REVIEW
Chern-Simons Currents and Chiral Fermions on the Lattice
High Energy Physics - Lattice · Submitted 1 Sep 1992 · 10 references
Pith's verdict: The constant-mass winding-number result is solid and the zeromode cross-check is genuine; the soft spots are the unproven claim about lattice multi-photon vertices and the adiabatic step at a one-plaquette wall.
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Referee report
The paper
The paper claims
The paper shows that the Chern-Simons term induced by Wilson fermions on a $(2n+1)$-dimensional lattice is a topological winding number that is piecewise constant in $m/r$, jumping at $m/r=0,2,4$ up to $2d$, and that this staircase matches,
The reading
What carries the argument
The load-bearing device is a rewriting of the induced Chern-Simons integral as a winding number. Gauge invariance gives the Ward identity $Lambda_mu(p,p) = -i partial_{p_mu} S^{-1}(p)$, which collapses the $(n+1)$-photon graph of equation (2) into the fully antisymmetrised trace of equation (4), $c_n propto int_{BZ} mathrm{Tr}[(S partial_{mu_1} S^{-1}) ... (S partial_{mu_{2n+1}} S^{-1})]$. Writing $S^{-1}(p) = N(p) V(p)$ with $V(p) = cos|vec{theta}(p)| + i hatsigma cdot vec{gamma} sin|vec{theta}(p)|$ unitary, the prefactor $N(p)$ drops out and $V(p)$ becomes a map from the Brillouin-zone torus to the sphere, whose homotopy class is the coefficient. Because a winding number can only change where the map degenerates, the coefficient must be piecewise constant, and the jumps can be computed from infinitesimal neighbourhoods of the Brillouin-zone corners alone, which is what yields the sum over $k$ in equation (9).
Extended reading notes
Core claim
The central result is equation (9): for Wilson fermions the induced Chern-Simons coefficient is $$c_n(m,r) = i(-1)^n 2(n+1)(2pi)^n n! sum_{k=0}^{d} (-1)^k binom{d}{k} frac{m-2rk}{|m-2rk|},$$ to be compared with the continuum answer $c_n(m) = i(-1)^n 2(n+1)(2pi)^n n! , m/|m|$. The lattice coefficient is a sum of step functions, so it is constant on each interval $2k < |m/r| < 2k+2$, vanishes for $m/r<0$ and for $m/r>2d$, and jumps at the $d+1$ critical values $m/r = 0, 2, ..., 2d$ where the inverse propagator has a zero at a corner of the Brillouin zone. Feeding this into a domain-wall mass profile, the ratio of the lattice to the continuum Chern-Simons current is $2(-1)^ell binom{d-1}{ell}$ for $2ell < |m_0/r| < 2(ell+1)$, which is exactly the multiplicity and chirality of zeromodes that the domain-wall spectrum is known to carry. For the usual window $0 < |m_0/r| < 2$ the current vanishes on one side of the wall and has twice the continuum magnitude on the other, so the total divergence across the wall still matches the continuum anomaly.
Load-bearing premise
The authors explicitly assume that far enough from the domain wall on either side the mass can be treated as constant, so that the current follows from varying the local Chern-Simons action with $c_n(m(s),r)$; the wall in the lattice model is only about one lattice spacing thick, so this adiabatic step is an assumption rather than a derivation, and the paper leans on the separate numerical check and the zeromode match to support it.
Editorial extensions
If this is right
- On a lattice the induced Chern-Simons current vanishes on the side of a domain wall where $m(s)/r < 0$, and on the other side carries $2(-1)^ell binom{d-1}{ell}$ times the continuum value when $2ell < |m_0/r| < 2(ell+1)$.
- For the standard domain-wall window $0 < |m_0/r| < 2$ the lattice current is zero on one side and twice the continuum magnitude on the other, so the total current flow into the wall still equals the continuum anomaly of a single chiral zeromode.
- The domain-wall model of ref. [1] reproduces the continuum anomaly for chiral fermions on a finite lattice even when the domain-wall height $m_0$ is of order one in lattice units.
- Because the coefficient is computed exactly for $m$ of order one, the same bookkeeping carries over to non-Abelian background gauge fields without changing the method.
Reading between the lines
- Because the coefficient is piecewise constant, the inflow argument only needs the mass to cross the critical values; a wall one lattice spacing thick should still give the right divergence, which is a stronger statement than the adiabatic derivation actually delivers.
- The winding-number rewriting should apply to any lattice action whose inverse propagator factorises as $N(p)V(p)$ with $V$ unitary, making it a general diagnostic for whether a given discretisation reproduces a chosen anomaly.
- A discretisation whose induced coefficient varied continuously with $m/r$ would signal a propagator zero inside the Brillouin zone, and hence either an unwanted doubler or a phase with no gap.
Original abstract
We compute the Chern-Simons current induced by Wilson fermions on a $d=2n+1$ dimensional lattice, making use of a topological interpretation of the momentum space fermion propagator as a map from the torus to the sphere, $T^{d}\to S^{d}$. These mappings are shown to fall in different homotopy classes depending on the value of $m/r$, where $m$ is the fermion mass and $r$ is the Wilson coupling. As a result, the induced Chern-Simons term changes discontinuously at $d+1$ different values for $m$, unlike in the continuum. This behavior is exactly what is required by the peculiar spectrum found for a recently proposed model of chiral lattice fermions as zeromodes bound to a domain wall.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
We compute the Chern-Simons current induced by Wilson fermions on a $d=2n+1$ dimensional lattice, making use of a topological interpretation of the momentum space fermion propagator as a map from the torus to the sphere, $T^{d}\to S^{d}$. These mappings are shown to fall in different homotopy classes depending on the value of $m/r$, where $m$ is the fermion mass and $r$ is the Wilson coupling. As a result, the induced Chern-Simons term changes discontinuously at $d+1$ different values for $m$, unlike in the continuum. This behavior is exactly what is required by the peculiar spectrum found for a recently proposed model of chiral lattice fermions as zeromodes bound to a domain wall.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We compute the Chern-Simons current induced by Wilson fermions on a $d=2n+1$ dimensional lattice, making use of a topological interpretation of the momentum space fermion propagator as a map from the torus to the sphere, $T^{d}\to S^{d}$. | contribution 0.91 | 0.95 | 0.10 | asserted 1.00 |
| s1 | These mappings are shown to fall in different homotopy classes depending on the value of $m/r$, where $m$ is the fermion mass and $r$ is the Wilson coupling. | result 0.92 | 0.86 | 0.12 | asserted 1.00 |
| s2 | As a result, the induced Chern-Simons term changes discontinuously at $d+1$ different values for $m$, unlike in the continuum. | result 1.00 | 0.84 | 0.50 | asserted 1.00 |
| s3 | This behavior is exactly what is required by the peculiar spectrum found for a recently proposed model of chiral lattice fermions as zeromodes bound to a domain wall. | interpretation 0.99 | 0.27 | 0.29 | asserted 1.00 |
What was examined, and what was not
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Figures
Figures reproduced from arXiv:hep-lat/9209003 by the authors.
References
How the literature uses it
34 arXiv papers cite this in 62 sentences
Papers citing it · 44 sentences without a confident label
Builds on it 1 paper
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In the present paper we generalize the approach of [ref] , where momentum space topology of the relativistic gapped models was considered (see also [ref] and [ref] ).
Generalized unparticles, zeros of the Green function, and momentum space topology of the lattice model with overlap fermions · Introduction · 1.00
Uses its method or tool 2 papers
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While it is tricky to directly implement CS action at the lattice scale, to implement it in the IR, we can use a lattice fermion $χ^a$ in the fundamental representation of $U(N)$, with $1<M_χ<3$ [ref] (see Appendix [ref]).
Strong-Weak Chern-Simons-Matter Dualities from a Lattice Construction · Lattice Constructions · 0.93
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To address this we borrow the techniques of Ref. [ref] to show that the Feynman integral is actually computing a winding number.
Index theorems, generalized Hall currents and topology for gapless defect fermions · Majorana fermion in $1+1$ dimensions · 1.00
Compares with it 2 papers
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There is an analogous calculation on the lattice which arrives at the same result [ref] .
Gauge field flow for chiral gauge theories on a slab · The setup · 0.83
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This is similar to the result obtained in [ref] for the different space-time dimension.
Momentum space topological invariants for the 4D relativistic vacua with mass gap · $Ñ_5$ for free Wilson fermions · 1.00
Cites it as background 13 papers
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Domain wall fermions were introduced to solve the problem of realizing chiral symmetry on the lattice by exploiting the fact that in infinite volume, massless fermion modes must exist on the spatial boundary between two topologically distinct phases [ref] .
Chiral gauge theory at the boundary between topological phases · Introduction · 0.82
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Though the physics of topological materials started 30 years ago with the pioneering work by Nielsen and Ninomiya [ref] , who demonstrated that the masslessness of elementary particles can be protected by the momentum space topology, the investigation of the topologically nontrivial vacua in relativistic quantum field theories is still in its infancy, with a rather few papers exploiting topological invariants in momentum space (see e.g. [ref] ).
Momentum space topological invariants for the 4D relativistic vacua with mass gap · Introduction · 0.96
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The connection between GW fermions and extra dimensions is well established — the first explicit solution to the GW equations being the overlap operator [ref] which was derived to describe edge states of domain wall fermions in one higher dimension [ref] .
Generalized Ginsparg-Wilson relations · Introduction · 0.90
Discussion
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