Constraints on stability and renormalization group flows in nonequilibrium matter

This paper establishes non-perturbative and perturbative stability constraints on nonequilibrium renormalization group flows by demonstrating that conditional mutual information serves as a monotonic scaling function and provides bounds for symmetry-breaking states under quantum channels.

The Gist

Imagine you are watching a complex system, like a crowd of people, a flock of birds, or a magnetic material, as it evolves over time. Physicists call the rules that govern how these systems change "Renormalization Group (RG) flows." Think of RG flow like zooming out with a camera: as you step back, the tiny details blur, and you start to see the big picture of how the system behaves.

In this paper, the authors (Yu-Hsueh Chen and Tarun Grover) use a concept from quantum information theory called Conditional Mutual Information (CMI) to set strict "traffic rules" for these systems. They want to know: Can a system change from one stable state to another? And if so, in which direction?

Here is a breakdown of their findings using everyday analogies:

1. The "Secret Handshake" (What is CMI?)

To understand CMI, imagine three rooms: Room A, Room B, and Room C.

• Room B is a large hallway separating Room A and Room C.
• CMI measures how much "secret information" or correlation exists between Room A and Room C that cannot be explained by what's happening in the hallway (Room B).

If A and C are whispering secrets to each other through the walls, ignoring the hallway, that's high CMI. If the hallway explains everything, or if A and C are totally disconnected, the CMI is low (or zero).

2. The "One-Way Street" Rule (Monotonicity)

The first major discovery is that as a system evolves (flows) from a microscopic scale (UV) to a macroscopic scale (IR), this "secret handshake" (CMI) has a strict rule: It can only go down, never up.

• The Analogy: Imagine a river flowing downhill. You can't swim upstream against the current. Similarly, a system cannot naturally evolve from a state with "low secret connections" to a state with "high secret connections."
• The Consequence: If a system is in a stable state with very little "hidden correlation" (low CMI), it is safe. It cannot spontaneously destabilize and turn into a chaotic state with high hidden correlations. However, a state with high hidden correlations can easily fall apart into a simpler, low-correlation state.

3. The "Mixture" Rule (Stability)

The second discovery deals with mixing different states together. Imagine you have a bowl of pure red marbles (State A) and a bowl of pure blue marbles (State B). If you mix them, you get a purple mixture.

The authors proved that the "secret connections" in the purple mixture cannot be stronger than the connections in the pure red or pure blue bowls, plus a small amount of "noise" introduced by the mixing process.

• The Analogy: If you take a very stable, ordered structure (like a perfect crystal) and shake it up a little bit (add noise or break its symmetry), it won't suddenly become more ordered or develop new, complex hidden connections. It will stay in the same "phase" of matter, provided the shaking isn't too violent.

4. Real-World Examples from the Paper

The authors tested these rules on several scenarios:

• Decoherence (The "Fading Memory" Test): They looked at a system where information is slowly lost to the environment (like a spinning top slowing down). They showed that as long as the "noise" isn't too strong, the system remains in its original stable state. It won't suddenly jump to a completely different, more complex state.
• Magnetic Spins (The "Falling Dominoes"): They studied a model where spins (tiny magnets) are either up or down. They showed that if you start with a perfectly ordered state and introduce a little bit of randomness, the system stays ordered. It doesn't spontaneously break into a chaotic mess unless the randomness is overwhelming.
• The "Flocking" Birds (Speculative): The authors suggest these rules might explain why certain groups of animals (like bird flocks) can form organized patterns even when they aren't in a perfect equilibrium. They argue that if a system starts with certain "non-local" connections, it might be able to reach a stable, organized state that a simple, local system could never achieve.

Summary

In simple terms, this paper uses the math of "information sharing" to prove that nature has a bias toward simplicity as systems evolve.

• You can easily go from Complex/Ordered $\rightarrow$ Simple/Disordered.
• You generally cannot go from Simple/Disordered $\rightarrow$ Complex/Ordered without external help.

This gives physicists a powerful new tool to predict which states of matter are stable and which ones are doomed to collapse, simply by measuring how much "hidden correlation" exists between different parts of the system.

Technical Summary

Technical Summary: Constraints on Stability and Renormalization Group Flows in Nonequilibrium Matter

Problem Statement
Non-perturbative constraints on renormalization group (RG) flows have proven essential for understanding the phase diagrams of strongly interacting equilibrium systems, often derived from information-theoretic inequalities. However, it remains an open question whether similar constraints can be established for the landscape of phenomena in general nonequilibrium settings, such as long-time evolved quantum systems or classical nonequilibrium steady states. The authors address the lack of general stability criteria and RG flow constraints for these nonequilibrium phases, specifically investigating how quantum information measures can bound the evolution of mixed states and the stability of symmetry-breaking phases against perturbations.

Methodology
The paper employs quantum information inequalities, specifically focusing on Conditional Mutual Information (CMI), defined as $I(A : C|B) = S(AB) + S(BC) - S(B) - S(ABC)$, where $S(X)$ is the von Neumann entropy of region $X$. The authors utilize the property of Strong Subadditivity (SSA), which guarantees the positivity of CMI and its monotonicity with respect to the size of the separating region $B$.

The methodology proceeds in two main theoretical directions:

1. RG Flow Analysis: The authors consider a system near a critical point described by a scaling variable $t$ and a dimensionless length $l_B$. They analyze the scaling function of CMI, $I(\infty|l_B, t) \equiv f(x)$, where $x = t l_B^{1/\nu}$. They assume the CMI is "UV finite" (the UV-divergent contributions cancel out, a property expected for systems admitting a local space-time action description, such as those described by MSRJD functionals).
2. Convex Decomposition: The authors derive an inequality bounding the CMI of a convex mixture of states, $\rho = \sum_\alpha p_\alpha \rho_\alpha$, in terms of the CMI of the individual components and a term related to the classical information of the mixture. This utilizes SSA to establish an analog of the Holevo bound for CMI.

Key Contributions and Results

1. Monotonicity of CMI along RG Flows:
   The authors prove that under the assumption of UV finiteness, the scaling function $f(x)$ associated with CMI is monotonic along the RG flow:
   $$\frac{df(x)}{d|x|} \leq 0$$
   This implies that as the system flows from the UV (small $|x|$) to the IR (large $|x|$), the CMI does not increase.

   • Stability Criterion: A fixed point with a smaller CMI cannot be destabilized toward a fixed point with a larger CMI. For instance, a trivial state with zero CMI cannot flow to a symmetry-broken state with non-zero CMI under perturbations that preserve the UV finiteness assumption.
   • Markov Length Implications: If an IR fixed point has an infinite Markov length (implying infinite CMI or non-decaying correlations), the UV fixed point cannot be a Gibbs state of a local Hamiltonian. This is applied to exotic multicritical points (e.g., those flowing to Directed Percolation and KPZ), suggesting the underlying state must possess infinite Markov length and cannot be a standard local Gibbs state.

2. Bounding CMI in Convex Mixtures:
   The authors derive the inequality:
   $$I_\rho(A : C|B) \leq \sum_\alpha p_\alpha I_{\rho_\alpha}(A : C|B) + S_\sigma(D|B)$$
   where $\sigma$ represents the classical-quantum state of the mixture.

   • Stability of Symmetry Breaking: This result is used to infer the perturbative stability of spontaneous symmetry breaking (SSB) states against quantum channels that explicitly break the symmetry. If the individual components (e.g., product states) have finite Markov lengths and the noise is sufficiently weak (bounded channel), the mixture retains a finite Markov length, remaining in the same phase of matter.

3. Illustrative Examples:

   • Classical SSB and Decoherence: The authors analyze a classical Ising-like state subjected to a spin-flip channel. They demonstrate that the CMI scaling function is monotonic and UV finite, confirming that the decohered state remains in the same phase as the original symmetry-broken state for small noise strengths.
   • Transverse Field Ising Model (TFIM): For pure states, the CMI reduces to mutual information. The authors show that for the 1+1D TFIM, the CMI diverges logarithmically at the critical point ($x \to 0$), consistent with conformal field theory predictions ($c=1/2$). They note that unlike the $c$-theorem, this divergence does not yield a non-trivial constraint for flows between 1+1D CFTs because the function is not bounded from above in the same manner.
   • Strong-to-Weak SSB (SWSSB): The paper examines the transition between trivial and SWSSB states in 2D. Numerical results using tensor networks show that the CMI at the critical point exceeds that of the IR fixed points, consistent with the monotonicity constraint. The scaling function exhibits a logarithmic divergence near the critical point, mapping to the disorder-averaged free energy of the Random Bond Ising Model.

Significance and Claims
The paper claims to establish a "non-perturbative stability criterion" for nonequilibrium phases based on information-theoretic principles. By demonstrating that CMI is monotonic along RG flows (under UV finiteness), the authors provide a tool to rule out certain RG trajectories, such as the destabilization of a low-CMI fixed point toward a high-CMI one.

The authors emphasize that these constraints apply to a broad class of systems, including those described by local space-time actions (like MSRJD functionals for classical steady states), thereby extending information-theoretic constraints beyond ground states of local Hamiltonians. They suggest that CMI monotonicity offers a guiding principle for exploring nonequilibrium phase diagrams, potentially explaining why certain equilibrium critical points (which might have divergent Markov lengths) cannot evolve into specific nonequilibrium fixed points without violating information-theoretic bounds.

The work is modest in its scope, explicitly excluding systems that do not admit a conventional thermodynamic limit (e.g., fractonic systems) where UV finiteness of CMI may fail. The authors frame their results as constraints derived from general inequalities rather than a complete classification of all nonequilibrium phases.