🔬 condensed matter

Markov length can diverge in systems whose universal physics is spatially Markovian

This paper demonstrates that states flowing to the same renormalization-group fixed point can exhibit qualitatively different conditional mutual information (CMI) behaviors, specifically showing that certain RG-irrelevant perturbations can generate algebraically decaying CMI with infinite Markov length at finite cutoffs despite the underlying universal physics being spatially Markovian in the continuum limit.

Original authors: Yu-Hsueh Chen, Tarun Grover

Published 2026-10-09

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Yu-Hsueh Chen, Tarun Grover

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the study of matter, physicists often look for the underlying rules that govern how things behave when they are very large or very far apart. Imagine trying to understand the weather: while local wind gusts might seem chaotic, the overall climate patterns follow predictable laws. Similarly, in complex systems made of billions of tiny particles, scientists use a framework called the renormalization group to sort different materials into "phases." This method groups together systems that, despite looking different up close, share the same fundamental behavior when viewed from a distance. A key idea in this framework is "universality," which suggests that if two systems flow toward the same mathematical destination, they should act the same way in the long run. For decades, this has been a reliable guide for understanding everything from magnets to fluids. However, a new study challenges a specific assumption about how information travels through these systems, revealing a subtle but important exception to the rule that has been taken for granted.

The researchers focused on a concept called conditional mutual information, which acts like a test for how much two distant parts of a system know about each other, given what a middle section already knows. In many standard systems, if you place a thick enough buffer zone between two regions, the information they share drops off so quickly that it becomes negligible. This rapid drop-off is known as being "spatially Markovian," meaning the past and future are effectively separated by the present. If this separation holds, the system is considered to have a finite "Markov length," a measure of how far information can travel before it fades away. The prevailing wisdom was that if a system belongs to a class of materials known to be Markovian, its information should always fade away exponentially fast, no matter how you tweak the system, as long as the tweaks are small.

The team, led by Yu-Hsueh Chen and Tarun Grover, discovered that this expectation is not always true. They showed that it is possible to take a system that is fundamentally Markovian and add a tiny, specific disturbance that breaks the usual symmetry of its motion. While this disturbance is too weak to change the system's large-scale phase or its universal properties, it does something surprising to the way information decays. Instead of fading away quickly, the shared information between distant regions begins to drop off much more slowly, following a power law. This means that even though the system is technically in a phase where information should be cut off, the "Markov length" effectively becomes infinite. The two distant regions remain connected by a faint, long-range thread of information that would normally be impossible in such a system.

Crucially, the researchers found that this long-range connection is a fragile artifact of the microscopic details of the system. They demonstrated that while the information decays slowly at any fixed, real-world scale, this slow decay is suppressed by the size of the smallest building blocks of the material. If you were to imagine zooming out to a perfect, continuous view where the atomic details disappear, this slow decay would vanish, and the system would return to its expected, rapid behavior. This distinction is vital: the system exhibits a long-range connection that is real and measurable at any finite scale, yet it is not a fundamental property of the phase itself. It is a "cutoff-suppressed" effect, meaning it exists only because the universe is made of discrete atoms rather than a smooth continuum.

To prove this, the scientists constructed specific mathematical models and ran detailed simulations. They started with a critical state of a classical system, which is a delicate balance point where correlations are already long-ranged. By introducing a perturbation that breaks the detailed balance—a condition where the forward and backward processes of the system are perfectly symmetric—they observed the emergence of these slow-decaying tails. They confirmed their findings using a solvable model involving two interacting fields with specific anisotropic properties, where they could calculate the information sharing exactly. The results showed that the information decayed according to a precise power law, with the strength of the decay depending on how far the system was from the critical point and the size of the lattice.

The study also applied these findings to a well-known system called Toom's cellular automaton, a grid-based model used to simulate non-equilibrium physics. The researchers predicted that the critical point of this system, which is expected to be Markovian, should exhibit these same cutoff-suppressed power-law tails. They further showed that a similar phenomenon occurs in the high-temperature phase of a long-range Ising model, where the system flows to a trivial state but still displays these unusual tails. In contrast, they noted that at the actual critical point of this long-range model, the power-law decay is genuine and survives even when the atomic details are removed, highlighting the difference between the two types of behavior.

This work refines our understanding of how phases of matter are defined. It suggests that the equivalence of states, which usually relies on them sharing the same long-distance properties, needs to be more nuanced when considering how information is processed. While the systems in question still belong to the same universal class as their unperturbed counterparts, they are distinguishable by the presence of these long, faint threads of information that persist at any finite scale. The findings imply that the "Markov length" can diverge even in systems whose universal physics is Markovian, creating a tension between the traditional view of universality and newer definitions of phases based on the ability to reverse local changes. The researchers conclude that while these long-range connections are not fundamental to the phase in the continuum limit, they are a real obstruction to local reversibility at any microscopic scale, adding a new layer of complexity to how we classify the behavior of matter.

Technical Summary: Markov Length Divergence in Spatially Markovian Systems

Problem Statement
The paper addresses a tension between the standard Renormalization Group (RG) definition of universality and the emerging framework for defining mixed-state phases based on local reversibility. In the RG paradigm, states flowing to the same fixed point share universal long-distance properties. However, the mixed-state phase framework relies on Conditional Mutual Information (CMI), $I(A:C|B)$ , to distinguish phases: states with exponentially decaying CMI are considered equivalent (finite Markov length), while those with algebraically decaying CMI (infinite Markov length) belong to distinct phases.

The central problem is whether states that flow to the same RG fixed point—specifically, those with "spatially Markovian" universal physics where CMI should vanish beyond an interaction range—can nevertheless exhibit algebraically decaying CMI. The authors investigate whether "irrelevant" perturbations that break detailed balance can induce such non-Markovian behavior, and if so, whether this behavior is a genuine feature of the continuum limit or an artifact of the lattice cutoff.

Methodology
The authors employ a combination of perturbative field theory, scaling arguments, and exact numerical solutions in solvable models:

Perturbative Expansion: Starting from a critical Gibbs state of a finite-range local classical Hamiltonian (where CMI vanishes beyond the interaction range), they introduce a detailed-balance-breaking perturbation, $\epsilon v[\phi]$ , to the Langevin dynamics. They compute the leading-order CMI using the McLennan–Zubarev expression for the steady state combined with a quadratic expansion of CMI around a classical Markov distribution.
Scaling Analysis: They analyze the scaling of CMI in terms of the lattice cutoff $a$ , the buffer size $l_B$ , the region size $l$ , and the crossover length scale $\xi_{\text{cross}} \sim a |p|^{-1/\omega}$ determined by the irrelevant perturbation's RG eigenvalue $\omega < 0$ .
One-Dimensional Rigorous Bound: For 1D translationally invariant systems, they derive an exact relationship between the decay rate of CMI and the behavior of the lattice cutoff, proving that CMI decays faster than $l_B^{-2}$ if and only if the CMI vanishes as $a \to 0$ .
Solvable Gaussian Models: They construct and solve Gaussian models, including a two-component Toom-like cellular automaton and a long-range Ising model, to non-perturbatively verify the scaling forms.

Key Contributions and Results

Cutoff-Suppressed Power-Law CMI: The primary result is the demonstration that an irrelevant, detailed-balance-breaking perturbation to a critical Gibbs state generically induces algebraically decaying CMI. Crucially, these power laws are cutoff-suppressed: while they imply an infinite Markov length at any fixed lattice cutoff, their amplitude vanishes as a positive power of the cutoff ( $a$ ) in the continuum limit. This distinguishes them from "genuine" power-law CMI (e.g., in 1+1-D directed percolation) which survives the continuum limit.
Scaling Form: For a generic perturbation with RG eigenvalue $\omega < 0$ , the CMI in the regime $l, l_B \gg \xi_{\text{cross}}$ scales as:
$\text{CMI} \sim \left( \frac{\xi_{\text{cross}}}{l_B} \right)^{2|\omega|} \Phi\left(\frac{l_B}{l}\right)$
where $\Phi$ decays as a power of the ratio $l_B/l$ . This confirms that the Markov length diverges at any fixed cutoff but the effect disappears as $\xi_{\text{cross}} \to 0$ (the continuum limit).
One-Dimensional Constraint: The authors prove that in 1D, CMI decays faster than $l_B^{-2}$ if and only if the fixed point is Markovian (i.e., CMI $\to 0$ as $a \to 0$ ). If CMI decays as $l_B^{-\eta}$ with $\eta < 2$ , the CMI is not UV-finite.
Specific Models:
- Toom's Cellular Automata: They predict that the Ising-symmetric critical point of Toom's cellular automata exhibits cutoff-suppressed power-law CMI due to the irrelevant nature of the Toom perturbation, with an estimated RG eigenvalue of $\omega \approx -1.3 \pm 0.2$ .
- Long-Range Ising Model: They show a contrast between the high-temperature paramagnetic phase (cutoff-suppressed CMI with $\eta > 2$ ) and the finite-temperature critical point (genuine power-law CMI with $\eta = 2$ that survives the continuum limit).
- Gaussian Systems: They derive general scaling forms for Gaussian systems with local critical kernels supplemented by irrelevant nonlocal contributions, confirming the crossover behavior between universal and non-universal tails.

Significance and Claims
The paper claims to refine the understanding of universality in the context of mixed-state phases. It demonstrates that the equivalence of states under RG flow does not strictly guarantee the equivalence of their CMI decay properties at finite cutoffs. Specifically:

Distinction of Power Laws: The work provides a rigorous criterion to distinguish between "genuine" non-Markovian physics (continuum-surviving power laws) and "accidental" non-Markovianity arising from irrelevant perturbations (cutoff-suppressed power laws).
Obstruction to Local Reversibility: The authors note that despite vanishing in the continuum limit, cutoff-suppressed CMI remains an obstruction to local reversibility at any fixed microscopic cutoff. This suggests a potential tension in defining mixed-state phases if one strictly requires exponential CMI decay at finite cutoffs, even for states that are RG-equivalent.
Predictive Power: The paper predicts that specific non-equilibrium critical points, such as the Ising-symmetric line in Toom's cellular automata, will exhibit these cutoff-suppressed tails, offering a concrete target for future numerical or experimental verification of this phenomenon.

The authors maintain a modest tone regarding the implications for phase classification, suggesting that while the phenomenon is robust, the resolution of the tension between RG universality and channel-based phase definitions may require relaxing the constraints on channel range or strength in the definition of mixed-state equivalence.

Technical Summary: Markov Length Divergence in Spatially Markovian Systems

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