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The "not-A", RSPT and Potts phases in an $S_3$-invariant chain

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Pith's one-line read A single S3-invariant three-state quantum chain has four gapped phases meeting at one multicritical point, with every phase transition in the three-state Potts universality class.

desk verdict A solid, creative paper on an S3-invariant chain with exact ground states and a plausible phase diagram; the main weakness is the unproven Potts universality of the anti-self-dual transitions. read the letter →

arxiv 1908.02767 v3 pith:VVCKNY36 submitted 2019-08-07 cond-mat.stat-mech cond-mat.str-el

Edward O'Brien , Eric Vernier , Paul Fendley This is my paper · ORCID
classification cond-mat.stat-mechcond-mat.str-el
keywords three-statePottsmodelS3symmetryquantumspinchainsymmetry-protectedtopologicalorderRSPTnot-Amulticriticalpointconformalfieldtheory
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 aims to establish the full phase diagram of a nearest-neighbour quantum chain with three states per site whose Hamiltonian is invariant under every permutation of the three states. The model combines the three-state Potts chain with a U(1)-invariant, self-dual chain, and the authors argue that four distinct gapped phases meet at the self-dual multicritical point: ordinary Potts order, Potts disorder, an unusual "not-A" ordered phase in which each ground state excludes one of the three spin states, and a representation symmetry-protected topological (RSPT) phase. A central quantitative claim is that every transition line lies in the three-state Potts universality class, so the same conformal field theory with central charge $c=4/5$ describes all four boundaries. If correct, the paper supplies a minimal lattice setting in which conventional order, topological order, and representation-protected edge modes are unified, with exact ground states available in each phase.

What carries the argument

Two devices carry the argument. The first is the projector rewrite of the Hamiltonian, $H_{\gamma,\hat\gamma}=\sum_j[P_{2j-1,2j}(\gamma)+P_{2j,2j+1}(\hat\gamma)]$ with $P_{a,a+1}(\gamma)=\gamma p_a+3\gamma^{-1}p_{a+1}-p_ap_{a+1}-p_{a+1}p_a$; because the $p_a$ obey the projector algebra (13), special parameter pairs make neighbouring projectors commute and give exact zero-energy ground states. The second is the continuum action (53), $S_B+\Gamma\int d^2z\,[\,f\cos\sqrt{6}\,\phi+J\cos\sqrt{6}\,\hat{\phi}\,]$, where $S_B$ is the free compact boson at radius $r=\sqrt{3/2}$. The two cosine terms are the only relevant operators invariant under the full $S_3$ symmetry, and duality exchanges them; the self-dual direction $f=J$ flows to the three-state Potts conformal field theory, which is what ties all four phase boundaries to the same universality class.

What would settle it

Compute the effective central charge and the magnetization scaling along the numerically located not-A/disorder and Potts/RSPT transition lines: if the central charge differs from $4/5$ or the scaling dimensions differ from the three-state Potts values, the central claim fails. A second check is to couple an extra $S_3$ doublet to one edge of the open chain; the paper predicts the fourfold degeneracy splits into two doublets, whereas a true symmetry-protected topological phase would stay fourfold degenerate.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that the two-parameter Hamiltonian (11) has a four-phase structure around the multicritical point $H_0$. The two ordered phases are distinguished by the sign of $M^3$, where $M=\langle g|\sigma_j|g\rangle$ in an $S_3$-breaking ground state: $M^3>0$ in the Potts ordered phase and $M^3<0$ in the not-A phase. The two phases without local order both have $M^3=0$ but differ in their open-chain ground-state degeneracy: one state for Potts disorder, four states for RSPT order. In the RSPT phase each edge transforms as a two-dimensional representation of $S_3$ rather than a projective representation, which is why the phase is called representation-protected rather than a full symmetry-protected topological phase. The field-theory section derives the perturbed compact-boson action (53), identifies the boson radius as $r=\sqrt{3/2}$, and uses the known flow of that theory to the $c=4/5$ Potts conformal field theory to conclude that all four transition lines, including the two not fixed by self-duality, lie in the three-state Potts universality class. Exact product and matrix-product ground states are found at one point in each phase, and the eight states match the eight conformal boundary conditions of the critical Potts CFT.

Load-bearing premise

The load-bearing premise is that the continuum action (53), containing only the two $S_3$-invariant cosine operators, is the complete relevant field-theory description of the lattice model near $H_0$; if another relevant operator survives the continuum limit, the location and universality class of the vertical transitions could change.

Editorial extensions

If this is right

  • If the phase diagram of Fig. 1 is correct, the Potts ordered and not-A phases cannot be connected without closing the gap; the sign of $M^3$ is a robust diagnostic separating them.
  • All four phase boundaries have central charge $c=4/5$, including the direct transition from Potts order to the RSPT phase, so measurements of entanglement entropy along any of the lines should find the three-state Potts value.
  • The exact ground states at the four special points realize the eight conformal boundary conditions of the three-state Potts CFT on a lattice, giving a concrete lattice playground for boundary-condition physics.
  • The fourfold ground-state degeneracy of the RSPT phase persists under any $S_3$-preserving deformation of the chain, but coupling an extra $S_3$ doublet to one edge splits the degeneracy into two doublets, a signature that separates RSPT from true SPT order.

Reading between the lines

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

  • The sign of $M^3$ could be used as a generic order parameter for "exclusion" order in other permutation-symmetric or clock models, where ordering means avoiding one state rather than preferring one.
  • The representation-versus-projective distinction suggests a hierarchy: other non-abelian symmetry groups in one dimension should support analogous representation-protected phases whose edge degeneracy equals the dimension of the group's representation, with the same fragility to auxiliary degrees of freedom.
  • Because the two anti-self-dual transition lines are not fixed by symmetry, their precise shape near the multicritical point is a quantitative field-theory prediction; high-precision numerics that resolve a bend at small couplings would point to a missing relevant operator.
  • The duality between the RSPT matrix-product state and the not-A product states may extend to a broader correspondence between edge-representation data and local exclusion rules in $\mathbb{Z}_N$-symmetric chains.
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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 / 3 minor

Summary. The paper analyzes the two-parameter S3-invariant spin-1 chain H(J,f,λ)=λH0+HP(J,f)+(2λ−f−J)L (Eq. 11), where H0 is the U(1)-invariant self-dual point and HP is the three-state Potts Hamiltonian. The authors construct exact ground states at four isolated points: the three ferromagnetic product states, the symmetric disordered state, the three 'not-A' product states |ĀĀ...>, and an AKLT-like matrix product state. They show that the sign of M^3 distinguishes the Potts-ordered and not-A phases, while open-boundary degeneracies (1 vs 4) distinguish the disordered and RSPT phases. A free-boson field theory with compact radius r=√(3/2) is derived from duality, and the authors argue from known c=1 to c=4/5 flows that all four transition lines are in the three-state Potts universality class. DMRG provides central-charge and gap data consistent with the phase diagram.

Significance. If the central claim is fully substantiated, this is a significant contribution: it provides a single nearest-neighbor model where four distinct gapped phases meet at a multicritical point, with exact frustration-free ground states in every phase, an explicit RSPT phase protected by a non-Abelian representation, and a lattice realization of the free-boson-to-Potts flow. The projector-algebra proofs at the four special points are clean and convincing; the derivation of the boson radius r=√(3/2) from the self-duality of the Z3-twisted partition function is parameter-free and self-contained; and the MPS/RSPT analysis gives concrete, falsifiable predictions (M^3 sign and edge degeneracy) that are checked numerically. The paper is clearly written and will be of interest to the exactly-solvable-models and symmetry-protected-topology communities.

major comments (2)
  1. [V.C, Eq. (53), footnote 45] The conclusion that the two anti-self-dual transition lines (not-A/disorder and Potts/RSPT) are in the three-state Potts universality class is load-bearing and rests on an identification that is asserted rather than derived. The field theory (53) is a U(1)-invariant free boson, but the cited integrable c=1 to c=4/5 flow of Refs. 18–20 was established for the Z4 parafermion orbifold. As footnote 45 acknowledges, these are different theories: the free boson has an additional U(1) current algebra and an exactly marginal radius operator, so the flow is not automatic. Since no lattice symmetry protects the vertical transitions, the Potts classification of these lines depends on this RG-flow identification. I ask the authors to either supply a derivation of the free-boson-to-Potts flow (or a precise mapping from (53) to the known Z4 case), or to restrict the universality claim accordingly.
  2. [VI, Fig. 3] The numerical evidence for Potts universality at the anti-self-dual transitions is limited to a single effective-central-charge estimate c≈0.8 at β=0.75 (Fig. 3). A central charge alone does not identify the Potts CFT, since other CFTs can share the same central charge. Additional checks, such as scaling dimensions, entanglement-spectrum degeneracies, or correlation-function exponents, would be needed to confirm the Potts operator content. Alternatively, the text should say that these transitions are 'consistent with' Potts universality rather than that they are 'in' it. The authors' own statement in Section VI that they could not confirm the predicted vertical approach of these lines near the multicritical point further limits the strength of the claim.
minor comments (3)
  1. [Fig. 4] The horizontal axis label appears truncated ('/' in the reproduced figure); it should read θ/π.
  2. [Table I] The column header 'M3' would be clearer as 'M^3' to match the notation M^3_g used in the text, and a brief definition of M^3 via Eq. (55) would help the reader.
  3. [V.B] The sentence that 'all the possible scaling dimensions for a compact boson are contained in the partition function' is imprecise, since the partition function displays only the operator content on the torus; the later discussion of local operators is clear, but this sentence could be rephrased.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: H0 and exact ground states are explicit, the boson radius is derived by duality, and the Potts-universality claim rests on external RG-flow results, not on fitting or self-citation.

full rationale

The paper's central derivation is self-contained in the sense required by the circularity rubric. The Hamiltonian (11) is explicitly defined; the multicritical H0 is written out in (10), and the earlier self-citation [12] is used only for side remarks about degeneracies, while the CFT radius r=√3/2 is either credited to Ref 43 or re-derived from the Z3-twisted partition-function duality in Section V.B. The four exact ground states follow from explicit Temperley-Lieb projector calculations (18)-(20), not from the phase diagram. The field-theory action (53) is obtained by listing the only relevant S3- and chirally-symmetric operators in the c=1 free boson; no parameter is fitted to the later transition data. The Potts-universality claim for all four lines is imported from the external integrable-flow literature (Refs 18-20), and the numerical check is an independent measurement of c≈0.8 at one anti-self-dual point. The one genuine caveat is footnote 45: the cited flow was established for the Z4 parafermion orbifold, and the authors assert that the same flow occurs for the free boson; this is an assumption about the validity of an external result, not a reduction of the prediction to its own input. Therefore no circular step can be quoted.

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

No fitted parameters appear. The boson radius r=√3/2 is derived from duality (Sec. V.B); the non-universal constant Γ in Eq. (53) is not determined but is not used in any prediction. The phase diagram axes are Hamiltonian couplings, not fitted numbers. No new particles or degrees of freedom are introduced; the not-A and RSPT phases are phases of the existing Hilbert space.

assumptions (4)
  • domain assumption Kramers-Wannier duality acts on operators as τ_j -> σ†_j σ_{j+1} and σ†_j σ_{j+1} -> τ_{j+1} (Eq. 7).
    This duality is standard for three-state Potts models and is used to relate the Potts and RSPT/not-A phases and to derive the boson radius.
  • domain assumption The continuum limit of H0 is a free compact boson with radius r = √(3/2).
    Identified from integrability (Ref. 43) and confirmed by a duality argument in Sec. V.B; this is the starting point for the field theory analysis.
  • domain assumption The only relevant S3-invariant operators near H0 are cos√6φ and cos√6φ̂ with dimension (3/4,3/4).
    This follows from the compact boson operator content and symmetry constraints; it underlies the prediction that all transitions are in the Potts universality class.
  • domain assumption The self-dual perturbation of the c=1 boson flows to the c=4/5 Potts CFT (Refs. 18-20).
    This external RG flow result is used to identify the self-dual transition lines as Potts critical.

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Cite this review

Pith. "Pith review of The "not-A", RSPT and Potts phases in an $S_3$-invariant chain." pith.science (2026). https://pith.science/paper/VVCKNY36

@misc{pith2026190802767,
  author       = {Pith},
  title        = {Pith review of: The "not-A", RSPT and Potts phases in an $S_3$-invariant chain},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VVCKNY36}},
  note         = {Machine review of arXiv:1908.02767}
}
abstract

We analyse in depth an $S_3$-invariant nearest-neighbor quantum chain in the region of a U(1)-invariant self-dual multicritical point. We find four distinct proximate gapped phases. One has three-state Potts order, corresponding to topological order in a parafermionic formulation. Also nearby is a phase with "representation" symmetry-protected topological (RSPT) order. Its dual exhibits an unusual "not-A" order, where the spins prefer to align in two of the three directions. Within each of the four phases, we find a frustration-free point with exact ground state(s). The exact RSPT ground state is similar to that of Affleck-Kennedy-Lieb-Tasaki, whereas its dual states in the not-A phase are product states, each an equal-amplitude sum over all states where one of the three spin states on each site is absent. A field-theory analysis shows that all transitions are in the universality class of the critical three-state Potts model. They provide a lattice realization of a flow from a free-boson field theory to the Potts conformal field theory.

Figures

Figures reproduced from arXiv: 1908.02767 by the authors.

Figure 1
Figure 1. FIG. 1. The phase diagram surrounding the multicritical point, where the axes are defined via [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 1
Figure 1. figure 1 [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. RG flow about the [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figures from the paper (3 more)
Figure 1
Figure 1. Figure 1: In this section we present numerics strongly support￾ing this picture, and indicating that there are no other phases in this region. We also locate the vertical critical lines describing the ordered/RSPT and disorder/not-A transitions. We use the Density Matrix Renorma…
Figure 4
Figure 4. Figure 4: FIG. 4. The energy gaps to the lowest-energy state in the [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. The order parameter [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]

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