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REVIEW 3 major objections 5 minor 81 references

Topology meets symmetry breaking: Hidden order, intrinsically gapless topological states and finite-temperature topological transitions

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Hidden-order topological phases survive at finite temperature.

desk verdict The exact decoupling in the closed-loop sector is solid and the finite-T SPT idea is genuinely new, but the robustness claim is proven only in the mu_tau -> infinity limit and the paper should scope it accordingly. read the letter →

arxiv 2506.03146 v2 pith:CG3HP4G2 submitted 2025-06-03 cond-mat.str-el cond-mat.quant-gasquant-ph

Reja H. Wilke , Henning Schlömer , Simon M. Linsel , Annabelle Bohrdt , Fabian Grusdt This is my paper
classification cond-mat.str-elcond-mat.quant-gasquant-ph
keywords hiddenordersymmetry-protectedtopologicalphasesfinite-temperaturephasetransitionssqueezedspaceloopgasIsinggaugetheoryBKTtransitionintrinsicallygapless
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

This paper argues that symmetry-protected topological (SPT) phases can be stable at nonzero temperature in two dimensions, when the protecting symmetry is a 1-form (loop) symmetry rather than an ordinary global symmetry. The construction couples a spin system to a fluctuating loop gas: a non-local unitary 'squeezed space' transformation decouples the spins from the loops, so the spins spontaneously break their global Z2 or U(1) symmetry in the new basis, while the bulk looks completely symmetric in the original basis. As a consequence, the paper predicts finite-temperature SPT transitions of Ising type (for the discrete Z2 case) and BKT type (for the continuous U(1) case), with the U(1) state being an intrinsically gapless SPT phase with a hidden Goldstone mode. If correct, this shows that certain topological orders can survive thermal fluctuations in two dimensions, and it connects the resulting phases to the Higgs-SPT phase of Z2 lattice gauge theory.

What carries the argument

The central object is the non-local unitary transformation $\hat{U}$, defined by flipping each spin according to the parity of negative $\hat{\tau}^z$ links along a path from a reference site (the 'squeezed space' transformation). In the closed-loop subspace, where the product of $\hat{\tau}^z$ around every plaquette equals $+1$ (enforced in the limit of an infinite plaquette coupling $\mu_\tau$), $\hat{U}$ is well-defined and exactly decouples the hidden-order Hamiltonian into two independent parts: a transverse-field Ising model (or XXZ model) for the spins and a perturbed toric code for the loop gas. Order parameters in squeezed space, such as the magnetization $M^*_S$ or the string correlator $C^*(j) = \langle \hat{S}^z_r (\prod_{l \in \mathcal{L}_j} \hat{\tau}^z_l) \hat{S}^z_j \rangle$, become non-local string operators in the original basis, which is what makes the (quasi) long-range order hidden in the bulk and detectable through edge correlations.

What would settle it

A finite-temperature simulation of the 2D HIO model at finite but large $\mu_\tau$, measuring the string order parameter $C^*(j) = \langle \hat{S}^z_r (\prod_{l \in \mathcal{L}_j} \hat{\tau}^z_l) \hat{S}^z_j \rangle$, would settle the claim: if this correlator decays exponentially with distance for any T>0 in the thermodynamic limit, the finite-temperature SPT phase exists only in the idealized closed-loop limit and is destroyed by thermally generated open strings.

Watch

Extended reading notes

Core claim

The central claim is that hidden-order SPT phases, in which a global Z2 or U(1) symmetry is spontaneously broken in squeezed space but hidden by a fluctuating loop gas, remain robust at nonzero temperature in two dimensions, producing finite-temperature SPT transitions of Ising and BKT type, with the U(1) variant being an intrinsically gapless SPT state. More precisely, the paper constructs an exact non-local unitary transformation U that decouples the Hamiltonian into a conventional spin model (a transverse-field Ising model or an XXZ model) plus a perturbed toric code acting on the loop variables. In the transformed ('squeezed') basis the spins show ordinary long-range order (or quasi-long-range order for U(1) at T>0), but in the original basis this order is hidden because the domain walls of the order parameter are bound to the fluctuating loops, turning long-range spin correlations into non-local string correlations. The paper then shows that, within the closed-loop subspace where the loop gas has a 1-form symmetry, this hidden order inherits the finite-temperature behavior of the underlying spin model, so the HO-SPT phase persists up to a critical temperature and is destroyed in an SPT transition that is invisible to local bulk order parameters. It further argues that the Z2 HO-SPT coincides with the Higgs-SPT phase of the Z2 Ising gauge theory, and that the hidden continuous case provides an example of an intrinsically gapless SPT phase at zero temperature.

Load-bearing premise

Everything about the finite-temperature hidden order relies on restricting to configurations where every flipped link closes into a loop with no open ends, which is guaranteed only in the limit of infinite loop stiffness; for any finite stiffness, thermally excited open strings break the 1-form symmetry, the unitary transformation becomes ill-defined, and the paper provides no order parameter that can still detect the hidden order.

Editorial extensions

If this is right

  • A finite-temperature SPT transition of Ising type exists in the 2D hidden-order model, protected by a 1-form symmetry, so topological order can survive thermal fluctuations even where the no-go theorem for global-symmetry SPT phases applies.
  • The hidden U(1) state is an intrinsically gapless SPT phase at T=0 with a Goldstone mode that is a string operator in the original basis, and it undergoes a hidden BKT transition at finite temperature with power-law-to-exponential edge correlations.
  • Since the Z2 HO-SPT phase coincides with the Higgs-SPT phase of the Z2 Ising gauge theory, the Higgs-SPT phase remains stable at finite temperature and can be detected through string order parameters.
  • Long-range edge correlations provide a direct experimental signature of the hidden order, even though bulk two-point correlations are short-ranged; this suggests quantum simulators of toric-code-type models are a natural place to look.
  • The thermal sequence LRO to HO-SPT to disordered shows a step-like destruction of long-range order in two steps, a scenario that may occur in correlated materials that lose their local order parameter before becoming fully symmetric.

Reading between the lines

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

  • If the construction extends to hidden SU(2) order, as the paper suggests, the finite-temperature pseudogap crossover in hole-doped cuprates could be viewed as an HO-SPT transition ending in a hidden quantum critical point.
  • The closed-loop assumption means the finite-temperature phase is defined only in the $\mu_\tau \to \infty$ limit; for finite $\mu_\tau$ a thermal crossover may replace the true transition, so an upper bound on the temperature window of hidden order would be a useful quantitative goal.
  • Because the Goldstone mode and other excitations become string operators, their spectral signatures may be invisible to local probes; non-local response functions or entanglement-based probes would be needed to observe them.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper constructs a class of exactly solvable spin–loop-gas models—the 1D and 2D HIO models with Z2 symmetry and the 2D HXYO model with U(1) symmetry—in which a global symmetry is spontaneously broken in a transformed “squeezed space” while remaining hidden in the original basis by a fluctuating loop gas. In the closed-loop sector B_box=1 (the limit µ_τ→∞), a non-local unitary U decouples the Hamiltonians exactly into a TFIM plus a toric code in a field (HIO) or an XXZ model plus a toric code in a field (HXYO). From this decoupling the authors identify hidden-order SPT phases, argue for a connection to the Higgs-SPT phase of Z2 Ising gauge theory, and predict finite-temperature SPT transitions of Ising and BKT type. The paper also presents 1D DMRG evidence for the HO phase away from the exactly solvable point λ=0.

Significance. If the finite-temperature claims could be established for the full Hamiltonian, this would be a significant conceptual advance: it would provide concrete microscopic models where SPT order survives at T>0 in 2D, including an intrinsically gapless SPT phase with a gapless Goldstone mode, and would connect hidden-order physics to Higgs-SPT phases and to possible pseudogap phenomenology. The strengths of the paper are the explicit and careful derivation of the unitary transformation and decoupled Hamiltonians (no fitted parameters), the exactness of the structure in the closed-loop sector at T=0, and the 1D DMRG support for robustness of the HO phase when λ≠0. The schematic finite-T phase diagrams follow from known Ising and BKT results in the decoupled basis, which is a reasonable strategy, but the central claim's domain of validity is narrower than the abstract suggests.

major comments (3)
  1. [Finite-temperature phase diagram, Eq. (5), footnote 1] The central claim that HO-SPT phases are robust to thermal fluctuations is proven only in the closed-loop sector B_box=1, enforced by µ_τ→∞. For any finite µ_τ and T>0, the thermal density of plaquette violations B_box=-1 is nonzero (~e^{-2µ_τ/T}); these open strings make the parity (-1)^{p_j} in Eq. (6) path-dependent, so U is not well-defined and the decoupling underlying Eq. (7) and Figs. 3 and 5 fails. The Discussion explicitly concedes that no order parameter is known without the 1-form symmetry. Hence the finite-T SPT transition is established for an idealized constrained Hamiltonian, not for the HIO/HXYO models with finite µ_τ, and the abstract's unconditional wording should be revised. I would need either a thermal open-string confinement argument or an explicit statement that finite-µ_τ stability is a conjecture.
  2. [Open strings & relation to Ising gauge theory] The stability argument for h_X>0 (Eq. (10)) is zero-temperature: the paper argues that open-string endpoints (B_box=-1) are confined by a linear force when h_S=0 (Fig. 4b) and that the HO phase persists until h_X,c>0. No finite-temperature analogue is given. Since h_X>0 explicitly breaks the 1-form symmetry, the finite-T behavior of the original Hamiltonian is precisely the regime in which the paper's exact tools fail; the claim “stable against general perturbations of the loop gas model” in the Discussion therefore goes beyond what is established.
  3. [Finite-temperature phase diagram, Figs. 3 and 5] Both finite-T phase diagrams are schematic and no two-dimensional numerical simulation is reported. The HIO boundaries are taken from the decoupled TFIM and dual TFIM/TC-F; the HXYO BKT transition is taken from the XXZ model in squeezed space. While this is a reasonable use of the exact decoupling, the original-basis order parameters are nonlocal strings and the paper itself notes the difficulty of probing the BKT-SPT transition in the bulk. A direct test in at least one finite-size 2D model (e.g., QMC for the closed-loop sector or DMRG on a cylinder) would materially strengthen the central claim.
minor comments (5)
  1. [Fig. 2 caption] The color scale for the values of λ is not defined in the caption of Fig. 2; please add a legend or color bar.
  2. [HIO finite-T discussion] The value T/J_S≈2.27 appearing in the HIO finite-T discussion is the Onsager critical temperature of the 2D Ising model; please state this explicitly with a reference.
  3. [Appendix A, Eqs. (A1) and (A6)] The notation (2 S^x_j)^{γ_j} should be defined explicitly as an operator power, since γ_j is not an integer.
  4. [Discussion] The suggestion that the cuprate pseudogap transition might realize a finite-T HO-SPT transition is speculative; please label it clearly as a conjecture distinct from the model results.
  5. [References] Reference [40] is cited as an arXiv preprint; if a journal version exists, please cite that instead.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: finite-T transitions follow from an exact decoupling, with no fitted parameters or load-bearing self-citations.

full rationale

The central finite-temperature claims are derived, not fitted: for the 2D HIO model, the unitary U defined in Eqs. (2) and (6) is shown in Appendix A to map the closed-loop sector (Eq. (5), B_box=1) of the original Hamiltonian exactly onto the sum of a transverse-field Ising model and a toric code in a field, Eq. (7); the HXYO model maps similarly onto an XXZ model plus toric code, Eq. (15). No parameter is adjusted to reproduce a target: the finite-T phase boundaries are the standard Ising and BKT transitions of the decoupled spin models, plus the known duality of the toric-code-in-a-field to a 2D TFIM. The paper explicitly states that the hidden order 'inherits the usual Ginzburg-Landau classification of SSB,' so identifying the HO-SPT transition with the squeezed-space spin transition is a transparent consequence of the exact mapping rather than a circular input. Self-citations (e.g., Refs. [43,64,65,66,70]) are used for context, percolation diagnostics, and experimental schemes, but none is load-bearing for the algebraic decoupling or the transition temperatures. A genuine limitation, flagged in the Discussion ('we are not aware of an order parameter which is able to detect the hidden SSB without the 1-form symmetry in place'), is that the finite-T robustness is established only in the mu_tau -> infinity closed-loop sector; this narrows the claim's physical scope but is not circularity.

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

No parameters are fitted to data. The model couplings (J_S, h_S, h_tau, J_tau, mu_tau, Delta, h_X, lambda) are Hamiltonian inputs; critical values in 2D are quoted from known results, not fitted here. The 'hidden order', 'hQCP', and 'HO-SPT' are phase labels, not new physical entities. The most consequential assumptions are the closed-loop limit and the conjectured equality with the Higgs-SPT phase.

assumptions (4)
  • ad hoc to paper The system is restricted to the closed-loop sector B_box = 1 on every plaquette, enforced by mu_tau -> infinity.
    Eq. (5) and the section 'Squeezed space and HIO model in 2D'; this sector is required for the unitary U to be well-defined (path-independence of the parity). All finite-T conclusions are derived inside this sector.
  • domain assumption The finite-temperature phase diagram of the toric code in a field is governed by a duality to a 2D transverse-field Ising model, with a finite-T Ising transition separating percolating and confined string configurations.
    Invoked in 'Finite-temperature phase diagram' of Section 2; relies on Wegner duality [68] and the percolation mapping [64].
  • standard math The unitary U decouples the full Hamiltonian into independent spin and link sectors, and this factorization extends to the thermal Gibbs state.
    Eqs. (7) and (15); exact within the closed-loop sector, and the thermal state factorizes under the unitary because the Hamiltonian is a sum of commuting sector Hamiltonians.
  • ad hoc to paper The HO-SPT phase is identical to the Higgs-SPT phase of the Z2 Ising gauge theory and remains stable for finite open-string coupling h_X.
    Section 'Open strings & relation to Ising gauge theory': 'We conjecture that the HO-SPT phase we constructed above is identical to the Higgs-SPT phase found by Verresen et al.' This conjecture is load-bearing for the claim that the phase extends beyond the exactly solvable limit.

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Pith. "Pith review of Topology meets symmetry breaking: Hidden order, intrinsically gapless topological states and finite-temperature topological transitions." pith.science (2026). https://pith.science/paper/CG3HP4G2

@misc{pith2026250603146,
  author       = {Pith},
  title        = {Pith review of: Topology meets symmetry breaking: Hidden order, intrinsically gapless topological states and finite-temperature topological transitions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CG3HP4G2}},
  note         = {Machine review of arXiv:2506.03146}
}
abstract

Since the discovery of phase transitions driven by topological defects, the classification of phases of matter has been significantly extended beyond Ginzburg and Landau's paradigm of spontaneous symmetry breaking (SSB). In particular, intrinsic and symmetry-protected topological (SPT) orders have been discovered in (mostly gapped) quantum many-body ground states. However, these are commonly viewed as zero-temperature phenomena, and their robustness in a gapless ground state or against thermal fluctuations remains challenging to tackle. Here we introduce an explicit construction for SPT-type states with hidden order associated with SSB: They feature (quasi) long-range correlations along appropriate edges, but short-range order in the bulk; ground state degeneracy associated with SSB; and non-local string order in the bulk. We apply our construction to predict two types of finite-temperature SPT transitions, in the Ising and BKT class respectively, where the usual signs of criticality appear despite the absence of a diverging correlation length in the bulk. While the state featuring hidden Ising order is gapped, the other SPT state associated with the BKT-SPT transition has hidden $U(1)$, or XY-order and constitutes an intrinsically gapless SPT state, associated with a gapless Goldstone mode. Specifically, in this work we discuss spins with global $\mathbb{Z}_2$ or $U(1)$ symmetry coupled to link variables constituting a loop gas model. By mapping this system to an Ising-gauge theory, we demonstrate that one of the SPT phases we construct corresponds to the Higgs-SPT phase at $T=0$ -- which we show here to remain stable at finite temperature. Our work paves the way for a more systematic search for hidden order SPT phases, including in gapless systems, and raises the question if a natural (finite-$T$) spin liquid candidate exists that realizes hidden order in the Higgs-SPT class.

Figures

Figures reproduced from arXiv: 2506.03146 by the authors.

Figure 1
Figure 1. FIG. 1: We predict a class of HO-SPT phases featuring SSB without a local bulk order parameter, in systems where [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Numerical study of the HO-SPT phase in the [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Schematic phase diagram of the 2D HIO model, [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Zero-temperature phase diagram of the double [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: Phase diagram of the HXYO model, Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Schematic phase diagram of the IGT with a [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: Hidden order transformation [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]

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