Classical-to-quantum crossover in the critical behavior of the transverse-field Sherrington-Kirkpatrick spin glass model

We study the critical behavior of the Sherrington-Kirkpatrick model in transverse field (at finite temperature) using Monte Carlo simulation and exact diagonalization (at zero temperature). We determine the phase diagram of the model by estimating the Binder cumulant. We also determine the correlation length exponent from the collapse of the scaled data. Our numerical studies here indicate that critical Binder cumulant (indicating the universality class of the transition behavior) and the correlation length exponent cross over from their “classical” to “quantum” values at a finite temperature (unlike the cases of pure systems, where such crossovers occur at zero temperature). We propose a qualitative argument supporting such an observation, employing a simple tunneling picture.

Figure 1

Monte Carlo results for the Binder cumulant ($g$) plotted as function of temperature $T$ and transverse field $\mathrm{\Gamma }$ are shown: (a) for classical SK model (at $\mathrm{\Gamma }=0$) and (c) and (e) for $T=0.65$ and $0.60$, respectively. The crossing points give the estimate for $T_c$ or ${\mathrm{\Gamma }}_c$. The statistical errors are indicated by the symbol sizes. Figures (b), (d), and (f) show the collapses of $g$ curves of (a), (c), and (e), respectively, when the variations of $g$ are plotted against $[T-T_c]N^{x_T}$ or $[\mathrm{\Gamma }-{\mathrm{\Gamma }}_c]N^{x_{\mathrm{\Gamma }}}$ [see Eq. (4)]. The scaling collapses give the values $x_T$ or $x_{\mathrm{\Gamma }}=0.31\pm 0.02$.

Figure 2

Monte Carlo results for the Binder cumulant ($g$) plots with transverse field $\mathrm{\Gamma }$ at temperatures $0.30$ and $0.25$ are shown in (a) and (c), respectively. The statistical errors are of the order of the symbol sizes. Figures (b) and (d) show the collapses of $g$ curves in (a) and (c), respectively. Again the variations of $g$ are plotted according to the scaling relation Eq. (4) and the collapses give the value $x_{\mathrm{\Gamma }}=0.50\pm 0.02$.

Figure 3

(a) The plot shows the Monte Carlo results for the variation of Binder cumulant ($g$) as a function of temperature $T$ at the transverse field $\mathrm{\Gamma }=1.5$. (b) The plot shows collapse of $g$ curves in (a) with $x_T=0.49$. The statistical errors of the data points are of the order of the symbol sizes.

Figure 4

The plot (a) shows the variation of Binder cumulant $g$ as a function of $\mathrm{\Gamma }$ for different system sizes for quantum SK model at $T=0$ (exact diagonalization results). The larger system sizes intersect at higher values of $\mathrm{\Gamma }$ signifying finite-size effect of the system. (b) The Binder cumulant curves for different system sizes ($N$) collapse following the scaling fit [to Eq. (4) for $M=0$] with an estimated ${\mathrm{\Gamma }}_c^e=1.62$ and exponent $x_{\mathrm{\Gamma }}=0.5$.

Figure 5

The plot shows critical transverse field $\left[{\mathrm{\Gamma }}_c(N,N^{\prime })\right]$ and critical Binder cumulant $\left[g_c(N,N^{\prime })\right]$ as a function of ${\left(N{N}^{\prime }\right)}^{-x_{\mathrm{\Gamma }}/2}$ and $1/\sqrt{N{N}^{\prime }}$, respectively, where $N$ and $N^{\prime }$ are two system sizes, obtained from exact diagonalization: (a) The extrapolated value of ${\mathrm{\Gamma }}_c$ is found to be $1.62$ and (b) $g_c$ tends to a null value for a infinite size system. For these two plots best fit line is also shown.

Figure 6

Consolidated phase diagram of the SK spin glass model in transverse field, obtained from the Monte Carlo simulation and exact diagonalization. The statistical errors are indicated if they are more than the symbol sizes. Here SG and PM denote, respectively, the spin glass and paramagnetic phases. The points at $T=0$ and $\mathrm{\Gamma }=0$ correspond to purely quantum and classical cases, respectively. The obtained critical behaviors are indicated ($g_c\simeq 0, \nu \simeq 1/4$ for low $T$-high $\mathrm{\Gamma }$ region, and $g_c\simeq 0.22, \nu \simeq 1/2$ for high $T$-low $\mathrm{\Gamma }$ region). The crossover point is around $T\simeq 0.45$ and $\mathrm{\Gamma }\simeq 1.33$.