Evidence of a second-order phase transition in the six-dimensional Ising spin glass in a field

The very existence of a phase transition for spin glasses in an external magnetic field is controversial, even in high dimensions. We carry out massive simulations of the Ising spin-glass in a field, in six dimensions (which, according to classical—but not generally accepted—field-theoretical studies, is the upper critical dimension). We obtain results compatible with a second-order phase transition and estimate its critical exponents for the simulated lattice sizes. The detailed analysis performed by other authors of the replica symmetric Hamiltonian, under the hypothesis of critical behavior, predicts that the ratio of the renormalized coupling constants remain bounded as the correlation length grows. Our numerical results are in agreement with this expectation.

Figure 1

Susceptibilities for the replicon ${\chi }_{\text{R}}$ (on the top), Eqs. (2) and (4), for the anomalous mode ${\chi }_{\text{A}}$ (on the middle), Eqs. (3) and (4), and for the ratio of the anomalous mode and the replicon one, ${\chi }_{\text{A}}/{\chi }_{\text{R}}$, (on the bottom), vs temperature $T$, as computed for our different system sizes in a magnetic field $h=0.075$. For $h=0$ one trivially shows that ${\chi }_{\text{R}}={\chi }_{\text{A}}$. Instead, for $h=0.075$, we find that ${\chi }_{\text{R}}$ rapidly grows with $L$ at low temperatures, while ${\chi }_{\text{A}}/{\chi }_{\text{R}}$ goes to zero. The different size dependence ensures that we are working far enough from the $h=0$ point in the dAT line. The two vertical lines are our estimates for the critical temperatures for $h=0.075$ (left vertical line) and for $h=0$ (right line). In particular, note from the middle panel that the values of ${\chi }_{\text{A}}$, as computed in $L=7$ and 8 lattices, are compatible: i.e., differences smaller than two standard deviations at, and below, our estimated critical temperature for $h=0.075$.

Figure 2

Probability density function of the overlap $P\left(q\right)$ for two different sizes: $L=6$ (left) and $L=8$ (right). We have computed $P\left(q\right)$ for the lowest available temperature for $L=8$ ($\beta \approx 0.389$) and the closest available temperature for $L=6$ ($\beta \approx 0.390$).

Figure 3

Second-moment correlation length ${\xi }_2$, Eq. (10), measured in units of the lattice size $L$ (on the top), and dimensionless ratio $R_{12}$, Eq. (12) (on the bottom), as a function of temperature $T$. Both quantities are shown as computed for the replicon propagator, Eq. (2), for all our lattice sizes at $h=0.075$. At the critical point, the curves for different sizes of the system intersect at the same $T$, meaning that both $\xi /L$ and $R_{12}$ are scale invariant at $T_c$. The presence of a crossing point for ${\xi }_2/L$ indicates that the anomalous behavior of the wave vector $\mathit{k}=\mathit{0}$ is less severe in space dimension $D=6$ than previously found at $D=4$ [24]. The difference between the crossing points found for both quantities should vanish as $L\to \infty$, since it is due to scaling corrections.

Figure 4

Plot of the three-, four-, and six-replica estimator of ${\lambda }_r$, Eqs. (C7), (C8), and (B9), respectively, as a function of the temperature for $L=6$ (bottom panel), for $L=7$ (middle panel), and for $L=8$ (top panel). We also plot the infinite temperature limits for three and four replicas.

Figure 5

Plot of the six-replica estimator of ${\lambda }_r$, Eq. (B9) and (C5), as a function of the temperature for all the lattice sizes $L$.

Figure 6

Three- and four-replica estimators of ${\lambda }_r$, Eqs. (C7) and (C8), as a function of the temperature. The six-replica estimator, shown in Appendix pp3, turned out to be compatible with—but much noisier than—the four-replica estimator.

Figure 7

Binder cumulant $g$ (on the top), Eq. (G2), and the second moment correlation length ${\xi }_2$ divided by the lattice size $L$ and by the logarithmic correction term ${(lnL)}^{1/6}$ (on the bottom) as a function of temperature for lattice sizes between $L=6$ and $L=9$. The presence of an intersection between the curves at temperatures $T_c^B$ and $T_c^{\xi }$ signals the existence of a continuous phase transition.

Figure 8

Second moment correlation length ${\xi }_2$ divided by the lattice size $L$ and by the logarithmic correction term ${(lnL)}^{1/6}$ (on the top) and the spin-glass susceptibility ${\chi }_{SG}$ divided by $L^{2-\eta }$ (using the mean-field result $\eta =0$) and by the logarithmic correction term ${(lnL)}^{2/3}$ (on the bottom) as a function of the argument of the scaling functions $f_{\xi }$ and $f_{\chi }$ for lattice sizes $L=8$ and $L=9$. Since Eqs. (G9) and (G10) hold asymptotically for large vales of $L$ we would expect the two curves to collapse into a single one only if the used values of the logarithmic correction critical exponents $\widehat{q}$ and $\widehat{a}$ are the correct one.

Figure 9

Plot of ${\chi }_{SG}\left(T_c\right)/L^2$ (on the top) and ${\xi }_2\left(T_c\right)/L$ (on the bottom) vs the lattice size $L$. The data show a monotonic growth, a strong indication of the presence of logarithmic corrections.

Figure 10

Plot of ${\chi }_{SG}\left(T_c\right)/\left[L^2{(lnL)}^{2/3}\right]$ (on the top) and ${\xi }_2\left(T_c\right)/\left[L{(lnL)}^{1/6}\right]$ (on the bottom) vs the lattice size $L$. Our data approach asymptotically to a constant value for $L\gg 1$, from which we conclude that they are compatible with the values $\widehat{a}=2/3$ and $\widehat{q}=1/6$. The constant behavior for the bottom plot was expected since the scaling of $\xi$ was used to determine the value of $T_c$.

Figure 11

Plot of ${\omega }_1$ (on the top), Eq. (B9), and the cummulant ${\mathrm{\Lambda }}_1$ (on the bottom), Eq. (H1), as a function of $T/T_c$.

Figure 12

Up: Plot of ${\omega }_2$ as a function of $T/T_c$. Down: Plot of ${\mathrm{\Lambda }}_2$ as a function of $T/T_c$.