Many-body localization in a non-Hermitian quasi-periodic system
MMany-body localization in a non-Hermitian quasi-periodic system
Liang-Jun Zhai , Shuai Yin , Guang-Yao Huang ∗ School of Mathematics and Physics, Jiangsu University of Technology, Changzhou 213001, China School of Physics, Sun Yat-Sen University, Guangzhou 510275, China and Institute for Quantum Information & State Key Laboratory of High Performance Computing,College of Computer, National University of Defense Technology, Changsha 410073, China (Dated: August 17, 2020)In the present study, the interplay among interaction, topology, quasiperiodicity, and non-Hermiticity is studied. The hard-core bosons model on a one-dimensional lattice with asymmetryhoppings and quasiperiodic onsite potentials is selected. This model, which preserves time-reversalsymmetry (TRS), will exhibit three types of phase transition: real-complex transition of eigenen-ergies, topological phase transition and many-body localization (MBL) phase transition. For thereal-complex transition, it is found that the imaginary parts of the eigenenergies are always sup-pressed by the MBL. Moreover, by calculating the winding number, a topological phase transitioncan be revealed with the increase of potential amplitude, and we find that the behavior is quitedifferent from the single-particle systems. Based on our numerical results, we conjecture that thesethree types of phase transition occur at the same point in the thermodynamic limit, and the MBLtransition of quasiperiodic system and disordered system should belong to different universalityclasses. Finally, we demonstrate that these phase transitions can profoundly affect the dynamics ofthe non-Hermitian many-body system.
I. INTRODUCTION
In recent years, as an extension of the noninteractingAnderson localization, a phenomenon termed as many-body localization (MBL) in the quantum many-body sys-tems has received a lot of attention . In such a phase,the system fails to act as a bath for its own subsystemsand thermalization does not occur. It has been estab-lished that the MBL phase has drastically different spec-tra and dynamical properties compared with the delo-calization (thermal) phase. Although MBL is usuallystudied for systems with random disorder, there is an-other type of system, the quasiperiodic system, whichalso supports MBL due to its unique features . Thequasiperiodic system breaks translational invariance bythe incommensurate period, and shows some randomlikeproperties similar to the disordered system. However,compared with the disordered system, the quasiperiodicsystem has a long-range correlation, and creates a dis-order in a more controlled way . Thus the quasiperi-odic system constitutes an intermediate phase betweena periodic system and a fully disordered system. In theprevious theoretical studies, it has been found that MBLphase transition in the Hermitian disordered system andquasiperiodic system belongs to two distinct universalityclasses , and MBL phase in the quasiperiodic systemis more stable as compared to the disordered system .Most recently, great interest has been devoted tostudying the MBL phenomena in the non-Hermitian sys-tems . The results showed that MBL signatures canbe restored even in the appearance of dissipation . Insome classes of non-Hermitian system with time-reversalsymmetry (TRS), there is a real-complex transition ofeigenenergies featuring parity-time (PT) symmetry .It is found that MBL can suppress the imaginary parts ofthe complex eigenenergies of the disordered system, and the real-complex transition occurs accompanied with theMBL phase transition . On the other hand, exotic topo-logical phases were unveiled in the non-Hermitian quan-tum systems . For the non-Hermitian single particlesystems, theoretical studies found that the localization-delocalization phase transition for both the disorderedand quasiperiodic systems has a topological nature, andthe localization and delocalization phases can be charac-terized by the winding number . For the Hermitianmany-body systems, it has been found that the interac-tions can destroy the topological phases or create newtopological phases which are topologically distinct fromthe trivial states , and MBL eigenstates can exhibitor fail to exhibit topological orders . However, for thenon-Hermitian many-body systems, there are few worksthat have been done to investigate the affection of theinteraction on the topological phase and the relations be-tween the topological and MBL phase transitions.With this background, the interplay among interac-tion, topology, quasiperiodicity, and non-Hermiticity isexamined in this paper. The study is applied to a hard-core bosons model on a one-dimensional lattice withasymmetry hoppings and quasiperiodic onsite potentials.The non-Hermiticity of the model comes from asymmetryhopping, but it still has the TRS. We find that the MBLphase transition, real-complex transition and topologicalphase transition coexist for this model, and the transi-tion points of these transitions are close. The obtainedcritical exponent of quasiperiodic system is different fromthe disordered system, which means that they belong todifferent universality classes. Based on our numerical re-sults, we conjecture that these three phase transitions oc-cur at the same point in the thermodynamic limit. Sincethe real-complex transition and MBL phase transitioncan profoundly affect the dynamics of the non-Hermitiansystem , the dynamical behaviors of the real part of a r X i v : . [ c ond - m a t . d i s - nn ] A ug eigenenergy and entanglement entropy are studied.The remainder of the paper is organized as follows.In Sec. II, the model of the non-Hermitian quasiperiodicsystem is presented. Numerical investigation is presentedin Sec. III. A summary is given in Sec. IV. II. MODEL
A non-Hermitian hard-core bosons model on a one-dimensional lattice is considered in the present study.The Hamiltonian readsˆ H = L (cid:88) i =1 [ − J ( e − g ˆ b † i +1 ˆ b i + e g ˆ b † i ˆ b i +1 ) + U ˆ n i ˆ n i +1 (1)+ W i ˆ n i ] . Here, ˆ b i and ˆ b † i are the annihilation and creation opera-tors of a hard-core boson, and ˆ n i = ˆ b † i ˆ b i is the particle-number operator at site i . J and g label the asymme-try hopping amplitude between the nearest-neighboring(NN) sites, and U is the interaction between NN sites.For a quasiperiodic system, the onsite potential is W i = W cos (2 παi + φ ), where W is the amplitude of the po-tential, and φ is the phase of the potential, and α isirrational for incommensurate potentials.For this model, the non-Hermiticity is controlled by theparameter g , but it still has the TRS. In the following,we assume J = 1, U = 2, g = 0 .
5, and the subspacewith fixed particle number M = L/ α is chosen as the inverse of thegolden ratio α = ( √ − /
2, which could be comparedwith the experimental results . The periodic boundarycondition is assumed in the following calculation. III. NUMERICAL RESULTS ANDDISCUSSIONSA. Real-complex transition of eigenenergies
Firstly, the real-complex transition of this model isstudied. As shown in Fig. 1, the eigenenergies of Hamil-tonian Eq. (1) with L = 12 and different W are plotted.Since the non-Hermitian Hamiltonian still has the TRS,it is found that the imaginary parts of the spectra aresymmetric around the real axis. With the increase of W ,the eigenenergies with nonzero imaginary part decrease.To measure the variation of the ratio of the complexeigenenergies with nonzero imaginary part, f Im is definedas f Im = D Im /D, (2)where the D Im is the number of eigenenergies withnonzero imaginary part, and D is the total number ofeigenenergies. Here, a cutoff of C = 10 − is used, that - 1 0 0 1 0- 101 - 4 0 - 2 0 0 2 0 4 0- 0 . 0 0 2- 0 . 0 0 10 . 0 0 00 . 0 0 10 . 0 0 2 W = 1 0 Im( E ) R e ( E ) W = 2 R e ( E ) Im( E ) FIG. 1. Eigenenergies of the Hamiltonian Eq. (1) with W = 2(left) and W = 10 (right). Here, the lattice size is L = 12. - 5 - 3 - 1 - 2 0 0 0 2 0 0 4 0 01 0 - 5 - 3 - 1 ( b ) ( W - W Rc ) L (cid:1) fImfIm W L ( a ) FIG. 2. (a) f Im as a function of W for L = 10 , ,
14 and 16,and (b) the rescaled curves according to Eq. (3). is, | Im E | ≤ C is identified to be a machine error. InFig. 2(a), f Im as a function of W for different L is plot-ted, and the results are obtained by averaging 500 choicesof φ for L = 10 ,
12, and 14, and 100 choices of φ for L = 16. Roughly speaking, when W ≤ W R C = 6 . f Im increases with the increase of L , while f Im decreases withthe increase of L for W ≥ W RC . The curves of f Im versus W can be rescaled by the following scaling function f Im ∝ ( W − W R C ) L /ν , (3)where ν = 0 .
7. As shown in Fig. 2(b), the rescaledcurves collapse onto each other, which confirms Eq. (3).These results demonstrate that in the thermodynamiclimit ( L → ∞ ) the model of Eq. (1) should have a real-complex transition at W = W R C , that is, when W < W R C the eigenenergies are almost complex, while the eigenen-ergies are almost real for W > W R C . Similar results havealso been found in the non-Hermitian Hamiltonian witha random onsite potential , but the scaling exponent ν for the disordered system is different from what we foundhere, which implies the real-complex transition shouldbe in different universality classes for the disordered andquasiperiodic systems. B. Topological phase transition
Different from the Hermitian systems, to study thetopological phases of non-Hermitian systems, not onlythe ground state but also the full complex spectra should - 4 0 - 2 0 2 0 4 0- 4 0- 2 02 04 0 (cid:2)(cid:1)(cid:4) (cid:3)(cid:1)(cid:4) - 0 . 40 . 4
R e ( b )
I m ( a )
R e I m
FIG. 3. (a) The Φ dependence of det H (Φ) / | det H (0) | in thecomplex plane for (a) W = 3 . W = 6 .
5. The latticesize is L = 10 and φ = π/ (cid:1) W p / 6 p / 2 (cid:1) ( b ) (cid:1) W ( a ) FIG. 4. (a)The W dependence of winding number ω with φ = 0 , π/ π/
2, (b) and the W dependence of averagedwinding number ω . The lattice size is L = 10. be taken into account . Therefore, a natural topolog-ical object arising from the complex energy plane is thewinding number, that is, a loop constituted by eigenener-gies which encircles a prescribed base point. The windingnumber is topologically stable and changes its value onlywhen the curve is crossing the base point. Recently, thewinding number has been defined to study the topologi-cal phase for non-Hermitian systems in the single particlepicture without interactions . Generalizing the ideaof defining the winding number to our interacting non-Hermitian systems, a parameter Φ is introduced througha gauge transformation ˆ b j → e i Φ L j ˆ b j and ˆ b † j → e − i Φ L j ˆ b † j ,which can be viewed as a magnetic flux Φ through non-Hermitian ring with length L is applied. The Hamilto-nian becomes H (Φ) = L (cid:88) j =1 [ − J ( e − g e − i Φ L ˆ b † j +1 ˆ b j + e g e i Φ L ˆ b † j ˆ b j +1 ) (4)+ U ˆ n j ˆ n j +1 + W i ˆ n j ] , and subsequently the winding number is defined as ω = (cid:90) π d Φ2 πi ∂ Φ ln det { H (Φ) − E B } . (5)Here, E B is the prescribed basis point which is not aneigenenergy of H (Φ). Different from the bulk-edge cor-respondence in the Hermitian systems, a positive (neg-ative) winding number ω implies a ω ( − ω ) independentedge modes localized at the left (right) boundary in the semi-infinite space. As demonstrated in Ref. , the wind-ing number does not depend on E B . The basis point ischosen as E B = 0 in the calculation, so that the loopensures the coexistence of the Im E <
E > H (Φ) / | det H (0) | toillustrate the loop winding around the base point . Dur-ing the variation of Φ from 0 to 2 π , det H (Φ) / | det H (0) | draws a closed loop in the complex plane, and if the loopwinds around the origin m times, the winding numberis ± m (+ means a counterclockwise winding, while − means a clockwise winding). In Fig. 3, the Φ dependenceof det H (Φ) / | det H (0) | with different W for L = 10 isplotted, and the phase is chosen as φ = π/
6. As seenin Fig. 3 (a), det H (Φ) / | det H (0) | draws a closed curvewith surrounding the origin four times in the complexplane for W = 3 .
5, while det H (Φ) / | det H (0) | draws aclosed curve without surrounding the origin for W = 6 . ω = 4 for W = 3 . ω = 0 for W = 6 . φ . Therefore,for a specific W , the winding number ω also changes withthe phase φ . In Fig. 4 (a), the W dependence of ω withdifferent φ for L = 10 is plotted. Although φ inducessome differences for these curves, some behaviors are incommon. On the one hand, ω decreases with an increaseof W , which is different from the single particle non-Hermitian system. For the single particle non-Hermitiansystem, the winding number is found as ω = ± ω > ω = 0) appears around W = 7. The av-erage winding number ω is plotted in Fig. 4 (b), and itis shown that the topological phase transition point isaround W T C = 7.It should be noted that the topological transition isnot equal to the disappearance of the imaginary part ofeigenenergies, although the imaginary parts of eigenener-gies are necessary to construct a close loop in the energyplane. The topological phase transition gives anotherviewpoint on the MBL energy spectra complemented tothe real-complex transition. This winding number, de-fined in the complex plane by the gauge transformation,serves as a collective indicator of the eigenenergies beingcomplex or real of the original Hamiltonian. C. MBL phase transition
To characterize the MBL in the non-Hermitian sys-tems, the nearest-level-spacing distribution of eigenener-gies has been generalized from the Hermitian systems .On the complex plane, the nearest-level spacings for aneigenenergy E a (before unfolding) are defined as the s PC(s)PC(s) s (cid:1) (cid:1) (cid:1) (cid:1) W = 1 4 W = 2 FIG. 5. (a) The nearest-level-spacing distribution (unfolded)for W = 2 (left) and W = 14 (right). The lattice size is 16,and φ = π/ G W L FIG. 6. G as a function of W for L = 10 , ,
14, and 16.The results are obtained by averaging 500 choices of φ for L = 10 ,
12 and 14, and 100 choices of φ for L = 16. minimum distance of | E a − E b | . For the delocaliza-tion phase, it has been demonstrated that the statisticsof the nearest-level spacing obey a Ginibre distribution P cGin ( s ) = cp ( cs ), where p ( s ) = lim N →∞ (cid:34) N − (cid:89) n =1 e n ( s ) e − s (cid:35) N − (cid:88) i =1 s n +1 n ! e n ( s ) , (6)with e n ( x ) = (cid:80) nm =0 x m m ! and c = (cid:82) ∞ sp ( s ) ds =1 . . Since the MBL tends to suppress the imag-inary part of the complex eigenenergies, these eigenen-ergies are almost real in the MBL phase, and thenearest-level-spacing distribution becomes the Poisso-nian as P RP o ( s ) = e − s . By taking the eigenenergies ly-ing within ±
10% of the real and imaginary parts fromthe middle of the spectra of Eq. (1), the nearest-level-spacing distributions (unfolded) for different W are plot-ted in Fig .5. It is shown that for W = 2 the distributionis a Ginibre distribution and the distribution is a Poissondistribution for W = 14. These results demonstrate thatthe non-Hermitian quasiperiodic system also has a MBLphase transition with the increase of W .
024 1 0 - 1
2 4 6 7 8 1 0 1 2 1 4 ( b )
ER(t) ( a ) W W = S(t) t W =
FIG. 7. Time evolution of E R ( t ) for W = 2 , , , , , , S ( t ) for g = 0 . g = 0 (dotted lines) (b). The lattice size is L = 12,and the initial state is taken as | ψ (cid:105) = | · · · (cid:105) . Based on the response of the system’s eigenstates to alocal perturbation, a dimensionless parameter G has beenintroduced to detect the MBL phases transition in theHermitian systems . It is shown that G decreases withthe increase of system size L in the MBL phase and growswith the increase of system size L in the many-body de-localization phase for the Hermitian systems, and thephase transition point appears when G ( L ) is independentof L . For non-Hermitian system, since the stability of theeigenstates under perturbations ˆ V is also important forthe complex eigenenergies, G ( L ) has been extended tostudy the MBL phase transition in non-Hermitian sys-tems . G ( L ) for the non-Hermitian systems is definedas G = ln (cid:104) ψ lα +1 | ˆ V | ψ rα (cid:105)| E (cid:48) α +1 − E (cid:48) α | , (7)where (cid:104) ψ lα | and | ψ rα (cid:105) are the left and right eigenvectors ofnon-Hermitian system, ˆ V is a perturbation operator, and E (cid:48) α = E α + (cid:104) ψ lα | ˆ V | ψ rα (cid:105) is the modified eigenenergy. Here,the states with E (cid:48) α stays real are only considered, andthe perturbation operator is selected as ˆ V = ˆ b + i +1 ˆ b i . InFig. 6, G as a function of W for different L is plotted. Itis found that in the many-body delocalization phase theabsolute value of G decreases with increase of L , whilethe absolute value of G grows with the increase of L inthe MBL phases, and the MBL phase transition occursat W MBL C = 6 ± . W R C , W T C ,and W MBL C are close, and the slight difference is ascribedto the finite-size effect. Therefore, based on the numericalresults, we can conjecture these transition points shouldcoincide in the thermodynamic limit. D. Effects on the dynamical behaviors
Finally, the effects of the phase transition on the dy-namical stability of the non-Hermitian systems are stud-ied. To illustrate this, the time evolution of real part ofenergy E R ( t ) and half-chain entanglement S ( t ) are stud-ied. E R ( t ) is defined as E R ( t ) = Re[ (cid:104) ψ r ( t ) | ˆ H | ψ r ( t ) (cid:105) ] . (8)Here, (cid:104) ψ r ( t ) | is the Hermitian conjugate of | ψ r ( t ) (cid:105) . Asdisplayed in Fig. 7 (a), the time evolution of E R ( t ) withdifferent W for L = 12 are plotted. It is found that for W < W R C and around W R C , E R ( t ) changes significantlyduring the evolution since the nonzero imaginary part ofthe eigenenergies can induce dynamical instability. For W is much larger than W R C , E R ( t ) is almost conservedduring the evolution, since the eigenvalues are almostreal.The half-chain entanglement S ( t ) is evaluated by thevon Neumann entropy, S ( t ) = − Tr( ρ ( t ) ln ρ ( t )) , (9)where ρ ( t ) = Tr L/ [ | ψ r ( t ) (cid:105)(cid:104) ψ r ( t ) | ] / (cid:104) ψ r ( t ) | ψ r ( t ) (cid:105) is thereduced density matrix of the right eigenstate. The timeevolution of S ( t ) for different W are plotted in Fig. 7(b),and the Hermitian case of g = 0 is also shown for a com-parison. For W = 14, the dynamical behaviors of S ( t ) for g = 0 . g = 0, since the eigenvaluesare almost real for g = 0 .
5. However, for W = 2, S ( t )first linearly grows for both values of g , but decreasesfor t (cid:39) g = 0 .
5. In addition, the long-timevalue of the entanglement of the many-body delocaliza-tion phase is larger than that of the MBL phase. Thereason is that entanglement entropy in the MBL phasestill obeys the area law rather than the volume law evenin the non-Hermitian system . IV. SUMMARY
In this paper, we have studied the real-complex tran-sition, topological phase transition and MBL phase tran- sition in a non-Hermitian quasiperiodic system havingthe TRS. Our numerical results showed that these threetypes of phase transitions coexist for this model, and inthe thermodynamic limit these three transition pointsshould coincide. These results demonstrated that theimaginary parts of the eigenenergies are always sup-pressed by the MBL, and the MBL phase transitionshould have a topological nature similar to that of thesingle particle systems. Moreover, the obtained criti-cal exponent for the real-complex transition is differentfrom that of the disordered system, which means the non-Hermitian many-body disordered system and the many-body quasiperiodic system should belong to different uni-versality classes. Finally, we find that these phase tran-sitions can affect the dynamical stability, but the dy-namical entanglement still obeys the area law for theMBL phase and volume law for the delocalized phase.Recently, the asymmetry hopping has been realized ex-perimentally in an ultracold atomic system , and theMBL in the quasiperiodic system has also been experi-mentally studied . Therefore, we expect our study canbe verified in these experiments. The diagonal energyshifts of non-Hermitian systems were demonstrated tohave a power-law distribution due to non-orthogonalityof right and left eigenvectors , which is different fromthe nearest-level-spacing distribution. It should be aninteresting work to study the diagonal energy shifts inthe non-Hermitian MBL and many-body delocalizationphases. We leave this for further studies. ACKNOWLEDGMENTS
L.J.Z. is supported by the Natural Science Foundationof Jiangsu Province, China (Grant No. BK20170309) andNational Natural science Foundation of China (Grant No.11704161). S.Y. is supported by the startup grant (No.74130-18841229) at Sun Yat-Sen University. ∗ [email protected] D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, Rev.Mod. Phys. , 021001 (2019). D. A. Abanin and Z. Papi´c, Ann. Phys. (Berlin) , 1700169(2017). R. Nandkishore and D. A. Huse, Annu. Rev. Condens.Matter Phys. , 15-38 (2015). M. Rispoli, A. Lukin, R. Schittko, S. Kim, M. Eric Tai, J.L´eonard, and M. Greiner, Nature (London) , 385-389(2019). T. Kohlert, S. Scherg, X. Li, H. P. L¨uschen, S. Das Sarma,I. Bloch, and M. Aidelsburger, Phys. Rev. Lett. , 170403 (2019). A. Morningstar and D. A. Huse, Phys. Rev. B , 224205(2019). A. Lukin, M. Rispoli, R. Schittko, M. E. Tai, A. M. Kauf-man, S. Choi, V. Khemani, J. L´eonard, and M. Greiner,Science , 256 (2019). D. M. Basko, I. L. Aleiner, and B. L. Altshuler, Ann. 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