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arXiv:1606.00816 · Journal of Physics A Mathematical and Theoretical

Quantum integrable multi-well tunneling models

L H Ymai, Arlei Prestes Tonel, Angela Foerster, Jon Links

Published May 10, 2017• 19 citations
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Abstract

Abstract In this work we present a general construction of integrable models describing boson tunneling in multi-well systems. We show how the models may be derived through the quantum inverse scattering method and solved by algebraic Bethe ansatz means. From the transfer matrix we find only two conserved operators. However, we construct additional conserved operators through an alternative approach. As a consequence the models admit multiple pseudovacua, each associated with a set of Bethe ansatz equations (BAEs). We show that all sets of BAEs are needed to obtain a complete set of eigenstates.

DOI
10.1088/1751-8121/aa7227

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Abstract In this work we present a general construction of integrable models describing boson tunneling in multi-well systems. We show how the models may be derived through the quantum inverse scattering method and solved by algebraic Bethe ansatz means. From the transfer matrix we find only two conserved operators. However, we construct additional conserved operators through an alternative approach. As a consequence the models admit multiple pseudovacua, each associated with a set of Bethe ansatz equations (BAEs). We show that all sets of BAEs are needed to obtain a complete set of eigenstates.
We introduce an integrable, four-well ring model for bosons where the tunneling couplings between nearest-neighbour wells are not restricted to be equal. We show how the model may be derived through the quantum inverse scattering method from a solution of the Yang–Baxter equation, and in turn solved by algebraic Bethe ansatz means. The model admits multiple pseudovacuum states. Numerical evidence is provided to indicate that all pseudovacua are required to obtain a complete set of Bethe eigenstates. The model has the notable property that there is a class of eigenstates which admit a simple, closed-form energy expression.
The generalisation of the quantum inverse scattering method is used to derive the Bethe ansatz equation of a system for a multicomponent nonlinear Schrodinger models with n species (or colours) of bosons or fermions. The matrix periodic boundary conditions for bosons and fermions are obtained and the relationship between them discussed. The eigenstates for the infinite conserved quantities of the system are constructed. Lastly, the author generalises these results to the non-linear Schrodinger model with the mirror-image delta potentials proposed by Schulz (1987).
We formulate the quantum inverse scattering method for the case of anyonic grading. This provides a general framework for constructing integrable models describing interacting hard-core anyons. Through this method we reconstruct the known integrable model of hard core anyons associated with the XXX model, and as a new application we construct the anyonic t − J model. The energy spectrum for each model is derived by means of a generalization of the algebraic Bethe ansatz. The grading parameters implementing the anyonic signature give rise to sector-dependent phase factors in the Bethe ansatz equations.
In this article we review recent developments in the one-dimensional Bethe ansatz solvable multi-chain quantum models. The algebraic version of the Bethe ansatz (the quantum inverse scattering method) permits us to construct new families of integrable Hamiltonians using simple generalizations of the well known constructions of the single-chain model. First we consider the easiest example (`basic' model) of this class of models: the antiferromagnetic two-chain spin-1/2 model with the nearest-neighbour and next-nearest-neighbour spin-frustrating interactions (zigzag chain). We show how the algebra of the quantum inverse scattering method works for this model, and what are the important features of the Hamiltonian (which reveal the topological properties of two dimensions together with the one-dimensional properties). We consider the solution of the Bethe ansatz for the ground state (in particular, commensurate-incommensurate quantum phase transitions present due to competing spin-frustrating interactions are discussed) and construct the thermal Bethe ansatz (in the form of the `quantum transfer matrix') for this model. Then possible generalizations of the basic model are considered: an inclusion of a magnetic anisotropy, higher-spin representations (including the important case of a quantum ferrimagnet), the multi-chain case, internal degrees of freedom of particles at each site, etc. We observe the similarities and differences between this class of models and related exactly solvable models: other groups of multi-chain lattice models, quantum field theory models and magnetic impurity (Kondo-like) models. Finally, the behaviour of non-integrable (less constrained) multi-chain quantum models is discussed.