---
title: Anomaly-Free 3-4-5-0 Chiral Fermion Model
url: https://www.emergentmind.com/topics/anomaly-free-3-4-5-0-model
type: topic
---

# Anomaly-Free 3-4-5-0 Chiral Fermion Model

Searching arXiv for the cited papers to ground the article in current records.
The anomaly-free 3-4-5-0 model is a \(1+1\) dimensional chiral fermion theory with four complex fermions whose chiral \(U(1)\) charges are \((3,4,5,0)\), arranged as two left-moving and two right-moving modes. In the continuum formulation used in the literature, it is an anomaly-free chiral theory with vanishing perturbative \(U(1)\) anomaly and vanishing gravitational anomaly, and it has been studied as a concrete test case for non-perturbative lattice regularization of chiral matter with onsite symmetry. The model was proposed as a target for a local quantum Hamiltonian on a \(1\)D spatial lattice with continuous time and later realized numerically through DMRG as the edge theory of a thin \(2+1\)D multi-layer Chern-insulator strip, where the mirror edge is gapped by specially designed multi-fermion interactions while the light chiral edge remains gapless [1307.7480], [2202.12355].

## 1. Definition and chiral content

The target continuum theory contains four complex fermions with action
\[
S=\int dt\,dx\;\sum_{I=1}^{4}\psi_I^\dagger (i\partial_t+i v_I\partial_x)\psi_I,
\]
with velocity assignment
\[
(v_1,v_2,v_3,v_4)=(+1,+1,-1,-1).
\]
In this convention, \(\psi_1,\psi_2\) are left-moving and \(\psi_3,\psi_4\) are right-moving, while the chiral \(U(1)\) charges are
\[
(q_1,q_2,q_3,q_4)=(3,4,5,0).
\]
This is the origin of the designation “3-4-5-0” [2202.12355].

A closely related notation used in the earlier non-perturbative regularization paper writes the same chiral content as \(3_L\)-\(5_R\)-\(4_L\)-\(0_R\), namely a left-moving fermion of charge \(3\), a right-moving fermion of charge \(5\), a left-moving fermion of charge \(4\), and a right-moving fermion of charge \(0\) [1307.7480]. The difference is not physical; it reflects a different ordering of fields.

| Mode | Chirality | \(U(1)\) charge |
|---|---|---|
| \(\psi_1\) or \(\psi_{L,3}\) | left-moving | \(3\) |
| \(\psi_2\) or \(\psi_{L,4}\) | left-moving | \(4\) |
| \(\psi_3\) or \(\psi_{R,5}\) | right-moving | \(5\) |
| \(\psi_4\) or \(\psi_{R,0}\) | right-moving | \(0\) |

The model is chiral because left- and right-movers carry different charges, so parity \(P\) and time reversal \(T\) are broken, even though the total number of left and right movers is equal [1307.7480]. This combination of chirality and anomaly cancellation is the key structural feature that makes the model useful as a testbed for lattice regularization of chiral fermions.

## 2. Anomaly cancellation and its theoretical role

The anomaly-free character of the model is expressed by the \(1+1\)D Abelian chiral anomaly condition. In the velocity convention above, the perturbative \(U(1)\) ’t Hooft anomaly is proportional to
\[
\sum_I v_I q_I^2.
\]
For the 3-4-5-0 charge assignment,
\[
\sum_I v_I q_I^2 = 3^2 + 4^2 - 5^2 - 0^2 = 9+16-25=0.
\]
Equivalently, in left-right notation,
\[
\mathcal{A}=\sum_j q_{L,j}^2-\sum_j q_{R,j}^2
\]
gives
\[
3^2-5^2+4^2-0^2=0.
\]
Thus the chiral \(U(1)\) anomaly cancels exactly [2202.12355], [1307.7480].

The model is also gravitational-anomaly-free. One formulation states this directly, while another gives the explicit balance
\[
c_L-c_R=2-2=0.
\]
Because the theory is anomaly-free, it should admit a consistent lattice regularization in \(1+1\)D [2202.12355], [1307.7480].

The 2013 formulation further embeds the target \(U(1)\) structure into an enlarged \(U(1)^2\) setting with charge vectors
\[
\mathbf{t}_1=(3,5,4,0),\qquad \mathbf{t}_2=(0,4,5,3),
\]
for which
\[
\mathbf{t}_i^{T}K\mathbf{t}_j=0,\qquad i,j\in\{1,2\}.
\]
In that construction, the UV lattice Hamiltonian naturally has an enlarged \(U(1)^2\) symmetry before a small perturbation reduces it back to the desired single \(U(1)\) [1307.7480]. This suggests that anomaly cancellation is being used not merely as a kinematic constraint, but as a design principle for identifying symmetry-compatible gapping interactions.

## 3. Non-perturbative lattice realization

The non-perturbative regularization program is based on a local quantum Hamiltonian on a \(1\)D spatial lattice with continuous time, implemented through a Chern-insulator or quantum-Hall-type lattice construction on a finite-width cylinder with two edges [1307.7480]. The later numerical work follows the Wang-Wen chiral fermion model and realizes the chiral fermions and their mirror partners on opposite boundaries of a thin strip of a \(2+1\)D lattice model of multi-layer Chern insulator, whose finite width makes the quantum system effectively \(1+1\)D [2202.12355].

In the DMRG implementation, the geometry is a two-leg ladder, treated as effectively \(1+1\)D with the transverse direction absorbed into internal degrees of freedom. The construction stacks four layers of Chern insulators with four complex fermions \(\psi_{i,I}\) at each lattice site \(i\), and the free Hamiltonian is
\[
H_{\text{free}}=\sum_{I=1}^{4}\sum_{i,j}\left(t_{I,ij}\psi_{I,i}^\dagger \psi_{I,j}+\text{h.c.}\right).
\]
The hopping pattern is chosen so that layers \(1,2\) have one chirality of edge modes and layers \(3,4\) are complex conjugates with opposite chirality. Specifically, nearest-neighbor hopping is purely imaginary with phase \(e^{i\pi/4}\), next-nearest-neighbor hoppings are real \(t_2\) or \(-t_2\), the pattern produces a \(\pi\) Berry flux per plaquette, and the parameters are \(t_1=1\), \(t_2=0.5\) [2202.12355].

This band structure has gapped bulk states and gapless edge states on both boundaries. One edge is the desired low-energy chiral sector, and the opposite edge is the mirror sector. The free theory is therefore vector-like and doubled at the lattice level, but the strategy is to gap only the mirror boundary while preserving the chiral \(U(1)\) symmetry. The 2013 paper emphasizes that this is a finite Hilbert space, short-range, onsite symmetry construction, so it is genuinely a \(1\)D lattice Hamiltonian with continuous time rather than a Euclidean spacetime discretization of the continuum action [1307.7480].

## 4. Mirror decoupling, gapping rules, and interaction design

A central issue is that symmetry-preserving bilinear mass terms are forbidden in the mirror sector because the four fermions carry distinct \(U(1)\) charges, so any bilinear mixing left- and right-movers would break the chiral symmetry [2202.12355]. The regularization therefore relies on specially designed multi-fermion interactions rather than conventional Dirac or Majorana masses.

In the DMRG realization, two six-fermion local interactions are added only on the mirror boundary \(B\):
\[
H_\text{int}=\sum_{i\in \text{B}} \Big[ g_1\big(\psi_{1,i}\psi_{2,i}^\dagger\psi_{2,i+1}^\dagger\psi_{3,i}\psi_{4,i}\psi_{4,i+1}+\text{h.c.}\big) +g_2\big(\psi_{1,i}\psi_{1,i+1}\psi_{2,i}\psi_{3,i}^\dagger\psi_{3,i+1}^\dagger\psi_{4,i}+\text{h.c.}\big) \Big].
\]
These interactions are local, \(U(1)\)-symmetric, and designed to satisfy the appropriate gapping conditions, namely self-bosonic and mutual-bosonic operator algebra conditions. The authors stress that they are the lowest-order operators satisfying the gapping criteria and are not generic quartic interactions like in earlier CGP-type approaches [2202.12355].

The earlier theoretical formulation expresses the same logic through bosonization. The four chiral fermions are bosonized into compact bosons \(\Phi_I\), with edge action
\[
S_\Phi = \frac{1}{4\pi}\int dt\,dx \left( K_{IJ}\,\partial_t\Phi_I\,\partial_x\Phi_J - V_{IJ}\,\partial_x\Phi_I\,\partial_x\Phi_J \right) + \int dt\,dx\, \left[ g_1\cos(\ell_1\cdot\Phi)+g_2\cos(\ell_2\cdot\Phi) \right],
\]
and fermionic canonical \(K\)-matrix
\[
K=K^f= \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix} \oplus \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix}.
\]
Example gapping vectors are
\[
\ell_1=(1,1,-2,2),\qquad \ell_2=(2,-2,1,1),
\]
or equivalently
\[
\ell_1=(3,-5,4,0),\qquad \ell_2=(0,4,-5,3),
\]
subject to
\[
\mathbf{t}_i^{T}\ell_j=0,\qquad \ell_i^{T}K^{-1}\ell_j=0.
\]
These are the symmetry-preserving condition and the topological gapping or null condition [1307.7480].

The 2013 work identifies the boundary fully gapping rules with the Haldane null-vector criterion and proves an equivalence:
\[
\boxed{ \text{U(1) anomaly cancellation} \;\Longleftrightarrow\; \text{existence of a symmetry-preserving fully gapped boundary} }.
\]
More concretely,
\[
\mathbf{t}^{T}K\mathbf{t}=0 \quad\Longleftrightarrow\quad \mathbf{L}^{T}K^{-1}\mathbf{L}=0.
\]
This is one of the defining theoretical results surrounding the anomaly-free 3-4-5-0 model [1307.7480].

## 5. Symmetric mass generation and the BKT transition

The mirror-sector gapping mechanism is symmetric mass generation. In the numerical study, the mirror fermions become gapped without any fermion bilinear condensate and without spontaneous breaking of the chiral \(U(1)\) symmetry [2202.12355]. Physically, both edges are gapless at weak coupling; increasing the interaction strength \(g\) drives a transition in the mirror sector; for \(g>g_c\), the mirror boundary develops a mass gap; and the gap opens without breaking \(U(1)\).

Evidence against bilinear condensation is provided by measuring correlations of Dirac mass operators \(\psi_I^\dagger\psi_J\) and Majorana mass operators \(\psi_I\psi_J\). In the gapped phase, all such mass correlations on edge \(B\) decay exponentially. This indicates no long-range bilinear order and therefore supports an interaction-driven and symmetric mass gap [2202.12355].

The paper identifies the transition as belonging to the Berezinskii-Kosterlitz-Thouless universality class. In the bosonized mirror-sector description,
\[
L=\frac{1}{4\pi}\left( \partial_t\varphi^\intercal K \partial_x\varphi + \partial_x\varphi^\intercal V \partial_x\varphi \right) +\sum_{\alpha=1,2} g_\alpha \cos(l_\alpha^\intercal \varphi),
\]
with
\[
K=\sigma^{30},\qquad l_1=(1,-2,1,2)^\intercal,\qquad l_2=(2,1,-2,1)^\intercal,
\]
the interaction is irrelevant at the free-fermion fixed point but becomes more important under RG because the Luttinger parameters renormalize. The transition occurs when the interaction scaling dimension reaches the marginal value \(\Delta_{\text{int}}=2\), which is the hallmark of a BKT transition in this setting [2202.12355].

The RG equations are written as
\[
\frac{d g_\alpha}{d\ell} = (2-\Delta_{l_\alpha})g_\alpha -\frac{1}{2}\sum_{l_\beta\pm l_\gamma=l_\alpha}g_\beta g_\gamma,
\]
\[
\frac{dV^{-1}}{d\ell} = \frac{1}{2}\sum_{\alpha}g_\alpha^2 \left( K^{-1}l_\alpha l_\alpha^\intercal K^{-1} - V^{-1}l_\alpha l_\alpha^\intercal V^{-1} \right),
\]
and the interaction scaling dimension is parameterized as
\[
\Delta_{\text{int}}=\Delta_{l_1}=\Delta_{l_2} = \sqrt{1+y_1^2+y_2^2}-3y_1-4y_2.
\]
The DMRG-extracted operator dimensions imply that \(\Delta_{\text{int}}\) decreases continuously from \(5\) at \(g=0\) to approximately \(2.17\pm 0.27\) near the transition, consistent with approaching marginality [2202.12355].

## 6. Numerical evidence, significance, and relation to broader anomaly-free model building

The DMRG calculations were performed on a two-leg ladder with \(20\) unit cells and bond dimensions
\[
\mathcal{D}=6000,\ 7000,\ 8000,
\]
with extrapolation to \(\mathcal{D}\to\infty\) [2202.12355]. From the ground-state energy and its derivative with respect to \(g\), the transition is identified near
\[
g_c \approx 5.7.
\]
The derivative \(\partial_g E_{\text{GS}}\) shows a smooth kink rather than a discontinuity, consistent with a continuous BKT-type transition.

The fermion correlators
\[
C_\psi(r)=\langle \psi_{I,i+r}^\dagger \psi_{I,i}\rangle
\]
show that edge \(A\) always has power-law decay, both below and above \(g_c\), whereas edge \(B\) has power-law decay for \(g<g_c\) and exponential decay for \(g>g_c\). This is direct evidence that only the mirror sector is gapped. The extracted fermion scaling dimension satisfies \(\Delta_\psi=1/2\) at \(g=0\), rises to about \(0.67\) near the transition in the mirror sector, and stays near \(1/2\) on edge \(A\), indicating that the light chiral sector remains essentially gapless [2202.12355].

Taken together, these results support the conclusion that the anomaly-free 3-4-5-0 theory can be realized on the lattice by embedding it as edge modes of a thin \(2+1\)D multi-layer Chern insulator, assigning the charges \((3,4,5,0)\), applying two specially designed six-fermion interactions only on the mirror boundary, driving the mirror sector into a symmetrically gapped phase via symmetric mass generation, and leaving the desired chiral edge gapless [2202.12355]. The earlier theoretical work had already argued, in more general terms, that any \(1+1\)D \(U(1)\)-anomaly-free chiral matter theory can be defined as a finite system on a \(1\)D lattice with onsite symmetry by using a quantum Hamiltonian with continuous time and properly designed interactions [1307.7480].

A common source of confusion is terminological rather than technical. In the literature surveyed here, “anomaly-free 3-4-5-0 model” refers specifically to the \(1+1\)D chiral fermion construction just described. Other papers also use the phrase “anomaly free” in unrelated contexts, such as \(U(3)\) family-symmetry yukawaon assignments, revised Sumino family-gauge-boson models, or general anomaly-free \(U(1)^m\) extensions of the Standard Model [1308.2129], [1608.04514], [2007.08733]. Those works share anomaly-cancellation logic, but they are not the \(1+1\)D chiral 3-4-5-0 lattice-regularization problem.

A plausible implication is that the 3-4-5-0 model has become a concrete bridge between abstract anomaly matching, bosonized gapping criteria, and explicit many-body numerics. Within the scope of the cited works, its importance lies not in a phenomenological application, but in providing a controlled example where anomaly cancellation, onsite symmetry, mirror decoupling, and symmetric mass generation can all be realized in a single non-perturbative framework [1307.7480], [2202.12355].

Source: https://www.emergentmind.com/topics/anomaly-free-3-4-5-0-model