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Intertwined order of generalized global symmetries

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Pith's one-line read A topological coupling locks the breaking of zero-form and one-form $\mathbb{Z}_N$ symmetries in 2+1 dimensions.

desk verdict A strong, checkable lattice model that realizes oblique phases and non-invertible symmetry breaking; the only real caveat is an explicitly stated assumption about nonzero-Theta robustness. read the letter →

arxiv 2412.02748 v2 pith:U2O2LLCQ submitted 2024-12-03 cond-mat.str-el hep-th

Benjamin Moy , Eduardo Fradkin This is my paper · ORCID
classification cond-mat.str-elhep-th
keywords generalizedglobalsymmetriesone-formsymmetryobliquephasessymmetry-protectedtopologicalordernon-invertibleZ_Nlatticegaugetheoryclockmodelcoupling
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 establishes that zero-form and one-form global symmetries can be forced to break together. The setting is a 2+1-dimensional Euclidean lattice model in which a $\mathbb{Z}_N$ clock model and a $\mathbb{Z}_N$ gauge theory interact only through a topological term, an analogue of the $\theta$ term; the topological term binds the charges of one symmetry to the disorder operators of the other. The central claim is that in every phase of this model, if the $\mathbb{Z}_N^{(0)}$ symmetry spontaneously breaks to a nontrivial subgroup, then the $\mathbb{Z}_N^{(1)}$ symmetry breaks to the same subgroup, giving 'oblique' phases labeled by $(N,p)$ with coexisting ordinary symmetry breaking, topological order, and SPT response. If correct, the model realizes every SPT phase protected by $\mathbb{Z}_N^{(0)}\times\mathbb{Z}_N^{(1)}$, and its gauged axion extension realizes a spontaneously broken non-invertible symmetry whose domain walls fuse non-group-like. The paper also constructs gapped and gapless boundary states for these phases, so it offers a microscopic route to intertwined generalized order.

What carries the argument

The load-bearing object is the topological interaction of Eq. (2.4), a lattice analogue of the $\theta$ term that implements the generalized Witten effect: it shifts quantum numbers to $(n+\frac{\Theta}{2\pi}m,\,m)$ for spins and monopoles and similarly for electric charges and vortices, so that vortices carry electric charge and monopoles carry fractional $\mathbb{Z}_N$ spin. The second engine is the pair of dualities $S$ and $T$ -- Kramers-Wannier-like duality and periodicity/SPT-stacking -- acting on the coupling $\tau=\frac{\Theta}{2\pi}+i\frac{2\pi}{Nge}$ and on the charges of condensed operators; tracking these charges locates the oblique phases and the SPTs in the phase diagram. Deep inside an oblique phase the effective field theory is the gauged BF-type action $S=\frac{iNp}{2\pi}\int c\wedge b+\cdots$, whose gauge-invariant operators produce the $\mathbb{Z}_{N/L}$ clock-shift algebra, the topological order, and the boundary anomalies.

What would settle it

Run a direct numerical simulation of the Villain lattice action, Eq. (2.1), at $\Theta/2\pi=-1/p$ with small couplings $e,g$ on a torus: if the ground-state degeneracy is not $[\gcd(N,p)]^3$, or if the Wilson-loop and order-parameter correlators do not show $\mathbb{Z}_N^{(0)}\times\mathbb{Z}_N^{(1)}\to\mathbb{Z}_{N/L}^{(0)}\times\mathbb{Z}_{N/L}^{(1)}$, the central claim fails.

Watch

Extended reading notes

Core claim

On its own terms, the paper claims that the phases of the model in Eq. (2.1) are controlled by the generalized Witten effect plus $\mathrm{SL}(2,\mathbb{Z})$ duality. Condensing bound states with spin charge $N$ and magnetic charge $p$, and loop operators with electric charge $N$ and vorticity $p$, produces the oblique phase $(N,p)$: the full symmetry $G=\mathbb{Z}_N^{(0)}\times\mathbb{Z}_N^{(1)}$ is broken to $H=\mathbb{Z}_{N/L}^{(0)}\times\mathbb{Z}_{N/L}^{(1)}$ with $L=\gcd(N,p)$, and the unbroken $H$ carries the SPT response $S_{\rm resp}=-\frac{iNk}{2\pi L}\int A\wedge B$, implying ground state degeneracy $L^{2g+1}$ on a genus-$g$ surface. The same mechanism yields, at $\Theta=2\pi p$ and strong coupling, every $\mathbb{Z}_N^{(0)}\times\mathbb{Z}_N^{(1)}$ SPT with response $\frac{iNp}{2\pi}\int A\wedge B$. Promoting $\Theta$ to a dynamical $\mathbb{Z}_N$ axion and gauging $G$ turns the axion symmetry into a non-invertible surface operator $U_p$ with fusion $U_{p_1}\times U_{p_2}=(\mathbb{Z}_N)^2\,U_{p_1+p_2}$; in the large-$J$ phase this non-invertible symmetry is spontaneously broken, with ground state degeneracy $\sum_{p=0}^{N-1}[\gcd(N,p)]^{2g+1}$.

Load-bearing premise

The analysis leans on the assumption that the phase diagram at $\Theta=0$, where the clock model and gauge theory decouple, continues to describe the model over a finite nonzero region of the topological coupling, so that the $S$-$T$ duality orbits of those phases are the actual phases of the model.

Editorial extensions

If this is right

  • If the central claim holds, the single lattice model of Eq. (2.1) microscopically realizes every $\mathbb{Z}_N^{(0)}\times\mathbb{Z}_N^{(1)}$ SPT with response $\frac{iNp}{2\pi}\int A\wedge B$, for every $p$ mod $N$.
  • In every oblique phase, the unbroken subgroup $H$ has both topological order and SPT response, so any gapped boundary that preserves $H$ must break the zero-form part of $H$ spontaneously; the paper gives explicit electric and magnetic boundary states realizing this.
  • For $L=\gcd(N,p)\ge 4$, the gapless boundary state of the SPT is a quantum critical point or critical line with enhanced $U(1)\times U(1)\times\mathbb{Z}_N^{(0)}\times\mathbb{Z}_N^{(1)}$ symmetry between two gapped boundary phases.
  • Gauging the $\mathbb{Z}_N$ axion model produces a non-invertible symmetry $U_p$ whose spontaneous breaking means its domain walls separate different oblique phases and obey the fusion rule $U_{p_1}\times U_{p_2}=(\mathbb{Z}_N)^2 U_{p_1+p_2}$.
  • The intertwined breaking pattern -- zero-form and one-form symmetries break to the same subgroup -- holds for every phase reachable by the general $\mathrm{SL}(2,\mathbb{Z})$ word of Eq. (2.39), not just the special point $\Theta/2\pi=-1/p$.

Reading between the lines

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

  • In the gauged axion phase, the fusion rule implies that the integer $p$ labeling domain walls is conserved mod $N$ even though the symmetry is not invertible; a numerical tensor-network calculation of the surface operator $U_p$ acting on the low-energy spectrum could test this selection rule directly.
  • If the intertwined-breaking pattern is generic, then scanning the phase diagram at rational values of $\Theta/2\pi$ given by the continued fraction (2.40) should reveal oblique phases with the same $\mathbb{Z}_N\to\mathbb{Z}_{N/L}$ locking; a Monte Carlo study of the Villain action at small couplings could look for the predicted $L^3$ degeneracy on a torus.
  • The half-gauging recipe used here -- turn one member of a mixed-anomalous pair into a non-invertible defect -- should apply to other mixed-anomaly pairs, e.g. two zero-form symmetries, yielding lattice models with non-invertible one-form symmetries and non-Abelian topological order.
  • The electric boundary condition for oblique phases, which preserves the bulk $G$ while breaking $H$ at the boundary, may serve as a template for anomaly-free boundary theories of symmetry-enriched topological orders, since the boundary operators realize precisely the clock-shift algebra needed to cancel the bulk anomaly.
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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

2 major / 5 minor

Summary. This paper studies a three-dimensional Euclidean lattice model, Eq. (2.1), in which a Z_N clock model and a Z_N gauge theory are coupled only through a topological interaction, a lattice analogue of a theta term. The central claim is that the zero-form symmetry Z_N^(0) and the one-form symmetry Z_N^(1) are intertwined throughout the phase diagram: in any phase, if Z_N^(0) is spontaneously broken to a nontrivial subgroup, then Z_N^(1) is broken to the same subgroup. Using S and T duality transformations, the authors identify oblique phases labeled by (N,p), with L = gcd(N,p), whose low-energy theory is a gauged BF-type action, Eq. (2.37), and whose unbroken subgroup H = Z_{N/L}^(0) × Z_{N/L}^(1) carries an SPT response, Eq. (4.6), with ground-state degeneracy L^{2g+1}. The paper also constructs gapped boundary states, a Hamiltonian lattice realization in Section 7, gapless boundary criticality in Section 8, and Z_N axion extensions whose gauged version exhibits a spontaneously broken non-invertible symmetry with fusion rule Eq. (9.22).

Significance. If the central claims hold, this is a substantial contribution to the theory of generalized symmetries in lattice models. The paper provides a microscopic lattice realization of every Z_N^(0) × Z_N^(1) SPT of the form Eq. (1.1), a classification of oblique phases with intertwined symmetry breaking and topological order, explicit boundary constructions, and a lattice example of non-invertible symmetry breaking. The derivations are largely explicit and checkable: the lattice duality in Appendix B is presented in detail, the Hamiltonian ground-state correlations in Eqs. (7.14), (7.16), and the boundary algebra Eq. (7.23) are exact commuting-projector computations, and the fusion rule Eq. (9.22) is derived from a half-gauging construction. These concrete, verifiable computations are a notable strength of the manuscript. The main weakness is that the universal intertwining claim rests on an assumption about the persistence of the Θ = 0 phase diagram to nonzero Θ, which is explicitly stated as an expectation rather than proven.

major comments (2)
  1. [§2.4–§3 (footnote 6)] The universal intertwining statement is not fully established by the argument given. Footnote 6 of Section 3 justifies the claim by "observing the action of the most generic transformation, Eq. (2.39), on the phases at Θ = 0," but Section 2.4 explicitly states "We expect these phases and transitions to extend to a finite region for nonzero Θ," and Section 5.1 says that determining the fate of the transitions for small nonzero Θ is beyond the scope of the work. The S/T-orbit argument therefore establishes the intertwining pattern only for phases reachable from the Θ = 0 decoupled phase diagram by duality transformations, under the assumption that these phases persist in a finite neighborhood of nonzero Θ. If a new phase nucleates at arbitrarily small |Θ|, or if the topological term drives a first-order transition that bypasses the intermediate phases, the statement "in every phase of our lattice model" at the end of Section 3 would need modification. The quantitative predictions at the self-dual points Θ/2π = -1/p with large eg are exact duality consequences and are not affected, but the universality claim should either be restricted to duality-reachable phases or supported by a separate argument controlling the Θ ≠ 0 region. This is a domain-of-validity gap, not an internal inconsistency.
  2. [§9.2, Eq. (9.27)] The ground-state degeneracy formula for the non-invertible symmetry breaking phase, D_GS = Σ_{p=0}^{N-1} [gcd(N,p)]^{2g+1}, is presented as a sum over the known degeneracies of the oblique phases for each p. However, the text does not derive from the dynamics of Eq. (9.13) that the Hilbert space decomposes as a direct sum over p sectors with no additional identifications or mixing. In particular, the p = 0 term requires the convention gcd(N,0) = N, and the unusual gauge transformations of Eq. (9.14) leave some room for redundancy between sectors. The authors should either provide an explicit derivation of the decomposition or state clearly that Eq. (9.27) follows from the assumption that the different axion vacua are exactly degenerate and decoupled. This is a load-bearing point for the claim that the non-invertible symmetry is spontaneously broken, since the extra degeneracy beyond the ordinary oblique phases is the main quantitative signature.
minor comments (5)
  1. [§3] The phrase "canoncial commutation relations" should read "canonical commutation relations."
  2. [§1, footnote 1] The phrase "A analogue of this criterion" should read "An analogue of this criterion."
  3. [§6.2.2] The heading "Z(0)_N p symmetry" is rendered ambiguously; it should be written as Z_{Np}^(0) symmetry to avoid confusion with a p-dependent subgroup.
  4. [§8, Eqs. (8.10)–(8.18)] The operator eO is notationaly confusing because the coupling e is also used for the gauge coupling; consider renaming it, for example O_v, to avoid confusion with the electric coupling.
  5. [§9.2, Eq. (9.27)] The sum over p is stated to run from 0 to N-1, and p is implicitly mod N; it would be helpful to say explicitly that L = gcd(N,p) is defined with the convention gcd(N,0) = N.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the oblique-phase predictions are derived from explicit duality transformations and an effective field theory, then checked in an independent commuting-projector Hamiltonian; the nonzero-Theta extension is an acknowledged assumption rather than a circular input.

full rationale

The paper's central derivation chain is self-contained in the relevant sense. Starting from the Shapere-Wilczek lattice action (Eq. 2.1), the authors introduce background fields and derive the S and T duality maps (Eqs. 2.26-2.28), including how local and loop quantum numbers transform; these operator maps are obtained in Appendices B and C rather than assumed. The oblique phases are then constructed by applying ST^p to the trivial large-ge phase, giving the effective action Eq. (2.37), from which the unbroken subgroup H = Z_{N/L}^{(0)} x Z_{N/L}^{(1)}, the response coefficient -iNk/(2pi L), and the ground-state degeneracy L^{2g+1} are computed. None of these quantities is fitted to a target; the same results are re-derived in Section 7 from a commuting-projector Hamiltonian whose ground state satisfies exact local constraints (Eq. 7.13), yielding the same L^{2g+1} degeneracy. The only self-citations ([37] for inspiration, and [38] as the external source of the self-dual model) are not load-bearing: the duality used is rederived in the paper and the phase-diagram argument does not rest on an unverified result of the authors' prior work. The statement in Section 2.4 that the Theta=0 phases 'extend to a finite region for nonzero Theta' and the Section 5.1 caveat that determining the fate of transitions at small nonzero Theta is 'beyond the scope of this work' are domain-of-validity assumptions, not reductions of the conclusions to their inputs. The footnote justifying the universal same-subgroup intertwining claim explicitly appeals to checking the action of Eq. (2.39) on the Theta=0 phases, which is a derivation (modulo the stated robustness assumption), not a circular definition. Accordingly, no circular step can be exhibited.

Assumptions & free parameters 0 free parameters · 3 assumptions · 1 invented entities

No parameters are fitted to data; the model couplings (e, g, J, N, p) are inputs. The main non-standard input is the assumption that Θ=0 phases persist to finite Θ. The Z_N axion is introduced as a new matter field without an independent experimental handle.

assumptions (3)
  • ad hoc to paper The Θ=0 phase diagram of the decoupled clock model and gauge theory (ordered/disordered, deconfined/confining) is correct and persists to a finite region of nonzero Θ.
    Section 2.4 states 'We expect these phases and transitions to extend to a finite region for nonzero Θ'; all Θ≠0 phases, including oblique phases, are obtained as S/T images of these Θ=0 phases. If this fails, the phase diagram collapses.
  • domain assumption One-form global symmetries cannot be spontaneously broken in (1+1)d, so SPT boundaries must break the zero-form symmetry or be gapless.
    Invoked in Sections 6.1 and 8, relying on Refs. [8,47]; load-bearing for the boundary-state classification.
  • standard math Standard lattice field theory techniques (Poisson summation, Villain form, duality manipulation) are valid for the action of Eq. (2.1).
    Used throughout Sections 2-3 and the appendices; not formally proved in this paper but standard in the lattice gauge theory literature.
invented entities (1)
  • Z_N axion field θ(R)
    purpose: Promote the theta angle Θ to a dynamical lattice field (Eq. 9.3) to generate mixed anomalies and, after gauging, a non-invertible symmetry U_p that spontaneously breaks.
    The axion is introduced by hand in Section 9.1 as a new matter field; it has no experimentally measured properties or outside falsifiable prediction. It is a model-building device, though it leads to internal consistency checks like the fusion rules (Eq. 9.22).

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Pith. "Pith review of Intertwined order of generalized global symmetries." pith.science (2026). https://pith.science/paper/U2O2LLCQ

@misc{pith2026241202748,
  author       = {Pith},
  title        = {Pith review of: Intertwined order of generalized global symmetries},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/U2O2LLCQ}},
  note         = {Machine review of arXiv:2412.02748}
}
abstract

We investigate the interplay of generalized global symmetries in 2+1 dimensions in a lattice model that couples a $\mathbb{Z}_N$ clock model to a $\mathbb{Z}_N$ gauge theory via a topological interaction. This coupling binds the charges of one symmetry to the disorder operators of the other, and when these composite objects condense, they give rise to emergent generalized symmetries with mixed 't Hooft anomalies. These anomalies result in phases with ordinary symmetry breaking, topological order, and symmetry-protected topological (SPT) order, where the different types of order are not independent but intimately related. We further explore the gapped boundary states of these exotic phases and develop theories for phase transitions between them. Additionally, we extend this lattice model to incorporate a non-invertible global symmetry, which can be spontaneously broken, leading to domain walls with non-trivial fusion rules.

Figures

Figures reproduced from arXiv: 2412.02748 by the authors.

Figure 1
Figure 1. Depiction of the topological interaction on the lattice, Eq. (2.4). This (2+1)d analogue of the theta term represents an interaction of the field strengths, fνλ, which are on plaquettes of the direct lattice, and spin phase variations, ∆µφ, which live on links of the dual lattice. we place the gauge fields aµ on links of the lattice, matter fields φ on sites of the dual lattice (i.e., centers of cubes on the direct … view at source ↗
Figure 2
Figure 2. The phase diagram at Θ = 0 for fixed κ = e/g as Im(τ ) = 2π/Nge is varied. In this limit, the lattice model reduces to a decoupled ZN clock model and ZN gauge theory. At large e and g, there is a trivial phase in which the clock model is disordered and the gauge theory is confining, fully preserving the global symmetry G = Z (0) N × Z (1) N . At small e and g, the clock model orders, and the gauge theory is deconfin… view at source ↗
Figure 3
Figure 3. The phases and transitions captured by the theory of Eq. (5.5), which has global symmetry G = Z (0) N × Z (1) N . The field ζ is the Z (0) N order parameter, and ˜ζ is a magnetic monopole. In the lattice model, Eq. (2.1), these phases and transitions appear in the vicinity of Θ = 0, where the ZN spin model is decoupled from the ZN lattice gauge theory. The Z (0) N and Z (1) N symmetries can independently be broken o… view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: The phase diagram for the field theory of Eq. (5.8). There are four phases: the (N, p) oblique phase, phases in which either Z (0) N or Z (1) N is broken but the other is preserved, and a trivial phase that preserves the full symmetry Z (0) N × Z (1) N . Here, ζ repres…
Figure 5
Figure 5. Figure 5: The phases and transitions described by the field theory of Eq. (5.13). There are two distinct oblique phases—the (N, p) phase and the (N, p + 1) phase. There are also intermediate phases in which either the Z (0) N or Z (1) N is broken while the other is preserved. Th…
Figure 6
Figure 6. Figure 6: Illustration of how to detect the anomaly for the SPT protected by Z (0) N × Z (1) N global symmetry when the SPT is on a manifold X with a boundary ∂X. The operator U0(Σ, Γ) that acts with the Z (0) N symmetry is supported on a surface Σ in the bulk and ends at the li…
Figure 7
Figure 7. Figure 7: Depictions of (7a) the plaquette operator BP , (7b) the dual link operator Aℓ˜, and (7c) the gauge charge operator Q(r). The darker lines indicate links of the lattice while the lighter lines are links of the dual lattice. (at the centers of plaquettes of the direct la…
Figure 8
Figure 8. Figure 8: Action of the symmetry operators when there is a lattice boundary. The ’t Hooft line T(ΓeR0 ) L along ΓeR0 , depicted by the dashed line, ends at a point R0 on the boundary. The symmetry operator U L for the zero-form symmetry acts on the boundary along the red line, w…
Figure 9
Figure 9. Figure 9: Examples of Z (0) N × Z (1) N SPT boundary transitions captured by our gapless state, Eq. (8.1), as a function of the Luttinger parameter K if different global symmetries are preserved. In all cases, the gapless phase is Eq. (8.1), and the phase for K < 2/π has Z (0) N…
Figure 10
Figure 10. Figure 10: Depiction of the degrees of freedom in the lattice model, Eq. (9.3), on a single cube of the direct lattice. The ZN gauge field kµ(r) is defined on links ℓ of the direct lattice, labeled by the blue points. The ZN spin variables k(R) are on sites R of the dual lattice…
Figure 11
Figure 11. Figure 11: Depiction of the domain wall for the ZN axion. This domain wall is the non-invertible defect of Eq. (9.12). The vertical direction is the y-direction, and the plaquettes at the bottoms of the blue pyramids define the y = 0 plane. The y = 1/2 plane intersects the red p…
Figure 12
Figure 12. Figure 12: Depiction of twisted boundary conditions for a one-form global symmetry. The square above represents the xy-plane, which is a torus, so the opposite sides of the square are identified. We introduce a nontrivial background field Bxy. If we put a Wilson loop along the n…
Figure 13
Figure 13. Figure 13: Statistical interaction in the Coulomb gas. The last term in the Coulomb gas action, Eq. (C.1), is a statistical interaction of spins and vortices. If mµ(r) is a straight line along the Euclidean time direction and n(R) is in the xy-plane, then Φµ(r − R) is the angle …

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