Exact mobility edges, PT -symmetry breaking and skin effect in one-dimensional non-Hermitian quasicrystals
EExact mobility edges, PT -symmetry breaking and skin effect in one-dimensionalnon-Hermitian quasicrystals Yanxia Liu, Yucheng Wang,
2, 3, 4
Xiong-Jun Liu,
3, 4
Qi Zhou, ∗ and Shu Chen
1, 6, 7, † Beijing National Laboratory for Condensed Matter Physics,Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China Shenzhen Institute for Quantum Science and Engineering, and Department of Physics,Southern University of Science and Technology, Shenzhen 518055, China International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China Collaborative Innovation Center of Quantum Matter, Beijing 100871, China Chern Institute of Mathematics and LPMC, Nankai University, Tianjin 300071, China School of Physical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China Yangtze River Delta Physics Research Center, Liyang, Jiangsu 213300, China (Dated: September 7, 2020)We propose a general analytic method to study the localization transition in one-dimensionalquasicrystals with parity-time ( PT ) symmetry, described by complex quasiperiodic mosaic latticemodels. By applying Avila’s global theory of quasiperiodic Schr¨odinger operators, we obtain exactmobility edges and prove that the mobility edge is identical to the boundary of PT -symmetrybreaking, which also proves the existence of correspondence between extended (localized) statesand PT -symmetry ( PT -symmetry-broken) states. Furthermore, we generalize the models to moregeneral cases with non-reciprocal hopping, which breaks PT symmetry and generally induces skineffect, and obtain a general and analytical expression of mobility edges. While the localized statesare not sensitive to the boundary conditions, the extended states become skin states when theperiodic boundary condition is changed to open boundary condition. This indicates that the skinstates and localized states can coexist with their boundary determined by the mobility edges. Introduction.-
The study of localization induced by dis-order is a long-standing research area in condensed mat-ter physics [1]. While localized and extended states cancoexist at different energies in three dimensions with theexistence of mobility edges, random disorder genericallycauses Anderson localization of the entire spectrum inone and two dimensions [2–4]. Recently, the interplayof non-Hermiticity and disorder attracted much atten-tion as the non-Hermiticity brings new perspective forthe localization problem by releasing the Hermiticity con-dition, e.g., non-Hermitian random matrices contains 38different classes according to Bernard-LeClair symmetryclasses [5–8], which generalizes the standard ten classesof Altland-Zirnbauer classification of random Hermitianmatrices. In terms of non-Hermitian random-matrix the-ory, the spectral statistics for non-Hermitian disorder sys-tems has also been unveiled to display some different fea-tures from the Hermitian systems [9–12].The Hatano-Nelson model is a prototype model de-scribing the interplay of the nonreciprocal hoppingand random disorder [13–16], which leads to a fi-nite localization-delocalization (LD) transition in one-dimensional (1D) non-Hermitian Anderson model. Theeffect of complex disorder potentials has also been studied[17, 18]. Besides the random disorder, the quasiperiodicsystems have also attracted intensive studies in recentyears [19–23], including the Aubry-Andr´e (AA) model[21–23] and its various extensions [22, 24–27]. By in-troducing either short-range (long-range) hopping pro-cesses [28–34]or modified quasiperiodic potentials [35–37], the quasiperiodic lattice models can display energy- dependent mobility edges. While the LD transition inthe AA model was analytically predicted in 1980’s byutilizing the self-duality property [21], its rigorous math-ematical proof was only given recently [38–41]. Non-Hermitian quasiperiodic lattices have also been studiedin various references [42–48], which mainly focused onthe non-Hermitian effect on the AA model. Very re-cently the mobility edges in the non-Hermitian quasicrys-tal with long-range hopping [49] or modified quasiperi-odic potentials [50, 51] have also been studied numeri-cally. Although previous works on quasicrystals with PT symmetry found numerical evidence that the localiza-tion transition point coincides with the PT -symmetry-breaking point [46, 49], the reason behind this observa-tion remains elusive. As nonreciprocal hopping generallyinduces non-Hermitian skin effect [47, 52–57], i.e., the ex-ponential accumulation of extended bulk states to edgeswhen the boundary condition is changed from the peri-odic to open boundary condition (PBC to OBC), it is notclear whether skin states and localized states can coexistin non-Hermitian quasicrystals and how they are relatedto the mobility edges?Aiming to address the above issues, in this letter wefirst propose a class of PT -symmetrical quasiperiodicmosaic lattices with exact non-Hermitian mobility edges,and rigorously prove the intrinsic relation between themobility edges and PT -symmetry breaking by apply-ing Avila’s global theory [41], one of his Fields Medalwork, to the non-Hermitian quasiperiodic system at thefirst time. Our method is a general and mathematicallyrigorous method going beyond the usual dual transfor- a r X i v : . [ c ond - m a t . d i s - nn ] S e p mation, which requires some special forms of Hamilto-nian [29, 30, 49] to get analytical expression of mobilityedges. Then we study the more general case with non-reciprocal hopping and obtain a concise but unified ana-lytical formula for the mobility edges, which also worksas the boundaries separating skin states and localizedstates. Our analytical results are crucial to gain exactunderstanding of the non-Hermitian mobility edges, PT -symmetry breaking and interplay of skin effect and local-ization in 1D quasicrystals with both complex quasiperi-odic potential and non-reciprocal hopping. Model with PT symmetry.- We consider a 1Dquasiperiodic mosaic model with complex quasiperiodicpotential described by H = (cid:88) j ( t | j (cid:105) (cid:104) j + 1 | + t | j + 1 (cid:105) (cid:104) j | + V j | j (cid:105)(cid:104) j | ) , (1)with V j = (cid:26) λ cos(2 πωj + θ ) , j = κm, , otherwise, (2)where θ = φ + ih describes a complex phase factor and κ is an integer. The quasi-cell has the κ lattice sites. If thequasi-cell number is taken as N , i.e. m = 1, 2, · · · , N , thesystem size will be L = κN . For convenience, we set t = 1as the unit of energy. By taking | ψ (cid:105) = (cid:80) j u j | j (cid:105) , the eigenequation is given by Eu j = u j +1 + u j − + V j u j . Withoutloss of generality, we take ω = (cid:0) √ − (cid:1) /
2, which canbe approached by ω = lim n →∞ F n − F n , where F n is theFibonacci numbers defined rcursively by F n +1 = F n + F n − with F = F = 1.The model (1) has PT symmetry for φ = 0 due to V j = V ∗− j , and its eigenvalues are real if the PT symme-try is preserved [58]. In the limit of λ = 0, all eigenstatesare extended with real eigenvalues. When λ increases,localization transition is expected to occur. To studythe localization transition, we shall study Lyapunov ex-ponent (LE) of the model, which can be exactly ob-tained by applying Avila’s global theory of quasiperiodicSchr¨odinger operators [41]. The LE can be computed as γ ( E, h ) = lim n →∞ n (cid:90) ln || T n ( φ + ih ) || dφ, (3)where || A || denotes the norm of the matrix A and T n isthe transfer matrix of a quasi-cell. Avila’s global theory[41] shows that as a function of h, κγ ( E, h ) is a convex,piecewise linear function, and their slopes are integers.Thus in the large- h limit [37, 59], κγ ( E, h ) = ln | λa κ | + | h | , where a κ = 1 √ E − (cid:34)(cid:32) E + √ E − (cid:33) κ − (cid:32) E − √ E − (cid:33) κ (cid:35) . (4)Moreover, Avila’s global theory can tell us that, the en-ergy E does not belong to the spectrum of the Hamilton ( a )2 γ ( E , h ) h h + ln | λE c | −ln | λ ( E + iϵ ) | h γ ( E + iϵ ,0) h ( b )2 γ ( E , h ) h + ln | λE c | −ln | λ ( E + iϵ ) | h γ ( E + iϵ ,0) FIG. 1: The blue line shows the Lyapunov exponent γ ( E c , h )for critical energy E c , where E c is the mobility edge for thesystem with h = h = − ln | λE c | . The red line shows theLyapunov exponent γ ( E + i(cid:15), h ). (a) No complex energy in theregime | E + i(cid:15) | < E c belong to the spectrum of the system with h = h . (b) In the regime | E + i(cid:15) | > E c , the complex energymight belong to the spectrum of the system with h = h ,when γ ( E + i(cid:15), h ) is an extreme point of γ ( E + i(cid:15), h ). H with h = h , if and only if γ ( E, h ) > γ ( E, h )is an affine functions in a neighborhood of h = h . Con-sequently, if the energy E lies in the spectrum of theHamilton H with h = h , we have κγ ( E, h ) = max { ln | λa κ ( E ) | + | h | , } . (5)Specially, when h = 0, κγ ( E,
0) = max { ln | λa κ ( E ) | , } . Note that γ ( E, h ) = 0 corresponds to the extendedstate, γ ( E, h ) > e | h | | a κ ( E c ) | = 1 | λ | , (6)where a κ is given by Eq.(4) and h = h .If κ = 2, then a ( E ) = E , and E c = 1 / ( λe | h | ) . (7)By the above discussion especially formula (5), one canconclude that the eigenstates with energies | E + i(cid:15) | < E c and | E + i(cid:15) | > E c correspond to the extended statesand localized states, respectively. Now we analysis thedistribution of the real energies and complex energies ofthe system with h = h . To that end, we should startwith the case h = 0. If h = 0, then the Hamiltonian isHermitian, and the spectrum is real. The complex energy E + i(cid:15) does not belong to the spectrum of the system with h = 0, so γ ( E + i(cid:15), >
0, where E and (cid:15) are real. If | E + i(cid:15) | < E c , it’s easy to see that γ ( E + i(cid:15), h ) is an affinefunctions in a neighborhood of h = h , so in this regime,complex energy E + i(cid:15) does not belong to the spectrumof the system with h = h either, as shown in Fig.1(a).The extended states only happens for real energies, whichpossess PT symmetry. On the other hand, E + i(cid:15) belongsto the spectrum of the system with h = h , if and only if | h | + ln | λ ( E + i(cid:15) ) | = 2 γ ( E + i(cid:15), , -202-202 I m ( E ) -4 -2 0 2 4 Re(E) -202 0 2 4 |E| -2-1012 I m ( E ) |E| |E| I P R h=1.8 (a)(c) (b)(d) h=2.8 h=0.8h=1.8 h=2.8h=0.8 FIG. 2: Eigenvalues in the space spanned by (a) Im( E ) andRe( E ) and (b) Im( E ) and | E | , (c) the Lyapunov exponent,and (d) IPR versus | E | for the system with λ = 0 . h =0 .
8, 1 . . N = 233. as shown in Fig.1(b), where | E + i(cid:15) | > E c . The complexenergies only correspond to the localized states, whichhave no PT -symmetry. We stress that our analytic rea-soning applies to general κ .To get an intuitive understanding of the above an-alytical results, we shall demonstrate numerical re-sults of the energy spectrum, LE and inverse par-ticipation ratio (IPR) for the system with κ = 2.For a finite-site lattice, the LE can be obtainedby numerically calculating γ ( E ) = ln (cid:0) max (cid:0) θ + i , θ − i (cid:1)(cid:1) ,where θ ± i ∈ R are the eigenvalues of the matrix Θ = (cid:16) T † κN ( E, φ, h ) T κN ( E, φ, h ) (cid:17) / (2 L ) . There are twoeigenvalues and the LE is taken to be the maximumone. The IPR of an eigenstate is defined as IPR ( i ) =( (cid:80) n (cid:12)(cid:12) u in (cid:12)(cid:12) ) / (cid:16)(cid:80) n (cid:12)(cid:12) u in (cid:12)(cid:12) (cid:17) , where the superscript i de-notes the i th eigenstate, and n labels the lattice site.While IPR (cid:39) (cid:39) /L for an extended eigenstate and tends to zero as L → ∞ .Figs.2(a)-(d) show numerical results of systems withfixed λ = 0 . h . Here the eigenvaluesare plotted in the complex space spanned by Im( E ) andRe( E ) in Fig.2(a), and also mapped to the space spannedby Im( E ) and | E | in Fig.2(b), in order to compare withFig.2(c) and Fig.2(d), which display LE and IPR ver-sus | E | , respectively. For h = 0 .
8, all eigenvalues arereal with | E | < | E c | , which indicates the correspondingeigenstates being extended states. In this case, no PT -symmetry breaking happens as all eigenstates with real eigenvalues fulfill the PT -symmetry. When h exceeds acritical value h c = − ln( λE max ) ≈ .
9, where E max isthe maximum eigenvalue of the corresponding HermitianHamiltonian, PT -symmetry breaking happens. Whileeigenvalues fulfilling | E | < | E c | are still real, they be-come complex when | E | > | E c | as displayed in Figs.2(a)and 2(b) for cases of h = 1 . .
8. Both the LEand IPR have a sudden increase when | E | > | E c | , whichconfirms mobility edges are consistent with the analyt-ical results. Although increasing h shall shrink E c , thereal extended states and complex localized states alwayscoexist even h → ∞ . Such a novel phenomenon wasnever predicted in literature. Our results demonstratethat the transition from extended to localized states and PT -symmetry breaking transition have the same bound-ary. So the LD transition can be also read out from thechange of spectrum structure. Model with non-reciprocal hopping.-
Now we considera more general case with Hamiltonian given by˜ H = (cid:88) j ( t L | j (cid:105) (cid:104) j + 1 | + t R | j + 1 (cid:105) (cid:104) j | + V j | j (cid:105)(cid:104) j | ) , (8)where t L = te − g and t R = te g are the left-hopping andright-hopping amplitude, respectively, and V j is given byEq.(2). The nonreciprocal hopping breaks the PT sym-metry of Hamiltonian, and also generally induces skineffect under OBC. The Hamiltonian ˜ H ( g ) under OBCcan be transformed to H via a similar transformation H = S ˜ H ( g ) S − , where S =diag (cid:0) e − g , e − g , · · · , e − Ng (cid:1) isa similarity matrix with exponentially decaying diago-nal entries and H = ˜ H ( g = 0) is just the Hamiltonian(1) under OBC. The eigenvectors of ˜ H and H satisfy (cid:12)(cid:12)(cid:12) ˜ ψ (cid:69) = S − | ψ (cid:105) . An extended states | ψ (cid:105) under the trans-formation S − becomes skin-mode states, which are ex-ponentially accumulated to one boundary. A localizedstate of H generally takes the form | u j | ∝ e − γ | j − j | ,where j is the index of the localization center, and γ is the Lyapunov exponent in (5). Then the correspond-ing wavefunction of ˜ H ( g ) takes the form of | u j | ∝ (cid:26) e − ( γ − g ) | j − j | j > j e − ( γ + g ) | j − j | j < j , which manifest different decaying behaviors on two sidesof the localization center. When | g | ≥ γ , delocalizationoccurs on one side and then skin modes emerge to theboundary on the same side. The transition point of lo-calized states and skin states is given by | h | + ln | λa κ | κ = | g | . (9)Since the localized state is not sensitive to the bound-ary condition, we can conclude the LD transition in theperiodic boundary system is also given by Eq.(9), whichgives rise to the mobility edge | λa κ ( E c ) | = e κ | g |−| h | . (10) |E| -0.500.5 I m ( E ) PBC Mobility edge OBC0 1 2 |E| I P R h | E | IPR(b)(a) (c)
FIG. 3: (a) Eigenvalues in the space spanned by Im E and | E | and (b) IPR versus | E | for the system with λ = 0 . g = 0 . h = 1 . N = 233 under PBC (red dots) and OBC (bluecrosses). (c) (b) IPR of different eigenstates as a function ofthe corresponding absolute value of eigenenergies and h with g = 0 . λ = 0 .
2. Dashed lines represent mobility edges.
For κ = 2, the mobility edge is given by | λE c | = e | g |−| h | . (11)Since the eigenvalue of ˜ H is generally complex, the mo-bility edge is only associated with the absolute value ofthe eigenvalue. While increasing h suppresses | E c | , | g | tends to enlarge the region of extended states. Particu-larly, when | h | = 2 | g | , | E c | = 1 / | λ | .The similar transformation suggests that the eigen-value of ˜ H under OBC is identical to H , i.e., the openboundary eigenvalue is irrelevant with g . On the otherhand, the spectrum under PBC depends on g , whichis clearly manifested by the periodic spectrum of E =2 t cos( k + ig ) in the limit of λ = 0. Since the similartransition only holds true under the OBC, the spectrumof ˜ H under the OBC and PBC are generally different [59].In Fig.3(a), we display the spectrum of the system underboth PBC and OBC. While eigenvalues with | E | > | E c | are shown to be almost the same under both PBC andOBC, the parts of spectra with | E | < | E c | are obviouslydifferent under different boundaries, which is a characterof the existence of skin effect [60, 61]. In Fig.3(b), weplot the IPR versus | E | for both systems under PBC andOBC. The IPRs have a sudden increase when | E | > | E c | and display almost the same distributions in this local-ized region under different boundary conditions. Due tothe existence of skin states in the region of E < | E c | , theIPRs of open boundary system take finite values and areobviously different from the periodic system. In Fig.3(c),we display IPRs of the corresponding eigenstates as afunction of h for the periodic system with λ = 0 . g = 0 .
2. The dashed lines in Fig.3 represent the mo-bility edges determined by Eq.(11), which separate theextended and localized states with the values of IPR be-low which approaching zero and above being finite. Thenumerical results from IPR agree well with the analyticalrelation given by Eq. (11).The insensitivity of the localized states to the bound- -50 0 5000.20.40.6 || -50 0 50 site n || =0 =2 =10 = < x > |E| -202 I m ( E ) (a) (d)(c)(b) FIG. 4: (a) The mean position of wavefunctions (cid:104) x (cid:105) for thesystem with λ = 0 . g = 0 . h = 2 . N = 55 and different υ . (b) Eigenvalues in the space spanned by Im E and | E | forthe same system under OBC and PBC. The dashed lines rep-resent the mobility edges. The distribution of wavefunction | ψ | corresponding to the minimum (c) and maximum (d) | E | with υ = 0, 2 , 10, ∞ . aries has suggested that the onset of localization transi-tion should be irrelevant to the boundary conditions. Tosee the effect of boundary clearly, we consider that thehopping term between the L -th and first site is replacedby h L = η ( t L | L (cid:105) (cid:104) | + t R | (cid:105) (cid:104) L | ), with the introductionof a boundary anisotropic parameter η ∈ [0 ,
1] [56]. Forconvenience, we take η = e − υ with υ = 0 ( ∞ ) corre-sponding to the PBC (OBC). In Fig.4(a), we display themean position (cid:104) x (cid:105) = (cid:104) ψ | ˆ x | ψ (cid:105) of eigenstates versus | E | un-der different boundary conditions, where ˆ x = (cid:80) n n | n (cid:105)(cid:104) n | is the position operator. Fig.4(b) shows the correspond-ing spectrum in the parameter space spanned by Im ( E )and | E | . While (cid:104) x (cid:105) near the center of the lattices indi-cates that the wavefunction distributes over the wholelattice (the extended state), its accumulation on one ofthe boundary corresponds to skin states. For the local-ized states, (cid:104) x (cid:105) can take arbitrary values within positionof lattices. It is shown that the extended states are sen-sitive to the boundary condition and become skin statesunder the OBC, whereas the localized states almost haveno change with the change of boundary anisotropic pa-rameter. This is also witnessed by wavefunction distri-butions shown in Figs.4(c) and 4(d). Summary and discussion.-
In summary, we proposeda general analytic method to study the LD tran-sition and PT -symmetry breaking for non-Hermitianquasiperiodic models. Specially, we studied 1D non-Hermitian quasiperiodic mosaic models with both com-plex quasiperiodic potential and non-reciprocal hoppingand obtained analytically the exact mobility edges uni-formly described by Eq.(10), which is the central resultof the present work. For the case with PT symmetry, weproved that the mobility edge is identical to the bound-ary of PT -symmetry breaking. In the presence of non-reciprocal hopping, while the localized states are not sen-sitive to the boundary conditions, extended states aredriven to skin states when the PBC is changed to OBC,and skin states can coexist with localized states with theirboundaries given by the mobility edges.While the mobility edges only exist for cases of κ ≥ κ = 1, where a = 1and the model reduces to the non-Hermitian AA models.From Eq.(10), we get the localization transition occurringat | λ | = e −| h | + | g | . It is clear that the transition point isirrelevant to the eigenvalue E , indicating that no mobil-ity edges exist. All eigenstates are extended (localized)when | λ | < e −| h | + | g | ( | λ | > e −| h | + | g | ), which recovers theresult of Ref.[46] for h (cid:54) = 0 and g = 0 and the result ofRef.[47] for g (cid:54) = 0 and h = 0. For h = g = 0, our modelreduces to its Hermitian limit [21, 37]. An interestinglimit case of our model is obtained in the double limit h → ∞ , λ → λe h → V finite, corresponding tothe quasiperiodic potential given by V j = V exp( − i πωj )for j = km and 0 otherwise in Eq.(8). In this limit, themobility edges are given by | a κ ( E c ) | = e κ | g | / | V | . Thediversity and solvability of our models provide a new zoofor analytically exploring the richness of non-Hermitianlocalization phenomena.This work is supported by NSFC under GrantsNos. 11974413, the National Key Research and De-velopment Program of China (2016YFA0300600 and2016YFA0302104) and the Strategic Priority ResearchProgram of CAS (XDB33000000). Q. Zhou was par-tially supported by support by NSFC grant (11671192,11771077), The Science Fund for Distinguished YoungScholars of Tianjin (No. 19JCJQJC61300) and NankaiZhide Foundation. Y. Wang and X.-J. Liu are sup-ported by National Nature Science Foundation of China(11825401, 11761161003, and 11921005), the NationalKey R&D Program of China (2016YFA0301604), Guang-dong Innovative and Entrepreneurial Research TeamProgram (No.2016ZT06D348), the Science, Technologyand Innovation Commission of Shenzhen Municipality(KYTDPT20181011104202253), and the Strategic Pri-ority Research Program of Chinese Academy of Science(Grant No. XDB28000000). ∗ Electronic address: [email protected] † Electronic address: [email protected][1] P. W. Anderson, Absence of diffusion in certain randomlattices, Phys. Rev. , 1492(1958).[2] E. Abrahams, P. W. Anderson, D. C. Licciardello, andT. V. Ramakrishnan, Scaling theory of localization: Ab-sence of quantum diffusion in two dimensions, Phys. Rev.Lett. , 673 (1979).[3] P. A. Lee and T. V. Ramakrishnan, Disordered electronic systems, Rev. Mod. Phys. , 287(1985).[4] F. Evers and A. D. Mirlin, Anderson transitions, Rev.Mod. Phys. , 1355 (2008).[5] D. Bernard and A. 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