Phase transitions in a non-Hermitian Aubry-André-Harper model
PPhase transitions in a non-Hermitian Aubry-Andr´e-Harper model
Stefano Longhi ∗
1, 2 Dipartimento di Fisica, Politecnico di Milano, Piazza L. da Vinci 32, I-20133 Milano, Italy IFISC (UIB-CSIC), Instituto de Fisica Interdisciplinar y Sistemas Complejos, E-07122 Palma de Mallorca, Spain ∗ The Aubry-Andr´e-Harper model provides a paradigmatic example of aperiodic order in a one-dimensional lattice displaying a delocalization-localization phase transition at a finite critical value V c of the quasiperiodic potential amplitude V . In terms of dynamical behavior of the system, thephase transition is discontinuous when one measures the quantum diffusion exponent δ of wavepacket spreading, with δ = 1 in the delocalized phase V < V c (ballistic transport), δ (cid:39) / V = V c (diffusive transport), and δ = 0 in the localized phase V > V c (dynamicallocalization). However, the phase transition turns out to be smooth when one measures, as adynamical variable, the speed v ( V ) of excitation transport in the lattice, which is a continuousfunction of potential amplitude V and vanishes as the localized phase is approached. Here weconsider a non-Hermitian extension of the Aubry-Andr´e-Harper model, in which hopping alongthe lattice is asymmetric, and show that the dynamical localization-delocalization transition isdiscontinuous not only in the diffusion exponent δ , but also in the speed v of ballistic transport.This means that, even very close to the spectral phase transition point, rather counter-intuitivelyballistic transport with a finite speed is allowed in the lattice. Also, we show that the ballisticvelocity can increase as V is increased above zero, i.e. surprisingly disorder in the lattice can resultin an enhancement of transport. ∗ [email protected] a r X i v : . [ c ond - m a t . d i s - nn ] F e b I. INTRODUCTION
One-dimensional lattices with aperiodic order, i.e. displaying a long-range periodicity intermediate between ordinaryperiodic crystals and disordered systems, provide fascinating models to study unusual transport phenomena in a widevariety of classical and quantum systems, ranging from condensed-matter systems to ultracold atoms, photonic andacoustic systems [1–7]. Quasiperiodicity gives rise to a range of unusual behavior including critical spectra, multifractaleigenstates, localization transitions at a finite modulation of the on-site potential, and mobility edges [8–27]. Typicaldynamical variables that characterize single-particle transport are the largest propagation speed v of excitation inthe lattice, which is bounded (to form a light cone) for short-range hopping according to the Lieb-Robinson bound[28], and the quantum diffusion exponent δ , which measures the asymptotic spreading of wave packet variance σ ( t )in the lattice according to the power law σ ( t ) ∼ t δ . Such dynamical quantities are highly relevant in experiments,since they can be readily measured detecting the temporal spreading of an initially localized wave packet [2, 4, 6, 29],and can thus provide insightful information about the underlying properties of the system. While there is not aone-to-one correspondence between spectral and dynamical properties of a quantum system [30, 31], a general rule ofthumb is that absolutely continuous spectrum and extended states yield ballistic transport ( v (cid:54) = 0 and δ = 1), pure-point spectrum and exponentially-localized states usually yield dynamical localization, i.e. suppression of transportand quantum diffusion ( v = 0 and δ = 0), while singular continuous spectrum and critical states yield diffusive (oranomalous diffusive) behavior ( v = 0, 0 < δ < V of the on-sitequasi-periodic potential is increased above a critical value V c [45, 46]. In terms of dynamical behavior of a wavepacket, measured by the exponent δ = δ ( V ) of wave spreading, the phase transition is discontinuous since δ ( V ) = 1for V < V c (ballistic transport), δ ( V ) (cid:39) / V = V c (almost diffusive transport), and δ ( V ) = 0in the localized phase V > V c (dynamical localization) [18, 21]; see Fig.1(a). However, in terms of the spreadingvelocity v = v ( V ), defined as v ∼ σ ( t ) /t , the phase transition turns out to be smooth, with v ( V ) continuous functionof potential amplitude V and v ( V ) = 0 for V ≥ V c [see Fig.1(a)]. This result corresponds to physical intuitionthat in terms of dynamical evolution of the system the transition from ballistic wave packet spreading to dynamicallocalization, as the potential amplitude V is increased above the critical value V c , is a continuous process.Recently, fresh and new perspectives on spectral localization, transport and topological phase transitions have beendisclosed in non-Hermitian lattices, where complex on-site potentials or asymmetric hopping are phenomenologicallyintroduced to describe system interaction with the surrounding environment [29, 47–74]. In particular, the interplayof aperiodic order and non-Hermiticity has been investigated in several recent works [62–74], revealing that the phasetransition of eigenstates, from exponentially localized to extended (under periodic boundary conditions), can be oftenrelated to the change of topological (winding) numbers of the energy spectrum [55, 64, 72]. However, the dynamicalbehavior of the system near the phase transition, probed by the diffusion exponent or propagation speed of excitation,remains so far largely unexplored.In this work we consider a non-Hermitian extension of the Aubry-Andr´e-Harper model, where non-Hermiticity isintroduced by asymmetric (non-recirpocal) hopping amplitudes like in the Hatano-Nelson model [47–49, 55]. Specialfocus is devoted to the limit of unidirectional hopping, where rigorous analytical results can be obtained. The analysisunveils distinct features of dynamical phase transition when the hopping is asymmetric: while in the Hermitian casethe the dynamical localization-delocalization transition is discontinuous in the diffusion exponent δ solely [Fig.1(a)],in the non-Hermitian case the phase transition is discontinuous both in the exponent δ and in the speed v of ballistictransport [Fig.1(b)]. This means that, rather counter intuitively, ballistic transport with a finite speed is allowed in thelattice even very close to the critical point. Such a surprising result is related to the different spectral measure of theabsolutely continuous spectrum near the phase transition point, which vanishes for symmetric hopping (Hermitiancase) but not for asymmetric hopping (non-Hermitian case). Also, for a sufficiently large asymmetry in hoppingamplitudes, we show that an increase of the potential V results in an enhancement (rather than attenuation) of wavespreading in the lattice, thus providing an example of disorder-enhanced transport. SPE C T RA L P HA SE T RAN S I T I O ND Y NA M I CA L P HA SE T RAN S I T I O N SYMMETRIC HOPPING J = J =J ASYMMETRIC HOPPING J > J
L R
L RV
We consider the non-Hermitian extension of the Aubry-Andr´e-Harper (AAH) model defined by the Hamiltonian i dψ n dt = J R ψ n +1 + J L ψ n − + 2 V cos(2 παn ) ψ n ≡ Hψ n (1)where J L , J R are the left and right hopping amplitudes on the tight-binding lattice, V is the amplitude of theon-site incommensurate potential, and α is irrational. Without loss of generality, we assume 0 ≤ J R ≤ J L and V ≥
0. Owing to the non-Hermitian nature of the Hamiltonian, the dynamics described by Eq.(1) does not preservethe norm. This typically occurs when dealing with the dynamics of classical systems, such as photonic, acoustic,mechanical, or electrical systems, where the non-unitary dynamics simply indicates that the system exchanges energywith the surrounding environment. On the other hand, in an open quantum system we require, after any infinitesimaltime step dt , to normalize the wave function, i.e. the state vector | ψ ( t ) (cid:105) of the system evolves according to thetwo-step process | ψ ( t + dt ) (cid:105) = exp( − iHdt ) | ψ ( t ) (cid:105) and | ψ ( t + dt (cid:105) ) = | ψ ( t + dt (cid:105) ) / (cid:107) ψ ( t + dt (cid:105) ) (cid:107) . This two-time stepprocedure physically corresponds to the dynamics of postselected quantum trajectories in open quantum systemsunder continuous measurements, where quantum jumps are neglected [55, 59, 75, 76].In this section we provide numerical results and some qualitative physical insights that highlight the different behaviorof dynamical phase transitions in the Hemitian (symmetric hopping J L = J R ) versus non Hermitian (asymmetrichopping J L > J R ) models. The special case J R = 0, corresponding to unidirectional hopping on the lattice, is treatedseparately in Sec.III. Figure 1 illustrates the main features of spectral and dynamical phase transitions in Hermitian(symmetric hopping) and non-Hermitian (asymmetric hopping) AAH models. A. Symmetric hopping (Hermitian lattice)
The Hermitian limit J R = J L = J corresponds to the usual AAH model, and the Hamiltonian H is referred to asthe almost Mathieu operator in the mathematical literature. In this case the spectral and dynamical properties of H are well understood [9, 11, 18, 43, 45, 46]. For irrational α , the energy spectrum is a Cantor set and, for almostevery α (e.g. with Diophantine properties), one has purely absolutely continuous spectrum with extended states for V < J , purely singular continuous spectrum with critical wave functions for V = J , and pure point spectrum withexponentially decaying wave functions for V > J . In the localized phase, all wave functions have the same localizationlength ξ = 1 /L , where L is the energy-independent Lypaunov exponent given by L = log VJ . (2)The dynamical behavior of the system is captured by the asymptotic behavior at long times t of the second-ordermoment of position operator σ ( t ) describing wave spreading, σ ( t ) = (cid:80) n n | ψ n ( t ) | (cid:80) n | ψ n ( t ) | (3)with typical initial state localized at the site n = 0 in the lattice, i.e. ψ n (0) = δ n. . The asymptotic spreading of σ ( t ) is described by a power law, i.e. σ ( t ) ∼ t δ where δ is dubbed the diffusion exponent. In the delocalized phase V < J , transport in the lattice is ballistic with the exponent δ ( V ) = 1. The excitation propagates bidirectionallyalong the lattice with the non-vanishing speed v ( V ) ∼ σ ( t ) t (4)which determines a full light cone pattern, as schematically shown in the bottom panel of Fig.1(a). On the other
150 0 150 0 150 0 150 0 150 - l a tt i c e s i t e n normalized time Jt potential amplitude V/J v e l o c i t y v (b)(a) V/J =0.2
V/J =0.6
V/J =0.9
V/J =1 V/J =1.2
FIG. 2. (color online) (a) Numerically-computed evolution of occupation amplitudes | ψ n ( t ) | , for single-site input excitation ψ n (0) = δ n, , in the symmetric hopping case J L = J R = J for a few increasing values of potential amplitude V and for α = ( √ − /
2. (b) Behavior of the spreading velocity v ( t ) = σ ( t ) /t versus potential amplitude V , computed for the largestpropagation time t = 150 /J . The spreading velocity v is expressed in units of J . hand, in the localized phase V > J transport is prevented (dynamical localization), so that δ ( V ) = v ( V ) = 0. Atthe phase transition point V = V c = J , transport is intermediate between ballistic and localized; previous works haveshown that transport in the lattice is nearly diffusive with an exponent δ ( V c ) (cid:39) . δ ( V ) near the critical point V = V c [Fig.1(a)]. However, when one considers the speed v ( V ) as an order parameter, the transition from ballisticto diffusive transport and localization is a smooth (continuous) process: v ( V ) decreases almost linearly with V andvanishes for V ≥ V c , with a discontinuity of the first derivative ( dv/dV ) at V = V c This behavior is characteristic for
250 0 250 0 250 0 250 0 250 l a tt i c e s i t e n normalized time J t potential amplitude
V/J v e l o c i t y v (b)(a) V/J =0.2
V/J =0.6
V/J =0.9
V/J =1 V/J =1.2
L LLLLLL
FIG. 3. (color online) Same as Fig.2, but for asymmetric hopping with J R /J L = 0 .
5. In the spreading dynamics, the amplitudes ψ n ( t ) have been normalized at each time step to (cid:112)(cid:80) n | ψ n ( t ) | . The spreading velocity v is expressed in units of J L .
250 0 250 0 250 0 250 0 250 l a tt i c e s i t e n normalized time J t potential amplitude
V/J v e l o c i t y v (b)(a) V/J =0.2
V/J =0.6
V/J =0.9
V/J =1 V/J =1.2
L LLLLLL
FIG. 4. (color online) Same as Fig.3, but for J R /J L = 0 . a second-order phase transition [Fig.1(a)]. Examples of numerically-computed wave spreading dynamics in the latticefor increasing values of V /J , and corresponding behavior of velocity v ( V ), are shown in Fig.2, clearly suggesting thatthe phase transition is first-order in v . The numerical results are obtained by solving the time-domain Schr¨odingerequation (1) using a variable-step fourth-order Runge-Kutta method assuming single-site excitation at initial time; asufficiently large number of lattice sites (typically 600 sites) has been considered to avoid edge effects up to the largestintegration time.To physically understand why the ballistic speed v diminishes and vanishes as the critical point is approached,let us consider a rational approximation α ∼ q/p of the irrational α , with p, q prime integers, so that the actualincommensurate potential is approximated by a superlattice with period p [11]. For example, for the inverse of thegolden ratio α = ( √ − / . ... , the sequence α n = q n /p n converges to α in the n → ∞ limit, where p n = 0 , , , , , , , , , , , , , ... are the Fibonacci numbers and q n = p n − . The Bloch wave functionsof the superlattice satisfy the periodicity condition ψ n + p = ψ n exp( ikp ), where − π/p ≤ k < π/p is the Bloch wavenumber, and the energy spectrum of H is thus approximated by a set of p energy bands with dispersion curves E l ( k )( l = 1 , , , ..., p ), separated by ( p −
1) energy gaps, obtained as solutions of a determinantal equation (see AppendixA). It is well known that, for large p , the sum ∆ W of the widths ∆ W l of the allowed energy bands, i.e. the Lebesguemeasure ∆ W = (cid:80) l ∆ W l of energy spectrum, is given by ∆ W = 4 | J − V | , vanishing as the critical point is attained[11]. The corresponding Bloch eigenstates turn out to be delocalized over the entire period p of the superlattice for V < J , whereas they tend to be tightly localized inside the superlattice period for
V > J , with a localization length ξ = 1 /L = 1 / log( V /J ). To estimate the spreading behavior of the wave packet in the delocalized phase, we expandthe initial state at time t = 0 as a superposition of Bloch eigenstates of various bands, and make the (rather crude)approximation that each band is equally excited. The various wave packets in different bands propagate independentlyeach other, and in the long time limit the wave packet displaying the largest spread is the one belonging to the band l = l with the largest bandwidth ∆ W l , which propagates at the largest group velocity v g given by v g ∼ p ∆ W l . Such a relation for the group velocity is justified as follows. The dispersion curve E l ( k ) of a tiny superlattice band,with k varying in the range ( − π/p, π/p ), can be approximated by the tight-binding curve E l ( k ) = (∆ W l /
2) cos( kp ),where ∆ W l is the full width of the band. The group velocity v g can be then estimated from the standard relation v g = | ( dE l /dk ) π/ p | , relating the excitation speed and band dispersion curve, i.e. v g (cid:39) p ∆ W l /
2. Hence, at largetimes one expects the wave packet to spread far away from the initial site n = 0 by the distance σ ( t ) ∼ (1 /p ) v g t ,where the factor (1 /p ) accounts for the excitation fraction of the l -th band. The wave packet thus undergoes ballisticspreading with a velocity v ( V ) ∼ ( v g /p ) ∼ ∆ W l /
2. Taking into account that ∆ W l ≤ ∆ W = 4 | J − V | , one thus has v ( V ) < ∼ | J − V | , indicating that the ballistic speed v ( V ) should vanish as V approaches the critical value V c = J .Note that, if all bands in the superlattice had the same bandwidth and were equally excited, they all would contributeto the asymptotic wave spreading and the above reasoning would give v ( V ) ∼ ∆ W/ | J − V | , corresponding toa linear decrease of v with potential amplitude V till to vanish at V = J , as observed in numerical simulations [seeFig.2(b)]. B. Asymmetric hopping (non-Hermitian lattice)
Let us now assume J R < J L , with a non-vanishing hopping J R >
0. The special case of unidirectional hopping,i.e. J R = 0, will be considered in more details in Sec.III. The spectral properties of the non-Hermitian AAH modelwith asymmetric hopping amplitudes have been studied in some recent works [64, 72]. A localization-delocalizationtransition is found at the critical value V = V c of the on-site potential given by V c = J L (5)with all eigenstates extended and complex energy spectrum under periodic boundary conditions for V < V c (delocalizedphase), and all eigenstates exponentially localized with real and pure-point spectrum for V > V c (localized phase).Interestingly, in the localized phase the energy spectrum of H is the same as the one of the associated HermitianAAH Hamiltonian H H φ n = J ( φ n +1 + φ n − ) + 2 V cos(2 παn ) φ n (6)with symmetric hopping amplitude J given by J = (cid:112) J R J L , (7)while the eigenfunctions ψ n of H are obtained from those φ n of H after multiplication by the term ∼ exp( hn ), i.e. ψ n = exp( nh ) φ n , where we have set h = 12 log (cid:18) J L J R (cid:19) . (8)Hence, in the localized phase the localization lengths of the wave functions ψ n are asymmetric for left and rightsides, which is reminiscent of the non-Hermitian skin effect [78–81] in lattices with asymmetric hopping under openboundary conditions. A distinctive feature of the spectral phase transition between Hermitian and non-Hermitianmodels is that in the latter case the Lebesgue measure ∆ W of energy spectrum at the critical point V = V c = J L does not vanish and reads ∆ W c = 4 | J L − (cid:112) J R J L | . (9)As we are going to discuss below, a non vanishing spectral measure at the phase transition point enables ballistictransport in the lattice in the delocalized phase with a velocity v ( V ) which does not vanish as V → V − c . Figures 3and 4 show a few numerical results of wave packet spreading in a non-Hermitian lattice and the numerically-computedbehavior of the ballistic velocity v ( V ) for two values of the ratio J R /J L . Note that, as the hopping is asymmetric,transport in the lattice is unidirectional [82, 83], so that in the delocalized phase excitation spreading is describedby a half light cone. The behavior of v ( V ) versus the amplitude V of on-site incommensurate potential unveils twomajor distinctive and somehow unexpected features:(i) For J R /J L smaller than ∼ .
5, as V is increased from zero the ballistic speed v first increases (rather thandecreases), i.e. wave spreading in the lattice becomes faster at larger disorder [see Fig.4(b)]. This result is rathercounterintuitive and provides a noteworthy example of disorder-enhanced propagation [32–41] in the non-Hermitianrealm.(ii) The ballistic speed v is discontinuous at the phase transition point V = V c , i.e. excitation can propagate alongthe lattice at a non-vanishing velocity even arbitrarily close to the critical point.To explain such a behavior, as in previous subsection let us consider the rational approximation α ∼ q/p of α , with p , FIG. 5. (color online) Energy spectrum in complex energy plane for the AAH Hamiltonian with asymmetric hopping and fora few increasing values of the on-site amplitude V . Curve 1: V /J L = 0; curve 2: V /J L = 0 .
3; curve 3:
V /J L = 0 .
6; curve4:
V /J L = 0 .
9; curve 5:
V /J L = 1 (phase transition point). In (a) J R /J L = 0 .
5, in (b) J R /J L = 0 .
3, whereas J R = 0 in (c)(unidirectional hopping(. The energy spectra in (a) and (b) have been numerically computed as the eigenvalues of the matrix M defined by Eq.(A5) in Appendix A for the rational approximation q/p = 89 /
144 of the inverse of the golden ratio. q irreducible integer numbers, in the large p limit. Let us indicate by E l ( k ) the dispersion curve of the l -th band of thesuperlattice described by the Hermitian Hamiltonian H , defined by Eq.(6). It can be readily shown that the p energybands of H , under periodic boundary conditions, are given by E l ( k + ih ), i.e they are obtained from the dispersioncurves of the associated Hermitian lattice [Eq.(6)] after the replacement k → k + ih , i.e. after complexification ofthe Bloch wave number, with h = (1 /
2) log( J L /J R ) (technical details are given in Appendix A). It should be notedthat in the current literature on the non-Hermitian skin effect complexification of the Bloch wave number k , knownas the generalized Brillouin zone, is relevant to determine the energy spectrum of a given Hamiltonian H under theopen boundary conditions in the thermodynamic limit [78, 80]: the energy spectrum of H under open boundaryconditions is obtained from the Hamiltonian H ( k ) in Bloch space after complexification of k , so as β ≡ exp( ik ) doesnot describe anymore a unit circle in complex plane. However, in our case complexification of k is not related tothe open boundary condition case: we consider two different superlattice models with Hamiltonians H and H inphysical space, the former non-Hermitian and the latter Hermitian, and show that the energy spectra (Bloch minibands) under periodic boundary conditions of H are obtained from those of H , under the same periodic boundaryconditions, after complexification of k . For a vanishing potential amplitude V = 0, we can take p = 1, so thatthe Hermitian lattice of Eq.(6) displays the single tight-binding band E ( k ) = 2 J cos k ; correspondingly, the energyspectrum of the non-Hermitian lattice with asymmetric hopping reads E ( k ) = 2 (cid:112) J R J L cos( k + ih ) = J R exp( ik ) + J L exp( − ik ) (10)describing an ellipse in the complex energy plane [see curve 1 in Figs.5(a) and (b)]. The largest velocity at which anexcitation propagates along the lattice is given by the group velocity v g = Re { ( dE/dk ) k m } at the Bloch wave number k m where the imaginary part of E ( k ) takes its largest value [82, 84], i.e. k m = − π/ v g = J R + J L for V = 0. Asthe potential amplitude V is increased, the energy spectrum in complex plane undergoes a smooth deformation froman ellipse, as shown by curve 2 in Figs.5(a) and (b). The deformation of the ellipse changes the dispersion relationand, for sufficiently small J R /J L , can result in an increase of the group velocity v g , above the value ( J L + J R ) foundat V = 0. Numerical results indicate that this happens for J R /J L < ∼ .
5, i.e it requires strong enough asymmetrybetween left and right hopping amplitudes. As the potential amplitude V is further increased, the closed energy loopin complex energy plane splits into multiple separated loops; see Figs.5(a) and (b). The number of splitted loopsincreases as the critical point is approached, and their radius shrinks so as the energy spectrum becomes entirely real,with a fractal structure, at the critical point V c = J L [curve 5 in Figs.5(a) and (b)]. For V ≥ V c , i.e. in the localizedphase, the energy spectra of non-Hermitian H and associated Hermitian H Hamiltonians do coincide. The groupvelocity v g that describes propagation of an excitation along the lattice in the delocalized phase close to the criticalpoint V c = J L can be estimated as v g (cid:39) p ∆ W l /
2, where ∆ W l is the width (real part of the energy) of the bandof the superlattice displaying the largest value of imaginary part (thus dominating the dynamics at long times). Anumerical inspection of the band structure near the critical point indicates that the band with the largest imaginarypart of energy also corresponds to the wider band in the real part of energy. Clearly, since (cid:80) l ∆ W l (cid:39) ∆ W c > W l ≥ ∆ W c /p and thus v g ≥∼ ∆ W c /
2, indicating that the speed of ballistic motionremains finite as the critical point is approached.
III. DYNAMICAL PHASE TRANSITION IN THE NON-HERMITIAN AUBRY-ANDR´E-HARPERMODEL WITH UNIDIRECTIONAL HOPPING
In this section we consider the special case corresponding to the AAH model with unidirectional hopping, i.e. J L = J > J R = 0. The Hamiltonian reads Hψ n = Jψ n − + 2 V cos(2 παn ) ψ n . (11) A. Energy spectrum and localization-delocalization transition
The energy spectrum of this model can be determined in an exact form, as shown in Appendix B (see also [65]).The main results can be summarized as follows:(i) For
V < V c ≡ J , the energy spectrum E is absolutely continuous and describes an ellipse in complex energy planedefined by the dispersion relation E ( ω ) = (cid:18) J + V J (cid:19) cos ω − i (cid:18) J − V J (cid:19) sin ω (12)with − π ≤ ω < π . The corresponding wave functions ψ n ( ω ) are extended (generalized eigenfunctions). Note thatthe ellipse shrinks into a segment on the real axis, from E = − J to E = 2 J , as V approaches the critical value J [Fig.5(c)].(ii) For V ≥ J , the energy spectrum E is real, pure point and dense in the interval ( − V, V ), with one-sided wavefunctions exponentially-localized with energy-independent localization length given by ξ = 1 / log( V /J ).Note that, compared to the lattice model with asymmetric but bidirectional hopping discussed in Sec. II.B, as theon-site potential amplitude V is increased to approach the critical value V c , the energy spectrum remains an ellipseand does not split into a set of loops [compare Figs.5(a,b) and Fig.5(c)]. Also, in the localized phase the spectrumis pure point but does not show the typical Cantor set structure of infinitely many small bands separated by smallgaps (the energy spectrum is the entire ( − V, V ) interval). The reason thereof is that, while in the asymmetrichopping case with J R > H in the localized phase is the same as the one of the associatedHermitian AAH Hamiltonian H , thus displaying a fractal structure, in the unidirectional hopping limit J R = 0 sucha correspondence becomes invalid and, as shown in Appendix B, the pure point energy spectrum is equidistributedin the full range ( − V, V ). B. Dynamical behavior
In this subsection we study the temporal behavior of a wave packet in the lattice with unidirectional hopping, andderive analytical expression of the largest velocity v for propagation of excitation along the lattice that defines the half l a tt i c e s i t e n normalized time Jt potential amplitude V/J v e l o c i t y v (b)(a) V/J =0.2
V/J =0.6
V/J =0.9
V/J =1 V/J =1.2
FIG. 6. (color online) Same as Fig.3, but for J R = 0, J L = J . The dashed line in (b) shows the behavior of velocity v predictedby the saddle point method [Eq.(25)]. light cone aperture of Fig.1(b). Clearly, for V > J the energy spectrum is real, all wave functions are exponentiallylocalized with the same localization length, so that spectral localization implies also dynamical localization [85].Therefore, for
V > J one has v ( V ) = 0 and δ ( V ) = 0. On the other hand, for V < J (delocalized phase) the spectrumis absolutely continuous and transport in the lattice is expected to be ballistic ( δ = 1) with largest propagationvelocity v = v ( V ), that we wish to calculate analytically. The most general wave packet at t = 0 can be decomposedas a superposition of generalized eigenfunctions ψ n ( ω ) of H by suitable spectral amplitudes F ( ω ), which depend onthe initial state. The solution to the Schr¨odinger equation i ( dψ n /dt ) = Hψ n at times t ≥ ψ n ( t ) = (cid:90) π − π dωF ( ω ) ψ n ( ω ) exp[ − iE ( ω ) t ] (13)where the dispersion relation E = E ( ω ) is given by Eq.(12) and where the form of generalized eigenfunctions ψ n ( ω )is given in Appendix B. To determine the largest propagation speed v = v ( V ) of the wave packet along the lattice,we follow the method outlined in Ref.[84], considering the asymptotic behavior at long times t of the wave functionalong the space-time line n = νt , with some fixed velocity ν >
0, i.e. we consider the asymptotic behavior of ψ ( t ) ≡ ψ n = vν ( t ) = (cid:90) π − π dωF ( ω ) ψ n = νt ( ω ) exp[ − iE ( ω ) t ] (14)as t → ∞ . For n >
0, the wave function ψ n ( ω ) has the form [see Eq.(B2) in Appendix B] ψ n ( ω ) = n (cid:89) l =1 JE ( ω ) − V cos(2 παl ) (15)where we assumed, without loss of generality, ψ ( ω ) = 1. We can formally write ψ νt ( ω ) = exp[ iνtS ( ω )] (16)where we have set S ( ω ) = iνt νt (cid:88) l =1 log (cid:18) E ( ω ) − V cos(2 παl ) J (cid:19) . (17)In the large νt limit, the sum on the right hand side of Eq.(17) can be approximated by an integral owing to theWeyl (cid:48) s equidistribution theorem and the properties of irrational rotations, namely one has S ( ω ) = i π (cid:90) π − π dk log (cid:18) E ( ω ) − V cos kJ (cid:19) = iγ ( ω ) (18)where γ ( ω ) = − iω is the right Lyapunov exponent associated to the wave function ψ n ( ω ) (see Appendix B for technicaldetails). Hence, in the large t limit we may assume ψ νt ( ω ) (cid:39) exp( iνtω ) and thus ψ ( t ) ∼ (cid:90) π − π dωF ( ω ) exp[ iνωt − iE ( ω ) t ] . (19)0The growth rate λ ( ν ) of the wave packet along the space-time line n = νt , given by λ ( ν ) = lim t →∞ log | ψ ( t ) | t , (20)can be finally calculated by the saddle-point method [84]. This yields λ ( v ) = − ν Im( ω s ) + Im( E ( ω s )) (21)where the saddle point ω = ω s is the root of the equation (cid:18) dEdω (cid:19) ω s = ν (22)in complex plane. In our model, since E ( ω ) = 2 V cos( ω + iρ ), with ρ = log( J/V ), it readily follows that ω s = − iρ − arcsin( ν/ V ) and λ ( ν ) = (cid:26) ρν ν < Vρν − ν arccosh (cid:0) ν V (cid:1) + √ ν − V ν > V. (23)The largest value λ m of the growth rate λ ( ν ) is given by λ m = 2 V sinh ρ = J − V J (24)and it is attained at the speed ν = v , given by v = 2 V cosh ρ = J + V J (25)with the saddle point ω s = − π/ v associated to the largest growth rate provides theaperture of the half light cone of Fig.1(b). Equation (25) clearly shows that such a velocity is an increasing function ofpotential amplitude V , varying form v = J at V = 0 to v = 2 J as the critical point V = J is approached from below.Clearly, for V > J one has v = 0 (dynamical localization), thus proving that the behavior of v ( V ) is discontinuousat the phase transition point V = J (first-order phase transition). Our analytical results have been confirmed bynumerical simulations of wave packet spreading in the lattice, which are illustrated in Fig.6. In particular, thenumerically-computed behavior of the velocity v , computed by the relation v = σ ( t ) /t , turns out to be in very goodagreement with Eq.(25) predicted by the saddle-point method. IV. CONCLUSION
The Aubry-Andr´e-Harper model is the simplest and most studied one-dimensional model of aperiodic order dis-playing a delocalization-localization transition as the potential amplitude is increased above a finite threshold value.The abrupt transition in the energy spectrum (spectral phase transition), from absolutely continuous spectrum withextended wave functions in the delocalized phase to pure point spectrum with exponentially-localized wave functionsin the localized phase, is associated to a distinct dynamical behavior of wave spreading in the lattice (dynamicalphase transition), with ballistic transport in the delocalized phase and dynamical localization (suppression of wavespreading) in the localized phase. In terms of the velocity v of wave spreading assumed as an order parameter, thedynamical phase transition is of second order, i.e. v = v ( V ) is a continuous function of the potential amplitude V ,vanishes in the localized phase and its first derivative is discontinuous at the critical point V = V c . Similar spectralphase transitions have been recently found in certain non-Hermitian extensions of the Aubry-Andr´e-Harper model[64, 65, 69, 70, 72], however the features of associated dynamical phase transition have been overlooked. In thiswork we unveiled distinct physical behavior in wave spreading and dynamical phase transitions in a non-HermitianAubry-Andr´e-Harper model with asymmetric hopping amplitudes, as compared to the Hermitian limit of symmetrichopping. Remarkably, we found that for sufficiently strong asymmetry in the hopping amplitudes the propagation ofan excitation along the lattice is enhanced (rather than inhibited) by disorder. Also, the dynamical phase transitionis of first-order in the velocity v , since v ( V ) turns out to be discontinuous at the critical point. Such results provideimportant advances to understand the nontrivial interplay between disorder and non-Hermiticity, which is currentlya hot area of research [29, 53–61, 64, 67, 69, 71, 73, 74]. The kind of non-Hermitian Hamiltonian with asymmetrichopping amplitudes, considered in this work, could be realized in synthetic matter using, for example, photonic sys-tems [53, 64, 86, 87], topoelectrical circuits [88], mechanical metamaterials [89] or in continuously-measured ultracoldatom systems with reservoir engineering [55, 90].1 ACKNOWLEDGMENTS
The author acknowledges the Spanish State Research Agency through the Severo Ochoa and Mara de MaeztuProgram for Centers and Units of Excellence in R&D (Grant No. MDM-2017-0711).
Appendix A: Energy spectrum for a commensurate potential
For rational α = q/p , i.e. for a commensurate potential V n = 2 V cos(2 παn ), the Hamiltonian H of the lattice, Hψ n = J R ψ n +1 + J L ψ n − + V n ψ n , (A1)describes a superlattice of period p . Under periodic boundary conditions, its energy spectrum is thus absolutelycontinuous and is formed by a set of p energy bands. According to the Bloch theorem, the eigenfunctions ψ n of H satisfy the Bloch condition ψ n + p = ψ n exp( ikp ) , (A2)where the Bloch wave number k varies in the range ( − π/p, π/p ), and can be assumed as a continuous variable for aninfinitely extended lattice. After setting A = ψ , A = ψ , ..., A p = ψ p , it then readily follows that the eigenvalueequation J R ψ n +1 + J L ψ n − + V n ψ n = Eψ n (A3)is satisfied provided that E A = M A (A4)where we have set A = ( A , A , ..., A p ) T and M is the p × p matrix given by M = V J R ... J L exp( − ikp ) J L V J R ... ... ... ... ... ... ... ... ... J L V p − J R J R exp( ikp ) 0 0 ... J L V p . (A5)The energy bands E = E l ( k ) ( l = 1 , , , ..., p ) are thus the p eigenvalues of the matrix M , which depend continuouslyon the Bloch wave number k . Clearly, for symmetric hopping J R = J L = J the Hamiltonian H is Hermitian and theenergies E l ( k ) of various bands are real. In the large p limit, a large set of narrow bands separated by small gaps isfound, which gives a Cantor set in the p → ∞ limit. The Lebesgue measure ∆ W of energy spectrum is defined as thesum of the widths ∆ W l of various bands, i.e. ∆ W = (cid:80) l ∆ W l , which converges toward 4 | J − V | as p → ∞ [11].For asymmetric hopping, the dispersion curves of energy bands take values in complex plane and describe rathergenerally one or more closed loops in complex energy plane (see Fig.5). Interestingly, the dispersion curves E l ( k ) of thenon-Hermitian lattice can be formally obtained from the one of an associated Hermitian lattice after complexificationof the Bloch wave number k . In fact, let us consider the p × p diagonal matrixΛ = ...
00 exp( − h ) 0 ... ... ... ... ... ... ... exp( − ph + h ) (A6)and the matrix M , obtained from M by the similarity transformation M = Λ M Λ − . (A7)Clearly, for any Bloch wave number k the matrices M and M have the same eigenvalues. If we assume h =(1 /
2) log( J L /J R ), from Eqs.(A5), (A6) and (A7) it readily follows that M = V J ... J exp( − iκp ) J V J ... ... ... ... ... ... ... ... ... J V p − JJ exp( iκp ) 0 0 ... J V p (A8)2where we have set J ≡ √ J L J R and κ ≡ k + ih. (A9)Equation (A8) clearly indicates that the energies E l of H with asymmetric hopping J L > J R , i.e. the eigenvalues of M , are the same than the energies of an Hermitian Hamiltonian H with symmetric hopping J = √ J L J R but withthe Bloch wave number complexified according to Eq.(A9). Appendix B: Energy spectrum of the unidirectional Aubry-Andr´e-Harper model
The energy spectrum of the unidirectional Aubry-Andr´e-Harper model is the set of complex numbers E such thatthe solutions to the recurrence equation Eψ n = Jψ n − + V n ψ n , (B1)with V n = 2 V cos(2 παn ), is not unbounded as n → ±∞ . To determine whether a given value E belongs to thespectrum, we have to study the asymptomatic behavior of ψ n in the large | n | limit. To this aim, let us distinguishtwo cases. First case.
Let us first consider a complex value of E with E / ∈ ( − V, V ). Since E (cid:54) = V n for any n , for a given value ψ of the wave function amplitude at site n = 0 from Eq.(B1) one has ψ n = (cid:32) n (cid:89) l =1 JE − V l (cid:33) ψ (B2)for n >
0, and ψ n = (cid:32) n (cid:89) l = − E − V l +1 J (cid:33) ψ (B3)for n <
0. The right and left Lyapunov exponents γ ± , that determine the asymptotic behavior of ψ n as n → ±∞ , aregiven by γ + = − lim n →∞ n log (cid:18) ψ n ψ (cid:19) (B4) γ − = lim n →−∞ n log (cid:18) ψ n ψ (cid:19) . (B5)Using Eqs.(B2) and (B3), it readily follows that γ + = − γ − ≡ γ , with γ = lim n →∞ n n (cid:88) l =1 log (cid:18) E − V cos(2 παl ) J (cid:19) . (B6)Note that, since γ − = − γ + , if the wave function were exponentially localized at n → ∞ , i.e. Re( γ + ) >
0, then itwould be exponentially delocalized as n → −∞ since Re( γ − ) <
0, and viveversa. Hence the wave function cannot beexponentially localized for any value of E outside the range ( − V, V ), i.e. E does not belong to the point spectrumof the Hamiltonain. However, provided that the real part of γ vanishes the wave function is extended but does notsecularly grow with | n | , i.e. E belongs to the continuous spectrum of the Hamiltonian whenever Re( γ ) = 0. Forirrational α , the limit on the right hand side of Eq.(B6) can be calculated using the Weyl (cid:48) s equidistribution theorem[65], yielding γ = 12 π (cid:90) π − π dk log (cid:18) E − V cos kJ (cid:19) . (B7)Since E is not inside the interval ( − V, V ), we may set E = 2 V cos θ , with θ a complex angle and Im( θ ) <
0. Theintegral on the right hand side of Eq.(B7) can be calculated in an exact form [65], yielding the following final formfor the Lyapunov exponent γ = log VJ + iθ. (B8)3Note that since Im( θ ) <
0, one has Re( γ ) > log( V /J ). Therefore, for
V > J one has Re( γ ) >
0, regardless of the valueof energy E , so that E does not belong to the continuous spectrum. On the other hand, for V ≤ J the continuousspectrum is not empty and is composed by the set of complex energies E such that Re( γ ) = 0, i.e. γ = − iω with ω arbitrary real parameter. Using Eq.(B8), the condition γ = − iω and Ansatz E = 2 V cos θ , yields E = 2 V cos (cid:18) i log VJ − ω (cid:19) = J exp( − iω ) + V J exp( iω ) . (B9)Note that, as ω varies in the range ( − π, π ), the continuous spectrum E describes an ellipse in complex energy plane,which shrinks toward the segment ( − V, V ) on the real energy axis as V approaches from below the critical value V c = J [Fig.5(c)]. Second case.
Let us now assume that E is real and inside the range ( − V, V ). Precisely, let us assume that E = 2 V cos(2 παn ) for some integer n . Note that, since α is irrational, the set of energies E obtained when n varies from −∞ to ∞ is dense and equidistributed in the range ( − V, V ). In this case the solution to Eq.(B1) readsexplicitly ψ n = n < n n = n (cid:81) nl = n +1 JE − V l n > n (B10)The right Lyapunov exponent reads γ + = − lim n →∞ n log ψ n = 12 π (cid:90) π dk log (cid:18) E − V cos kJ (cid:19) . (B11)For E ∈ ( − V, V ), the integral on the right hand side of Eq.(B11) turns out to be independent of E and equals tolog( V /J ), i.e. γ + = log (cid:18) VJ (cid:19) . (B12)For V < J , γ + < ψ n exponentially grows as n → ∞ , i.e. E does not belong to the point spectrum of theHamiltonian. On the other hand, for V > J one has γ + > E = 2 V cos(2 παn ), dense in the range( − V, V ): hence E belongs to the point spectrum of the Hamiltonian. 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