PT-symmetry breaking with divergent potentials: lattice and continuum cases
PPT -symmetry breaking with divergent potentials: lattice and continuum cases
Yogesh N. Joglekar , Derek D. Scott , and Avadh Saxena Department of Physics, Indiana University Purdue University Indianapolis (IUPUI), Indianapolis, Indiana 46202, USA and Theoretical Division and Center for Nonlinear Studies,Los Alamos National Laboratory, Los Alamos, New Mexico 87544, USA
We investigate the parity- and time-reversal ( PT )-symmetry breaking in lattice models in thepresence of long-ranged, non-hermitian, PT -symmetric potentials that remain finite or becomedivergent in the continuum limit. By scaling analysis of the fragile PT threshold for an openfinite lattice, we show that continuum loss-gain potentials V α ( x ) ∝ i | x | α sign( x ) have a positive PT -breaking threshold for α > −
2, and a zero threshold for α ≤ −
2. When α < PT -symmetry breaking in coupled waveguides, and show that theemergence of localized states dramatically shortens the relevant time-scale in the PT -symmetrybroken region. I. INTRODUCTION
Since Bender and co-workers’ seminal work on non-hermitian Hamiltonians a decade and a half ago, therehas been tremendous progress in the field of parity andtime-reversal ( PT -) symmetric quantum theory [1–3].For continuum, PT -symmetric, non-hermitian Hamilto-nians on an infinite line, they showed that the eigenvaluespectrum is purely real when the strength of the “non-hermiticity” is small, and becomes complex when it islarge. Traditionally, the region of the parameter spacewhere the eigenvalues of the PT -symmetric Hamiltonianare purely real, (cid:15) λ = (cid:15) ∗ λ , and the eigenfunctions are simul-taneous eigenfunctions of the combined PT -operation, f λ ( x ) = f ∗ λ ( − x ), is called the PT -symmetric region. Inthe early years, significant theoretical progress was madetowards the development of a self-consistent quantumtheory via a Hamiltonian-dependent inner product, un-der which the eigenfunctions become orthonormal in the PT -symmetric phase [2]. This progress was accompaniedby mathematical advances in the field of pseudohermi-tian operators - operators that are not hermitian underthe standard inner product, but may be self-adjoint un-der an appropriately defined metric [4]. Most of theseinvestigations were focused on continuum Hamiltonianson an infinite line.During the past five years, discrete PT -symmetricHamiltonians on finite lattices and continuum PT -symmetric Hamiltonians on a finite line have been exten-sively studied due to their experimental relevance. PT symmetry breaking is a non-perturbative phenomenon that occurs when the strength of the non-hermitianpotential is equal to the relevant hermitian energyscale. A perturbation-theory characterization of the PT -symmetry breaking criterion was developed in Ref. [5],which showed that the eigenvalues of a PT -symmetricHamiltonian H = H + iγV remain real when thestrength γ of the non-hermitian potential V is smallerthan the radius of convergence γ R for the perturbationexpansion; this radius is determined by the hermitianpiece H . It also demonstrated that coupled optical waveguides with balanced loss and gain provide an idealcandidate to visualize the effects of approaching the PT -breaking transition [5]. Since then, it has become clearthat PT -symmetric Hamiltonians naturally arise as “ef-fective Hamiltonians” for open systems with balancedloss and gain, and PT -symmetry breaking experimen-tally manifests as a transition from a quasiequilibriumstate to a state with broken reciprocity. Experimentaldemonstrations of PT -symmetry breaking in optics [6–9], and the natural emergence of PT -symmetric effectivepotentials in driven condensed matter systems [10–12]have complemented theoretical studies of PT -symmetrybreaking in lattice models [13–18] and continuum modelson a finite line [19–25]. Special attention has been paidto the number of eigenvalues that become complex [26],their location in the energy spectrum [27, 28], the ex-tended or localized nature of the corresponding eigen-states [29], and the experimental consequences of the spa-tial extent of the states that break the PT symmetry [30].A salient difference between the lattice and continuummodels is as follows. In all cases, continuum models ona finite line have shown a positive PT -symmetry break-ing threshold [19–25]. In contrast, most lattice modelshave shown a vanishing PT -symmetry breaking thresh-old that goes to zero as the number of lattice sites N diverges [15, 27, 28, 31]. This remarkable discrepancysuggests that understanding the differences between lat-tice and continuum models is crucial for a detailed un-derstanding of the PT -symmetry breaking phenomenon,particularly because all of its realizations have been insmall lattices with N (cid:46)
100 sites.Here, we investigate PT -symmetry breaking in N -sitelattices with extended loss-gain potentials characterizedby strength γ > α , and their contin-uum counterparts on a finite segment. The paper isorganized as follows. In the next section, we presentthe tight-binding model, and discuss the results for the PT -symmetric threshold γ P T ( N, α ) on the lattice andtheir continuum implications. In particular, we showthat some divergent continuum potentials on a finite seg-ment have a positive PT -symmetry breaking threshold.In Sec. III we discuss the signatures of PT -symmetry a r X i v : . [ qu a n t - ph ] A ug breaking in such lattices, and show that they are consis-tent with the expectations based on the extended natureof PT -broken eigenstates. In Sec. IV we show that lo-calized states with complex energies within the latticeenergy band emerge at much larger loss-gain strength γ c (cid:29) γ P T , and discuss their significance. We concludethe paper with Sec. V.
II. TIGHT-BINDING MODEL
Consider an N -site tight-binding lattice with site-to-site distance a and nearest-neighbor tunneling J >
0. Itshermitian tunneling Hamiltonian is given by H = − J N − (cid:88) k =1 (cid:16) a † k +1 a k + a † k a k +1 (cid:17) , (1)where a † k ( a k ) represents the creation (annihilation) oper-ator for a state | k (cid:105) localized at site k . We keep coupledoptical waveguides in mind for an experimental realiza-tion of this lattice; thus, a † k represents the creation op-erator for the single-mode electric field in the waveguide1 ≤ k ≤ N . The parity operator on an open lattice isgiven by P : a † n → a † ¯ n where site ¯ n = N + 1 − n is theparity-symmetric counterpart of site n . The action of thetime-reversal (or motion-reversal) operator is T : i → − i .We note that the Hamiltonian Eq.(1) represents a latticewith open boundaries, and thus its eigenfunctions ψ n ( m )satisfy the constraint ψ n ( m = 0) = 0 = ψ n ( m = N + 1).In the continuum limit, this boundary condition trans-lates into Dirichlet boundary condition with a vanishingwave function.The spectrum of the tight-binding model is given by E n = − J cos( k n ), and the corresponding extended, nor-malized eigenfunctions consistent with the open bound-ary condition are ψ n ( m ) = (cid:104) m | ψ n (cid:105) = sin( k n m ). Here k n = nπ/ ( N + 1) with 1 ≤ n ≤ N . Note that thespectrum is symmetric about zero, E n = − E ¯ n , theeigenfunctions have equal weights on parity-symmetricsites, and the eigenstates at energies ± E n are relatedby (cid:104) m | ψ n (cid:105) = ( − m (cid:104) m | ψ ¯ n (cid:105) . These symmetries of thespectrum and eigenfunctions remain valid in the pres-ence of pure loss-gain potentials in the PT -symmetricregion [32].We consider a class of extended loss-gain potentialsparameterized by α , V α = iγ N (cid:88) k =1 | k − n c | α sign( k − n c ) a † k a k . (2)Here, γ > n c =( N + 1) / k ≤ n c , is the “lossregion” and the second half of the lattice, k > n c , isthe “gain region”. In coupled optical waveguides, sucha potential is implemented by a site-dependent complex index of refraction n k = n Rk + in Ik with a symmetric realpart, n Rk = n R ¯ k , and an antisymmetric imaginary part, n Ik = − n I ¯ k . At this point, we remind the reader that ex-tended potentials on a lattice, Eq.(2), were investigatedand deemed unstable due to the vanishing PT -symmetricthreshold that is obtained in the limit N (cid:29) N, J → ∞ and a → N a → L defines the length of the finite segment and Ja → (cid:126) / m defines the mass of the non-relativistic quantum particleconfined in this segment. With this notation, it followsthat the continuum potential becomes V α ( x ) = i Γsign( x ) (cid:12)(cid:12)(cid:12) xL (cid:12)(cid:12)(cid:12) α = V ∗ α ( − x ) , (3)where the potential strength Γ is given byΓ = lim N →∞ γ (cid:18) N (cid:19) α . (4)Note that the continuum potential Eq.(3) is not analyticat x = 0 except when α is an odd integer. Then, for α > V α ( x ) reduces to cases considered in earlier inves-tigations [1–3, 23–25], although none of those works con-sider divergent potentials α <
0. The continuum problemcorresponding to the Hamiltonian H + V α is given by theSchr¨odinger equation − (cid:126) m ∂ x ψ q ( x ) + V α ( x ) ψ q ( x ) = E q ψ q ( x ) (5)subject to boundary conditions ψ q ( x = ± L ) = 0. Atthis point, we remind the reader that a positive PT -breaking threshold was found for the spectrum of Eq.(5)when α = 0 [19], α = 1 [10, 11, 23], and α = 3 , PT -symmetric Hamiltonian H α = H + V α cannot be obtained analytically, we nu-merically obtain the threshold γ P T ( N, α ) below whichall eigenvalues of the discrete N × N Hamiltonian arepurely real. The left-hand panel in Fig. 1 shows the de-pendence of the threshold on the lattice size. We seethat γ P T ( N, α ) /J decreases in a power-law fashion as N increases, and that the power-law exponent is deter-mined by α > −
2. Thus, for α > −
2, the PT -symmetricthreshold vanishes with increasing N and the resultant PT -symmetric phase is fragile [31], but in a very specificmanner, γ P T ( N, α ) J → A α (cid:18) N (cid:19) α +2 , (6)for N (cid:29)
1. It follows from Eqs.(4) and (6) that thedimensionless continuum threshold for V α ( x ) is equal tothe power-law prefactor,Γ P T ( α ) E L = lim N →∞ γ P T J (cid:18) N (cid:19) α +2 = A α , (7) − − − − − − − Inverse lattice size 1/N T h r e s ho l d s t r eng t h (cid:97) P T ( N , (cid:95) ) / J (cid:95) = − (cid:95) = − (cid:95) = 0 (cid:95) =1 (cid:95) =2 − − (cid:95) C on t i nuu m t h r e s ho l d (cid:75) P T ( (cid:95) ) / E L FIG. 1. (Color online) Left-hand panel: the lattice PT -threshold γ PT ( N, α ) /J shows a power-law dependence on the inverselattice size 1 /N for different loss-gain potentials V α . These results are obtained with 100 ≤ N ≤ PT ( α ) /E L , obtained from the scaling data, shows thatcontinuum models, Eq.(3), have a positive threshold for α > −
2. This includes divergent potentials such as V ( x ) = i Γ L/x . where E L = (cid:126) / mL is the continuum energy scale fora particle on a finite segment. This continuum thresholdΓ P T ( α ) /E L , obtained from the scaling data, is shown inthe right-hand panel of Fig. 1. It shows that Γ P T ( α ) /E L is a positive, monotonically increasing function of α thatgoes to zero as α → − + . These results imply that,surprisingly, divergent potentials including V ( x ) = i Γ L/x have a positive, finite PT - breaking threshold. When α ≤ −
2, we find that the numerically obtainedlattice threshold γ P T ( α ) /J is independent of the latticesize N . Therefore, the corresponding continuum thresh-old obtained via Eq.(7) vanishes, Γ P T ( α ≤ − /E L = 0.To quantify the applicability of the scaling proposed inEq.(6) to finite lattices, in Table I we list the mean valueof A α and its variance obtained from the size-dependent γ P T ( N, α ) for lattice sizes varying in steps of 200 from N = 200 to N = 2000. We see that for α ≥ −
1, thevariance in A α is less than 1% of its mean value; for α = − .
5, the variance is larger, but only due to finite-size effects that become dominant as the exponent α +2 → α → −
2. Thus, the scaling trend postulatedin Eq.(6) holds well down to N ∼ few hundred. Thesecond column in Table I also indicates that when α = { , , , } , the mean A α exactly matches the continuumthreshold results obtained in the literature.Results in Fig. 1 and Table I reconcile the nonzero PT -threshold, i.e. Γ P T ( α ) /E L >
0, in a continuum modelwith a vanishing PT -threshold, i.e. γ P T ( α ) /J →
0, inthe corresponding lattice model. The existence of a pos-itive continuum threshold, particularly for α ≥
0, followsfrom perturbation theory. The spectrum of the hermi-tian Hamiltonian H is given by (cid:15) n = E L ( nπ/ andhas a finite minimum gap. PT -symmetry breaking ispreceded by the closing of the finite gap between adja- exponent α mean A α A α variance ratio-1.5 0.3259 0.0109 0.033-1.0 1.1092 0.0079 0.007-0.5 2.4046 0.0118 0.0050.0 4.4436 [19] 0.0231 0.0050.5 7.5454 0.0253 0.0031.0 12.2470 [10] 0.0510 0.0041.5 20.2661 0.0866 0.0042.0 34.4561 0.2430 0.0072.5 40.9759 0.2454 0.0063.0 50.9557 [24] 0.3314 0.0073.5 69.0344 0.4891 0.0074.0 89.8517 0.7237 0.0084.5 102.1642 0.8520 0.0085.0 121.6964 [25] 1.1069 0.009TABLE I. The mean and variance of A α obtained from van-ishing lattice thresholds γ PT ( N, α ) /J with 200 ≤ N ≤ cent eigenvalues , which, in turn requires a finite strengthof the non-hermitian potential [33]. This finite thresh-old mandates, via Eq.(4), that the corresponding lattice-model threshold γ P T /J must vanish algebraically withincreasing lattice size. These findings are not surpris-ing for α ≥
0, when the continuum potential V α ( x ) isbounded over the entire line. However, our analysis alsopredicts that divergent PT -potentials, too, have a posi-tive threshold when − < α <
0. (Such potentials havenot been investigated in the literature.) In the followingsection, we investigate the signatures of PT symmetrybreaking in such potentials. III. PT BREAKING SIGNATURES
The lattice Hamiltonian H α can be realized in an ar-ray of coupled optical waveguides. The tunneling J isdetermined by the waveguide cross-section and the dis-tance between adjacent waveguides. It is easily tunedwith present-day technology [34–36], as is the loss poten-tial in the first half of the lattice, k ≤ n c , engineered viathe imaginary part of the index of refraction. The fabri-cation of an extended, position-dependent gain potentialhas not yet been experimentally demonstrated, althoughit may be relatively straightforward to implement in thediscrete parity-time synthetic lattices [9, 30]. In this sec-tion, we present the signatures of PT -symmetry break-ing in the time-evolution of an initially normalized wavepacket, and discuss their relationship with the spatialstructure of the PT -broken eigenfunctions.Since H α is a single-particle, time-independent Hamil-tonian, it is straightforward to obtain the time-evolvedstate | ψ ( t ) (cid:105) = G ( t ) | ψ (0) (cid:105) where G ( t ) = exp( − iH α t/ (cid:126) ) isthe non-unitary time evolution operator. The site-andtime-dependent intensity is then obtained as I ( k, t ) = |(cid:104) k | ψ ( t ) (cid:105)| , and the total intensity I ( t ) = (cid:80) k I ( k, t ) isnot conserved. Note that since the finite-dimensionalHamiltonian H α ( γ ) is diagonalizable for γ (cid:54) = γ P T , thenet intensity I ( t ) oscillates but remains bounded when γ < γ P T and increases exponentially with time at longtimes when γ > γ
P T . At the PT -breaking point, theHamiltonian is defective, and can be reduced to a Jor-dan canonical form with at least one non-trivial Jordanblock. Therefore, at long times, the net intensity scales asa power-law, I ( t ) ∝ t p − , where p ≥ PT -breaking, degenerate eigenvalue; at shorter times, the in-tensity I ( t ) is a polynomial of order 2( p −
1) whose exactform is determined by the Hamiltonian at the exceptionalpoint γ = γ P T . Thus, for any finite lattice, at the PT -breaking threshold, the net intensity at long times scalesas an even power of time or, equivalently, the distancealong the waveguide. We will see in the next section thatthe relevant time-scale that codifies “long time” is cru-cially determined by the extended vs. localized nature ofeigenfunctions with complex energies.We use an initial state centered at lattice site k , (cid:104) k | ψ (0) (cid:105) = 1 A e − ( k − k ) / σ , (8)where A ( σ, N, k ) = (cid:80) Nk =1 exp (cid:2) − ( k − k ) /σ (cid:3) ensuresthat the state is normalized. The results shown in Fig. 2are for σ = 1 and k = n c , but one obtains qualitativelysimilar results for broad wave packets centered at arbi-trary locations. The left-hand column in Fig. 2 showsthe site- and time-dependent intensity for α = +1 (toppanel), α = 0 (center panel), and α = − PT -potential strength is just belowthe threshold, γ/γ P T ( N, α ) = 0 . T s = 2 π/ ∆ av where the average level spacingis ∆ av ∼ J/N ( (cid:126) = 1). We have chosen the time-range t/T s ≤
150 to cover one bounded intensity oscillation.The top panel shows that when α = +1, the wave packetundergoes amplification near the center of the lattice.The region of maximal intensity is spread out broadlyfor α = 0 (center panel), whereas when α = −
1, themaximum amplification does not occur near the centerof the lattice. In each case maximum site-intensity inthe loss region, k ≤ n c , lags the maximum site-intensityin the gain region, k > n c . The right-hand column showsthat just above the threshold, γ/γ P T ( N, α ) = 1 . α -dependence.These intensity profiles have two surprising features.The first is that the amplification in the gain regionis faithfully transferred to the loss region. Thus, thelag between the intensity maxima is the primary dis-tinguisher between gain and loss regions. The secondis that the maximum site intensity does not occur inthe region of maximum gain potential. When α > α <
0, the loss-gain maxima occur at thecenter of the lattice, whereas the intensity maxima aredisplaced outward. This counterintuitive behavior is un-derstood by focusing on the eigenfunctions that breakthe PT -symmetry and dominate the non-unitary timeevolution. At γ = 0, the ground state wave function ψ ( k ) = sin( πk/ ( N + 1)), and the excited state wavefunction ψ ( k ) = sin(2 πk/ ( N + 1)) are orthogonal. As γ → γ P T , the corresponding γ -dependent eigenfunctionsbecome degenerate, and show a maximum at the cen-ter of the lattice for α > α <
0. Indeed, at long times, the intensity pro-file I ( k, t ) is determined by the intensity profile of theseeigenfunctions and is therefore (mostly) independent ofthe choice of the initial state | ψ (0) (cid:105) .For all potentials V α considered in this paper, the PT -symmetry is first broken via the ground-state and first-excited-state eigenfunctions that are extended over theentire lattice. When the potential strength exceeds thefragile threshold γ P T ( N, α ) ∝ J (2 /N ) α +2 and contin-ues to increase, generically, the fraction of eigenvaluesthat become degenerate and then complex increases, andthe corresponding extended eigenfunctions become PT asymmetric [23, 33]. In the following section, we will in-vestigate the emergence of localized states with complexenergies at loss-gain potential strengths comparable tothe lattice bandwidth, γ ∼ J (cid:29) γ P T ( N, α ). IV. BOUND STATES IN THE CONTINUUM
The discrete spectrum of the Hamiltonian H isbounded by ± J and becomes a continuous band whenthe lattice is infinite. A remarkable property of theHamiltonian H α is that, no matter how “strong” the loss-gain potential is, the real part of the energy spectrum
20 40 60 80 100 120 1402060100 w a v egu i de i nde x ( N = )
20 40 60 80 100 120 1402060100 normalized time
20 40 60 80 100 120 1402060100 00.20.400.500.51 (cid:95) =0 (cid:95) =1 (cid:95) = −
20 40 60 80 1002060100 W a v egu i de i nde x ( N = )
20 40 60 80 1002060100 Normalized time
20 40 60 80 1002060100 2402468051015 (cid:95) =0 (cid:95) =1 (cid:95) = − FIG. 2. (Color online) Signatures of PT -breaking transition in extended potentials with α = 1 (top row), α = 0 (center row),and α = − T s = 2 π/ ∆ av .Left-hand column: below the threshold, γ/γ PT ( N, α ) = 0 . does not occur in the region withlargest loss-gain potential. Right-hand column: after the threshold, γ/γ PT ( N, α ) = 1 . N = 100lattice, Eq.(8) with σ = 1 and k = n c . of H α remains confined to this band while the imagi-nary part of complex energies increases with the loss-gainstrength for γ (cid:29) γ P T . Just as bound states occur in thepresence of a hermitian potential, they do in the pres-ence of a non-hermitian, PT -symmetric potential whenits strength γ exceeds a threshold γ c . The crucial dif-ference in the latter case, though, is that the (real partof) energy of such localized states lies in the band ± J .It is possible to analytically obtain the threshold γ c fora single pair of PT impurities in an infinite lattice [29].However, in the present case with extended potentials V α , we locate this threshold numerically.Figure 3 shows the bound-state threshold γ c ( α ) /J foran infinite lattice, obtained from the data for lattice sizesranging from N = 100 to N = 4000. When α >
0, thepotential does not have any bound states. As α decreases,the potential deepens near the center and bound stateslocalized near the lattice-center emerge. We find that therequisite threshold γ c ( α ) /J decreases monotonically withdecreasing α ≤ −
1. This is consistent with the fact thatfor large, negative α , the potential is concentrated nearthe lattice center, and therefore the threshold strengthnecessary to support a bound state is lowered. When − < α <
0, the cumulative gain V c ( α ) = γ N (cid:88) k>n c k − n c ) | α | (9)diverges as N → ∞ for any value of γ >
0. There-fore the threshold strength γ c ( α ) required for a boundstate vanishes in this limit. We emphasize that the bound-state threshold value is much larger than the PT -symmetry breaking threshold for the same lattice, γ c ( α ) /J ∼ (cid:29) γ P T ( α, N ) /J .The inset in Fig. 3 shows the emergence of a boundstate at α = −
1, when the bound-state threshold is γ c /J = 1. The horizontal axis represents fractional lo-cation along the lattice, and the vertical axis denotesthe site intensity; the solid lines represent results for N = 200 and the dashed lines correspond to the N = 400case. When γ/γ c = 0 .
95, the eigenstate site-intensity isnonzero over the entire gain region. (The eigenstate withthe complex-conjugate energy has site-intensity that isnonzero over the entire loss region.) As is expected foran extended state, when the lattice size is doubled from N = 200 (red solid line) to N = 400 (red dashed line) thesite intensity is reduced by a factor of two. This changesdramatically as γ crosses the bound-state threshold. At γ/γ c = 1 .
05, the corresponding eigenstate now becomeslocalized near the lattice center in the gain region. Wenote that in contrast to the extended state, the localizedstate site-intensity profiles for N = 200 (blue solid line)and N = 400 (blue dashed line) have the same height,and the fractional width of the profile is halved as N isdoubled. This is the key signature of a localized state.Lastly, we demonstrate the dramatic effect of the emer-gence of a localized state with complex energy on thetime-scale that determines the “long-time” behavior ofnet intensity I ( t ) in the PT -symmetry broken phase.All results in Fig. 4 are obtained for an N = 100 lat-tice with α = −
1, and a broad initial state at the centerof the lattice, Eq.(8), with σ = 10. Panel (a) in Fig. 4 − − − − (cid:95) B ound − s t a t e i n t he c on t i nuu m t h r e s ho l d (cid:97) c ( (cid:95) ) / J fractional location (cid:97) =1.05 (cid:97) =0.95 (cid:97) =0.95 (cid:97) =1.05 (cid:95) = − FIG. 3. (Color online) Bound-state threshold γ c ( α ) /J foran infinite lattice, obtained from lattice sizes N =100-4000.There are no localized states for a positive α . The thresholdis vanishingly small for − < α ≤
0, and when α < − α . The inset showsthe emergence of a localized eigenstate for α = − γ c = J . An extended eigenstate at γ/γ c = 0 .
95 (red solid or dashedline) becomes localized past the threshold, γ/γ c = 1 .
05 (bluesolid or dashed line). The vertical axis in the inset is theeigenstate site intensity. N e t I n t en s i t y I ( t ) time (in units of 1/E L ) time (in units of 1/J) time (in units of 1/J) w a v egu i de i nde x
20 40 60 80 10020406080100 (cid:97) / (cid:97) c =0.998 (cid:97) / (cid:97) c =1.006 − (cid:97) / (cid:97) PT (N, (cid:95) )=1 + (a) (b)(c) FIG. 4. (Color online) Panel (a): net intensity I ( t ) showsexponential growth at long times tE L (cid:29) PT -symmetry is broken by extended eigenfunctions. Panel(b): for γ PT ( N, α ) (cid:28) γ < γ c ( α ), the net intensity I ( t )shows a staircase structure over time-scale 1 /J (cid:28) /E L =(1 /J )( N/ (blue solid line). This staircase changes toa straight line over the same time-scale when the bound-state threshold is crossed, γ > γ c ( α ) (red dot-dashed line).Panel (c): log I ( k, t ) shows that step in I ( t ) corresponds tothe return of partial waves from the lattice edges at time ∼ N/ J = 50 /J marked by the vertical dashed white line. shows that the net intensity I ( t ) increases exponentiallyat times tE L (cid:38)
20 where 1 /E L represents the time-scaleassociated with extended states that break the PT sym-metry. Recall that this time-scale is much longer than thebound-state time-scale, 1 /E L = (1 /J )( N/ (cid:29) (1 /J ).Panel (b) in Fig. 4 shows the behavior of net intensity I ( t ) over time-scale 1 /J . Below the bound-state thresh-old, γ/γ c = 0 .
998 (blue solid line), the intensity on thelogarithmic scale shows a step-like structure. This step-structure is replaced by a straight line just above thethreshold, γ/γ c = 1 .
006 (red dot-dashed line). Thus, thepresence of a localized state with complex energy dramati-cally shortens the time-scale for exponential intensity be-havior from ∼ /E L to ∼ /J (cid:28) /E L . Panel (c) inFig. 4 shows the logarithm of site and time-dependentintensity I ( k, t ) just below the bound-state threshold, γ/γ c = 0 . I ( k, t ) and panel(b) I ( t ) show that the step-structure in the intensity cor-responds to the return of such partial waves. Below thebound-state threshold, such staircase structure in the netintensity I ( t ) is exhibited at times T n = nN/ J . V. DISCUSSION
In this paper, we have investigated PT -symmetrybreaking in the presence of extended potentials on a lat-tice, some of which map onto divergent potentials on a fi-nite segment in the continuum limit. We have shown thatthe vanishing PT -breaking threshold in lattice modelswith extended loss-gain potentials guarantees a positive,finite threshold in their continuum counterparts. In ad-dition, we have found that divergent, loss-gain potentialssuch as V ( x ) = i Γ L/x on a finite segment have a positive PT -breaking threshold. We have shown that the emer-gence of localized states in PT potentials dramaticallyshortens the time-scale necessary for the net intensity toexhibit an exponential-in-time behavior.Our results elucidate the connection between latticeand continuum models. They raise similar questionsabout PT -symmetry breaking in an infinite lattice andits counterpart on an infinite line, as well as a lattice withcontinuous, local degree of freedom and its field-theorycounterpart [37]. They also hint at the existence of ana-lytical solutions for special values of α , such as α = − α = −
1. Addressing these questions will deepen ourunderstanding of PT -symmetry breaking and its observ-able consequences in experimentally accessible finite lat-tice systems such as optical waveguide arrays. ACKNOWLEDGMENTS
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