Bogolyubov inequality for the ground state and its application to interacting rotor systems

TL;DR

This paper extends the Bogolyubov inequality to certain operators at zero temperature and applies it to interacting rotor systems, demonstrating conditions under which spontaneous magnetization vanishes in low-dimensional lattices.

Contribution

It formulates and proves a zero-temperature Bogolyubov inequality for a class of operators and applies it to show absence of long-range order in low-dimensional rotor systems.

Findings

Spontaneous magnetization is zero in 1D and 2D systems under certain conditions.

The inequality is extended from matrices to specific operators at zero temperature.

Heuristic arguments suggest a size-independent energy gap in small-interaction regimes.

Abstract

We have formulated and proved the Bogolyubov inequality for operators at zero temperature. So far this inequality has been known for matrices, and we were able to extend it to certain class of operators. We have also applied this inequality to the system of interacting rotors. We have shown that if: {\em i)} the dimension of the lattice is 1 or 2, {\em ii)} the interaction decreases sufficiently fast with a distance, and {\em iii)} there is an energy gap over the ground state, then the spontaneous magnetization in the ground state is zero, i.e. there is no LRO in the system. We present also heuristic arguments (of perturbation-theoretic nature) suggesting that one- and two-dimensional system of interacting rotors has the energy gap independent of the system size if the interaction is sufficiently small.