Linking phase transitions and quantum entanglement at arbitrary temperature

In this work, we establish a general theory of phase transitions and quantum entanglement in the equilibrium state at arbitrary temperatures. First, we derive a set of universal functional relations between the matrix elements of a 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 the reduced density matrix while the second-order phase transitions are witnessed by the first derivatives of the reduced density matrix elements. Near the 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 bipartite entanglement at arbitrary temperatures. Our general results are demonstrated in an experimentally relevant many-body spin model.