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Physical Review A

Clustering of levels in a solvable many-body problem with a degenerate ground state

Mohsen Razavy, R. B. Ludwig

Published Mar 1, 1986• 3 citations
Publisher

Abstract

The level spacing of the Bassichis-Foldy model of an interacting boson system is studied as a function of the coupling F between the two levels of opposite momenta. For a specific value of this coupling F=${F}_{T}$, the ground state of the system becomes degenerate, and the histogram of the energy differences between neighboring levels shows a sharp peak near the origin. The dependence of the energy levels corresponding to the minimum spacing on F follows the classical separatrix energy for a wide range of the parameters of the system, except when F approaches ${F}_{T}$. Around ${F}_{T}$ the ground-state energy changes very rapidly as a function of F unlike the slow variation of the classical separatrix energy. By studying the classical description of the motion in terms of the action-angle variables, one can show an additional symmetry of the Hamiltonian for F very close to ${F}_{T}$. Thus, in this model the nonlinear classical Hamiltonian can be used as an indicator of the degeneracy of the system, even though the quantum-mechanical system does not have the symmetry of the classical problem.

DOI
10.1103/physreva.33.1519

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The level spacing of the Bassichis-Foldy model of an interacting boson system is studied as a function of the coupling F between the two levels of opposite momenta. For a specific value of this coupling F=FT{F}_{T}FT​, the ground state of the system becomes degenerate, and the histogram of the energy differences between neighboring levels shows a sharp peak near the origin. The dependence of the energy levels corresponding to the minimum spacing on F follows the classical separatrix energy for a wide range of the parameters of the system, except when F approaches FT{F}_{T}FT​. Around FT{F}_{T}FT​ the ground-state energy changes very rapidly as a function of F unlike the slow variation of the classical separatrix energy. By studying the classical description of the motion in terms of the action-angle variables, one can show an additional symmetry of the Hamiltonian for F very close to FT{F}_{T}FT​. Thus, in this model the nonlinear classical Hamiltonian can be used as an indicator of the degeneracy of the system, even though the quantum-mechanical system does not have the symmetry of the classical problem.
The Bassichis–Foldy model of a simple interacting boson is solved numerically and the results are compared with those obtained by the Bogoliubov approximation and by the Brueckner–Sawada t-matrix formalism. In the normal region, contrary to the widely held view, the Brueckner–Sawada approximation for the energy of the ground state is not reliable for strong, well-behaved, repulsive forces. The Bogoliubov approximation, on the other hand, remains valid for a wide range of values of the coupling constant. In the inverted region, the attractive force causes a population inversion in the levels of the system. For this case a modified Brueckner–Sawada approximation is developed. This method is applied to the calculation of the transition point and the energies of the ground and the first excited states of the system. Here most of the predictions of the modified Brueckner–Sawada approximation are quite accurate. By a simple change in the Bassichis–Foldy model it is shown that even, for two bosons there can be a phase transition. In this model, the derivative of the ground state energy with respect to the coupling constant is discontinuous at the transition point.
Excited-state quantum phase transitions (ESQPTs) strongly influence the spectral properties of collective many-body quantum systems, changing degeneracy patterns in different quantum phases. Level degeneracies in turn affect the system's dynamics. We analyze the degeneracy dependence on the size of two-level boson models with a u(n+1)u(n+1)u(n+1) dynamical algebra, where nnn is the number of collective degrees of freedom. Below the ESQPT critical energy of these models, the energy gap between neighboring levels that belong to different symmetry sectors gets close to zero as the system size increases. We report and explain why this gap goes to zero exponentially for systems with one collective degree of freedom but algebraically in models with more than one degree of freedom. As a consequence, we show that the infinite-time average of out-of-time-order correlators is an ESQPT order parameter in finite systems with n=1n=1n=1, but in systems with n>1n>1n>1, this average only works as an order parameter in the mean-field limit.
A general spin system with a nonaxially symmetric Hamiltonian containing Jx, Jz-linear and Jz-quadratic terms, widely used in many-body fermionic and bosonic systems and in molecular magnetism, is considered for the variations of general parameters describing intensity interaction changes of each of its terms. For this model Hamiltonian, a semiclassical energy surface (ES) is obtained by means of the coherent-state formalism. An analysis of this ES function, based on catastrophe theory, determines the separatrix in the control parameter space of the system Hamiltonian: the loci of singularities representing semiclassical phase transitions. Here we show that distinct regions of qualitatively different spectrum structures, as well as a singular behavior of quantum states, are ruled by this separatrix: here we show that the separatrix not only describes ground-state singularities, which have been associated with quantum phase transitions, but also reveals the structure of the excited spectrum, distinguishing different quantum phases within the parameter space. Finally, we consider magnetic susceptibility and heat capacity of the system at finite temperature, in order to study thermal properties and thermodynamical phase transitions in the perspective of the separatrix of this Hamiltonian system.
The ground state of a one-dimensional system of many bosons interacting through a repulsive $\ensuremath{\delta}$-function potential is studied when the coupling parameter $\ensuremath{\gamma}$ of the system is small. The procedure is based on the variational analysis in the Bijl-Dingle-Jastrow wave-function space. The ground-state energy, determined in a series form in powers of ${\ensuremath{\gamma}}^{\frac{1}{2}}$ and truncated after the third leading term, is found to be in fair agreement with exact results of Lieb and Liniger over a range of the coupling parameter extending up to $\ensuremath{\gamma}=6$. Two slightly different approaches are considered, and they are shown to be equivalent.
We present some characteristic properties of the ground state of a two-level unit linearly coupled to a bosonic field, exactly deduced without previously solving the eigenvalue problem for the physical system. Such properties are valid in the whole range of the characteristic parameters appearing in the Hamiltonian modelling this system. Our treatment is based on operatorial methods which combine symmetry considerations with some general peculiarities of the lowest energy state. In particular we present rigorous conclusions concerning the expectation value of the two-level unit energy and the mean value of the interaction energy for arbitrary strenght of the coupling constant. The relevance of these rigorous results in connection with an approximate variational determination of the ground state of the system is evidenced.
We consider identical quantum bosons with weak contact interactions in a two-dimensional isotropic harmonic trap. When the interactions are turned off, the energy levels are equidistant and highly degenerate. At linear order in the coupling parameter, these degenerate levels split, and we study the patterns of this splitting. It turns out that the problem is mathematically identical to diagonalizing the quantum resonant system of the two-dimensional Gross-Pitaevskii equation, whose classical counterpart has been studied previously in the mathematical literature on turbulence. Our purpose is to explore the implications of the symmetries and energy bounds of this resonant system, previously studied for the classical case, for the quantum level splitting. Simplifications in computing the splitting spectrum numerically result from exploiting the symmetries. The highest energy state emanating from each unperturbed level is explicitly described by our analytics. We furthermore discuss the energy-level spacing distributions in the spirit of quantum chaos theory. After separating the eigenvalues into blocks with respect to the known conservation laws, we observe the Wigner-Dyson statistics within specific large blocks, which leaves little room for further integrable structures in the problem beyond the symmetries that are already explicitly known.