Matrix Order Indices in Statistical Mechanics

TL;DR

This paper introduces matrix order indices that relate matrix norm and trace, providing a versatile tool to quantify and classify various orders and phase transitions in many-body systems, including finite ones.

Contribution

It proposes a new concept of matrix order indices applicable to any matrix, enabling analysis of order in finite and nonequilibrium systems without relying on thermodynamic limits.

Findings

Indices characterize long-range and mid-range order.

They distinguish phases in finite systems.

They classify crossover phase transitions.

Abstract

A new notion is introduced of matrix order indices which relate the matrix norm and its trace. These indices can be defined for any given matrix. They are especially important for matrices describing many-body systems, equilibrium as well as nonequilibrium, for which the indices present a quantitative measure of the level of ordering. They characterize not only the long-range order, but also mid-range order. In the latter case, when order parameters do not exist, the matrix indices are well defined, providing an explicit classification of various mid-range orders. The matrix order indices are suitable for describing phase transitions with both off-diagonal and diagonal order. Contrary to order parameters whose correct definition requires the thermodynamic limit, the matrix indices do not necessarily need the latter. Because of this, such indices can distinguish between different phases of finite systems, thus, allowing for the classification of crossover phase transitions.