Fermionic Villain model with exact lattice chiral symmetries

Shu-Heng Shao; Zhiyao Lu

REVIEW 3 major objections 4 minor 95 references

Fermionic Villain model with exact lattice chiral symmetries

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The fermionic Villain model realizes exact U(1)_L × U(1)_R chiral symmetry with 't Hooft anomaly of a Dirac fermion on the lattice, and gauging anomaly-free symmetries yields exactly solvable chiral gauge theories matching continuum…

desk verdict A genuinely fermionic lattice Hamiltonian with exact U(1)L × U(1)R chiral symmetry and exact spectra for gauged models; the continuum identification leans on standard bosonization, but the construction is solid and deserves a serious referee. read the letter →

arxiv 2608.02728 v1 pith:2HXNSMI3 submitted 2026-08-03 hep-th cond-mat.str-elhep-lat

Zhiyao Lu , Shu-Heng Shao This is my paper · ORCID

classification hep-thcond-mat.str-elhep-lat

keywords fermionicVillainmodelchiralsymmetrylatticegaugetheory'tHooftanomalyMajoranachainsymmetricmassgenerationT-dualitySchwinger

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

The paper constructs the fermionic Villain model: a 1+1d lattice Hamiltonian that couples a bosonic Villain compact scalar to a Kitaev chain of Majorana fermions through two Gauss-law constraints. It claims this model realizes the ordinary, compact group $U(1)_L \times U(1)_R$ as an exact locality-preserving symmetry of a genuinely fermionic system, which Nielsen–Ninomiya-type no-go theorems forbid for finite-dimensional lattice systems. The lattice charges $Q_L$ and $Q_R$ are shown to be quantized correctly, their densities to produce the continuum Schwinger term, and a $2\pi$ rotation of $U(1)_L$ to pump one unit of fermion number, establishing the level-$\pm1$ ’t Hooft anomaly at the operator level. For two copies, the anomaly-free $U(1)_{3450}$ symmetry is realized with symmetric boundary conditions, and six-fermion interactions are mapped by an exact fermionic T-duality to ordinary fermion bilinears, giving an analytic demonstration of symmetric mass generation. Gauging anomaly-free symmetries yields lattice versions of the Schwinger model and the $U(1)_{3450}$ gauge theory whose dispersion relations and masses agree with the continuum; a sympathetic reader would care because this is a genuinely fermionic lattice realizing chiral global symmetries, not merely a bosonized description.

What carries the argument

The central object is the fermionic Villain model itself: site bosons $(\phi_j,p_j)$ and link Villain gauge fields $(w_{j,j+1},\tilde\phi_{j,j+1})$ from the bosonic Villain Hamiltonian, tensored with a pair of Majorana fermions $\gamma_{j,j+1},\gamma'_{j,j+1}$ on every link, and constrained by two Gauss laws that identify bosonic exponentials with fermion bilinears, $e^{i\pi p_j + \frac{i}{2}\tilde\phi_{j-1,j}-\frac{i}{2}\tilde\phi_{j,j+1}} = i\gamma'_{j-1,j}\gamma_{j,j+1}$ and $e^{2\pi i w_{j,j+1}} = i\gamma_{j,j+1}\gamma'_{j,j+1}$. These constraints implement a fermionic Villain gauge transformation under which the Majoranas flip sign and $w_{j,j+1}$ shifts by half-integers, converting the bosonic compact scalar into a fermionic theory. The exact chiral charges $Q_L,Q_R$ are linear combinations of $p_j$ and $w_{j,j+1}$; the Schwinger term in their density commutators and the spectral-flow identity Eq. (18) carry the anomaly. The fermionic T-duality, mapping the model at radius $R$ to radius $2/R$ while translating the Majoranas by one link, is the engine behind both the analytic symmetric mass generation and the exact solvability of the gauged theories.

What would settle it

On a finite chain (e.g., $N\ge 8$ sites), exactly diagonalize the Hamiltonian (5) subject to the Gauss-law constraints (2) in both Ramond and Neveu–Schwarz sectors, and compare the low-lying spectrum and the quantum numbers $(Q_M,Q_W)$ with the fermionized compact-boson predictions of Appendix A; a mismatch in the charge sectors, the level spacing, or the $R=\sqrt{2}$ free-fermion degeneracies would falsify the continuum-limit claim. More targeted is the operator identity (18), that a $2\pi$ rotation of $U(1)_L$ over a segment equals $i(-1)^{j_2-j_1+1}\psi_{L,j_1-1/2}\psi^\dagger_{L,j_2+1/2}$, which can be checked by explicit commutation relations on the lattice.

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Extended reading notes

Core claim

The central discovery is that the obstructions to lattice chiral fermions vanish once the local Hilbert space is infinite-dimensional: the fermionic Villain Hamiltonian is a massless Dirac fermion in disguise, but with an exact $U(1)_L \times U(1)_R$ action generated by $Q_L = \sum_j (\tfrac12 p_j + w_{j,j+1})$ and $Q_R = \sum_j (\tfrac12 p_j - w_{j,j+1})$, where $p_j$ is the site momentum of the Villain boson and $w_{j,j+1}$ the integer Villain gauge field. The Gauss laws of the model tie the compactness of $\phi$ to the occupation numbers of the link Majoranas, so the theory is genuinely fermionic and not a bosonized reformulation. The lattice charge densities obey $[\rho_{L,j},\rho_{L,j'}] = \frac{i}{4\pi}(\delta_{j+1,j'}-\delta_{j-1,j'})$, the exact lattice Schwinger term, and the spectral-flow calculation shows a $2\pi$ $U(1)_L$ twist inserts a single left-moving fermion. The authors further prove an exact fermionic T-duality $R \leftrightarrow 2/R$ that translates the Majoranas by half a link, and use it to show that the six-fermion deformations preserving the anomaly-free $U(1)_{3450}$ global symmetry become, in the dual frame, ordinary fermion bilinear mass terms, thereby demonstrating symmetric mass generation without numerics. Finally, gauging arbitrary anomaly-free Abelian global symmetries by the background-field/Gauss-law/kinetic-term procedure produces lattice chiral gauge theories; the paper solves the spectrum of the lattice Schwinger model and the $U(1)_{3450}$ gauge theory, recovering the continuum masses $m_{\rm Schwinger}=e_{\rm cont}R/\sqrt{2\pi}$ and, at zero Thirring coupling, $m_{3450}=5e_{\rm cont}/\sqrt{\pi}$.

Load-bearing premise

The load-bearing premise is that the continuum limit of the fermionic Villain Hamiltonian, including its Gauss-law constraints, is the massless Dirac fermion with a Thirring interaction; this relies on the exact spectrum of the bosonic Villain model being the $c=1$ compact boson CFT at radius $R$, and on the standard fermionization map, a step the paper states but does not itself derive from the lattice dynamics.

Editorial extensions

If this is right

The model provides a lattice on which the level-$\pm1$ chiral anomaly of $U(1)_L \times U(1)_R$ is realized exactly, so anomaly inflow, spectral flow, and Schwinger-term physics can be studied in finite volume without noncompact or bosonized devices.
Gauging the anomaly-free $U(1)_{3450}$ symmetry gives an exactly solvable lattice chiral gauge theory whose mass spectrum, including $m_{3450}=5e_{\rm cont}/\sqrt{\pi}$ at zero Thirring coupling and an infrared free massless Dirac fermion, agrees with the continuum.
The same gauging procedure applies to any integer charges $n^{(I)}_{L,R}$ satisfying the anomaly cancellation condition $\sum_I (n^{(I)}_L)^2 = \sum_I (n^{(I)}_R)^2$, so the construction yields a broad class of solvable Abelian lattice chiral gauge theories.
The T-duality proof of symmetric mass generation is analytic rather than numerical: the six-fermion terms preserving $U(1)_{3450}\times U(1)_{0543}$ become fermion bilinears under a duality, showing that the 3450 model can be trivially gapped while preserving the symmetry.

Reading between the lines

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

If the continuum-limit identification is accepted, the same boson-plus-Majorana construction should transplant to 3+1d, where a bosonic Villain Hamiltonian already realizes chiral symmetries; the natural test is whether the 3+1d analogue admits an exact fermionic charge operator once vortex loops are decorated by Kitaev chains.
Because $T^2$ is a one-site lattice translation, the exact lattice symmetry is not simply $U(1)_L\times U(1)_R$; the continuum discrete chiral symmetry $C_R$ is emanant from the translation-like $T$, so the mod-8 anomaly of $C_R$ could be probed by finite-size energy splittings in the fermionic Villain chain.
The solvability of the gauged models suggests that the 3450 mass formula and the infrared free-fermion result are robust lattice predictions; a Euclidean or Monte Carlo simulation of the bosonized versions should reproduce $m_{3450}=5e_{\rm cont}/\sqrt{\pi}$ in the continuum limit, providing a testable cross-check of the fermionization step.

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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: this is a genuinely fermionic lattice Hamiltonian that realizes the exact U(1)L × U(1)R chiral symmetry and its 't Hooft anomaly in 1+1d, and the payoff is analytic: symmetric mass generation for the 3450 model and exact spectra for the lattice Schwinger model and the 3450 gauge theory. The construction is a Villain boson coupled to a Kitaev chain of Majoranas, with two Gauss laws that turn the compact scalar into a fermionic theory. I checked the core algebra: charge quantization from the Gauss laws, the Schwinger term, the spectral-flow argument, and the fermionic T-duality that maps the six-fermion deformation to a bilinear mass. All of it hangs together. The spectrum in Appendix A is explicit and clean: gauge-fix the Majoranas, and the fermionic model becomes the bosonic Villain model with specific sector restrictions, which is exactly the standard fermionization of a c=1 compact boson. The reference list is appropriate; the self-cited fermionization framework is the standard one, not a hidden assumption.

The one real soft spot is flagged in your note: the final identification with the Dirac fermion, including the radius-to-coupling relation 2/R^2 = 1 + g/π, is imported from continuum bosonization rather than derived from lattice operators. The sector table and the charge quantization are derived on the lattice, but the step 'this fermionized c=1 boson is the Thirring model with that coupling' is taken from Refs. [31,36]. That is a standard and extremely well-tested dictionary, so I don't consider it a deal-breaker, but it is load-bearing: the Schwinger and 3450 mass formulas in Section IV depend on it. A referee will reasonably ask for a sharper statement of what is being assumed and what is proven, and maybe for a direct lattice demonstration that no extra Z2 orbifold lurks in the mapping.

The second soft spot is smaller: the symmetric boundary Hamiltonian in Section III.C is sketched rather than solved. The claim that the boundary terms gap the edge modes is supported by a T-duality argument, but the details of the bounded spectrum are not shown. That is a gap in presentation, not evidence of a flaw. There are also no numerical checks, though the analytic results are strong enough that I don't think numerics are required for the main claims.

Who should read this: anyone working on lattice chiral fermions, symmetric mass generation, or 2d gauge theories. It is a substantial methodological step, and the exact solvability of the gauged models makes it a useful reference for future work. I would send it to a serious referee. The construction is new, the calculations are careful, and the central argument holds up; the requested revisions will be about tightening the dictionary and filling in the boundary details, not about fixing a broken result.

Referee Report

3 major / 4 minor

Summary. The paper introduces a 'fermionic Villain model' in 1+1 dimensions: Villain bosons on sites and links are coupled to a Kitaev chain of Majorana fermions via two Gauss-law constraints (Eq. (2)). The authors claim that this genuinely fermionic lattice Hamiltonian exactly realizes the U(1)_L × U(1)_R global symmetry and the associated chiral anomaly of a massless Dirac fermion, with the bosonic fields evading Nielsen–Ninomiya-type no-go theorems. They construct local fermion operators (Eq. (4)), compute the Schwinger term and spectral flow, and establish a fermionic T-duality (Eq. (19)). For two copies, they realize the anomaly-free U(1)_3450 symmetry, construct symmetric boundary conditions, and demonstrate symmetric mass generation by mapping six-fermion terms to bilinear masses. They then gauge general anomaly-free Abelian symmetries and derive exact mass spectra for the lattice Schwinger model and the 3450 gauge theory, matching continuum results.

Significance. If the claims hold, this is a significant advance: it provides a local, genuinely fermionic lattice Hamiltonian with exact chiral global symmetry and anomaly, circumventing no-go theorems by using continuous bosonic fields. The analytic solvability is a major strength: the spectrum reduction in App. A, the Schwinger-term computation, and the dispersion relations are derived from the Hamiltonian rather than fitted. The paper also gives a concrete route to lattice chiral gauge theories. The main weakness is that the identification of the continuum limit with the Thirring model relies on standard c=1 bosonization/fermionization results that are cited rather than rederived at the lattice level; this leaves a gap in the central claim, though the exact lattice-level symmetry and anomaly statements are independent and convincing.

major comments (3)

[§II.B and App. A] The central claim that the lattice Hamiltonian (5) flows to the massless Dirac fermion with Thirring coupling (6) is not derived entirely at the lattice level. Appendix A reduces the fermionic spectrum to that of the bosonic Villain model with sector restrictions (Table I), but the final identification with the Dirac fermion uses the standard c=1 bosonization dictionary (Refs. [31,36]), including the relation 2/R^2 = 1 + g/π and the assignment of half-integer Ramond charges. Since the mass formulas in Secs. IV.B–C and the interpretation of the exact chiral symmetry as that of a Dirac fermion depend on this identification, please either provide additional lattice-level evidence (e.g., the current algebra or fermion two-point functions in the scaling limit) or state explicitly that the continuum identification is an assumption following from standard bosonization, and soften the wording of the central claim accordingly.
[§III.C, Eqs. (31)–(36)] The construction of symmetric boundary conditions is only sketched. The paper asserts that the boundary Hamiltonian in Eq. (36) removes the edge-mode degeneracy and yields a well-defined spectrum, but no computation or proof is given. Since the existence of symmetric boundary conditions is one of the main results, please provide an explicit low-energy spectrum or at least a detailed argument for the gap and for the absence of residual zero modes.
[§IV.A, Eqs. (38)–(41)] In the general gauging procedure, the Gauss law Eq. (41) has exp(2πi E_{j,j+1}) equal to a product of fermion operators, so the electric field E is not necessarily integer-valued. The later treatment of the Schwinger model transforms to exp(2πiE)=1, but for the general construction the quantization of E and the role of the Z2 subgroup of the gauge group should be clarified. This is needed to substantiate the claim that the procedure gauges a compact U(1) symmetry and to ensure that the four Gauss laws (38)–(41) are mutually consistent beyond the two examples studied.

minor comments (4)

[§II.D, Eq. (19)] The fermionic T-duality is stated to map R to 2/R, but the action on the Thirring coupling g in Eq. (6) is not given. Adding g' in terms of g would help readers verify the duality in the continuum limit.
[Eqs. (29) and App. D] The notation for the capitalized fields in Eqs. (29) and (D1)–(D4) is introduced quickly; a short summary table of the duality frames would improve readability.
[App. D] The statement that the ground state is 'gapped and non-degenerate' after gauge fixing would benefit from a few more steps, especially regarding the residual Gauss law relating different integers m^{(I)}.
[References] Reference [78] is listed as 'to appear'; please update if possible.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: lattice spectra and anomaly computations are derived explicitly, and continuum identifications rest on external bosonization results.

full rationale

The paper's derivation chain is self-contained at the lattice level. Appendix A obtains the fermionic Villain spectrum by gauge-fixing the Majorana Gauss laws, reducing it to the bosonic Villain spectrum with the sector restrictions in Table I; the bosonic spectrum is taken from an external exact solution (Ref. [19], not by the present authors), and the sector restrictions are derived from the lattice Gauss laws (Eqs. (A5)-(A9)), not assumed. The final identification of the fermionized c=1 CFT with the massless Dirac fermion plus Thirring interaction uses the standard continuum bosonization dictionary of Refs. [31,36], including the radius-coupling relation 2/R^2=1+g/pi; these are independent external inputs, not fitted values or the paper's own outputs. The chiral charges, Schwinger term, and spectral flow are computed directly from the lattice operators (Eqs. (9), (15), (18)) and then compared with continuum results, rather than being derived from them. The Schwinger and 3450 mass formulas follow from explicit dispersion relations (Eqs. (54), (59)-(60)) with no free parameters tuned to the continuum answers. The self-citation to Ref. [32] only corroborates the standard fermionization sector map that is independently exhibited in Table I, so it is not load-bearing. No step reduces by construction to its own input.

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

The model introduces a new lattice construction, but it does not introduce new fundamental constants or entities. It relies on standard fermionization and CFT identifications, which are broadly accepted but not proven inside the paper.

free parameters (3)

R = free parameter
Radius of the bosonic Villain model, related to the Thirring coupling; not fitted to data, but varied to define the model.
lambda = free parameter
Coefficient of the six-fermion deformation, used to drive symmetric mass generation; taken large but not fitted.
e = free parameter
Gauge coupling introduced in the gauged models; related to continuum coupling but not fitted to data.

assumptions (3)

domain assumption The bosonic Villain model has the spectrum of a c = 1 compact boson CFT at radius R.
The continuum limit identification of the fermionic model relies on this correspondence, which is standard but not proven within the paper.
domain assumption The fermionization of the c = 1 CFT is the massless Dirac fermion with a Thirring interaction.
Used to claim the lattice Hamiltonian flows to the Dirac fermion theory.
domain assumption The T-duality transformations preserve the Gauss laws and map the model to itself.
Explicitly verified in the paper via the Gauss law constraints, but the continuum interpretation relies on standard T-duality results.

invented entities (1)

Fermionic Villain model
purpose: A lattice Hamiltonian coupling Villain bosons to Kitaev Majorana fermions to realize chiral symmetry.
This is a new model; its physical validity is argued through consistency checks, not through an independent external prediction.

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Cite this review

Pith. "Pith review of Fermionic Villain model with exact lattice chiral symmetries." pith.science (2026). https://pith.science/paper/2HXNSMI3

@misc{pith2026260802728,
  author       = {Pith},
  title        = {Pith review of: Fermionic Villain model with exact lattice chiral symmetries},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2HXNSMI3}},
  note         = {Machine review of arXiv:2608.02728}
}

abstract

We present a fermionic lattice Hamiltonian that exactly realizes the $\mathrm{U(1)}_\mathrm{L}\times \mathrm{U(1)}_\mathrm{R}$ global symmetry and the associated chiral anomalies of a massless Dirac fermion in 1+1 dimensions. The construction couples Villain bosons to a Kitaev chain of Majorana fermions, where the bosonic fields are essential for evading Nielsen-Ninomiya-type no-go theorems. For two copies of our model, we realize the anomaly-free $\mathrm{U(1)}_{3450}$ global symmetry and construct symmetric boundary conditions. We analytically demonstrate symmetric mass generation by mapping the symmetry-preserving six-fermion interactions to fermion bilinear terms using fermionic T-duality. Finally, by gauging general anomaly-free global symmetries, we obtain a broad class of lattice chiral gauge theories. As nontrivial applications, we compute the mass spectra of the lattice Schwinger model and the 3450 gauge theory, finding agreement with the corresponding continuum results.

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Works this paper leans on

95 extracted references · 14 canonical work pages

[1]

Bosonic Villain model 10
[2]

Fermionic Villain model with exact lattice chiral symmetries

Fermionic Villain model 10 B. Schwinger term 10 C. Spectral flow 11 D. More on the six-fermion deformation 11 E. Dispersion relations 12 References 12 I. INTRODUCTION Chiral symmetry plays several central roles in quantum field theory. In quantum chromodynamics, the sponta- neous breaking of the chiral global symmetry controls much of low-energy hadron ph...

work page Pith review arXiv 2026
[3]

No Go Theorem for Regularizing Chiral Fermions,

H. B. Nielsen and M. Ninomiya, “No Go Theorem for Regularizing Chiral Fermions,”Phys. Lett. B105(1981) 219–223

1981
[4]

In contrast, the model of Ref

We stress that our fermionic operators are local operators. In contrast, the model of Ref. [12] is bosonic and its fermion operators are non-local operators attached to topological lines. B. A Hamiltonian for the Dirac fermion The simplest Hamiltonian for the fermionic Villain model is H= 1 2R2 NX j=1 p2 j +R2 2 NX j=1 ϕj+1−ϕj 2π +wj,j+1 2 .(5) Even thoug...
[5]

Constraints on the Existence of Chiral Fermions in Interacting Lattice Theories

Y. Shamir, “Constraints on the existence of chiral fermions in interacting lattice theories,”Phys. Rev. Lett. 71(1993) 2691–2694,arXiv:hep-lat/9306023

work page Pith review arXiv 1993
[6]

Bosonic Villain model We first recall the spectrum of the bosonic Villain Hamiltonian, which was solved exactly in Ref. [19]. While the Hamiltonian for the bosonic model is formally the same as Eq. (5), the Gauss laws are different [13, 14]: exp 2πipj +i ˜ϕj−1,j−i ˜ϕj,j+1 = 1,(A1a) exp (2πiwj,j+1) = 1.(A1b) The Hamiltonian can be diagonalized as H= 1 4N Q...
[7]

(2a) exp iπpj + i 2 ˜ϕj−1,j− i 2 ˜ϕj,j+1 =iγ′ j−1,jγj,j+1 (A5) to flip the fermion numbers on the two links (j−1,j) and (j,j+ 1)

Fermionic Villain model On each link, the fermion state is determined by its eigenvalue of the fermion number operator nj,j+1 = 1 2(1−iγ j,j+1γ′ j,j+1) = 0,1.(A4) We use the first Gauss law in Eq. (2a) exp iπpj + i 2 ˜ϕj−1,j− i 2 ˜ϕj,j+1 =iγ′ j−1,jγj,j+1 (A5) to flip the fermion numbers on the two links (j−1,j) and (j,j+ 1). Hence, we can gauge fix all bu...
[8]

Absence of Neutri- nos on a Lattice. 1. Proof by Homotopy Theory,

H. B. Nielsen and M. Ninomiya, “Absence of Neutri- nos on a Lattice. 1. Proof by Homotopy Theory,”Nucl. Phys. B185(1981) 20. [Erratum: Nucl.Phys.B 195, 541 13 (1982)]

1981

Show all 95 references

[9]

Absence of Neutrinos on a Lattice. 2. Intuitive Topological Proof,

H. B. Nielsen and M. Ninomiya, “Absence of Neutrinos on a Lattice. 2. Intuitive Topological Proof,”Nucl. Phys. B193(1981) 173–194

1981
[10]

A modified Villain formulation of fractons and other exotic theories,

P. Gorantla, H. T. Lam, N. Seiberg, and S.-H. Shao, “A modified Villain formulation of fractons and other exotic theories,”J. Math. Phys.62no. 10, (2021) 102301, arXiv:2103.01257 [cond-mat.str-el]

2021 arXiv
[11]

A PROOF OF THE NIELSEN-NINOMIYA THEOREM,

D. Friedan, “A PROOF OF THE NIELSEN-NINOMIYA THEOREM,”Commun. Math. Phys.85(1982) 481–490

1982
[12]

Bosonization versus the Nielsen-Ninomiya theorem,

S. U. Baig, S. Chen, A. Cherman, and M. Neuzil, “Bosonization versus the Nielsen-Ninomiya theorem,” arXiv:2607.09935 [hep-th]

arXiv
[13]

A No-Go Result for Imple- menting Chiral Symmetries by Locality-Preserving Uni- taries in a Three-Dimensional Hamiltonian Lattice Model of Fermions,

L. Fidkowski and C. Xu, “A No-Go Result for Imple- menting Chiral Symmetries by Locality-Preserving Uni- taries in a Three-Dimensional Hamiltonian Lattice Model of Fermions,”Phys. Rev. Lett.131no. 19, (2023) 196601, arXiv:2306.10105 [cond-mat.str-el]

2023 arXiv
[14]

Anomalous Symmetries of Quantum Spin Chains and a Generalization of the Lieb–Schultz–Mattis Theorem,

A. Kapustin and N. Sopenko, “Anomalous Symmetries of Quantum Spin Chains and a Generalization of the Lieb–Schultz–Mattis Theorem,”Commun. Math. Phys. 406no. 10, (2025) 238,arXiv:2401.02533 [math-ph]

2025 arXiv
[15]

Anomalies in quantum spin systems and Nielsen-Ninomiya type Theorems,

R. Liu, “Anomalies in quantum spin systems and Nielsen-Ninomiya type Theorems,”arXiv:2602.13948 [math-ph]
[16]

Abelian gauge the- ories on the lattice:θ-Terms and compact gauge theory with(out) monopoles,

T. Sulejmanpasic and C. Gattringer, “Abelian gauge the- ories on the lattice:θ-Terms and compact gauge theory with(out) monopoles,”Nucl. Phys. B943(2019) 114616, arXiv:1901.02637 [hep-lat]

2019 arXiv
[17]

Chiral Lattice Gauge Theories from Symmetry Disentanglers,

R. Thorngren, J. Preskill, and L. Fidkowski, “Chiral Lattice Gauge Theories from Symmetry Disentanglers,” arXiv:2601.04304 [hep-th]
[18]

ONE-DIMENSIONAL STRING THEORY ON A CIRCLE,

D. J. Gross and I. R. Klebanov, “ONE-DIMENSIONAL STRING THEORY ON A CIRCLE,”Nucl. Phys. B344 (1990) 475–498

1990
[19]

The first Gauss law implements a fermionic Villain gauge transformation: ϕj∼ϕ j +πmj, w j,j+1∼w j,j+1− mj+1−mj 2 , γj,j+1∼(−1) mjγj,j+1, γ ′ j,j+1∼(−1) mj+1γ′ j,j+1, (3) withm j∈Z

in the context of the lattice chiral gauge theory, and can be understood as the gauging of aZ 2 (momentum) symmetry of the bosonic Villain model using fermions [27]. The first Gauss law implements a fermionic Villain gauge transformation: ϕj∼ϕ j +πmj, w j,j+1∼w j,j+1− mj+1−mj ...
[20]

Lieb-Schultz-Mattis, Lut- tinger, and ’t Hooft - anomaly matching in lat- tice systems,

M. Cheng and N. Seiberg, “Lieb-Schultz-Mattis, Lut- tinger, and ’t Hooft - anomaly matching in lat- tice systems,”SciPost Phys.15no. 2, (2023) 051, arXiv:2211.12543 [cond-mat.str-el]

2023 arXiv
[21]

Lattice quantum Vil- lain Hamiltonians: compact scalars, U(1) gauge theo- ries, fracton models and quantum Ising model dualities,

L. Fazza and T. Sulejmanpasic, “Lattice quantum Vil- lain Hamiltonians: compact scalars, U(1) gauge theo- ries, fracton models and quantum Ising model dualities,” JHEP05(2023) 017,arXiv:2211.13047 [hep-th]

2023 arXiv
[22]

Lattice chiral symmetry from bosons in 3+1d,

Z. Lu, S. Seifnashri, and S.-H. Shao, “Lattice chiral symmetry from bosons in 3+1d,”arXiv:2604.06307 [hep-th]

arXiv
[23]

Non- invertible bosonic chiral symmetry on the lattice,

L. Fidkowski, C. Xu, and C. Zhang, “Non- invertible bosonic chiral symmetry on the lattice,” arXiv:2510.17969 [cond-mat.str-el]
[24]

Quan- tized Axial Charge of Staggered Fermions and the Chiral Anomaly,

A. Chatterjee, S. D. Pace, and S.-H. Shao, “Quan- tized Axial Charge of Staggered Fermions and the Chiral Anomaly,”Phys. Rev. Lett.134no. 2, (2025) 021601, arXiv:2409.12220 [hep-th]

2025 arXiv
[25]

Exact lattice chiral symmetry in 2D gauge theory,

E. Berkowitz, A. Cherman, and T. Jacobson, “Exact lattice chiral symmetry in 2D gauge theory,”Phys. Rev. D110no. 1, (2024) 014510,arXiv:2310.17539 [hep-lat]

2024 arXiv
[26]

Exactly Solvable 1+1d Chiral Lattice Gauge Theories,

S. Seifnashri, “Exactly Solvable 1+1d Chiral Lattice Gauge Theories,”arXiv:2601.14359 [hep-th]

arXiv
[27]

Gauge Invariance and Mass. 2.,

J. S. Schwinger, “Gauge Invariance and Mass. 2.,”Phys. Rev.128(1962) 2425–2429

1962
[28]

gamma(5) invariance,

K. Johnson, “gamma(5) invariance,”Phys. Lett.5(1963) 253–255

1963
[29]

Gapped Boundary Phases of Topological Insulators via Weak Coupling,

N. Seiberg and E. Witten, “Gapped Boundary Phases of Topological Insulators via Weak Coupling,”PTEP 2016no. 12, (2016) 12C101,arXiv:1602.04251 [cond-mat.str-el]

2016 arXiv
[30]

Strong Cou- pling Calculations of Lattice Gauge Theories: (1+1)- Dimensional Exercises,

T. Banks, L. Susskind, and J. B. Kogut, “Strong Cou- pling Calculations of Lattice Gauge Theories: (1+1)- Dimensional Exercises,”Phys. Rev. D13(1976) 1043

1976
[31]

A Web of 2d Duali- ties:Z 2 Gauge Fields and Arf Invariants,

A. Karch, D. Tong, and C. Turner, “A Web of 2d Duali- ties:Z 2 Gauge Fields and Arf Invariants,”SciPost Phys. 7(2019) 007,arXiv:1902.05550 [hep-th]

2019 arXiv
[32]

A New fermion Hamiltonian for lattice gauge theory,

M. Creutz, I. Horvath, and H. Neuberger, “A New fermion Hamiltonian for lattice gauge theory,” Nucl. Phys. B Proc. Suppl.106(2002) 760–762, arXiv:hep-lat/0110009

2002 arXiv
[33]

Unpaired Majorana fermions in quan- tum wires,

A. Kitaev, “Unpaired Majorana fermions in quan- tum wires,”Phys. Usp.44no. 10S, (2001) 131–136, arXiv:cond-mat/0010440

2001 arXiv
[34]

Lat- tice T-duality from non-invertible symmetries in quan- tum spin chains,

S. D. Pace, A. Chatterjee, and S.-H. Shao, “Lat- tice T-duality from non-invertible symmetries in quan- tum spin chains,”SciPost Phys.18no. 4, (2025) 121, arXiv:2412.18606 [cond-mat.str-el]

2025 arXiv
[35]

Bosonization on Higher Genus Rie- mann Surfaces,

L. Alvarez-Gaume, J. B. Bost, G. W. Moore, P. C. Nel- son, and C. Vafa, “Bosonization on Higher Genus Rie- mann Surfaces,”Commun. Math. Phys.112(1987) 503

1987
[36]

Spin TQFTs and fermionic phases of matter,

D. Gaiotto and A. Kapustin, “Spin TQFTs and fermionic phases of matter,”Int. J. Mod. Phys. A 31no. 28n29, (2016) 1645044,arXiv:1505.05856 [cond-mat.str-el]

2016 arXiv
[37]

Fermionic SPT phases in higher dimensions and bosonization,

A. Kapustin and R. Thorngren, “Fermionic SPT phases in higher dimensions and bosonization,”JHEP10(2017) 080,arXiv:1701.08264 [cond-mat.str-el]

2017 arXiv
[38]

Comments on the N=2, N=3, N=4 Superconformal Algebras in Two- Dimensions,

A. Schwimmer and N. Seiberg, “Comments on the N=2, N=3, N=4 Superconformal Algebras in Two- Dimensions,”Phys. Lett. B184(1987) 191–196

1987
[39]

Topological Transi- tion on the Conformal Manifold,

W. Ji, S.-H. Shao, and X.-G. Wen, “Topological Transi- tion on the Conformal Manifold,”Phys. Rev. Res.2no. 3, (2020) 033317,arXiv:1909.01425 [cond-mat.str-el]

2020 arXiv
[40]

Duality Defect of the Monster CFT,

Y.-H. Lin and S.-H. Shao, “Duality Defect of the Monster CFT,”J. Phys. A54no. 6, (2021) 065201, arXiv:1911.00042 [hep-th]

2021 arXiv
[41]

Fermionic minimal models,

C.-T. Hsieh, Y. Nakayama, and Y. Tachikawa, “Fermionic minimal models,”Phys. Rev. Lett. 126no. 19, (2021) 195701,arXiv:2002.12283 [cond-mat.str-el]

2021 arXiv
[42]

Two More Fermionic Minimal Models,

J. Kulp, “Two More Fermionic Minimal Models,”JHEP 03(2021) 124,arXiv:2003.04278 [hep-th]

2021 arXiv
[43]

The Quantum Sine-Gordon Equation as the Massive Thirring Model,

S. R. Coleman, “The Quantum Sine-Gordon Equation as the Massive Thirring Model,”Phys. Rev. D11(1975) 2088

1975
[44]

Field theory commutators,

J. S. Schwinger, “Field theory commutators,”Phys. Rev. Lett.3(1959) 296–297

1959
[45]

The effect of interactions on 2D fermionic symmetry-protected topological phases with Z2 symmetry,

Z.-C. Gu and M. Levin, “The effect of interactions on 2D fermionic symmetry-protected topological phases with Z2 symmetry,”Phys. Rev. B89(2014) 201113, arXiv:1304.4569 [cond-mat.str-el]

2014 arXiv
[46]

Anomalies and Bounds on Charged Operators,

Y.-H. Lin and S.-H. Shao, “Anomalies and Bounds on Charged Operators,”Phys. Rev. D100no. 2, (2019) 025013,arXiv:1904.04833 [hep-th]

2019 arXiv
[47]

Twist gap and global symmetry in two dimen- sions,

N. Benjamin, H. Ooguri, S.-H. Shao, and Y. Wang, “Twist gap and global symmetry in two dimen- sions,”Phys. Rev. D101no. 10, (2020) 106026, arXiv:2003.02844 [hep-th]

2020 arXiv
[48]

Infinite-Order Lattice Chiral Anomalies and CPT,

E. Lew-Smith, S. D. Pace, and S.-H. Shao, 14 “Infinite-Order Lattice Chiral Anomalies and CPT,” arXiv:2606.12510 [hep-th]

arXiv
[49]

Target space duality in string theory,

A. Giveon, M. Porrati, and E. Rabinovici, “Target space duality in string theory,”Phys. Rept.244(1994) 77–202, arXiv:hep-th/9401139

1994 arXiv
[50]

A new class of (2 + 1)-dimensional topological superconductors with{Z} 8 topological classification,

X.-L. Qi, “A new class of (2 + 1)-dimensional topological superconductors with{Z} 8 topological classification,” New Journal of Physics15no. 6, (June, 2013) 065002, arXiv:1202.3983 [cond-mat.str-el]

2013 arXiv
[51]

Interacting topological phases and modular invariance,

S. Ryu and S.-C. Zhang, “Interacting topological phases and modular invariance,”Phys. Rev. B85no. 24, (June,
[52]

A lattice non-perturbative definition of an SO(10) chiral gauge theory and its induced stan- dard model,

X.-G. Wen, “A lattice non-perturbative definition of an SO(10) chiral gauge theory and its induced stan- dard model,”Chin. Phys. Lett.30(2013) 111101, arXiv:1305.1045 [hep-lat]

2013 arXiv
[53]

Nonperturbative regulariza- tion of (1+1)-dimensional anomaly-free chiral fermions and bosons: On the equivalence of anomaly matching conditions and boundary gapping rules,

J. Wang and X.-G. Wen, “Nonperturbative regulariza- tion of (1+1)-dimensional anomaly-free chiral fermions and bosons: On the equivalence of anomaly matching conditions and boundary gapping rules,”Phys. Rev. B 107no. 1, (2023) 014311,arXiv:1307.7480 [hep-lat]

2023 arXiv
[54]

Fermionic Symmetry Protected Topological Phases and Cobordisms,

A. Kapustin, R. Thorngren, A. Turzillo, and Z. Wang, “Fermionic Symmetry Protected Topological Phases and Cobordisms,”JHEP12(2015) 052,arXiv:1406.7329 [cond-mat.str-el]

2015 arXiv
[55]

Notes on 8 Majorana Fermions,

D. Tong and C. Turner, “Notes on 8 Majorana Fermions,”SciPost Phys. Lect. Notes14(2020) 1, arXiv:1906.07199 [hep-th]

2020 arXiv
[56]

Majorana chain and Ising model - (non-invertible) translations, anomalies, and em- anant symmetries,

N. Seiberg and S.-H. Shao, “Majorana chain and Ising model - (non-invertible) translations, anomalies, and em- anant symmetries,”SciPost Phys.16no. 3, (2024) 064, arXiv:2307.02534 [cond-mat.str-el]

2024 arXiv
[57]

Backfiring bosonisation,

P. Boyle Smith and Y. Zheng, “Backfiring bosonisation,” JHEP03(2026) 221,arXiv:2403.03953 [hep-th]

2026 arXiv
[58]

Symmetry-protected topolog- ical orders for interacting fermions: Fermionic topolog- ical nonlinearσmodels and a special group supercoho- mology theory,

Z.-C. Gu and X.-G. Wen, “Symmetry-protected topolog- ical orders for interacting fermions: Fermionic topolog- ical nonlinearσmodels and a special group supercoho- mology theory,”Phys. Rev. B90no. 11, (2014) 115141, arXiv:1201.2648 [cond-mat.str-el]

2014 arXiv
[59]

Quantization of Anomalous Two- dimensional Models,

I. G. Halliday, E. Rabinovici, A. Schwimmer, and M. S. Chanowitz, “Quantization of Anomalous Two- dimensional Models,”Nucl. Phys. B268(1986) 413–426

1986
[60]

Symmetric Mass Generation in the 1+1 Dimensional Chiral Fermion 3-4-5-0 Model,

M. Zeng, Z. Zhu, J. Wang, and Y.-Z. You, “Symmetric Mass Generation in the 1+1 Dimensional Chiral Fermion 3-4-5-0 Model,”Phys. Rev. Lett.128no. 18, (2022) 185301,arXiv:2202.12355 [cond-mat.str-el]

2022 arXiv
[61]

Relationship between Symmetry Protected Topological Phases and Boundary Conformal Field Theories via the Entanglement Spectrum,

G. Y. Cho, K. Shiozaki, S. Ryu, and A. W. W. Ludwig, “Relationship between Symmetry Protected Topological Phases and Boundary Conformal Field Theories via the Entanglement Spectrum,”J. Phys. A50no. 30, (2017) 304002,arXiv:1606.06402 [cond-mat.str-el]

2017 arXiv
[62]

A Solution to the 1+1D Gauged Chiral Fermion Problem,

J. Wang and X.-G. Wen, “A Solution to the 1+1D Gauged Chiral Fermion Problem,”Phys. Rev. D99 no. 11, (7, 2018) 111501,arXiv:1807.05998 [hep-lat]

2018 arXiv
[63]

Chiral Gauge Theories on the Lattice,

E. Eichten and J. Preskill, “Chiral Gauge Theories on the Lattice,”Nucl. Phys. B268(1986) 179–208

1986
[64]

On the de- coupling of mirror fermions,

C. Chen, J. Giedt, and E. Poppitz, “On the de- coupling of mirror fermions,”JHEP04(2013) 131, arXiv:1211.6947 [hep-lat]

2013 arXiv
[65]

Staggered Fermions with Chiral Anomaly Cancellation,

L.-X. Xu, “Staggered Fermions with Chiral Anomaly Cancellation,”arXiv:2501.10837 [hep-lat]

arXiv
[66]

Symmetric Mass Generation,

J. Wang and Y.-Z. You, “Symmetric Mass Generation,” Symmetry14no. 7, (2022) 1475,arXiv:2204.14271 [cond-mat.str-el]

2022 arXiv
[67]

Comments on symmetric mass generation in 2d and 4d,

D. Tong, “Comments on symmetric mass generation in 2d and 4d,”JHEP07(2022) 001,arXiv:2104.03997 [hep-th]

2022 arXiv
[68]

Anomalous symme- tries end at the boundary,

R. Thorngren and Y. Wang, “Anomalous symme- tries end at the boundary,”JHEP09(2021) 017, arXiv:2012.15861 [hep-th]

2021 arXiv
[69]

Remarks on boundaries, anomalies, and noninvertible symmetries,

Y. Choi, B. C. Rayhaun, Y. Sanghavi, and S.-H. Shao, “Remarks on boundaries, anomalies, and noninvertible symmetries,”Phys. Rev. D108no. 12, (2023) 125005, arXiv:2305.09713 [hep-th]

2023 arXiv
[70]

Boundary conformal field theory and symmetry protected topolog- ical phases in 2 + 1 dimensions,

B. Han, A. Tiwari, C.-T. Hsieh, and S. Ryu, “Boundary conformal field theory and symmetry protected topolog- ical phases in 2 + 1 dimensions,”Phys. Rev. B96no. 12, (2017) 125105,arXiv:1704.01193 [cond-mat.str-el]

2017 arXiv
[71]

’t Hooft anomalies and boundaries,

K. Jensen, E. Shaverin, and A. Yarom, “’t Hooft anomalies and boundaries,”JHEP01(2018) 085, arXiv:1710.07299 [hep-th]

2018 arXiv
[72]

Mixed global anoma- lies and boundary conformal field theories,

T. Numasawa and S. Yamaguchi, “Mixed global anoma- lies and boundary conformal field theories,”JHEP11 (2018) 202,arXiv:1712.09361 [hep-th]

2018 arXiv
[73]

Boundary States for Chiral Symmetries in Two Dimensions,

P. B. Smith and D. Tong, “Boundary States for Chiral Symmetries in Two Dimensions,”JHEP09(2020) 018, arXiv:1912.01602 [hep-th]

2020 arXiv
[74]

Boundary RG flows for fermions and the mod 2 anomaly,

P. B. Smith and D. Tong, “Boundary RG flows for fermions and the mod 2 anomaly,”SciPost Phys.10 no. 1, (2021) 010,arXiv:2005.11314 [hep-th]

2021 arXiv
[75]

What Symmetries are Preserved by a Fermion Boundary State?,

P. B. Smith and D. Tong, “What Symmetries are Preserved by a Fermion Boundary State?,” arXiv:2006.07369 [hep-th]

2006 arXiv
[76]

Majorana fermions, exact mapping between quantum impurity fixed points with four bulk fermion species, and solu- tion of the ’unitarity puzzle’,

J. M. Maldacena and A. W. W. Ludwig, “Majorana fermions, exact mapping between quantum impurity fixed points with four bulk fermion species, and solu- tion of the ’unitarity puzzle’,”Nucl. Phys. B506(1997) 565–588,arXiv:cond-mat/9502109

1997 arXiv
[77]

Lieb-Schultz-Mattis anomalies as ob- structions to gauging (non-on-site) symmetries,

S. Seifnashri, “Lieb-Schultz-Mattis anomalies as ob- structions to gauging (non-on-site) symmetries,”Sci- Post Phys.16no. 4, (2024) 098,arXiv:2308.05151 [cond-mat.str-el]

2024 arXiv
[78]

On the Absence of Sym- metric Simple Conformal Boundary Conditions,

P. Wei and Y. Zheng, “On the Absence of Sym- metric Simple Conformal Boundary Conditions,” arXiv:2512.23976 [hep-th]
[79]

Monopoles, scattering, and generalized symmetries,

M. van Beest, P. Boyle Smith, D. Delmastro, Z. Ko- margodski, and D. Tong, “Monopoles, scattering, and generalized symmetries,”JHEP03(2025) 014, arXiv:2306.07318 [hep-th]

2025 arXiv
[80]

Non-Invertible Symme- tries and Boundaries for Two-Dimensional Fermions,

G. Arias-Tamargo, P. Boyle Smith, R. Mouland, and M. L. Vel´ asquez Cotini Hutt, “Non-Invertible Symme- tries and Boundaries for Two-Dimensional Fermions,” arXiv:2605.13952 [hep-th]

arXiv
[81]

A Twist on Scattering from Defect Anomalies,

A. Antinucci, C. Copetti, G. Galati, and G. Rizi, “A Twist on Scattering from Defect Anomalies,” arXiv:2605.13961 [hep-th]

arXiv
[82]

Non-invertible Symmetries in Weyl Fermions, and Applications to Fermion-Boundary Scattering Problem,

P. Wei and Y. Zheng, “Non-invertible Symmetries in Weyl Fermions, and Applications to Fermion-Boundary Scattering Problem,”arXiv:2605.19363 [hep-th]

arXiv
[83]

Monopole catalyzed baryon de- cay: A Boundary conformal field theory approach,

I. Affleck and J. Sagi, “Monopole catalyzed baryon de- cay: A Boundary conformal field theory approach,”Nucl. Phys. B417(1994) 374–402,arXiv:hep-th/9311056

1994 arXiv
[84]

Generalized BKT Transitions and Persistent Or- der on the Lattice,

E. Berkowitz, S. Buesing, S. Chen, A. Cherman, and S. Sen, “Generalized BKT Transitions and Persistent Or- der on the Lattice,”PoSLATTICE2024(2025) 384, arXiv:2409.00502 [hep-lat]

2025 arXiv
[85]

Arf- Brown-Kervaire invariant on a lattice,

S. Araki, H. Fukaya, T. Onogi, and S. Yamaguchi, “Arf- Brown-Kervaire invariant on a lattice,”Phys. Rev. D114 no. 1, (2026) 014503,arXiv:2512.11424 [hep-lat]

2026
[86]

Appendix A: Spectrum In this Appendix, we show that the fermionic Vil- lain model Eq

for coordinating submission of their related work on a fermionic model involving rotors and fermions. Appendix A: Spectrum In this Appendix, we show that the fermionic Vil- lain model Eq. (5) reproduces the continuum spectrum of a massless Dirac fermion field with a four-fermi...
[87]

Y. Choi, N. Seiberg, S. Seifnashri, and W. Zhang , to appear. 15
[88]

Discrete chiral symmetry and mass shift in the lattice Hamiltonian approach to the Schwinger model,

R. Dempsey, I. R. Klebanov, S. S. Pufu, and B. Zan, “Discrete chiral symmetry and mass shift in the lattice Hamiltonian approach to the Schwinger model,”Phys. Rev. Res.4no. 4, (2022) 043133,arXiv:2206.05308 [hep-th]

2022 arXiv
[89]

Quantum electro- dynamics in two-dimensions,

J. H. Lowenstein and J. A. Swieca, “Quantum electro- dynamics in two-dimensions,”Annals Phys.68(1971) 172–195

1971
[90]

Vacuum po- larization and the absence of free quarks,

A. Casher, J. B. Kogut, and L. Susskind, “Vacuum po- larization and the absence of free quarks,”Phys. Rev. D 10(1974) 732–745

1974
[91]

Charge Shielding and Quark Confinement in the Massive Schwinger Model,

S. R. Coleman, R. Jackiw, and L. Susskind, “Charge Shielding and Quark Confinement in the Massive Schwinger Model,”Annals Phys.93(1975) 267

1975
[92]

Phases of 2d gauge theories and symmetric mass generation,

R. Mouland, D. Tong, and B. Zan, “Phases of 2d gauge theories and symmetric mass generation,”JHEP04 (2026) 154,arXiv:2509.12305 [hep-th]

2026 arXiv
[94]

Dharanikota and L

T. Dharanikota and L. Fidkowski 1+1d Lattice Dirac Fermions from Non-Onsite Vector and Axial Symmetries
[95]

APPLIED CONFORMAL FIELD THEORY,

P. H. Ginsparg, “APPLIED CONFORMAL FIELD THEORY,” inLes Houches Summer School in Theoret- ical Physics: Fields, Strings, Critical Phenomena. 9, 1988.arXiv:hep-th/9108028

1988 arXiv
[2012]

245132,arXiv:1202.4484 [cond-mat.str-el]

arXiv

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

Bosonic Villain model 10

[2] [2]

Fermionic Villain model with exact lattice chiral symmetries

Fermionic Villain model 10 B. Schwinger term 10 C. Spectral flow 11 D. More on the six-fermion deformation 11 E. Dispersion relations 12 References 12 I. INTRODUCTION Chiral symmetry plays several central roles in quantum field theory. In quantum chromodynamics, the sponta- neous breaking of the chiral global symmetry controls much of low-energy hadron ph...

work page Pith review arXiv 2026

[3] [3]

No Go Theorem for Regularizing Chiral Fermions,

H. B. Nielsen and M. Ninomiya, “No Go Theorem for Regularizing Chiral Fermions,”Phys. Lett. B105(1981) 219–223

1981

[4] [4]

In contrast, the model of Ref

We stress that our fermionic operators are local operators. In contrast, the model of Ref. [12] is bosonic and its fermion operators are non-local operators attached to topological lines. B. A Hamiltonian for the Dirac fermion The simplest Hamiltonian for the fermionic Villain model is H= 1 2R2 NX j=1 p2 j +R2 2 NX j=1 ϕj+1−ϕj 2π +wj,j+1 2 .(5) Even thoug...

[5] [5]

Constraints on the Existence of Chiral Fermions in Interacting Lattice Theories

Y. Shamir, “Constraints on the existence of chiral fermions in interacting lattice theories,”Phys. Rev. Lett. 71(1993) 2691–2694,arXiv:hep-lat/9306023

work page Pith review arXiv 1993

[6] [6]

Bosonic Villain model We first recall the spectrum of the bosonic Villain Hamiltonian, which was solved exactly in Ref. [19]. While the Hamiltonian for the bosonic model is formally the same as Eq. (5), the Gauss laws are different [13, 14]: exp 2πipj +i ˜ϕj−1,j−i ˜ϕj,j+1 = 1,(A1a) exp (2πiwj,j+1) = 1.(A1b) The Hamiltonian can be diagonalized as H= 1 4N Q...

[7] [7]

(2a) exp iπpj + i 2 ˜ϕj−1,j− i 2 ˜ϕj,j+1 =iγ′ j−1,jγj,j+1 (A5) to flip the fermion numbers on the two links (j−1,j) and (j,j+ 1)

Fermionic Villain model On each link, the fermion state is determined by its eigenvalue of the fermion number operator nj,j+1 = 1 2(1−iγ j,j+1γ′ j,j+1) = 0,1.(A4) We use the first Gauss law in Eq. (2a) exp iπpj + i 2 ˜ϕj−1,j− i 2 ˜ϕj,j+1 =iγ′ j−1,jγj,j+1 (A5) to flip the fermion numbers on the two links (j−1,j) and (j,j+ 1). Hence, we can gauge fix all bu...

[8] [8]

Absence of Neutri- nos on a Lattice. 1. Proof by Homotopy Theory,

H. B. Nielsen and M. Ninomiya, “Absence of Neutri- nos on a Lattice. 1. Proof by Homotopy Theory,”Nucl. Phys. B185(1981) 20. [Erratum: Nucl.Phys.B 195, 541 13 (1982)]

1981