Spin glasses in a field: Three and four dimensions as seen from one space dimension

We study the existence of a line of transitions of an Ising spin glass in a magnetic field—known as the de Almeida-Thouless line—using one-dimensional power-law diluted Ising spin-glass models. We choose the power-law exponent to have values that approximately correspond to three- and four-dimensional nearest-neighbor systems and perform a detailed finite-size scaling analysis of the data for large linear system sizes, using both a new approach proposed recently [Phys. Rev. Lett. 103, 267201 (2009)], as well as traditional approaches. Our results for the model corresponding to a three-dimensional system are consistent with there being no de Almeida-Thouless line, although the new finite-size scaling approach does not rule one out. For the model corresponding to four space dimensions, the new and traditional finite-size scaling analyses give conflicting results, indicating the need for a better understanding of finite-size scaling of spin glasses in a magnetic field.

Figure 1

Data used in a standard FSS for $\sigma =0.784$, which is a proxy for four dimensions. The left column is for $H=0$, the right column for $H=0.1$. The top row shows the finite-size correlation length divided by $L$ as a function of temperature $T$ for different system sizes. The bottom row shows results for the scale-invariant spin-glass susceptibility, ${\chi }_{{}_{\mathrm{SG}}}/L^{2\sigma -1}$. The intersections seen in the left-hand column indicate a transition in zero field. The intersection temperatures (shown by the dashed vertical line) are $T_c\simeq 1.34$ from ${\xi }_L/L$ data in (a), and $T_c\simeq 1.36$ from ${\chi }_{\mathrm{SG}}/L^{2\sigma -1}$ data in (c). The small difference is likely due to sensitive corrections to scaling in the susceptibility (see Ref. 33 for details). By contrast, the lack of intersections in the right-hand column indicates no transition in a magnetic field.

Figure 2

Data used in the modified FSS analysis for $\sigma =0.784$. As in Fig. 1, the left column is for $H=0$ and the right column for $H=0.1$. The top row shows the inverse spin-glass susceptibility ${\chi }_{{}_{\mathrm{SG}}}{(k\to 0)}^{-1}$ extrapolated from results for $k\ne 0$. The bottom row shows data for the dimensionless ratio $R_{12}$ defined in Eq. (11). The dashed line for ${\chi }_{{}_{\mathrm{SG}}}{(k\to 0)}^{-1}$ [top row, panels (a) and (b)] indicates the extrapolated transition temperature $T_c$ determined from the fits in Fig. 3. The $R_{12}$ data for $H=0$ [panel (c)] show intersections but the trend of intersection temperatures is not smooth for this range of sizes. The dotted vertical line shows the transition temperature estimated from the intersections of ${\chi }_{{}_{\mathrm{SG}}}/L^{2\sigma -1}$ in Fig. 1 ($T_c=1.36$) and so is just a guide to the eye. The $R_{12}$ data for $H=0.1$ [panel (d)] show no sign of a transition in the range of temperatures studied.

Figure 3

Values of $T^{☆}\left(L\right)$, the temperature where ${\chi }_{{}_{\mathrm{SG}}}{(k\to 0)}^{-1}$ goes through zero, for different system sizes for $H=0$ (left) and $H=0.1$ (right) for $\sigma =0.784$. The data are fitted to Eq. (10) with $T_c$, $A$, and $\lambda$ as adjustable parameters. In both cases a finite value of $T_c$ (the intercept) is found. For $H=0$ the goodness-of-fit parameter[39, 40] $Q$ is not very small, indicating a satisfactory fit. For $H=0.1$ the value of $Q$ is smaller because the point for the largest size is well below the fit.

Figure 4

Data used in the standard FSS for $\sigma =0.896$, which is a proxy for three space dimensions. The left column is for $H=0$ and the right column for $H=0.1$. The top row shows the finite-size correlation length divided by $L$ as a function of $T$ for different system sizes. The bottom row shows results for the scale-invariant spin-glass susceptibility, ${\chi }_{{}_{\mathrm{SG}}}/L^{2\sigma -1}$. The intersections in the data for ${\chi }_{{}_{\mathrm{SG}}}/L^{2\sigma -1}$ for $H=0$ [panel (c)] indicate a transition at $T_c\simeq 0.795$, which is shown by a dashed vertical line. The zero-field results for ${\xi }_L/L$ [panel (a)] show stronger finite-size effects and the evidence for intersections is not so clear. The same result was found recently by Baños et al.,[20] who were able to study somewhat larger sizes for almost the same model. Their FSS analysis of all their data supports a zero-field transition. In panel (a) the dotted vertical line indicates the transition temperature found for the data in panel (c) and so is just a guide to the eye. The data in panel (a) are consistent with intersections tending to this value for $L\to \infty$. By contrast, the lack of intersections in the right-hand column indicates no transition in a magnetic field.

Figure 5

Data used in the modified FSS analysis for $\sigma =0.896$. As in Fig. 4, the left column is for $H=0$ and the right column for $H=0.1$. The top row shows the inverse spin-glass susceptibility ${\chi }_{{}_{\mathrm{SG}}}(k\to 0)$ extrapolated from results for $k\ne 0$. The bottom row shows data for the dimensionless ratio $R_{12}$ defined in Eq. (11). The dashed vertical line in panel (a) marks the value for $T_c$ obtained by extrapolating the data for $T^{☆}\left(L\right)$ [where ${\chi }_{{}_{\mathrm{SG}}}{(k\to 0)}^{-1}$ vanishes] according to Eq. (10); see Fig. 6. The result is $T_c=0.81\left(3\right)$, in agreement with the value of $0.795$ from the intersections of the scaled ${\chi }_{{}_{\mathrm{SG}}}$ in Fig. 4c. The data for $R_{12}$ in zero field in panel (c) show intersections, although the intersection temperatures do not vary smoothly. The dotted vertical line shows $T_c$ obtained from the scaled ${\chi }_{{}_{\mathrm{SG}}}$ and is just a guide to the eye. The results for $R_{12}$ for $H=0.1$ in panel (d) do not provide clear evidence for a transition.

Figure 6

Values of $T^{☆}\left(L\right)$, the temperature where ${\chi }_{{}_{\mathrm{SG}}}{(k\to 0)}^{-1}$ goes through zero, for different system sizes for $H=0$ (left) and $H=0.1$ (right) for $\sigma =0.896$. The data are fitted to Eq. (10) with $T_c$, $A$, and $\lambda$ as adjustable parameters. The error bars on the parameters in the fit to the $H=0.1$ data are very large (this is discussed in the text and in the caption to Fig. 8). Note that the $H=0.1$ data are consistent with $T_c=0$. The goodness-of-fit parameter[40] indicates a satisfactory fit in both cases.

Figure 7

Log-log plot of the data for $T^{☆}\left(L\right)$ for $\sigma =0.896$ and $H=0.1$ assuming $T_c=0$. The fit works very well according to the goodness-of-fit parameter $Q$, which is $0.81$.

Figure 8

Cumulative distribution of the transition temperature $T_c$ obtained from a bootstrap analysis of the data for $\sigma =0.896$ and $H=0.1$. The estimate of $T_c$ from the global fit ($T_c=-0.37$, indicated by the dashed vertical line) agrees well with the median (50th percentile) of the bootstrap estimates, which is indicated by a horizontal dashed line. Also indicated by horizontal dashed lines are the 16th and 84th percentiles which would correspond to one standard deviation if the distribution of $T_c$'s were Gaussian, which is clearly not the case. Only 30$\%$ of the bootstrap fits have a positive $T_c$.