Bethe free-energy approximations for disordered quantum systems

Given a locally consistent set of reduced density matrices, we construct approximate density matrices which are globally consistent with the local density matrices we started from when the trial density matrix has a tree structure. We employ the cavity method of statistical physics to find the optimal density matrix representation by slowly decreasing the temperature in an annealing algorithm, or by minimizing an approximate Bethe free energy depending on the reduced density matrices and some cavity messages originated from the Bethe approximation of the entropy. We obtain the classical Bethe expression for the entropy within a naive (mean-field) approximation of the cavity messages, which is expected to work well at high temperatures. In the next order of the approximation, we obtain another expression for the Bethe entropy depending only on the diagonal elements of the reduced density matrices. In principle, we can improve the entropy approximation by considering more accurate cavity messages in the Bethe approximation of the entropy. We compare the annealing algorithm and the naive approximation of the Bethe entropy with exact and approximate numerical simulations for small and large samples of the random transverse Ising model on random regular graphs.

Figure 1

Comparing the phase transition points in the transverse Ising model ($J_{ij}=1$ and $h_i=h$) on a random regular graph of degree $K=3$ obtained by the annealing algorithm (for different density matrix representations) and the exact solution of the path integral quantum cavity (PIQC) method [13] in the thermodynamic limit. We take density matrices with one-spin interactions (MF), and with two-spin interactions between nearest neighbors (NN) and next-nearest neighbors (nNN) from the quantum interaction graph ${\mathcal{E}}_q$.

Figure 2

The magnetization density $m_x$ at zero temperature in the random transverse Ising model on a random regular graph of degree $K=3$ obtained by the annealing algorithm and the exact numerical simulations for a small system of size $N=20$. Inset: the error $\Delta E_0\equiv E_0^{\mathrm{ann}}-E_0^{\mathrm{exact}}$ in estimating the ground-state energy by the annealing algorithm. Here the $J_{ij}=1$ and the transverse fields $h_i$ are random numbers uniformly distributed in $[0,h]$. We take density matrices with two-spin interactions between nearest neighbors (NN) from the quantum interaction graph ${\mathcal{E}}_q$.

Figure 3

(a) The free energy $F$ and (b) the magnetization density $m_x$ in the random transverse Ising model on a random regular graph of degree $K=3$ obtained by minimizing the approximate Bethe free energy with $S_{\mathrm{Bethe}}^{\left(1\right)}$ (denoted by $B{P}^{\left(1\right)}$) and the exact numerical simulations for a small system of size $N=12$. Here the $J_{ij}=1$ and the transverse fields $h_i$ are random numbers uniformly distributed in $[0,h]$. The $B{P}^{\left(1\right)}$ data are shown in the region where the entropy is non-negative.

Figure 4

Comparing (a) the magnetization density $m_x$ and (b) the phase transition points $T_c$ in the random transverse Ising model on a random regular graph of degree $K=3$ obtained by minimizing the approximate Bethe free energy with $S_{\mathrm{Bethe}}^{\left(1\right)}$ (denoted by $B{P}^{\left(1\right)}$) and the operator quantum cavity method (OQC) of Refs. [14, 17] for a system of size $N=1000$. Here the $J_{ij}=1$ and the transverse fields $h_i$ are random numbers uniformly distributed in $[0,h]$. The $B{P}^{\left(1\right)}$ data are shown in the region where the entropy is non-negative.