Many-body localization of 1D disordered impenetrable two-component fermions
M. S. Bahovadinov, D. V. Kurlov, B. L. Altshuler, G. V. Shlyapnikov
MMany-body localization of 1D disordered impenetrable two-component fermions
M. S. Bahovadinov,
1, 2, ∗ D. V. Kurlov, B. L. Altshuler,
3, 1 and G. V. Shlyapnikov
1, 4, 5, † Russian Quantum Center, Skolkovo, Moscow 143025, Russia Physics Department, National Research University Higher School of Economics, Moscow, 101000, Russia Physics Department, Columbia University, 538 West 120th Street, New York, New York 10027, USA Universit´e Paris-Saclay, CNRS, LPTMS, 91405 Orsay, France Van der Waals-Zeeman Institute, Institute of Physics, University of Amsterdam,Science Park 904, 1098 XH Amsterdam, The Netherlands
We study effects of disorder on eigenstates of 1D two-component fermions with infinitely strongHubbard repulsion. We demonstrate that the spin-independent (potential) disorder reduces theproblem to the one-particle Anderson localization taking place at arbitrarily weak disorder. Incontrast, a random magnetic field can cause reentrant many-body localization-delocalization tran-sitions. Surprisingly weak magnetic field destroys one-particle localization caused by not too strongpotential disorder, whereas at much stronger fields the states are many-body localized. We presentnumerical support of these conclusions.
Introduction . Many-body localization (MBL) emergesfrom an interplay between disorder and interparticle in-teractions, extending the phenomenon of Anderson lo-calization (AL) to interacting many-body systems [1, 2].Numerous works on the MBL systems revealed their dis-tinctive properties, such as vanishing steady transport,absence of thermalization, area-law scaling of entangle-ment entropy, and the presence of quasi-local integrals ofmotions [3–19]. For recent reviews, see e.g. [20–22]MBL has been extensively studied in systems of inter-acting one-dimensional (1D) spinless fermions and vari-ous spin chain models [23–26]. All these models possessonly a single local degree of freedom. However, this istypically not the case in more realistic systems. For in-stance, the Hubbard model for two-component fermionspossesses two local degrees of freedom, charge and spin,both of which can be coupled to disorder. Recent numer-ical studies deal with the disordered 1D Hubbard modelin the regime of a finite interaction strength [27–32]. Theresults suggest that a sufficiently strong random poten-tial localizes the charge degree of freedom, whereas spinexcitations apperently exhibit a subdiffusive transport,i.e. remain delocalized [33–36]. A symmetric situationis observed in the presence of a sufficiently strong ran-dom magnetic field coupled to spins: spin excitations arelocalized whereas charge excitations are extended [37].This Letter is devoted to the 1D Hubbard model fortwo-component fermions in the limit of infinitely strongon-site repulsion, U = ∞ . Theoretical considerationssupported by the numerical analysis allowed us to demon-strate that arbitrarily weak potential disorder leads tothe localization of both charge and spin excitations. In-deed, each site of the 1D lattice in this limit can bein 3 (rather than 4 as in the case of U < ∞ ) distinctstates: empty and single occupied with 2 possible spinorientations. The absence of the double occupation re-duces the possible dynamics to the motion of ’emptyspaces’ (holons). The holons behave themselves like non-interacting fermions [38] even in the presence of a spin- independent disorder [39] and thus are subject to theconventional single-particle AL. A weak magnetic disor-der causes an effective interaction between the holons,which can result in the delocalization, unless the disor-der is sufficiently strong. Model and its symmetries . The Hamiltonian of theconventional 1D Hubbard model with an infinitely strongon-site repulsion on a ring with L sites is, H = − t L (cid:88) i =1 (cid:88) σ = ↑ , ↓ P (cid:16) c † i,σ c i +1 ,σ + H.c. (cid:17) P . (1)Here c j,σ is the annihilation operator of a fermion inthe spin state σ in the site j . The operator P = (cid:81) i (1 − n i, ↑ n i, ↓ ) with n i,σ = c † i,σ c i,σ , projects out thestates with doubly-occupied sites prohibited due to theinfinite onsite repulsion, and t is the nearest neighborhopping amplitude. Below we impose periodic bound-ary conditions ( c L +1 = c ) and set t = 1. The integrableHamiltonian (1) was fully analyzed by means of the BetheAnsatz [40]. The disorder is represented by the additionalterm in the Hamiltonian: H D = L (cid:88) i =1 ε i ( n i, ↑ + n i, ↓ ) + L (cid:88) i =1 h i n i, ↑ − n i, ↓ , (2)where ε i and h i are the random potential and magneticfield, respectively. We assume that both are uniformlydistributed: ε i ∈ [ − W, W ] and h i ∈ [ − B, B ].The disorder breaks translational and SU(2) symme-tries, but the Hamiltonian H = H + H D still pos-sesses a number of symmetries. First of all, H con-serves the number of particles with a given spin projec-tion N {↑ , ↓} = (cid:80) i n i, {↑ , ↓} . Moreover, the infinite repul-sion does not allow particles with opposite spins to ex-change their positions, i.e. the spin pattern is conservedup to cyclical transmutations. Localization measures.
Quantum states of themodel (1,2) with a fixed set of conserved quantities form a r X i v : . [ c ond - m a t . d i s - nn ] F e b a N H -dimensional Hilbert space, with N H < L . Wewill analyze many-body eigenstates in the basis of stateswith a given occupation (0-empty, ↑ , ↓ -occupied by a par-ticle with a particular spin direction) on each lattice site(computational basis): | s (cid:105) = | s (cid:105) ⊗ | s (cid:105) ⊗ ... ⊗ | s L (cid:105) and | s i (cid:105) ∈ {| (cid:105) , | ↑(cid:105) , | ↓(cid:105)} .Several quantitative measures have been proposed tocharacterize localization properties of the wavefunctions ψ α ( s ) = (cid:104) s | α (cid:105) . Analysis of the scaling of the participa-tion entropies S q with N H allows one to determine a setof fractal dimensions D q : S q = 1 q − (cid:32) N H (cid:88) s =1 | ψ α ( s ) | q (cid:33) N H →∞ −−−−−→ D q ln ( N H ) . (3)The eigenstates | α (cid:105) localized (LO) on a finite set of | s (cid:105) have S q independent of N H and thus D q = 0 for any q >
0. Extended ergodic (EE) states with | ψ α ( s ) | ∼ N H− are characterized by D q = 1. The states with 0 < D q < → NEE and NEE → EE at two different strengths of thedisorder.The fractal dimensions D q may be not very useful inthe numerical identification of the LO → NEE transition,because it is difficult to distinguish D q = 0 from 0 In our exact diagonalization ofthe Hamiltonian H = H + H D in the computationalbasis, we choose L to be multiple of 4, L ∈ { , , } ,and restrict ourselves by the considiration of sector with FIG. 1. L =4 ring with one holon (empty site) and spinpattern | ↑↓↑(cid:105) . A clockwise detour (0 (cid:8) 4) of the holon leadsto a cyclical transmutation | ↑↓↑(cid:105) ⇒ | ↓↑↑(cid:105) . The systemreturns to the original state only after 3 detours: | ↑↓↑(cid:105) ⇒| ↓↑↑(cid:105) ⇒ | ↑↑↓(cid:105) ⇒ | ↑↓↑(cid:105) . Therefore, φ = ± π . Note that fora purely potential disorder the states 0, 4 and 8 have the samepotential energy, i.e. φ can be viewed as a quasimomentumof the system. In general, φ is a multiple of 2 π/ Z . N ↑ = N ↓ = L . The size of this sector is (cid:0) LL/ (cid:1) × (cid:0) L/ L/ (cid:1) .The conservation of the spin pattern further reduces thedimension of the Hilbert space to N H = Z (cid:0) LL/ (cid:1) , where Z is the number of distinct spin sequences connected by thecyclical transmutations for a given spin pattern. Belowwe only consider the Ne´el spin pattern ( Z = 2) with thecorresponding N H ∈ { , , } . We employthe shift-invert exact diagonalization algorithm to obtain m ∈ { , , } states from the central part of the en-ergy spectrum. We average KL , D and D over these m states and then perform averaging over N d realizationsof the disorder (it varies from N d ∼ for the smallestsystem size up to N d ∼ for the largest one). For laterconvenience, we parametrize the strengths of the randompotential and magnetic field as, B = ρ D cos( θ ) , W = ρ D sin( θ ) , (5)where ρ D > ρ D and θ , ranging from the random potential ( θ = π/ B = 0) to the random magnetic field ( θ = 0 and W = 0). Potential disorder. In the Bethe Ansatz solution of thedisorder-free problem governed by the Hamiltonian (1)each holon is characterized by momentum k j . The onlydifference of the holons from free spinless fermions on aring is the quasi-periodic rather than periodic boundarycondition that effects the quantization, k j L = 2 πn j + φ , n j ∈ Z . Qualitatively, the extra phase φ can be under-stood as follows. The infinite Hubbard repulsion pre-serves the spin ordering but only up to cyclical transmu-tations. Each detour of a holon around the ring causessuch a transmutation. To recover the original patternthe holon should make a number of detours equal to thenumber of non-equivalent cyclical transmutations of thepattern Z (see Fig. 1). For the spin pattern fixed upto the cyclical permutations the many-body states of theclean system | { k j } , φ (cid:105) are characterized by the momentaof the holons { k j } and the total ”quasimomentum” φ ofthe particles. The quantity φ should be a multiple of2 π/ Z . For the Ne´el pattern with an even number of par-ticles we have Z = 2 allowing only φ = 0 and φ = π .All of the above applies to the disordered model H + D D ρ D 228 81624242432 K L L=12, N H =1848L=16, N H =25740L=20, N H =369512 Region I Region II KL = 2 (d)(e)(f) (a) (b)(c) FIG. 2. The fractal dimensions D (a), D (b), and theKullback-Leibler divergence KL (c) versus the total disorderstrength ρ D for the case of θ = 0 (random magnetic field only,no potential disorder) for L = 12 (black filled circles), L = 16(red open diamonds) and L = 20 (blue filled triangles). Insets(d)-(f) show the data near the critical point ρ D (cid:39) . 57. InRegion I the states are extended, whereas in Region II theyare localized. The boundary of the regions is determined fromthe drifted intersection point of D curves. Error bars smallerthan the symbol sizes are omitted. H D as long as the disorder is purely potential ( B = 0 ,θ = π in Eq. (5)). The potential disorder alters onlyone-particle states of the holons. The plane waves aresubstituted by the eigenfunctions of a particle subjectto the random potential { ε i } with quasiperiodic bound-ary conditions. These one-particle states are known tobe localized for arbitrarily small W . It is important toemphasize that as long as the random magnetic field isabsent the disorder does not lead to any interaction be-tween the holons and the occupation numbers of the one-particle states are conserved. The many-body wave func-tion is characterized by a macroscopically large set { µ j } φ instead of { k j } , where each { µ j } φ labels a localized one-particle wave function with a fixed φ . This means thatfrom the many-body point of view the problem remainsintegrable even in the presence of the potential disorder.The Bethe Ansatz based theory of the Hubbard modelwith potential disorder will be published elsewhere [39]. Random magnetic field. In the presence of a spin-dependent disorder ( θ < π/ φ , i.e. gives rise to the off-diagonal matrixelements of the Hamiltonian, (cid:104) (cid:8) µ (cid:48) j (cid:9) , φ (cid:48) | H | { µ j } , φ (cid:105) (cid:54) = 0.We thus come to the conclusion that the random mag-netic field plays a two-fold role. First, like the potentialdisorder, it tends to transform the plane waves into local- ized one-particle states. On the other hand, it hybridizesdifferent | µ j (cid:105) - states and can cause the many-body de-localization in the | { µ j } , φ (cid:105) -basis.Below we present the results of numerical studies of theMBL transition in the computational basis {| s (cid:105)} ratherthan in the {| { µ j } , φ (cid:105)} basis, i.e. in the clean limit( H = H , ρ D = 0) the many- body eigenstates | { k j } , φ (cid:105) are apparently extended, whereas the one-particle local-ization of the holon states | µ j (cid:105) due to the potential disor-der implies MBL in the computational basis. We there-fore have to check numerically the following predictions:i) If the disorder is purely magnetic, θ = 0 , the many-body states remain extended for a finite but weak enoughdisorder because the problem is not a 1D one-particle one.With increasing the randomness ρ D the system eventu-ally becomes many-body localized, i.e. it undergoes theMBL transition; ii) For the purely potential disorder, θ = π/ 2, the eigenstates are always localized in the {| s (cid:105)} - basis. Sufficiently strong magnetic disorder can destroythis localization if ρ D is small enough.Fig. 2 presents the results of numerical diagonalizationof the Hamiltonian (1), (2) in the absence of the potentialdisorder ( W = 0, θ = 0, ρ D = B ) for different values ofthe system size L . One can see that for a weak disorderthe fractal dimensions D , are close to the ergodic value1. Moreover, they increase with L , i.e. it is plausiblethat the quantum states remain not only extended butalso ergodic. The deviations from D , = 1 can be viewedas a finite size effect. If the disorder is strong, ρ D (cid:38) . D , decrease as L increases andit is likely that D , → L → ∞ , which correspondsto the LO phase. At the transition between extendedand LO regimes the curves D , ( ρ D ) for different (largeenough) L should cross at a critical disorder ρ D = ρ CD .It looks like this happens and ρ CD ≈ . D , increase with L but are much smaller than 1 and decrease as the dis-order gets stronger. One may assume that in the limit L → ∞ the fractal dimensions remain between 1 and 0.This would signal the existence of the NEE phase. How-ever it is impossible to exclude that D , L →∞ −−−−→ 0. Thedivergence KL is sensitive to the transition to the LOstate but not to the EE → NEE transition. We foundthat KL is practically L -independent for ρ D < ρ CD as itshould be if the states are extended. The KL ( L ) - depen-dence at ρ D > ρ CD is another evidence of the LO phaseat large disorder. Note that with the accuracy limitedby the finite size of the system the analysis of the datafor both the fractal dimensions D , and KL yields thesame value of the critical disorder. Two-component disorder. Fig. 3 presents the numeri-cal calculation of localization properties in the presenceof both random magnetic field and random potential( W > , B > , < θ < π/ D , and KL on θ at a fixed total disorder D D L=12, N H =1848L=16, N H =25740L=20, N H =369512 θ/π 246 8 K L (a)(b)(c) (d)(e)(f)KL=2 FIG. 3. The fractal dimensions D (a), D (b), and theKullback-Leibler divergence KL (c) versus the angle θ [seeEq. (5)] at the total disorder strength ρ D = 2 for L = 12(black filled circles), L = 16 (red open squares), and L = 20(blue open triangles). Right panel (d)-(f) shows the datanear θ = π/ 2. Error bars smaller than the marker size arenot shown. ●●●●● ● ● ● ● ● ● ● ● ● ● ●●● W LocalizedExtended FIG. 4. A sketch of the phase diagram in B − W space,based on the finite-size calculations of the fractal dimensions D . The colorbar indicates the value of D for L = 16. ρ D = √ W + B = 2. First note that at θ = π/ 2, i.e. inthe potential limit, KL increases with the system size L and both D and D decrease as L grows. This behav-ior is an indication of the LO phase, which we expect.This localization is well pronounced being compared tothe extended phase, exhibited in purely magnetic disor-der. The last can be seen from Fig. 2 and is also clearfrom the dependence of D , and KL on θ close to θ = 0in Fig. 3. It is difficult to find any deviation of KL fromthe ergodic value KL = 2, whereas both D and D areclose to 1 and increase with L . The most interesting and somewhat unexpected is thefact that this extended behavior persists at ρ D = 2 foralmost all values of θ . Deviations are visible only when π/ − θ becomes smaller than 0 . π (which correspondsto B < . D , and KL indicatesthe transition to the LO phase at θ C ≈ . π , i.e. B C (cid:39) . 01. This tiny field is sufficient to destroy the LOphase created by the potential disorder W ≈ 2. This canbe seen from Fig. 4 presented in terms of the potentialdisorder strength W and magnetic field strength B . Inparticular, the LO phase can be recovered by increasingthe amplitude of the random magnetic field up to B (cid:39) B C < B (cid:46) H + H D are extended. It is possiblethat close to the edges of this interval the states loose er-godicity and become NEE. Unfortunately, the sizes L ofthe system available for our computations are too smallto justify this guess. However, the reentrance to the lo-calized regime with increasing B is hard to dispute.For ρ D (cid:29) ρ D (cid:39) . 57. In the other limit, ρ D (cid:28) 1, thestates are localized at θ = π . Even a weak random mag-netic field induces effective interholon interaction, whichcompletely delocalizes the states. This implies θ c → . π with ρ D → 0. Our final results for the phase diagram arepresented in Fig. 4. Conclusions. Compared to the potential static disor-der, the random magnetic field has a dramatically differ-ent effect on the quantum many-body states of the 1DHubbard model with infinitely strong on-site repulsion.The potential disorder does not violate the integrabilityof the problem - the holons remain free spinless fermionsalbeit subject to the random potential. Accordingly, theone-particle wave functions are localized at arbitrarilyweak disorder. To complete the mapping onto the stan-dard free fermion problem one has to take into accountthe quasiperiodic boundary conditions (QBC) for thesefunctions. The QBC reflect the cyclical permutation inthe spin sequence caused by the motion of the holons.Each stationary state of the system is characterized by aparticular occupation of single-holon states as well as bythe phase φ of the QBC, which can take several valuesdetermined by the spin pattern.The infinite on-site repulsion preserves the spin pat-tern up to cyclical permutations irrespective of the typeof disorder. In the potential case distinct cyclical trans-mutations do not change the potential energy giving risethe “quasimomentum”-phase φ of the QBC. Violation ofthis symmetry by the random magnetic field hybridizesthe states with different φ and different holon quantumnumbers making the system non-integrable and possiblyextended in the MBL sense.A very strong disorder of any type practically excludesthe cyclical transmutations and many-body states are lo-calized in the computational basis. For a purely poten-tial disorder it means that the eigenenergies are degen-erate in φ . If the potential disorder is weak but finite,a gradual reduction of the random magnetic field froma large value eventually causes the many-body delocal-ization. However a further decrease should recover thelocalized phase.These predictions are supported by the Bethe Ansatzbased analytics [39] and clearly demonstrate the reen-trant MBL behavior caused by the random magneticfield. Unfortunately, the size of the system available forthe exact diagonalization turns out to be too small forthe analysis of the ergodicity of the extended states. TheHamiltonian (1),(2) can be implemented in systems of ul-tracold atoms or existing multi-qubit arrays. Being qual-itatively transparent with well-developed analytical ap-proaches, this model has advantages over e.g. spin chainsas a laboratory for investigating MBL and searching forQuantum Supremacy [42].We thank V.E. Kravtsov, V. Gritsev, O. Gamayun,and V. Smelyanskiy for useful discussions. 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