Generalized Aubry-André self-duality and Mobility edges in non-Hermitian quasi-periodic lattices
GGeneralized Aubry-Andr´e self-duality and Mobility edges in non-Hermitianquasi-periodic lattices
Tong Liu, ∗ Hao Guo, † Yong Pu, ‡ and Stefano Longhi
3, 4, § Department of Applied Physics, School of Science,Nanjing University of Posts and Telecommunications, Nanjing 210003, China Department of Physics, Southeast University, Nanjing 211189, China 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 - Palma de Mallorca, Spain (Dated: July 14, 2020)We demonstrate the existence of generalized Aubry-Andr´e self-duality in a class of non-Hermitianquasi-periodic lattices with complex potentials. From the self-duality relations, the analytical expres-sion of mobility edges is derived. Compared to Hermitian systems, mobility edges in non-Hermitianones not only separate localized from extended states, but also indicate the coexistence of complexand real eigenenergies, making it possible a topological characterization of mobility edges. An exper-imental scheme, based on optical pulse propagation in synthetic photonic mesh lattices, is suggestedto implement a non-Hermitian quasi-crystal displaying mobility edges.
PACS numbers: 71.23.An, 71.23.Ft, 05.70.Jk
I. INTRODUCTION
Anderson localization , i.e. the absence of diffusion ofquantum or classical waves in disordered systems, is amilestone in condensed matter physics and beyond .According to the scaling theory , in one-dimensional(1D) and two-dimensional systems with random disorderall eigenstates are exponentially localized, no matter howsmall the strength of disorder is, while mobility edges,separating localized and extended states in energy spec-trum, are observed in three dimensional (3D) systems .However, it is well known that lattices with correlateddisorder can undergo a metal-insulator transition evenin 1D (see, e.g., and references therein). A paradig-matic example is provided by the famous Aubry-Andr´emodel , where the localization-delocalization transitioncan be derived from a symmetry (self-duality) argument .A hallmark of this model is the sharp nature of the local-ization transition and the absence of mobility edges, i.e.all single-particle eigenstates in the spectrum suddenlybecome exponentially localized above a threshold levelof disorder. Recent works reported on mobility edges incertain quasi-periodic 1D lattices displaying a generalized Aubry-Andr´e self-duality , making it possible the ob-servation of mobility edges in 1D systems withoutresorting to 3D models. The role of particle interactionand many-body localization in such systems have beeninvestigated as well . However, such previous studieshave been limited to consider Hermitian models.Non-Hermitian lattices show exotic physical phe-nomena without any Hermitian counterparts, suchas exceptional points, breakdown of bulk-boundarycorrespondence based on Bloch band invariants, andnon-Hermitian skin effect . Remarkably, disorder canbehave differently in Hermitian versus non-Hermitiansystems (see e.g. and references therein). A seminalwork dealing with disorder in non-Hermitian lattices is the Hatano-Nelson model , in which an asymmet-ric hopping caused by an imaginary gauge field results ina localization-delocalization transition and the existenceof mobility edges . Since this pioneering study, severalnon-Hermitian models with either random or incom-mensurate disorder have been investigated, in whichnon-Hermiticity is introduced by either asymmetric hop-ping amplitudes or complex on-site potentials . Incertain models, the topological nature of the localizationtransition and self-duality have been discussed .However, we emphasize that the self-dual symmetry fallsinto two categories. The first category is the generalized Aubry-Andr´e self-dual symmetry. Models possessingsuch a generalized symmetry show mobility edges, andtheir analytical form can be obtained from the self-dualrelations. Such models are rare and there are only fewknow examples for Hermitian systems . The secondcategory is the “simple” self-dual symmetry. This is amuch more common kind of symmetry which is foundin many models , including some non-Hermitiansystems. However, this type of symmetry cannot beused to derive analytical form of mobility edges, and sofar available non-Hermitian models displaying mobilityedges resort to numerical results. Here a majorquestion arises: can a generalized
Aubry-Andr´e self-dualsymmetry and exact form of mobility edges be foundbeyond Hermitian quasi-crystals?In this work we address this open question and in-troduce exactly-solvable non-Hermitian models inquasi-periodic lattices with complex potentials dis-playing mobility edges and a generalized Aubry-Andr´eself-duality. A photonic implementation of the proposedmodels, based on pulse propagation in synthetic fibermesh lattices, is also presented. a r X i v : . [ c ond - m a t . d i s - nn ] J u l EE minmax delocalized phase localized phase t V V FIG. 1: (Color online) The real part of eigenvalues of Eq. (1)and IPR as a function of V with the parameter s = 1. Thetotal number of sites is set to be L = 500. Different colours ofthe eigenvalue curves indicate different magnitudes of the IPRof the corresponding wave functions. The black eigenvaluecurves denote the delocalized states, and the bright yelloweigenvalue curves denote the localized states. The blue solidlines represent the boundary between spatially localized anddelocalized states, i.e., the mobility edge E m = t ( V − E max ) and lower ( E min )boundaries of the energy spectrum are independent of V . II. EXACTLY-SOLVABLE NON-HERMITIANMODELS DISPLAYING GENERALIZEDAUBRY-ANDR´E SELF-DUALITYA. Model I
As a first example of generalized Aubry-Andr´e self-duality, let us consider a non-Hermitian 1D model withexponentially-decaying hopping amplitude and quasi-periodic complex on-site potential, defined by the eigen-value equation( E + t ) ψ n = t (cid:88) n (cid:48) e − s | n − n (cid:48) | ψ n (cid:48) + V e i παn ψ n , (1)where s > V is the complex potential strength, α is irra-tional, and ψ n is the amplitude of wave function at the n th lattice. We choose the parameters α = ( √ − / t = e s . When the on-site potential is replaced by V cos(2 παn ), this model becomes the Hermitian quasi-periodic lattice studied in Ref. , which displays mobilityedges given by E m = cosh( s ) V − e s . To determine theexpression of mobility edges in the non-Hermitian case,we first introduce a function W n ( s ) defined as E + t − V e i παn = ( E + t ) W n ,e s = E + tV , W n ( s ) = e s − e i παn e s , (2) so that Eq. (1) takes the form( E + t ) W n ( s ) ψ n = t (cid:88) n (cid:48) e − s | n − n (cid:48) | ψ n (cid:48) . (3)After multiplying both sides of Eq. (3) by W k ( s ) e ik παn ,summing over n and setting φ k = (cid:80) n e πiαkn W n ( s ) ψ n ,one obtains( E + t ) W k ( s ) φ k = t (cid:88) k (cid:48) e − s | k − k (cid:48) | φ k (cid:48) . (4)The detailed derivation of Eq. (4) is given in AppendixA. Remarkably, when s = s , Eq. (1) has the same formas Eq. (4), i.e. a generalized Aubry-Andr´e self-duality isfound. Following the Aubry-Andr´e work , we conjecturethat the localization-delocalization transition is locatedat the self-dual point E + tV = e s . Thus the non-Hermitianquasi-periodic lattice defined by Eq. (1) displays a mo-bility edge at the energy E m = t ( V − . (5)To verify our conjecture, we calculated analytically theLyapunov exponent µ ( E ) for the eigenstates of Eq. (1),and found that the mobility edge obtained from Lya-punov exponent analysis is precisely given by Eq. (5);technical details are given in Appendix B. Remarkably,the self-duality argument enables to analytically com-pute mobility edges without solving the spectral prob-lem. The following bounds E min < E < E max for theenergy spectrum E of the delocalized phase can be de-rived (Appendix B), where E min = − tt + 1 , E max = 2 tt − , (6)indicating that the delocalized modes correspond to realenergies. Conversely, for the localized modes the energyspectrum is complex. In other words, the mobility edge E m not only discriminates between localized and delocal-ized states, but also between real and complex energies.A major implication of this result is that a topological number can be introduced to predict the existence of amobility edge. This entails to compute winding numbers w ( E B ) that count the number of times the complex spec-tral trajectory encircles a base energy E B as an externalphase in the Hamiltonian is varied . As shown inAppendix C, the knowledge of the winding numbers atthe two base energies E B = E min and E B = E max issufficient to topologically predict the existence of the mo-bility edge.Our theoretical predictions have been verified by a nu-merical analysis of Eq. (1) on a finite lattice containing L sites under periodic boundary conditions. The localiza-tion properties of eigenstates are measured by the inverseparticipation ratio (IPR) . For a normalized wave func-tion, it is defined asIPR n = L (cid:88) j =1 (cid:12)(cid:12) ψ nj (cid:12)(cid:12) , delocalized phase localized phase FIG. 2: (Color online) The real part of eigenvalues of Eq. (7)and IPR as a function of V with the tuning parameter a = 0 . L = 500. Differentcolours of the eigenvalue curves indicate different magnitudesof the IPR of the corresponding wave functions. The blackeigenvalue curves denote the delocalized states, and the brightyellow eigenvalue curves denote the localized states. The bluesolid lines represent the boundary between spatially localizedand delocalized states, i.e., the mobility edge E m = a +1 /a . Inthe inset, it is clearly shown that the mobility edge E m = 2 . where n is the index of energy level. It is well known thatthe IPR of a delocalized state scales like L − , thus van-ishing in the thermodynamic limit, while it is finite fora localized state. In Fig. 1 we show the numerically-computed IPR diagram in the (Re( E ) , V ) plane on apseudo color map, clearly demonstrating a mobility edgealong the line defined by Eq. (5). The mobility edge alsoseparates real and complex energies. B. Model II
As a second example, let us consider a nearest-neighborhopping model with tunable complex on-site potentialdefined by the eigenvalue equation Eψ n = ψ n +1 + ψ n − + V − ae i παn ψ n , (7)where 0 < a < and displaying gen-eralized self-duality. As compared to model I, model II ismore feasible for an experimental implementation since itdoes not require hopping control, but it is not amenablefor a full analytical treatment (Lypaunov exponent cal-culation). As we are going to show, model II displaysgeneralized self-duality, which is enough to analytically predict mobility edges. To this aim, let us multiply bothsides of Eq. (7) by e i παnm and sum over n . After setting φ m = (cid:80) n e i παnm ψ n , from Eq. (7) one obtains[ E − παm )] φ m = (cid:88) n e i παnm V − ae i παn ψ n . (8)We introduce some functions defined as follows e s = 1 a , W n ( s ) = e s − e i παn e s ,E = 2 cosh( s ) , Ω m ( s ) = cosh( s ) − cos(2 παm )sinh( s ) , (9)so that Eq. (8) can be written as2 sinh( s )Ω m ( s ) φ m = V (cid:88) r (cid:48) e −| m − r (cid:48) | s φ r (cid:48) . (10)Multiplying both sides of Eq. (10) by Ω k ( s ) e i παmk ,summing over m and after setting ϕ k = (cid:80) m e i παmk Ω m ( s ) φ m , Eq. (10) takes the form2 sinh( s )Ω k ( s ) ϕ k = V (cid:88) k (cid:48) e −| k − k (cid:48) | s ϕ k (cid:48) . (11)Finally, let us multiply both sides of Eq. (11) by e i παkq ,summing over k and setting µ q = (cid:80) k e i παkq ϕ k , Eq.(11) can be transformed as2 sinh( s )sinh( s ) cosh( s ) µ q = µ q − + µ q +1 + V e s e s − e i παq µ q . (12)Note that when s = s , Eq. (7) has the same form asEq. (12). From the self-dual relations e s = 1 /a and E = 2 cosh( s ), we obtain the mobility edge energy E m = a + 1 /a. (13)Interestingly, for a fixed value of the tuning parameter a , E m is independent of the potential strength V . Theproperty of “being constant” of the mobility edge is aremarkable result, not reported in the literature yet. Wechecked the predictions of the theoretical analysis by di-rect numerical simulations of Eq. (7). The numericalresults, shown in Fig. 2, confirm our theoretical predic-tions with excellent accuracy. From the analysis of theenergy spectrum, we find the same scenario like modelI, i.e. extended (localized) eigenstates correspond to real(complex) energies (see the inset of Fig. 2). A windingnumber, revealing the topological signature of the mobil-ity edge, can be also introduced for this model as well,as shown in Appendix C. III. PROPOSAL OF EXPERIMENTALIMPLEMENTATION
Photonic systems have been recently shown to pro-vide an experimentally-accessible platform to implement
EOM AOM SOA 3x3coupler 3x3coupler 2x2coupler input pulseSOA nn -1 n +1 detection L LL+ Δ LL- Δ Ln Δ t τ p ( g ) ( G ) FIG. 3: (Color online) Schematic of a synthetic mesh pho-tonic lattice with complex potential, based on pulse propaga-tion in fiber loops. An optical pulse propagating in the mainfiber loop of length L is splitted into three pulses, after eachtransit, via a three-arm fiber interferometer with unbalancedarms of length L and L ± ∆ L , with L (cid:28) L . Time pulseseparation ∆ t = ∆ L/c , introduced by arm length unbalance,provides the time slot of the lattice mesh. The interferome-ter is coupled to the main fiber loop via two 3 × τ p < ∆ t/ × a ( m ) n at successive transits m in the loop can be monitoredby the output port of the coupler. The synthetic complexpotential is obtained by placing amplitude (AOM) and phase(EOM) modulators in the central arm of the interferometerand driven by independent step-wise waveforms h ( AOM ) n and h ( EOM ) n . Two semiconductor optical amplifiers (SOA), withgain parameters g and G , are also included in the central armof the interferometer and in the main loop. non-Hermitian lattices and to observe a wide variety ofnon-Hermitian phenomena, such as parity-time symme-try breaking, exceptional points, non-Hermitian skin ef-fect etc. To observe mobility edges in non-Hermitianquasi-periodic potentials, we focus our attention tomodel II discussed in previous section, which is moreamenable for an experimental realization. We considerdiscrete-time quantum walk of optical pulses in syn-thetic photonic mesh lattices , realized in coupledfiber rings with unbalanced path-lengths. Such syntheticlattices have been experimentally used to demonstrateda wealth of phenomena, such as Bloch oscillations ,parity-time symmetric phase transitions , Andersonlocalization , and the non-Hermitian skin effect .By proper combination of amplitude and phase modu-lators in the fiber loops , they can engineer ratherarbitrary non-Hermitian potentials. A schematic of thesynthetic photonic lattice is shown in Fig 3. A shortpulse is launched, via a 2 × L , which is coupled to athree-arm fiber interferometer of mismatched lengths L (central arm) and L ± ∆ L (upper and lower arms) by3 × L (cid:28) L . The cen-tral arm of the interferometer includes a semiconductoroptical amplifier (SOA), an electro-optic phase modula- tor (EOM) and an acousto-optic amplitude modulator(AOM). A second SOA is also placed in the main loopof length L . At each transit in the main loop, a pulseentering into the interferometer is splitted, at the outputport, into three pulses with time delays − ∆ t , 0 and ∆ t ,where ∆ t = ∆ L/c is the time delay introduced by theunbalanced arms in the interferometer. The successivepulse splitting emulates a discrete-time quantum walk,where the complex amplitude a ( m ) n of pulse occupyingthe n -th time slot (discrete space distance) at the m -thround trip evolves according a linear map . In par-ticular, by tailoring the EOM and AOM signals, one canimplement a non-Hermitian Hamiltonian with nearest-neighbor hopping and a rather arbitrary complex on-sitepotential V n , such as the quasi-periodic potential in Eq.(7). Details are given in Appendix D. Pulse evolutionmeasurements at successive transits in the loop, detectedby the output port of the 2 × m , of the normalized pulse amplitudes | a ( m ) | / √ P m , with P m = (cid:80) n | a ( m ) n | , as obtained fromthe map, defined by Eq. (D-3) of Appendix D, for theinitial condition a (0) n = δ n, (a single pulse is injected intothe main fiber loop) and for V = 0 . a = 0 .
5. Accordingto Fig.2, for such parameter values all eigenmodes are de-localized and the energy spectrum entirely real. On theother hand, in the presence of a mobility edge some eigen-states are localized with corresponding complex energies.In this case the localized modes with the highest growthrate (i.e. imaginary part of the eigenenergy) will domi-nate, resulting in a frozen spreading of | a ( m ) | as the timestep m increases. This case is illustrated in Fig. 4(b),corresponding to V = 1 and a = 0 . IV. CONCLUSIONS
In this work we unveiled a class of non-Hermitianquasi-periodic lattices displaying generalized Aubry-Andr´e self-duality and provided for the first time theanalytic form of mobility edges in any non-Hermitian dis-ordered system. An experimental scheme to observe mo-bility edges, accessible with current photonic technolo-gies, has been proposed. The self-dual symmetry hasa simple structure but a profound significance in non-Hermitian models since it predicts both the boundary ofcritical states and the transition from real to complex en-ergy spectrum, thus enabling to introduce a topological signature of mobility edges. Our results push the con-cept of generalized self-duality beyond known Hermitianmodels, providing a major tool to explore the rich physics site number n time step mtime step m t i m e s t e p m t i m e s t e p m PP mm (a)(b) FIG. 4: (Color online) Behavior of normalized pulse ampli-tudes | a ( m ) n | / √ P m (with P m = (cid:80) n | a ( m ) n | ) on a pseudocolormap for the complex potential of Model II with (a) V = 0 . a = 0 . V = 1, a = 0 . P m versus time steps m for again parameter G = 0 .
656 in (a), and G = 0 .
31 in (b). of non-Hermitian systems with correlated disorder.
Acknowledgments
This work is supported by the National NaturalScience Foundation of China (Grants No. 61874060,11674051, and U1932159), Natural Science Foundationof Jiangsu Province (Grant No. BK2020040057), andNUPTSF (Grant No. NY217118).
Appendix A: Derivation of Eq. (4)
In this Appendix we provide some technical detailsleading to Eq. (4) given in the main text. Multiplyingboth sides of Eq. (3) by W k ( s ) e ik παn and summing over n , one obtains ( E + t ) W k ( s ) φ k = S rh (A1)where we have set S rh = W k ( s ) (cid:88) n e ik παn t (cid:88) n (cid:48) e − s | n − n (cid:48) | ψ n (cid:48) . (A2)With the substitution r = n − n (cid:48) , one obtains S rh = t (cid:88) n (cid:48) e ik παn (cid:48) W k ( s ) (cid:88) r e ik παr e − s | r | ψ n (cid:48) . (A3)Using the identity W k ( s ) − = e s e s − e i παk = (cid:88) r e −| r | s e ir παk , (A4) one has S rh = t (cid:88) n (cid:48) e in (cid:48) παk ψ n (cid:48) . (A5)Clearly, since W n (cid:48) ( s ) − W n (cid:48) ( s ) = 1, and can also write S rh = t (cid:88) n (cid:48) W n (cid:48) ( s ) − W n (cid:48) ( s ) e in (cid:48) παk ψ n (cid:48) = t (cid:88) n (cid:48) (cid:88) r e −| r | s e ir παn (cid:48) W n (cid:48) ( s ) e in (cid:48) παk ψ n (cid:48) = t (cid:88) n (cid:48) (cid:88) r e −| r | s e i ( r + k )2 παn (cid:48) W n (cid:48) ( s ) ψ n (cid:48) . (A6)Finally, after the substitution k (cid:48) = r + k , one obtains S rh = t (cid:88) k (cid:48) e −| k − k (cid:48) | s (cid:88) n (cid:48) e in (cid:48) παk (cid:48) W n (cid:48) ( s ) ψ n (cid:48) = t (cid:88) k (cid:48) e −| k − k (cid:48) | s φ k (cid:48) . (A7)Substitution of Eq. (A7) into Eq. (A1) yields Eq. (4)given in the main text. Appendix B: Lyapunov exponent analysis (Model I)
In this Appenix we provide exact analytical computa-tion of the energy-dependent Lyapunov exponent for thesolutions to Eq. (1) with real eigenvalue E . To this aim,let us assume periodic boundary conditions on a ring ofsize L with αL ∼ integer, i.e., ψ n + L = ψ n , and thentake the L → ∞ limit. We consider the discrete Fouriertransform φ n = 1 √ L L (cid:88) l =1 e − πiαln ψ l ,ψ n = 1 √ L L (cid:88) l =1 e πiαln φ l , (B1)so that in the dual space Eq. (1) yields the followingdifference equation for φ n , Eφ n = Ω n φ n + V φ n − , (B2)where e have set Ω n = t (cid:80) l (cid:54) =0 e −| l | s e − πiαln =2 t Re { β n − β n } , β n = e − s − πiαn . The explicit expressionof Ω n reads Ω n = 2 t t cos(2 παn ) − − t cos(2 παn ) + t , (B3)For an arbitrary integer n , a formal solution with theeigenvalue E = Ω n of Eq. (B2) is given by φ n ∝ n < n n = n VE − Ω n n > n (B4)Let us calculate the Lyapunov exponent µ of the eigen-function (B4) in dual space with the eigenvalue E = Ω n µ ( E ) = − lim n →∞ n − n log (cid:12)(cid:12)(cid:12)(cid:12) φ n φ n (cid:12)(cid:12)(cid:12)(cid:12) , (B5)with µ ( E ) > (cid:80) n | φ n | < ∞ , i.e., µ ( E ) > localization in dual space and the delo-calization in real space. From Eq. (B4) and (B5), oneobtains µ ( E ) = lim n →∞ n − n n (cid:88) k = n +1 log (cid:12)(cid:12)(cid:12)(cid:12) Ω k − Ω n V (cid:12)(cid:12)(cid:12)(cid:12) . (B6)After setting F ( q ) ≡ t cos( q ) − − t cos( q ) + t (B7)so that Ω k = 2 tF ( q = 2 παk ), using the Weyl (cid:48) s equidis-tribution theorem of irrational rotations one can write µ ( E ) = 12 π (cid:90) π − π dq log (cid:12)(cid:12)(cid:12)(cid:12) t F ( q ) − F ( q ) V (cid:12)(cid:12)(cid:12)(cid:12) . (B8)with q = 2 παn , i.e., µ ( E ) = log (cid:18) tV (cid:19) + 12 π (cid:90) π − π dq log | F ( q ) − F ( q ) | . (B9)Taking into account that12 π (cid:90) π − π dq log | F ( q ) − F ( q ) | = 12 π Re (cid:26)(cid:90) π − π dq log( t cos( q ) − − t cos( q ) + t − σ ) (cid:27) , (B10)with σ ≡ Ω n / (2 t ) = E/ (2 t ), the integral on the rightside of Eq. (B10) can be computed in a closed form togive12 π (cid:90) π − π dq log | F ( q ) − F ( q ) | = log (cid:18) σ + 12 t (cid:19) (B11)so that µ ( E ) = log (cid:18) E/t + 1 V (cid:19) . (B12)The real energy E belongs to the spectrum of the Hamil-tonian, with delocalized eigenstate in real space, pro-vided that µ ( E ) >
0. Using Eq. (B12), this condi-tion yields
E > E m , where we have set E m = t ( V −
1) = exp( s )( V − E = E m corresponds to the mobility edge given byEq. (5), derived using the self-duality argument. Thismeans that the mobility edge E = E m , besides separat-ing localized and delocalized eigenstates, provides alsothe boundary for the energy spectrum to remain real.The Lyapunov exponent analysis also provides the upper and lower boundaries of the energy spectrum in the delo-calized phase, E min and E max , shown in Fig. 1. In fact,since E = 2 t Ω n = 2 tF ( q = 2 παn ), from Eq. (B7) onehas E min = min − π ≤ q<π { tF ( q ) } = − tt + 1 (B13)and E max = max − π ≤ q<π { tF ( q ) } = 2 tt − . (B14) Appendix C: Topological signature of mobility edges
The appearance of mobility edges, separating extendedand localized states, can be characterized by a topologi-cal invariant, given by a winding number that measuresthe times the spectral trajectory of the system encirclea given base energy E B when an additional phase in thepotential is varied . Here we discuss in details howthe winding numbers can be introduced for the two mod-els discussed in main text. Model I.
As discussed in the main text, the existence of amobility edge in Model I separates delocalized states withreal energies and localized states with complex energies.In particular, for a given value of t = exp( s ), a mobilityedge with energy E m = t ( V −
1) inside the energy spec-trum, E min < E m < E max , is found provided that thepotential amplitude V is bounded as V < V < V , with(Fig. 1) V = 1 − t + 1 , V = 1 + 2 t − . (C1)To provide a topological characterization of the existenceof the mobility edge, we consider Eq. (1) on a ring lat-tice comprising L sites with periodic boundary conditionsand add an extra-phase term θ to the potential, i.e. weconsider the eigenvalue equation Eψ n = t (cid:88) n (cid:48) (cid:54) = n e − s | n − n (cid:48) | ψ n (cid:48) + V e i παn + iθ ψ n ≡ H ( θ ) ψ n (C2)with Hamiltonian H = H ( θ ). For a given base energy E B , we can introduce a winding number w as follows [2,3] w ( E B ) = lim L →∞ πi (cid:90) π dθ ∂∂θ log det (cid:26) H (cid:18) θL (cid:19) − E B (cid:27) (C3)which counts the number of times the complex spectraltrajectory encircles the base energy point E B when thephase θ varies from zero to 2 π . Clearly, we expect w ( E B )to vanish when the energy spectrum is entirely real, butalso when the energy spectrum can be partially complexbut the base energy E B is smaller (larger) than the realpart of any eigenvalue. Therefore, an appropriate topo-logical characterization of the mobility edge requires tointroduce two winding numbers w = w ( E B = E min ) potential amplitude V w i nd i ng nu m be r w w w FIG. 5: (Color online) Numerically-computed behavior of thewinding numbers w and w versus potential amplitude V forModel I with s = 1. The energy-dependent mobility edge isfound between the two vertical dashed lines at V = V and V = V , where the winding numbers w and w undergo anabrupt change from 0 to 1 and w = w ( E B = E max ), where E min , E max are thelower and upper edges of the energy spectrum in the de-localized phase (Fig. 1). The numerically-computed be-havior of w and w versus V , shown in Fig. 5, clearlyindicates that the topological number W = w (1 − w )is non-vanishing and equals one solely for V < V < V ,i.e when there is an energy-dependent mobility edge. Model II.
For Model II, we consider the spectral problemfor the Hamiltonian H = H ( θ ), which includes a phaseshift θ in the potential, given by Eψ n = ψ n +1 + ψ n − + V − ae πiαn + iθ ψ n ≡ H ( θ ) ψ n . (C4)For a given base energy E B , we can introduce a wind-ing number w ( E B ) according to Eq. (C3). As noticed inthe main text, for a given tuning parameter a the mo-bility edge E m is independent of the potential strength V and given by E m = a + 1 /a . Owing to such a pe-culiar property, a single winding number can provide atopological signature of the mobility edge, which is ob-tained by assuming a base energy infinitesimally largerthan E m , i.e. E B = E + m . As the potential strength V is increased above zero, the winding number w ( E B ) isequal to one in the range V < V < V , where V and V are intersections of the mobility edge energy E = E m with the boundaries of the real part of the energy spec-trum. This behavior is illustrated in Fig. 6, which showsthe numerically-computed behavior of winding number w versus V . Note that below V (above V ), where alleigenstates are extended (localized), the winding numbervanishes. Appendix D: Photonic implementation of Model II
In this section we present the main model that describethe discrete-time quantum walk of optical pulses in thesynthetic photonic mesh lattice realized by the fiber op-tical setup described in Fig. 3. The analysis is rather potential amplitude V w i nd i ng nu m be r w V R e ( E ) E m V V V V FIG. 6: (Color online) Numerically-computed behavior of thewinding number w versus potential amplitude V for ModelII with a = 0 .
5, corresponding to a flat mobility edge E m =2 .
5. The base energy used to compute the winding numberis E B = 2 .
51. The inset shows the behavior of the real partof the energy spectrum versus V . The crossing point V = V ( V = V ) corresponds to the potential amplitude below(above) which all eigenstates are extended (localized). standard and is an extension of derivations provided inprevious works (see, for instance, which largely inspiredour setup). At each transit in the main loop, a pulse en-tering into the interferometer is splitted, at the outputport, into three pulses with time delays − ∆ t , 0 and ∆ t ,where ∆ t = ∆ L/c is the time delay introduced by theunbalanced arms in the interferometer. The successivepulse splitting emulates a discrete-time quantum walk,where the complex amplitude a ( m ) n of pulse occupyingthe n -th time slot (discrete space distance) at the m -th round trip evolves according a linear map . Thesymmetric 3 × S (3) = 1 √ iθ ) exp( iθ )exp( iθ ) 1 exp( iθ )exp( iθ ) exp( iθ ) 1 (D1)with θ = − π/
3, while the 2 × S (2) = 1 √ (cid:18) ii (cid:19) . (D2)From Fig. 3 and using Eqs. (D1) and (D2), the followingmap can be readily obtained a ( m +1) n = exp( G )3 √ (cid:104) exp(2 iθ ) (cid:16) a ( m ) n +1 + a ( m ) n − (cid:17) + U n a ( m ) n (cid:105) (D3)where we have set U n = exp( g − h ( AOM ) n − ih ( EOM ) n ) . (D4)In the above equations, G and g are the gain coeffi-cients of the SOA in the main loop and in the centralarm of the interferometer, respectively, whereas h ( AOM ) n −20 −15 −10 −5 0 5 10 15 2000.511.5−20 −15 −10 −5 0 5 10 15 20−1−0.500.51 site number n E O M a m p l i t u d e A O M a m p l i t u d e (a)(b) FIG. 7: (Color online) Patterns of (a) AOM and (b) EOMsignals ( h ( AOM ) n and h ( EOM ) n ) that realize the complex poten-tial V n of model II for parameter values V = 1 and a = 0 . g is set to g = 1. In(b), the constant bias − θ to h ( EOM ) n has been omitted forthe sake of simplicity. and h ( EOM ) n are the amplitudes of AOM and EOM mod-ulators, respectively, at the n -th time slot. To create atime-domain analog of a complex optical U n , the EOMand AOM modulators are independently driven with anwaveform generator. The waveform is a specially de-signed stepwise pulse pattern that enables to generatearbitrary phase and amplitude distributions along thefast coordinate (the time slot n ) but constant alongthe slow coordinate , i.e. periodic at time intervals T = ( L + L ) /c . The spectrum and localization proper-ties of the eigenstates of the map Eq. (D3) are obtainedby taking the Ansatz a ( m ) n = µ m ψ n (D5)which yields the matrix spectral problem Eψ n = ψ n +1 + ψ n − + V n ψ n (D6) where we have set V n = U n exp( − iθ ) = exp (cid:104) − iθ + g − h ( AOM ) n − ih ( EOM ) n (cid:105) (D7)and E = 3 √ µ exp( − G − iθ ), i.e. µ = E √ G + 2 iθ ) . (D8)Note that the gain G supplied by the SOA in the mainloop controls the growth/decay rate | µ | of the pulse trainat successive transits, however it does not affect the shapeof the complex potential V n . The gain G should betuned to keep the system below the instability (lasing)threshold yet close to the threshold point in order tohave enough signal at the detection output to monitorthe pulse spreading dynamics in the lattice for severaltransits .To reproduce the Model II, i.e. to realize the complexpotential V n = V / [1 − a exp(2 πiαn )] with 0 < a < h ( AOM ) n and h ( EOM ) n of the modulatorsshould be set as follows h ( EOM ) n = − θ + atan a sin(2 παn )1 − a cos(2 παn ) (D9) h ( AOM ) n = g + log (cid:112) a − a cos(2 παn ) V . (D10)As an example, Fig. 7 shows the patterns of the EOMand AOM amplitudes corresponding to parameter values V = 1, a = 0 . g = 0 .
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