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

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

Conditional Mutual Information and Phase Distinction

Conditional mutual information (CMI) plays a central role in defining mixed-state phases based on local reversibility. This framework sharply distinguishes exponentially decaying from algebraically decaying CMI, since the latter obstructs equivalence to a state with finite Markov length. The paper shows that states that flow to the same renormalization-group (RG) fixed point can nevertheless exhibit qualitatively different CMI. Specifically, an RG-irrelevant perturbation that breaks detailed balance generically generates algebraically decaying CMI, which is cutoff-suppressed and vanishes in the scaling limit where the ratio of lattice cutoff to macroscopic length scales is sent to zero. This feature allows one to distinguish continuumsurviving power-law CMI from cutoff-suppressed powerlaw CMI, which remains an obstruction to local reversibility at any fixed microscopic cutoff.

Mechanism of Cutoff-Suppressed Power-Law CMI

A generic detailed-balance-breaking perturbation of Langevin dynamics whose unperturbed steady state is a scale-invariant critical local Gibbs state induces a cutoff-suppressed power-law CMI. This result is derived by combining the quadratic expansion of the CMI around a classical Markov distribution with the first-order McLennan–Zubarev expression for the steady state. The resulting CMI scaling in one dimension shows that it decays faster than l−2B if and only if the CMI vanishes as a lattice cutoff a → 0 at fixed l and lB.

Scaling Regimes for Gaussian Systems

For Gaussian distributions, the CMI scaling in the annular geometry is determined by three regimes distinguished by their relation to the crossover length scale ξcross. The resulting scaling forms are given by:

1. When ξcross ≪ l ≪ lB, one finds I(A: CB) ∼ (ξcross/lB)2ω l/lB d+hloc.

2. When l ≪ ξcross ≪ lB, the exponent for lB/l now depends on the nonlocal scale-invariant theory as I(A: CB) ∼ (ξcross/lB)ω l/lB d+hnloc.

3. When l ≪ lB ≪ ξcross, the nonlocal part controls both restricted inverses at the relevant distances, and thus I(A: CB) ∼ l/lB d+hnloc.

Application to Specific Models

The paper applies these findings to several models. For the Ising-symmetric critical point of Toom’s cellular automata, the perturbation generically produces cutoff-suppressed power-law CMI. In the long-range Ising model, it is shown that the critical point has genuine power-law CMI that survives the continuum limit, whereas the high-temperature paramagnetic phase supports a cutoff-suppressed power-law CMI. Furthermore, for a two-component Toom-like model perturbed with a D3ϕ term, the predicted scaling is I(A: CB) ∼ (ξcross/lB)2(l/lB)4.

Constraints on Markovian Physics

The result constrains CMI of fixed points whose long-distance physics is spatially Markovian, implying that if CMI is UV finite and not cutoff-suppressed, it must scale as I(A: CB) ∼ (l/lB)2. This suggests that any powerlaw exponent η < 2 would imply that the CMI is not UV-finite. The paper also notes that slow relaxation dynamics can generate cutoff-suppressed power-law mutual information (MI) and CMI, even when the steady state itself has short-ranged correlations.

Conclusion and Future Directions

The divergence of Markov length in systems whose universal physics is Markovian exposes a tension between conventional RG notions of universality and channel-based notions of mixed-state phase equivalence. The paper predicts a cutoffsuppressed power-law CMI tail at the critical point on Toom’s symmetric line, despite its expected Markovian universal physics. It also highlights an upcoming work that constructs one-dimensional examples where an arbitrarily weak local quantum channel generates power-law CMI with η > 2. The discussion suggests relaxing the channel range in mixed-state phase equivalence from polylogarithmic to sublinear range, or enlarging it to include short-depth channels with algebraic spatial tails if their strength is cutoff-suppressed.

Numerical Verification

Numerical verification in the Gaussian regime for d = 2 shows a crossover between regimes described by Eqs. (7) and (8). The data collapse supports Eq.(C11), showing the predicted scaling behavior across different length scales. The numerical results confirm the predicted l−6B decay and the collapse of I(A: CB)(lB/l)4 versus lB/ξcross onto a curve of slope −2. The analysis also verifies that even when the unperturbed steady state has short-ranged correlations, slow relaxation dynamics can generate cutoff-suppressed power-law CMI.

Long-Range Ising Model Results

For the long-range Ising model in d spatial dimensions, the CMI tail l−ηB with the geometry in Fig. 1(b) has exponent η > 2 for sufficiently small β. When considering the critical point with a quadratic kernel K(q) = µ + qσ, for d = 1, the CMI in the lB >> l limit is found to be I(A: CB) = σ2(l/lB)2/8 + O((l/lB)3). This shows that the CMI has a universal tail with η = 2 that survives the continuum limit, reflecting the nonlocality of the critical Gaussian theory.

Gaussian System Scaling Exponents

The scaling exponents for CMI in Gaussian systems are derived from Eqs. (6)–(8). The analysis shows that the CMI scales with various combinations of length scales, such as I(A: CB) ∼ (ξcross/lB)2ω l/lB d+hloc. This scaling is consistent with the intuition that if ξcross is greater than all the length scales at which one probes the system, scale-invariant behavior associated with the non-local kernel is observed. The analysis also shows that in any spatial dimension, Eqs.(6)-(8), with hloc = 0, hnloc = σ, describe the crossover from genuine power-law CMI at the critical point (ξcross = ∞) to cutoff-suppressed power-law CMI in the high-temperature phase at distances lB >> ξcross.

End Matter Summary

The goal of this End Matter is to present a simple argument that yields the scaling exponents of the CMI I(A: CB) given in Eqs. (6)–(8) in the main text for a Gaussian distribution P[ϕ] ∝ exp−1/2 P ij phi i K ij phi j. The CMI for a Gaussian distribution is expressed as I(A: CB) = −1/2 Tr log(1 − T). By decomposing the kernel K into a local scale-invariant part Kloc and an irrelevant nonlocal part Knloc, the leading term in the lB >> l limit is I(A: CB) = ξ2ωcross / (lB)2 Φ(lB/l).

1. textbf Conditional Mutual Information for Phase Classification: Enables discrimination between states flowing to same RG fixed points that exhibit qualitatively different CMI by distinguishing exponentially decaying from algebraically decaying CMI. This allows AI systems to classify mixed-state phases based on a more refined notion of local reversibility, moving beyond simple exponential decay criteria.

2. textbf Cutoff-Suppressed Power-Law Identification: AI can specifically identify and distinguish continuumsurviving power-law CMI from cutoff-suppressed powerlaw CMI, which has the property that it vanish[es] as a positive power of the lattice cutoff in the continuum limit. This provides a fingerprint to determine if a system's long-distance physics is genuinely Markovian or merely exhibits transient non-Markovian behavior.

3. textbf Adaptive Phase Equivalence: AI models can utilize channel-based notions of mixed-state phase equivalence, recognizing that channels of polylogarithmic range cannot connect a state with power-law CMI to one with exponentially decaying CMI through a locally reversible path. This allows for more nuanced classification of states that might otherwise be incorrectly grouped under conventional RG universality.

4. textbf Predictive Power-Law Exponent Calculation: For systems like the Ising-symmetric critical point of Toom’s cellular automata, AI can predict the exponent of such a power-law tail is universal and encodes the irrelevant RG eigenvalue, which is crucial for characterizing nonuniversal CMI tails in these models.

5. textbf Model-Specific CMI Scaling Prediction: Given a Gaussian distribution with a local scale-invariant kernel decomposed into local and irrelevant nonlocal parts, AI can predict the scaling forms of CMI across three regimes: (i) when xicross ≪ l ≪ lB, (ii) when l ≪ ξcross ≪ lB, and (iii) when l B ≪ ξcross. This enables predictive modeling of correlation decay based on the relative sizes of microscopic, crossover, and macroscopic length scales.

Conditional mutual information (CMI) probes spatial non-Markovianity and plays a central role in a recently developed framework for defining mixed-state phases based on local reversibility. This framework sharply distinguishes exponentially decaying from algebraically decaying CMI, since the latter obstructs equivalence to a state with finite Markov length. Here we show that states that flow to the same renormalization-group (RG) fixed point can nevertheless exhibit qualitatively different CMI. Starting from a critical Gibbs state of a finite-range local classical Hamiltonian, whose CMI vanishes beyond the interaction range, we show that an RG-irrelevant perturbation that breaks detailed balance generically generates algebraically decaying CMI. Crucially, these power-law tails are cutoff-suppressed: although they lead to an infinite Markov length at every fixed lattice cutoff, they vanish as a positive power of the cutoff in the continuum limit. We construct solvable models that exhibit this phenomenon and predict that the Ising-symmetric critical point of Toom's cellular automata generically exhibits such behavior. We also find analogous cutoff-suppressed CMI in the high-temperature paramagnetic phase of the long-range Ising paramagnet, in contrast with the genuine power-law CMI at the finite-temperature critical point in this same model that survives the continuum limit. We also derive a general result that in one spatial dimension, CMI decays faster than the inverse square of the buffer length if and only if the long-distance physics is Markovian.