Pith. sign in

REVIEW 4 major objections 4 minor 121 references

This paper derives information-theoretic constraints on renormalization group flows in nonequilibrium systems, showing that conditional mutual information (CMI) can only remain constant or decrease along a flow, and that CMI bounds the stab

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-04 06:09 UTC pith:76BE27XF

load-bearing objection Useful SSA-based CMI monotonicity for mixed-state RG flows, but the main stability claim leans on an unproven UV-finiteness assumption, and the abstract promises an example the text does not deliver. the 4 major comments →

arxiv 2602.04961 v2 pith:76BE27XF submitted 2026-02-04 cond-mat.stat-mech cond-mat.softcond-mat.str-elhep-thquant-ph

Non-perturbative constraints on stability and renormalization group flows in nonequilibrium matter

Yu-Hsueh Chen , Tarun Grover This is my paper
classification cond-mat.stat-mech cond-mat.softcond-mat.str-elhep-thquant-ph MSC 82B2881P4582B26
keywords conditional mutual informationrenormalization groupnonequilibrium steady statesstrong subadditivityMarkov lengthdecoherencespontaneous symmetry breakingmixed-state phases
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper derives two information-theoretic constraints on out-of-equilibrium phases. First, when conditional mutual information (CMI)—a measure of correlations between two regions that are not mediated by the region between them—is finite in the scaling limit, its crossover function must be monotonically non-increasing as the relevant scaling variable grows. Thus, under RG flow, a fixed point with smaller CMI cannot be destabilized toward one with larger CMI. Second, the CMI of a convex mixture of states is bounded by the weighted sum of component CMIs plus a classical decoding-error term; this implies that spontaneous symmetry-broken states remain in the same phase under sufficiently weak symmetry-breaking local noise. The authors demonstrate these constraints in classical and quantum examples and use them to argue that certain nonequilibrium fixed points cannot be Gibbs states of local Hamiltonians.

Core claim

The central claim is that strong subadditivity of von Neumann entropy, applied to a slab geometry, forces the scaling function of conditional mutual information to satisfy df(x)/d|x| ≤ 0 along a renormalization-group flow, provided CMI is UV finite (i.e., depends only on x = t l_B^{1/ν} in the scaling limit). Since CMI measures correlations not mediated by the intermediate region, this says a fixed point with lower CMI is stable against perturbations that would drive it to one with higher CMI. The authors also prove a convex-decomposition bound on CMI and use it to show that classical symmetry-broken states retain finite Markov length under weak symmetry-breaking channels, meaning they remai

What carries the argument

The central object is conditional mutual information I(A:C|B)=S(AB)+S(BC)-S(B)-S(ABC), whose non-negativity follows from strong subadditivity. The key identity is the monotonicity of CMI as the separating slab B thickens (Eq. 1), which after assuming UV finiteness becomes a monotonicity constraint on the crossover scaling function f(x). The second tool is a bound Iρ(A:C|B) ≤ Σ pα Iρα(A:C|B) + Sσ(D|B), which lets the authors control CMI of mixtures via error-decoding entropy. These are used to infer stability of phases and to rule out certain RG flows.

Load-bearing premise

CMI is UV finite in the scaling limit—that is, I(∞|l_B,t) depends only on the combination x = t l_B^{1/ν} and not separately on the cutoff, l_B, or l—so that the monotonicity in l_B translates into monotonicity in x.

What would settle it

A concrete counterexample would be a local nonequilibrium steady state, expressible by a local space-time action, where numerical or analytic computation shows the CMI scaling function f(x) increases with |x| over some range while l/l_B and l|t|^ν are large. Alternatively, if a system with conventional thermodynamic limit is found where CMI in the scaling limit depends separately on l_B and x, the UV-finiteness premise fails.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • A fixed point with zero CMI (e.g., a Gibbs state of a local Hamiltonian, or a product state) cannot be destabilized toward a state with nonzero CMI.
  • A SWSSB fixed point with CMI log 2 cannot be driven to a Z4 SWSSB state with CMI log 4; similarly, SWSSB states cannot destabilize toward a Z2×Z2 SPT phase.
  • If the IR fixed point has infinite Markov length (e.g., 1+1D directed percolation), the UV fixed point cannot be a Gibbs state of a local Hamiltonian.
  • In 2D, a continuous flocking transition fixed point cannot be a Gibbs state due to Mermin-Wagner and CMI monotonicity.
  • The anomalous dimension of CMI in 1D with finite l must be non-negative.
  • The convex-decomposition bound implies that classical symmetry-broken states remain in the same phase under sufficiently weak p-bounded local noise, with a threshold that can be estimated from decoding error probabilities.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The monotonicity of CMI could serve as a general 'c-theorem-like' irreversibility principle for nonequilibrium RG flows, complementing the F-theorem without requiring Lorentz invariance.
  • The convex-decomposition bound suggests a practical numerical diagnostic: measuring CMI scaling near decoherence-driven transitions can test whether a mixed state is a Gibbs state.
  • The UV-finiteness assumption may fail in systems with fractonic or subsystem symmetries; a concrete test would be to search for counterexamples where df/d|x| > 0 appears exactly in such systems.
  • The paper's application to flocking predicts that the Markov length diverges at continuous flocking transitions; this is testable in numerical simulations of active-matter models.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper derives two information-theoretic constraints for mixed states and applies them to phase stability and RG flows. The first result starts from strong subadditivity (SSA): CMI in a slab geometry is monotonically decreasing in the separator length l_B, Eq. (1). Assuming CMI is UV finite in the scaling limit, the crossover function f(x) with x = t l_B^{1/ν} satisfies d f(x)/d|x| ≤ 0, Eq. (2), which is interpreted as a non-perturbative stability criterion: a fixed point with smaller CMI cannot flow to one with larger CMI. The second result, Eq. (3), bounds the CMI of a convex mixture by the averaged CMIs of its components plus a conditional entropy term, and is used to argue perturbative stability of classical symmetry-broken states under local decoherence. Applications include an exactly solvable one-dimensional asymmetric decoherence model, the transverse-field Ising model, and a two-dimensional trivial-to-SWSSB transition. Appendices A--C give the SSA proofs and the exact CMI calculation; Appendices D--E review SWSSB and provide additional numerics.

Significance. If the UV-finiteness assumption holds, Eq. (2) is a nontrivial, non-perturbative constraint on RG flows that does not require Lorentz invariance or detailed balance, and it has no equilibrium counterpart of comparable generality. The SSA derivations in Appendices A and B are correct, the exact CMI calculation in Illustration 1 is a useful controlled test, and the SWSSB numerics use a publicly available tensor-network implementation. These are genuine strengths. However, the advertised scope is broader than what is established: the key scaling assumption is not proved for general nonequilibrium steady states, and one promised nonequilibrium example is missing. The paper is therefore a valuable conditional contribution, but the central claim needs either additional support or a more carefully stated scope.

major comments (4)
  1. [Derivation of Eq. (2), paragraph beginning "Let's now consider the limit"] Eq. (2), d f(x)/d|x| ≤ 0, follows from Eq. (1) only after assuming CMI is UV finite, so that I(∞|l_B,t) depends solely on x = t l_B^{1/ν}. This assumption is load-bearing and is not proven for generic nonequilibrium steady states. The locality argument based on MSRJD actions is plausible but not a proof, and none of the examples is a genuine NESS with nonzero current. The stability conclusions derived from Eq. (2) inherit this conditionality.
  2. [Abstract and Introduction] The abstract promises an example entitled "area-law CMI in anisotropic conserved dynamics", but no such example appears in the full text or in the appendices. This is not a cosmetic omission: it is the only advertised illustration involving genuinely nonequilibrium conserved dynamics, and it would have directly supported the UV-finiteness assumption for the class named in the title. Either the example should be supplied or the promise removed.
  3. [Consequences of Eq. (2), paragraph beginning "Conversely, if the IR fixed point"] The statement "if the IR fixed point has infinite Markov length, the UV fixed point cannot be a Gibbs state of a local Hamiltonian" does not follow from Eq. (2). Infinite Markov length means only that CMI does not decay exponentially; it may decay algebraically to zero. In that case monotonicity I_IR ≤ I_UV is satisfied by a Gibbs UV fixed point with I_UV = 0. To exclude a Gibbs UV fixed point one needs a positive fixed-point CMI in the IR, not merely an infinite Markov length. The same issue affects the inference that a zero fixed-point CMI implies a Gibbs state via Hammersley–Clifford: the theorem requires conditional independence at finite l_B, not merely I → 0 as l_B → ∞.
  4. [Abstract] The abstract states that monotonicity "implies that the CMI scaling exponent cannot increase along the RG flow," but no theorem about a scaling exponent is derived in the body. The only exponent inequality obtained is η ≥ 0 for the finite-l correction I ∼ [(αl − l_B)/l_B]^η f(x), which is not the fixed-point CMI scaling exponent. Either derive the claimed exponent monotonicity or delete/soften the sentence.
minor comments (4)
  1. [Paragraph containing Eq. (2)] The differentiability assumption on f(x) is stronger than needed for Eq. (2); the inequality is naturally one-sided in |x|. The later discussion of a discontinuous limit x→0 in Illustration 1 should be reconciled with the continuity assumption stated just before Eq. (2).
  2. [Illustration 1, Eq. (5)] The text says f(x) is monotonically increasing for x∈(−∞,0), which is consistent with df/d|x| ≤ 0 but may confuse readers reading Eq. (2) literally. Please state explicitly that monotonicity in |x| corresponds to f increasing toward x=0^- in this example.
  3. [Figure captions, Figs. 2-4] The captions do not state whether l/l_B and l|p−p_c|^ν are large enough to satisfy the limits in which Eq. (2) is claimed. Adding these values would help the reader judge the apparent violations in Fig. 4.
  4. [References to Supplemental Material] Ref. [39] points to "Supplemental Material" for Eq. (3), the SWSSB review, and the finite-r calculation, but these appear in the visible Appendices C and D. Please update the cross-references so the reader is directed to the correct appendices.

Circularity Check

0 steps flagged

No significant circularity: main inequalities follow from SSA plus an explicit UV-finiteness assumption; self-citations are applications, not inputs.

full rationale

The central results Eq. (2) and Eq. (3) are derived from strong subadditivity, an external theorem. Eq. (2) additionally relies on the stated assumption that CMI is UV finite in the scaling limit, i.e. I(∞|l_B,t)=f(t l_B^{1/ν}); this is an explicit hypothesis, not a conclusion disguised as a definition, and the monotonicity d f/d|x|≤0 is a nontrivial consequence of Eq. (1) combined with the scaling form. No parameter is fitted to enforce Eq. (2); Illustrations 1–3 use exact calculations (p_c=1, ν=1 for the one-dimensional decohered SSB model), exact TFIM exponents (h_c=1, ν=1), and literature values for the RBIM Nishimori point (p_c=0.109, ν=1.5). The self-citations (Refs [77], [95], [32]) are used as applications (DP Markov length), as a numerical method with public code, and as part of a multi-author local-reversibility criterion; none of these is an input to the derivation of Eq. (2) or Eq. (3), so their self-referential status does not make the central claims circular. The paper is transparent that the UV-finiteness assumption is not proven for general nonequilibrium steady states; it argues from local space-time actions and excludes fractonic/subsystem-symmetric systems. The abstract's promised example 'area-law CMI in anisotropic conserved dynamics' is absent from the full text; this is a completeness gap in supporting the assumption for the title's class of systems, not a circularity. Overall, the derivation chain is self-contained conditional on the stated assumption, and no specific equation reduces to its own input.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

The paper introduces no new fitting parameters or entities. Its central result rests on SSA plus a UV-finiteness domain assumption, together with cited results on Gibbs-state Markov length and product-state stability. The continuity-at-x=0 assumption is stated but violated by one of the paper's own examples, which is the main ledger liability.

axioms (5)
  • standard math Strong subadditivity of von Neumann entropy (SSA)
    Used to prove Eq.(1) (monotonicity in l_B) and Eq.(3) (convex mixture bound). SSA is a theorem in quantum information, taken as given.
  • domain assumption UV finiteness of CMI in the scaling limit for local space-time actions
    Stated in the main text: 'assuming CMI is finite in the scaling limit... it can depend only on x.' The authors argue from locality and cite evidence for ground states, but this is not proven for all nonequilibrium systems; they exclude fractonic/subsystem-symmetric systems.
  • ad hoc to paper Continuity of CMI(x) at x=0 along the RG-relevant flow
    Assumed after Eq.(2) to identify fixed-point CMI values; later an example shows a discontinuity at x=0 (Illustration 1), which the authors handle by placing the UV fixed point at x=0^-.
  • domain assumption Gibbs states of local Hamiltonians have finite Markov length (Ref [35])
    Used to conclude that a finite-temperature Gibbs fixed point has zero CMI at l_B >> ξ, so it cannot flow to a state with nonzero CMI. This is an external theorem.
  • domain assumption Stability of trivial product states under p-bounded channels (Ref [84])
    Used in the convex-mixture application to ensure CMI of individual components decays exponentially; cited to prior work.

pith-pipeline@v1.3.0-alltime-deepseek · 19598 in / 21620 out tokens · 211409 ms · 2026-08-04T06:09:42.919942+00:00 · methodology

0 comments
read the original 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 derive a monotonicity constraint on its crossover scaling function. Under certain assumptions, this implies that the CMI scaling exponent cannot increase along the RG flow. 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, area-law CMI in anisotropic conserved dynamics, and even transitions in pure quantum states. We also discuss implications for classical nonequilibrium steady states.

Figures

Figures reproduced from arXiv: 2602.04961 by Tarun Grover, Yu-Hsueh Chen.

Figure 1
Figure 1. Figure 1: FIG. 1. We primarily focus on the geometry shown in (a), [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a) The CMI of the TFIM as a function of [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The CMI for the trivial-to-SWSSB transition in two [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. To prove that [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Phase diagram of the mixed state in Eq.( [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. (a) An alternate geometry used to compute the CMI of the trivial-to-SWSSB transition (in contrast, in the main text [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

121 extracted references · 10 linked inside Pith

  1. [1]

    A. B. Zamolodchikov, Irreversibility of the flux of the renormalization group in a 2d field theory, Jetp Letters 43, 565 (1986)

    1986

  2. [2]

    ground-state degeneracy

    I. Affleck and A. W. W. Ludwig, Universal noninteger “ground-state degeneracy” in critical quantum systems, Phys. Rev. Lett.67, 161 (1991)

    1991

  3. [3]

    Friedan and A

    D. Friedan and A. Konechny, Boundary entropy of one-dimensional quantum systems at low temperature, Phys. Rev. Lett.93, 030402 (2004)

    2004

  4. [4]

    J. L. Cardy, Is there a c-theorem in four dimensions?, Physics Letters B215, 749 (1988)

    1988

  5. [5]

    Komargodski and A

    Z. Komargodski and A. Schwimmer, On renormalization group flows in four dimensions, Journal of High Energy Physics2011, 99 (2011)

    2011

  6. [6]

    R. C. Myers and A. Sinha, Seeing a c-theorem with holography, Phys. Rev. D82, 046006 (2010)

    2010

  7. [7]

    D. L. Jafferis, I. R. Klebanov, S. S. Pufu, and B. R. Safdi, Towards the f-theorem:n= 2 field theories on the three-sphere, Journal of High Energy Physics2011, 102 (2011)

    2011

  8. [8]

    Casini and M

    H. Casini and M. Huerta, Renormalization group run- ning of the entanglement entropy of a circle, Phys. Rev. D85, 125016 (2012)

    2012

  9. [9]

    R. A. Patil and A. W. Ludwig, Shannon entropy of the measurement record at measurement-dominated criti- cality and rg flow: A c-theorem for effective central charge and a g-theorem for effective boundary entropy, arXiv preprint arXiv:2507.07959 (2025)

    arXiv 2025

  10. [10]

    Zamolodchikov, Renormalization group and pertur- bation theory about fixed points in two-dimensional field theory, Sov

    A. Zamolodchikov, Renormalization group and pertur- bation theory about fixed points in two-dimensional field theory, Sov. J. Nucl. Phys.(Engl. Transl.);(United States)46(1987)

    1987

  11. [11]

    A. W. Ludwig and J. L. Cardy, Perturbative evaluation of the conformal anomaly at new critical points with applications to random systems, Nuclear Physics B285, 687 (1987)

    1987

  12. [12]

    D. A. Huse, Exact exponents for infinitely many new multicritical points, Phys. Rev. B30, 3908 (1984)

    1984

  13. [13]

    Appelquist, A

    T. Appelquist, A. G. Cohen, and M. Schmaltz, A new constraint on strongly coupled field theories, Physical Review D60, 045003 (1999)

    1999

  14. [14]

    Anselmi, D

    D. Anselmi, D. Freedman, M. T. Grisaru, and A. Jo- 6 hansen, Non-perturbative formulas for central functions of supersymmetric gauge theories, Nuclear Physics B 526, 543 (1998)

    1998

  15. [15]

    M. A. Luty, J. Polchinski, and R. Rattazzi, The a- theorem and the asymptotics of 4d quantum field the- ory, Journal of High Energy Physics2013, 1 (2013)

    2013

  16. [16]

    I. R. Klebanov, S. S. Pufu, and B. R. Safdi, F- theorem without supersymmetry, Journal of High En- ergy Physics2011, 1 (2011)

    2011

  17. [17]

    Grover, Entanglement monotonicity and the stability of gauge theories in three spacetime dimensions, Physi- cal review letters112, 151601 (2014)

    T. Grover, Entanglement monotonicity and the stability of gauge theories in three spacetime dimensions, Physi- cal review letters112, 151601 (2014)

    2014

  18. [18]

    Casini and M

    H. Casini and M. Huerta, A finite entanglement entropy and the c-theorem, Physics Letters B600, 142 (2004)

    2004

  19. [19]

    Casini, I

    H. Casini, I. S. Landea, and G. Torroba, The g-theorem and quantum information theory, Journal of High En- ergy Physics2016, 1 (2016)

    2016

  20. [20]

    Casini, I

    H. Casini, I. S. Landea, and G. Torroba, Irreversibility in quantum field theories with boundaries, Journal of High Energy Physics2019, 1 (2019)

    2019

  21. [21]

    Casini, I

    H. Casini, I. S. Landea, and G. Torroba, Entropicg theorem in general spacetime dimensions, Phys. Rev. Lett.130, 111603 (2023)

    2023

  22. [22]

    Harper, H

    J. Harper, H. Kanda, T. Takayanagi, and K. Tasuki, g theorem from strong subadditivity, Physical Review Letters133, 031501 (2024)

    2024

  23. [23]

    Grover, Certain general constraints on the many-body localization transition, arXiv preprint arXiv:1405.1471 (2014)

    T. Grover, Certain general constraints on the many-body localization transition, arXiv preprint arXiv:1405.1471 (2014)

    Pith/arXiv arXiv 2014

  24. [24]

    I. H. Kim, Long-range entanglement is necessary for a topological storage of quantum information, Phys. Rev. Lett.111, 080503 (2013)

    2013

  25. [25]

    B. Shi, K. Kato, and I. H. Kim, Fusion rules from en- tanglement, Annals of Physics418, 168164 (2020)

    2020

  26. [26]

    Huang, J

    J.-L. Huang, J. McGreevy, and B. Shi, Knots and en- tanglement, SciPost Phys.14, 141 (2023)

    2023

  27. [27]

    I. H. Kim, X. Li, T.-C. Lin, J. McGreevy, and B. Shi, Conformal geometry from entanglement, SciPost Phys. 18, 102 (2025)

    2025

  28. [28]

    Lin and J

    T.-C. Lin and J. McGreevy, Conformal field theory ground states as critical points of an entropy function, Phys. Rev. Lett.131, 251602 (2023)

    2023

  29. [29]

    Li, T.-C

    X. Li, T.-C. Lin, and J. McGreevy, A systematic search for conformal field theories in very small spaces, arXiv preprint arXiv:2509.04596 (2025)

    Pith/arXiv arXiv 2025

  30. [30]

    T.-H. Yang, B. Shi, and J. Y. Lee, Topological mixed states: Axiomatic approaches and phases of matter, arXiv preprint arXiv:2506.04221 (2025)

    arXiv 2025

  31. [31]

    Sang and T

    S. Sang and T. H. Hsieh, Stability of mixed-state quan- tum phases via finite markov length, Phys. Rev. Lett. 134, 070403 (2025)

    2025

  32. [32]

    S. Sang, L. A. Lessa, R. S. Mong, T. Grover, C. Wang, and T. H. Hsieh, Mixed-state phases from local re- versibility, arXiv preprint arXiv:2507.02292 (2025)

    Pith/arXiv arXiv 2025

  33. [33]

    Coser and D

    A. Coser and D. P´ erez-Garc´ia, Classification of phases for mixed states via fast dissipative evolution, Quantum 3, 174 (2019)

    2019

  34. [34]

    S. Sang, Y. Zou, and T. H. Hsieh, Mixed-state quantum phases: Renormalization and quantum error correction, Phys. Rev. X14, 031044 (2024)

    2024

  35. [35]

    Chen and C

    C.-F. Chen and C. Rouz´ e, Quantum gibbs states are locally markovian, arXiv preprint arXiv:2504.02208 (2025)

    Pith/arXiv arXiv 2025

  36. [36]

    Clifford and J

    P. Clifford and J. M. Hammersley, Markov fields on fi- nite graphs and lattices (1971)

    1971

  37. [37]

    M. S. Leifer and D. Poulin, Quantum graphical mod- els and belief propagation, Annals of Physics323, 1899 (2008)

    2008

  38. [38]

    Brown and D

    W. Brown and D. Poulin, Quantum markov net- works and commuting hamiltonians, arXiv preprint arXiv:1206.0755 (2012)

    Pith/arXiv arXiv 2012

  39. [39]

    See Supplemental Material for details

  40. [40]

    D. V. Fursaev, Entanglement entropy in critical phe- nomena and analog models of quantum gravity, Phys. Rev. D73, 124025 (2006)

    2006

  41. [41]

    Ryu and T

    S. Ryu and T. Takayanagi, Aspects of holographic en- tanglement entropy, Journal of High Energy Physics 2006, 045 (2006)

    2006

  42. [42]

    S. N. Solodukhin, Entanglement entropy, conformal in- variance and extrinsic geometry, Physics Letters B665, 305 (2008)

    2008

  43. [43]

    M. A. Metlitski, C. A. Fuertes, and S. Sachdev, Entan- glement entropy in theo(n) model, Phys. Rev. B80, 115122 (2009)

    2009

  44. [44]

    Grover, A

    T. Grover, A. M. Turner, and A. Vishwanath, Entan- glement entropy of gapped phases and topological order in three dimensions, Phys. Rev. B84, 195120 (2011)

    2011

  45. [45]

    Liu and M

    H. Liu and M. Mezei, A refinement of entanglement en- tropy and the number of degrees of freedom, Journal of High Energy Physics2013, 1 (2013)

    2013

  46. [46]

    Casini, M

    H. Casini, M. Huerta, R. C. Myers, and A. Yale, Mutual information and the f-theorem, Journal of High Energy Physics2015, 1 (2015)

    2015

  47. [47]

    Van Raamsdonk, Finite entropy sums in quantum field theory, arXiv preprint arXiv:2508.21276 (2025)

    M. Van Raamsdonk, Finite entropy sums in quantum field theory, arXiv preprint arXiv:2508.21276 (2025)

    Pith/arXiv arXiv 2025

  48. [48]

    A prototypical example is provided by a 1+1-D CFT with central chargec, where the divergent part of the entanglement entropyS X for a regionXconsisting of Ndisjoint pieces is 2N× c 6 log(1/a), reflecting the ad- ditive contribution from the 2Nentangling boundary points [50]. This expression remains true even when the CFT is slightly perturbed and acquires...

  49. [49]

    Holzhey, F

    C. Holzhey, F. Larsen, and F. Wilczek, Geometric and renormalized entropy in conformal field theory, Nuclear Physics B424, 443 (1994)

    1994

  50. [50]

    Calabrese and J

    P. Calabrese and J. Cardy, Entanglement entropy and quantum field theory, Journal of Statistical Mechanics: Theory and Experiment2004, P06002 (2004)

    2004

  51. [51]

    M. A. Levin and X.-G. Wen, String-net condensation: A physical mechanism for topological phases, Phys. Rev. B71, 045110 (2005)

    2005

  52. [52]

    Vardhan, A

    S. Vardhan, A. Y. Wei, and Y. Zou, Petz recovery from subsystems in conformal field theory, Journal of High Energy Physics2024, 1 (2024)

    2024

  53. [53]

    Balasubramanian, N

    V. Balasubramanian, N. Jokela, A. P¨ onni, and A. V. Ramallo, Information flows in strongly coupled abjm theory, Journal of High Energy Physics2019, 1 (2019)

    2019

  54. [54]

    P. C. Martin, E. D. Siggia, and H. A. Rose, Statisti- cal dynamics of classical systems, Phys. Rev. A8, 423 (1973)

    1973

  55. [55]

    H.-K. Janssen, On a lagrangean for classical field dy- namics and renormalization group calculations of dy- namical critical properties, Zeitschrift f¨ ur Physik B Con- densed Matter23, 377 (1976)

    1976

  56. [56]

    C. d. Dominicis, Techniques de renormalisation de la 7 th´ eorie des champs et dynamique des ph´ enomenes cri- tiques, in J. Phys., Colloq, Vol. 37 (1976) p. 247

    1976

  57. [57]

    U. C. T¨ auber, Critical dynamics: A field theory ap- proach to equilibrium and non-equilibrium scaling be- havior (Cambridge University Press, 2014)

    2014

  58. [58]

    H. Ma, A. T. Schmitz, S. A. Parameswaran, M. Her- mele, and R. M. Nandkishore, Topological entanglement entropy of fracton stabilizer codes, Physical Review B 97, 125101 (2018)

    2018

  59. [59]

    Paramekanti, L

    A. Paramekanti, L. Balents, and M. P. A. Fisher, Ring exchange, the exciton bose liquid, and bosonization in two dimensions, Phys. Rev. B66, 054526 (2002)

    2002

  60. [60]

    Chamon, Quantum glassiness in strongly correlated clean systems: An example of topological overprotec- tion, Phys

    C. Chamon, Quantum glassiness in strongly correlated clean systems: An example of topological overprotec- tion, Phys. Rev. Lett.94, 040402 (2005)

    2005

  61. [61]

    Haah, Local stabilizer codes in three dimensions with- out string logical operators, Phys

    J. Haah, Local stabilizer codes in three dimensions with- out string logical operators, Phys. Rev. A83, 042330 (2011)

    2011

  62. [62]

    Vijay, J

    S. Vijay, J. Haah, and L. Fu, Fracton topological order, generalized lattice gauge theory, and duality, Phys. Rev. B94, 235157 (2016)

    2016

  63. [63]

    Pretko, Subdimensional particle structure of higher ranku(1) spin liquids, Phys

    M. Pretko, Subdimensional particle structure of higher ranku(1) spin liquids, Phys. Rev. B95, 115139 (2017)

    2017

  64. [64]

    Gorantla, H

    P. Gorantla, H. T. Lam, N. Seiberg, and S.-H. Shao, Low-energy limit of some exotic lattice theories and uv/ir mixing, Phys. Rev. B104, 235116 (2021)

    2021

  65. [65]

    Seiberg and S.-H

    N. Seiberg and S.-H. Shao, ExoticU(1) symmetries, du- ality, and fractons in 3+1-dimensional quantum field theory, SciPost Phys.9, 046 (2020)

    2020

  66. [66]

    Haah, Bifurcation in entanglement renormalization group flow of a gapped spin model, Physical Review B 89, 075119 (2014)

    J. Haah, Bifurcation in entanglement renormalization group flow of a gapped spin model, Physical Review B 89, 075119 (2014)

    2014

  67. [67]

    Shirley, K

    W. Shirley, K. Slagle, and X. Chen, Foliated fracton order in the checkerboard model, Physical Review B 99, 115123 (2019)

    2019

  68. [68]

    A. Dua, P. Sarkar, D. J. Williamson, and M. Cheng, Bi- furcating entanglement-renormalization group flows of fracton stabilizer models, Physical Review Research2, 033021 (2020)

    2020

  69. [69]

    Y. F. Zhang and S. Gopalakrishnan, Stability of mixed- state phases under weak decoherence, arXiv preprint arXiv:2511.01976 (2025)

    arXiv 2025

  70. [70]

    Kardar, G

    M. Kardar, G. Parisi, and Y.-C. Zhang, Dynamic scaling of growing interfaces, Physical Review Letters56, 889 (1986)

    1986

  71. [71]

    Bassler and B

    K. Bassler and B. Schmittmann, Critical dynamics of nonconserved ising-like systems, Physical review letters 73, 3343 (1994)

    1994

  72. [72]

    Hwa and M

    T. Hwa and M. Kardar, Dissipative transport in open systems: An investigation of self-organized criticality, Physical review letters62, 1813 (1989)

    1989

  73. [73]

    Hwa and M

    T. Hwa and M. Kardar, Avalanches, hydrodynamics, and discharge events in models of sandpiles, Physical Review A45, 7002 (1992)

    1992

  74. [74]

    J. Y. Lee, C.-M. Jian, and C. Xu, Quantum criticality under decoherence or weak measurement, PRX Quan- tum4, 030317 (2023)

    2023

  75. [75]

    L. A. Lessa, R. Ma, J.-H. Zhang, Z. Bi, M. Cheng, and C. Wang, Strong-to-weak spontaneous symme- try breaking in mixed quantum states, arXiv preprint arXiv:2405.03639 (2024)

    Pith/arXiv arXiv 2024

  76. [76]

    Ginelli, V

    F. Ginelli, V. Ahlers, R. Livi, D. Mukamel, A. Pikovsky, A. Politi, and A. Torcini, From multiplicative noise to directed percolation in wetting transitions, Physical Re- view E68, 065102 (2003)

    2003

  77. [77]

    Chen and T

    Y.-H. Chen and T. Grover, Local reversibility and di- vergent markov length in 1+ 1-d directed percolation, arXiv preprint arXiv:2512.07220 (2025)

    arXiv 2025

  78. [78]

    Zeng and D.-L

    B. Zeng and D.-L. Zhou, Topological and error- correcting properties for symmetry-protected topolog- ical order, Europhysics Letters113, 56001 (2016)

    2016

  79. [79]

    Levin and X.-G

    M. Levin and X.-G. Wen, Detecting topological order in a ground state wave function, Phys. Rev. Lett.96, 110405 (2006)

    2006

  80. [80]

    Kitaev and J

    A. Kitaev and J. Preskill, Topological entanglement en- tropy, Physical review letters96, 110404 (2006)

    2006

Showing first 80 references.