Sums over geometries and improvements on the mean field approximation

The saddle points of a Lagrangian due to Efetov are analyzed. This Lagrangian was originally proposed as a tool for calculating systematic corrections to the Bethe approximation, a mean-field approximation which is important in statistical mechanics, glasses, coding theory, and combinatorial optimization. Detailed analysis shows that the trivial saddle point generates a sum over geometries reminiscent of dynamically triangulated quantum gravity, which suggests new possibilities to design sums over geometries for the specific purpose of obtaining improved mean-field approximations to $D$-dimensional theories. In the case of the Efetov theory, the dominant geometries are locally treelike, and the sum over geometries diverges in a way that is similar to quantum gravity’s divergence when all topologies are included. Expertise from the field of dynamically triangulated quantum gravity about sums over geometries may be able to remedy these defects and fulfill the Efetov theory’s original promise. The other saddle points of the Efetov Lagrangian are also analyzed; the Hessian at these points is nonnormal and pseudo-Hermitian, which is unusual for bosonic theories. The standard formula for Gaussian integrals is generalized to nonnormal kernels.

Figure 1

(a) a simple example target geometry; node 1 is a box, node 2 a circle. (b) the two vertices in the perturbation theory for this target geometry.

Figure 2

The second order geometries for our example. (a) the spanning trees. (b) the disconnected geometry, weight $1/2$. (c) the connected geometry, weight $3/2$.

Figure 3

The third order geometries for our example. The weights are 3 (a), 1 (b), and $1/3$ (c).