Pith. sign in

REVIEW 2 major objections 6 minor 5 cited by

A single symmetry-derived action captures dissipation and noise for photons in any insulating medium.

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

2026-08-03 13:01 UTC pith:Y4SYEXDM

load-bearing objection A solid, careful SK-EFT construction for dissipative photons; the 'most general' claim is slightly overbroad because the paper's own classification admits a·B but omits it without a stated assumption. the 2 major comments →

arxiv 2601.00605 v2 pith:Y4SYEXDM submitted 2026-01-02 hep-th cond-mat.mes-hallcond-mat.str-elhep-phnucl-th

Effective field theory for dissipative photons from higher-form symmetries

Genki Yoshimura , Yukinao Akamatsu , Yuji Hirono This is my paper
classification hep-th cond-mat.mes-hallcond-mat.str-elhep-phnucl-th
keywords effective field theoryhigher-form symmetriesdissipative photonsclosed-time-path formalismdynamical KMS symmetryfluctuation-dissipation relationentropy currentinsulating media
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 tries to establish that the low-energy behavior of photons in a warm insulating medium is fixed by symmetry alone: treating the photon as the gapless mode of a spontaneously broken one-form symmetry, and writing the most general closed-time-path action compatible with unitarity and with the discrete equilibrium (KMS) symmetry, yields a unique effective Lagrangian up to fourth order. That Lagrangian contains the familiar electric and magnetic response terms plus two relaxation coefficients and a parity-violating magnetoelectric coupling, together with noise terms whose variance is slaved to the damping. From this one action the authors derive stochastic Maxwell equations, the fluctuation–dissipation relation, the reciprocal relations between electric and magnetic response, and an entropy current with non-negative divergence. If correct, this gives a model-independent, symmetry-based derivation of dissipative electrodynamics in insulating media at finite temperature.

Core claim

In D=3+1, assuming spatial translation and rotation invariance, the most general effective Lagrangian of order m+n≤4 for the spontaneously broken one-form symmetry U(1)_r×U(1)_a→1 is eq. (3.32): L_eff = e·ε(E−τ_E Ė+v²γ Ḃ) − b·μ^{-1}(B+τ_B Ḃ+γ Ė) + (i/β)(τ_E e·εe + τ_B b·μ^{-1}b). Here (A,a) are the doubled gauge fields on the two time branches; (E,B) and (e,b) are their field strengths; ε and μ are the permittivity and inverse permeability; τ_E and τ_B are two independent relaxation times; γ is a magnetoelectric coefficient that vanishes if parity is unbroken; β is the inverse temperature. The dynamical KMS symmetry fixes the relative coefficients, and the paper proves that this action repr

What carries the argument

The construction rests on three pieces of machinery. (1) The photon is the gapless mode of a spontaneously broken one-form symmetry; after the usual doubling of fields on the forward and backward time branches, the low-energy variables are a pair of one-form gauge fields (A,a), the 'r/a' fields of the closed-time-path formalism. (2) The dynamical KMS symmetry — the discrete antilinear transformation that encodes thermal equilibrium — acts by shifting the a-type field by iβ times a time derivative of the A field; this single constraint enforces the fluctuation–dissipation relation and reciprocal relations. (3) A classification lemma (Appendix A) determines which local terms are invariant unde

Load-bearing premise

The 'most general' conclusion rests on the assumptions that no degrees of freedom beyond the doubled gauge fields enter the low-energy theory and that the appendix's term classification is complete.

What would settle it

Measure the low-frequency noise spectrum of electric displacement in a clean isotropic insulator: if the noise variance divided by the damping coefficient differs from β, or if a static magnetoelectric response appears despite isotropy, then the effective Lagrangian (3.32) is not the most general one and the symmetry argument needs revision.

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

If this is right

  • The EFT is equivalent to stochastic Maxwell equations: the electric flux density and magnetic field obey the deterministic divergence and curl equations plus Gaussian white noise with variances 2ετ_E/β and 2τ_B/(βμ).
  • Photons in the medium acquire a finite lifetime; at low momentum the frequency is ω(k)=±vk − i(τ_E+τ_B)v²k²/2 plus a helicity-dependent v²γk² correction, so both damping and circular birefringence are predicted.
  • The entropy production rate is β(τ_E Ė² + τ_B Ḃ²) up to terms vanishing on shell, so the second law follows from unitarity and the KMS symmetry.
  • Dissipative electromagnetic duality holds: the dual theory has ε↔μ and τ_E↔τ_B with γ unchanged, so electric and magnetic relaxation swap roles.
  • In the unbroken phase the same framework predicts purely diffusive magnetic modes, ω(k)=−iτ_B v²k²−2iλ_h γτ_B v⁴k³+…, instead of propagating light.

Where Pith is reading between the lines

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

  • The 'no additional degrees of freedom' assumption means the Lagrangian is universal only if phonons, magnons, or other low-energy modes are heavier than the photon; if a future experiment finds a static magnetoelectric response in a clean isotropic insulator, that would signal such hidden degrees of freedom rather than a failure of the symmetry principle.
  • The same coset-plus-KMS algorithm can likely be applied to higher-rank gauge theories, such as fracton models, to derive dissipative terms for restricted-mobility excitations.
  • The two relaxation times τ_E and τ_B are independent parameters, not fixed by symmetry; measuring both in a single material from optical dispersion and noise would directly test the claim that (3.32) is exhaustive.
  • The chiral magnetic term a·B, allowed when parity-violating symmetries are spontaneously broken, drives an exponential instability of the linearized modes; the paper leaves open how nonlinearity regularizes it, suggesting a route to inverse cascade in this EFT.

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

2 major / 6 minor

Summary. The paper develops a Schwinger–Keldysh effective field theory for photons in insulating media at finite temperature, based on the spontaneous breaking of a doubled U(1) 1-form symmetry, U(1)_+ × U(1)_- → 1. It combines the generalized coset construction with dynamical KMS symmetry, and in D = 3+1 constructs an effective Lagrangian up to power-counting order m+n ≤ 4. The authors derive equivalent Langevin equations, dispersion relations, fluctuation-dissipation relations, Onsager reciprocity, a non-negative entropy current, and a dissipative electromagnetic duality. They also discuss the unbroken phase, where diffusive symmetry emerges, and relate the SK structure to strong and weak symmetries.

Significance. If the central classification claim is correct, the paper provides a useful and systematic framework for dissipative photon EFTs, connecting higher-form symmetry breaking with Schwinger–Keldysh techniques. The explicit enumeration of symmetry-invariant terms, the transparent application of dynamical KMS symmetry, and the construction of an entropy current are valuable technical contributions. The paper also gives clear physical applications: noise and dissipation in the Langevin representation, dispersion relations with relaxation times, and electromagnetic duality. However, the claim that Eq. (3.32) is the most general Lagrangian under the stated assumptions is currently undermined by the omission of a term that the authors themselves identify as allowed by their symmetry and KMS analysis.

major comments (2)
  1. [§3.3, Eq. (3.32), Table 2, footnote 9, Appendix B] The asserted ‘most general’ Lagrangian in Eq. (3.32) omits the term a·B, even though this term is allowed under the paper’s own criteria. Eq. (3.27a) lists a·B as an invariant candidate, Table 2 marks it as dynamical-KMS-allowed for Θ = T, CT (and allowed with a Θ-odd coefficient when Θ is spontaneously broken), and Appendix B shows explicitly that its KMS variation, Eq. (B.9), is a total derivative. The term is linear in the a-type field and real, so it does not violate the unitarity conditions (2.9b)–(2.9c). Footnote 9 merely says the term is not included, without stating an additional symmetry or physical assumption that forbids it. Hence Eq. (3.32) is not the most general Lagrangian consistent with the stated symmetries, power counting, unitarity, and dynamical KMS symmetry. The omission also changes physical content: according to Appendix B it produces a CME-type contribution and a
  2. [§3.3, after Eq. (3.29)] The text states that all m+n ≤ 4 candidate terms are enumerated in Eq. (3.27), followed by the KMS constraint. Yet the later effective Lagrangian in Eq. (3.32) drops one of the KMS-allowed terms without a stated symmetry reason. This is not merely a presentation issue: the ‘most general’ claim appears in the abstract, in Section 3.3, and in Section 7. The authors should either amend the theorem to include the a·B term or explicitly add a ‘we assume unbroken parity / zero chiral chemical potential’ condition at the start of the construction, and then consistently repeat this condition in the summary. This is a load-bearing point for the paper’s main result.
minor comments (6)
  1. [§2, Eq. (2.25)] The notation ∂−t is unusual and could be defined more clearly when it first appears; it seems to mean the derivative with respect to the transformed time argument.
  2. [Appendix C, Eq. (C.8)] In the fourth line of Eq. (C.8), the left-hand side appears to be ilde a_i, not ilde A_i. Please correct this typographical error.
  3. [§5.1, paragraph before Eq. (5.8)] There is a typo: “seel also” should be “see also”.
  4. [§7, Summary] The paragraph beginning “Examples include systems lacking spatial rotational or translational symmetry” is duplicated immediately afterwards with the wording “rotational or parity symmetry”. One copy should be removed.
  5. [§3.3, Table 2] The entries for C, P, CP are useful, but the table caption could clarify whether ‘+’ and ‘−’ refer only to the KMS condition or also to the discrete symmetry eigenvalue. This is clear in the text, but a sentence in the caption would help.
  6. [§4.1, Eq. (4.2)] After Eq. (4.2), the notation D and H is introduced, but the reader must infer that D is the electric flux density and H the magnetic field. A short explicit identification would improve readability.

Circularity Check

0 steps flagged

No circular reduction; the EFT derivation is self-contained. The omitted a·B term is a completeness gap, not circularity.

full rationale

The paper's construction is not circular in the sense relevant here. The dynamical KMS symmetry is derived from the thermal KMS condition of the generating functional in Section 2.1, not assumed in the form of the final fluctuation-dissipation relations. The effective Lagrangian (3.32) is assembled by enumerating rotationally invariant, U(1)+ x U(1)- symmetric terms under the stated power counting (Section 3.3, Tables 1-2, Appendix A), then imposing unitarity (2.9) and dynamical KMS symmetry. The FDR, Onsager reciprocity, and entropy current in Section 4 follow from those structural inputs; they are not fitted parameters renamed as predictions. The transport coefficients tau_E, tau_B, gamma, etc. remain undetermined by the derivation, so the claim is a symmetry classification, not a fit masquerading as a prediction. There is minor self-citation (Refs. [14] and [50] include coauthor Hirono), but [14] is used only as background for the higher-form coset construction and the relevant classification is re-derived in Appendix A, while [50] appears only in a future-directions remark; neither is load-bearing. The main caveat is a correctness/completeness issue rather than circularity: footnote 9 says 'we do not include the term a.B' even though Tables 2 and Appendix B show that term is dynamical-KMS-compatible for Theta = T, CT within the same power counting, so the 'most general' claim is overbroad unless an additional parity/no-CME assumption is explicitly stated. That gap does not reduce Eq. (3.32) to its inputs by construction.

Axiom & Free-Parameter Ledger

6 free parameters · 7 axioms · 0 invented entities

The paper is an EFT construction; its input axioms are the spontaneously broken higher-form symmetry pattern, the no-extra-dofs assumption, KMS symmetry, and standard differential geometry. The transport coefficients are undetermined Wilson coefficients rather than fitted numbers.

free parameters (6)
  • epsilon (electric permittivity)
    Wilson coefficient in eq. (3.32); not fitted in this paper, but undetermined by symmetry.
  • mu (magnetic permeability)
    Wilson coefficient in eq. (3.32); not fitted.
  • tau_E (electric relaxation time)
    Dissipative coefficient in eq. (3.32); constrained only by tau_E >= 0.
  • tau_B (magnetic relaxation time)
    Dissipative coefficient in eq. (3.32); constrained only by tau_B >= 0.
  • gamma (parity-odd magnetoelectric coupling)
    Appears in eq. (3.32); vanishes if parity symmetry is unbroken.
  • kappa (nonlinear magnetoelectric coefficient, unbroken phase)
    Appears in eq. (5.25) for the unbroken phase; not fitted.
axioms (7)
  • domain assumption U(1) 1-form symmetry is spontaneously broken in the insulating medium, with the photon as the resulting Nambu-Goldstone mode.
    Invoked in Section 3.1 to identify A and a as the low-energy degrees of freedom; central to the coset construction.
  • domain assumption No additional light degrees of freedom beyond the doubled gauge fields.
    Stated in Section 3.3; excludes phonons, magnons, etc. that would alter the 'most general' claim.
  • domain assumption Spacetime translational and spatial rotational symmetry in the medium rest frame u^mu.
    Stated in Section 3.3; simplifies the classification; anisotropies are postponed to future work.
  • standard math Local terms with two factors of A cannot be invariant under U(1)^{[1]} (classification lemma).
    Proved in Appendix A; underpins the claim that only a wedge F^n-type terms survive.
  • domain assumption Dynamical KMS symmetry is the correct encoding of thermal equilibrium.
    Review in Section 2; used to fix the noise-dissipation relation and entropy current.
  • domain assumption The discrete symmetry Theta is not spontaneously broken unless otherwise stated.
    Used in Section 2.1 and Appendix C.3; broken Theta is handled by introducing an order parameter c.
  • standard math Poincare lemma and Stokes' theorem for differential forms.
    Used in Section 4.5 duality and in Appendix A.

pith-pipeline@v1.3.0-alltime-deepseek · 35470 in / 18650 out tokens · 182589 ms · 2026-08-03T13:01:52.521974+00:00 · methodology

0 comments
read the original abstract

Recent developments in generalized symmetries have provided new insights into quantum field theories. Within this framework, photons can be understood as Nambu-Goldstone modes associated with a spontaneously broken higher-form symmetry. In this work, we develop an effective field theory that builds on this symmetry structure to describe the real-time dynamics of photons in insulating media at finite temperature. Combining the Schwinger-Keldysh formalism with the generalized coset construction, we formulate a symmetry-based effective action that incorporates both conservative and dissipative effects. The effective theory implements the dynamical Kubo-Martin-Schwinger symmetry, ensuring consistency with the fluctuation-dissipation relation and Onsager's reciprocal relations. Within this framework, we derive the entropy current associated with dissipative photon dynamics and demonstrate the non-negativity of its divergence, in accordance with the second law of thermodynamics. We also clarify the symmetry origin of the gauge redundancy in the unbroken phase within the Schwinger-Keldysh framework, relating it to strong and weak realizations of higher-form symmetries. Our results provide a model-independent effective description of photon dynamics in insulating media at finite temperature.

discussion (0)

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

Forward citations

Cited by 5 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Bottom-up open EFT for non-Abelian gauge theory with dynamical color environment

    hep-th 2026-05 unverdicted novelty 7.0

    Develops a local open EFT for non-Abelian gauge theories using dynamical color-frame variables and color-current sectors in Schwinger-Keldysh formalism, yielding nonlocal dissipative kernels and naturally incorporatin...

  2. Schwinger-Keldysh Path Integral for Gauge theories

    hep-th 2026-04 unverdicted novelty 7.0

    Constructs a manifestly diagonal-BRST-invariant Schwinger-Keldysh path integral for open non-Abelian gauge theories with arbitrary physical initial states, yielding Ward-Takahashi-Slavnov-Taylor identities and a Keldy...

  3. Schwinger-Keldysh Path Integral for Gauge theories

    hep-th 2026-04 unverdicted novelty 6.0

    A manifestly BRST-invariant Schwinger-Keldysh path integral is derived for non-Abelian gauge theories with generic initial states, enabling perturbative Ward-Takahashi-Slavnov-Taylor identities and Open EFT expansions...

  4. Effective Field Theory for Superconducting Phase Transitions

    hep-th 2026-04 unverdicted novelty 6.0

    An effective field theory for superconducting phase transitions is constructed via Schwinger-Keldysh formalism, reproducing Ginzburg-Landau equations upon truncation while showing overdamped Higgs modes and complex re...

  5. Gauging Open EFTs from the top down

    hep-th 2025-12 unverdicted novelty 6.0

    Derives gauge-invariant influence functionals for photons and Stueckelberg fields in open U(1) gauge EFTs via BRST on the in-in contour after integrating out matter.

Reference graph

Works this paper leans on

59 extracted references · 47 linked inside Pith · cited by 4 Pith papers

  1. [1]

    Landau,On the theory of phase transitions,Zh

    L.D. Landau,On the theory of phase transitions,Zh. Eksp. Teor. Fiz.7(1937) 19

    1937

  2. [2]

    Landau and E.M

    L.D. Landau and E.M. Lifshitz,Statistical Physics, vol. 5 ofCourse of Theoretical Physics, Elsevier, Oxford, 3rd (reprint) ed. (2013)

    2013

  3. [3]

    Weinberg,The quantum theory of fields

    S. Weinberg,The quantum theory of fields. Vol. 2: Modern applications, Cambridge University Press (8, 2013), 10.1017/CBO9781139644174

    doi:10.1017/cbo9781139644174 2013

  4. [4]

    Chaikin and T.C

    P.M. Chaikin and T.C. Lubensky,Principles of Condensed Matter Physics, Cambridge University Press, Cambridge, UK (1995), 10.1017/CBO9780511813467

    doi:10.1017/cbo9780511813467 1995

  5. [5]

    Coleman, J

    S.R. Coleman, J. Wess and B. Zumino,Structure of phenomenological Lagrangians. 1., Phys.Rev.177(1969) 2239

    1969

  6. [6]

    Callan, Curtis G., S.R

    J. Callan, Curtis G., S.R. Coleman, J. Wess and B. Zumino,Structure of phenomenological Lagrangians. 2.,Phys.Rev.177(1969) 2247

    1969

  7. [7]

    Gaiotto, A

    D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett,Generalized Global Symmetries,JHEP 02(2015) 172 [1412.5148]. – 44 –

    Pith/arXiv arXiv 2015

  8. [8]

    Bhardwaj, L.E

    L. Bhardwaj, L.E. Bottini, L. Fraser-Taliente, L. Gladden, D.S.W. Gould, A. Platschorre et al.,Lectures on generalized symmetries,Phys. Rept.1051(2024) 1 [2307.07547]

    Pith/arXiv arXiv 2024

  9. [9]

    Schafer-Nameki,ICTP lectures on (non-)invertible generalized symmetries,Phys

    S. Schafer-Nameki,ICTP lectures on (non-)invertible generalized symmetries,Phys. Rept. 1063(2024) 1 [2305.18296]

    Pith/arXiv arXiv 2024

  10. [10]

    Gomes,An introduction to higher-form symmetries,SciPost Phys

    P.R.S. Gomes,An introduction to higher-form symmetries,SciPost Phys. Lect. Notes74 (2023) 1 [2303.01817]

    Pith/arXiv arXiv 2023

  11. [11]

    Brennan and S

    T.D. Brennan and S. Hong,Introduction to Generalized Global Symmetries in QFT and Particle Physics,2306.00912

    Pith/arXiv arXiv

  12. [12]

    Lake,Higher-form symmetries and spontaneous symmetry breaking,1802.07747

    E. Lake,Higher-form symmetries and spontaneous symmetry breaking,1802.07747

    Pith/arXiv arXiv

  13. [13]

    Hofman and N

    D.M. Hofman and N. Iqbal,Goldstone modes and photonization for higher form symmetries, SciPost Phys.6(2019) 006 [1802.09512]

    Pith/arXiv arXiv 2019

  14. [14]

    Hidaka, Y

    Y. Hidaka, Y. Hirono and R. Yokokura,Counting Nambu-Goldstone Modes of Higher-Form Global Symmetries,Phys. Rev. Lett.126(2021) 071601 [2007.15901]

    Pith/arXiv arXiv 2021

  15. [15]

    Watanabe and H

    H. Watanabe and H. Murayama,Unified Description of Nambu-Goldstone Bosons without Lorentz Invariance,Phys. Rev. Lett.108(2012) 251602 [1203.0609]

    Pith/arXiv arXiv 2012

  16. [16]

    Hidaka,Counting rule for Nambu-Goldstone modes in nonrelativistic systems,Phys

    Y. Hidaka,Counting rule for Nambu-Goldstone modes in nonrelativistic systems,Phys. Rev. Lett.110(2013) 091601 [1203.1494]

    Pith/arXiv arXiv 2013

  17. [17]

    Minami and Y

    Y. Minami and Y. Hidaka,Spontaneous symmetry breaking and Nambu-Goldstone modes in dissipative systems,Phys. Rev. E97(2018) 012130 [1509.05042]

    Pith/arXiv arXiv 2018

  18. [18]

    Hidaka and Y

    Y. Hidaka and Y. Minami,Spontaneous symmetry breaking and Nambu–Goldstone modes in open classical and quantum systems,PTEP2020(2020) 033A01 [1907.08241]

    Pith/arXiv arXiv 2020

  19. [19]

    Kamenev,Field Theory of Non-Equilibrium Systems, Cambridge University Press (2011), 10.1017/CBO9781139003667

    A. Kamenev,Field Theory of Non-Equilibrium Systems, Cambridge University Press (2011), 10.1017/CBO9781139003667

    doi:10.1017/cbo9781139003667 2011

  20. [20]

    Crossley, P

    M. Crossley, P. Glorioso and H. Liu,Effective field theory of dissipative fluids,JHEP09 (2017) 095 [1511.03646]

    Pith/arXiv arXiv 2017

  21. [21]

    Glorioso, M

    P. Glorioso, M. Crossley and H. Liu,Effective field theory of dissipative fluids (II): classical limit, dynamical KMS symmetry and entropy current,JHEP09(2017) 096 [1701.07817]

    Pith/arXiv arXiv 2017

  22. [22]

    Glorioso and H

    P. Glorioso and H. Liu,The second law of thermodynamics from symmetry and unitarity, 1612.07705

    Pith/arXiv arXiv

  23. [23]

    Haehl, R

    F.M. Haehl, R. Loganayagam and M. Rangamani,Adiabatic hydrodynamics: The eightfold way to dissipation,JHEP05(2015) 060 [1502.00636]

    Pith/arXiv arXiv 2015

  24. [24]

    Haehl, R

    F.M. Haehl, R. Loganayagam and M. Rangamani,Schwinger-Keldysh formalism. Part I: BRST symmetries and superspace,JHEP06(2017) 069 [1610.01940]

    Pith/arXiv arXiv 2017

  25. [25]

    Haehl, R

    F.M. Haehl, R. Loganayagam and M. Rangamani,Effective Action for Relativistic Hydrodynamics: Fluctuations, Dissipation, and Entropy Inflow,JHEP10(2018) 194 [1803.11155]

    Pith/arXiv arXiv 2018

  26. [26]

    Jensen, R

    K. Jensen, R. Marjieh, N. Pinzani-Fokeeva and A. Yarom,A panoply of Schwinger-Keldysh transport,SciPost Phys.5(2018) 053 [1804.04654]

    Pith/arXiv arXiv 2018

  27. [27]

    Harder, P

    M. Harder, P. Kovtun and A. Ritz,On thermal fluctuations and the generating functional in relativistic hydrodynamics,JHEP07(2015) 025 [1502.03076]. – 45 –

    Pith/arXiv arXiv 2015

  28. [28]

    Glorioso and H

    P. Glorioso and H. Liu,Lectures on non-equilibrium effective field theories and fluctuating hydrodynamics,1805.09331

    Pith/arXiv arXiv

  29. [29]

    Lessa, R

    L.A. Lessa, R. Ma, J.-H. Zhang, Z. Bi, M. Cheng and C. Wang,Strong-to-Weak Spontaneous Symmetry Breaking in Mixed Quantum States,PRX Quantum6(2025) 010344 [2405.03639]

    Pith/arXiv arXiv 2025

  30. [30]

    D. Gu, Z. Wang and Z. Wang,Spontaneous symmetry breaking in open quantum systems: strong, weak, and strong-to-weak,2406.19381

    Pith/arXiv arXiv

  31. [31]

    Huang, M

    X. Huang, M. Qi, J.-H. Zhang and A. Lucas,Hydrodynamics as the effective field theory of strong-to-weak spontaneous symmetry breaking,Phys. Rev. B111(2025) 125147 [2407.08760]

    Pith/arXiv arXiv 2025

  32. [32]

    Akyuz, G

    C.O. Akyuz, G. Goon and R. Penco,The Schwinger-Keldysh coset construction,JHEP06 (2024) 004 [2306.17232]

    Pith/arXiv arXiv 2024

  33. [33]

    Vardhan, S

    S. Vardhan, S. Grozdanov, S. Leutheusser and H. Liu,Effective field theories of dissipative fluids with one-form symmetries,JHEP05(2025) 184 [2408.12868]

    Pith/arXiv arXiv 2025

  34. [34]

    Salcedo, T

    S.A. Salcedo, T. Colas and E. Pajer,An Open Effective Field Theory for light in a medium, JHEP03(2025) 138 [2412.12299]

    Pith/arXiv arXiv 2025

  35. [35]

    Kaplanek, M

    G. Kaplanek, M. Mylova and A.J. Tolley,Gauging Open EFTs from the top down, 2512.17089

    Pith/arXiv arXiv

  36. [36]

    Baker-Jarvis, B

    J. Baker-Jarvis, B. Riddle and M.D. Janezic,Dielectric polarization evolution equations and relaxation times,Physical Review E75(2007) 056612

    2007

  37. [37]

    Glorioso and D.T

    P. Glorioso and D.T. Son,Effective field theory of magnetohydrodynamics from generalized global symmetries,1811.04879

    Pith/arXiv arXiv

  38. [38]

    Buˇ ca and T

    B. Buˇ ca and T. Prosen,A note on symmetry reductions of the lindblad equation: transport in constrained open spin chains,New Journal of Physics14(2012) 073007

    2012

  39. [39]

    Albert and L

    V.V. Albert and L. Jiang,Symmetries and conserved quantities in lindblad master equations, Phys. Rev. A89(2014) 022118

    2014

  40. [40]

    Akamatsu, M

    Y. Akamatsu, M. Asakawa and S. Kajimoto,Dynamics of in-medium quarkonia in SU(3) and SU(2) gauge theories,Phys. Rev. D105(2022) 054036 [2108.06921]

    Pith/arXiv arXiv 2022

  41. [41]

    Hongo, N

    M. Hongo, N. Sogabe, M.A. Stephanov and H.-U. Yee,Schwinger-Keldysh effective action for hydrodynamics with approximate symmetries,2411.08016

    Pith/arXiv arXiv

  42. [42]

    Nielsen and S

    H.B. Nielsen and S. Chadha,On How to Count Goldstone Bosons,Nucl. Phys. B105(1976) 445

    1976

  43. [43]

    Sogabe and N

    N. Sogabe and N. Yamamoto,Triangle Anomalies and Nonrelativistic Nambu-Goldstone Modes of Generalized Global Symmetries,Phys. Rev. D99(2019) 125003 [1903.02846]

    Pith/arXiv arXiv 2019

  44. [44]

    You,Quantum liquids: Emergent higher-rank gauge theory and fractons,Annual Review of Condensed Matter Physics16(2025) 83

    Y. You,Quantum liquids: Emergent higher-rank gauge theory and fractons,Annual Review of Condensed Matter Physics16(2025) 83

    2025

  45. [45]

    Gromov and L

    A. Gromov and L. Radzihovsky,Colloquium: Fracton matter,Rev. Mod. Phys.96(2024) 011001 [2211.05130]

    Pith/arXiv arXiv 2024

  46. [46]

    Pretko and L

    M. Pretko and L. Radzihovsky,Fracton-Elasticity Duality,Phys. Rev. Lett.120(2018) 195301 [1711.11044]

    Pith/arXiv arXiv 2018

  47. [47]

    Pretko,Subdimensional Particle Structure of Higher Rank U(1) Spin Liquids,Phys

    M. Pretko,Subdimensional Particle Structure of Higher Rank U(1) Spin Liquids,Phys. Rev. B95(2017) 115139 [1604.05329]. – 46 –

    Pith/arXiv arXiv 2017

  48. [48]

    Pretko,Generalized Electromagnetism of Subdimensional Particles: A Spin Liquid Story, Phys

    M. Pretko,Generalized Electromagnetism of Subdimensional Particles: A Spin Liquid Story, Phys. Rev. B96(2017) 035119 [1606.08857]

    Pith/arXiv arXiv 2017

  49. [49]

    P´ erez, S

    A. P´ erez, S. Prohazka and A. Seraj,Fracton Infrared Triangle,Phys. Rev. Lett.133(2024) 021603 [2310.16683]

    Pith/arXiv arXiv 2024

  50. [50]

    Hirono, M

    Y. Hirono, M. You, S. Angus and G.Y. Cho,A symmetry principle for gauge theories with fractons,SciPost Phys.16(2024) 050 [2207.00854]

    Pith/arXiv arXiv 2024

  51. [51]

    Kharzeev, L.D

    D.E. Kharzeev, L.D. McLerran and H.J. Warringa,The Effects of topological charge change in heavy ion collisions: ’Event by event P and CP violation ’,Nucl. Phys. A803(2008) 227 [0711.0950]

    Pith/arXiv arXiv 2008

  52. [52]

    Fukushima, D.E

    K. Fukushima, D.E. Kharzeev and H.J. Warringa,The Chiral Magnetic Effect,Phys.Rev. D78(2008) 074033 [0808.3382]

    Pith/arXiv arXiv 2008

  53. [53]

    Joyce and M.E

    M. Joyce and M.E. Shaposhnikov,Primordial magnetic fields, right-handed electrons, and the Abelian anomaly,Phys. Rev. Lett.79(1997) 1193 [astro-ph/9703005]

    Pith/arXiv arXiv 1997

  54. [54]

    Akamatsu and N

    Y. Akamatsu and N. Yamamoto,Chiral Plasma Instabilities,Phys. Rev. Lett.111(2013) 052002 [1302.2125]

    Pith/arXiv arXiv 2013

  55. [55]

    Boyarsky, J

    A. Boyarsky, J. Frohlich and O. Ruchayskiy,Self-consistent evolution of magnetic fields and chiral asymmetry in the early Universe,Phys. Rev. Lett.108(2012) 031301 [1109.3350]

    Pith/arXiv arXiv 2012

  56. [56]

    Tashiro, T

    H. Tashiro, T. Vachaspati and A. Vilenkin,Chiral Effects and Cosmic Magnetic Fields,Phys. Rev. D86(2012) 105033 [1206.5549]

    Pith/arXiv arXiv 2012

  57. [57]

    Hirono, D

    Y. Hirono, D. Kharzeev and Y. Yin,Self-similar inverse cascade of magnetic helicity driven by the chiral anomaly,Phys. Rev.D92(2015) 125031 [1509.07790]

    Pith/arXiv arXiv 2015

  58. [58]

    Yamamoto,Scaling laws in chiral hydrodynamic turbulence,Phys

    N. Yamamoto,Scaling laws in chiral hydrodynamic turbulence,Phys. Rev. D93(2016) 125016 [1603.08864]

    Pith/arXiv arXiv 2016

  59. [59]

    Hattori, Y

    K. Hattori, Y. Hirono, H.-U. Yee and Y. Yin,MagnetoHydrodynamics with chiral anomaly: phases of collective excitations and instabilities,Phys. Rev. D100(2019) 065023 [1711.08450]. – 47 –

    Pith/arXiv arXiv 2019