Dynamic response function in Ising systems below $T_c$

With the aid of Nelson’s method we derive expressions for the frequency-, momentum-, and temperature-dependent response function below a critical point in the Ising-type system. The scaling function is given within the renormalization-group formalism at one-loop order for zero external field. The comparison with the corresponding expressions above $T_c$ and with the mean-field approximation is made. We also discuss the dynamic correlation function, focusing on the deviations from the Gaussian expression. The comparison with field-theoretical calculations is also given.

Figure 1

The diagrams contributing to $O$($\epsilon$) to the self-energy below $T_c$.

Figure 2

The real part of the susceptibility for $q=0$ and $\omega /\Gamma =0.01$. The dashed line is the mean-field approximation (MFA).

Figure 3

The real part of the susceptibility for $q=0.1$ and $\omega /\Gamma =0.05$. The dashed line is the mean-field approximation (MFA).

Figure 4

The real part of the susceptibility for $q=0.1$ and $\omega /\Gamma =0.01$. The dashed line is the mean-field approximation (MFA).

Figure 5

The real part of the susceptibility for $q=0.01$ and $\omega /\Gamma =0,0.001,0.002,0.005$, and $0.01$.

Figure 6

The temperature dependence of the dynamic correlation function $C(q,\omega ,\tau )$ for$\epsilon =1$ and values of $q$ and $\omega /\Gamma$ shown in Fig. 7.

Figure 7

An enlargement of the previous figure near the critical point, showing small maxima above $T_c$.

Figure 8

Deviations of the scaling function $\mathrm{Ĉ}(x,y)$ from the Van Hove shape ${\mathrm{Ĉ}}_0(x,y)$ for the high-temperature phase shown as a function of the reduced frequency $y$ for the reduced momentum $x$ equal 0, 0.5, 1, and 1.25.

Figure 9

Deviations of the scaling function $\mathrm{Ĉ}(x,y)$ from the Van Hove shape ${\mathrm{Ĉ}}_0(x,y)$ for the low-temperature phase shown as a function of the reduced frequency $y$ for the reduced momentum $x$ equal to 0, 0.5, 1, and 1.25.

Figure 10

The scaling function $\mathrm{Ĉ}(x,y)$ as a function of the reduced frequency $y$ for several values of the reduced momentum $x$ in the low-temperature phase. The solid line is the result obtained in this paper (PE); dot-dashed line corresponds to the Calabrese et al. (Ref. 5) approach (CMPV); the dashed curve is the result based on Täuber and Schwabl’s work (Refs. 18, 41) (TS). The dotted line is the Gaussian approximation.

Figure 11

Deviations of the scaling function $\mathrm{Ĉ}(x,y)$ from the Van Hove shape ${\mathrm{Ĉ}}_0(x,y)$ for different approaches in the low-temperature phase shown as a function of the reduced frequency $y$ for $x=0.005$. Solid line: PE; dashed: TS; dot-dashed: CMPV approximation.

Figure 12

Deviations of the scaling function $\mathrm{Ĉ}(x,y)$ from the Van Hove shape for different approaches in the low-temperature phase shown as a function of the reduced frequency $y$ for $x=0.5$. Solid line: PE; dashed: TS; dot-dashed: CMPV approximation.

Figure 13

Deviations of the scaling function $\mathrm{Ĉ}(x,y)$ from the Van Hove shape for the different approaches in the low-temperature phase shown for $x=2$. Solid line: PE; dashed: TS; dot-dashed: CMPV approximation.

Figure 14

Deviations of the scaling function $\mathrm{Ĉ}(x,y)$ from the conventional shape for the different approaches in the low-temperature phase shown for $x=10$. Solid line: PE; dashed: TS; dot-dashed: CMPV approximation.