Quantum Correlations, Separability, and Quantum Coherence Length in Equilibrium Many-Body Systems

Nonlocality is a fundamental trait of quantum many-body systems, both at the level of pure states, as well as at the level of mixed states. Because of nonlocality, mixed states of any two subsystems are correlated in a stronger way than what can be accounted for by considering the correlated probabilities of occupying some microstates. In the case of equilibrium mixed states, we explicitly build two-point quantum correlation functions, which capture the specific, superior correlations of quantum systems at finite temperature, and which are directly accessible to experiments when correlating measurable properties. When nonvanishing, these correlation functions rule out a precise form of separability of the equilibrium state. In particular, we show numerically that quantum correlation functions generically exhibit a finite quantum coherence length, dictating the characteristic distance over which degrees of freedom cannot be considered as separable. This coherence length is completely disconnected from the correlation length of the system—as it remains finite even when the correlation length of the system diverges at finite temperature—and it unveils the unique spatial structure of quantum correlations.

Figure 1

(a) A quantum Hamiltonian acting coherently on the subsystems $A$ and $B$ correlates quantum mechanically the uncertainties on the local observables $O_A$ and $O_B$, generating a nonvanishing quantum correlation function. (b) The opposite scenario is the one in which the two subsystems are locally in an equilibrium state governed by two separated Hamiltonians, correlated to a common classical noise source; in this case, the quantum correlation function $⟨\delta O_A\delta O_B{⟩}_Q$ vanishes identically. (c) When two sites $i$ and $j$ of a lattice system are separated by a distance far exceeding the quantum coherence length ${\xi }_Q$, the sites in between mediate their correlations in a similar way as a classical statistical field coupling the two sites.

Figure 2

Left panels: quantum correlation function $g_Q$ of Bose fields for (a) hard-core bosons with $L=64$ and different temperatures $t=k_{B}T/J$, and for (b) quantum rotors with $u=U/(2J\overline{n})=1$, $L=32$, and different temperatures $t=k_{B}T/(2J\overline{n})$. The $g_Q$ function is compared to the two-point quantum discord $D(r)$ for hard-core bosons, and to the total correlations $g(r)$ for quantum rotors. Here, $r$ is the short-hand notation for ($r$, 0). The solid lines are exponential fits $A^{\prime }{e}^{-d(x|L)/{\xi }_Q}d(x|L{)}^{-{\eta }^{\prime }}$ for $g_Q$, where $d(x|L)=(L/\pi )\sin (\pi x/L)$ is the cord length. All data refer to the superfluid phase. Right panels: quantum coherence length ${\xi }_Q$ and inseparability length $l_I$ (see text) versus temperature for (c) hard-core bosons, and for (d) quantum rotors with $u=1$. The solid and dashed lines indicate $t^{-1}$ and $t^{-1/2}$ power laws.