Ising spin glasses in dimension five

Ising spin-glass models with bimodal, Gaussian, uniform, and Laplacian interaction distributions in dimension five are studied through detailed numerical simulations. The data are analyzed in both the finite-size scaling regime and the thermodynamic limit regime. It is shown that the values of critical exponents and of dimensionless observables at criticality are model dependent. Models in a single universality class have identical values for each of these critical parameters, so Ising spin-glass models in dimension five with different interaction distributions each lie in different universality classes. This result confirms conclusions drawn from measurements in dimension four and dimension two.

Figure 1

Gaussian 5D ISG model. Even $L$ Binder cumulants $g(\beta ,L)$ against inverse temperature $\beta ,L=4, 6,8$, and 10 (top to bottom on the left).

Figure 2

Gaussian 5D ISG model. Even $L$ data for the observable $h(\beta ,L)$ against $\beta ,L=4,6,8$, and 10 (top to bottom on the left).

Figure 3

Gaussian 5D ISG. $P_{\mathrm{skew}}(\beta ,L)$ against $1/L$ for fixed $\beta ,\beta =0.424,0.422,0.420, 0.419,0.418,0.416$, and 0.414 (top to bottom). Dashed line, estimated criticality.

Figure 4

Gaussian 5D ISG. Peak location $y={\beta }_{max}$ against inverse peak height $x=1/D_{max}$ for the derivatives $\partial P_W/\partial \beta ,\partial h/\partial \beta$, and $\partial g/\partial \beta$ (top to bottom). Sizes $L=3,4,5,6,7,8,9$, and 10 (increasing to the left). For each observable the points extrapolate to $y\left(x\right)={\beta }_c$ at the intercept; see text.

Figure 5

Gaussian 5D ISG. Effective exponent $\gamma (\tau ,L)$ as a function of $\tau$ with ${\beta }_c=0.419$. Points, simulation data for $L=10,9,8,7,6$, 5, and 4 (left to right); dashed curve, fit; continuous (green) curve on the right, calculated by summing the HTSE tabulation of [12].

Figure 6

Gaussian 5D ISG. Effective exponent $\nu (\tau ,L)$ as function of $\tau$ with ${\beta }_c=0.419$. Points, simulation data for $L=10,9,8,7,6,5$, and 4 (left to right); continuous (green) curve, fit.

Figure 7

Gaussian 5D ISG. The ratio $\chi (\beta ,L)/{[\xi (\beta ,L)/\beta ]}^2$ against $\xi (\beta ,L)/\beta$ for $L=10,9,8,7,6,5$, and 4 (right to left). Continuous green curve, fit. No value is assumed for ${\beta }_c$.

Figure 8

Gaussian 5D ISG. Privman-Fisher-like scaling of the $\chi (\beta ,L)$ data following the form used in [26], with assumed parameters ${\beta }_c=0.419, \nu =0.72,\eta =-0.19$, and no adjustments. $L=10$, pink squares; $L=8$, black circles; $L=6$, red triangles; $L=4$, blue inverted triangles. Upper branch, $\beta >{\beta }_c$; lower branch, $\beta <{\beta }_c$.

Figure 9

Bimodal 5D ISG. $\xi (\beta ,L)/L$ against $1/L$ for fixed $\beta ,\beta =0.395,0.392,0.390, 0.389,0.388,0.387,0.385$, and 0.382 (top to bottom). $L=10,9,8,7,6,5,4$, and 3 (right to left). Dashed line, estimated criticality, $\beta =0.3885$.

Figure 10

Bimodal 5D ISG. $P_W(\beta ,L)$ against $1/L$ for fixed $\beta ,\beta =0.385,0.387,0.388, 0.389,0.390$, and 0.392 (top to bottom). $L=10,9,8, 7,6,5,4$, and 3 (left to right). Dashed line, estimated criticality, $\beta =0.3885$.

Figure 11

Bimodal 5D ISG. $g(\beta ,L)$ against $1/L$ for fixed $\beta ,\beta =0.392,0.390,0.389, 0.388,0.387,0.385$ (top to bottom). $L=10,9,8, 7,6,5,4,3$ (left to right). Dashed line, estimated criticality, $\beta =0.3885$.

Figure 12

Bimodal 5D ISG. $h(\beta ,L)$ against $1/L$ for fixed $\beta ,\beta =0.392,0.390,0.389, 0.388,0.387,0.385$ (top to bottom). $L=10,9,8, 7,6,5,4,3$ (left to right). Dashed line, estimated criticality, $\beta =0.3885$.

Figure 13

Bimodal 5D ISG. Peak location $y={\beta }_{max}$ against inverse peak height $x=1/D_{max}$ for the derivative sets $\partial h(\beta ,L)/\partial \beta , \partial P_W(\beta ,L)/\partial \beta ,\partial P_{\mathrm{skew}}(\beta ,L)/\partial \beta$, and $\partial g(\beta ,L)/\partial \beta$ (top to bottom). Sizes $L=3,4, 5,6,7,8,9$, and 10 (increasing to the left). For each observable the points extrapolate to $y\left(x\right)={\beta }_c$ at the intercept; see text.

Figure 14

Bimodal 5D ISG. Effective exponent $\gamma (\tau ,L)$ as a function of $\tau$ with ${\beta }_c=0.3885$. Points, simulation data for $L=10,9,8,7,6$, 5 (left to right); continuous (blue) curve on the right, calculated by summing the HTSE tabulation of [12]; dashed line, fit.

Figure 15

Bimodal 5D ISG. Effective exponent $\nu (\tau ,L)$ as function of $\tau$ with ${\beta }_c=0.3885$. Points, simulation data for for $L=10,9,8,7$, 6, and 5 (left to right); red arrow, exact limit; dashed line, fit.

Figure 16

Bimodal 5D ISG. The ratio $\chi (\beta ,L)/{[\xi (\beta ,L)/\beta ]}^2$ against $\xi (\beta ,L)/\beta$ for $L=10,9,8,7,6,5$, and 4 (right to left). Continuous (green) curve, fit. No value is assumed for ${\beta }_c$.

Figure 17

Bimodal 5D ISG. Privman-Fisher-like scaling of the $\chi (\beta ,L)$ data following the form used in [26], with assumed parameters ${\beta }_c=0.3885, \nu =0.77,\eta =-0.25$ and no adjustments. $L=10$, pink squares; $L=8$, black circles; $L=6$, red triangles; $L=4$, blue inverted triangles. Upper branch, $\beta >{\beta }_c$; lower branch, $\beta <{\beta }_c$.

Figure 18

Uniform 5D ISG. The Binder cumulant $g(\beta ,L)$ against $1/L$ for fixed $\beta ,\beta =0.405, 0.4025,0.400,0.3975$, and 0.395 (top to bottom). $L=9, 8,7,6,5,4$, and 3 (left to right). Dashed line, estimated criticality, $\beta =0.400$.

Figure 19

Uniform 5D ISG. The observable $h(\beta ,L)$ against $1/L$ for fixed $\beta ,\beta =0.405, 0.4025,0.4000,0.3975,0.3950$ (top to bottom). $L=9, 8,7,6,5,4,3$ (left to right). Dashed line, estimated criticality, $\beta =0.400$.

Figure 20

Uniform 5D ISG. Peak location $y={\beta }_{max}$ against inverse peak height $x=1/D_{max}$ for the derivative sets $\partial h(\beta ,L)/\partial \beta$ (top) and $\partial g(\beta ,L)/\partial \beta$ (bottom). Sizes, $L=3,4, 5,6,7,8$, and 9 (increasing to the left). For both observables the points extrapolate to $y\left(x\right)={\beta }_c$ at the intercept; see text.

Figure 21

Uniform 5D ISG. Effective exponent $\gamma (\tau ,L)$ as function of $\tau$ with ${\beta }_c=0.400$. Points, simulation data for $L=9,8,7,6,5$, and 4 (left to right). Red arrow, exact limit; continuous (green) curve, fit; continuous (red) curve on the right, almost hidden under the fit curve, calculated by summing the HTSE tabulation of [12].

Figure 22

Uniform 5D ISG. Effective exponent $\nu (\tau ,L)$ as function of $\tau$ with ${\beta }_c=0.400$. Points, simulation data for for $L=9,8,7,6$, and 5 (left to right). Red arrow, exact limit; continuous (green) curve, fit.

Figure 23

Uniform 5D ISG. The ratio $\chi (\beta ,L)/{[\xi (\beta ,L)/\beta ]}^2$ against $\xi (\beta ,L)/\beta$ for $L=9,8,7,6,5,4$, and 3 (right to left). Continuous (green) curve, fit. No value is assumed for ${\beta }_c$.

Figure 24

Uniform 5D ISG. Privman-Fisher-like scaling of the $\chi (\beta ,L)$ data following the form used in [26], with assumed parameters ${\beta }_c=0.3885, \nu =0.77,\eta =-0.25$, and no adjustments. $L=9$, pink squares; $L=8$, black circles; $L=6$, red triangles; $L=4$, blue inverted triangles. Upper branch, $\beta >{\beta }_c$; lower branch, $\beta <{\beta }_c$.

Figure 25

Laplacian 5D ISG. The parameter $g(\beta ,L)$ against $1/L$ for fixed $\beta ,\beta =0.4600, 0.4575,0.4550,0.4525,0.4500$ (top to bottom). $L=8, 7,6,5,4,3$ (left to right). Dashed line, estimated criticality, $\beta =0.455$.

Figure 26

Laplacian 5D ISG. The parameter $h(\beta ,L)$ against $1/L$ for fixed $\beta ,\beta =0.4600, 0.4575,0.4550,0.4525,0.4500$ (top to bottom). $L=8, 7,6,5,4,3$ (left to right). Dashed line, estimated criticality, $\beta =0.455$.

Figure 27

Laplacian 5D ISG. The parameter $P_W(\beta ,L)$ against $1/L$ for fixed $\beta ,\beta =0.450, 0.4525,0.455,0.4575$, and 0.460 (top to bottom). $L=8, 7,6,5,4$, and 3 (left to right). Dashed line, estimated criticality, $\beta =0.455$.

Figure 28

Laplacian 5D ISG. Peak location $y={\beta }_{max}$ against inverse peak height $x=1/D_{max}$ for the derivative sets $\partial W_q(\beta ,L)/\partial \beta , \partial h(\beta ,L)/\partial \beta$, and $\partial g(\beta ,L)/\partial \beta$ (top to bottom). Sizes $L=3,4, 5,6,7$, and 8 (increasing to the left). For each observable, the points extrapolate to $y\left(x\right)={\beta }_c$ at the intercept; see text.

Figure 29

Laplacian 5D ISG. Effective exponent $\gamma (\tau ,L)$ as a function of $\tau$ with ${\beta }_c=0.455$. Points, simulation data for $L=8,7,6,5$, and 4 (left to right); red arrow, exact limit; continuous (green) curve, fit.

Figure 30

Laplacian 5D ISG. Effective exponent $\nu (\tau ,L)$ as function of $\tau$ with ${\beta }_c=0.455$. Points, simulation data for for $L=8,7,6,5$, and 4 (left to right); red arrow, exact limit; continuous (green) curve, fit.

Figure 31

Laplacian 5D ISG. The ratio $\chi (\beta ,L)/{[\xi (\beta ,L)/\beta ]}^2$ against $\xi (\beta ,L)/\beta ,L=8,7,6,5$, and 4 (right to left). Continuous (green) curve, fit. No value is assumed for ${\beta }_c$.

Figure 32

Laplacian 5D ISG. Privman-Fisher-like scaling of the $\chi (\beta ,L)$ data following the form used in [26], with assumed parameters ${\beta }_c=0.455, \nu =0.69,\eta =-0.21$, and no adjustments. $L=9$, pink squares; $L=8$, black squares; $L=6$, red circles; $L=5$, green triangles; $L=4$, blue inverted triangles. Upper branch, $\beta >{\beta }_c$; lower branch, $\beta <{\beta }_c$.

Figure 33

Bimodal 5D ISG model. The sample specific grand mean standard error $\widehat{\epsilon }$ compared to the random sample grand mean standard error $\widehat{\sigma }$ for observable $q^2$, plotted versus inverse temperature $\beta ,L=4,6, 8,10$ (left to right).

Figure 34

Bimodal 5D ISG model. The grand mean standard error $\widehat{\sigma }$ of observable $q^2$ for a random sample set compared to its grand mean $\left[\langle q^2\rangle \right]$, plotted versus inverse temperature $\beta ,L=4,6,8,10$ (left to right). Red line at ${\beta }_c=0.3885$.