Linking Phase Transitions and Quantum Entanglement at Arbitrary Temperature

TL;DR

This paper develops a universal theory linking phase transitions and quantum entanglement at any temperature, showing how reduced density matrices and free energy derivatives signal different orders of phase transitions and their universal behaviors.

Contribution

It introduces a general framework connecting phase transitions with entanglement measures via reduced density matrices at arbitrary temperatures.

Findings

First order transitions signaled by reduced density matrix elements.

Second order transitions indicated by derivatives of these elements.

Universal scaling behaviors near second order transition points.

Abstract

In this work, we establish a general theory of phase transitions and quantum entanglement in the equilibrium state at arbitrary temperatures. First, we derived a set of universal functional relations between the matrix elements of two-body reduced density matrix of the canonical density matrix and the Helmholtz free energy of the equilibrium state, which implies that the Helmholtz free energy and its derivatives are directly related to entanglement measures because any entanglement measures are defined as a function of the reduced density matrix. Then we show that the first order phase transitions are signaled by the matrix elements of reduced density matrix while the second order phase transitions are witnessed by the first derivatives of the reduced density matrix elements. Near second order phase transition point, we show that the first derivative of the reduced density matrix elements present universal scaling behaviors. Finally we establish a theorem which connects the phase transitions and entanglement at arbitrary temperatures. Our general results are demonstrated in an experimentally relevant many-body spin model.