Constraints on stability and renormalization group flows in nonequilibrium matter

Abstract

We derive constraints on renormalization group (RG) flows and stability of phases in nonequilibrium systems using quantum information inequalities. These constraints involve conditional mutual information (CMI), which quantifies correlations between spatially separated regions not mediated by their surroundings. First, assuming CMI is UV finite, we show that the scaling function associated with CMI is monotonic along the RG flow. This implies a non-perturbative stability criterion: a fixed point with smaller CMI cannot be destabilized toward one with larger CMI. Second, we bound the CMI of a convex mixture of states in terms of the CMI of individual components. We use this inequality to infer perturbative stability of spontaneous symmetry breaking states against quantum channels that explicitly break symmetry. We illustrate these constraints through several examples, including decoherence-driven transitions in classical symmetry-broken states, strong-to-weak symmetry breaking criticality in two dimensions, and even transitions in pure quantum states. We also discuss implications for classical nonequilibrium steady states.

Figures (7)

• Figure 1: We primarily focus on the geometry shown in (a), where the full system in $d$ spatial dimensions of size $\alpha l \times l^{d-1}$ ($\alpha$ is an arbitrary order one number) is partitioned into three regions: $A$, $B$, and $C$. $B$ is a slab of size $l_B \times l^{d-1}$ that separates $A$ and $C$ of equal sizes. (b), (c): Other geometries for which our results can be straightforwardly generalized to (see main text).
• Figure 2: (a) The conditional mutual information(CMI) $I(A:C|B)=I(l|l_B,p)$ of the mixed state discussed in Illustration 1 as a function of $p$ for various $l_B$ with $l = 10^8 (\approx \infty)$. The inset shows the data collapse of the scaling ansatz $I(l|l_B,p)=f[(p-p_c)l_B^{1/\nu}]$ with $p_c = 1$ and $\nu = 1$. (b) The corresponding RG flow.
• Figure 3: (a) The CMI of the TFIM as a function of $l_B$ for several values of $h$. The inset shows a data collapse using the scaling ansatz $I(A:C|B) = f[(h-h_c)l_B^{1/\nu}]$ with $h_c = 1$ and $\nu = 1$. (b) The CMI as a function of $\log l_B$ for several values of $h < h_c$. The insets show the same data plotted as a function of $\log(|h-h_c|\,l_B^{1/\nu})$.
• Figure 4: The CMI for the trivial-to-SWSSB transition in two spatial dimension as a function of $p$ for various $l_B$. The inset shows the data collapse with the scaling ansatz $I(A:C|B)=f[(p-p_c)l_B^{1/\nu}]$ with $p_c= 0.109$ and $\nu = 1.5$.
• Figure 5: To prove that $I(A:C|B)\geq I(A':C'|B")$, we first introduce an intermediate configuration $AB'C'$ in (b) that interpolates between $ABC$ in (a) and $A'B"C'$ in (c), and then establish the chain of inequalities $I(A:C|B)\geq I(A:C'|B')\geq I(A':C'|B")$.