Possible ergodic-nonergodic regions in the quantum Sherrington-Kirkpatrick spin glass model and quantum annealing

We explore the behavior of the order parameter distribution of the quantum Sherrington-Kirkpatrick model in the spin glass phase using Monte Carlo technique for the effective Suzuki-Trotter Hamiltonian at finite temperatures and that at zero temperature obtained using the exact diagonalization method. Our numerical results indicate the existence of a low- but finite-temperature quantum-fluctuation-dominated ergodic region along with the classical fluctuation-dominated high-temperature nonergodic region in the spin glass phase of the model. In the ergodic region, the order parameter distribution gets narrower around the most probable value of the order parameter as the system size increases. In the other region, the Parisi order distribution function has nonvanishing value everywhere in the thermodynamic limit, indicating nonergodicity. We also show that the average annealing time for convergence (to a low-energy level of the model, within a small error range) becomes system size independent for annealing down through the (quantum-fluctuation-dominated) ergodic region. It becomes strongly system size dependent for annealing through the nonergodic region. Possible finite-size scaling-type behavior for the extent of the ergodic region is also addressed.

Figure 1

Monte Carlo results for the plots of the area-normalized order parameter distribution $P\left(\right|q\left|\right)$ for given sets of transverse field $\mathrm{\Gamma }$ and temperature $T$ values: (a) for $T=0.20$ and $\mathrm{\Gamma }=1.00$; (b) for $T=0.40$ and $\mathrm{\Gamma }=0.80$. Insets: Extrapolations of $P\left(0\right)$ with $1/N$. In the first case the extrapolated values of $P\left(0\right)$ and $W$ tend to 0 in the large-system-size limit, whereas in the other case the values of such quantities remain finite even in the thermodynamic limit.

Figure 2

Monte Carlo results for plots of the peak-normalized order parameter distribution $P\left(\right|q\left|\right)$ for given sets of transverse field $\mathrm{\Gamma }$ and temperature $T$ values: (a) for $T=0.15$ and $\mathrm{\Gamma }=1.00$; (b) for $T=0.20$ and $\mathrm{\Gamma }=1.00$. Insets: Extrapolations of $P\left(0\right)$ with $1/N$. In both cases the extrapolated value of $P\left(0\right)$ tends to 0 in the large-system-size limit.

Figure 3

Monte Carlo results for plots of the peak-normalized order parameter distribution $P\left(\right|q\left|\right)$ for given sets of transverse field $\mathrm{\Gamma }$ and temperature $T$ values: (a) for $T=0.30$ and $\mathrm{\Gamma }=0.80$; (b) for $T=0.40$ and $\mathrm{\Gamma }=0.80$. Insets: Extrapolations of $P\left(0\right)$ with $1/N$. In these cases extrapolated values of $P\left(0\right)$ for the large-system-size limit remain finite.

Figure 4

Exact diagonalization results (at zero temperature) for the variation of $P\left(\right|Q\left|\right)$ as a function of $\left|Q\right|$ for quantum SK spin glass for four system sizes $N$ at $T=0$ and $\mathrm{\Gamma }=0.3$. (a) The area under the $P\left(\right|Q\left|\right)$ curves for each $N$ is normalized to 1; (b) the peaks of all $P\left(\right|Q\left|\right)$ curves are normalized to their maximum values. In both plots, the extrapolated values of $P\left(\right|Q\left|\right)$ for large $N$ (see insets) are plotted against $\left|Q\right|$ as well. Insets: Typical extrapolations of $P\left(\right|Q\left|\right)$, for $\left|Q\right|=0.0$ and 0.1, and $W$ with $N$.

Figure 5

(a) Schematic of the quantum SK model (cf. [12]). Here SG and PM denote the spin glass and paramagnetic phases, respectively. Our numerical simulations indicate that the spin glass phase is further divided into two regions: the ergodic SG(E) region and the nonergodic SG(NE) region. The filled red circle on the SG-PM phase boundary line indicates the quantum-classical crossover point in the critical behavior of the model [12, 13]. We anneal by tuning both $T$ and $\mathrm{\Gamma }$ following different linear paths passing through both SG(E) and SG(NE) regions (e.g., as indicated by the two diagonal straight lines). (b) Variation of annealing time $\tau$ with $S$, the length of the arc along the phase boundary starting from the pure quantum critical point ($T=0, \mathrm{\Gamma }\simeq 1.6$), up to the crossing point of the annealing line on the phase boundary. One does not get any system-size dependence of $\tau$ up to $S\simeq 0.55$ (corresponding to $T\simeq 0.46, \mathrm{\Gamma }\simeq 1.35$; indicated in both panels by vertical arrows). As the annealing line passes through the SG(NE) region (beyond the quantum-classical crossover point), $\tau$ is seen to acquire a strong system-size dependence.