Sublattice signatures of transitions in a PT -symmetric dimer lattice
SSublattice signatures of transitions in a
P T -symmetric dimer lattice
Andrew K. Harter and Yogesh N. Joglekar
Abstract
Lattice models with non-hermitian, parity and time-reversal ( PT ) sym-metric Hamiltonians, realized most readily in coupled optical systems, have beenintensely studied in the past few years. A PT -symmetric dimer lattice consistsof dimers with intra-dimer coupling ν , inter-dimer coupling ν (cid:48) , and balanced gainand loss potentials ± i γ within each dimer. This model undergoes two independenttransitions, namely a PT -breaking transition and a topological transition. We nu-merically and analytically investigate the signatures of these transitions in the time-evolution of states that are initially localized on the gain-site or the loss-site. Finite, discrete systems have always been an important testing ground in that theyare often amenable to straightforward numerical approach, while retaining the com-plex and interesting features of their infinite and continuum counterparts. Latticemodels, where a quantum particle occupies discrete locations and only tunnels be-tween adjacent sites, successfully describe physical properties of a number of crys-talline, condensed matter systems [1, 2] as well as light propagation in arrays ofcoupled optical waveguides [3] in the paraxial approximation [4]. A dimer model,where the tunneling strength alternates between two values, was first explored bySu, Schrieffer, and Heeger (SSH) in the context of solitons in polyacetylene [5, 6].Since then, the one-dimensional SSH model has been extensively studied because it
Andrew K. HarterIndiana University Purdue University Indianapolis(IUPUI), Indianapolis, Indiana 46202 USA, e-mail: [email protected]
Yogesh N. JoglekarIndiana University Purdue University Indianapolis (IUPUI), Indianapolis, Indiana 46202 USA, e-mail: [email protected] a r X i v : . [ qu a n t - ph ] M a y Andrew K. Harter and Yogesh N. Joglekar exhibits topologically non-trivial edge states [7] and its generalizations lead to bandstructures with nonzero Chern numbers [8, 9].Realizations of an SSH model in coupled optical waveguides instead of thenature-given long acetylene chains are advantageous [10]. In the former, the ratioof tunneling strengths, and the size and parity of the dimer chain can be varied overa wide range, and the entire bandwidth of the SSH band structure is accessible;andone can model non-hermitian, gain and loss potentials because the absorption andamplification of electromagnetic waves are both easily implemented [11]. Experi-mental realizations of non-uniform waveguide lattices have been demonstrated withlattice sites N ∼ −
100 [12], single-site or wide-beam input [13], and single-photon source inputs [14]; in particular, edge states and their adiabatic transfer inquasi-periodic waveguide lattices have been experimentally investigated [15].The past five years have seen a surge of interest, driven primarily by experi-ments on optical systems [16–26], in open systems that are faithfully described byan effective, non-hermitian Hamiltonian that is invariant under combined parity andtime-reversal ( PT ) operations [27, 28]. Typically, a PT -symmetric Hamiltonian H is comprised of a hermitian, kinetic energy term H and a non-hermitian, PT -symmetric potential term V = PT V PT (cid:54) = V † that represents balanced, spatiallyseparated gain and loss. Although H is not hermitian, its spectrum is purely realwhen the strength of the non-hermitian potential is small, and changes into complex-conjugate pairs when it exceeds a threshold called the PT -breaking threshold [28].In contrast with the traditional hermitian case, the non-hermitian, PT -symmetricHamiltonian is defective at the PT -breaking threshold [29,30]. A PT -symmetricSSH model, or equivalently a dimer model, has gain and loss of equal strengths onalternate sites [31], and is mathematically equivalent to a dimer model which hasonly a loss term on every other site. This purely lossy dimer model shows a quan-tized mean displacement that, under certain constraints, has a topological origin.This transition is driven by the ratio of inter-dimer and intra-dimer tunneling am-plitudes, and befitting a topological transition, is independent of the strength of theloss potential and robust over a broad range of model parameters [32].In this paper, we discuss the properties of PT -symmetric dimer model overa wide range of parameters, such that it undergoes both the PT -breaking transi-tion and the topological transition. The PT -breaking transition in a PT -dimermodel was studied by Zheng et al. [31], and the topological transition in a purelylossy dimer model was predicted by Rudner and Levitor [32]. Neither, however,investigated the interplay between these two transitions.The plan of the paper is as follows. In Sect. 2 we present the key properties ofa PT -symmetric dimer model, as they relate to the two transitions it undergoes.In Sect. 3 we present numerical results for the time evolution of a wave packet thatis initially localized on the gain site or a loss site. Since the intensities on the gainsites are orders of magnitude higher than those on the loss sites, particularly in the PT -broken phase, we separately consider the intensity distributions on the gain-sublattice and the loss-sublattice. We show that these distributions undergo a qual-itative change across the topological transition. In Sect. 4, we obtain approximate,analytical expressions for the two sublattice intensity distributions. We conclude the ublattice signatures of transitions in a PT -symmetric dimer lattice 3 paper with a brief discussion in Sect. 5. Our results show that the signatures of thetopological transition imprint themselves on the sublattice intensity distributions inthe broken PT -symmetric phase. PT -symmetric dimer model In this section, we establish the notation and recall results for the PT -breakingtransition in a dimer lattice [31], and the topological transition in a purely lossydimer lattice [32]. Let us consider a PT -symmetric dimer lattice, where eachdimer consists of a gain site ( G ) with potential + i γ and a loss site L with poten-tial − i γ . The dimer is labeled by the index m where − M ≤ m ≤ M denotes a finitelattice with N = M + ν denotes the tunneling within a dimer, and ν (cid:48) de-notes the tunneling between two adjacent dimers. In this case, the parity operator P exchanges the gain and the loss sites within each dimer whereas the time-reversaloperator T corresponds to complex conjugation, thus interchanging the gain withthe loss. Figure 1 shows a schematic of such a lattice. m − G L ν ν (cid:31) G L ν ν (cid:31) G L ν m m + 1 γ i − γ i − γ i − γ i γ i γ i Fig. 1: Schematic of a PT -symmetric dimer lattice. The gain-sites G , shown byopen circles, have a gain potential + i γ while the loss sites L , shown by black solidcircles, have the decay potential − i γ . The dashed rectangular box indicates the cen-tral, m =
0, dimer. The tunneling within a dimer is given by ν and the inter-dimertunneling is ν (cid:48) .The non-hermitian, PT -symmetric Hamiltonian H = H + V for the lattice isgiven by H = − M − ∑ m = − M (cid:0) ν | mG (cid:105)(cid:104) mL | + ν (cid:48) | mL (cid:105)(cid:104) m + G | + h . c . (cid:1) , (1) V = + i γ M ∑ m = − M ( | mG (cid:105)(cid:104) mG | − | mL (cid:105)(cid:104) mL | ) , (2)where | mG (cid:105) and | mL (cid:105) denote single-particle states localized on the gain and losssites of dimer m , h.c. denotes the hermitian conjugate, and we have considered alattice with open boundary conditions. In the Fourier space, this Hamiltonian is Andrew K. Harter and Yogesh N. Joglekar block-diagonalized into 2 × H k n = (cid:20) i γ − ν ∗ k n − ν k n − i γ (cid:21) = i γσ z − ( ν + ν (cid:48) cos k n ) σ x − ν (cid:48) sin k n σ y . (3)Here σ i are the Pauli matrices, ν k n = ν + ν (cid:48) exp ( ik n ) , * denote complex conjuga-tion, and k n = n π / ( N + ) (1 ≤ n ≤ N ) are the eigenmomenta consistent with openboundary conditions. For periodic boundary conditions, the corresponding eigenmo-menta are given by k n = π n / N with | n | ≤ ( N / ) . The spectrum of the Hamiltonian H k is given by ± ε k = ± (cid:112) ( | ν k | − γ ) ; therefore, the PT -breaking threshold forthe dimer lattice is given by γ PT = min k ( | ν k | ) and becomes, in the infinite-latticelimit [31], γ PT = | ν k | k = π = | ν − ν (cid:48) | . (a) Gain sublattice intensity profile (b) Loss sublattice intensity profile Fig. 2: The gain (a) and loss (b) sublattice intensities for an N =
41 dimer latticewith ν (cid:48) / ν = γ / ν = .
5. The vertical axis shows the dimerindex m with − ≤ m ≤
20, and the horizontal axis denotes normalized time ν t .Note the order of magnitude difference between intensities on the gain sublatticeand the loss sublattice.In the PT -symmetric phase, the eigenvalues ε k are real for all k , the non-unitarytime evolution generated by the Hamiltonian is periodic, and the total intensity I ( t ) = (cid:104) ψ ( t ) | ψ ( t ) (cid:105) of an initially normalized wave packet | ψ ( t ) (cid:105) remains boundedas a function of time. When the gain-loss strength exceeds γ PT but is smaller thanmax k | ν k | = ν + ν (cid:48) , some Fourier components of the initial state grow exponentiallywhile others remain bounded, leading to a total intensity I ( t ) that oscillates withan amplitude that increases exponentially with time. For γ > | ν k | k = = ν + ν (cid:48) , allFourier components grow exponentially and so does the net intensity. Figure 2 showthe intensities for the gain sublattice I G ( m , t ) = |(cid:104) mG | ψ ( t ) (cid:105)| and the lossy sub-lattice I L ( m , t ) = |(cid:104) mL | ψ ( t ) (cid:105)| , for a 41-site dimer lattice with ν (cid:48) / ν =
1, gain-lossstrength γ / ν = .
5, and an initial state localized on the gain site of the central dimer, | ψ ( ) (cid:105) = δ m | mG (cid:105) . We remind the reader that since the Hamiltonian H = H + V ublattice signatures of transitions in a PT -symmetric dimer lattice 5 is not hermitian, the time-evolved state | ψ ( t ) (cid:105) = exp ( − iHt ) | ψ ( ) (cid:105) does not have aconstant norm. (We use ¯ h = PT -symmetric dimer lattice. For alossy lattice, each dimer has one neutral site ( N ) and one lossy site ( L ). The Hamil-tonian for the lossy lattice with open boundary conditions is given by H L ( ν , ν (cid:48) , γ ) = H + V L ( γ ) where V L ( γ ) = − i γ M ∑ m = − M | mL (cid:105)(cid:104) mL | . (4)The eigenvalues of the non-hermitian, non- PT -symmetric Hamiltonian H Lk have apurely decaying part for all eigenmomenta k , and therefore any typical initial state iseventually completely absorbed. The mean displacement of the wave packet beforeit is absorbed is determined solely by the intensities on the loss sublattice, ∆ m ( ν , ν (cid:48) , γ ) = ∑ m m (cid:90) ∞ dt γ I L ( m , t ) . (5)Prima facie, Eq.(5) represented a complicated global measure of the intensity distri-bution on the loss sublattice; it depends on the initial state, the decay rate γ , and thetwo tunneling amplitudes ν , ν (cid:48) that characterize the dimer lattice. For an initial statelocalized on the neutral site in the central dimer, m =
0, however, it can be shown -through some non-trivial algebra [32] - that the mean displacement ∆ m is equal tothe winding number of the k -space tunneling amplitude ν ∗ k = ν + ν (cid:48) exp ( − ik ) [32].Since the winding number is a topological quantity that changes discontinuouslyand is robust against small disorder perturbations, it follows that the mean displace-ment, defined by Eq.(5), is quantized and robust. It changes sharply from 0 to -1 asthe inter-dimer tunneling strength ν (cid:48) exceeds the intra-dimer tunneling strength ν ,and is independent of the decay rate γ > ν (cid:48) is small, the wave packet is primarily absorbed on the loss site within theinitial dimer; on the other hand, when the inter-dimer coupling ν (cid:48) becomes large, theloss-site corresponding to absorption is in the dimer to the left, with index m = − γ , since Eq.(5)integrates over all possible times, the final result is independent of the decay rate.Being topological in its origin, the analytical result for ∆ m is independent of the lossstrength and small disorder, but is valid only for the specific initial state in an infinitelattice [32]. Experimentally, the transition is substantially softened and broadeneddue to the finite size and disorder effects [33]. It follows from Fig. 1 that a dimerlattice with ν (cid:48) > ν after time reversal and shift by half-a-cell is equivalent to a dimerlattice with ν < ν (cid:48) .These two lattices - a PT -symmetric dimer lattice [31] and the purely lossydimer lattice ´a la Rudner and Levitov [32] - are equivalent to each other because Andrew K. Harter and Yogesh N. Joglekar their respective Hamiltonians differ only by a non-hermitian shift proportional tothe identity, H ( ν , ν (cid:48) , γ ) = i γ · + H L ( ν , ν (cid:48) , γ ) . Therefore, we can define a scaledmean-displacement by considering the intensities on the lossy sublattice [34], ∆ m PT ( ν , ν (cid:48) , γ ) = ∑ m m (cid:90) ∞ dt γ e − γ t I L ( m , t ) . (6)It follows that the scaled mean displacement ∆ m PT undergoes a topologicaltransition at ν (cid:48) = ν which corresponds to a vanishing PT -breaking threshold, γ PT = | ν − ν (cid:48) | =
0. Therefore, in a PT -symmetric dimer, the topological tran-sition in ∆ m PT always occurs in the PT -broken phase. Note that for a generalinitial state, the intensities I G ( m , t ) and I L ( m , t ) on both sub-lattices increase ex-ponentially with time in the PT -symmetry broken phase. However, the integral inEq.(6) converges. In the following section, we numerically investigate the signaturesof this transition in the site- and time-dependent intensities I G ( m , t ) and I L ( m , t ) onthe gain and loss sublattices respectively. The two independent transitions in the PT -symmetric dimer lattice are driven bytwo dimensionless parameters, namely the tunneling ratio ν (cid:48) / ν which governs thetopological transition, and the gain-loss strength γ / ν which determines the PT -breaking phase boundary. Fig. 3 and Fig. 4 show the gain and loss sublattice inten-sities for ν (cid:48) / ν = { , . , , . , } and γ / ν = { , . , } . In each frame, the verticalaxis denotes the dimer index m ranging from -20 to 20, and the horizontal axis de-notes normalized time ν t ranging from 0 to 10. Note that when ν (cid:48) / ν = PT -symmetric dimers, and there-fore the wave packet remains confined to the central dimer alone; as ν (cid:48) / ν increases,the lateral spread of the wave packet across the lattice also increases.First, let us consider the time evolution of a wave packet initially localized on thegain site of the central dimer, | ψ ( ) (cid:105) = δ m | mG (cid:105) . Panel (a) in Fig. 3 shows the gain-sublattice intensity I G ( m , t ) and panel (b) shows the corresponding loss-sublatticeintensity I L ( m , t ) . Note that the topological transition occurs across the central row, ν (cid:48) = ν , whereas the PT -breaking transition occurs across the two dot-dashed greylines, given by γ / ν = | − ν (cid:48) / ν | . Therefore, in both panels, we see that the sub-lattice intensities are bounded and oscillatory in the PT -symmetric phase (PT-S).In the PT -broken phase (PT-B), the gain-sublattice distribution I G ( m , t ) shows asingle Gaussian whose intensity is maximum at ν (cid:48) = ν because it corresponds toa vanishing PT -breaking threshold. The loss-sublattice distribution I L ( m , t ) alsoshows a single Gaussian, except at ν (cid:48) = ν , when the intensity shows a symmetric,bimodal distribution, marked by the white oval in panel (b).The time-evolution of a state initially localized on the loss-site of the centraldimer, | ψ ( ) (cid:105) = δ m | mL (cid:105) is shown in Fig. 4. Note that in this case, the mean dis- ublattice signatures of transitions in a PT -symmetric dimer lattice 7(a) Gain sublattice intensity I G ( m , t ) (b) Loss sublattice intensity I L ( m , t ) Fig. 3: Evolution of gain-sublattice (a) and loss-sublattice (b) intensities for 0 ≤ ν (cid:48) / ν ≤ ≤ γ / ν ≤
1, and an initial state on the gain-site of the central dimer, | ψ ( ) (cid:105) = δ m | mG (cid:105) . Vertical axis in each frame denotes the dimer index m and thehorizontal axis denotes normalized time ν t . The dot-dashed gray lines denote the PT -symmetric phase boundary γ PT / ν = | − ν (cid:48) / ν | . In the PT -symmetric phase(PT-S), the intensities are bounded and oscillatory. In the PT -broken phase (PT-B), they are Gaussian except for the loss-sublattice distribution I L ( m , t ) at the topo-logical transition ν = ν (cid:48) , denoted by a white oval in (b). Andrew K. Harter and Yogesh N. Joglekar(a) Loss sublattice intensity I L ( m , t ) (b) Gain sublattice intensity I G ( m , t ) Fig. 4: Loss-sublattice (a) and gain-sublattice (b) intensities for an initial state lo-calized on the loss site, | ψ ( ) (cid:105) = δ m | mL (cid:105) . Vertical axis in each frame denotes thedimer index m and the horizontal axis denotes normalized time ν t . The dot-dashedgray lines denote the PT -symmetric phase boundary γ PT / ν = | − ν (cid:48) / ν | . Bothintensities I L ( m , t ) and I G ( m , t ) show Gaussian behavior except at ν (cid:48) = / ν , markedby white ovals in both panels. ublattice signatures of transitions in a PT -symmetric dimer lattice 9 placement ∆ m PT does not undergo any change as ν (cid:48) / ν is varied; it remains zero,meaning the particle is primarily absorbed on the loss-site it is initially locatedon [33]. The dash-dotted gray lines in both panels denote the PT -symmetry break-ing threshold γ / ν = | − ν (cid:48) / ν | . Both panels show that in the PT -symmetric phase(PT-S), the time-evolution is oscillatory and the net intensities on the gain and theloss sublattices are comparable to each other.Panel (a) shows that in the PT -broken phase (PT-B), the loss-sublattice inten-sity profile I L ( m , t ) has a symmetric, trimodal distribution , marked by a white ovalin (a) when ν (cid:48) / ν =
1. This is in sharp contrast to the results for ν (cid:48) / ν (cid:54) =
1, when thedistribution consists of a single Gaussian. Panel (b) shows that in the PT -brokenphase, the average gain-site intensity is orders of magnitude higher than the averageloss-site intensity. The gain intensity I G ( m , t ) shows a symmetric, bimodal distribu-tion exactly at ν (cid:48) / ν = ν (cid:48) / ν (cid:54) =
1, the intensity distribution has a singleGaussian peak.Thus, the key numerical observations can be summarized as follows. At ν (cid:48) / ν =
1, when the winding number of ν k = ν + ν (cid:48) exp ( ik ) changes from 0 to 1, for aninitial state on the gain sublattice, the gain intensity I G ( m , t ) shows a single Gaussianpeak, whereas the loss intensity I L ( m , t ) shows a two-peak structure. When the initialstate is localized on the loss sublattice, the gain intensity I G ( m , t ) shows a two-peak structure whereas the loss intensity I L ( m , t ) shows a structure with three peaks.When ν (cid:48) / ν (cid:54) =
1, both gain and loss intensities show a single Gaussian peak at longtimes in the PT -broken region. In the next section, we will analytically investigatethis behavior. PT -broken region In this section, we will develop approximate analytical expressions for the real-space, time-dependent wave functions for the two sublattices in the PT -brokenregion. As discussed in Sect. 2, the PT -symmetric dimer Hamiltonian is most eas-ily diagonalized in the Fourier space, and the first emergence of complex-conjugateeigenvalues occurs at k = π . In the PT -broken phase, the 2 × G k ( t ) = exp ( − iH k t ) = cosh ( Γ k t ) − i H k Γ k sinh ( Γ k t ) , (7)where Γ k = (cid:112) γ − | ν k | > Γ k t (cid:29)
1, the Fourier-space time-evolution operator becomes G k ( t ) = exp ( Γ k t )( − iH k / Γ k ) /
2. Therefore, equivalently, the real space propagator is given by G mn ( t ) = π (cid:90) π dk e i ( m − n ) k + Γ k t (cid:18) − i H k Γ k (cid:19) . (8) Note that Eq.(8) is valid in the PT -broken phase even if the eigenvalues of H k arereal for momenta away from k = π . These momenta, with real eigenvalues ε k , lead toa time evolution operator G k ( t ) with bounded norm, and therefore their contributionto Eq.(8) is vanishingly small at long times Γ k t (cid:29)
1. Since the largest contribution tothe integral arises from a vanishingly small neighborhood of k = π + p , the integrandin Eq.(8) is estimated by approximating Γ π + p ≈ Γ − Dp / Γ = γ − γ PT and D = νν (cid:48) / Γ , leading to G m ( t ) = ( − ) m e Γ t π (cid:90) ∞ − ∞ d p e ipm − Dt p / (cid:34) + γΓ − i Γ ( ν − ν (cid:48) e + ip ) − i Γ ( ν − ν (cid:48) e − ip ) − γΓ − Dp / (cid:35) . (9)Here, without loss of generality, we have chosen n = p to the entire real line becausein the long-time limit, Dt (cid:29)
1, the integrand contains a Gaussian sharply peaked at p =
0. We have retained the p dependence in the denominator of one of the matrixelements because the matrix element otherwise vanishes at the topological transitionboundary ν = ν (cid:48) . It is now straightforward to carry out the Gaussian integrals andobtain explicit expressions for the time-dependent wave function at long times inthe PT -broken phase.For an initial state localized on the gain-sublattice, | ψ ( ) (cid:105) = δ n | nG (cid:105) (Fig. 3), weobtain the following expressions for the gain and loss sublattice wave functions, ψ G ( m , t ) ∼ ( − ) m e Γ t √ π Dt (cid:16) + γΓ (cid:17) exp (cid:20) − m Dt (cid:21) , (10) ψ L ( m , t ) ∼ i ( − ) m e Γ t Γ √ π Dt (cid:26) ν exp (cid:20) − m Dt (cid:21) − ν (cid:48) exp (cid:20) − ( m + ) Dt (cid:21)(cid:27) . (11)Note that both wave functions grow exponentially with the amplification rate Γ ≤ γ .It follows from Eq.(10) that the wave function ψ G ( m , t ) describes a classical, dif-fusing particle with diffusion constant D = νν (cid:48) / Γ . This result is expected because,in the PT -broken phase, where the wave packet intensity increases exponentiallywith time, we should recover the classical behavior [35]. For the loss sublattice, wefind that ψ L ( m , t ) is the difference of two diffusing Gaussians with centers at m = m = − ν = ν (cid:48) , the losssublattice wave function ψ L ( m , t ) shows a symmetric, two-peak structure.For an initial state localized on the loss-sublattice, | ψ ( ) (cid:105) = δ n | nL (cid:105) (Fig. 4), thewave functions are given by ψ G ( m , t ) ∼ i ( − ) m e Γ t Γ √ π Dt (cid:26) ν exp (cid:20) − m Dt (cid:21) − ν (cid:48) exp (cid:20) − ( m − ) Dt (cid:21)(cid:27) , (12) ψ L ( m , t ) ∼ ( − ) m e Γ t √ π Dt e − m / Dt (cid:20) − γΓ (cid:18) + Γ t − m νν (cid:48) t (cid:19)(cid:21) . (13) ublattice signatures of transitions in a PT -symmetric dimer lattice 11 It follows from Eq.(12) that the gain-sublattice intensity distribution is the differ-ence of two diffusing Gaussians centered at m = m = +
1, weighted by thetunneling strengths. In particular, when ν = ν (cid:48) , we obtain the symmetric, bimodaldistribution seen in panel (b) of Fig. 4. Eq.(13) implies that the loss-sites wave func-tion ψ L ( m , t ) is a diffusing Gaussian centered at m =
0. However, only at ν = ν (cid:48) ,the leading order term in the square bracket vanishes . It generates a multiplicativefactor ( − m / Dt ) that accompanies the diffusive Gaussian. This implies that theloss-sublattice intensity vanishes at m ∗ = ± (cid:112) ( Dt ) = ± (cid:112) ( ν t / γ ) , and gives rise tothe three-peak structure seen in panel (a) of Fig. 4.These results can be easily generalized to an arbitrary state on the central dimer, | ψ ( ) (cid:105) = δ n ( cos θ | nG (cid:105) + sin θ e i φ | nL (cid:105) ) . When the initial state has no transverse mo-mentum, φ (cid:54) = π , the system does not undergo a topological transition [34],whereas when φ = π , it has a quantized scaled mean displacement. In this paper, we have investigated the interplay between two transitions that arepredicted to take place in a PT -symmetric dimer (or SSH) model.The PT -symmetry breaking transition is governed by the gain-loss strength γ relative tothe the tunneling modulation strength | ν − ν (cid:48) | , whereas the topological transition inthe scaled mean displacement ∆ m PT is governed by the ratio of the inter-dimer tointra-dimer tunneling ν (cid:48) / ν .We have shown that the gain and loss sublattice intensity profiles, I G ( m , t ) and I L ( m , t ) respectively, show distinct features at the intersection of the topologicaltransition ν = ν (cid:48) and the PT -symmetry breaking transition γ PT = | ν − ν (cid:48) | . Thesefeatures can be understood through long-time behavior of gain-site and loss-sitewave functions, which also capture the classical, diffusive behavior that is expectedin the PT -broken phase. Although experimental realization of an active SSHmodel, where half the waveguides have a constant amplification, is challenging,it is feasible with the current sample fabrication technology; therefore, we expectthat all of its attendant properties, including symmetric, edge-localized states willbe observable in it.In this work, we have not considered the effects of nonlinearity [36]. In the PT -broken phase, the nonlinearity manifests itself in two ways. First, it intro-duces a state-dependent potential V G ( m ) ∝ | ψ G ( m , t ) | on each gain site and a cor-responding potential V L ( m ) on each loss site; physically, this potential representsthe intensity-dependent change in the local index of refraction [37–39]. Second, asthe site-dependent intensity increases, the model with constant, local-intensity in-dependent gain and loss coefficients becomes less reliable [40]. Thus, our findingsare valid in a range of parameters where the effects of nonlinearity are mitigated.They suggest that the interplay between PT -symmetry breaking transition, andtopological transitions in one or two dimensional PT -symmetric models leads tointeresting results. Acknowledgements
The authors thank Avadh Saxena for useful discussions. This work was sup-ported by NSF DMR-1054020.
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