Sherrington-Kirkpatrick model for spin glasses: Solution via the distributional zeta function method

We discuss the Sherrington-Kirkpatrick mean-field version of a spin glass within the distributional zeta function method (DZFM). In the DZFM, since the dominant contribution to the average free energy is written as a series of moments of the partition function of the model, the spin-glass multivalley structure is obtained. Also, an exact expression for the saddle points corresponding to each valley and a global critical temperature showing the existence of many stables or at least metastable equilibrium states is presented. Near the critical point, we obtain analytical expressions of the order parameters that are in agreement with phenomenological results. We evaluate the linear and nonlinear susceptibility and we find the expected singular behavior at the spin-glass critical temperature. Furthermore, we obtain a positive definite expression for the entropy and we show that ground-state entropy tends to zero as the temperature goes to zero. We show that our solution is stable for each term in the expansion. Finally, we analyze the behavior of the overlap distribution, where we find a general expression for each moment of the partition function.

Figure 1

Numerical solutions of (32) and (33) for $J/k_{B}T=1$, $J_0/k_{B}T=0.1$, $h=0$, and different $k$. The continuum blue curve describes the behavior of $q_k$ and the dashed red curve describes $m_k$. We have a transition in these parameters for a given $k$.

Figure 2

Behavior of (38) for different $k$, $k_{B}T/J\ll 1$, and large $N$.

Figure 3

Numerical solutions up to $O\left({\beta }^2\right)$ for (32) near the critical point for different values of $k$. We can identify the critical temperature at $\beta J=1$.

Figure 4

Numerical solutions for (32) and for the complete expansion. We can identify the critical temperature at $\beta J=1$. Inset shows the behavior with this expansion near and beyond the point $\beta J=1$.

Figure 5

Entropy for the numerical solutions for (32) and fixed $J\ll 1$.

Figure 6

Behaviors of (57) for (a) $J_0/J=2$, (b) $J_0/J=4$, and (c) $J_0/J=8$, and (16) for $J_0/J=16$. Each curve correspond to $k=1$ (indigo, dashed line), $k=2$ (blue, x marker), $k=3$ (dark cyan, diamond marker), $k=4$ (green, left-triangle marker), $k=5$ (yellow, dashed line, circle marker), $k=6$ (orange, -down-triangle marker), and $k=7$ (red, dashed line, star marker).

Figure 7

Behavior of (58) for $J_0/J=3$ and $k=3$ (dot-dashed line), $k=4$ (dashed line), and $k=5$ (continuous line) as functions of $k_{B}T/J$. Note the appearance of the discontinuity at the spin-glass critical temperature. Inset shows the typical behavior of the nonlinear susceptibility (59).

Figure 8

Behavior of (58) for $J_0/J=4$ (lime, circle), $J_0/J=5$ (orange, star), and $J_0/J=6$ (red, x) and $k=3$ (dot-dashed line), $k=4$ (dashed line), and $k=5$ (continuous line) in functions of $k_{B}T/J$. Note the appearance of the discontinuity at the spin-glass critical temperature. Inset shows the typical behavior of the nonlinear susceptibility (59).

Figure 9

Regions of validity of regularized free energy (40) and subsequent regularized quantities for (a) $J=1$, (b) $J=2$, (c) $J=3$, and (d) $J=4$.

Figure 10

Numerical solutions of (32) for (a) $k=0$ (replica symmetric ansatz), (b) $k=1$, (c) $k=2$, (d) $k=3$, (e) $k=4$, and (f) $k=5$. Here, the curves projected on each plane are level curves used to delimit the stability regions.