Schwinger-Dyson equations and disorder

Using simple models in $D=0+0$ and $D=0+1$ dimensions we construct partition functions and compute two-point correlations. The exact result is compared with saddle point approximation and solutions of Schwinger-Dyson equations. When integrals are dominated by more than one saddle point we find Schwinger-Dyson equations do not reproduce the correct results unless the action is first transformed into dual variables.

Figure 1

Comparison of perturbative (dashed lines), Schwinger-Dyson (dashed-dotted line), and exact, numerical evaluation of the two-point correlation (solid line) for the action with a unique classical vacuum ($ϵ=1$). On this scale the NLO SDE solution is indistinguishable and coincides with the exact result.

Figure 2

Saddle-point approximation computed to a varying order in $\lambda$ (dashed lines) compared with the exact result for $ϵ=-1$ obtained by numerical integration (solid line). The $O({\lambda }^4)$ perturbative results is indistinguishable form the exact result. For easier comparison we subtracted the leading $6/\lambda$ term from $⟨x^2⟩$. On this scale the solution of the SDE tends to $-6/\lambda \to -\infty$ as $\lambda \to 0$.

Figure 3

A typical action used here in a model study of multiple, quasidegenerate vacua. For large $\Lambda$ and/or small $g$ the number of minima contributing to the partition function is proportional to ${\Lambda }^2/g^2$. For the case shown in this figure, besides the central minimum, there are additional $N=32$ local minima.

Figure 4

Comparison between exact (solid line) and Schwinger-Dyson, Eq. (24), approximation (dashed line) to $⟨x^2⟩$ for the action given by Eq. (18). We used $g^2/{\Lambda }^2=0.01$ (cf. Fig. 3). The dashed-dotted lines correspond to the two limits given by Eqs. (19, 20), were, as discussed in the text Schwinger-Dyson approximation becomes exact. For the range of couplings shown, the saddle point approximation of Eq. (25) is indistinguishable from the exact result and involves summation over $N=33$ saddle points.

Figure 5

Comparison between exact, numerical computation and the saddle point approximation to $⟨x^2⟩$ for different values of $g^2/{\Lambda }^2$. As $\Lambda$ increases for fixed $g$ so does the number of saddle points. The three cases shown correspond to $N=3$, 7, 33 points, respectively [cf. Eq. (25)]. As $\Lambda$ increases, saddle points become degenerate and Eq. (25) becomes an increasingly better approximation.

Figure 6

Comparison between exact (dashed line) and approximate evaluation of the correlation function as a function of temperature. In the approximation Eq. (29) only the $q=0$ term is retained.