Bimodal and Gaussian Ising spin glasses in dimension two

An analysis is given of numerical simulation data to size $L=128$ on the archetype square lattice Ising spin glasses (ISGs) with bimodal ($\pm J$) and Gaussian interaction distributions. It is well established that the ordering temperature of both models is zero. The Gaussian model has a nondegenerate ground state and thus a critical exponent $\eta \equiv 0$, and a continuous distribution of energy levels. For the bimodal model, above a size-dependent crossover temperature $T^{*}\left(L\right)$ there is a regime of effectively continuous energy levels; below $T^{*}\left(L\right)$ there is a distinct regime dominated by the highly degenerate ground state plus an energy gap to the excited states. $T^{*}\left(L\right)$ tends to zero at very large $L$, leaving only the effectively continuous regime in the thermodynamic limit. The simulation data on both models are analyzed with the conventional scaling variable $t=T$ and with a scaling variable ${\tau }_b=T^2/(1+T^2)$ suitable for zero-temperature transition ISGs, together with appropriate scaling expressions. The data for the temperature dependence of the reduced susceptibility $\chi ({\tau }_b,L)$ and second moment correlation length $\xi ({\tau }_b,L)$ in the thermodynamic limit regime are extrapolated to the ${\tau }_b=0$ critical limit. The Gaussian critical exponent estimates from the simulations, $\eta =0$ and $\nu =3.55\left(5\right)$, are in full agreement with the well-established values in the literature. The bimodal critical exponents, estimated from the thermodynamic limit regime analyses using the same extrapolation protocols as for the Gaussian model, are $\eta =0.20\left(2\right)$ and $\nu =4.8\left(3\right)$, distinctly different from the Gaussian critical exponents.

Figure 1

Gaussian 2D ISG. Logarithmic derivative of the specific heat $\partial ln{C}_v(T,L)/\partial \beta$ against $T$. Sizes $L=64,48,32,24,16,12$ top to bottom in the dip. Curve: extrapolation.

Figure 2

Bimodal 2D ISG. Logarithmic derivative of the specific heat $\partial ln{C}_v(T,L)/\partial \beta$ against $T$. Full points: simulation data $L=96,48,24,12,8$ (black, green, red, pink, cyan) top to bottom. Open points: Pfaffian data; red polygons $L=512$, blue right triangles $L=256$, black left triangles $L=128$, brown diamonds $L=64$ (all data from Ref. [5]), green down triangles $L=50$, red up triangles $L=24$, pink circles $L=12$, all data from Ref. [4]. Dashed diagonal red line $y=-3x$, green diagonal line $y=-4+2x$.

Figure 3

Gaussian 2D ISG. Derivative $\partial ln\chi (T,L)/\partial ln\xi (T,L)$ against $1/\xi (T,L)$ for $L=128,96,64,48,32,24$ left to right. In this and all following figures both Gaussian and bimodal, the color coding is black, pink, red, blue, green, brown, cyan, olive for $L=128,96,64,48,32,24,16,12$.

Figure 4

Bimodal 2D ISG. Derivative $\partial ln\chi (T,L)/\partial ln\xi (T,L)$ against $1/\xi (T,L)$ for $L=128,96,64,48,32,24$ left to right. Same color coding as in Fig. 3.

Figure 5

Gaussian 2D ISG. The derivative $\partial ln\chi (T,L)/\partial ln{\tau }_b$ against ${\tau }_b$. Sizes $L=128,96,64,48,32,24,16$ left to right. Same color coding as in Fig. 3. Dashed line: extrapolation. Red arrow: exact infinite temperature value. Blue arrow: Gaussian critical value.

Figure 6

Bimodal 2D ISG. The derivative $\partial ln\chi (T,L)/\partial ln{\tau }_b$ against ${\tau }_b$. Sizes $L=128,96,64,48,32,24,16$ left to right. Same color coding as in Fig. 3. Dashed line: extrapolation. Red arrow: exact infinite temperature value.

Figure 7

Gaussian 2D ISG. The derivative $\partial ln\left[T\xi (T,L)\right]/\partial ln{\tau }_b$ against ${\tau }_b$. Sizes $L=128,96,64,48,32,24$ left to right. Same color coding as in Fig. 3. Red curve: fit.

Figure 8

Bimodal 2D ISG. The derivative $\partial ln\left[T\xi (T,L)\right]/\partial ln{\tau }_b$ against ${\tau }_b$. Sizes $L=128,96,64,48,32,24,16$ left to right. Same color coding as in Fig. 3. Dashed line: extrapolation. Red arrow: exact infinite temperature value.

Figure 9

Gaussian 2D ISG. The derivative $\partial lng(T,L)/\partial ln{\tau }_b$ against ${\tau }_b$, where $g(T,L)$ is the Binder cumulant. $L=128,96,64,48,32,24$ left to right. Same color coding as in Fig. 3. Green curve: fit.

Figure 10

Bimodal 2D ISG. The derivative $\partial lng(T,L)/\partial ln{\tau }_b$ against ${\tau }_b$, where $g(T,L)$ is the Binder cumulant. Sizes $L=128,96,64,48,32,24$ left to right. Same color coding as in Fig. 3. Dashed line: extrapolation.

Figure 11

Gaussian 2D ISG. The derivative $\partial ln\chi (T,L)/\partial ln\left[T\xi \right(T,L\left)\right]$ against ${\tau }_b$. Sizes $L=128,96,64,48,32,24$ left to right. Same color coding as in Fig. 3. Dashed line: extrapolation. Red arrow: exact infinite temperature value. Blue arrow: Gaussian critical value.

Figure 12

Bimodal 2D ISG. The derivative $\partial ln\chi (T,L)/\partial ln\left[T\xi \right(T,L\left)\right]$ against ${\tau }_b$. Sizes $L=128,96,64,48,32,24,16$ left to right. Same color coding as in Fig. 3. Dashed line: extrapolation. Red arrow: exact infinite temperature value.

Figure 13

Gaussian 2D ISG. $L^{2}g\left({\tau }_b\right)$ against ${\tau }_b$. Sizes $L=128,96,64,48,32,24,16,12$ top to bottom. Same color coding as in Fig. 3. $g(T,L)$ is the Binder cumulant. Line slope $-3.5$.

Figure 14

Bimodal 2D ISG. $L^{2}g\left({\tau }_b\right)$ against ${\tau }_b$. Sizes $L=128,96,64,48,32,24,16,12$ top to bottom. Same color coding as in Fig. 3. $g(T,L)$ is the Binder cumulant. Line slope $-4.2$.

Figure 15

The ThL $\partial ln\chi \left({\tau }_b\right)/\partial ln{\tau }_b$ data for the Gaussian model from Fig. 5 (lower, black, squares) with the polynomial fit (lower, green, circles, smooth), and for the bimodal model from Fig. 6, (upper, red, squares) with the polynomial fit (upper, blue, circles, smooth).

Figure 16

The ThL $\partial ln\left[T\xi \left({\tau }_b\right)\right]/\partial ln{\tau }_b$ data for the Gaussian model from Fig. 7 (lower, black, squares) with the polynomial fit (lower, green, circles, smooth), and for the bimodal model from Fig. 8 (upper, red, squares) with the polynomial fit (upper, blue, circles, smooth).

Figure 17

The ThL $\partial lng\left({\tau }_b\right)/\partial ln{\tau }_b$ data for the Gaussian model from Fig. 9 (lower, black, squares) with the polynomial fit (lower, green, circles, smooth), and for the bimodal model from Fig. 10 (upper, red, squares) with the polynomial fit (upper, blue, circles, smooth).

Figure 18

The ThL $\partial ln\chi \left({\tau }_b\right)/\partial ln\left[T\xi \left({\tau }_b\right)\right]$ data for the Gaussian model from Fig. 11 (upper, black, squares) with the polynomial fit (upper, green, circles, smooth), and for the bimodal model from Fig. 12 (lower, red, squares) with the polynomial fit (lower, blue, circles, smooth).

Figure 19

The Gaussian model normalized correlation length $y=T\xi ({\tau }_b,L){\left({\tau }_b\right)}^{{\nu }_b}$ with ${\nu }_b=1.28$ against $x={\tau }_b$, see text. Sizes $L=128$ black squares, $L=64$ red circles, $L=32$ green triangles, $L16$ cyan diamonds, curves left to right. Diagonal dashed line $y\left(x\right)=0.65(1+0.61x)$. The exact infinite temperature limit is $x=1,y=1$.