Cavity method for quantum spin glasses on the Bethe lattice

We propose a generalization of the cavity method to quantum spin glasses on fixed connectivity lattices. Our work is motivated by the recent refinements of the classical technique and its potential application to quantum computational problems. We numerically solve for the phase structure of a connectivity $q=3$ transverse field Ising model on a Bethe lattice with $\pm J$ couplings and investigate the distribution of various classical and quantum observables.

Figure 1

The three cavity operations. Although we have not labeled them in the figure, the new links are $J_{01}$, $J_{02}$, etc.

Figure 2

Iteration of cavity rods. There are periodic boundary conditions in the imaginary-time (vertical) direction. Other cavity operations are analogous.

Figure 3

Merging of two cavity rods onto a “broken” central rod. Here $N_t=3$; but instead of imposing periodic boundary conditions everywhere we depict $\sigma \left(0\right)$ and $\sigma \left(N_t\right)$ as independent spins. Each vertical link corresponds to a $\Gamma$ coupling while each horizontal link corresponds to a $J_{ij}/2N_t$ coupling. The dashed lines connecting the cavity rods indicate identification of the top and bottom spins for each rod and the wiggly line indicates an effective action. By imposing different values on the top and bottom spins of the broken rod and summing out the rest, the elements of the reduced density matrix ${\rho }_0$ may be determined.

Figure 4

(a) Phase diagram at $q=3$. The solid phase transition curve has been calculated at $N_t=10$, $N_{\text{rods}}=2500$, and $N_{\text{iter}}=1000N_{\text{rods}}$ on a fine mesh in the $\left(\tau ,B_t\right)$ plane. The vertical dotted line is the asymptotic critical line for large $B_t$ at $N_t=10$ (i.e., $\tau ={\tau }_c/N_t$). The points marked $x$ with error bars indicate $N_t\to \infty$ fits based on Fig. 6. The dashed transition curve is a weighted quadratic fit through the estimated low-temperature points and the $N_t=10$ points in the range $0.5<\tau <1$. This leads to an estimated $B_t^c=1.775\pm 0.03$. As this fit is clearly heuristic, we have suggested a much larger range for our estimate of $B_t^c$ in the figure. The stars and stripes indicate points in the phase space which we have investigated in more detail below. (b) The average von Neumann entropy $S_{\text{vonN}}$ (in bits) of a central spin as a function of $\left(\tau ,B_t\right)$ at $N_t=8$. The dashed line indicates the estimated region of validity of the discretization approximation $\left(B_t\le N_t\tau /2\right)$.

Figure 5

(Color online) These graphs all correspond to the horizontal slice at $B_t=1$ shown in Fig. 4. The order parameter $q_{\text{EA}}$ and all effective action variances undergo mean–field-like transitions at the critical temperature (e g. $q_{\text{EA}}\sim {\left|\tau -{\tau }_c\right|}^1$). This allows us to estimate the critical temperature precisely despite softening due to critical slowing down near the phase transition. These curves were calculated at $N_t=8$. However, the results are stable to increasing $N_t$ to 10 to within an error of $\pm 0.005$ in ${\tau }_c$.

Figure 6

(Color online) The phase transition curve in the high-field low-temperature regime at various $N_t$. The vertical dashed lines indicate the finite discretization critical asymptotes. The estimated curve for $N_t\to \infty$ in Fig. 4a is given by fitting $B_t\left(N_t\right)=a/N_t^2+b/N_t^4+c$ to the points calculated at each temperature and then sending $N_t$ to infinity in the result. These fits suffer from a paucity of data but are suggestive nonetheless.

Figure 7

(Color online) The distribution of the field term of the cavity rod action at the four different points in phase space labeled by the stars on Fig. 4. The distinctive distribution of the low-temperature replica symmetric classical spin glass is reproduced in the bottom left corner while the three other points all lie closer to the phase boundary in the $\tau$ or $B_t$ directions.

Figure 8

(Color online) Histograms of nearest-neighbor (top row) and next-nearest-neighbor (bottom row) interactions in imaginary time at $\tau =0.25$ and $B_t=1$ (in the SG phase). The second and third column provide the conditional distribution of the interactions given $\left|h\right|$ is small or large.

Figure 9

(Color online) (a) The average next-to-nearest-neighbor interaction in time $C^{\left(2\right)}\left(2\Delta t\right)$ of a cavity rod action at $B_t=1$, varying the temperature. The bars indicate the variance of $C^{\left(2\right)}\left(2\Delta t\right)$. Notice the zero variance above the critical temperature. (b) Single-site imaginary-time correlation function ${⟨\sigma \left(0\right)\sigma \left(t\right)⟩}_J$ at $\tau =0.25$ for various $B_t$ passing through the transition at $B_t^c\sim 1.85$.