Quantum Glass of Interacting Bosons with Off-Diagonal Disorder

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.

Figure 1

Phase diagrams at various temperatures for $M=5–12$ (circles) and extrapolated to $M=\infty$ (squares). Disordered ($D$) and glassy ($G$) phases are marked. Lines are to guide the eye.

Figure 2

Phase diagram in terms of $U-kT$ variables at $\mu /U=0.5$ for $M=5–12$ (circles) and extrapolated to $M=\infty$ (squares; exemplary fit in the inset). Disordered ($D$) and glassy ($G$) phases are marked.

Figure 3

Phase diagram at $kT/U=0.05$ with the Hilbert-space basis enlarged to four states, for $M=7–9$ (circles). (Bottom inset) Dynamic self-interactions ${\mathcal{R}}_{k{k}^{\prime }}$ for $\mu /U=0.5$ and $J=J_c$ vs $|k-k^{\prime }|$ at various temperatures; lines are to guide the eye. (Top inset) A schematic view of the proposed experimental realization of the considered system; a plaquette $\mathcal{P}$ and tunneling site $J_{\ell {\ell }^{\prime }}$ are marked.

Figure 4

(a) Color map of compressibility $\kappa$ (in units of $U$) at $kT/U=0.05$. Dashed line marks the critical boundary. Results for $\mu /U>1.35$ are obtained for a five-state basis to account for higher $⟨\widehat{n}⟩$. (b),(c) Bosonic filling factor $⟨\widehat{n}⟩$ (dashed line, right axis) and compressibility $\kappa$ (solid line, left axis) vs $\mu /U$ for two values of $J/U$ indicated by dotted lines in (a).

Figure 5

(a) Color map of the order-parameter glass susceptibility ${\chi }_G$ near the glass boundary from Fig. 3. (Inset) An exemplary fit of the power-law critical behavior. (b) Universal critical exponent $\gamma$ of ${\chi }_G$.