Quasi-exactly Solvable Models Research Articles

Entanglement entropy of inhomogeneous XX spin chains with algebraic interactions

We introduce a family of inhomogeneous XX spin chains whose squared couplings are a polynomial of degree at most four in the site index. We show how to obtain an asymptotic approximation for the Rényi entanglement entropy of all such chains in a constant magnetic field at half filling by exploiting their connection with the conformal field theory of a massless Dirac fermion in a suitably curved static background. We study the above approximation for three particular chains in the family, two of them related to well-known quasi-exactly solvable quantum models on the line and the third one to classical Krawtchouk polynomials, finding an excellent agreement with the exact value obtained numerically when the Rényi parameter α is less than one. When α ≥ 1 we find parity oscillations, as expected from the homogeneous case, and show that they are very accurately reproduced by a modification of the Fagotti-Calabrese formula. We have also analyzed the asymptotic behavior of the Rényi entanglement entropy in the non-standard situation of arbitrary filling and/or inhomogeneous magnetic field. Our numerical results show that in this case a block of spins at each end of the chain becomes disentangled from the rest. Moreover, the asymptotic approximation for the case of half filling and constant magnetic field, when suitably rescaled to the region of non-vanishing entropy, provides a rough approximation to the entanglement entropy also in this general case.

Alternative quantisation condition for wavepacket dynamics in a hyperbolic double well

We propose an analytical approach for computing the eigenspectrum and corresponding eigenstates of a hyperbolic double well potential of arbitrary height or width, which goes beyond the usual techniques applied to quasi-exactly solvable models. We map the time-independent Schrödinger equation onto the Heun confluent differential equation, which is solved by using an infinite power series. The coefficients of this series are polynomials in the quantisation parameter, whose roots correspond to the system’s eigenenergies. This leads to a quantisation condition that allows us to determine a whole spectrum, instead of individual eigenenergies. This method is then employed to perform an in depth analysis of electronic wave-packet dynamics, with emphasis on intra-well tunneling and the interference-induced quantum bridges reported in a previous publication Chomet et al (2019 New J. Phys. 21 123004). Considering initial wave packets of different widths and peak locations, we compute autocorrelation functions and Wigner quasiprobability distributions. Our results exhibit an excellent agreement with numerical computations, and allow us to disentangle the different eigenfrequencies that govern the phase-space dynamics.

Inhomogeneous XX spin chains and quasi-exactly solvable models

We establish a direct connection between inhomogeneous XX spin chains (or free fermion systems with nearest-neighbors hopping) and certain QES models on the line giving rise to a family of weakly orthogonal polynomials. We classify all such models and their associated XX chains, which include two families related to the Lamé (finite gap) quantum potential on the line. For one of these chains, we numerically compute the Rényi bipartite entanglement entropy at half filling and derive an asymptotic approximation thereof by studying the model’s continuum limit, which turns out to describe a massless Dirac fermion on a suitably curved background. We show that the leading behavior of the entropy is that of a c = 1 critical system, although there is a subleading log(log N) correction (where N is the number of sites) unusual in this type of models.

Quasi-Exactly Solvable Differential Models: A Canonical Polynomials Approach

A quasi-exactly solvable model refers to any second order differential equation with polynomial coefficients of the form A(x)y’’(x)+B(x)y’(x)+C(x)y(x)=0 where a pair of exact polynomials {y(x), C(x)} with respective degrees {deg[y]=n, deg[C]=p} are to be found simultaneously in terms of the coefficients of two given polynomials {A(x), B(x)}. The existing methods for solving quasi-exactly solvable models require the solution of a system of nonlinear algebraic equations of which the dimensions depend on n, the degree of the exact polynomial solution y(x). In this paper, a new method employing a set of polynomials, called canonical polynomials, is proposed. This method requires solving a system of nonlinear algebraic equations of which the dimensions depend only on p, the degree of C(x), and do not vary with n. Several examples are implemented to testify the efficiency of the proposed method.

Integrable coupled Liénard-type systems with balanced loss and gain

A Hamiltonian formulation of generic many-particle systems with space-dependent balanced loss and gain coefficients is presented. It is shown that the balancing of loss and gain necessarily occurs in a pair-wise fashion. Further, using a suitable choice of co-ordinates, the Hamiltonian can always be reformulated as a many-particle system in the background of a pseudo-Euclidean metric and subjected to an analogous inhomogeneous magnetic field with a functional form that is identical with space-dependent loss/gain co-efficient. The resulting equations of motion from the Hamiltonian are a system of coupled Liénard-type differential equations. Partially integrable systems are obtained for two distinct cases, namely, systems with (i) translational symmetry or (ii) rotational invariance in a pseudo-Euclidean space. A total number of m+1 integrals of motion are constructed for a system of 2m particles, which are in involution, implying that two-particle systems are completely integrable. A few exact solutions for both the cases are presented for specific choices of the potential and space-dependent gain/loss co-efficients, which include periodic stable solutions. Quantization of the system is discussed with the construction of the integrals of motion for specific choices of the potential and gain–loss coefficients. A few quasi-exactly solvable models admitting bound states in appropriate Stoke wedges are presented.

Exact solutions for a class of quasi-exactly solvable models: A unified treatment

The exact solution of the Schrodinger equation for the four quasi-exactly solvable potentials is presented using the functional Bethe ansatz method. It is shown that all models give rise to the same basic differential equation which is quasi-exactly solvable. The eigenvalues, eigenfunctions and the allowed potential parameters are given for each of the four models in terms of the roots of a set of algebraic Bethe ansatz equations.

Magnetic quantum dots and rings in two dimensions

We consider the motion of electrons confined to a two dimensional plane with an externally applied perpendicular inhomogeneous magnetic field, both with and without a Coulomb potential. We find that as long as the magnetic field is slowly-decaying, bound states in magnetic quantum dots are indeed possible. Several example cases of such magnetic quantum dots are considered in which one can find the eigenvalues and eigenfunctions in closed form, including two hitherto unknown quasi-exactly solvable models treated with confluent and biconfluent Heun polynomials. It is shown how a modulation of the strength of the magnetic field can exclude magnetic vortex-like states, rotating with a certain angular momenta and possessing a definite spin orientation, from forming. This indicates one may induce localization-delocalization transitions and suggests a mechanism for spin-separation.

Planar hydrogen-like atom in inhomogeneous magnetic fields: Exactly or quasi-exactly solvable models

We use a simple mathematical method to solve the problem of a two-dimensional hydrogen-like atom in the inhomogeneous magnetic fields B = (k/r)z and B = (k/r3)z. We construct a Hamiltonian that takes the same form as the Hamiltonian of a hydrogen-like atom in the homogeneous magnetic fields and obtain the energy spectrum by comparing the Hamiltonians. The results show that the whole spectrum of the atom in the magnetic field B = (k/r)z can be obtained, and the problem is exactly solvable in this case. We find analytic solutions of the Schrodinger equation for the atom in the magnetic field B = (k/r3)z for particular values of the magnetic strength k and thus present a quasi-exactly solvable model.

Haydock’s recursive solution of self-adjoint problems. Discrete spectrum

Haydock’s recursive solution is shown to underline a number of different concepts such as (i) quasi-exactly solvable models, (ii) exactly solvable models, (iii) three-term recurrence solutions based on Schweber’s quantization criterion in Hilbert spaces of entire analytic functions, and (iv) a discrete quantum mechanics of Odake and Sasaki. A recurrent theme of Haydock’s recursive solution is that the spectral properties of any self-adjoint problem can be mapped onto a corresponding sequence of polynomials {pn(E)} in energy variable E. The polynomials {pn(E)} are orthonormal with respect to the density of states n0(E) and energy eigenstate |E〉 is the generating function of {pn(E)}. The generality of Haydock’s recursive solution enables one to see the different concepts from a unified perspective and mutually benefiting from each other. Some results obtained within the particular framework of any of (i) to (iv) may have much broader significance.

Novel quasi-exactly solvable models with anharmonic singular potentials

We present new quasi-exactly solvable models with inverse quartic, sextic, octic and decatic power potentials, respectively. We solve these models exactly by means of the functional Bethe ansatz method. For each case, we give closed-form solutions for the energies and the wave functions as well as analytical expressions for the allowed potential parameters in terms of the roots of a set of algebraic equations.

Unified derivation of exact solutions for a class of quasi-exactly solvable models

We present a unified treatment of exact solutions for a class of four quantum mechanical models, namely, the anharmonic singular potential, the generalized quantum isotonic oscillator, the soft-core Coulomb potential, and the non-polynomially modified oscillator. We show that all four cases are reducible to the same basic ordinary differential equation, which is quasi-exactly solvable. A systematic and closed form solution to the basic equation is obtained via the Bethe ansatz method. Using the result, general exact expressions for the energies and the allowed potential parameters are given explicitly for each of the four cases in terms of the roots of a set of algebraic equations. A hidden sl(2) algebraic structure is also discovered in these models.

Exceptional orthogonal polynomials, QHJ formalism and SWKB quantization condition

We study the quantum Hamilton–Jacobi (QHJ) equation of the recently obtained exactly solvable models, related to the newly discovered exceptional polynomials, and show that the QHJ formalism reproduces the exact eigenvalues and the eigenfunctions. The fact that the eigenfunctions have zeros and poles in complex locations leads to an unconventional singularity structure of the quantum momentum function p(x), the logarithmic derivative of the wavefunction, which forms the crux of the QHJ approach to quantization. A comparison of the singularity structure for these systems with the known exactly solvable and quasi-exactly solvable models reveals interesting differences. We find that the singularity structure of the momentum function for these new potentials lies between the above two distinct models, sharing similarities with both of them. This prompted us to examine the exactness of the supersymmetric Wentzel–Kramers–Brillouin (SWKB) quantization condition. The interesting singularity structure of p(x) and of the superpotential for these models has important consequences for the SWKB rule and in our proof of its exactness for these quantal systems.

Prepotential approach to quasinormal modes

In this paper we demonstrate how the recently reported exactly and quasi-exactly solvable models admitting quasinormal modes can be constructed and classified very simply and directly by the newly proposed prepotential approach. These new models were previously obtained within the Lie-algebraic approach. Unlike the Lie-algebraic approach, the prepotential approach does not require any knowledge of the underlying symmetry of the system. It treats both quasi-exact and exact solvabilities on the same footing, and gives the potential as well as the eigenfunctions and eigenvalues simultaneously. We also present three new models with quasinormal modes: a new exactly solvable Morse-like model, and two new quasi-exactly solvable models of the Scarf II and generalized Pöschl–Teller types.

An approach to one-dimensional elliptic quasi-exactly solvable models

One-dimensional Jacobian elliptic quasi-exactly solvable second-order differential equations are obtained by introducing the generalized third master functions. It is shown that the solutions of these differential equations are generating functions for a new set of polynomials in terms of energy with factorization property. The roots of these polynomials are the same as the eigenvalues of the differential equations. Some one-dimensional elliptic quasi-exactly quantum solvable models are obtained from these differential equations.

Few-particle systems: quasi-exactly solvable models

Several quasi-exactly solvable models of few-particle systems are discussed. In particular, the Hooke atom (also known as harmonium) and its extensions appropriate to a description of non-Born-Oppenheimer one- and two-electron diatomic molecules are presented. Also several methods of construction of quasi-exactly solvable Schrodinger equations are described.

Partially Solvable Spin Chains and QES Spin Models

In this paper we prove an extension of the usual freezing trick argument which can be applied to a number of quasi-exactly solvable spin models of Calogero–Sutherland type. In order to illustrate the application of this method we analyze a partially solvable spin chain presenting near-neighbors interactions which was introduced and studied in J. Phys. A: Math. Theor. 40 (2007) 1857–1883; Nucl. Phys. 789 (2008) 452–482. Our discussion focuses on the existence of integer eigenvalues.

Quasi-exact solvability beyond the sl(2) algebraization

We present evidence to suggest that the study of one dimensional quasi-exactly solvable (QES) models in quantum mechanics should be extended beyond the usual $\sla(2)$ approach. The motivation is twofold: We first show that certain quasi-exactly solvable potentials constructed with the $\sla(2)$ Lie algebraic method allow for a new larger portion of the spectrum to be obtained algebraically. This is done via another algebraization in which the algebraic hamiltonian cannot be expressed as a polynomial in the generators of $\sla(2)$. We then show an example of a new quasi-exactly solvable potential which cannot be obtained within the Lie-algebraic approach.

Exact and quasiexact solvability of second order superintegrable quantum systems. II. Relation to separation of variables

We make explicit the intimate relationship between quasiexact solvability, as expounded, for example, by Ushveridze [Quasi-exactly Solvable Models in Quantum Mechanics (IOP, Bristol, 1993)], and the technique of separation of variables as it applies to specific superintegrable quantum Hamiltonians. It is the multiseparability of superintegrable systems that forces the existence of interesting families of polynomial solutions characteristic of quasiexact solvability that enables us to solve these systems in distinct ways and that gives us the basis of a classification theory. This connection is generalized in terms of the understanding of the role of finite solutions of quantum Hamiltonians.

Quasi-exactly solvable quasinormal modes

We consider quasinormal modes with complex energies from the point of view of the theory of quasi-exactly solvable (QES) models. We demonstrate that it is possible to find new potentials which admit exactly solvable or QES quasinormal modes by suitable complexification of parameters defining the QES potentials. Particularly, we obtain one QES and four exactly solvable potentials out of the five one-dimensional QES systems based on the sl(2) algebra.

Quasi-Exactly Solvable N-Body Spin Hamiltonians with Short-Range Interaction Potentials

We review some recent results on quasi-exactly solvable spin models presenting near-neighbors interactions. These systems can be understood as cyclic generalizations of the usual Calogero-Sutherland models. A nontrivial modification of the exchange operator formalism is used to obtain several infinite families of eigenfunctions of these models in closed form.