Interacting Symmetry-Protected Topological Phases Out of Equilibrium

Max McGinley; Nigel R. Cooper

REVIEW 2 major objections 5 minor 83 references

Interacting Symmetry-Protected Topological Phases Out of Equilibrium

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper classifies interacting symmetry-protected phases under finite-time symmetric unitary evolution: only the unitary subgroup of the symmetries survives, so time-reversal-only edge peaks broaden under low-frequency noise.

desk verdict A strong, likely-correct classification of interacting SPT phases out of equilibrium, with a real completeness gap and a testable noise-broadening prediction. read the letter →

arxiv 1908.06875 v2 pith:NKYMR4HP submitted 2019-08-19 cond-mat.str-el cond-mat.mes-hallcond-mat.quant-gasquant-ph

Max McGinley , Nigel R. Cooper This is my paper · ORCID

classification cond-mat.str-elcond-mat.mes-hallcond-mat.quant-gasquant-ph

keywords symmetry-protectedtopologicalphasesnon-equilibriumdynamicsgroupcohomologytime-reversalsymmetryantiunitarysymmetriesmatrixproductstatesedgemodesquantumnoise

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Topological phases are usually defined by what can be adiabatically connected to what. This paper replaces adiabatic deformation with finite-time unitary evolution generated by a symmetry-respecting Hamiltonian, and asks which short-ranged entangled states remain distinct under that more permissive relation. The answer is a non-equilibrium topological classification: in $d$ spatial dimensions it is the image of the restriction map $\mathrm{Res}^{d+1}: H^{d+1}[G,U_T(1)] \to H^{d+1}[G_T,U(1)]$, and in one dimension it reduces to the pair $(\omega,\alpha)$ with $\omega^2=\alpha^2=1$. The consequence is that antiunitary (time-reversal) symmetries, which protect many equilibrium phases, cannot protect topology under generic time evolution, because unitary evolution breaks them at intermediate times. This matters experimentally: the sharp zero-frequency spectroscopic peak of protected edge modes broadens under low-frequency noise precisely when the system is trivial in the non-equilibrium classification, as could be tested in the Rydberg-atom chain realization of an SPT phase.

What carries the argument

The central object is the restriction functor $\mathrm{Res}^{d+1}: H^{d+1}[G,U_T(1)] \to H^{d+1}[G_T,U(1)]$; its image is the non-equilibrium classification. Here $G$ is the full on-site symmetry group, $G_T$ is its unitary subgroup, and $U_T(1)$ is the $G$-module of phases on which antiunitary elements act by complex conjugation. In one dimension the same object reduces to the pair $(\omega,\alpha)$, the projective factor system and the one-dimensional representation, constrained by $\omega^2=\alpha^2=1$ once time reversal is included. The restriction map does the work: it keeps precisely the topological data that survive when antiunitary symmetries are dynamically broken at intermediate times, while discarding the $\beta(T)$ and $\gamma(g)$ data that require $T$ to be a good symmetry at every instant. The Hochschild-Serre spectral sequence, a cohomological bookkeeping device relating the cohomology of a group to that of a subgroup, is the computational tool used to evaluate the image in higher dimensions.

What would settle it

If two short-ranged entangled states with identical restricted data $\omega,\alpha$ were shown to be impossible to connect by any finite-time symmetry-respecting unitary circuit (for instance by an exhaustive tensor-network search), the classification would over-count; conversely, an experiment in the Rydberg chain with the symmetry-lowering perturbation added, in which the zero-frequency edge peak stayed sharp under low-frequency noise, would refute the predicted broadening $\gamma\sim V^4\tau_n/E_g^4$.

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Extended reading notes

Core claim

The paper's central discovery is that the set of equivalence classes of symmetry-respecting short-ranged entangled wavefunctions under finite-time unitary evolution generated by a $G$-symmetric Hamiltonian is not the equilibrium classification but the image of the cohomological restriction map from the full group $G$ to its unitary subgroup $G_T$. Dynamically induced symmetry breaking is the mechanism: an antiunitary generator $T$ fails to commute with the evolution operator because $T e^{-i\hat H t}T^{-1}=e^{+i\hat H t}$, so at intermediate times only the unitary subgroup $G_T$ is realized, and the invariants $\beta(T)$ and $\gamma(g)$ that record how time reversal is projectively represented become undefined. Only the data that remain well-defined throughout, namely $\omega$ and $\alpha$ with $\omega^2=\alpha^2=1$, can obstruct the connection. Computed through the Hochschild-Serre spectral sequence for dimensions 0 through 3 and checked against two exactly solvable models, this yields the tables of non-equilibrium classifications. Physically, the classification predicts that noise-induced broadening of edge-mode spectral peaks occurs exactly when the phase is trivial in this sense.

Load-bearing premise

The classification is complete only if any two short-ranged entangled states (states locally deformable to a product state) that carry the same surviving topological labels can always be connected by a finite-time evolution whose Hamiltonian respects the full symmetry group; the paper shows the labels never change along such evolutions, but it does not prove the connecting evolution always exists.

Editorial extensions

If this is right

Systems whose equilibrium protection comes only from time reversal become mutually connectable after generic finite-time dynamics, so their non-equilibrium classification collapses to the trivial group in most bosonic symmetry classes.
In interacting fermionic chains, the non-equilibrium classification of the BDI symmetry class (time-reversal-symmetric superconducting chains) reduces from $\mathbb{Z}_8$ to $\mathbb{Z}_2$, matching the earlier free-fermion result and identifying which Majorana edge features survive.
After a quench, the entanglement spectrum of an SPT state remains gapless only if the initial state is non-trivial in the non-equilibrium classification; otherwise it generically becomes gapped.
For low-frequency noise, the zero-frequency edge-mode peak keeps its sharp form in non-trivial classes, whereas in trivial classes it becomes a Lorentzian of width $\gamma\sim V^4\tau_n/E_g^4$.
The classification extends to spatial dimensions 0 through 3 for the listed symmetry groups, and weak topological indices out of equilibrium remain products of the non-equilibrium classifications in lower dimensions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The restriction-image recipe is likely to generalize beyond the printed tables: for any symmetry group extension, the out-of-equilibrium classification can in principle be computed from spectral-sequence data, and the same logic should apply to beyond-cohomology phases once their algebraic invariants are known.
For periodically driven systems, the present classification is a generic-drive baseline; any Floquet-engineering protocol that appears to restore antiunitary protection would be a genuinely new phenomenon outside this classification, not a contradiction of it.
The noise-broadening result offers a practical diagnostic: comparing the edge-peak width with and without symmetry-lowering perturbations, as the Rydberg chain experiment already permits, would certify whether the edge degeneracy is protected by unitary symmetries alone.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The paper is worth taking seriously. McGinley and Cooper put forward a non-equilibrium classification of interacting SPT phases under finite-time unitary evolution by symmetry-respecting Hamiltonians. The headline result—the classification is the image of the restriction map H^{d+1}[G,U_T(1)] → H^{d+1}[G_T,U(1)]—is natural and, I think, correct in essence. The main physical takeaway is that antiunitary symmetries drop out of the classification, and the zero-frequency edge-mode peak broadens under low-frequency noise exactly when the phase is trivial in this new sense. That is a sharp, testable statement.

The genuinely new parts are the interacting extension (earlier free-fermion results are reproduced), the tables for bosonic and fermionic groups, and the noise-broadening criterion. The 1D argument in Section III is clean: the projective-representation data that survive time evolution are exactly those compatible with the unitary subgroup. The higher-dimensional construction via the image of Res is a sensible cohomological guess, and the two exactly solvable 2D models in Appendix B give the right kind of support.

The main soft spot is a completeness gap. They prove the invariants are preserved under symmetric finite-time unitary evolution, but they do not prove that any two states with the same restricted data can be connected by such an evolution. Eq. (13) is asserted as the classification, not derived. In 1D the MPS structure likely closes this gap, but in higher dimensions it is a real omission. The paper would be stronger if it stated this explicitly and gave a proof or at least a careful argument. The noise calculation in Section V and Appendix C is also approximate: adiabatic and small-noise. The scaling result γ ~ V^4/(τ_n E_g^4) is plausible, but not rigorous. I would not treat it as a theorem.

The citation pattern looks fine. Self-citations to Refs. 29 and 30 are appropriate because the earlier free-fermion results are the non-interacting limit of the same construction. No fabricated entities, no fitted parameters.

Who is this for? Anyone working on nonequilibrium topology, Floquet or quench dynamics, or decoherence of topological edge modes. It deserves a serious referee, not a desk reject. The referee should push on the completeness question, but the central idea is solid enough to warrant full review.

Referee Report

2 major / 5 minor

Summary. The manuscript proposes a nonequilibrium topological classification for interacting symmetry-protected topological (SPT) phases. Two short-ranged entangled states are considered equivalent if they can be connected by finite-time unitary evolution generated by a Hamiltonian that respects the full symmetry group G. The authors argue that antiunitary symmetries are generically broken along such evolutions, leaving only the unitary subgroup G_T to protect topology, and propose that the nonequilibrium classification is the image of the restriction map Res^{d+1}: H^{d+1}[G,U_T(1)] → H^{d+1}[G_T,U(1)] (Eq. (13)). The framework is applied to 1D via projective representations and to higher dimensions via group cohomology and the Hochschild-Serre spectral sequence, yielding Tables I and II for common symmetry groups. The paper also derives a physical consequence: low-frequency classical noise broadens the zero-frequency edge-mode spectroscopic peak only when the system is trivial in this classification, and discusses a Rydberg-atom experiment where this can be tested.

Significance. If the proposed classification is correct, it provides a general and interacting framework for SPT order out of equilibrium, unifying previous free-fermion results and making concrete, falsifiable predictions. The cohomological formulation is natural, the consistency checks with exactly solvable 2D models and with the free-fermion limit are valuable, and the noise-broadening prediction is experimentally accessible. The paper is clearly written and the main idea is conceptually important.

major comments (2)

[Section IV, Eq. (13) and Section III] The central claim that the nonequilibrium classification is given by the image of the restriction map is not proven in the completeness direction. The authors argue that the restricted G_T data are invariant along finite-time evolutions generated by G-respecting Hamiltonians, which establishes that the classification is contained in im Res. However, they do not prove that any two full-G symmetric SRE states whose restricted G_T data coincide can be connected by such an evolution. In Section III, the statement that states differing only in β(T) and γ(g) 'can still be connected' is asserted without a construction or proof. Similarly, the sentence following Eq. (13) states without proof that each element of the image represents a collection of wavefunctions that can be mutually connected. This gap is load-bearing because Eq. (13) is the paper's main result. Please provide a proof of completeness (e.g., an explicit local circuit with gates generated by G-respecting Hamiltonians, or a general argument that the equivalence relation coincides with equality of the restricted invariants), or explicitly state that Eq. (13) is a conjecture supported by the examples.
[Appendix A and Tables I-II] The computation of the image of Res is carried out in detail only for the example group Z_n × Z_m × Z_2^T. The entries for the other symmetry groups in Tables I and II are stated to follow 'in similar ways' or from the triviality of certain homomorphisms, without sufficient detail for the reader to verify the results. Since these tables constitute the concrete predictions of the paper, the derivation should be documented for each nontrivial row, for instance by giving the relevant spectral sequence pages or the argument that the image is trivial. This is particularly important because the completeness of the 1D image claim (that the image equals the set of (ω,α) with ω²=1 and α²=1) is also not explicitly verified for the groups in Table I.

minor comments (5)

[Eq. (13)] The displayed formula for im(Res^n) uses ker(d^{n+1}) and ker(d_T^{n+1}), but the cohomology group H^n is defined from ker(d^n) and im(d^{n-1}); the indices should be corrected to make the restriction map on cohomology consistent.
[Section III, paragraph on finite-time evolution] The phrase 'sub-extensive evolution time' and the bound t < L_sys/v_LR could be misleading, since a finite-depth local unitary circuit can be implemented in a time independent of the system size. Clarify that the defining requirement is representability by a finite-depth circuit, with the Lieb-Robinson bound as a consequence rather than the definition.
[Section V.A, Eq. (24)] The Lorentzian form Γ_B(ω) = |B_01| γ/(ω²+γ²) has an unspecified normalization; a normalized Lorentzian would have an additional prefactor such as 1/π or a time-integral factor. Please specify the normalization convention or state that the overall prefactor is not important for the argument.
[Section V.A] The symbol θ_B(t) is used in Eq. (23) but the text does not explicitly define it as the Berry phase difference; please define it in the main text for readability.
[Table I] The fermionic rows in Table I (e.g., Z_2^F × Z_2^T, BDI) would benefit from a footnote explaining how the interacting Z8 reduction to Z2 is obtained, since the main text covers this only briefly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the non-equilibrium classification is derived from an independent equivalence relation plus standard cohomology; self-citations are consistency checks only.

full rationale

The paper defines non-equilibrium topological equivalence as finite-time unitary evolution generated by a G-symmetric Hamiltonian, and then derives the classification as the image of the restriction map Res^{d+1}: H^{d+1}[G,U_T(1)] -> H^{d+1}[G_T,U(1)] (Eq. 13). This is not circular: the equivalence relation is an independent physical definition, and the image-of-restriction characterization is a nontrivial consequence of the equilibrium cohomology classification of SPT phases (external Ref. [35]), the observation that antiunitary symmetries need not be respected by intermediate states (derived in the text from the non-invariance of the factor i under antiunitary operators), and the Hochschild-Serre spectral sequence (standard references [59,70,71]). The 1D data (omega, alpha) with omega^2 = alpha^2 = 1 follow from the MPS/projective-representation analysis of Refs. [42-46], not from the paper's own conclusions. Citations to the authors' earlier free-fermion papers [29,30] are used only as consistency checks or to name dynamically induced symmetry breaking, and the underlying mechanism is re-derived here. The spectral-broadening prediction is calculated explicitly for Z2 x Z2 and Z2 x Z2^T and then extended by physical reasoning; it is not a fitted parameter renamed as a prediction. The main unproved step is the completeness/converse direction: the paper asserts, rather than proves, that any two full-G symmetric SRE states with equal restricted G_T data can be connected by finite-time G-symmetric unitary evolution, supporting this only with two exactly solvable 2D models. This is a proof-gap/correctness risk, not circularity, because the claimed classification is not identical to the definition by construction. No specific reduction of the central claim to its inputs or to a self-citation chain can be exhibited.

Assumptions & free parameters 0 free parameters · 4 assumptions · 1 invented entities

The central claim is parameter-free. It rests on the equilibrium cohomology classification, the finite-depth/SRE assumption, the premise that antiunitary symmetries are dynamically broken, and an unproved completeness statement. No data are fitted, and no new physical constituents are introduced; the non-equilibrium class is a mathematical label.

assumptions (4)

domain assumption Bosonic SPT phases in d spatial dimensions are classified by the group cohomology H^{d+1}[G,U_T(1)] (Chen et al., Ref. 35).
The entire non-equilibrium classification is built from the equilibrium cohomology classification, so the central claim inherits the validity of that result for all relevant G.
domain assumption Short-ranged entangled wavefunctions remain describable by area-law-entangled tensor networks throughout finite-time local evolution.
Finite-time evolution is equated with finite-depth unitary circuits via Lieb-Robinson bounds; the classification applies only while states remain area-law entangled, as stated in Section V.C.
domain assumption At intermediate times an antiunitary symmetry is generically broken, so the state only respects the unitary subgroup G_T.
The relation T e^{-iHt} T^{-1} = e^{+iHt} is used to argue dynamically induced symmetry breaking; this is the physical premise that makes the non-equilibrium classification coarser than the equilibrium one.
ad hoc to paper Every pair of states in the image of Res can be connected by a finite-time evolution with a G-respecting Hamiltonian, making Eq. (13) the complete classification.
This is the unproved converse needed for the classification. It is not a background theorem; the paper asserts it and gives examples in Appendix B.

invented entities (1)

Non-equilibrium topological class (image of Res) independent evidence
purpose: Labels equivalence classes of SRE wavefunctions under finite-time, symmetry-respecting unitary evolution; replaces equilibrium SPT labels out of equilibrium.
Although a mathematical construct rather than a physical object, it generates a falsifiable prediction: zero-frequency edge peaks broaden under low-frequency noise only for trivial classes, testable in a Rydberg chain.

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Pith. "Pith review of Interacting Symmetry-Protected Topological Phases Out of Equilibrium." pith.science (2026). https://pith.science/paper/NKYMR4HP

@misc{pith2026190806875,
  author       = {Pith},
  title        = {Pith review of: Interacting Symmetry-Protected Topological Phases Out of Equilibrium},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NKYMR4HP}},
  note         = {Machine review of arXiv:1908.06875}
}

read the original abstract

The topological features of quantum many-body wave functions are known to have profound consequences for the physics of ground-states and their low-energy excitations. We describe how topology influences the dynamics of many-body systems when driven far from equilibrium. Our results are succinctly captured by a nonequilibrium topological classification that can be used to predict universal aspects of generic isolated quantum systems as they evolve unitarily in time. By analogy to the classifications used to describe systems in equilibrium, we consider two short-ranged entangled wave functions to be topologically equivalent if they can be interconverted via finite-time unitary evolution governed by a symmetry-respecting Hamiltonian. We demonstrate that this definition captures the salient features of these systems in a broad range of nonequilibrium scenarios. As well as providing conceptual insights into the constraints imposed by topology on many-body dynamics, we discuss the practical implications of our findings. In particular, we show that the characteristic zero-frequency spectroscopic peaks associated with topologically protected edge modes will be broadened by external noise only when the system is trivial in the nonequilibrium classification.

Figures

Figures reproduced from arXiv: 1908.06875 by the authors.

**Figure 2.** FIG. 2. Illustration of the ground state of Hamiltonian [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗

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Reference graph

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The equivalence class of the projective representa- tion of G, ω(g1,g 2)∈H2(G, C)

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(3) Here,G is the group of 1D linear representations of G

(Translational invariance only) The 1D linear rep- resentationα(g)∈G . (3) Here,G is the group of 1D linear representations of G. The analysis when G features antiunitary (time- reversal) elements follows similarly. In this case, G will factorize as G = GT× ZT 2 or G = GT ⋊ ZT 2 , where the group ZT 2 = {1,T} contains the time-reversal symme- try (TRS) ge...

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(Translational invariance only) The 1D linear rep- resentationα(g), which must satisfy α2 = 1

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The projective representation of ZT 2 , β(T ) =±1

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In that context, one must also include phases which result from a spontaneous breaking of the ZP 2 fermion parity symmetry, which results in boundary Majorana modes

The projective commutation relation between sym- metry transformations in G0 and ZT 2 , γ(g)∈G /G2 (6) Note that while the above refers to bosonic models, fermionic models can also be captured using the same arguments through a Jordan-Wigner transformation [45]. In that context, one must also include phases which result from a spontaneous breaking of the ...

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The equivalence class of the projective representa- tion of G, ω(g1,g 2), which must satisfy ω2 = 1

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