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Acta Physica Polonica A

Quantum Glass of Interacting Bosons with Off-Diagonal Disorder Has a Severe Sign Problem

Anna M. Piekarska, T. K. Kopeć

Published Jan 1, 2019• 2 citations
Publisher

Abstract

We study a disordered system of interacting bosons, described by the Bose-Hubbard model with the disorder introduced via Gaussian-distributed hopping integrals.After applying the replica trick and the Trotter formula, we arrive at an expression for order parameters involving averaging with respect to the effective Hamiltonian, that has to be numerically evaluated.We show that the Monte Carlo algorithm fails because of the sign problem, which is severe in this case, hence cannot be removed via standard methods.It cannot be therefore excluded that this problem belongs to the NP-complete complexity class, i.e., of problems without known polynomial-time solution but with a possibility of validating a given solution candidate at such a computational complexity.

DOI
10.12693/aphyspola.135.78

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We study a disordered system of interacting bosons, described by the Bose-Hubbard model with the disorder introduced via Gaussian-distributed hopping integrals.After applying the replica trick and the Trotter formula, we arrive at an expression for order parameters involving averaging with respect to the effective Hamiltonian, that has to be numerically evaluated.We show that the Monte Carlo algorithm fails because of the sign problem, which is severe in this case, hence cannot be removed via standard methods.It cannot be therefore excluded that this problem belongs to the NP-complete complexity class, i.e., of problems without known polynomial-time solution but with a possibility of validating a given solution candidate at such a computational complexity.
We study disordered interacting bosons described by the Bose-Hubbard model with Gaussian-distributed random tunneling amplitudes. It is shown that the off-diagonal disorder induces a spin-glass-like ground state, characterized by randomly frozen quantum-mechanical U(1) phases of bosons. To access criticality, we employ the "n-replica trick," as in the spin-glass theory, and the Trotter-Suzuki method for decomposition of the statistical density operator, along with numerical calculations. The interplay between disorder, quantum, and thermal fluctuations leads to phase diagrams exhibiting a glassy state of bosons, which are studied as a function of model parameters. The considered system may be relevant for quantum simulators of optical-lattice bosons, where the randomness can be introduced in a controlled way. The latter is supported by a proposition of experimental realization of the system in question.
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.
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.
We apply the Worldvolume Hybrid Monte Carlo (WV-HMC) method [arXiv:2012.08468] to the two-dimensional Hubbard model, which is known to suffer from a severe sign problem when the system is doped (away from half filling). We show that the method predicts physical observables with controlled statistical errors on an 8×88 \times 88×8 lattice at temperature T/t=1/6.4≈0.156T/t = 1/6.4 \approx 0.156T/t=1/6.4≈0.156 and interaction strength U/t=8.0U/t = 8.0U/t=8.0 (ttt is the hopping amplitude), for which the standard determinant quantum Monte Carlo fails.
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.
We apply the Worldvolume Hybrid Monte Carlo (WV-HMC) method [arXiv:2012.08468] to the two-dimensional Hubbard model, which is known to suffer from a severe sign problem when the system is doped (away from half filling). We show that the method predicts physical observables with controlled statistical errors on an 8×88 \times 88×8 lattice at temperature T/t=1/6.4≈0.156T/t = 1/6.4 \approx 0.156T/t=1/6.4≈0.156 and interaction strength U/t=8.0U/t = 8.0U/t=8.0 (ttt is the hopping amplitude), for which the standard determinant quantum Monte Carlo fails.