Robust and fragile -symmetric phases in a tight-binding chain
We study the phase-diagram of a parity and time-reversal () symmetric tight-binding chain with sites and hopping energy , in the presence of two impurities with imaginary potentials located at arbitrary (-symmetric) positions on the chain where . We find that except in the two special cases where impurities are either the farthest or the closest, the -symmetric region - defined as the region in which all energy eigenvalues are real - is algebraically fragile. We analytically and numerically obtain the critical impurity potential and show that as except in the two special cases. When the symmetry is spontaneously broken, we find that the maximum number of complex eigenvalues is given by . When the two impurities are the closest, we show that the critical impurity strength in the limit approaches () provided that is even (odd). For an even the symmetry is maximally broken whereas for an odd , it is sequentially broken. Our results show that the phase-diagram of a -symmetric tight-binding chain is extremely rich and that, in the continuum limit, this model may give rise to new -symmetric Hamiltonians.
Introduction: Lattice models, including tight-binding chains, have been a cornerstone of theoretical explorations due to their analytical and numerical tractability (1), the absence of divergences associated with the ultraviolet cutoff (2); (3), the availability of exact solutions (4), and the ability to capture counter-intuitive physical phenomena including the bound-states in repulsive potentials (5). In recent years, sophisticated optical lattice systems have increasingly permitted the experimental exploration of lattice models (6); (7). These lattice models are largely based on a Hermitian Hamiltonian. Over the past decade, it has become clear that non-Hermitian Hamiltonians with -symmetry can have purely real energy spectra (8); (9) and that, when they do, with an appropriately re-defined inner product, their eigenvectors can be appropriately orthonormalized (10). The theoretical work has been accompanied, most recently, by experiments in optical physics where spontaneous symmetry breaking in a classical system has been observed in waveguides with symmetric complex refractive index (11). In recent years, lattice models with a symmetric non-Hermitian “hopping” between adjacent levels of a simple-Harmonic oscillator (12), tridiagonal -symmetric models (13), and tight-binding models with a Hermitian hopping and -symmetric complex on-site potential (14); (15); (16) have been investigated. These models are of physical interest because they lead to unitary scattering even in the absence of a Hermitian Hamiltonian (17).
In this paper, we analytically and numerically investigate the phase-diagram of a one-dimensional tight-binding chain with hopping energy and two imaginary potentials as a function of the size of the chain and the positions of the two impurities within the chain (16); (17). Our main results are as follows: i) We show that except in the two special cases (the impurities are the closest or the farthest), in the thermodynamic limit , the critical potential strength vanishes, where is the relative position of the impurity. Thus, for a generic impurity location, the -symmetric phase in this system is algebraically fragile (18). ii) The “degree of symmetry breaking”, defined as the fraction of eigenvalues that become complex, is given by . iii) When the impurities are the closest, the critical potential strength is given by when is even and when is odd, as . iv) By considering the continuum limit of this problem, we argue that the -symmetric chain maps onto a -symmetric continuum Hamiltonian with an imaginary viscous drag term.
Tight-binding Model: We start with the Hamiltonian for an -site tight-binding chain with and two impurities with imaginary potentials at sites ,
where () is the creation (annihilation) operator at site , and is the reflection-counterpart of site . We only consider the single-particle sector and, since periodic-boundary conditions are incompatible with the parity operator, our system has open boundary conditions. In the lattice model, the action of the parity operator is given by with , and the action of the anti-linear time-reversal operator is given by . The potential term in the Hamiltonian, Eq.(1), is odd under individual and operations, and hence the Hamiltonian is -symmetric. The phase-diagram of this system with the impurities at the end and its equivalent Hermitian counterpart have been investigated in detail (15).
A general single-particle eigenfunction of the Hamiltonian can be written as where, according to the Bethe ansatz, the coefficients have the form
Note that open-boundary conditions, along with the requirement , constrain the eigenfunction coefficients to a sinusoidal form in the regions and and give . By considering the eigenvalue equation at points and their reflection-counterparts, we find that the quasimomenta obey the equation
The symmetry is unbroken provided that Eq.(3) has real solutions. When , the distinct solutions are given by the well-known result for a tight-binding chain with open boundary conditions, where . Since , if is a solution of Eq.(3), then so is . It also follows that when is odd, is a solution irrespective of the value of , and that the corresponding eigenvector has zero energy. When , Eq.(3) reduces to the criterion for quasimomentum obtained in Ref. (15); in that case, as , the two central become degenerate, the symmetry is spontaneously broken, and the system develops real and two complex (conjugate) eigenvalues.
The left panel in Fig. 1 shows the typical plot of quasimomentum values as a function of , for a chain with sites and the first impurity at =4 (thin solid red) or = 8 (thick dashed blue). Since the plot is symmetric about , we focus only on the left-half and note that since is even, there is no solution present at . As the impurity potential is increased, the adjacent quasimomenta and the corresponding energy levels become degenerate, leading to the symmetry breaking (14); (15). We see that the critical potential for is greater than that for , and, in contrast with the case, the two levels that become degenerate occur in a pair, with one near the origin and other near the zone-boundary . Therefore when , there are four complex eigenvalues. Since the symmetry breaking can be associated with the development of dissipative channels, we define the “degree of -symmetry breaking” as the fraction of eigenvalues that become complex. For a general , as is increased beyond , we find that a total of complex eigenvalues emerge and thus the degree of -symmetry breaking is given by . The right panel in Fig. 1 shows the typical evolution of critical potential strength with , for , obtained by numerically solving Eq.(3). The scaling suggests that the critical potential strength for the infinite chain approaches zero, . Thus, the -symmetric phase, which exists in the region , is algebraically fragile. This result can be qualitatively understood as follows: in the limit and with , Eq.(3) can be approximated by . If , this equation has only real solutions, and hence the symmetry is broken. We emphasize that this argument is invalid when and the corresponding critical potential is given by (15). It is also invalid when , the impurities are closest to each other and, as we discuss below, the critical is nonzero when . Incidentally, we point out that the corresponding -symmetric phase in a tridiagonal Hamiltonian with non-Hermitian “hopping” is not algebraically fragile (13).
Closest Impurities and the Even-Odd Effect: We now consider the case of closest impurities. Note that due to the -symmetric requirement, when is even the impurities are nearest neighbors with , whereas when is odd the impurities are next-nearest-neighbors with . We will first focus on the case with an even . In this case, the condition from Eq. (3) reduces to the following equation
When Eq.(4) has distinct solutions given by . As increases the adjacent approach each other and when , Eq.(4) has doubly-degenerate solutions given by where . When , it is clear that Eq.(4) has no real solutions. Thus the symmetry is maximally broken and all eigenvalues simultaneously become complex. When is odd, the impurities are at sites where is the site at the center of the chain. The equation then reduces to
This equation has all real solutions provided where corresponds to the first degenerate quasimomentum. Therefore, we find that as is increased from zero the -symmetry breaks at in the limit when adjacent near the origin (and their counterparts near the zone boundary) become degenerate. Hence, for , there are four complex eigenvalues. On the other hand, Eq.(5) has only one real solution, , when where is the degenerate quasimomentum closest to the zone center . Hence, the number of complex eigenvalues increases monotonically and when , it saturates to (19). Figure 2 shows the quasimomenta for a chain with and lattice sites obtained from Eq.(5), and confirms these results. Since the symmetric nature of the potential dictates the minimum distance between the impurities when is odd or even, the phase-diagram of the chain is sensitive to it even as .
Numerical Results: We start this section with results for the critical potential strength as a function of the relative impurity site location obtained by numerical diagonalization of the Hamiltonian Eq.(1) for various lattice sizes , even and odd. The left panel in Fig. 3 shows that, for an even , apart from finite-size effects that are prominent near and are also present in solutions to Eq. (3), the critical potential strength is vanishingly small except at and . In both special cases . The right panel in the same figure shows results for odd . When or equivalently , we recover the result (15). As in the case with even , we find that is suppressed with increasing everywhere except when or equivalently . These results are consistent with those obtained through the analytical treatment earlier.
We now briefly explore the change in a (typical) eigenfunction as a function of impurity potential in the case of nearest-neighbor impurities (even ). In the -symmetric phase, an eigenfunction is given by for and for where is a quasimomentum that satisfies Eq.(4). Using the eigenfunction constraints and Eq.(4), it follows that
where the angle satisfies . Figure 4 shows the amplitude and the phase of the ground-state wavefunction of a chain with sites and nearest-neighbor impurities. The top (blue) panel shows that when , the wavefunction amplitude is even about the center of the chain and the phase is given by , as is expected from Eq.(6). The bottom (red) panel shows that when , the broken -symmetry is reflected in the asymmetrical wavefunction amplitudes and in the position-dependent phase factor . These are generic features of the broken -symmetry phase. We also note that in the continuum limit, the eigenfunction becomes discontinuous at the center of the chain while the probability amplitude remains continuous.
Conclusion: We have investigated the phase diagram of an -site one-dimensional chain with a pair of complex -symmetric impurities located at sites within it. A remarkable feature of such a Hamiltonian, in contrast to a tridiagonal Hamiltonian with real entries (12), is that in the -symmetric region, its spectrum remains confined within the energy band of the model in the absence of impurities; as the impurity potential is increased, the level spacing between adjacent energy levels decreases. Our results show that the -symmetric phase of such a chain is algebraically fragile except when the impurities are farthest from each other or are closest to each other. In the latter case, we find that the -symmetric phase survives when (even ) or (odd ). We note that such a chain offers tremendous tunability due to its variable critical impurity strength for a finite , and the corresponding variable fraction of complex eigenvalues which translates into the number of dissipative channels in both classical (11) and quantum systems. Thus, a physical realization of such a model (16) may offer the ability to engineer the level-spacings and the dissipation in this system.
We conclude by briefly pointing out the continuum limit of this system. In the continuum limit, the lattice spacing vanishes and the number of lattice sites diverges such that the length of the chain remains constant. Note that since is a constant, where is the -symmetric potential, the corresponding continuum Schrödinger eigenvalue equation is given by
where the dimensionless impurity strength for nearest-neighbor impurities (even ), for next-nearest-neighbor impurities (odd ), and the eigenfunctions obey boundary conditions . Note that the continuum Hamiltonian is -symmetric, but not Hermitian, due to the non-commuting parts, and , of the “potential” term. Our results imply that the -symmetric phase of this Hamiltonian survives as long as , irrespective of whether the number of sites in the chain is odd or even. Indeed, Eq.(7) suggests a new class of -symmetric Hamiltonians with a “viscous drag potential” term of the form that is not Hermitian but is -symmetric provided is an even function of . In the lattice model, such a potential will correspond to -symmetric impurities at multiple locations. Detailed investigation of such models will improve our understanding of -symmetry breaking in discrete and (classical) continuum systems, that can be realized in optical lattices and waveguides with complex refractive index, respectively (11).
Y.J. is thankful to Los Alamos National Laboratory where part of this work was carried out. M.B. thanks the D.J. Angus Scientech Foundation for a Summer Fellowship.
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