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arXiv:cond-mat/0203512

New Class of Solvable and Integrable Many-Body Models

R. W. Richardson

Published Mar 25, 2002• 1 citation
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Abstract

Integrability conditions for systems of bosons or fermions with seniority conserving hamiltonians are derived. The conditions are shown to be invariant under a large class of transformations of the interaction matrix elements. Previously published integrable models are shown to satisfy these conditions and the existence of a new class of integrable models is demonstrated. The number of free parameters in the interaction in these models equals the number of single particle levels plus 3 . Equations for the energy eigenstates and eigenvalues are derived and the eigenvalues of the complete set of two-body integrals of the motion are given for the new class. Some two-body correlations in these eigenstates are derived from the integrals of the motion.

DOI
10.48550/arxiv.cond-mat/0203512

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Integrability conditions for systems of bosons or fermions with seniority conserving hamiltonians are derived. The conditions are shown to be invariant under a large class of transformations of the interaction matrix elements. Previously published integrable models are shown to satisfy these conditions and the existence of a new class of integrable models is demonstrated. The number of free parameters in the interaction in these models equals the number of single particle levels plus 3 . Equations for the energy eigenstates and eigenvalues are derived and the eigenvalues of the complete set of two-body integrals of the motion are given for the new class. Some two-body correlations in these eigenstates are derived from the integrals of the motion.
We prove that a wide class of two-level boson systems interacting through k -body forces is integrable in the semiclassical limit irrespective of the value of the parameters of the model. This class of models includes number-conserving Hamiltonians, and interactions which do not conserve the particle number. Applications to two-component and atom–molecule Bose–Einstein condensates at zero temperature as well as to the bosonic k -body embedded ensembles of Gaussian random matrices in the dense limit are discussed.
In this paper, we show that there is a large class of fermionic systems for which it is possible to find, for any dimension, a finite closed set of eigenoperators and eigenvalues of the Hamiltonian.Then, the hierarchy of the equations of motion closes and analytical expressions for the Green's functions are obtained in terms of a finite number of parameters, to be self-consistently determined.Several examples are given.In particular, for these examples it is shown that in the one-dimensional case it is possible to derive by means of algebraic constraints a set of equations which allow us to determine the self-consistent parameters and to obtain a complete exact solution.
A class of quantum many-body models of arbitrary dimension and arbitrary statistics of particles, for which exact eigenstates may be obtained, is found. It is assumed that: (i) models contain two (or 2m) kinds of particles with 'symmetric' matrix elements of pairwise interaction (all potentials coincide with each other to within a sign and wavefunctions of free particles of two components coincide to within a phase factor; pairwise interactions are otherwise arbitrary); (ii) there exists the degeneracy of (the sum) of free-particle spectra. Exact many-body eigenstates correspond to a condensation of non-interacting composite particles ('excitons') which are not exactly bosons, into a single quantum state, and to excitations over the condensate. The origin of the possibility of exact solution is the symmetry under the continuous rotations in the isospin space of two components, to which Bogolubov canonical transformations with parameters u, nu independent of momentum correspond. The class of such models comprises, in particular, two-dimensional electron-hole systems in a strong magnetic field.
The subject of the four manuscripts which make up this dissertation is the concept of integrability in quantum mechanical systems with two degrees of freedom, including the relation between quantum integrability and classical integrability in such systems. In the first manuscript, I consider a two-spin model both classically and quantum mechanically. In the six-dimensional parameter space of this model, classical integrability is satisfied on a five-dimensional hypersurface, and level crossings occur on four-dimensional manifolds that are completely embedded in the integrability hypersurface except for some lower-dimensional sub-manifolds. Under mild assumptions, the classical integrability condition can be reconstructed from a purely quantum mechanical study of level degeneracies in finite-dimensional invariant blocks of the Hamiltonian matrix. In the second manuscript, I analyze the functional dependence [special characters omitted] of the Hamiltonian operator on the action operators for an integrable quantum system, and I compare it with the corresponding functional relationship H (p1, q1; p2, q 2) = HC(J1, J2) in the classical limit of that system. It is demonstrated that the functional dependence is not the same quantum mechanically and (semi-)classically. The former is shown to converge toward the latter in some asymptotic regime associated with the classical limit, but the convergence may be non-uniform. This analysis is carried out for an integrable one- parameter two-spin model. Additional results presented for the (integrable) circular billiard model exhibit similar quantum corrections to the predictions of semiclassical quantization. In the third and the fourth manuscripts, I investigate the relations between symmetry, integrability, and the assignment of quantum numbers to eigenstates for two-spin and spin-boson models. I calculate quantum invariants in the form of expectation values for selected operators and monitor their dependence on the Hamiltonian parameters along loops within, without, and across the integrability region in parameter space. I find clear-cut signatures of integrability and nonintegrability in the observed traces of quantum invariants evaluated in invariant Hilbert subspaces. The results support the notion that quantum integrability depends on the existence of action operators as constituent elements of the Hamiltonian.