Non-relativistic Lee Model on two Dimensional Riemannian Manifolds

TL;DR

This paper extends the non-relativistic Lee model to two-dimensional Riemannian manifolds, focusing on renormalization, ground state energy bounds, and the complexities of mean field approximation in this lower dimension.

Contribution

It provides the first detailed analysis of the mean field approximation for the two-dimensional Lee model on Riemannian manifolds, highlighting differences from the three-dimensional case.

Findings

Ground state energy is bounded from below on compact and Cartan-Hadamard manifolds.

Renormalization methods from 3D models are applicable in 2D with similar calculations.

Mean field approximation in 2D requires more complex analysis than in 3D.

Abstract

This work is a continuation of our previous work (JMP, Vol. 48, 12, pp. 122103-1-122103-20, 2007), where we constructed the non-relativistic Lee model in three dimensional Riemannian manifolds. Here we renormalize the two dimensional version by using the same methods and the results are shortly given since the calculations are basically the same as in the three dimensional model. We also show that the ground state energy is bounded from below due to the upper bound of the heat kernel for compact and Cartan-Hadamard manifolds. In contrast to the construction of the model and the proof of the lower bound of the ground state energy, the mean field approximation to the two dimensional model is not similar to the one in three dimensions and it requires a deeper analysis, which is the main result of this paper.