Quantum Many-Body Lattice C-R-T Symmetry: Fractionalization, Anomaly, and Symmetric Mass Generation

Juven Wang; Yang-Yang Li; Yi-Zhuang You

REVIEW 3 major objections 5 minor 123 references

Quantum Many-Body Lattice C-R-T Symmetry: Fractionalization, Anomaly, and Symmetric Mass Generation

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper proves that in every spatial dimension, eight copies of staggered Majorana fermions—or four copies of staggered Dirac fermions with $U(1)$ broken to $\mathbb{Z}_4^F$—can be gapped by explicit symmetry-preserving stabilizer…

desk verdict A useful lattice construction of SMG with explicit stabilizers, but the general-dimension minimality claim needs a fuller proof before it can be taken as established. read the letter →

arxiv 2412.19691 v2 pith:PRTPESUQ submitted 2024-12-27 cond-mat.str-el hep-lathep-thquant-ph

Yang-Yang Li , Juven Wang , Yi-Zhuang You This is my paper · ORCID

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

keywords symmetricmassgenerationC-R-TsymmetryfractionalizationprojectiverepresentationstaggeredfermionslatticeanomalyMajoranaDirac

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

Massless Majorana and Dirac fermions stay gapless when charge conjugation, mirror reflection, time reversal, and internal symmetries forbid every fermion-bilinear mass term. The paper establishes how many copies of such fermions are needed for interactions to open a gap while preserving those symmetries—a process called symmetric mass generation (SMG). The answer is uniform in spatial dimension: eight copies of staggered Majorana fermions, or four copies of staggered Dirac fermions with the vector $U(1)$ broken to $\mathbb{Z}_4^F$, can be gapped symmetrically. The proof assigns explicit many-body symmetry operators on the lattice, identifies the projective (anomalous) phases that obstruct gapping below those copy numbers, and then exhibits commuting four-fermion stabilizer interactions with a unique ground state. This matters because SMG is a symmetry-preserving route to massive fermions, relevant to lattice regularization of chiral gauge theories and to classifying interacting fermion phases.

What carries the argument

The central object is the lattice invariant group $G \cong D_{2L_1} \times \cdots \times D_{2L_d} \times \mathbb{Z}_2$, presented by translations $T_i$, reflections $R_i$, time reversal $T$, and fermion parity $(-)^F$, with relations such as $T_i^{L_i}=1$, $R_i^2=1$, $T^2=1$, $(-)^F=(T_iR_i)^2$, $R_iT_i=(-)^F T_i^\dagger R_i$, $TT_i=(-)^F T_iT$, and $T_iT_j=(-)^F T_jT_i$. The machinery is the comparison between this invariant group and the group generated by the explicit many-body operators implementing the same geometric actions on staggered Majorana or Dirac operators. When the many-body group reproduces the invariant group only up to projective phases—for example $(-)^F(-)^F=-1$, or $R_iT_i=-(-)^F T_i^\dagger R_i$—the system is anomalous and cannot be symmetrically gapped. The dimensional-reduction step isolates the invariant subgroup inherited from the $(d-1)$-dimensional system, then inserts translational defects to make $L_i$ odd and reduce to the $0+1$d subgroup generated by $(-)^F$ and $T$, whose projective class is $\mathbb{Z}_8$ for Majorana fermions and $\mathbb{Z}_4$ for Dirac fermions. Finally, at the anomaly-free copy numbers the paper supplies on-site commuting stabilizers—four independent 4-Majorana terms per site—and proves uniqueness of the ground state by noting the eigenspace degeneracies are 1, 4, 6, 4, 1 for the four stabilizers.

What would settle it

A concrete falsifier: run exact diagonalization on a finite staggered lattice in $d=2$ or $d=3$ with all lattice lengths odd and 7 copies of Majorana fermions (or 3 copies of Dirac fermions), searching over local C-R-T-internal-symmetric four-fermion interactions; finding any interaction with a unique, symmetry-preserving gapped ground state would contradict the claimed minimal flavor numbers. A second check is to compute the full projective anomaly of the even-$L$ lattice invariant group and look for a nontrivial class that vanishes already in the odd-$L$ $0+1$d subgroup.

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Extended reading notes

Core claim

The paper's central claim is that symmetric mass generation for staggered lattice fermions is controlled by the projective representation of the lattice C-R-T-internal symmetry group, and that the minimal anomaly-free copy number is dimension-independent: 8 copies of staggered Majorana fermions and 4 copies of staggered Dirac fermions (with $U(1)$ broken to $\mathbb{Z}_4^F$) always admit an explicit SMG interaction. For fewer copies, the many-body symmetry operators realize the invariant group projectively—group relations acquire extra minus signs or phases—and these projective phases are anomalies that obstruct a unique symmetric gapped ground state. At the critical copy number the projective phases cancel; the paper writes down on-site four-fermion stabilizers, such as $\chi_1\chi_2\chi_3\chi_4+\chi_1\chi_2\chi_5\chi_6+\chi_1\chi_3\chi_5\chi_7+\chi_2\chi_3\chi_5\chi_8$ for Majoranas and $\psi_1\psi_2\psi_3\psi_4+\psi_1^\dagger\psi_2^\dagger\psi_3^\dagger\psi_4^\dagger$ for Dirac fermions, whose ground state is unique in the code space. The proof that this works in all dimensions proceeds by dimensional reduction: with all lattice lengths odd, an invariant subgroup isomorphic to the $(d-1)$-dimensional invariant group survives, so the full anomaly descends to the $0+1$d $\mathbb{Z}_8$ (Majorana) or $\mathbb{Z}_4$ (Dirac) subgroup; the paper verifies by direct computation that at the claimed copy numbers the full lattice invariant group is reproduced without any projective phase.

Load-bearing premise

The argument relies on dimensional reduction being exhaustive: it assumes that every possible obstruction to a symmetric gapped unique ground state shows up in the $0+1$d subgroup obtained by making all lattice lengths odd, so a surviving obstruction in the full lattice invariant group would invalidate the classification.

Editorial extensions

If this is right

The minimal flavor numbers for symmetric mass generation on the staggered lattice are exactly 8 Majorana copies and 4 Dirac copies in every spatial dimension, so any attempt to gap fewer copies must break one of the C-R-T-internal symmetries or leave the ground state degenerate.
At exactly those copy numbers the full lattice invariant group is reproduced without projective phases, so the explicit on-site stabilizer Hamiltonian has a unique ground state that preserves all C-R-T-internal symmetries.
Dimensional reduction turns the SMG classification into a $0+1$d statement: the $\mathbb{Z}_8$ projective anomaly of time reversal with fermion parity for Majoranas, and the $\mathbb{Z}_4$ anomaly for Dirac fermions with $U(1)$ broken to $\mathbb{Z}_4^F$, determine the answer in all spatial dimensions.
The result reproduces the known minimal flavor numbers from staggered-fermion anomaly analysis and from Kähler-Dirac-fermion or cobordism classifications, expressed as a dimension-independent $\mathbb{Z}_8$ (Majorana) or $\mathbb{Z}_4$ (Dirac) classification of SMG.
Any SMG interaction built from the on-site stabilizers can be translated across the lattice to give a full, explicitly gapped Hamiltonian with the same unique-ground-state property.

Reading between the lines

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

A testable extension of the construction is to use the same on-site stabilizers as a starting point for finite-size numerics: because the stabilizers commute, the full many-body spectrum and the symmetry charges of all excitations can be computed exactly, giving a sharp signature of the SMG transition in small systems.
The dimensional-reduction principle suggests that the minimal flavor numbers should be insensitive to the detailed lattice geometry, since any space-group symmetry whose translational generators have the same projective action on the low-energy modes would reduce to the same $0+1$d subgroup.
One could try to turn the dimensional-reduction step into a general classification scheme: given any fermionic lattice model with translations, reflections, and time reversal, compute the projective class of the $0+1$d subgroup and search for on-site stabilizers in the anomaly-free sector; the paper's results are the first worked example of this scheme.

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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

Here's my take. The paper supplies a genuinely new lattice many-body derivation of the SMG flavor numbers for staggered fermions: explicit symmetry operators, a dimensional-reduction argument, and commuting stabilizer interactions that produce a unique ground state. That construction side is solid work. The 8/4 copy counts agree with the Kähler–Dirac and cobordism classifications, which is reassuring independent evidence.

What's less solid is the general-dimension classification. The key step (Sec. IIF3, and the Dirac analogue in IIID4) states that with eight lattice copies the full invariant group is faithfully reproduced, but the formula it checks against—Eq. (138)—is only the (d−1)-dimensional subgroup generated by T_i, R_i for i≥2. The full d-dimensional presentation (Eq. (137)) includes T_1, R_1 and mixed relations with direction 1. For d=2 and d=3 the full group is checked explicitly, but for general d only the subgroup is invoked. So the claim that 8 copies are anomaly-free under the complete lattice symmetry is not actually proven as written. The dimensional-reduction step reduces to the 0+1d subgroup by taking all L_i odd, and it shows there that 4 copies have T^2 = −1. That rules out some, but not all, obstructions for fewer copies; it does not rule out an anomaly carried by T_1/R_1 alone. The minimality claim therefore rests on an unverified completeness assumption.

There are also smaller gaps: several central calculations are delegated to 'straightforward calculation,' and the stabilizer interaction terms are only shown to be invariant under the 0+1d subgroup (ZF2 × ZT2, or ZF4 × ...), not explicitly under all the translations and reflections. Since the interaction is a uniform sum over sites, translation invariance is automatic, but reflection invariance with the staggered signs is a separate check that I'd like to see done. The appendix lists many terms but no code or raw search output, so independent reproduction is work.

None of this is fatal. The on-site interacting Hamiltonian is real evidence for sufficiency, and the consistency with existing classifications carries weight. But as it stands, the general-dimension part of the paper is a plausible conjecture with strong supporting evidence, not a complete proof.

For whom: people working on lattice chiral gauge theory and interacting fermion SPTs will want to read this and will use the explicit operators. It deserves serious peer review. A good referee will ask for the missing full-group verification (ideally a computer algebra check) and for the invariance of the gapping interaction under all symmetries. I'd send it.

Referee Report

3 major / 5 minor

Summary. This paper studies the many-body lattice realization of C-R-T-internal symmetry for staggered Majorana and Dirac fermions. It assigns explicit many-body operators for translations, reflections, time reversal, charge conjugation, and ZF4 symmetry, and computes their projective phases as lattice anomalies. Using a dimensional-reduction procedure in which lattice lengths are made odd, it argues that 8 copies of staggered Majorana fermions or 4 copies of staggered Dirac fermions are anomaly-free and can be gapped by explicit on-site stabilizer interactions with a unique ground state. The paper also gives unitary transformations between staggered and free-fermion bases and tabulates SMG classifications in all spatial dimensions.

Significance. If fully established, the paper would provide a lattice many-body derivation of known minimal flavor numbers for symmetric mass generation, with explicit symmetry operators and gapping interactions, and would connect the projective-anomaly approach to Kähler-Dirac and cobordism results. Strengths include the explicit operator formulas, the extensive stabilizer search for 0+1d, the explicit on-site code Hamiltonians, and the consistency with the known Z8/Z4 classifications. The principal weakness is that the general-dimension proof relies on an unproven completeness assumption for dimensional reduction, and this affects the central general-d claim.

major comments (3)

[Sec. IIF3, Eq. (138) vs Eq. (137)] The claim that 'with eight copies of lattice, we can faithfully reproduce the invariant group defined in Eq. [138]' verifies only the subgroup generated by T_i and R_i for i=2,...,d, which is isomorphic to the (d-1)-dimensional invariant group; it omits T_1, R_1, and all mixed relations involving direction 1. The full d-dimensional presentation is Eq. (137). For d=2 and d=3 the full group check is shown (Secs. IID6 and IIE5), but for general d no such calculation is given. Since the abstract's central claim is that 'in general spatial dimensions ... 8 copies ... admit SMG', this missing check is load-bearing.
[Sec. IIF2, Eqs. (137)–(142)] The dimensional-reduction step sets all L_i odd and then examines only the 0+1d subgroup ZF2 x ZT2, finding T^2=-1 for four lattice copies. This shows that a particular obstruction survives the reduction, but it does not prove that every possible SMG obstruction of the full lattice invariant group is captured by this subgroup. In particular, an obstruction carried by T_1/R_1 or by mixed relations involving direction 1 could in principle survive even after the 0+1d subgroup becomes anomaly-free. Without a proof of completeness of this reduction, the 8-copy minimality claim and the anomaly-free full-symmetry claim are not fully established.
[Sec. IIID4, Eqs. (234)–(239)] The Dirac generalization has the same gap. The text states that 'through straightforward calculation, we can prove that these symmetries form exactly the original invariant group without anomalies' for four lattice copies, but the calculation is not shown, and the preceding dimensional reduction only verifies a lower-dimensional subgroup. The reader cannot verify from the manuscript that the full d-dimensional presentation in Eq. (226) is reproduced, which is needed for the claimed 4-copy staggered Dirac result in general dimensions.

minor comments (5)

[Sec. IIC10] The 1+1d eight-copy case is handled by asserting that doubling the four-copy operators cancels the Z2 anomalies, but no explicit eight-copy operators or group relations are shown; a short verification would make the argument self-contained.
[Sec. IIB, Eq. (15) and Appendix A] The search for stabilizers is described as 'straightforward' and the final set is listed, but the search procedure itself is not specified; including a reproducible algorithm or a short code snippet would strengthen the claim of exhaustiveness.
[Sec. IIB, Eq. (34)] The statement 'T^L = 1 for L=0,2 mod 8' includes L=0, which is not relevant for a finite chain; the condition should be restricted to L≥2.
[Appendix B] There is a typo in the sentence before Eq. (B4): 'we'll we'll choose' should be 'we'll choose'.
[Secs. IIC4–IIC10] The same symbol T is used for both translation and time-reversal operations, which makes formulas such as Eq. (39) difficult to read; a distinct symbol for translation would improve clarity.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction: the SMG copy numbers are fixed by explicit projective-representation checks and stabilizer interactions; the main weakness is an unproven dimensional-reduction completeness step, not circularity.

full rationale

The SMG copy-number claims are not circular. The paper constructs explicit many-body symmetry operators and computes their projective phases directly, e.g. T^2 = -1 in the ZF2 x ZT4 subgroup at four staggered-Majorana copies with all L_i odd (Eqs. (141)-(142)) and cancellation at eight copies (Eq. (145)). Sufficiency is established by explicit commuting-stabilizer interactions with a unique ground state (Eqs. (15), (146), (239)), which is an independent check rather than a fitted prediction. The cited prior classifications (Refs. [8,9]) supply the free-fermion invariant-group presentations used as targets, and those are self-citations; however, the paper's own lattice operator algebra and explicit gapped Hamiltonians carry the SMG argument, so the self-citations are inputs rather than the predicted result. The one significant weakness is a proof gap, not circularity: for general d, Sec. IIF3 checks only the (d-1)-dimensional invariant subgroup of Eq. (138) and the 0+1d time-reversal anomaly, not the full d-dimensional presentation of Eq. (137); the claim that the full lattice invariant group is faithfully reproduced for all d is therefore asserted rather than fully demonstrated. This affects completeness but does not make the derivation reduce to its own inputs.

Assumptions & free parameters 1 free parameters · 5 assumptions · 2 invented entities

The paper's central claim rests on standard Clifford algebra and Jordan-Wigner mathematics, plus the domain assumption that lattice symmetry fractionalization is the right diagnostic for SMG. The most fragile input is the dimensional reduction assumption that the 0+1d subgroup captures all obstructions. There are no fitted numerical parameters; the only hand-chosen element is the particular stabilizer interaction, which is not unique.

free parameters (1)

Choice of four stabilizer interactions = chi1 chi2 chi3 chi4, chi1 chi2 chi5 chi6, chi1 chi3 chi5 chi7, chi2 chi3 chi5 chi8
The specific stabilizer set is selected by brute-force search among many symmetry-preserving four-fermion terms (Appendix A). It is not forced by the anomaly argument, and different choices would also work, but the construction depends on the existence of some such set.

assumptions (5)

standard math Jordan-Wigner representation exists for Majorana operators on a finite lattice with the stated complex-conjugation convention.
Used from Sec. IIB onward to construct many-body symmetry operators in the Hilbert space of dimension 2^(N/2).
standard math Clifford algebra representation dimensions and the group presentations G ~= D_{2L} x ... x Z_2 are correct.
Invoked in Eq. (32), Eq. (68), Eq. (103), and Eq. (137), with GAP cited as the source for the presentations.
domain assumption The staggered fermion Hamiltonian with pi-flux is a valid lattice regularization of massless Majorana or Dirac fermions with the stated internal symmetry content.
The entire lattice symmetry action is defined to preserve the staggered Hamiltonian (Sec. IID1, Eq. (58); Sec. IIID1, Eq. (219)).
domain assumption A projective realization of the invariant symmetry group obstructs symmetric mass generation, while a faithful realization with an explicit commuting interaction is sufficient.
This is the conceptual bridge between symmetry fractionalization and gapping, stated in Sec. IIB and used in all classification claims.
ad hoc to paper Dimensional reduction by setting all L_i odd, via translational defects, exposes all anomalies relevant for SMG.
The paper relies on this to reduce arbitrary spatial dimension to the 0+1d case (Sec. IIF2), but does not prove that no anomaly can hide in the full lattice group once the 0+1d subgroup is anomaly-free.

invented entities (2)

Enlarged Hilbert space H ⊕ H_tw with sigma_0 / sigma_1 grading
purpose: To define well-behaved time-reversal and reflection operators when some L_i are odd and the naive lattice operator fails to commute with the Hamiltonian.
This is a mathematical bookkeeping device for twisted boundary conditions, not a new physical sector. Its use is standard in the Lieb-Schultz-Mattis and topological defect literature (Refs. 26,27).
Translational topological defects T_i
purpose: To change lattice boundary conditions and expose the anomaly structure in reduced dimensions.
The defects are borrowed from Refs. 26 and 27 and are used as a technical tool; no new independent evidence is provided outside the construction itself.

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Pith. "Pith review of Quantum Many-Body Lattice C-R-T Symmetry: Fractionalization, Anomaly, and Symmetric Mass Generation." pith.science (2026). https://pith.science/paper/PRTPESUQ

@misc{pith2026241219691,
  author       = {Pith},
  title        = {Pith review of: Quantum Many-Body Lattice C-R-T Symmetry: Fractionalization, Anomaly, and Symmetric Mass Generation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PRTPESUQ}},
  note         = {Machine review of arXiv:2412.19691}
}

read the original abstract

Charge conjugation (C), mirror reflection (R), and time reversal (T) symmetries, along with internal symmetries, are essential for massless Majorana and Dirac fermions. These symmetries are sufficient to rule out potential fermion bilinear mass terms, thereby establishing a gapless free fermion fixed point phase, pivotal for symmetric mass generation (SMG) transition. In this work, we systematically study the anomaly of C-R-T-internal symmetry in all spacetime dimensions by analyzing the projective representation (i.e. the fractionalization) of the C-R-T-internal symmetry group in the quantum many-body Hilbert space on the lattice. By discovering the fermion-flavor-number-dependent C-R-T-internal symmetry's anomaly structure, we demonstrate an alternative way to derive the minimal flavor number for SMG, which shows consistency with known results from K\"ahler-Dirac fermion or cobordism classification. Our findings reveal that, in general spatial dimensions, either 8 copies of staggered Majorana fermions or 4 copies of staggered Dirac fermions admit SMG. By directly searching for 4-fermion interactions that form commuting stabilizers respecting all symmetry constraints, we can prove the explicit SMG gapping retained a unique ground state in the codespace. Furthermore, we establish the correspondence between the symmetry operators of staggered fermions and free fermions, which is instrumental in facilitating the analysis of symmetry fractionalization at the field theory level.

Figures

Figures reproduced from arXiv: 2412.19691 by the authors.

**Figure 1.** Staggered fermion model on 2d lattice with staggered hopping amplitudes. There is an effective π-flux in each plaquette. Each unit cell contains A and B sublattices. By setting L1 odd, the anomalies in the 1d chain (shaded in red) are exposed. By further setting L2 odd, the anomalies in the 0d site (shaded in blue) are exposed. For higher space dimension d > 1, a convenient choice of lattice realization for Majoran… view at source ↗

**Figure 2.** Staggered fermion model on 3d lattice with staggered [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

123 extracted references · 32 canonical work pages

[1]

To do this, it’s intuitive for us to decompose them into Majorana operatorsψ = (χ(1) + iχ(2))/2, ψ† = (χ(1) − iχ(2))/2

With FullU(1)Symmetry We’ll start up with the full invariant group in the many-body Hilbert space and promote our previous Dirac operator ψ and ψ† into a matrix. To do this, it’s intuitive for us to decompose them into Majorana operatorsψ = (χ(1) + iχ(2))/2, ψ† = (χ(1) − iχ(2))/2. A convenient choice for the Majorana operator basis for different copies is...
[2]

With the intrinsic doublingνstag(2) = 23/dimRχCℓ(3,0) = 2, we automatically start with 2 copies on the lattice model

Many-Body Symmetry with 2 Copies: Even-L1, Even-L2, Even-L3 We start our discussion withL1, L2, andL3 even, where all reflection symmetriesZRi 2 (i = 1, 2, 3) is well-defined. With the intrinsic doublingνstag(2) = 23/dimRχCℓ(3,0) = 2, we automatically start with 2 copies on the lattice model. Translational symmetry onx−, y−, and z− directions ZTi Li is ge...
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Many-Body Symmetry with 2 Copies: Odd-L1, Even-L2, Even-L3 To expose the hidden anomalies, we reduceL1 to be odd by setting T1 topological defect. With the full invariant group above, we find that there’s an invariant subgroup T Li i = 1,R 2 i = 1,T 2 = 1,(−) F = (TiRi)2, (−)F (−)F = 1,R iTi = (−)F T † i Ri, TT i = (−)F TiT,R iT=T Ri,∀i= 2,3, TiTj = (−)F ...
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Many-Body Symmetry with 8 Copies: Odd-L1, Odd-L2, Odd-L3 According to the discussion in1 + 1d and2 + 1d, we expect to find the0 + 1d anomaly by settingL1, L2, and L3 odd. To do this, we first setT1, T2, and T3 topological defects to the Hamiltonian: H= i 2 X ν,l2,l3 ( L1−2X l1=0 χν,l1,l2,l3 χν,l1+1,l2,l3 +(−)l2+l3 χν,L1−1,l2,l3 χν,0,l2,l3 ) + i 2 X ν,l1,l...
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Our next task is to check the whole invariant group withL 1,L 2, andL 3 even

Many-Body Symmetry with 16 Copies: Even-L1, Even-L2, Even-L3 With sixteen copies of root states (eight copies of lattice system), the anomaly in the0 + 1d invariant subgroup cancels. Our next task is to check the whole invariant group withL 1,L 2, andL 3 even. Translational symmetries onx−, y−, andz− directions ZT1 L1, ZT2 L2, and ZT3 L3 are generated by ...
[6]

Dimensional Reduction with2 d+2/dimRχCℓ(d,0) Copies: Odd-L Following the discussion from1 + 1d to3 + 1d, we’ll do the same dimensional reduction process by setting trans- lational defects [26, 27] Ti on an even-Li (∀i = 1, ..., d) lattice to get an odd-Li (∀i = 1, ..., d) case, with Hamilto- nian: H= i 2 X lj ̸=li ( Li−2X li=0 (−) P µ<i lµ χlχ ˜Ti(l) + (−...
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8×2d/dimRχCℓ(d,0) copies of root states), the anomaly in the 0+1d invariant subgroup cancels

Classification for Symmetric Mass Generation With eight copies of the lattice system (i.e. 8×2d/dimRχCℓ(d,0) copies of root states), the anomaly in the 0+1d invariant subgroup cancels. Our next task is to check the whole invariant group with allLi even. Translational symmetry on each directionZTi Li is gener- ated by unitary operatorTi defined as: Ti = Y ...
[8]

Many-Body Symmetry with 1 Copy The U (1)symmetry with unitary operator U (θ)is de- fined by the action U(θ)ψ=e iθψU(θ),U(θ)ψ † =e −iθψ†U(θ),(157) which can be realized by setting charge operatorQ = ψ†ψ. The operator U (θ)in the many-body Hilbert space can be written as: U(θ) =e−iθψ†ψ = 1−(1−e −iθ)ψ†ψ = 1 +e−iθ 2 − 1−e −iθ 2 iχ(1)χ(2), (158) where we’ve us...

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Many-Body Symmetry with 2 Copies With 2 copies of the Dirac fermion system, the operator form of charge conjugationC changes, so we need to check the anomaly pattern again with 2 copies. To be more specific, the operators are defined as follows: U (1)symmetry is characterized ...
[10]

However, for n = 1, 2(i.e

WithZ F 2 orZ F 4 Symmetry For n≥ 3, the result is trivial, given by classification Z2n since the Z2n anomaly cancels with ν = 2n copies. However, for n = 1, 2(i.e. with ZF 2 and ZF 4 symmetry), the mixed Z2 anomaly between charge conjugation sym- metry ZC 2 and time-reversal ...
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The two Dirac fermions are set to be ψL and ψR, describing the left- and right-moving Dirac modes at low energy

Free Dirac Chain and Lattice Realization In the continuum model, we can use 2 Dirac fermions to describe the Hamiltonian. The two Dirac fermions are set to be ψL and ψR, describing the left- and right-moving Dirac modes at low energy. The Hamiltonian of the free Dirac chain is...
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For convenience, we set the condition Kχ(i) ν,lK = (−)i+1χν,l, where ν = 1, 2, ...labels different copies (hereν= 1) andl= 0,1, ...labels the sites

Many-Body Symmetry with 1 Copy: Even-L We’re now ready to do the second quantization process to the C-R-T -internal symmetry group on the lattice, and we’ll first focus onL=even case where reflectionR is well-defined. For convenience, we set the condition Kχ(i) ν,lK = (−)i+1χν...
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Many-Body Symmetry with 1 Copy: Odd-L In the odd-L case where reflectionR is ill-defined, we’ll again focus on the 0+1d invariant subgroup and promote our original Hilbert space into enlarged one, and the corresponding Hamiltonian becomes ˜H=H⊕H tw = H0 0H tw .(203) In the 0+1...
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Many-Body Symmetry with 2 Copies: Even-L Since all operators in the discussion of 1 copy (even-L) are bosonic operators, we can simply duplicate them to form the operator with 2 copies: Translational symmetry ZT L is generated by unitary operatorT: T= 2Y i=1 2Y ν=1 χ(i) ν,0 L−...
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Many-Body Symmetry with 2 Copies: Odd-L Though for even-L, the invariant group is free of anoma- lies, we’ll find in this section that there are still anomalies underlying in the odd-L case with 2 copies. Again, we can assign symmetries in the0 + 1d invariant subgroup: ZF 4 sy...
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Indeed, we can set on-site charge-4e superconducting interaction or other interactions given in Appendix

Many-Body Symmetry with 4 Copies By doubling the system with 2 copies, theZ2 anomalies successfully cancel, and there are no more obstructions towards gapping. Indeed, we can set on-site charge-4e superconducting interaction or other interactions given in Appendix. B. The expl...
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Staggered Fermion Model For higher space dimensiond >1, we can again use the staggered fermion model as lattice realization for Dirac fermion for convenience. The staggered fermion is described by the Hamiltonian: H= i 2 X l,ˆµ (−) P ν<µ lν ψ† l ψl+ˆµ+h.c.,(219) with the same ...
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Dimensional Reduction with2 d+1/dimCψCℓ(d) Copies: Odd-L To get an odd-Li (∀i = 1, ..., d) system where Ri is ill-defined, we can set translational defects [26, 27] Ti on an even-Li (∀i = 1, ..., d) lattice. The Hamiltonian is then given by: H= i 2 X lj ̸=li ( Li−2X li=0 (−) P...
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Our next task is to check if the whole invariant group with all Li even faithfully reproduce the original lattice invariant group

Classification for Symmetric Mass Generation The Z2 anomalies above cancels with 4 copies of lattice system (i.e.4 × 2d/dimCψCℓ(d) copies of root states). Our next task is to check if the whole invariant group with all Li even faithfully reproduce the original lattice invarian...
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Invariant Group on Lattice From the group presentation T L = 1,R 2 = 1,T 2 = 1,(−) F = (TR)2, (−)F (−)F = 1,RT= (−) F T †R, TT= (−) F TT,RT=T R, (C1) we can also derive the following relations: Commutation relation between translationT and re- flectionR: RT=RT(−) F (−)F = ((−)...
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Projective Invariant Group on Lattice In the projective invariant group of 1 copy: T L = 1,R 2 = 1,T 2 = 1,(−) F = (TR)2, (−)F (−)F =−1,RT=−(−) F T †R, TT=−(−) F TT,RT=T R, (C8) 36 these relations may attach an additional minus sign: Commutation relation between translationT a...

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

To do this, it’s intuitive for us to decompose them into Majorana operatorsψ = (χ(1) + iχ(2))/2, ψ† = (χ(1) − iχ(2))/2

With FullU(1)Symmetry We’ll start up with the full invariant group in the many-body Hilbert space and promote our previous Dirac operator ψ and ψ† into a matrix. To do this, it’s intuitive for us to decompose them into Majorana operatorsψ = (χ(1) + iχ(2))/2, ψ† = (χ(1) − iχ(2))/2. A convenient choice for the Majorana operator basis for different copies is...

[2] [2]

With the intrinsic doublingνstag(2) = 23/dimRχCℓ(3,0) = 2, we automatically start with 2 copies on the lattice model

Many-Body Symmetry with 2 Copies: Even-L1, Even-L2, Even-L3 We start our discussion withL1, L2, andL3 even, where all reflection symmetriesZRi 2 (i = 1, 2, 3) is well-defined. With the intrinsic doublingνstag(2) = 23/dimRχCℓ(3,0) = 2, we automatically start with 2 copies on the lattice model. Translational symmetry onx−, y−, and z− directions ZTi Li is ge...

[3] [3]

Many-Body Symmetry with 2 Copies: Odd-L1, Even-L2, Even-L3 To expose the hidden anomalies, we reduceL1 to be odd by setting T1 topological defect. With the full invariant group above, we find that there’s an invariant subgroup T Li i = 1,R 2 i = 1,T 2 = 1,(−) F = (TiRi)2, (−)F (−)F = 1,R iTi = (−)F T † i Ri, TT i = (−)F TiT,R iT=T Ri,∀i= 2,3, TiTj = (−)F ...

[4] [4]

Many-Body Symmetry with 8 Copies: Odd-L1, Odd-L2, Odd-L3 According to the discussion in1 + 1d and2 + 1d, we expect to find the0 + 1d anomaly by settingL1, L2, and L3 odd. To do this, we first setT1, T2, and T3 topological defects to the Hamiltonian: H= i 2 X ν,l2,l3 ( L1−2X l1=0 χν,l1,l2,l3 χν,l1+1,l2,l3 +(−)l2+l3 χν,L1−1,l2,l3 χν,0,l2,l3 ) + i 2 X ν,l1,l...

[5] [5]

Our next task is to check the whole invariant group withL 1,L 2, andL 3 even

Many-Body Symmetry with 16 Copies: Even-L1, Even-L2, Even-L3 With sixteen copies of root states (eight copies of lattice system), the anomaly in the0 + 1d invariant subgroup cancels. Our next task is to check the whole invariant group withL 1,L 2, andL 3 even. Translational symmetries onx−, y−, andz− directions ZT1 L1, ZT2 L2, and ZT3 L3 are generated by ...

[6] [6]

Dimensional Reduction with2 d+2/dimRχCℓ(d,0) Copies: Odd-L Following the discussion from1 + 1d to3 + 1d, we’ll do the same dimensional reduction process by setting trans- lational defects [26, 27] Ti on an even-Li (∀i = 1, ..., d) lattice to get an odd-Li (∀i = 1, ..., d) case, with Hamilto- nian: H= i 2 X lj ̸=li ( Li−2X li=0 (−) P µ<i lµ χlχ ˜Ti(l) + (−...

[7] [7]

8×2d/dimRχCℓ(d,0) copies of root states), the anomaly in the 0+1d invariant subgroup cancels

Classification for Symmetric Mass Generation With eight copies of the lattice system (i.e. 8×2d/dimRχCℓ(d,0) copies of root states), the anomaly in the 0+1d invariant subgroup cancels. Our next task is to check the whole invariant group with allLi even. Translational symmetry on each directionZTi Li is gener- ated by unitary operatorTi defined as: Ti = Y ...

[8] [8]

Many-Body Symmetry with 1 Copy The U (1)symmetry with unitary operator U (θ)is de- fined by the action U(θ)ψ=e iθψU(θ),U(θ)ψ † =e −iθψ†U(θ),(157) which can be realized by setting charge operatorQ = ψ†ψ. The operator U (θ)in the many-body Hilbert space can be written as: U(θ) =e−iθψ†ψ = 1−(1−e −iθ)ψ†ψ = 1 +e−iθ 2 − 1−e −iθ 2 iχ(1)χ(2), (158) where we’ve us...