Original electric-vertex formulation of the symmetric eight-vertex model on the square lattice is fully nonuniversal

The partition function of the symmetric (zero electric field) eight-vertex model on a square lattice can be formulated either in the original “electric” vertex format or in an equivalent “magnetic” Ising-spin format. In this paper, both electric and magnetic versions of the model are studied numerically by using the corner transfer matrix renormalization-group method which provides reliable data. The emphasis is put on the calculation of four specific critical exponents, related by two scaling relations, and of the central charge. The numerical method is first tested in the magnetic format, the obtained dependencies of critical exponents on the model's parameters agree with Baxter's exact solution, and weak universality is confirmed within the accuracy of the method due to the finite size of the system. In particular, the critical exponents $\eta$ and $\delta$ are constant as required by weak universality. On the other hand, in the electric format, analytic formulas based on the scaling relations are derived for the critical exponents ${\eta }_{\mathrm{e}}$ and ${\delta }_{\mathrm{e}}$ which agree with our numerical data. These exponents depend on the model's parameters which is evidence for the full nonuniversality of the symmetric eight-vertex model in the original electric formulation.

Figure 1

Admissible configurations of the eight-vertex model, with the corresponding notation of the Boltzmann vertex weight.

Figure 2

Mapping from the electric-vertex formulation with edge states $\varphi ,\chi ,\tau ,\kappa$ to the magnetic Ising representation with site spin variables ${\sigma }_1,{\sigma }_2,{\sigma }_3,{\sigma }_4$.

Figure 3

The CTMRG renormalization process. The density matrix $\rho$ is composed of four transfer matrices $C_1$, $C_2$, $C_3$, and $C_4$; each straight line represents $L$ matrix site indices which are either fixed (“free” lines adjacent to $C_1$ and $C_4$) or summed out [common lines of pairs $(C_1,C_2)$, $(C_2,C_3)$, and $(C_3,C_4)$]. The expansion process of the corner transfer matrix $C$ and the half-row transfer matrix $H$ from the previous iteration, see text.

Figure 4

The symmetric eight-vertex model with $c=d$: the dependence of the effective critical index ${\nu }^{\mathrm{eff}}$ on the inverse system size $1/L$, for four values of the critical vertex weight $b_c=0.15,0.25,0.35$, and 0.45. As $1/L$ goes to 0, the linear $a+b/L$ fittings of ${\nu }^{\mathrm{eff}}\left(L\right)$ give the asymptotic $\nu$ values denoted by crosses. Baxter's exact values are represented by dotted lines.

Figure 5

The symmetric eight-vertex model with $c\ne d$: the dependence of ${\nu }^{\mathrm{eff}}$ on $1/L$, for four choices of the critical vertex weights (4.2). The asymptotic $\nu$ values are denoted by crosses, Baxter's exact values are represented by dotted lines.

Figure 6

The symmetric eight-vertex model with $c=d$: the dependence of the effective critical index ${\eta }^{\mathrm{eff}}$ on the inverse system size $1/L$, for four values of the critical vertex weight $b_c=0.15,0.25,0.35$, and 0.45. In all cases, as $1/L\to 0,{\eta }^{\mathrm{eff}}$ goes to $1/4$.

Figure 7

The symmetric eight-vertex model with $c\ne d$: the dependence of ${\eta }^{\mathrm{eff}}$ on $1/L$, for four choices of the critical vertex weights (4.2). In all cases, ${\eta }^{\mathrm{eff}}$ goes asymptotically to $1/4$.

Figure 8

The symmetric eight-vertex model with $c=d$: the dependence of the effective critical exponent ${\delta }^{\mathrm{eff}}$ on the applied magnetic field $H$, for four values of the critical vertex weight $b_c=0.15,0.25,0.35$, and 0.45. The actual $\delta$ values, identified with the minimum points of the plots, are close to the exact Baxter's result, $\delta =15$.

Figure 9

The symmetric eight-vertex model with $c\ne d$: the dependence of ${\delta }^{\mathrm{eff}}$ on the magnetic field $H$, for four choices of the critical vertex weights (4.2).

Figure 10

The symmetric eight-vertex model with $c=d$: the effective exponent ${\beta }^{\mathrm{eff}}$ vs the deviation from the critical point $\mathrm{\Delta }T=T_c-T$, for four values of the critical vertex weight $b_c=0.15,0.25,0.35$, and 0.45. The $\beta$ values are identified with the maximum points of the plots, Baxter's exact values are represented by dotted lines. The inset documents an almost constant dependence of the rescaled exponent $\widehat{\beta }=\beta /\nu \sim 1/8$ on $b_c$.

Figure 11

The symmetric eight-vertex model with $c\ne d$: the dependence of ${\beta }^{\mathrm{eff}}$ on $\mathrm{\Delta }T$ for four choices of the critical vertex weights (4.2). Baxter's exact values are represented by dotted lines. The inset shows the dependence of $\widehat{\beta }$ on $b_c$.

Figure 12

The symmetric eight-vertex model with $c=d$: the dependence of the magnetic effective central charge $c^{\mathrm{eff}}$ on $1/L$, for four values of the critical vertex weight $b_c=0.15,0.25,0.35$, and 0.45. As $1/L\to 0$, all curves tend to the central charge $c=1$.

Figure 13

The symmetric eight-vertex model with $c=d$: the electric effective exponent ${\beta }_{\mathrm{e}}^{\mathrm{eff}}$ vs the deviation from the critical point $\mathrm{\Delta }T=T_c-T$, for four values of the critical vertex weight $b_c=0.15,0.25,0.35$, and 0.45. The ${\beta }_{\mathrm{e}}$ values are identified with the maximum points of the plots, Baxter's exact values are represented by dotted lines.

Figure 14

The symmetric eight-vertex model with $c\ne d$: the dependence of ${\beta }_{\mathrm{e}}^{\mathrm{eff}}$ on $\mathrm{\Delta }T$ for four choices of the critical vertex weights (4.2). Baxter's exact values are represented by dotted lines.

Figure 15

The symmetric eight-vertex model with $c=d$: the dependence of the effective critical exponent ${\eta }_{\mathrm{e}}^{\mathrm{eff}}$ on the inverse system size $1/L$, for four values of the critical vertex weight $b_c=0.15,0.25,0.35$, and 0.45. The suggested values obtained from (5.1) are represented by dotted lines.

Figure 16

The symmetric eight-vertex model with $c\ne d$: the dependence of ${\eta }_{\mathrm{e}}^{\mathrm{eff}}$ on $1/L$, for four choices of the critical vertex weights (4.2). The suggested values obtained from (5.1) are represented by dotted lines.

Figure 17

The symmetric eight-vertex model with $c=d$: the dependence of the effective critical exponent ${\delta }_{\mathrm{e}}^{\mathrm{eff}}$ on the the applied electric field $E$, for four values of the critical vertex weight $b_c=0.15,0.25,0.35$, and 0.45. The ${\delta }_{\mathrm{e}}$ values are identified with the minimum points of the plots, the suggested values (5.1) are represented by dotted lines.

Figure 18

The symmetric eight-vertex model with $c\ne d$: the dependence of ${\delta }_{\mathrm{e}}^{\mathrm{eff}}$ on the electric field $E$ for four choices of the critical vertex weights (4.2). The suggested values (5.1) are represented by dotted lines.

Figure 19

The symmetric eight-vertex model with $c=d$: the dependence of the effective electric central charge $c_{\mathrm{e}}^{\mathrm{eff}}$ on $1/L$ for four values of the critical vertex weight $b_c=0.15,0.25,0.35$, and 0.45. As $1/L\to 0$, all curves tend to $c_{\mathrm{e}}=1$.