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

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.

Emergence of a superglass phase in the random hopping Bose-Hubbard model

“…The derivation of the stability condition for the replicasymmetric solution, similar to the one done by de Almeida and Thouless [49] but more complex, may be found in Ref. [50]. Compared to the spin glass case, we deal with more independent variables (five deviations from the symmetric solution instead of two) and additional Trotter dimensions, which results in matrix elements becoming matrix blocks.…”