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arXiv:2202.06610 · Journal of Statistical Mechanics Theory and Experiment

Stability of the replica-symmetric solution in the off-diagonally-disordered Bose–Hubbard model

Anna M. Piekarska, T. K. Kopeć

Published Jul 1, 2022• 2 citations
arXivHTMLPDF

Abstract

Abstract We study a disordered system of interacting bosons described by the Bose–Hubbard Hamiltonian with random tunneling amplitudes. We derive the condition for the stability of the replica-symmetric solution for this model. Following the scheme of de Almeida and Thouless, we determine if the solution corresponds to the minimum of free energy by building the respective Hessian matrix and checking its positive semidefiniteness. Thus, we find the eigenvalues by postulating the set of eigenvectors based on their expected symmetry, and require the eigenvalues to be non-negative. We evaluate the spectrum numerically and identify matrix blocks that give rise to eigenvalues that are always non-negative. Thus, we find a subset of eigenvalues coming from decoupled subspaces that is sufficient to be checked as the stability criterion. We also determine the stability of the phases present in the system, finding that the disordered phase is stable, the glass phase is unstable, while the superfluid phase has both stable and unstable parts.

DOI
10.1088/1742-5468/ac7e4b

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Abstract We study a disordered system of interacting bosons described by the Bose–Hubbard Hamiltonian with random tunneling amplitudes. We derive the condition for the stability of the replica-symmetric solution for this model. Following the scheme of de Almeida and Thouless, we determine if the solution corresponds to the minimum of free energy by building the respective Hessian matrix and checking its positive semidefiniteness. Thus, we find the eigenvalues by postulating the set of eigenvectors based on their expected symmetry, and require the eigenvalues to be non-negative. We evaluate the spectrum numerically and identify matrix blocks that give rise to eigenvalues that are always non-negative. Thus, we find a subset of eigenvalues coming from decoupled subspaces that is sufficient to be checked as the stability criterion. We also determine the stability of the phases present in the system, finding that the disordered phase is stable, the glass phase is unstable, while the superfluid phase has both stable and unstable parts.
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Abstract We investigate a system of interacting bosons with random intersite tunnelling amplitudes. We describe these by introducing Gaussian-distributed hopping integrals into the standard Bose–Hubbard model. This system has been recently shown to exhibit a quantum phase transition to a glassy state. The latter is characterized by a quenched disorder of boson wave-function phases. In this aspect, the system resembles quantum spin-glass systems that attracted much attention. By exploiting this analogy, we employ the well-established methodology originated by Sherrington and Kirkpatrick, which bases on the replica trick and the Trotter–Suzuki expansion. This treatment transforms the original quantum problem into an effective classical one with an additional time-like dimension. Here, we focus on autocorrelation functions of canonical variables of the effective system in the time-like domain. Deep in the disordered phase, we find a highly dynamical nature of correlations in agreement with the expected short memory of the system. This behaviour weakens while approaching and passing the phase boundary, where in the glassy phase asymptotically non-vanishing correlations are encountered. Thus, the state features infinite memory, which is consistent with the quenched nature of glassy disorder with random but frozen boson phases.
The zero temperature phase diagram for ultracold bosons in a random 1D potential is obtained through a site decoupling mean-field scheme performed over a Bose-Hubbard (BH) Hamiltonian, whose hopping term is considered as a random variable. As for the model with random on-site potential, the presence of disorder leads to the appearance of a Bose glass phase. The different phases—i.e., Mott insulator, superfluid, and Bose glass—are characterized in terms of condensate fraction and superfluid fraction. Furthermore, the boundary of the Mott lobes is related to an off-diagonal Anderson model featuring the same disorder distribution as the original BH Hamiltonian.
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We study an experimentally feasible system of strongly correlated bosons with random hoppings described by the infinite-range Bose-Hubbard model on a lattice with hopping integrals given by independent random variables of Gaussian distribution with nonzero mean. We solve this quantum model in the thermodynamic limit, employing the replica method and the Trotter-Suzuki formula. We find and describe a superglass phase that emerges at the interface between glass and superfluid phases. Both glassy and long-range orderings are present in the superglass and compete with each other, as revealed by the anticorrelation of their order parameters. We present phase diagrams in various cross sections of the multidimensional space of system parameters. In selected parameter subspaces, we compare the results to those of nondisordered, diagonally disordered, and once celebrated spin-glass systems.