particle(but noninteracting: U = 0) case. A. Determinant formula for the closed chain withlead
Start with the Hamiltonian (1) in the g = 0 , U = 0,open boundary conditions case—call it H [1: N ]open = t N − (cid:88) j =1 [ c † j c j +1 + c † j +1 c j ] + N (cid:88) j =1 h j n j . (9)(We write H [ k : l ]open = t (cid:80) l − j = k [ c † j c j +1 + c † j +1 c j + n j n j +1 ] + (cid:80) lj = k h j n j for the Hamiltonian on sites k through l withopen boundary conditions; we will have occasion to usenot only H [1: N ]open but also H [2: N ]open .)Add a “lead” site with chemical potential µ connectedto both ends of the chain by a hopping amplitude t : H = H [1: N ]open + µ n + t ( c † c + c † c )+ t ( e ( N +1) g c † N c + e − ( N +1) g c † c N ) . (10)Since the chain now has periodic boundary conditions, wecan no longer gauge away the imaginary vector potential`a la (2); it is convenient to work in a gauge in which all ofthe vector potential lives on the bond between the leadand site N . We ultimately plan to take N (cid:29) g (cid:38) t e − ( N +1) g c † c N .For the purposes of finding a precise determinant for-mula, we take the lead to be weakly connected to therest of the chain: t (cid:28) t . We will discuss relaxing thisassumption below. H then has matrix representation EI − H ( g )= E − µ t t E − (cid:15) tt E − (cid:15) tt E − (cid:15) tt e ( N +1) g t E − (cid:15) (11)and determinantdet( EI − H ( g )) = ( E − µ ) det( EI − H open ) − t det( EI − H [2: N ]open )+ ( − N +2 t t N − e ( N +1) g (12)(expanding in minors along the first column). Since wetake t small we can ignore the t term compared to the t e gN term. If we take E to be an eigenvalue of H ( g )this is det( EI − H [1: N ]open )= ( − N t N − ( E − µ ) − t e ( N +1) g . (13) B. Open-chain Green’s function
We can re-write the determinant in (13) in terms ofthe open-chain Green’s function. This has (1 , N ) matrixelement G [1: N ]1 N ( E ) = [( EI − H [1: N ]open ) − ] N = [adj ( EI − H [1: N ]open )] N det( EI − H [1: N ]open ) (14)for G [1: N ]1 N ( E ) = ( − N +1 [det( EI − H [1: N ]open )] − t N − . (15) FIG. 3.
Sketch of the lead spectrum as in Sec. III C: E α is a level of the open chain, µ the chemical potential ofthe lead, and E the energy at which we measure the Green’sfunction. Unlabeled vertical lines are other eigenenergies ofthe open chain. With this relation (13) becomes( G (1: N ]1 N ) − = − ( E − µ ) − t e ( N +1) g . (16)We can extract the tunneling probability for an eigen-state | α (cid:105) of H [1: N ]open from the Green’s function (cid:104) | α (cid:105) (cid:104) α | N (cid:105) = ( E − E α ) G [1: N ]1 N ( E ) (17)by identifying poles, so Eq. (16) is e − ( N +1) g = (cid:104) | α (cid:105)(cid:104) α | N (cid:105) ( E α − E ) − ( E − µ ) − t . (18) C. Critical tilt g c Eq. 18 has three free parameters: µ , t , and g . E is not a (continuously tunable) parameter: it is fixed by µ , t , g , since it is an eigenvalue of the non-HermitianHamiltonian with lead site. But we can choose these pa-rameters to strongly constrain the non-Hermitian eigen-value E , and hence relate g c , the tilt at which eigenstate α coalesces with the lead state and develops an imaginarypart, to (cid:104) | α (cid:105)(cid:104) α | N (cid:105) .Suppose we wish to probe the eigenstate α of H [1: N ]open .Then choose0 Critical tilt g c against inverse localizationlength ξ − α via exact diagonalization for 1000 disorder re-alizations of (10) at system size N = 40 and lead parametersas in Eq. (19). (We consider eigenstate α = 20 of each real-ization.) This confirms our analytical relationship (21). where we define an eigenstate localization length ξ − n ≡ ln (cid:104) α | c † c N | α (cid:105) .We show g c and ξ α for eigenstate α = 20 of a chainwith N = 40 with 1000 disorder realizations in Fig. 4, andsee good agreement. We first diagonalize the open chain;we then take µ = E − . h/L and t = 0 . h/L ,in accordance with Eq. (19), and find g c in the result-ing closed chain. The variation comes from γ : E is notalways exactly E = ( E α + µ ). D. Closed chain without lead Even though the chain with lead has periodic boundaryconditions, in the sense that there are exactly two pathsbetween any two sites, it is not obvious that the results ofSec. III C will carry over to ordinary chains with periodicboundary conditions. Eq. 21, which connects g c and ξ α for some eigenstate, requires a carefully fine-tuned leadsite. Can we do better? Can we take a generic leadsite—that is, a straightforward periodic chain?If we are willing to relax our demands for rigor, we canmake some estimates. Take the Hamiltonian (1) with pe-riodic boundary conditions acting on one particle. Singleout one arbitrary site for treatment as the “lead”, andreturn to (12). Once again take E to be an eigenvalue of the non-Hermitian Hamiltonian H ( g ), so (12) becomes0 = ( E − µ ) det( EI − H [1: N ]open ) − t det( EI − H [2: N ]open )+ ( − N +1 t t N − e ( N +1) g . (22)Take t = t —the supposed lead site is just a normallattice site, after all—and write the determinants interms of the (1 , N ) components of the Green’s functions G [1: N ] , G [2: N ] of H [1: N ]open , H [2: N ]open . This becomes0 = ( E − µ ) (cid:104) | G [1: N ] | N (cid:105) − + t (cid:104) | G [2: N ] | N (cid:105) − + e ( N +1) g . (23)Now work at g c . Once again write E α for the eigenvaluenearest µ ; even though t is no longer small, we expect E − µ ∼ E α − E ∼ 12 ( E α − µ ) ∼ √ L (cid:28) , (24)so we can ignore the G [1: N ] term. If we assume (cid:104) | G [2: N ] | N (cid:105) − ∼ e − ( N − ξ − (25)then g c ∼ ξ − α . (26) IV. MANY-PARTICLE CASE ANDINTERACTION BROADENINGA. Many-particle non-interacting case Now let the same Hamiltonian (1) act on manyparticles—in fact on the half-filling sector—but take itto be noninteracting ( U = 0). Its eigenstates will beSlater determinants (cid:81) α ∈ A c † α | (cid:105) with eigenvalues E A = (cid:80) α ∈ A E α . When two single-particle states pass throughan exceptional point, developing imaginary parts to theirenergies, they therefore take with them a whole class ofmany-particle states.To quantify this effect consider first increasing g through g c , the tilt at which the first two single-particlestates go through an exceptional point. (In the exampleabove, of an open chain with a lead site, these will bethe lead site and the open-chain level n .) Call those twostates α and α , and occupy a set A of additional lev-els, not including α , α , with more particles. Since theenergy difference of the many body state is the same asthat of the delocalizing orbitals, E α A ( g ) − E α A ( g ) = E α ( g ) − E α ( g ) (27) every such set gives a pair of levels that coalesce at g c .As we tune g through g c , then, all n = 2 · (cid:18) N − N/ − (cid:19) (28) . . . . . g . . . . . . f i m ag N = 6 N = 8 N = 10 N = 12 FIG. 5. Emergence of imaginary part of energy withincreasing tilt g . y axis is fraction of eigenvalues havingimaginary component for one realization of (1) at interactionstrength U = 0, disorder width W = 7, and system size N asindicated. Dotted lines show our analytical results (28), (29)for the first few bifurcations. levels with either α or α occupied will coalesce withthe states with α and α occupation switched. (Recallthat we assume a half filled system with an even number N of sites.) These states will re-emerge with imaginaryparts, simply because the energies are the sum of thesingle-particle energies E α . (Note that if both α and α are occupied the resulting energy is real, because E α = E ∗ α .)Consider now increasing g through the tilt at which thesecond pair of single-particle eigenstates passes throughan exceptional point. At this tilt n = 2 · (cid:18) N − N/ − (cid:19) + 2 · (cid:18) N − N/ − (cid:19) = 4 · (cid:18) N − N/ − (cid:19) (29)eigenstates will develop imaginary parts (these two ex-pressions have α and α either fully occupied or fullyunoccupied).Fig. 5 shows the fraction of eigenenergies that developa complex eigenenergy as a function of disorder for aparticular disorder and no interactions. B. Many-particle interacting case Turn now to the interacting case, and consider disorderstrong enough that the Hamiltonian (1) is fully localizedfor 0 ≤ U ≤ , g = 0. In terms of (cid:96) -bits that interactingHamiltonian is H = (cid:88) E j ˜ n j + (cid:88) jk J (2) ˜ n j ˜ n k (30)+ (cid:88) j j j J (3) j j j ˜ n j ˜ n j ˜ n j + · · · . . . . g . . . . . . f i m ag U = 2 − U = 2 − U = 2 − U = 2 − U = 2 − U = 2 − U = 2 − U = 2 − U = 0 FIG. 6. Effect of weak interaction on emergence ofimaginary parts of energy . y axis is fraction of eigenval-ues having imaginary component for one realization of (1) atdisorder width W = 8, system size N = 10, and interactionstrength U as indicated. Dotted lines again show the analyt-ical results (28), (29). In the MBL phase ( U = 1 , W = 8)interactions smear out the discrete steps characteristic of thenoninteracting case, which result from coalescence of single-particle eigenstates. In the single-particle sector this reduces to H ( g ) = (cid:80) α E α n α ( g ).One can imagine running the same procedure as inthe previous part. As one increases g , the single-particleeigenvalues develop imaginary parts—but this cannotlead to simultaneous coalescence of many eigenvalues.The interaction terms (cid:80) αβ J (2) αβ ˜ n α ˜ n β + . . . mean thatnow E α A ( g ) − E α A ( g ) (cid:54) = E α − E α , (31)in contrast to (27), in which the energy difference wasindependent of the additional orbitals A . The (cid:96) -bit in-teractions of Eq. (30) therefore smooth the sharp step-like coalescence of many body states; the degree of thissmoothing probes the strength of those interactions. V. PHASE DIAGRAM OF THERANDOM-FIELD XXZ MODEL In this section we make use of the relationship g c ∼ ξ − to probe the phase diagram of the model (1) using thecritical tilt. We show that it is consistent with previousstudies. A. Fixed interaction Considering the critical tilt g c for each eigenstate givesus the localization length as a function of energy. Wemeasure g c ; rα for each eigenstate j of each disorder re-alization r with precision 0 . 05; we then average beforeinverting to estimate a localization length: ξ j := [¯ g c ] − = (cid:34) N realizations (cid:88) realizations r g c ; rj (cid:35) − . (32)In Fig. 7 we show ξ j as a function of eigenstate fraction j (cid:0) NN/ (cid:1) − . We mark ξ j = cL (33)with c chosen via finite-size scaling; this gives a heuristicestimate of the phase transition.The resulting phase diagram is broadly consistent withthat of 16. We see an apparent mobility edge for 1 (cid:46) W (cid:46) 4, as well as full localization (per our criterion (33))for W c (cid:39) 4. (Our critical disorder is different because wework at a different interaction strength.) We also see aslight asymmetry in (cid:60) E ↔ −(cid:60) E , again consistent with16.Finite-size scaling gives a better estimate for W c , aswell as an estimate for the correlation length exponent ν . In addition to averaging over disorder realizations,we average over 10 eigenstates n = 0 . (cid:0) NN/ (cid:1) through n + 9 = 0 . (cid:0) NN/ (cid:1) + 9 near the middle of the spectrum: ξ = [¯ g c ] − = N realizations (cid:88) realizations r n +9 (cid:88) j = n g c ; rj − . (34)This gives cleaner statistics, but does not appreciablychange the scaling parameters we extract. By seekinga scaling collapse (Fig. 8—cf App. C), we find W c ≈ ν ≈ 1. Our system sizes are very small, so we donot claim this scaling collapse reflects the ultimate large-system properties of the transition. (In looking for ul-timate large-system behavior, we would need to in ad-dition check for Kosterlitz-Thouless behavior. )Nevertheless, even at these small sizes our collapse is notconsistent with the result of Hamazaki et al., who find acorrelation-length exponent ν = . .Like us, Ref. 38, Hamazaki et al., investigates a PT-breaking transition in a localized many-body system,with a finite non-Hermitian tilt. Our measurements,however, differ from those of Ref. 38 both ontologicallyand operationally. Ontologically, Hamazki et al., treatthe non-Hermitian Hamiltonians as objects of study intheir own right. They fix g (cid:54) = 0 and look for a phase tran-sition as a function of disorder width, W . We, by con-trast, use non-Hermitian Hamiltonians as indicators ofthe properties of the underlying g = 0 Hermitian Hamil-tonian. Particularly, we are seeking to explore the prop-erties of the delocalization transition of the g = 0 system,and our scaling plots refer to the g = 0 transition only.The g = 0 transition could well have different univer-sal properties than the g > each eigenvalue and average.A straightforward interpretation of our finite-size scal-ing (Fig. 8) implies that our localization length ξ = g − c diverges at the transition. This is in striking contrastto the avalanche theory of the localization transition,which posits a finite typical localization length at thetransition The reason for the contrast is that the localizationlength used as a parameter in the avalanche picturemeasures the decay of matrix elements; the avalancheresults from the competition between that decay andHilbert space growth. Our g − c , in contrast, measuresthe competition directly: it is a quantity with dimen-sions of length measuring a competition between matrixelements of the end-to-end hopping c † c N and the many-body Hilbert space dimension, characterized by the gapsbetween eigenstates. In the language of 43 Sec. IV A, our g − c is ξ = g − c = [ s − l ∗ ] − (35)where s is the entropy density and l ∗ is the localiza-tion length associated with operator matrix elements.To see this, recall that before an eigenvalue can developan imaginary part, it must become degenerate with an-other eigenvalue. So if we imagine gauging all of theflux to one bond and adding that bond perturbatively,we find that the change in energy induced by the term te gN c † c N must be comparable to the gap between theeigenstate in question and one of its neighbors. This isprecisely our argument leading up to Eq. (5), with now | n (cid:105) a many-body eigenstate and ∆ E the gap between | n (cid:105) and another (many-body, interacting) eigenstate nearbyin energy. From the expression (35) it is clear that ourlocalization length ξ = g − c can diverge even when thelocalization length l ∗ associated with operator matrix el-ements is finite, and that our g − c should diverge at thecritical value of l ∗ predicted by either the straightforwardlogic of 43 or the more detailed logic of the avalanche pic-ture. g − c also immediately measures coherent end-to-endtransport in a finite segment of a chain. We showed thisexplicitly in a non-interacting chain, but even in an in-teracting chain we can see by rearranging Eq. (5) to e Ng c ∼ (cid:104) n | c † N c | n (cid:105) ∆ E − (36)that g c measures the magnitude of something having theform of an end-to-end Green’s function. (Note once againthat here | n (cid:105) and ∆ E are eigenstates and gaps of themany-body interacting Hamiltonian).We expect that the origin of the critical divergence of g − c is best understood in the context of long-range reso-nant structures; it may provide a useful diagnosticof those resonances. FIG. 7. Phase diagram of the disordered, interacting Hamiltonian (1) at system size N = 12, interaction strength U = 1 ( left ) and U = 3 ( right ), half filling, extracted from critical tilts. The color scale is ξ ≡ [¯ g c ] − ; we show it as a functionof disorder width W and eigenstate fraction with eigenstates sorted from lowest to highest. See Fig. 11 for localization lengthas a function of energy. Red dots indicate ξ = 0 . N ( U = 1, left) or ξ = 0 . N ( U = 3, right), which is consistent with thecrossing in the scaling collapse of Fig. 8. Compare to 16 Fig. 1; see Fig. 11 for the same data plotted against energy. B. Re-entrance in interaction strength Now fix the disorder width at W = 2 and vary the in-teraction strength U . (We show the resulting localizationlengths in Fig. 10.) At U = 0, the system is Andersonlocalized; as U increases we see the system delocalize(except near the band edges). But for U (cid:38) U ; in the more so-phisticated avalanche picture, we would expect increasinginteractions to increase the initial density of thermalized“bare spots”.But this reentrance is consistent with the work of Beraet al. They study the random-field XXZ model, aswe do; they characterize MBL by probing the extent towhich eigenstates of the many-body interacting Hamil-tonian can be approximated by Slater determinants ofsingle-particle states. They find (in their Fig. 1b) a reen-trance in interaction very similar to ours. VI. CONCLUSION In this manuscript we showed that an imaginary vectorpotential can provide direct access to localization lengthsof noninteracting as well interacting localized systems.The crucial quantity is the “critical tilt”, i.e., the vectorpotential at which an eigenenergy develops an imaginary part. We argue that the critical tilt measures the local-ization length of the underlying Hermitian system. Weshow this explicitly for the non-interacting limit with alead site connecting the two ends of an open system. Im-portantly, we show that the localization length of an opendisordered chain is given directly by the critical tilt (orcritical imaginary flux per bond) of a ring made of theopen chain plus a tunneling site. We then argue that theconnection remains for ordinary periodic boundary con-ditions, and that interactions cause a “broadening” in theappearance of imaginary eigenvalues. Finally, we use thecorrespondence to extract the localization length mostrelevant for transport properties in interacting, many-body localized, systems.By using the critical tilt to measure localization length,we study the MBL transition. We find re-entrance in theinteraction strength U , which is a priori surprising butconsistent with prior work , and with the MBL transi-tion found by Giudici et al. in U (1) lattice gauge theo-ries, where 1D Coulomb interactions cooperate with dis-order to localize the system, rather than competing. Wealso find a critical exponent ν ≈ 1, in agreement withother studies of the critical length exponent.Our work has finite overlap with the work by Hamazakiet al. , which studies directly the disorder-tuned PTbreaking transition of a disordered system with a finitetilt. We emphasize the difference between our works: Weare seeking to characterize the tilt-free MBL transition,whereas Ref. 38 studies the finite tilt transition and ob-tain a critical length exponent of ν = 1 / 2. Indeed, ourresults suggest that the two transitions—finite tilt and − − 10 0 10 20( W − W c ) N /ν . . . . ξ / N U = 1; ν = 1 . , W c = 4 . N = 8 N = 10 N = 12 − 50 0 500 . . − 20 0 20( W − W c ) N /ν . . . . . . ξ / N U = 3; ν = 1 . , W c = 6 . N = 8 N = 10 N = 12 − 50 0 500 . . . FIG. 8. Finite-size scaling collapse for the localizationlength ξ = [¯ g c ] − at interaction strength U = 1 ( top ) and U = 3 ( bottom ) with critical disorder width W c = 4 andcorrelation-length exponent ν = 1. We average over 10 eigen-states near the middle of the spectrum. We find a crossing at ξ = 0 . N and ξ = 0 . N (where N is system size) for U = 1and U = 3, respectively. The grey horizontal lines indicatethose crossings. We show putative scaling collapse for dif-ferent W c , ν in Appendix C. These scalings result from theaverage of 99 ( U = 1) or 98 ( U = 3) disorder realizations.Errorbars come from nonparametric bootstrap. zero-tilt—are in different universality classes, as we con-firm earlier observations on finite systems of ν = 1 atthe transition. This raises the possibility that the non-Hermitian system could provide differentiation betweenthe different length scales explored, e.g., in Ref. 23.Our per-eigenstate critical tilt measures a localizationlength of each eigenstate. It could be recast as the criticaltilt in each energy window, [ E, E + δE ), as is done in Figs.11 and 12 in App. B. This is in some sense the MBL-sidemirror image of the slow thermalization rates measuredby Pancotti et al. on the ETH side of the transition.Pancotti et al. characterize the distribution of opera-tor decay rates of the most nearly conserved local oper-ators in terms of extreme value statistics; these anoma-lously slow decay rates probe the least thermal states onthe ETH side of the MBL transition—those states thattake the longest to decay to equilibrium. As disorder in-creases they find a crossover from tight Gumbel statisticsto heavy-tailed Fr´echet statistics. Measuring critical tiltin an energy window, by contrast, would measure the lo-calization lengths of the least localized states on the MBL . . . . . g . . . . C oa l e s ce n ce r a t e ∂ ¯ f i m ag ∂ g N = 10 , W = 1 . N = 12 , W = 1 . N = 14 , W = 1 . N = 10 , W = 8 . N = 12 , W = 8 . N = 14 , W = 8 . FIG. 9. Distribution of coalescence points : rate ofchange ∂∂g ¯ f imag of fraction of eigenenergies having imaginaryparts as a function of g for interaction strength U = 1 anddisorder width W = 1 (dots), W = 8 (solid). The averageis over N realizations = 100 disorder realizations for N = 10 , N realizations = 10 disorder realizations for N = 14sites; errorbars are √ N realizations std ∂f imag ∂g . In the ETH phase( W = 1) the distribution is peaked near g = 0; in MBL phase( W = 8) it is peaked at some finite g max ∼ g c . This disorderaverage does not display a critical g c , because for any g therewill be disorder realizations with critical g c < g . The be-havior of ∂∂g ¯ f imag (cid:12)(cid:12)(cid:12) MBL near g = 0 therefore indirectly probesrare “quasi-thermal” disorder realizations. This provides a di-agnostic for the unrenormalized parameters of the avalanchepicture . side of the transition. It would be interesting to charac-terize the distribution of g c across disorder realizationsin terms of extreme value statistics. This would be thesubject of future work.It is also interesting to consider the critical tilt in lightof the avalanche picture of De Roeck et al. . In theavalanche narrative, one adds interactions to an Ander-son insulator via a quasi-perturbative RG scheme; re-gions where interactions cannot be treated perturbativelyare treated as thermal inclusions. They take the micro-scopic system to be parametrized by two parameters, anAnderson localization length and a density of these initialthermal inclusions. This is the basis for the RG picturein Ref. 22. In this picture it is not enough to considerthe critical g c in some energy window: this corresponds(we expect) to the localization length of the least localized eigenstate. But a single delocalized eigenstate should notbe enough to destabilize a surrounding localized region.Rather, one needs a finite fraction of eigenstates to be de-localized. In this picture g finite-frac (for systems of somefixed size N ) corresponds to the localization length that isthe key variable in the avalanche picture RG flow; in prin-ciple, computing g finite-frac as a function of system sizewill allow one to probe the flow of that variable, provid-0 FIG. 10. Re-entrance of the localized phase as a func-tion of interaction strength: Localization length (ex-tracted from critical tilt, ξ = [¯ g c ] − ) as a function of eigen-state fraction and interaction strength U for N = 12-site dis-ordered chains with width W = 2, at half-filling. See Fig. 10for localization length as a function of energy. There is anintermediate delocalized regime between U ≈ . − 4. Strongrepulsion in the large U limit freezes the system ; cf thework of Giudici et al. on lattice gauge theories, in which con-finement plays a similar role . The low-energy delocalizedregime for U (cid:29) U = 2. See Fig. 12 forthe same data plotted against energy. ing a sensitive test of the avalanche picture. Because—foropen boundary conditions—eigenstates of the tilted sys-tem are gauge-equivalent to eigenstates of the underlyingHermitian system, tensor network techniques maygive access to these quantities for large systems.The finite-fraction tilt g finite-frac will also probe theunrenormalized “bare spot” probability: that is, theprobability that a subsystem will be thermal. Recall(Fig. 9) that the distribution of g c extends all the wayto zero, even for large disorder width. This is becausesome (anomalous) disorder realizations have eigenstatesstretching across the system. If a particular disorderwidth has small g finite-frac , it is effectively thermal—thatis, it is a “bare spot”, in the language of the avalanchepicture. More careful measurements of the distribution of g c , and the analogous distribution for g finite-frac , at smallsystem size will therefore also characterize the unrenor-malized, microscopic inputs into the avalanche picture. ACKNOWLEDGMENTS We thank Bernd Rosenow, Sarang Gopalakrishnan,and Vadim Oganesyan for many helpful conversations;we also thank Naomichi Hatano and an anonymous re-viewer for commentary that prompted us to sharpen ourunderstanding and arguments.GR is grateful for funding from NSF grant 1839271as well as to the Simons Foundation, the Packard Foun-dation, and the IQIM, an NSF frontier center partiallyfunded by the Gordon and Betty Moore Foundation. Theauthors thank FAU Erlangen-N¨urnberg’s Prof. Dr. KaiP. Schmidt for setting up and accompanying the teamof researchers involved in this work. 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Appendix A: Jordan-Wigner tranform For convenience we note that the Hamiltonian (1) hasJordan-Wigner transform H = t (cid:88) j [ e g c † j c j +1 + e − g c † j +1 c j ]+ U (cid:88) j n j n j +1 + (cid:88) j h j n j (A1)= t (cid:88) j [ e g S + j S − j +1 + e − g S + j +1 S − j ]+ U (cid:88) j S zj S zj +1 + (cid:88) j h j n j (A2) ∝ t (cid:88) j 12 [ e g S + j S − j +1 + e − g S + j +1 S − j ]+ 12 U (cid:88) j S zj S zj +1 + (cid:88) j h j S zj (A3)which in the g = 0 case reduces to H ∝ (cid:88) j (cid:20) t ( S xj S xj +1 + S yj S yj +1 ) + U S zj S zj (cid:21) + (cid:88) j h j S zj (A4)(hence the choice of factors of 2).2 FIG. 11. Phase diagram of the disordered, interacting Hamiltonian (1) at system size N = 12, interaction strength U = 1 ( left ) and U = 3 ( right ), half filling, extracted from critical tilts. The color scale is ξ ≡ [¯ g c ] − ; x axis is still disorderwidth W ; y axis is now energy (rescaled by bandwidth); cf Fig. 7. Red dots indicate ξ = 0 . N ( U = 1, left) or ξ = 0 . N ( U = 3, right), which is consistent with the crossing in the scaling collapse of Fig. 8. Appendix B: Phase diagrams as a function of energy In Fig. 7 we plotted the localization length ξ (ex-tracted from the tilt) as a function of disorder widthand the “eigenstate fraction”—where in the sorted listof eigenstates a particular eigenstate falls. In Figs 11, 12we plot the localization length ξ as a function of energy(normalized by the bandwidth of each disorder realiza-tion). To be more precise, we1. average the energies for eigenstate α ∈ . . . (cid:0) NN/ (cid:1) (at fixed disorder width W and interactionstrength), and2. average the critical tilt g c for eigenstate α and ex-tract the localization length.We plot the localization length (averaged in this sense)against the disorder width W or interaction strength U and the energy (averaged in this sense).This changes the shape of the phase diagram, becausethe density of states is heuristically ρ ( E ) ∝ σ √ πN e − E / σ √ N , σ ∼ (cid:112) N ( t + W c )(B1)(before bandwidth normalization).The rescaling highlights certain “glitches” (e.g. inFig. 11 near (cid:60) ( E ) ≈ . real-space renormalization group arguments in our explana-tion above.) Appendix C: Finite-size scaling In Sec. V A we claimed that our finite-size scalings gave W c = 4 , U = 1 , ν = 1 for bothinteraction strengths. In Figs 13-16 we show (putative)finite-size scalings for a variety of W c , ν , so the readercan judge the accuracy and precision of our claims.3 FIG. 12. Re-entrance of the localized phase as a func-tion of interaction strength and energy: Localizationlength (extracted from critical tilt, ξ = [¯ g c ] − ) as a func-tion of eigenstate energy (rescaled by bandwidth) and interac-tion strength U for N = 12-site disordered chains with width W = 2, at half-filling (cf Fig. 10). There is an intermediatedelocalized regime between U ≈ . − 4. Strong repulsion inthe large U limit freezes the system ; cf the work of Giudiciet al. on lattice gauge theories, in which confinement plays asimilar role . The low-energy delocalized regime for U (cid:29) U = 2. − 100 0 100 200( W − W c ) N /ν . . . . ξ / N U = 1; ν = 0 . , W c = 4 . N = 8 N = 10 N = 12 − 500 0 5000 . . − 20 0 20( W − W c ) N /ν . . . ξ / N U = 1; ν = 0 . , W c = 4 . N = 8 N = 10 N = 12 − 100 0 1000 . . − − 10 0 10 20( W − W c ) N /ν . . . . ξ / N U = 1; ν = 1 . , W c = 4 . N = 8 N = 10 N = 12 − 50 0 500 . . − − W − W c ) N /ν . . . . . ξ / N U = 1; ν = 1 . , W c = 4 . N = 8 N = 10 N = 120 500 . . − − W − W c ) N /ν . . . . ξ / N U = 1; ν = 1 . , W c = 4 . N = 8 N = 10 N = 120 250 . . FIG. 13. Putative finite-size scaling for U = 1 , W c = 4, and avariety of ν . − − 10 0 10 20( W − W c ) N /ν . . . . . . ξ / N U = 1; ν = 1 . , W c = 3 . N = 8 N = 10 N = 120 500 . . − − 10 0 10 20( W − W c ) N /ν . . . . . . ξ / N U = 1; ν = 1 . , W c = 3 . N = 8 N = 10 N = 120 500 . . − − 10 0 10 20( W − W c ) N /ν . . . . ξ / N U = 1; ν = 1 . , W c = 4 . N = 8 N = 10 N = 12 − 50 0 500 . . − − 10 0 10 20( W − W c ) N /ν . . . . . ξ / N U = 1; ν = 1 . , W c = 4 . N = 8 N = 10 N = 12 − 50 0 500 . . FIG. 14. Putative finite-size scaling for U = 1 , ν = 1 . 0, and avariety of W c . − − 10 0 10 20( W − W c ) N /ν . . . . . ξ / N U = 3; ν = 1 . , W c = 5 . N = 8 N = 10 N = 12 − 50 0 500 . . . − − 10 0 10 20( W − W c ) N /ν . . . . . ξ / N U = 3; ν = 1 . , W c = 5 . N = 8 N = 10 N = 12 − 50 0 500 . . . − 20 0 20( W − W c ) N /ν . . . . . . ξ / N U = 3; ν = 1 . , W c = 6 . N = 8 N = 10 N = 12 − 50 0 500 . . . − − 10 0 10 20( W − W c ) N /ν . . . . ξ / N U = 3; ν = 1 . , W c = 6 . N = 8 N = 10 N = 12 − 50 0 500 . . . − − 10 0 10 20( W − W c ) N /ν . . . . ξ / N U = 3; ν = 1 . , W c = 6 . N = 8 N = 10 N = 12 − 50 00 . . . FIG. 15. Putative finite-size scaling for U = 3 , ν = 1 . 0, and avariety of W c − − 100 0 100 200( W − W c ) N /ν . . . . . . ξ / N U = 3; ν = 0 . , W c = 6 . N = 8 N = 10 N = 12 − 500 0 5000 . . . − − 25 0 25 50( W − W c ) N /ν . . . ξ / N U = 3; ν = 0 . , W c = 6 . N = 8 N = 10 N = 12 − 100 0 1000 . . . − 20 0 20( W − W c ) N /ν . . . . . . ξ / N U = 3; ν = 1 . , W c = 6 . N = 8 N = 10 N = 12 − 50 0 500 . . . − 20 0 20( W − W c ) N /ν . . . . . ξ / N U = 3; ν = 1 . , W c = 6 . N = 8 N = 10 N = 12 − 50 00 . . . − 10 0 10( W − W c ) N /ν . . . . ξ / N U = 3; ν = 1 . , W c = 6 . N = 8 N = 10 N = 12 − 25 00 . . . FIG. 16. Putative finite-size scaling for U = 3 , W c = 6, and avariety of= 6, and avariety of