Mutual information and spontaneous symmetry breaking

We show that the metastable, symmetry-breaking ground states of quantum many-body Hamiltonians have vanishing quantum mutual information between macroscopically separated regions and are thus the most classical ones among all possible quantum ground states. This statement is obvious only when the symmetry-breaking ground states are simple product states, e.g., at the factorization point. On the other hand, symmetry-breaking states are in general entangled along the entire ordered phase, and to show that they actually feature the least macroscopic correlations compared to their symmetric superpositions is highly nontrivial. We prove this result in general, by considering the quantum mutual information based on the two-Rényi entanglement entropy and using a locality result stemming from quasiadiabatic continuation. Moreover, in the paradigmatic case of the exactly solvable one-dimensional quantum $XY$ model, we further verify the general result by considering also the quantum mutual information based on the von Neumann entanglement entropy.

Figure 1

A many-body quantum system is partitioned in three distinguishable subsystems, $A,B$, and the remainder $E$, so that the total Hilbert space acquires the tensor product structure $\mathcal{H}={\mathcal{H}}_A\otimes {\mathcal{H}}_B\otimes {\mathcal{H}}_E$. The quantity $d(A,B)$ is the distance between the two regions $A\text{and}B$, and $l$ is the distance defining the new effective support after an adiabatic deformation of operators with initial support on ${\mathcal{H}}_A\text{and}{\mathcal{H}}_B$.

Figure 2

Behavior of the mutual information $\left[{\mathcal{I}}_2^{\left(s\right)}\left(\infty \right)\right]$ between two spins at infinite distance in the symmetric ground state of the one-dimensional $XY$ model (thermodynamic limit) as a function of the anisotropy $\gamma$ and transverse field $h$ in the ferromagnetic phase $0\le h\le h_c=1$. Within the ferromagnetic phase, it is always $m_{\nu }>0$, and hence ${\mathcal{I}}_2^{\left(s\right)}\left(\infty \right)>0$. On the other hand, $m_{\nu }=0$ either at $h_c=1$ or at $\gamma =0$. Only at these points ${\mathcal{I}}_2^{\left(s\right)}\left(\infty \right)=0$.

Figure 3

Behavior of the mutual information based on the two-Rényi entropy, ${\mathcal{I}}_2\left(A\right|B)$ (left), and the von Neumann entropy, $\mathcal{I}\left(A\right|B)$ (right), as functions of the logarithm of the interspin distance $r$ for different superpositions in the ferromagnetic phase of the one-dimensional $XY$ model at $(\gamma =h=0.5)$. Black circles, $u=1,v=0$ (symmetric state); red squares, $u=cos\left(0.05\pi \right),v=sin\left(0.05\pi \right)$; green diamonds, $u=cos\left(0.1\pi \right),v=sin\left(0.1\pi \right)$; brown up-triangles, $u=cos\left(0.15\pi \right),v=sin\left(0.15\pi \right)$; blue down-triangles, $u=cos\left(0.2\pi \right),v=sin\left(0.2\pi \right)$; and violet empty circles, $u=cos\left(0.25\pi \right),v=sin\left(0.25\pi \right)$ (maximally symmetry-breaking ground state). The two definitions feature the same qualitative behavior; in particular, they both vanish in and only in the maximally symmetry-breaking ground states.

Figure 4

Behavior of the mutual information based on the two-Rényi entropy, ${\mathcal{I}}_2\left(A\right|B)$ (left), and the von Neumann entropy, $\mathcal{I}\left(A\right|B)$ (right), as functions of the ground-state superposition parameter $\theta$, with the parametrization $u=cos\theta ,v=sin\theta$, for two extreme values of the distance $r$ between $A$ and $B$. Dashed black curve: $r=1$. Solid red curve: $r=\infty$. The absolute minimum is always realized at $\theta =\pi /4$, corresponding to the maximally symmetry-breaking ground state. At this point, both measures of mutual information vanish exactly for $r=\infty$.