Boundary-dependent Self-dualities, Winding Numbers and Asymmetrical Localization in non-Hermitian Quasicrystals
aa r X i v : . [ c ond - m a t . d i s - nn ] S e p Boundary-dependent Self-dualities, Winding Numbers and Asymmetrical Localizationin non-Hermitian Quasicrystals
Xiaoming Cai State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics,Wuhan Institute of Physics and Mathematics, IAPMST,Chinese Academy of Sciences, Wuhan 430071, China (Dated: September 15, 2020)We study a non-Hermitian Aubry-Andr´e-Harper model with both nonreciprocal hoppings andcomplex quasiperiodical potentials, which is a typical non-Hermitian quasicrystal. We introduceboundary-dependent self-dualities in this model and obtain analytical results to describe its Asym-metrical Anderson localization and topological phase transitions. We find that the Anderson local-ization is not necessarily in accordance with the topological phase transitions, which are character-istics of localization of states and topology of energy spectrum respectively. Furthermore, in thelocalized phase, single-particle states are asymmetrically localized due to non-Hermitian skin effectand have energy-independent localization lengths. We also discuss possible experimental detectionsof our results in electric circuits.
Introduction.–
Anderson localization (AL) has been afascinating topic in condensed matter physics, ever sincethe classic work of Anderson in 1958 [1]. In one dimen-sion, it is a known fact that an infinitesimal un-correlateddisorder localizes all single-particle states [2]. However,AL phase transitions can exist in one dimensional (1D)quasicrystals, such as the Aubry-Andr´e-Harper (AAH)model [3], which has attracted a continuous interestboth theoretically and experimentally for the past threedecades [4–8]. The system undergoes a sudden AL phasetransition at a finite strength of the quasiperiodical po-tential, which is guaranteed by a self-duality mappingbetween extended and localized phases. Further searchesfor generalized self-dualities have been brought to mod-ified AAH models with exact energy-dependent mobil-ity edges [9–12]. Besides AL, the AAH model is also ofthe topological nature [13, 14] and supports the Thoulesspumping [15].On the other hand, the ability to engineer non-Hermitian Hamiltonians, demonstrated in a series of re-cent experiments [16–22], sparked a great interest instudying intriguing features and applications of non-Hermitian systems [23–25]. In general, the non-Hermiticity is achieved by introducing nonreciprocal hop-ping or/and gain and loss, which leads to exotic phe-nomena, such as parity-time ( PT ) phase transitions [26],exceptional points [27, 28], new topological invariants[29, 30], non-Hermitian skin effect and revised bulk-edgecorrespondence [31–38]. In the presence of disorders,non-Hermitian systems can exhibit unique AL proper-ties, such as purely imaginary disorder induced AL [39–41] and non-Hermitian skin effect induced finite-strengthlocalization-delocalization transition [42–44] . Specifi-cally, non-Hermitian AAH models reveal remarkable im-pacts of the (quasi)periodical on-site potentials on the PT symmetry breaking [45–49], butterfly spectrum[46,50], topological edge states [48–53] and localization prop-erties of eigenstates [49–51, 54–59]. The interplay be- tween nonreciprocal hopping and (quasi)periodicity givesrise to boundary-dependent topologies [50, 60] and local-ization properties [61]. Very recently, topological natureof AL phase transition in non-Hermitian AAH modelshas also come to light [57, 61, 62].So far, most studies on AL in non-Hermitian AAHmodels are based on numerical simulations. Few ana-lytical results are available due to the complex natureof energy spectrum. In this Letter, we study topologi-cal properties and AL of non-Hermitian quasicrystals byconsidering a non-Hermitian AAH model with both non-reciprocal hoppings and complex quasiperiodical poten-tials. We report boundary-dependent self-duality map-pings, which is absent so far. We also provide analyticalresults on topological phase transitions and Asymmet-rical AL. The AL phase transition is not necessarily inaccordance with the topological phase transitions, whichare characteristics of two aspects, localization of statesand topology of energy spectrum, respectively. Numer-ical calculations are also carried out to further confirmthese results. Non-Hermitian Aubry-Andr´e-Harper model.–
We con-sider a 1D tight-binding non-Hermitian AAH model de-scribed by the following Hamiltonian H = X j [ te − η − iφ a † j a j +1 + te η + iφ a † j +1 a j + V j a † j a j ] , (1)where a † j ( a j ) is the creation (annihilation) operator of aparticle at site j . t L ( R ) ≡ te ∓ η is the left(right)-hoppingamplitude with η characterizing the asymmetry. φ cor-responds to an applied magnetic flux or artificial gaugefield. V j = 2 V cos(2 πβj + δ − ih ) is an on-site complexquasiperiodical potential with β an irrational number,e.g., the inverse of the golden ratio ( √ − / β = F n /F n +1 with F n the n -th Fibonacci number, and the total num-ber of lattice sites L = F n +1 . Without loss of generality,we will take all parameters to be positive and real. Inthe Hermitian limit ( η = h = 0) the system has a self-duality mapping and AL transition occurs at the self-duality symmetry point V /t = 1. Extended states for
V /t <
V /t > γ = ln( V /t ) [3], i.e.,the inverse of localization lengths. Two limit cases with η = φ = 0 and h = δ = 0 were considered in Ref.[61, 62],respectively. In general, non-Hermitian models may suf-fer from the non-Hermitian skin effect with boundary-dependent spectra and single-particle states [34], and weneed to study the properties under different boundariesseparately. Self-duality, topology, and asymmetrical localizationunder the periodic boundary condition.–
Under the peri-odic boundary condition (PBC), we introduce a dualityFourier transformation a j = 1 √ L L X k =1 b k e − ik (2 πβj +2 δ ) − kh ,a † j = 1 √ L L X k =1 ˜ b k e ik (2 πβj +2 δ )+2 kh , (2)where b k (˜ b k ) is the annihilation (creation) operator inmomentum space. It takes the Hamiltonian in real spaceinto momentum space within the same form of Eq.(1),but with simultaneous interchanges t ↔ V , η ↔ h and φ ↔ δ , which defines the self-duality [63]. Nu-merical spectra under PBC support the self-duality andexamples are shown in Fig.1(a). An interesting caseis obtained in the simultaneous double limit: t → η → ∞ with te η → t ′ finite, and V → h → ∞ with V e h → V ′ finite. The Hamiltonian reduces to H ′ = P j [ t ′ e iφ a † j +1 a j + V ′ e − i (2 πβj + δ ) a † j a j ], which is anon-Hermitian half of the classic Hermitian AAH model.The self-duality ensures the transition point V ′ /t ′ = 1.Due to connection to the two dimensional quan-tum Hall system, topological properties of 1D superlat-tices and quasicrystals in Hermitian [13, 14] and non-Hermitian [57, 61, 62] systems have attracted great in-terest recently, where some parameter (such as δ or φ inour case) is treated as an additional dimension. In thesame spirit, here we adopt the winding numbers of energyspectra [57, 61, 62] υ τ = lim L →∞ πi Z π/L dτ ∂τ log[det( H )] , (3)with τ = φ or δ , which refer to the widely used windingnumbers evaluated by applying a magnetic flux φ and thephase δ in the on-site potential respectively. Analyticallyderived in the Supplementary Material [63], under thePBC, two winding numbers of energy spectra are relatedand υ φ = 1 − υ δ = θ ( te η V e h − , (4) -4 -3 -2 -1 0 1 2 3 4-3-2-1012345 0.5 1.0 1.5 2.0 2.5 3.001 t=0.5 V=1.0 h =0.2 h=0.8 t=0.5 V=1.0 h =0.8 h=0.2 t=1.0 V=0.5 h =0.8 h=0.2 t=1.0 V=0.5 h =0.2 h=0.8 z =2e t=0.5 V=0.4 h =0.5 h=1.1-ln(0.4) z =0.5e -0.6 ReE
ImE (a) V t=2 h =3+ln(0.5) h=3 t=1 h =3 h=3 t=1 h =3+ln(1.5) h=3 t=1 h =3+ln(2) h=3 (b) n f z( V ) n f FIG. 1. (Color online) (a) Spectra in the complex energyplane for systems under the periodic boundary condition.(b) Winding number υ φ vs V , numerically computed usingEq.(3). Inset in (b): υ φ vs ζ ≡ V e h /te η , with the transitionpoint ζ = 1. Other parameters: L = 987, and φ = δ = 0. with θ ( x ) the step function. The topological phase transi-tion is also verified by numerical calculations of windingnumbers directly using Eq.(3). Examples of numerical υ φ vs V are presented in Fig.1(b). After rescaling, allcurves collapse with the precise topological phase tran-sition point ζ ≡ V e h /te η = 1. Eq.(4) suggests a moregeneral self-duality ζ ↔ /ζ , which is not supported bynumerical results [see Fig.1(a)].With an irrational β , the quasiperiodical potential actsas a quasirandom disorder which induces the localiza-tion of single-particle eigenstates. We denote the righteigenstates of H by | Ψ Rs i = P j ψ s ( j ) a † j | i with s theindex of eigenstates. The localization can be charac-terized by the inverse of the participation ratio (IPR) P s = P j | ψ s ( j ) | / [ P j | ψ s ( j ) | ] . For a localized statethe IPR approaches to around 1, whereas for an extendedstate the IPR is of the order 1 /L . In order to charac-terize the localization of the system we define the meaninverse of the participation ratio (MIPR) P = P s P s /L .We show MIPRs vs V for different systems under thePBC in Fig.2(a) and corresponding rescaled ones in theinset. All curves collapse with the AL phase transitionpoint ζ = 1, which is the same as the topological phasetransition point. No mobility edge is encountered. Alleigenstates are extended when ζ <
1, whereas localizedwhen ζ >
1. One can also define (M)IPR for the lefteigenstates, which give the same conclusion.In order to explore details of the localization, inFig.2(b) we show two typical distributions of right eigen-states in extended and localized phases, respectively. Dis-tinctively, under the PBC the localized state has anasymmetrical exponential decay. We adopt the asym-
MIPR P
V/t h =3+ln(0.5) h=3 h =1.2 h=1.2 h =2+ln(1.5) h=2 h =3+ln(2.0) h=3 MIPR P (a) z V=0.98 (b) | y( j ) | j V=1.02 g R V/t h =3+ln(0.5) h=3 h =0 h=0 h =1.2 h=1.2 h =2+ln(1.5) h=2 h =3+ln(2) h=3 (c) z g R g L - g L ( z=1 ) g L z h =3+ln(0.5) h=3 h =0 h=0 h =1.2 h=1.2 h =3+ln(1.5) h=2 h =3+ln(2) h=3 (d) z FIG. 2. (Color online) Anderson localization in the non-Hermitian Aubry-Andr´e-Harper model under the periodicboundary condition. (a) Mean inverse of the participationratios (MIPRs) vs V . Inset in (a): Corresponding collapsedMIPRs vs ζ ≡ V e h /te η . (b) Two typical distributions | ψ ( j ) | of right eigenstates for systems in extended and localizedphases respectively. η = h = 0 .
2, in numerical calculationof (b). (c) Mean right side Lyapunov exponents γ R , i.e., theinverse of right side localization lengths, vs V . Inset in (c):Corresponding collapsed γ R vs ζ . (d) Mean left side Lya-punov exponents γ L vs ζ . Inset in (d): γ L shifted by the sizeof jumping. In (c), (d) and insets in them, we also show theLyapunov exponent for the Hermitian Aubry-Andr´e-Harpermodel (red dash lines). Other parameters: L = 987, t = 1and φ = δ = 0. metrical wave functions ψ s ( j ) ∝ ( e − γ Rs ( j − j ) , j > j ,e − γ Ls ( j − j ) , j < j , (5)which manifest different exponential decaying behaviourson both sides of the localization center j with twoLEs γ R ( L ) s . Extracted by fitting the numerical datawith Eq.(5), the mean right and left side LEs γ R ( L ) = P s γ R ( L ) s /L are presented in Fig.2(c)(d), along with theLE for classic Hermitian AAH model. All mean rightside LEs γ R collapse into a single curve with the ALphase transition point ζ = 1. Based on the known LE γ = ln( V /t ) for the Hermitian AAH model, the right sideLE γ R = ln( V e h /te η ) for the non-Hermitian AAH modelunder the PBC. On the other side, the mean left side LEexperiences a sudden jump at ζ = 1. The size of jump-ing only depends on and approximately equals 2 η . Aftershifting the jump, all mean left side LEs collapse into theLE for Hermitian AAH model, which are shown in theinset of Fig.2(d). Both left and right side LEs are energy- independent. In a word, under the PBC right eigen-states in the localized phase have LEs γ R = ln( V e h /te η ), γ L = γ + γ R , and γ ≃ η . Skin effect induced novel physics under the open bound-ary condition.–
Under the open boundary condition(OBC), the model with a non-zero η suffers from thenon-Hermitian skin effect [29]. In the absence of disor-der, all right eigenstates are exponentially localized at theright(left) end when η > ( < )0. In general, we introducean asymmetric similarity transformation a j = e ( η + iφ ) j b j and a † j = e − ( η + iφ ) j ˜ b j . The Hamiltonian Eq.(1) is mappedto H = X j [ t ˜ b j b j +1 + t ˜ b j +1 b j + V j ˜ b j b j ] . (6)Then Hamiltonians H and H have the exact same spec-tra and winding numbers υ φ ( δ ) under the OBC. Note thathere we focus on bulk properties of the system, and thesetwo winding numbers are not useful to predict topologicaledge states. Please refer to the Supplementary Materialfor a brief study of edge states [63].For the moment, we concentrate on the Hamiltonian H , which was originally introduced in Ref.[62] (see alsoin the Supplementary Material [63]). The model supportsa topological AL phase transition at V e h /t = 1. Thewinding numbers [63] υ φ = 0 , υ δ = θ ( V e h /t − , (7)because H does not depend on φ . This model, with η = 0, does not suffer from the skin effect. All bulkstates are extended when V e h /t <
1. When
V e h /t > γ =ln( V e h /t ) [63].Back to the Hamiltonian H , under the OBC the modeldoes not have a self-duality when h = 0. When V e h /t < υ δ = 0, and the spectrum mainly consists of three”bands” without loop. While when V e h /t > υ δ = 1,and the spectrum consists of loops where the complexspectral trajectory encircles the origin [see Fig.3(a) forexamples]. There can not be any self-duality transfor-mation mapping between spectra with different topolo-gies. However, when h = 0, based on the relation tothe Hermitian AAH model, one can find a self-dualitytransformation [63] a j = 1 √ L e i ( φ − δ ) j + ηj X k b k e − ik (2 πβj + δ + φ ) − kη ,a † j = 1 √ L e − i ( φ − δ ) j − ηj X k ˜ b k e ik (2 πβj + δ + φ )+ kη . (8)It interchanges t and V in Hamiltonian H ( h = 0). Thespectrum of H ( h = 0) is the same as for the HermitianAAH model, with the self-duality critical point V /t = 1,which is not the localization critical point [61].On the other hand, following the asymmetric similar-ity transformation between H and H , under the OBCa right eigenstate of H satisfies ψ s ( j ) = e ( η + iφ ) j ψ ′ s ( j )with ψ ′ s ( j ) the corresponding right eigenstate of H . Itclearly shows how the skin effect affects states in differ-ent phases: For extended states of H , the correspondingwave functions ψ s ( j ) are localized at the right end withleft side LEs γ L = η ; For localized states of H , wavefunctions ψ s ( j ) have the form ψ s ( j ) ∝ (cid:26) e − ( γ − η )( j − j ) , j > j ,e − ( γ + η )( j − j ) , j < j . (9)Then the condition γ − η > γ − η = 0 gives the AL phasetransition point ζ ≡ V e h /te η = 1 . (10)It is the same as the transition point under the PBC, butnot the transition point for spectrum or topology underthe OBC. Furthermore, in the localized phase, the rightand left side LEs γ R = γ − η = ln( V e h /te η ) ,γ L = γ + η = 2 η + γ R . (11)The ALs under both boundary conditions are identical,proving the insensitivity of the localized states to bound-aries.The above analytical results are in good agreementwith numerical ones. In Fig.3(b) we show MIPRs un-der the OBC. Due to the skin effect induced boundary-localization nature of right eigenstates, the AL phasetransition point should correspond to the most extendedcase with the smallest MIPR. Therefore, there are deepdives in MIPRs around ζ = 1. In Fig.3(c) we show meanleft side LEs which show a jump to 2 η at the transitionpoint ζ = 1. When ζ < γ L = h owing to the skineffect, except there is a dive just before the transitionpoint which also indicates the delocalization of states.The dive begins at V e h /t = 1, which is the topologicalphase transition point or the AL phase transition pointfor the model H . This skin effect induced phase wascalled skin phase [61]. In Fig.3(d) we show mean rightside LEs, which show a clear transition at ζ = 1 andcorrect LEs in localized phase in spite of the unreliablenumerical LEs in skin phase due to the right-boundary-localization nature of right eigenstates. Experimental realization.–
The physics of non-Hermitian AAH model [Eq.(1)] can be simulated by elec-tric circuits [57], which recently have turned out to bepowerful platforms to simulate non-Hermitian and/ortopological phases [64]. The single-particle eigenvalueproblem is simulated by the Kirchhoff’s current law I a = P Lb =1 J ab V b , where J the Laplacian of circuit acts as theeffective Hamiltonian, and I a and V a are the current and -2 -1 0 1 2-0.020.000.020.04 (d)(c) (b) ImE
V=0.8
ImE
ReE (a) -3 -2 -1 0 1 2 3-0.10.00.1
ReE V=1.2 h =0.04 h=0.12 h =0.06 h=0.10 h =0.04 h=0.04 h =0.06 h=0.02 h =0.08 h=0.01 MIPR P P zg L V/t h =0.10 h=0.02 h =0.10 h=0.06 h =0.10 h=0.10 h =0.10 h=0.12 g L z Ve h /t=1 V/t g R V/t h =0.10 h=0.02 h =0.10 h=0.06 h =0.10 h=0.10 h =0.10 h=0.12 g R z FIG. 3. (Color online) Spectra and Anderson localizationfor the non-Hermitian Aubry-Andr´e-Harper model under theopen boundary condition. (a) Two typical spectra in complexenergy plane with different winding numbers υ δ . η = h = 0 . V . Inset in (b): CorrespondingMIPRs vs ζ . (c,d) Mean left/right side Lyapunov exponents γ L ( R ) vs V . Insets in (c,d): Corresponding collapsed γ L ( R ) vs ζ . Numerical results before transition points in (d) andinset are not reliable, because of the skin effect induced right-boundary-localization nature of right eigenstates. Other pa-rameters: L = 233, t = 1 and φ = δ = 0. voltage at node a . On-site complex potentials are pro-vided by grounding nodes with proper resistors [65], andasymmetrical hopping amplitudes are realized by neg-ative impedance converters with current inversion (IN-ICs) [66]. Furthermore, the boundary-dependent spectracould be obtained by measuring two-node impedances[50].In summary, we have analytically studied the non-Hermitian AAH model with both nonreciprocal hoppingsand complex quasiperiodical potentials. We first reportboundary-dependent self-dualities between localized andextended phases. We also provide analytical results ontopological phase transitions and asymmetrical AL. TheAL phase transition is not necessarily in accordance withthe topological phase transitions, which are character-istics of two aspects, localization of states and topol-ogy of energy spectrum, respectively. Under weak dis-orders, the skin effect dominates and the system exhibitsboundary-dependent behaviours. 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