restoration via increased loss-gain in -symmetric Aubry-Andre model
In systems with “balanced loss and gain”, the -symmetry is broken by increasing the non-hermiticity or the loss-gain strength. We show that finite lattices with oscillatory, -symmetric potentials exhibit a new class of -symmetry breaking and restoration. We obtain the phase diagram as a function of potential periodicity, which also controls the location complex eigenvalues in the lattice spectrum. We show that the sum of -potentials with nearby periodicities leads to -symmetry restoration, where the system goes from a -broken state to a -symmetric state as the average loss-gain strength is increased. We discuss the implications of this novel transition for the propagation of a light in an array of coupled waveguides.
Introduction. Open systems with balanced loss and gain have gained tremendous interest in the past three years since their experimental realizations in optical (1); (2); (3); (4), electrical (5), and mechanical (6) systems. Such systems are described by non-hermitian Hamiltonians that are invariant under combined parity and time-reversal () operations (7). Apart from their mathematical appeal, such non-hermitian Hamiltonians show non-intuitive properties such as unidirectional invisibility (4); (8); (9) and are thus of potential technological importance.
Historically, Hamiltonians on an infinite line were the first to be investigated (10); (11). The range of parameters where the spectrum of the Hamiltonian is purely real, , and the eigenfunctions are simultaneous eigenfunctions of the operator, , is called the -symmetric phase. The emergence of complex conjugate eigenvalues when the parameters are not in this region is called -symmetry breaking. A positive threshold for -symmetry breaking implies that the system transitions (2); (3); (4); (5); (6) from a quasi-equilibrium state at a small but nonzero non-hermiticity, to loss of reciprocity as the strength of the balanced loss-gain term crosses the threshold. Although -symmetric Hamiltonian studies started with continuum Hamiltonians, all of their realizations are in finite lattices where the continuum, effective-mass approximation may not apply. This observation has led to tremendous interest in -symmetric lattice models (12); (13); (14); (15); (16); (17); (18) that can be realized in coupled waveguide arrays (19); (20).
A universal feature of all such systems is that -symmetry is broken by increasing the balanced loss-gain strength and restored by reducing it. Here, we present a tight-binding model that can exhibit exactly opposite behavior, via a family of -symmetric, periodic potentials.
A remarkable property of lattice models, absent in the continuum limit, is the effects of a periodic potential. The spectrum of a charged particle in constant magnetic field in two dimensions consists of Landau levels (21); (22); (23); a similar particle on a two-dimensional lattice displays a fractal, Hofstadter butterfly spectrum (24); (25); (26); (27). In one dimensional lattices, a fractal spectrum emerges in the presence of a hermitian, periodic potential, and this model, known as the Aubry-Andre model (28), shows localization transition in a clean system when the strength of the incommensurate potential exceeds the nearest-neighbor hopping (29). Here, we consider a -symmetric Aubry-Andre model on an -site lattice with hopping and complex potential where is the lattice center and . Since , it is sufficient to consider the family of potentials with (when the problem reduces to the Aubrey-Andre model (24); (28)). We then consider the effect of two such potentials with .
Our salient results are follows: i) For a single potential , the threshold loss-gain strength shows local maxima along the axis; it is suppressed by a nonzero real modulation . ii) The discrete index of pair of eigenvalues that become complex can be tuned stepwise by varying . iii) For , generically, the phase diagram in the plane shows a re-entrant -symmetric phase: a broken -symmetry is restored by increasing the non-hermiticity and broken again when become sufficiently large. This behavior is absent in the extensively studied continuum Hamiltonians with complex potentials (30); (31); (32), and is a result of competition between the two lattice potentials and .
We emphasize that means the gain-regions of the two potentials mostly align as do their respective loss-regions. Thus, the competition between and is not in their loss-gain profiles, but, as we will show below, due to the relative locations of -symmetry breaking energy-levels in the spectrum.
phase diagram for a single potential. The tight-binding hopping Hamiltonian for an -site lattice with open boundary conditions is . Its particle-hole symmetric energy spectrum is given by and the corresponding normalized eigenfunctions are where and . The properties that relate eigenvalues and eigenfunctions at indices remain valid in the presence of pure loss-gain potential (33). The eigenvalue equation for an eigenfunction of the non-Hermitian, -symmetric Hamiltonian is given by ()
with . Since this difference equation is not analytically soluble for an arbitrary , we numerically obtain the spectrum and the -symmetry breaking threshold using different discretizations along the -axis. Due to the symmetry of the potential, it follows that the exact threshold loss-gain strength satisfies .
We consider a purely loss-gain potential, present results for an even lattice, and point out the salient differences that arise when lattice size is odd or when . The left-hand panel in Fig. 1 shows the -symmetric threshold for an lattice obtained by using discretization (blue squares), an lattice with (red markers), and an lattice with (black stars). There is a monotonic suppression of the threshold strength with increasing , and, crucially, the general shape of the phase diagram depends upon the size of relative to . A scaling of this threshold suppression for lattice sizes , shown in the inset, implies that where is a constant. Thus, the threshold strength is suppressed linearly and vanishes in the thermodynamic limit (34); (35). However, this algebraically fragile nature of the -phase is not an impediment since -systems to-date are only realized in small lattices with . The right-hand panel in Fig. 1 shows the phase diagram for case with discretization . The results for irrational values of and other discretizations lie on the same curve. The phase diagram shows local maxima located at and the two maxima at the end points, is symmetric about the center and has a local minimum in the threshold at . In addition, the function has minima at and smoothly oscillates over a period as shown in the inset (solid red circles). These results are generic for any lattice size with discretization .
When is odd, the non-Hermitian potential vanishes at and the spectrum of the Hamiltonian is purely real. For an odd lattice, a similarly obtained phase diagram shows local maxima that are distributed equally on the two sides of , along with a substantial enhancement in the threshold strength as . Adding a real potential modulation , to the loss-gain potential, in general, suppresses the threshold strength.
The phase diagram can be understood as follows: for a small , and the enhanced -breaking threshold, , is consistent with a linear-potential threshold (36). For an even lattice, the average of the gain-potential is given by . The threshold is greatest when the change in the average strength is maximum as is varied, . In the limit and , it implies that the maxima of occur at . On the other hand, is smallest when the change in the average strength is minimum, , and gives the locations of minima as . A similar analysis applies to odd lattices, where the average potential is given by .
Next, we focus on the location of the symmetry breaking. Due to the particle-hole symmetric spectrum of , two pairs of levels and become complex simultaneously. Figure 2 plots the indices of eigenvalues that become complex as a function of for an lattice with discretization . It shows that at small , the eigenvalues at the band edges become complex, whereas, as , the pairs of eigenvalues that become complex move to the center of the band. Thus the average range of s with the same location for -symmetry breaking is . It follows that by choosing an appropriate , one is able to control the location of -symmetry breaking in the energy spectrum. As we will see next, this control allows us to introduce competition between potentials with two different, but close, values of .
phase diagram with two potentials. We now consider the -symmetric phase of the Hamiltonian with two potentials, , in the plane, where both axes are scaled by their respective threshold values . Panel (a) in Fig. 3 shows the numerically obtained phase diagram for an lattice with (blue stars and squares). It shows that, from a -broken phase (point 1), it is possible to enter the -symmetric phase by increasing the non-hermiticity (point 2). We emphasize that increasing increases the average gain- (and loss-) strength , and yet drives the system into a -symmetric phase from a -broken phase. Increasing further, eventually, drives the system into a broken phase again (point 3).
While a -broken phase implies an exponential time-dependence of the net intensity or the norm of a state, in the -symmetric phase the net intensity oscillates within a bound that is determined by the proximity of the Hamiltonian to the phase boundary (20). Panel (b) in Fig. 3 shows the dramatic consequences of restoration on the site- and time-dependent intensity of a state that is initially localized on site ; the time is measured in units of . The top-subpanel shows that intensity has a monotonic amplification in regions with gain sites, leading to a striated pattern (point 1). Center subpanel shows that by increasing , oscillatory behavior in the intensity is restored (point 2). Bottom subpanel shows that increasing further breaks the symmetry again (point 3). Thus, we are able to restore -symmetry by increasing the non-hermiticity, and achieve amplification by both reducing or increasing the average gain-strength. This novel behavior is absent in all lattice models with a single potential.
Since each potential breaks the symmetry for , naively, one may expect that the phase boundary for is given by . This is indeed the case for , shown by red dashed line in panel (a), even though the potential periodicities differ by a factor of two.
What is the key difference between the two sets of parameters, one of which shows a re-entrant -symmetric phase? It is the indices of eigenvalues that become complex due to and . The red rectangle in Fig. 2 shows that for , eigenvalues become complex. In such a case, the two potentials and act in a cooperative manner effectively adding their strengths. Therefore, the -phase boundary is a straight line. In contrast, the blue oval in Fig. 2 shows that for , energy levels approach each other, become degenerate, and then complex as ; for , the energy levels that become complex as are . Thus, the level is lowered by potential and raised by the potential from its hermitian-limit value. This introduces competition between the two potentials and even though their gain-regions largely overlap and so do their respective loss regions.
This correspondence between completing potentials and -restoration is further elucidated in Fig. 4. In conjunction with Fig. 2, it shows that re-entrant -symmetric phase occurs when the two potentials compete (panels b-e, g, h). This restoration of -symmetry can be due to increased loss-gain strength in (panels d, e), (panels b, c, g), or both (panel h). On the other hand, when the two potentials break the same set of eigenvalues, the phase boundary is a line (panels a, f, k).
Discussion. Competing potentials, a common theme in physics, often stabilize phases that would be unstable in the presence of only one of them (37). A trivial definition of competing -potentials is that the gain-region of one strongly overlaps with the loss-region of another, thus reducing the average gain (and loss) strength.
Here, we have unmasked the subtle competition between potentials whose gain regions largely overlap, based on the location of -symmetry breaking induced by each. This competition results in -restoration and subsequent -breaking, leading to selective intensity suppression and oscillations at large loss-gain strength. Its hints were seen in a continuum model with complex -function and constant potentials, but that continuum model is neither easily experimentally realizable nor can it tune between cooperative and competitive behavior (39). The -symmetric Aubry-Andre model provides a family of potentials with tunable competition or cooperation among them, and is thus ideal for investigating the consequences of such competition; even lattices as small as that can be realized via coupled optical waveguides (19); (20) or cold atoms (38) may provide a comprehensive understanding of interplay between loss-gain strengths and -symmetry breaking.
This work was supported by NSF Grant No. DMR-1054020.
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