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

• The paper demonstrates an anomaly-free chiral fermion theory with charges (3,4,5,0) that exactly cancels U(1) and gravitational anomalies.
• It introduces a 1D lattice Hamiltonian realization using specifically designed six-fermion interactions to gap the mirror sector and preserve chiral symmetry.
• DMRG numerical evidence reveals a Berezinskii-Kosterlitz-Thouless transition, confirming symmetric mass generation while keeping the desired chiral edge gapless.

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 (Wang et al., 2013, Zeng et al., 2022).

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)$0 charges are

$U(1)$1

This is the origin of the designation “3-4-5-0” (Zeng et al., 2022).

A closely related notation used in the earlier non-perturbative regularization paper writes the same chiral content as $U(1)$2-$U(1)$3-$U(1)$4-$U(1)$5, namely a left-moving fermion of charge $U(1)$6, a right-moving fermion of charge $U(1)$7, a left-moving fermion of charge $U(1)$8, and a right-moving fermion of charge $U(1)$9 (Wang et al., 2013). The difference is not physical; it reflects a different ordering of fields.

Mode	Chirality	$(3,4,5,0)$0 charge
$(3,4,5,0)$1 or $(3,4,5,0)$2	left-moving	$(3,4,5,0)$3
$(3,4,5,0)$4 or $(3,4,5,0)$5	left-moving	$(3,4,5,0)$6
$(3,4,5,0)$7 or $(3,4,5,0)$8	right-moving	$(3,4,5,0)$9
$U(1)$0 or $U(1)$1	right-moving	$U(1)$2

The model is chiral because left- and right-movers carry different charges, so parity $U(1)$3 and time reversal $U(1)$4 are broken, even though the total number of left and right movers is equal (Wang et al., 2013). 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 $U(1)$5D Abelian chiral anomaly condition. In the velocity convention above, the perturbative $U(1)$6 ’t Hooft anomaly is proportional to

$U(1)$7

For the 3-4-5-0 charge assignment,

$U(1)$8

Equivalently, in left-right notation,

$U(1)$9

gives

$1$0

Thus the chiral $1$1 anomaly cancels exactly (Zeng et al., 2022, Wang et al., 2013).

The model is also gravitational-anomaly-free. One formulation states this directly, while another gives the explicit balance

$1$2

Because the theory is anomaly-free, it should admit a consistent lattice regularization in $1$3D (Zeng et al., 2022, Wang et al., 2013).

The 2013 formulation further embeds the target $1$4 structure into an enlarged $1$5 setting with charge vectors

$1$6

for which

$1$7

In that construction, the UV lattice Hamiltonian naturally has an enlarged $1$8 symmetry before a small perturbation reduces it back to the desired single $1$9 (Wang et al., 2013). 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 $2+1$0D spatial lattice with continuous time, implemented through a Chern-insulator or quantum-Hall-type lattice construction on a finite-width cylinder with two edges (Wang et al., 2013). 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$1D lattice model of multi-layer Chern insulator, whose finite width makes the quantum system effectively $2+1$2D (Zeng et al., 2022).

In the DMRG implementation, the geometry is a two-leg ladder, treated as effectively $2+1$3D with the transverse direction absorbed into internal degrees of freedom. The construction stacks four layers of Chern insulators with four complex fermions $2+1$4 at each lattice site $2+1$5, and the free Hamiltonian is

$2+1$6

The hopping pattern is chosen so that layers $2+1$7 have one chirality of edge modes and layers $2+1$8 are complex conjugates with opposite chirality. Specifically, nearest-neighbor hopping is purely imaginary with phase $2+1$9, next-nearest-neighbor hoppings are real $$S=\int dt\,dx\;\sum_{I=1}^{4}\psi_I^\dagger (i\partial_t+i v_I\partial_x)\psi_I,$$0 or $$S=\int dt\,dx\;\sum_{I=1}^{4}\psi_I^\dagger (i\partial_t+i v_I\partial_x)\psi_I,$$1, the pattern produces a $$S=\int dt\,dx\;\sum_{I=1}^{4}\psi_I^\dagger (i\partial_t+i v_I\partial_x)\psi_I,$$2 Berry flux per plaquette, and the parameters are $$S=\int dt\,dx\;\sum_{I=1}^{4}\psi_I^\dagger (i\partial_t+i v_I\partial_x)\psi_I,$$3, $$S=\int dt\,dx\;\sum_{I=1}^{4}\psi_I^\dagger (i\partial_t+i v_I\partial_x)\psi_I,$$4 (Zeng et al., 2022).

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 $$S=\int dt\,dx\;\sum_{I=1}^{4}\psi_I^\dagger (i\partial_t+i v_I\partial_x)\psi_I,$$5 symmetry. The 2013 paper emphasizes that this is a finite Hilbert space, short-range, onsite symmetry construction, so it is genuinely a $$S=\int dt\,dx\;\sum_{I=1}^{4}\psi_I^\dagger (i\partial_t+i v_I\partial_x)\psi_I,$$6D lattice Hamiltonian with continuous time rather than a Euclidean spacetime discretization of the continuum action (Wang et al., 2013).

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 $$S=\int dt\,dx\;\sum_{I=1}^{4}\psi_I^\dagger (i\partial_t+i v_I\partial_x)\psi_I,$$7 charges, so any bilinear mixing left- and right-movers would break the chiral symmetry (Zeng et al., 2022). 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 $$S=\int dt\,dx\;\sum_{I=1}^{4}\psi_I^\dagger (i\partial_t+i v_I\partial_x)\psi_I,$$8: $$S=\int dt\,dx\;\sum_{I=1}^{4}\psi_I^\dagger (i\partial_t+i v_I\partial_x)\psi_I,$$9 These interactions are local, $(v_1,v_2,v_3,v_4)=(+1,+1,-1,-1).$0-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 (Zeng et al., 2022).

The earlier theoretical formulation expresses the same logic through bosonization. The four chiral fermions are bosonized into compact bosons $(v_1,v_2,v_3,v_4)=(+1,+1,-1,-1).$1, with edge action

$(v_1,v_2,v_3,v_4)=(+1,+1,-1,-1).$2

and fermionic canonical $(v_1,v_2,v_3,v_4)=(+1,+1,-1,-1).$3-matrix

$(v_1,v_2,v_3,v_4)=(+1,+1,-1,-1).$4

Example gapping vectors are

$(v_1,v_2,v_3,v_4)=(+1,+1,-1,-1).$5

or equivalently

$(v_1,v_2,v_3,v_4)=(+1,+1,-1,-1).$6

subject to

$(v_1,v_2,v_3,v_4)=(+1,+1,-1,-1).$7

These are the symmetry-preserving condition and the topological gapping or null condition (Wang et al., 2013).

The 2013 work identifies the boundary fully gapping rules with the Haldane null-vector criterion and proves an equivalence: $(v_1,v_2,v_3,v_4)=(+1,+1,-1,-1).$8 More concretely,

$(v_1,v_2,v_3,v_4)=(+1,+1,-1,-1).$9

This is one of the defining theoretical results surrounding the anomaly-free 3-4-5-0 model (Wang et al., 2013).

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 $\psi_1,\psi_2$0 symmetry (Zeng et al., 2022). Physically, both edges are gapless at weak coupling; increasing the interaction strength $\psi_1,\psi_2$1 drives a transition in the mirror sector; for $\psi_1,\psi_2$2, the mirror boundary develops a mass gap; and the gap opens without breaking $\psi_1,\psi_2$3.

Evidence against bilinear condensation is provided by measuring correlations of Dirac mass operators $\psi_1,\psi_2$4 and Majorana mass operators $\psi_1,\psi_2$5. In the gapped phase, all such mass correlations on edge $\psi_1,\psi_2$6 decay exponentially. This indicates no long-range bilinear order and therefore supports an interaction-driven and symmetric mass gap (Zeng et al., 2022).

The paper identifies the transition as belonging to the Berezinskii-Kosterlitz-Thouless universality class. In the bosonized mirror-sector description,

$\psi_1,\psi_2$7

with

$\psi_1,\psi_2$8

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 $\psi_1,\psi_2$9, which is the hallmark of a BKT transition in this setting (Zeng et al., 2022).

The RG equations are written as

$\psi_3,\psi_4$0

$\psi_3,\psi_4$1

and the interaction scaling dimension is parameterized as

$\psi_3,\psi_4$2

The DMRG-extracted operator dimensions imply that $\psi_3,\psi_4$3 decreases continuously from $\psi_3,\psi_4$4 at $\psi_3,\psi_4$5 to approximately $\psi_3,\psi_4$6 near the transition, consistent with approaching marginality (Zeng et al., 2022).

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

The DMRG calculations were performed on a two-leg ladder with $\psi_3,\psi_4$7 unit cells and bond dimensions

$\psi_3,\psi_4$8

with extrapolation to $\psi_3,\psi_4$9 (Zeng et al., 2022). From the ground-state energy and its derivative with respect to $U(1)$00, the transition is identified near

$U(1)$01

The derivative $U(1)$02 shows a smooth kink rather than a discontinuity, consistent with a continuous BKT-type transition.

The fermion correlators

$U(1)$03

show that edge $U(1)$04 always has power-law decay, both below and above $U(1)$05, whereas edge $U(1)$06 has power-law decay for $U(1)$07 and exponential decay for $U(1)$08. This is direct evidence that only the mirror sector is gapped. The extracted fermion scaling dimension satisfies $U(1)$09 at $U(1)$10, rises to about $U(1)$11 near the transition in the mirror sector, and stays near $U(1)$12 on edge $U(1)$13, indicating that the light chiral sector remains essentially gapless (Zeng et al., 2022).

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 $U(1)$14D multi-layer Chern insulator, assigning the charges $U(1)$15, 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 (Zeng et al., 2022). The earlier theoretical work had already argued, in more general terms, that any $U(1)$16D $U(1)$17-anomaly-free chiral matter theory can be defined as a finite system on a $U(1)$18D lattice with onsite symmetry by using a quantum Hamiltonian with continuous time and properly designed interactions (Wang et al., 2013).

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 $U(1)$19D chiral fermion construction just described. Other papers also use the phrase “anomaly free” in unrelated contexts, such as $U(1)$20 family-symmetry yukawaon assignments, revised Sumino family-gauge-boson models, or general anomaly-free $U(1)$21 extensions of the Standard Model (Koide et al., 2013, Koide, 2016, Costa, 2020). Those works share anomaly-cancellation logic, but they are not the $U(1)$22D 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 (Wang et al., 2013, Zeng et al., 2022).