Topological Holography for Mixed-State Phases and Phase Transitions

Ran Luo; Yi-Nan Wang; Zhen Bi

REVIEW 3 major objections 6 minor 123 references

Topological Holography for Mixed-State Phases and Phase Transitions

T0 review · 3 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Mixed-state phases of a quantum system with symmetry G are classified by anyon condensation in a doubled topological order, Z(Vec_{G×G}), with Hermiticity and positivity of the density matrix pruning the admissible patterns.

desk verdict Serious paper; the positivity constraint on condensable algebras is the real contribution, and the realizability gap the authors concede in footnote 15 is the main thing to push on in review. read the letter →

arxiv 2507.06218 v1 pith:ASKNPWD3 submitted 2025-07-08 cond-mat.str-el hep-thquant-ph

Ran Luo , Yi-Nan Wang , Zhen Bi This is my paper

classification cond-mat.str-elhep-thquant-ph

keywords SymTFTmixed-statephasesstrong-to-weaksymmetrybreakingaveragesymmetry-protectedtopologicalphaseanyoncondensationDrinfeldcentercanonicalpurificationholography

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

Symmetry topological field theory (SymTFT) treats a system's global symmetry as a topological bulk in one higher dimension, so that gapped phases become choices of anyon condensation on the boundary. This paper extends that holographic picture from pure states to mixed states — density matrices — which is the natural setting for decohered, noisy, or monitored quantum matter. Its central proposal is that the SymTFT of an open quantum system with a finite global symmetry $G$ is the Drinfeld center $Z(\mathrm{Vec}_{G\times G})$ of the doubled symmetry category, obtained by canonically purifying the density matrix into a doubled Hilbert space. Because a genuine density matrix is Hermitian and positive semidefinite, not every condensation pattern that topology allows is physically realizable, and the paper derives two extra selection rules — J-symmetry and a positivity inequality — that cut the admissible list down to the physical phases. Applying the rules reproduces strong-to-weak symmetry breaking (SWSSB) and average symmetry-protected topological (ASPT) phases, enumerates them for $G = Z_2, Z_4, Z_2\times Z_2, S_3, D_4$, and places the phase transitions between mixed-state phases at non-Lagrangian condensable algebras described by known conformal field theories.

What carries the argument

The engine of the construction is canonical purification, which maps a density matrix $\rho = \sum_i p_i |\psi_i\rangle\langle\psi_i|$ to the pure state $|\rho_c\rangle\!\rangle = \sum_i \sqrt{p_i}\, |\psi_i\rangle |\bar{\psi}_i\rangle$ in the doubled Hilbert space $\mathcal{H}\otimes\mathcal{H}^*$, promoting the symmetry $G$ to $(G\times G)\rtimes Z_2$. The extra factor $J$ exchanges the two copies and encodes Hermiticity; because it can never be spontaneously broken it is treated as a structural constraint rather than promoted to a full TQFT. The bulk is the doubled Drinfeld center $Z(\mathrm{Vec}_{G\times G})$, whose simple anyons are pairs $a\bar{b}$ labeled by a conjugacy class of $G$ and an irreducible representation of its centralizer in each copy. A candidate phase is a condensable Lagrangian algebra, encoded as an integer matrix $N_{a,b}$; the new selection rules are J-symmetry ($N^T = N$) and positivity ($N_{a,b}\neq 0$ forces $N_{a,a}\neq 0$ and $N_{b,b}\neq 0$), the latter derived from the bound $|\mathrm{Tr}(\sqrt{\rho}\,W_e \sqrt{\rho}\,W_m^\dagger)|^2 \le \mathrm{Tr}(\sqrt{\rho}\,W_e \sqrt{\rho}\,W_e^\dagger)\,\mathrm{Tr}(\sqrt{\rho}\,W_m \sqrt{\rho}\,W_m^\dagger)$. The same machinery fixes the critical points: the intersection of two Lagrangian algebras is a non-Lagrangian condensable algebra whose deconfined subcategory selects the boundary CFT, and the effective-field-theory reduction of the slab produces the response actions, such as $-i\pi\int A\cup B$ for the $Z_4$ intrinsic ASPT phase.

What would settle it

A direct test is to try to realize a screened pattern in a finite lattice model: for the $Z_2$ symmetry, the paper predicts the algebra $1 \oplus e\bar{m} \oplus m\bar{e} \oplus f\bar{f}$ is unphysical while $1 \oplus e\bar{e} \oplus m\bar{m} \oplus f\bar{f}$ is the SWSSB phase, so one can simulate a decohered toric code or Ising chain and ask whether the purified state's string correlators ever reproduce the forbidden pattern's order without forcing a non-positive density matrix. Finding any local channel that realizes the forbidden pattern, or any admissible algebra whose purification provably has a negative eigenvalue, would falsify the claim that the screen is complete.

Watch

Extended reading notes

Core claim

The paper's central claim is that the open SymTFT of a $(1+1)$-dimensional mixed state carrying a strong global symmetry $G$ is the modular tensor category $Z(\mathrm{Vec}_{G\times G}) \cong Z(\mathrm{Vec}_G) \boxtimes Z(\mathrm{Vec}_G)$, and that gapped phases are classified by Lagrangian algebras $A = \oplus_{a,b} N_{a,b}\, a\bar{b}$ subject to five conditions: modular invariance, invariance under the anti-unitary exchange $J$ that swaps the two copies (Hermiticity), a positivity rule — if the composite anyon $a\bar{b}$ condenses then $a\bar{a}$ and $b\bar{b}$ must also condense — the quantum-dimension (Lagrangian) condition, and a stability condition. The positivity rule follows from the Cauchy–Schwarz inequality applied to the canonically purified state, and it is what rules out patterns that would otherwise look admissible, such as the $Z_2$ algebra $1 \oplus e\bar{m} \oplus m\bar{e} \oplus f\bar{f}$. Condensation patterns that factorize across the two copies give back the pure-state phases (SSB and SPT), while patterns that couple the copies give inherently mixed-state phases: SWSSB appears as paired diagonal condensation $a\bar{a}$, and ASPT phases appear together with SWSSB, diagnosed by decorated domain walls, projective boundary symmetry action, or a derived topological response term. The paper further claims that transitions between these gapped phases sit at the non-Lagrangian condensable algebra $A' = A_1 \cap A_2$; that a transition involving an inherent mixed-state phase has an inherently mixed-state critical point whose order and disorder parameters display power-law behavior only in Wightman (doubled-space) correlators; and that gauging within the open SymTFT swaps phases by changing the topological boundary condition, with the SWSSB phase invariant under full $G\times G$ gauging.

Load-bearing premise

The classification rests on the assumption that every condensation pattern passing the five-rule screen can be realized as a positive semidefinite density matrix by combining coherent anyon condensation with quantum channels — a correspondence the paper itself flags as only established for fixed-point wavefunctions in the strong-decoherence limit.

Editorial extensions

If this is right

For any finite group $G$, the five-rule screen on integer matrices $N_{a,b}$ is proposed as a complete enumeration of gapped mixed-state phases; the paper executes it for $Z_2$ (3 physical phases), $Z_4$ (7), $Z_2\times Z_2$ (16), $S_3$ (8), and $D_4$ (38, of which 27 are inherently mixed-state and 3 are intrinsic ASPTs).
The framework ties SWSSB and ASPT together: every ASPT phase arises through strong-to-weak symmetry breaking, and the intrinsic ASPTs occur in symmetry settings with no pure-state SPT ancestor, so they genuinely require a mixed state.
Mixed-state phase transitions are governed by the intersection of Lagrangian algebras; in the $Z_2$ case both the trivial-to-SWSSB and SSB-to-SWSSB transitions retain a toric-code sector, predicting Ising-CFT power laws in the decohered transverse-field Ising chain at maximal decoherence, with some correlators visible only through Wightman (nonlinear-in-$\rho$) probes.
Gauging in the open SymTFT is a boundary-condition change: full $Z_2\times Z_2$ gauging leaves SWSSB invariant while permuting SSB and SPT, while gauging the diagonal weak $Z_2$ swaps SSB and SWSSB and is unphysical (violates J-symmetry) when applied to the trivial phase.
The effective-field-theory derivation assigns concrete topological response functions to the novel phases — for example $-i\pi\int A\cup B$ for the $Z_4$ intrinsic ASPT — and the same construction extends to continuous $U(1)$ symmetry, producing SWSSB $U(1)_s \to U(1)_w$ and a $U(1)$ analogue of the intrinsic ASPT.

Reading between the lines

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

If the five-rule screen is complete, the enumerated phase counts (for instance 38 for $D_4$) become a benchmark that independent classifications — by finite-depth local quantum channels, tensor-network purifications, or Lindbladian steady states — must reproduce; any mismatch would reveal either a missing constraint or an unphysical entry in the list.
The screen turns classification into a finite combinatorial search over integer matrices for each group, suggesting the paper's method can be automated into a computational pipeline that produces the full mixed-state phase diagram of any finite group, including the higher-group and fermionic extensions flagged in the outlook.
The prediction that some transition observables appear only in Wightman or Rényi-type correlators gives quantum simulators a concrete experimental target: at a decoherence-driven transition, linear order-parameter correlations may look featureless while nonlinear-in-density-matrix probes reveal the underlying CFT scaling.
The realizability argument implies a hierarchy of reachability: diagonal condensation patterns come from generic noise, while off-diagonal patterns require coherent-noise channels that may need tuning; checking whether the intrinsic ASPT phases survive generic, untuned decoherence would test how many of the listed phases are practically reachable.

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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

Short version: worth reading, and the actual contribution is the positivity constraint on condensable algebras, not the doubled-space setup, which is honestly attributed to earlier work. The constraint is clean, the exclusion of the M2 pattern for Z2 is a concrete nontrivial consequence, and the lattice model plus the Z4 EFT response give independent checks. The main soft spot is the one the stress test flags: realizability of every admissible algebra is argued, not proven, and footnote 15 concedes the channel-to-canonical-purification mapping only clearly holds for fixed-point wavefunctions in the strong-decoherence limit. That gap is real, but it is not hidden—the authors state it themselves—and it bites only in the converse direction of the classification.

What is genuinely new: the Cauchy-Schwarz positivity inequality (Eqs. 7-11) is correct and general; the rule that condensing a\bar{b} forces a\bar{a} and b\bar{b} is a sharp necessary condition; the S3 and D4 enumerations are extensive; the intrinsic ASPT for Z4 with its boundary response term (Eq. 77) is a useful diagnostic. The overlap with the doubled-space papers [78,79,81,96] is properly acknowledged, and the note added flags the two parallel preprints without a comparison—a minor missed opportunity.

Soft spots, in proportion. (1) The stability condition (Eq. 18) is imported from pure-state domain-wall arguments with no mixed-state derivation; since the five rules are the algorithm, each borrowed rule is load-bearing. (2) Realizability: the Abelian off-diagonal argument is fairly convincing, but the non-Abelian cases rest on coherent noise channels without a lattice check, and the only explicit lattice realization is the diagonal class for Z2. A referee should ask whether the classification can overcount, and ideally ask for a second lattice check beyond Z2. (3) No code is shipped for the enumerations, and the D4 appendix is a wall of matrices; checkable but laborious.

Most readers in mixed-state phases or SymTFT will get value from this; it organizes a lot of results under one algorithm. Send it to serious peer review, conditional on the authors sharpening what is proven versus conjectured about realizability. The writing is dense but the thinking is clear.

Referee Report

3 major / 6 minor

Summary. The manuscript develops a symmetry topological field theory (SymTFT) framework for (1+1)-dimensional mixed states with a global symmetry G. Using canonical purification, a density matrix is embedded into a doubled Hilbert space and the bulk is proposed to be the Drinfeld center Z(Vec_{G×G}) ≅ Z(Vec_G) ⊠ Z(Vec_G), with an anti-unitary exchange symmetry J implementing Hermiticity. The paper formulates five rules—modularity, J-symmetry, positivity, dimensionality, and stability—that admissible condensable algebras should satisfy, and translates them into matrix constraints on N_{a,b}. It then classifies gapped mixed-state phases for Z2, Z4, Z2×Z2, S3, and D4, identifying pure-state SSB/SPT phases as well as inherent mixed-state phases such as SWSSB, ASPT, and intrinsic ASPT. Phase transitions are treated through non-Lagrangian condensable algebras, with lattice realizations for the Z2 and Zq cases; gauging is discussed both categorically and via BF effective actions, including a U(1) extension. The paper also derives topological response terms for several ASPT phases.

Significance. If the realizability and stability assumptions are justified, this is a valuable systematic step toward classifying mixed-state phases with symmetries. The core positivity inequality (Eqs. 7–11) is clean and checkable, and the explicit Z2 lattice calculations in Sec. IV.B provide an independent consistency check. The detailed enumeration for D4, the modular data in the appendices, and the BF-response derivations are useful resources for follow-up work. However, the classification's completeness rests on the Sec. II.C claim that every admissible condensable algebra is physically realizable, which the authors themselves qualify in footnote 15; because this point is load-bearing, the central claim is not yet fully established.

major comments (3)

[II.C and footnote 15] The five rules of Sec. II.B are presented as a complete classification algorithm for gapped mixed-state phases for any finite group G, so the converse claim of Sec. II.C—that every admissible condensable algebra is realizable as a positive semidefinite density matrix—is load-bearing. Footnote 15 (ref. [104]) concedes that the mapping from quantum channels to canonical purification is "less straightforward" and is expected to hold only for fixed-point wavefunctions in the strong-decoherence limit. Moreover, no derivation is supplied for the off-diagonal class (coherent noise E_{a+b}) or for the non-Abelian S3/D4 examples, and the only explicit lattice check (Sec. IV.B) realizes the diagonal class for Z2. If some admissible algebras are not realizable, the Z4, S3, and D4 lists overcount phases. Please provide a derivation of the purification mapping for the off-diagonal and non-Abelian channels, or construct a lattice/channel realization for at least one such phase, or explicitly state the classification as conditional on realizability.
[II.B, Eq. (18)] Rule 5 (stability) is imposed without derivation in the mixed-state context. The cited argument [103] concerns invariance of ground-state degeneracy under local Hamiltonian perturbations for gapped boundaries; the manuscript does not explain why the same inequality controls stability of a mixed-state phase under local quantum channels or Lindbladian perturbations, nor does it establish whether the condition is necessary, sufficient, or merely convenient. Since Eq. (18) is one of the five filters that select the phase lists in Sec. III and Appendix D, this gap should be closed: either derive Eq. (18) from a mixed-state stability criterion or replace it with a justified condition and re-run the classifications.
[II.B, Eqs. (14)–(15)] The proposed enumeration treats modular invariance of the partition vector |L>, together with integer entries and first entry 1, as sufficient for a Lagrangian algebra. Modular invariance is necessary but is not generally sufficient for the existence of a commutative separable algebra structure, especially when multiplicities N_α > 1 occur in the non-Abelian examples. The manuscript does not state that associativity and separability were verified for the matrices in Sec. III.C and Appendix D, nor does it cite a theorem that for Z(Vec_{G×G}) this criterion is complete. Please either supply such a theorem or reference, or provide a verification argument and the computational search details so that the D4 38-phase list can be independently checked.

minor comments (6)

[Appendix C.2] The labels used for the S3 gauging results are inconsistent with Sec. III.C: the matrix called MSWSSB_11 in Appendix C.2 is the one called MSWSSB_4 in Sec. III.C, and similarly for MSWSSB_7/MSWSSB_3 and MSWSSB_6/MSWSSB_1. Please reconcile the numbering.
[Eq. (5) and Sec. III.A] The label M2 is used both for a positivity-forbidden Z2 Lagrangian algebra in Eq. (5) and for a valid Z4 SWSSB phase in Sec. III.A; renaming one of these would avoid confusion.
[Eqs. (56) and (60)] The relative sign of the b∪δb̂ term differs between the general action in Eq. (56) and the Z2 action in Eq. (60); for Z2 the two forms agree modulo 2πi, but the general-N convention should be fixed and stated.
[IV.A] The notation M_{1⊕e¯e} and M_{1⊕m¯m} is introduced without defining that these denote the non-Lagrangian condensable algebras at the intersections A1∩M and A2∩M; please define this notation explicitly, and fix the typo "equvalently" below Eq. (39).
[IV.B] The statement at p=1/2 that the channel corresponds to infinite effective evolution time should be made precise, since the subsequent argument identifies p=1/2 with projective measurement of all Xi; the phase diagram in Fig. 5 would also benefit from explicit axis and region labels in the caption.
[II.C, footnote 15] The caveat in footnote 15 qualifies the central realizability claim and should be moved into the main text of Sec. II.C rather than left as a footnote.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the mixed-state phase classification is an enumeration from explicit constraints, and the realizability gap flagged in footnote 15 is a limitation rather than a circular reduction.

full rationale

The paper's central derivation — classifying mixed-state gapped phases as Lagrangian algebras in Z(Vec_{G×G}) satisfying modularity, J-symmetry, positivity, dimensionality, and stability (Sec. II.B) — is an enumeration from stated constraints plus the modular data of Z(Vec_G), not a quantity fitted from or defined by the outputs. The positivity rule is derived from the Cauchy–Schwarz inequality on density-matrix expectation values (Eqs. 6–11), and the phase lists for Z4, Z2×Z2, S3, and D4, the Wightman-correlation predictions at criticality (Eqs. 41–42), and the EFT response Sresponse = −iπ ∫ A∪B (Eq. 77) are computed consequences, not inputs. Self-citations such as [76] and [84] appear only as background for the ASPT and SWSSB concepts; they are not load-bearing for the classification, and the doubled-space ansatz is attributed to external prior work [78,79,96]. The genuinely weak step is the sufficiency claim in Sec. II.C that every algebra passing the positivity screen is physically realizable: footnote 15 concedes that the channel-to-purification mapping is 'less straightforward' and expected to hold only for fixed-point wavefunctions in the strong-decoherence limit. This is an unproven completeness assumption, which could mean the classification overcounts phases, but it is not a circular reduction — realizability is not used as an input to the enumeration, and the lattice check in Sec. IV.B independently confirms the Z2 diagonal class. No circular step can be exhibited from the paper's own equations or citations.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No numerical fitting to data occurs in this paper; the classification is a combinatorial enumeration with no free constants. The only adjustable objects are the finite groups G themselves, which are inputs chosen as examples.

assumptions (6)

domain assumption Canonical purification maps a mixed state rho to a pure state |rho_c>> in H x H*, with symmetry promoted to (G x G) ⋊ Z2.
Standard quantum information tool; stated in Sec. II, Eq. (1).
ad hoc to paper The open SymTFT for a strong G-symmetric mixed state is the Drinfeld center Z(Vec_{G x G}) ≅ Z(Vec_G) ⊠ Z(Vec_G).
This is the central proposal of the paper (Sec. II), motivated by canonical purification but not derived from a more fundamental principle.
ad hoc to paper The anti-unitary J symmetry is a structural constraint that is never spontaneously broken and need not be dynamically realized in the bulk.
Stated in Sec. II.A; if J could be broken, the classification would miss phases.
ad hoc to paper A physical mixed-state phase corresponds to a condensable algebra A in Z(Vec_{G x G}) satisfying modularity, J-symmetry, positivity, dimensionality, and stability.
The five rules are assembled in Sec. II.B (Eqs. 14-18); the stability condition (Eq. 18) is borrowed from pure-state domain wall arguments [103] without a mixed-state derivation.
domain assumption Anyon condensation with Wightman correlators detects SWSSB, and decorated domain wall patterns diagnose ASPT.
The SWSSB diagnostic is cited to refs [94,95]; the ASPT diagnostic is motivated by boundary projective representations and EFT response in Secs. III and VI.
domain assumption A phase transition between two gapped phases is described by the intersection of their Lagrangian algebras, with the residual theory constrained by modular invariance.
This categorical Landau picture is cited to refs [27,28] and used in Sec. IV.

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Cite this review

Pith. "Pith review of Topological Holography for Mixed-State Phases and Phase Transitions." pith.science (2026). https://pith.science/paper/ASKNPWD3

@misc{pith2026250706218,
  author       = {Pith},
  title        = {Pith review of: Topological Holography for Mixed-State Phases and Phase Transitions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ASKNPWD3}},
  note         = {Machine review of arXiv:2507.06218}
}

read the original abstract

We extend the symmetry topological field theory (SymTFT) framework to open quantum systems. Using canonical purification, we embed mixed states into a doubled (2+1)-dimensional topological order and employ the slab construction to study (1+1)-dimensional mixed-state phases through condensable algebras in the doubled SymTFT. Hermiticity and positivity of the density matrix impose additional constraints on allowable anyon condensations, enabling a systematic classification of mixed-state phases - including strong-to-weak symmetry breaking (SWSSB) and average symmetry-protected topological (ASPT) phases. We present examples of mixed-state phase transitions involving SWSSB and show how gauging within the open SymTFT framework reveals connections among different mixed-state phases.

Figures

Figures reproduced from arXiv: 2507.06218 by the authors.

**Figure 1.** In this construction, we do not promote the Z2 symmetry J – which exchanges the original Hilbert space H and its dual H∗ – to a full TQFT. There are two key reasons for this. First, from a technical standpoint, it remains unclear how to consistently incorporate anti-unitary symmetries such as J into a bulk topological field theory. Second, J reflects a structural constraint of the density matrix: it encodes Hermiti… view at source ↗

**Figure 2.** FIG. 2. Constraints on condensation patterns due to positiv [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3. SWSSB and Wightman correlation function of [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (6 more)

**Figure 4.** Figure 4: FIG. 4. Symmetry localization and projective representation [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

**Figure 5.** Figure 5: FIG. 5. Phase diagram of the decohered transverse field Ising [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]

**Figure 6.** Figure 6: FIG. 6. In the [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]

**Figure 7.** Figure 7: FIG. 7. The web of gauging. The dark red arrows re [PITH_FULL_IMAGE:figures/full_fig_p016_7.png]

**Figure 8.** Figure 8: FIG. 8. Picture of SymTFT. On the right is a theory [PITH_FULL_IMAGE:figures/full_fig_p026_8.png]

**Figure 9.** Figure 9: FIG. 9. The image of solid torus. We label the two directions of loops as the above. The dark red loop gives Γ [PITH_FULL_IMAGE:figures/full_fig_p030_9.png]

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Works this paper leans on

123 extracted references · 49 canonical work pages

[104]

For canonical purification, this correspondence is less straightforward

Strictly speaking, the mapping between a quantum channel and an evolution operator is well-defined in the Choi–Jamio lkowski doubled-space representation. For canonical purification, this correspondence is less straightforward. Nonetheless, for fixed-point wavefunc- tions of topologically ordered states and in the strong decoherence limit, we expect quant...
[103]

Gapped Domain Walls, Gapped Boundaries and Topo- logical Degeneracy,

Tian Lan, Juven C. Wang, and Xiao-Gang Wen, “Gapped Domain Walls, Gapped Boundaries and Topo- logical Degeneracy,” Phys. Rev. Lett. 114, 076402 25 (2015), arXiv:1408.6514 [cond-mat.str-el]

arXiv 2015
[1]

Topological boundary Ttop on M d×{0}, where all information of topological operators are pushed to
[2]

Physical boundary Tphy on M d×{1}, where all the dynamical information stays. The topological boundary determines the symmetry category and the charges (representations) of the symmetry, while the physical boundary determines which charge operators can tunnel through the SymTFT, leading to the spontaneous breaking of the symmetry in the IR sector. 26 S y ...
[3]

A T -matrix giving the information of anyon spin, calculated by T([g],ρ),([h],σ) = δ[g],[h]δρ,σ Trρ(g) Trρ(e) . (A2) The fusion rule of anyon lines is given by ( α, β, γare labels of simple anyons, 0 = ([e], 1) labels the trivial anyon) Wα⊗ Wβ = ∑ γ N γ αβWγ N γ αβ = N−1 ∑ δ=0 S∗ δαS∗ δβ Sδγ S0δ . (A3) The quantum dimension of each simple anyon is d([g],ρ...
[4]

Appendix B: Modular Data In this section we present the modular data in the modular tensor categories we encountered in this paper

if the phase A cannot be deformed into a gapped SPT phase, then this is an intrinsically gapless SPT (igSPT) phase. Appendix B: Modular Data In this section we present the modular data in the modular tensor categories we encountered in this paper
[5]

The pure-state phase of non-trivial Z2,a2× Z2,ab SPT, given by the Lagrangian algebra ASSB 4 = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜ ⎝ 1 0 ¯ eG ¯eB 0 ¯ eGB 0 0 2 ¯ mR 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 eG 0 eG¯eG eG¯eB 0 eG¯eGB 0 0 2 eG ¯mR 0 0 0 0 0 0 0 0 0 0 0 0 0 eB 0 eB ¯eG eB ¯eB 0 eB ¯eGB 0 0 2 eB ¯mR 0 0 0 ...

work page
[6]

The pure-state phase of Z2,a2× Z2,ab trivial SPT, given by the Lagrangian algebra ASSB 5 = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜ ⎝ 1 0 ¯ eG 0 0 0 ¯ eRB ¯eRGB 0 0 0 0 0 0 0 0 0 0 2 ¯ mRB 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 eG 0 eG¯eG 0 0 0 eG¯eRB eG¯eRGB 0 0 0 0 0 0 0 0 0 0 2 eG ¯mRB 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ...

work page

Show all 123 references

[7]

The pure-state phase with Z2,ab symmetry, given by the Lagrangian algebra ASSB 6 = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜ ⎝ 1 0 ¯ eG 0 0 0 0 0 ¯ mR 0 0 0 ¯ mB 0 0 0 0 0 ¯ mRB 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 eG 0 eG¯eG 0 0 0 0 0 eG ¯mR 0 0 0 eG ¯mB 0 0 0...
[8]

The pure-state phase with Z4,a symmetric state ASSB 7 = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜ ⎝ 1 0 0 ¯ eB ¯eRG 0 0 ¯ eRGB 0 0 0 0 0 0 2 ¯ mRG 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 eB 0 0 eB ¯eB eB ¯eRG 0 0...
[9]

The non-trivial D4 SPT phase, given by the Lagrangian algebra ASPT 8 = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜ ⎝ 1 0 0 ¯ eB 0 0 0 0 ¯ mR 0 ¯ mG 0 0 0 ¯ mRG 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 eB 0 0 eB ¯eB ...
[10]

The pure-state phase of D4 trivial SPT, given by the Lagrangian algebra ASPT 9 = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜ ⎝ 1 0 0 0 0 0 0 ¯ eRGB 0 0 0 0 0 0 ¯ mRG 0 ¯ mGB 0 ¯ mRB 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ...
[11]

The pure-state phase of non-trivial Z2,a2× Z2,b SPT, given by the Lagrangian algebra ASSB 10 = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜ ⎝ 1 ¯ eR 0 ¯ eB 0 0 ¯ eRB 0 0 0 2 ¯ mG 0 0 0 0 0 0 0 0 0 0 0 eR eR¯eR 0 eR¯eB 0 0 eR¯eRB 0 0 0 2 eR ¯mG 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0...
[12]

The SWSSB phase with Zs 2,a2× Zw 2,ab ASPT, given by the Lagrangian algebra MSWSSB11 = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜ ⎝ 1 0 ¯ eG ¯eB 0 ¯ eGB 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 eR¯eR 0 0 eR¯eRG 0 eR¯eRB eR¯eRGB 0 0 0 0 0 0 0 0 0 0 0 0 0 0 eG 0 eG¯eG eG¯eB 0 eG¯eGB ...
[13]

The SWSSB phase with Dw 4 within which Zs 2,a2 is strong with ASPT, given by Lagrangian algebra MSWSSB12 = ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ 1 0 0 ¯ eB 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 eR¯eR 0 0 0 0 eR¯eRB 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 eG¯eG 0 0 eG¯eGB 0 0 0 ...
[14]

The SWSSB phase with Zw 2,a2× Zw 2,ab symmetry, given by Lagrangian algebra MSWSSB13 = ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ 1 0 ¯ eG 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 eR¯eR 0 0 eR¯eRG 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0eG 0 eG¯eG 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ...
[15]

The SWSSB phase with Dw 4 weak symmetry, within which Z2,a2 is strong, and formulates an intrinsic ASPT. This phase is given by Lagrangian algebra MSWSSB14 = ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ 1 0 0 0 0 ¯ eGB 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 eR¯eR 0 0 0 0 0 eR¯eRGB 0...
[16]

The mixed-state phase with Dw 4 symmetry MSWSSB15 = ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00eR¯eR 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 eG¯eG 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 eB¯eB 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ...
[17]

The SWSSB phase with Zs 2,a2× Zw 2,b formulating an ASPT, given by Lagrangian algebra MSWSSB16 = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜ ⎝ 1 ¯ eR 0 ¯ eB 0 0 ¯ eRB 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 eR eR¯eR 0 eR¯eB 0 0 eR¯eRB 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 eG¯eG 0 eG¯eRG eG...
[18]

The SWSSB phase with Zs 2,a2× Zw 2,b symmetry, given by Lagrangian algebra MSWSSB17 = ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ 1 ¯eR 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0eR eR¯eR 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 eG¯eG 0 eG¯eRG 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00...
[19]

The SWSSB phase with Dw 4 weak symmetry, within which theZ2,a2 is strong, and formulates an intrinsic ASPT. 44 This phase is given by the Lagrangian algebra MSWSSB18 = ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ 1 0 0 0 0 0 ¯ eRB 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 eR¯eR 0 eR¯eB 0...
[20]

The SWSSB phase with Zw 2,a2 symmetry, given by Lagrangian algebra MSWSSB19 = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜ ⎝ 1 ¯ eR ¯eG 0 ¯ eRG 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 eR eR¯eR eR¯eG 0 eR¯eRG 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 eG eG¯eR eG¯eG 0 eG¯eRG 0 0 0 0 0 0 0 0 0...
[21]

The SWSSB phase with Zw 4,a symmetry, given by Lagrangian algebra MSWSSB20 = ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ 1 0 0 0 ¯ eRG 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 eR¯eR eR¯eG 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 eG¯eR eG¯eG 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 ...
[22]

The SWSSB phase with Zw 4,a within which the Zs 2,a2 is strong, given by Lagrangian algebra MSWSSB21 = ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ 1 0 0 ¯ eB ¯eRG 0 0 ¯ eRGB 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 eR¯eR eR¯eG 0 0 eR¯eGB eR¯eRB 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 eG¯eR eG¯e...
[23]

The SWSSB phase with Dw 4 weak symmetry, within which Zs 2,a2 is strong, given by the Lagrangian algebra MSWSSB22 = ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ 1 0 0 0 0 0 0 ¯ eRGB 0 0 0 0 0 0 0 0 0 0 0 0 0 00 eR¯eR 0 0 0 eR¯eGB 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 eG¯eG 0 0 0 ...
[24]

The SWSSB phase with Zs 2,a2× Zw 2,b, given by Lagrangian algebra MSWSSB23 = ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ 1 ¯ eR 0 0 0 ¯ eGB 0 ¯ eRGB 0 0 0 0 0 0 0 0 0 0 0 0 0 0 eR eR¯eR 0 0 0 eR¯eGB 0 eR¯eRGB 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 eG¯eG eG¯eB eG¯eRG 0 eG¯eRB 0 0 0 0...
[25]

The SWSSB phase with Zs 2,a2× Zw 2,ab, given by Lagrangian algebra MSWSSB24 = ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ 1 0 ¯ eG 0 0 0 ¯ eRB ¯eRGB 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 eR¯eR 0 eR¯eB eR¯eRG eR¯eGB 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 eG 0 eG¯eG 0 0 0 eG¯eRB eG¯eRGB 0 0 0...
[26]

The SWSSB phase with Zw 4,a weak symmetry, within which Zs 2,a2 is strong, formulating an intrinsic ASPT. This phase is given by Lagrangian algebra MSWSSB25 = ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ 1 0 0 0 ¯ eRG ¯eGB ¯eRB 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 eR¯eR eR¯eG eR¯eB...
[27]

The SWSSB phase with Zw 2,ab, given by Lagrangian algebra MSWSSB26 = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜ ⎝ 1 0 ¯ eG 0 0 0 0 0 0 0 0 0 ¯ mB 0 0 0 0 0 0 0 0 0 0 eR¯eR 0 0 eR¯eRG 0 0 0 0 0 0 0 eR ¯mB 0 0 0 0 0 0 0 0 0 eG 0 eG¯eG 0 0 0 0 0 0 0 0 0 eG ¯mB 0 0 0 0 0 0 0 0 0...
[28]

The SWSSB phase with Zw 2,b, given by Lagrangian algebra MSWSSB27 = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜ ⎝ 1 ¯ eR 0 0 0 0 0 0 0 0 0 0 ¯ mB 0 0 0 0 0 0 0 0 0 eR eR¯eR 0 0 0 0 0 0 0 0 0 0 eR ¯mB 0 0 0 0 0 0 0 0 0 0 0 eG¯eG 0 eG¯eRG 0 0 0 0 0 0 0 eG ¯mB 0 0 0 0 0 0 0 0 0 ...
[29]

The SWSSB phase with Zs 2,a2 × Zs 2,b strong symmetry and additional Zw 2,ab weak symmetry, formulating an 48 ASPT. This phase is given by Lagrangian algebra MSWSSB28 = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜ ⎝ 1 0 0 ¯ eB 0 0 0 0 0 0 ¯ mG 0 0 0 0 0 0 0 0 0 0 0 0 eR¯eR 0 0...
[30]

The SWSSB phase with Zs 2,b× Zw 2,a2 , formulating an ASPT. This phase is given by Lagrangian algebra MSWSSB29 = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜ ⎝ 1 ¯ eR 0 0 0 0 0 0 0 0 ¯ mG 0 0 0 0 0 0 0 0 0 0 0 eR eR¯eR 0 0 0 0 0 0 0 0 eR ¯mG 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0...
[31]

The SWSSB phase with Zs 2,b× Zw 2,a2 symmetry, given by Lagrangian algebra MSWSSB30 = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜ ⎝ 1 ¯ eR 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ¯ mGB 0 0 0 0 0 eR eR¯eR 0 0 0 0 0 0 0 0 0 0 0 0 0 0 eR ¯mGB 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ...
[32]

The SWSSB phase with Dw 4 weak symmetry, within which Zs 2,a2× Zs 2,b is strong, given by Lagrangian algebra MSWSSB31 = ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜ ⎝ 1 0 0 0 0 0 0 ¯ eRGB 0 0 0 0 0 0 0 0 ¯ mGB 0 0 0 0 0 0 eR¯eR 0 0 0 eR¯eGB 0 0 0 0 0 0 0 0 0 0 eR¯mGB 0 0 0 0 0 ...
[33]

The SWSSB phase with Dw 4 weak symmetry, within which Zs 2,a2 × Zs 2,ab is strong, formulating an ASPT. This 50 phase is given by the Lagrangian algebra MSWSSB32 = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜ ⎝ 1 0 0 ¯ eB 0 0 0 0 ¯ mR 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ...
[34]

The SWSSB phase with Zs 2,ab×Zw 2,a2 symmetry, formulating an ASPT. This phase is given by Lagrangian algebra MSWSSB33 = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜ ⎝ 1 0 ¯ eG 0 0 0 0 0 ¯ mR 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 eG 0 eG¯eG 0 0 ...
[35]

The SWSSB phase with Zs 2,ab× Zw 2,a2 symmetry, given by Lagrangian algebra MSWSSB34 = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜ ⎝ 1 0 ¯ eG 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ¯ mRB 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 eG 0 eG¯eG 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 eG ¯mRB...
[36]

The SWSSB phase with Dw 4 weak symmetry, within which Zs 2,a2× Zs 2,ab is strong, given by Lagrangian algebra MSWSSB35 = ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜ ⎝ 1 0 0 0 0 0 0 ¯ eRGB 0 0 0 0 0 0 0 0 0 0 ¯ mRB 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 eG¯eG 0 0...
[37]

The SWSSB phase with Dw 4 weak symmetry, within which Zs 4,a is strong, formulating an ASPT. This phase is 52 given by Lagrangian algebra MSWSSB36 = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜ ⎝ 1 0 0 ¯ eB 0 0 0 0 0 0 0 0 0 0 ¯ mRG 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ...
[38]

The SWSSB phase with Dw 4 weak symmetry, within which Zs 4,a is strong, given by Lagrangian algebra MSWSSB37 = ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜ ⎝ 1 0 0 0 0 0 0 ¯ eRGB 0 0 0 0 0 0 ¯ mRG 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0...
[63]

Simple anyons are labeled by ([g], ρ), a conjugacy class in G and an irreducible representation of its centralizer (denoted N(Cg))
[64]

An S-matrix giving the information of anyon braiding calculated by S([g],ρ),([h],σ) = 1 ∣G∣ ∑ a∈[g] ∑ b∈[h]∩N(Ca) Trρ(xbx−1)∗⋅ Trσ(yay−1)∗ , (A1) where x and y are solutions to a= x−1gx and b= y−1hy
[65]

Mixed-state phases from local reversibility,

Shengqi Sang, Leonardo A. Lessa, Roger S. K. Mong, Tarun Grover, Chong Wang, and Timothy H. Hsieh, “Mixed-state phases from local reversibility,” (2025), arXiv:2507.02292 [quant-ph]

2025 arXiv
[66]

Stability of mixed-state quantum phases via finite markov length,

Shengqi Sang and Timothy H. Hsieh, “Stability of mixed-state quantum phases via finite markov length,” Phys. Rev. Lett. 134, 070403 (2025)

2025
[67]

Topological mixed states: Axiomatic approaches and phases of matter,

Tai-Hsuan Yang, Bowen Shi, and Jong Yeon Lee, “Topological mixed states: Axiomatic approaches and phases of matter,” (2025), arXiv:2506.04221 [cond- mat.str-el]

2025
[68]

Strong-to-weak spontaneous breaking of 1-form symmetry and intrinsically mixed topological order,

Carolyn Zhang, Yichen Xu, Jian-Hao Zhang, Cenke Xu, Zhen Bi, and Zhu-Xi Luo, “Strong-to-weak spontaneous breaking of 1-form symmetry and intrinsically mixed topological order,” Phys. Rev. B 111, 115137 (2025)

2025
[69]

Higher-form anomaly and long-range entanglement of mixed states,

Leonardo A. Lessa, Shengqi Sang, Tsung-Cheng Lu, Timothy H. Hsieh, and Chong Wang, “Higher-form anomaly and long-range entanglement of mixed states,” (2025), arXiv:2503.12792 [quant-ph]. 24

2025 arXiv
[70]

Intrin- sic mixed-state topological order,

Zijian Wang, Zhengzhi Wu, and Zhong Wang, “Intrin- sic mixed-state topological order,” PRX Quantum 6, 010314 (2025)

2025
[71]

Anyon condensation in mixed-state topological order,

Ken Kikuchi, Kah-Sen Kam, and Fu-Hsiang Huang, “Anyon condensation in mixed-state topological order,” (2024), arXiv:2406.14320 [hep-th]

2024
[72]

Symmetry Protected Topological Order in Open Quantum Systems,

Caroline de Groot, Alex Turzillo, and Norbert Schuch, “Symmetry Protected Topological Order in Open Quantum Systems,” Quantum 6, 856 (2022), arXiv:2112.04483 [quant-ph]

2022 arXiv
[73]

Average Symmetry- Protected Topological Phases,

Ruochen Ma and Chong Wang, “Average Symmetry- Protected Topological Phases,” Phys. Rev. X 13, 031016 (2023), arXiv:2209.02723 [cond-mat.str-el]

2023 arXiv
[74]

Fidelity Strange Correlators for Average Symmetry-Protected Topological Phases,

Jian-Hao Zhang, Yang Qi, and Zhen Bi, “Fidelity Strange Correlators for Average Symmetry-Protected Topological Phases,” (2022), arXiv:2210.17485 [cond- mat.str-el]

2022
[75]

Symmetry protected topological phases under deco- herence,

Jong Yeon Lee, Yi-Zhuang You, and Cenke Xu, “Symmetry protected topological phases under deco- herence,” Quantum 9, 1607 (2025), arXiv:2210.16323 [cond-mat.str-el]

2025 arXiv
[76]

Topological phases with average symmetries: The decohered, the disordered, and the in- trinsic,

Ruochen Ma, Jian-Hao Zhang, Zhen Bi, Meng Cheng, and Chong Wang, “Topological phases with average symmetries: The decohered, the disordered, and the in- trinsic,” Phys. Rev. X 15, 021062 (2025)

2025
[77]

Fractonic higher-order topological phases in open quantum systems,

Jian-Hao Zhang, Ke Ding, Shuo Yang, and Zhen Bi, “Fractonic higher-order topological phases in open quantum systems,” Phys. Rev. B 108, 155123 (2023)

2023
[78]

Symmetry-Protected Topological Phases of Mixed States in the Dou- bled Space,

Ruochen Ma and Alex Turzillo, “Symmetry-Protected Topological Phases of Mixed States in the Dou- bled Space,” PRX Quantum 6, 010348 (2025), arXiv:2403.13280 [quant-ph]

2025 arXiv
[79]

Tensor network formulation of symmetry protected topologi- cal phases in mixed states,

Hanyu Xue, Jong Yeon Lee, and Yimu Bao, “Tensor network formulation of symmetry protected topologi- cal phases in mixed states,” (2024), arXiv:2403.17069 [cond-mat.str-el]

2024 arXiv
[80]

Locally purified density opera- tors for symmetry-protected topological phases in mixed states,

Yuchen Guo, Jian-Hao Zhang, Hao-Ran Zhang, Shuo Yang, and Zhen Bi, “Locally purified density opera- tors for symmetry-protected topological phases in mixed states,” Phys. Rev. X 15, 021060 (2025)

2025
[81]

Holographic View of Mixed-State Symmetry-Protected Topological Phases in Open Quantum Systems,

Shijun Sun, Jian-Hao Zhang, Zhen Bi, and Yizhi You, “Holographic View of Mixed-State Symmetry-Protected Topological Phases in Open Quantum Systems,” PRX Quantum 6, 020333 (2025), arXiv:2410.08205 [quant- ph]

2025 arXiv
[82]

Mixed-state quantum anomaly and multipartite entan- glement,

Leonardo A. Lessa, Meng Cheng, and Chong Wang, “Mixed-state quantum anomaly and multipartite entan- glement,” Phys. Rev. X 15, 011069 (2025)

2025
[83]

Quantum criticality under decoherence or weak mea- surement,

Jong Yeon Lee, Chao-Ming Jian, and Cenke Xu, “Quantum criticality under decoherence or weak mea- surement,” PRX Quantum 4, 030317 (2023)

2023
[84]

Strong-to-Weak Spontaneous Symmetry Breaking in Mixed Quantum States,

Leonardo A. Lessa, Ruochen Ma, Jian-Hao Zhang, Zhen Bi, Meng Cheng, and Chong Wang, “Strong-to-Weak Spontaneous Symmetry Breaking in Mixed Quantum States,” (2024), arXiv:2405.03639 [quant-ph]

2024 arXiv
[85]

Spontaneous strong symmetry breaking in open systems: Purification perspective,

Pablo Sala, Sarang Gopalakrishnan, Masaki Oshikawa, and Yizhi You, “Spontaneous strong symmetry breaking in open systems: Purification perspective,” Phys. Rev. B 110, 155150 (2024), arXiv:2405.02402 [quant-ph]

2024 arXiv
[86]

Spontaneous symmetry breaking in open quantum systems: strong, weak, and strong-to-weak,

Ding Gu, Zijian Wang, and Zhong Wang, “Spontaneous symmetry breaking in open quantum systems: strong, weak, and strong-to-weak,” (2024), arXiv:2406.19381 [quant-ph]

2024 arXiv
[87]

Instability of steady-state mixed-state symmetry- protected topological order to strong-to-weak sponta- neous symmetry breaking,

Jeet Shah, Christopher Fechisin, Yu-Xin Wang, Joseph T. Iosue, James D. Watson, Yan-Qi Wang, Brayden Ware, Alexey V. Gorshkov, and Cheng-Ju Lin, “Instability of steady-state mixed-state symmetry- protected topological order to strong-to-weak sponta- neous symmetry breaking,” (...

2024
[88]

Fluctuation-dissipation theorem and information geometry in open quantum systems,

Jian-Hao Zhang, Cenke Xu, and Yichen Xu, “Fluctuation-dissipation theorem and information geometry in open quantum systems,” (2024), arXiv:2409.18944 [quant-ph]

2024 arXiv
[89]

Strong-to-weak spon- taneous symmetry breaking meets average symmetry- protected topological order,

Yuchen Guo and Shuo Yang, “Strong-to-weak spon- taneous symmetry breaking meets average symmetry- protected topological order,” Phys. Rev. B 111, L201108 (2025), arXiv:2410.13734 [cond-mat.str-el]

2025 arXiv
[90]

Error threshold of syk codes from strong-to-weak parity sym- metry breaking,

Jaewon Kim, Ehud Altman, and Jong Yeon Lee, “Error threshold of syk codes from strong-to-weak parity sym- metry breaking,” (2024), arXiv:2410.24225 [quant-ph]

2024 arXiv
[91]

Strong-to-weak spontaneous symmetry breaking and average symmetry protected topological order in the doubled Hilbert space,

Yoshihito Kuno, Takahiro Orito, and Ikuo Ichinose, “Strong-to-weak spontaneous symmetry breaking and average symmetry protected topological order in the doubled Hilbert space,” Phys. Rev. B 111, 174110 (2025), arXiv:2503.10311 [quant-ph]

2025 arXiv
[92]

Many-body quantum catalysts for transforming between phases of matter,

David T. Stephen, Rahul Nandkishore, and Jian-Hao Zhang, “Many-body quantum catalysts for transforming between phases of matter,” PRX Quantum 6, 020353 (2025)

2025
[93]

Hydrodynamics as the effective field the- ory of strong-to-weak spontaneous symmetry breaking,

Xiaoyang Huang, Marvin Qi, Jian-Hao Zhang, and An- drew Lucas, “Hydrodynamics as the effective field the- ory of strong-to-weak spontaneous symmetry breaking,” Phys. Rev. B 111, 125147 (2025)

2025
[94]

Diagnosing strong-to-weak sym- metry breaking via wightman correlators,

Zeyu Liu, Langxuan Chen, Yuke Zhang, Shuyan Zhou, and Pengfei Zhang, “Diagnosing strong-to-weak sym- metry breaking via wightman correlators,” Communi- cations Physics 8, 274 (2025)

2025
[95]

Efficient detection of strong-to-weak spontaneous symmetry breaking via the r´ enyi-1 correla- tor,

Zack Weinstein, “Efficient detection of strong-to-weak spontaneous symmetry breaking via the r´ enyi-1 correla- tor,” Phys. Rev. Lett. 134, 150405 (2025)

2025
[96]

Bilayer construction for mixed state phenomena with strong, weak symmetries and symmetry breakings,

Shuangyuan Lu, Penghao Zhu, and Yuan-Ming Lu, “Bilayer construction for mixed state phenomena with strong, weak symmetries and symmetry breakings,” (2024), arXiv:2411.07174 [cond-mat.str-el]

2024 arXiv
[97]

Entan- glement phases as holographic duals of anyon conden- sates,

Kasper Duivenvoorden, Mohsin Iqbal, Jutho Haege- man, Frank Verstraete, and Norbert Schuch, “Entan- glement phases as holographic duals of anyon conden- sates,” Phys. Rev. B 95, 235119 (2017)

2017
[98]

Fault-tolerant quantum computation by anyons,

A. Yu. Kitaev, “Fault-tolerant quantum computation by anyons,” Annals of Physics 303, 2–30 (2003)

2003
[99]

Family of non- abelian kitaev models on a lattice: Topological con- densation and confinement,

H. Bombin and M. A. Martin-Delgado, “Family of non- abelian kitaev models on a lattice: Topological con- densation and confinement,” Phys. Rev. B 78, 115421 (2008)

2008
[100]

Hamil- tonian and algebraic theories of gapped boundaries in topological phases of matter,

Iris Cong, Meng Cheng, and Zhenghan Wang, “Hamil- tonian and algebraic theories of gapped boundaries in topological phases of matter,” Communications in Mathematical Physics 355, 645–689 (2017)

2017
[101]

205 (AMS Math- ematical Surveys and Monographs, 2015)

Pavel Etingof, Shiomo Gelaki, Dmitri Nikshych, and Victor Ostric, Tensor Categories, Vol. 205 (AMS Math- ematical Surveys and Monographs, 2015)

2015
[102]

Finite group modular data,

Antoine Coste, Terry Gannon, and Philippe Ruelle, “Finite group modular data,” Nuclear Physics B 581, 679–717 (2000)

2000
[105]

The finite group velocity of quantum spin systems,

Elliott H. Lieb and Derek W. Robinson, “The finite group velocity of quantum spin systems,” Communica- tions in Mathematical Physics 28, 251–257 (1972)

1972
[106]

Lieb-robinson bound and locality for general markovian quantum dynamics,

David Poulin, “Lieb-robinson bound and locality for general markovian quantum dynamics,” Phys. Rev. Lett. 104, 190401 (2010)

2010
[107]

Speed limits and locality in many-body quantum dynamics,

Chi-Fang (Anthony) Chen, Andrew Lucas, and Chao Yin, “Speed limits and locality in many-body quantum dynamics,” Reports on Progress in Physics 86, 116001 (2023)

2023
[108]

The Perron–Frobenius theorem states that for a matrix with non-negative real components, the eigenvector with the largest eigenvalue also has non- negative real components

To use the theorem, consider the following manipulation which does not change the eigenstates H ′ = −HTFIM + c1, where c > 0 is a large enough so that entries of H ′ are positive. The Perron–Frobenius theorem states that for a matrix with non-negative real components, the eige...
[109]

Exploring critical systems under mea- surements and decoherence via keldysh field theory,

Ruochen Ma, “Exploring critical systems under mea- surements and decoherence via keldysh field theory,” (2023), arXiv:2304.08277 [quant-ph]

2023 arXiv
[110]

Intrinsically gapless topological phases,

Ryan Thorngren, Ashvin Vishwanath, and Ruben Ver- resen, “Intrinsically gapless topological phases,” Phys. Rev. B 104, 075132 (2021)

2021
[111]

Anomalies and gauging of U(1) symmetries,

Andrea Antinucci and Francesco Benini, “Anomalies and gauging of U(1) symmetries,” Phys. Rev. B 111, 024110 (2025), arXiv:2401.10165 [hep-th]

2025 arXiv
[112]

SymTh for non-finite symmetries,

Fabio Apruzzi, Francesco Bedogna, and Nicola Dondi, “SymTh for non-finite symmetries,” (2024), arXiv:2402.14813 [hep-th]

2024
[113]

SymTFTs for Continuous non-Abelian Sym- metries,

Federico Bonetti, Michele Del Zotto, and Ruben Mi- nasian, “SymTFTs for Continuous non-Abelian Sym- metries,” (2024), arXiv:2402.12347 [hep-th]

2024
[114]

Holographic Duals of Symmetry Broken Phases,

Andrea Antinucci, Francesco Benini, and Giovanni Rizi, “Holographic Duals of Symmetry Broken Phases,” Fortsch. Phys. 72, 2400172 (2024), arXiv:2408.01418 [hep-th]

2024 arXiv
[115]

SymTFTs for U(1) symmetries from descent,

Finn Gagliano and Inaki Garcia Etxebarria, “SymTFTs for U(1) symmetries from descent,” (2024), arXiv:2411.15126 [hep-th]

2024 arXiv
[116]

Symmetry Topological Field Theory for Fla- vor Symmetry,

Qiang Jia, Ran Luo, Jiahua Tian, Yi-Nan Wang, and Yi Zhang, “Symmetry Topological Field Theory for Fla- vor Symmetry,” (2025), arXiv:2503.04546 [hep-th]

2025 arXiv
[117]

Higher- Matter and Landau-Ginzburg Theory of Higher- Group Symmetries,

Ruizhi Liu, Ran Luo, and Yi-Nan Wang, “Higher- Matter and Landau-Ginzburg Theory of Higher- Group Symmetries,” SciPost Phys. 18, 052 (2025), arXiv:2406.03974 [hep-th]

2025 arXiv
[118]

Lec- ture notes on generalized symmetries and applications,

Ran Luo, Qing-Rui Wang, and Yi-Nan Wang, “Lec- ture notes on generalized symmetries and applications,” Phys. Rept. 1065, 1–43 (2024), arXiv:2307.09215 [hep- th]

2024 arXiv
[119]

Drinfeld centers and morita equivalence classes of fusion 2-categories,

Thibault D. D´ ecoppet, “Drinfeld centers and morita equivalence classes of fusion 2-categories,” (2024), arXiv:2211.04917 [math.QA]

2024 arXiv
[120]

Anomaly in Open Quan- tum Systems and its Implications on Mixed-State Quantum Phases,

Zijian Wang and Linhao Li, “Anomaly in Open Quan- tum Systems and its Implications on Mixed-State Quantum Phases,” PRX Quantum 6, 010347 (2025), arXiv:2403.14533 [quant-ph]

2025 arXiv
[121]

Sym- SETs and self-dualities under gauging non-invertible symmetries,

Da-Chuan Lu, Zhengdi Sun, and Zipei Zhang, “Sym- SETs and self-dualities under gauging non-invertible symmetries,” (2025), arXiv:2501.07787 [hep-th]

2025 arXiv
[122]

The Symmetry Taco: Equivalences between Gapped, Gapless, and Mixed- State SPTs,

Marvin Qi, Ramanjit Sohal, Xie Chen, David T. Stephen, and Abhinav Prem, “The Symmetry Taco: Equivalences between Gapped, Gapless, and Mixed- State SPTs,”
[123]

SymTFT Approach for Mixed States with Non-Invertible Symmetries,

Sakura Schafer-Nameki, Apoorv Tiwari, Alison War- man, and Carolyn Zhang, “SymTFT Approach for Mixed States with Non-Invertible Symmetries,”
[124]

Finite group modular data,

Antoine Coste, Terry Gannon, and Philippe Ruelle, “Finite group modular data,” Nucl. Phys. B 581, 679– 717 (2000), arXiv:hep-th/0001158. Appendix A: Categorical Landau Paradigm In this appendix we briefly review the categorical Landau paradigm for (1+ 1)D phases [38, 39, 41, 4...

2000 arXiv
[125]

if L contains any charge anyon, this is a spontaneously symmetry breaking (SSB) phase
[126]

Each gapless phase corresponds to a non-Lagrangian condensable algebra A on the physical boundary

if L contains no charge anyon, this is an SPT phase. Each gapless phase corresponds to a non-Lagrangian condensable algebra A on the physical boundary
[127]

if A contains any charge anyon, this is a gapless SSB (gSSB) phase
[128]

if A contains charge anyon and cannot be deformed into gapped SSB phase, then this is an intrinsically gapless SSB (igSSB) phase
[129]

if A contains no charge anyon, this is a gapless SPT (gSPT) phase
[130]

They are more frequently referred to as {1, e, m, f} respectively

Z(VecZ2) There are four simple anyons labelled {([0], r+),([0], r−),([1], r+),([1], r−)} (B1) where r+ is the trivial 1-dimensional representation and r− is the non-trivial one. They are more frequently referred to as {1, e, m, f} respectively. The S matrix is S = 1 2 ⎛ ⎜⎜⎜ ⎝ ...
[131]

Z(VecZ4) There are 16 simple anyons, labelled by {eamb} , a, b∈ Z4 , (B4) where e= ([0], ϕ1), ϕ1 ∶ 1↦ eπi/2) and m= ([1], ϕ0) with ϕ0 being the trivial representation. The modular S and T matrices are S = 1 4 ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜ ⎝ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ...
[132]

Z(VecS3) There are 8 simple anyons are labelled {([e], r+),([e], r−),([e], E),([a], φ0),([a], φ1),([a], φ2),([b], ϕ0),([b], ϕ1)} , (B7) and referred to as {W1, W2, W3, W4, W5, W6, W7, W8} . (B8) The S matrix is S = 1 6 ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜ ⎝ 1 1 2 2 2 2 3 3 1 1 2 2 2 2 −3 −3 2 2 4 −2...
[133]

Z(VecD4) The group D4 is presented as D4 = ⟨a, b∣a4 = b2 = e, ab= ba3⟩ . (B11) There are 22 simple anyons labeled [42] {([e], 1),([e], 1b),([e], 1ab),([a2], 1a),([e], 1a),([a2], 1b),([a2], 1ab),([a2], 1),([ab],+−),([ab],−−),([b],+−), ([b],−−),([e], E),([a2], E),([a], 1),([a],−...
[134]

changing topological boundary condition
[135]

To explicitly relate the two definitions, we derive the symmetry sectors of the partition vector for the physical theory, and observe how it transforms under gauging

summing over defect sectors. To explicitly relate the two definitions, we derive the symmetry sectors of the partition vector for the physical theory, and observe how it transforms under gauging. In some cases where gauging maintains the symmetry category invariant, gauging ca...
[136]

Gauging Z2 × Z2 As previously stated, there are three ways to describe this gauging, the first two are

Gauging in Z2 × Z2 a. Gauging Z2 × Z2 As previously stated, there are three ways to describe this gauging, the first two are
[137]

change topological boundary from LS = 1⊕ e⊕ ¯e⊕ e¯e to A1 = 1⊕ m⊕ ¯m⊕ m ¯m
[138]

For the third perspective, we first construct the sectors and physical partition vector

automorphism of MTC, by relabeling simple anyons 1↔ 1 , e↔ m , ¯e↔ ¯m , (C7) and thus LS↦ A1. For the third perspective, we first construct the sectors and physical partition vector. Since Z(VecG×G)≅ Z(VecG)⊠ Z(VecG) , (C8) we can do the decomposition ∣χT ⟩=(ZT [00]∣00⟩+ ZT [0...
[139]

This is due to the choice of boundary condition, which we analyze in Sec.V A
[140]

As an example we demonstrate the procedure of gauging Zd

Gauging in S3 × S3 The gauging of non-Abelian symmetries is more complicated than Abelian ones, for it cannot be seen as automor- phisms of MTC anymore. As an example we demonstrate the procedure of gauging Zd
[141]

Here we present three examples of mixed-state gauging: gauging Zd 2, gauging Zd 3 and gauging Sd 3

In this specific case, the previous technique of regrouping anyons does not apply either, since Sd 3 is no longer a normal subgroup of S3× S3. Here we present three examples of mixed-state gauging: gauging Zd 2, gauging Zd 3 and gauging Sd 3 . The simple anyons of Z(VecS3) lab...
[142]

Gauging Zd 3 is equivalent to switching the topological boundary condition from LS = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜ ⎝ 1 1 2 0 0 0 0 0 1 1 2 0 0 0 0 0 2 2 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟ ⎠ into MSWSSB 11 = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜ ⎝...
[143]

Gauging Zd 2 is equivalent to switching the topological boundary condition from LS = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜ ⎝ 1 1 2 0 0 0 0 0 1 1 2 0 0 0 0 0 2 2 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟ ⎠ into MSWSSB 7 = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜ ⎝ ...
[144]

The pure-state phase of D4 symmetry SSB, given by the Lagrangian algebra LS = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜ ⎝ 1 ¯ eR ¯eG 0 ¯ eRG 0 0 0 0 0 0 0 2 ¯ mB 0 0 0 0 0 0 0 0 0 eR eR¯eR eR¯eG 0 eR¯eRG 0 0 0 0 0 0 0 2 eR ¯mB 0 0 0 0 0 0 0 0 0 eG eG¯eR eG¯eG 0 eG¯eRG 0 0 0...
[145]

The pure-state phase with Z2,a2 ≡ {e, a2}⊂ D4 symmetry preserved, given by the Lagrangian algebra ASSB1 = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜ ⎝ 1 ¯ eR ¯eG ¯eB ¯eRG ¯eGB ¯eRB ¯eRGB 0 0 0 0 0 0 0 0 0 0 0 0 0 0 eR eR¯eR eR¯eG eR¯eB eR¯eRG eR¯eGB eR¯eRB eR¯eRGB 0 0 0 0 0 ...
[146]

The pure-state phase with Z2,a2× Z2,b symmetry preserved, given by Lagrangian algebra ASSB 2 = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜ ⎝ 1 ¯ eR 0 0 0 ¯ eGB 0 ¯ eRGB 0 0 0 0 0 0 0 0 2 ¯ mGB 0 0 0 0 0 eR eR¯eR 0 0 0 eR¯eGB 0 eR¯eRGB 0 0 0 0 0 0 0 0 2 eR ¯mGB 0 0 0 0 0 0 0 0...
[147]

The pure-state phase with Z2,b ≡ {e, b} symmetry preserved, given by the Lagrangian algebra ASSB 3 = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜ ⎝ 1 ¯ eR 0 0 0 0 0 0 0 0 ¯ mG 0 ¯ mB 0 0 0 ¯ mGB 0 0 0 0 0 eR eR¯eR 0 0 0 0 0 0 0 0 eR ¯mG 0 eR ¯mB 0 0 0 eR ¯mGB 0 0 0 0 0 0 0 0 0...

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [104]

For canonical purification, this correspondence is less straightforward

Strictly speaking, the mapping between a quantum channel and an evolution operator is well-defined in the Choi–Jamio lkowski doubled-space representation. For canonical purification, this correspondence is less straightforward. Nonetheless, for fixed-point wavefunc- tions of topologically ordered states and in the strong decoherence limit, we expect quant...

[2] [103]

Gapped Domain Walls, Gapped Boundaries and Topo- logical Degeneracy,

Tian Lan, Juven C. Wang, and Xiao-Gang Wen, “Gapped Domain Walls, Gapped Boundaries and Topo- logical Degeneracy,” Phys. Rev. Lett. 114, 076402 25 (2015), arXiv:1408.6514 [cond-mat.str-el]

arXiv 2015

[3] [1]

Topological boundary Ttop on M d×{0}, where all information of topological operators are pushed to

[4] [2]

Physical boundary Tphy on M d×{1}, where all the dynamical information stays. The topological boundary determines the symmetry category and the charges (representations) of the symmetry, while the physical boundary determines which charge operators can tunnel through the SymTFT, leading to the spontaneous breaking of the symmetry in the IR sector. 26 S y ...

[5] [3]

A T -matrix giving the information of anyon spin, calculated by T([g],ρ),([h],σ) = δ[g],[h]δρ,σ Trρ(g) Trρ(e) . (A2) The fusion rule of anyon lines is given by ( α, β, γare labels of simple anyons, 0 = ([e], 1) labels the trivial anyon) Wα⊗ Wβ = ∑ γ N γ αβWγ N γ αβ = N−1 ∑ δ=0 S∗ δαS∗ δβ Sδγ S0δ . (A3) The quantum dimension of each simple anyon is d([g],ρ...

[6] [4]

Appendix B: Modular Data In this section we present the modular data in the modular tensor categories we encountered in this paper

if the phase A cannot be deformed into a gapped SPT phase, then this is an intrinsically gapless SPT (igSPT) phase. Appendix B: Modular Data In this section we present the modular data in the modular tensor categories we encountered in this paper

[7] [5]

The pure-state phase of non-trivial Z2,a2× Z2,ab SPT, given by the Lagrangian algebra ASSB 4 = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜ ⎝ 1 0 ¯ eG ¯eB 0 ¯ eGB 0 0 2 ¯ mR 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 eG 0 eG¯eG eG¯eB 0 eG¯eGB 0 0 2 eG ¯mR 0 0 0 0 0 0 0 0 0 0 0 0 0 eB 0 eB ¯eG eB ¯eB 0 eB ¯eGB 0 0 2 eB ¯mR 0 0 0 ...

work page

[8] [6]

The pure-state phase of Z2,a2× Z2,ab trivial SPT, given by the Lagrangian algebra ASSB 5 = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜ ⎝ 1 0 ¯ eG 0 0 0 ¯ eRB ¯eRGB 0 0 0 0 0 0 0 0 0 0 2 ¯ mRB 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 eG 0 eG¯eG 0 0 0 eG¯eRB eG¯eRGB 0 0 0 0 0 0 0 0 0 0 2 eG ¯mRB 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ...

work page