Symmetric mass generation of interacting chiral fermions on a one-dimensional lattice without fermion doubling

Abstract: Symmetric mass generation is the interaction-induced opening of a fermion gap without spontaneous symmetry breaking. The anomaly-free 3-4-5-0 model of Wang and Wen provides a minimal one-dimensional setting for this phenomenon, but a direct lattice realization faces two obstacles: fermion doubling for local chiral discretizations and perturbative irrelevance of the six-fermion gapping interaction. We address both obstacles. First, we formulate the model on a strictly one-dimensional tangent-fermion lattice, where a nonlocal hopping produces a single chiral branch without a mirror partner while retaining an efficient tensor-network representation. Second, we add a Hubbard-type density-density interaction (Luttinger parameter $K$) that reduces the scaling dimension of the 3-4-5-0 interaction from $5$ to $5K$, making it relevant for $K<2/5$. Density-matrix renormalization group calculations show the opening of an excitation gap in this regime without the appearance of a degenerate ground state, the hallmark of symmetric mass generation.

• The paper demonstrates SMG by realizing a fermionic gap through six-fermion interactions without spontaneous symmetry breaking.
• It employs a tangent-fermion lattice to avoid fermion doubling and uses a Hubbard repulsion to tune the effective Luttinger parameter.
• DMRG simulations validate a nondegenerate ground state and smooth momentum occupation, confirming the robustness of the SMG mechanism.

Symmetric Mass Generation of Interacting Chiral Fermions on a One-Dimensional Lattice Without Fermion Doubling

Introduction and Context

The paper presents a lattice realization of symmetric mass generation (SMG) for interacting chiral fermions in one spatial dimension, targeting the anomaly-free 3-4-5-0 model originally formulated by Wang and Wen. Unlike conventional mass generation mechanisms such as the Anderson-Higgs paradigm, SMG induces a fermionic gap purely through interactions without spontaneous symmetry breaking. The research addresses prominent obstacles: the fermion-doubling problem in chiral discretizations and the perturbative irrelevance of the six-fermion interaction responsible for gapping.

Model Construction and Overcoming Fermion Doubling

The authors employ a tangent-fermion lattice construction based on Stacey's nonlocal discretization, generating a tangent dispersion $E(k) = 2t_0 \tan(k/2)$ and circumventing the fermion-doubling problem. This approach ensures a single chiral branch per flavor and facilitates efficient tensor-network representations. The 3-4-5-0 model comprises two right-moving fermions (charges 3 and 4) and two left-moving fermions (charges 5 and 0), satisfying anomaly cancellation ($N_L = N_R$, $3^2 + 4^2 = 5^2 + 0^2$).

The six-fermion interactions preserve the protecting $U(1)$ symmetry due to the specific charge assignments and interaction vectors, as detailed algebraically in the paper. Importantly, the tangent dispersion's nonlocal nature is operationally local when recast as a generalized eigenvalue problem, maintaining computational tractability.

Making the Interaction Relevant: Luttinger Parameter and Hubbard Repulsion

The SMG interaction is highly irrelevant in the RG sense at weak coupling (scaling dimension $D_{3450} = 5 > 2$). To render it relevant at accessible system sizes, a Hubbard-type density-density interaction is introduced. Tuning the effective Luttinger parameter $K$ via this auxiliary interaction decreases the scaling dimension to $D_{3450} = 5K$. The six-fermion perturbation becomes RG relevant for $K < 2/5$. This controlled regime allows for robust scaling analysis and direct DMRG numerical investigation, with the Hubbard term serving solely as a tuning knob, not a gap-opening mechanism.

Bosonization Analysis and Symmetry Considerations

Bosonization recasts the theory into two decoupled Tomonaga-Luttinger liquids under a suitable charge-rotation basis, separating neutral from charged modes. The vertical operators responsible for the SMG, $\cos(2\pi/5\,\phi_p)$, are constructed from neutral fields, and their relevance depends explicitly on $K$. Pinning these cosines does not break the $N_L = N_R$0 symmetry; the ground state remains nondegenerate, a phenomenon traced to the primitivity of the interaction lattice spanned by the vectors associated with the 3-4-5-0 interaction.

The algebraic-topological examination of ground-state degeneracy, utilizing the structure of integer matrix maps between tori, further strengthens the argument that the gap-opening does not produce spontaneous symmetry breaking, as observed in classic sine-Gordon scenarios.

Numerical Results: DMRG Signatures of SMG

The authors employ large-scale DMRG simulations with matrix product states, using anti-periodic boundary conditions for system sizes up to $N_L = N_R$1 sites and bond dimensions up to $N_L = N_R$2. Two principal numerical signatures are observed when both the SMG interaction and sufficiently low $N_L = N_R$3 are present:

1. Excitation Gap Without Degenerate Ground State: The gap in the excitation spectrum persists as $N_L = N_R$4 increases, distinguishing it from gapless and spontaneously symmetry-breaking gapped systems, where the lowest excited state merges with the ground state.
2. Smoothing of Momentum Occupation Factor: The occupation factor $N_L = N_R$5, which shows power-law singularities in gapless Luttinger liquids, exhibits smoothed $N_L = N_R$6-dependence consistent with gapped fermions where $N_L = N_R$7 and the SMG interaction is present.

Detailed error analysis demonstrates that only the simultaneous presence of both the 3-4-5-0 interaction and sufficiently renormalized $N_L = N_R$8 produces the finite-size scaling expected from a symmetric, non-symmetry-breaking mass gap.

Theoretical and Practical Implications

This work provides a direct path for realizing and analyzing SMG in strictly one-dimensional lattice systems, obviating the need for mirror fermions or extra spatial dimensions. The tangent fermion formalism facilitates a transparent mapping between lattice and field-theoretic degrees of freedom, advantageous for both analytical and numerical studies. The separation between gapping (from the SMG six-fermion term) and scaling-dimension tuning (via Hubbard repulsion) offers a general strategy for probing interaction-induced mass gaps in other strongly correlated chiral systems.

Potential future directions include:

• Extension to other anomaly-free chiral models with similar lattice constructions.
• Investigation of dynamic properties, transport, and non-equilibrium phenomena in the SMG regime.
• Application to lattice simulations of exotic gauge theories and boundary physics in topological phases.

Conclusion

The paper establishes a rigorous lattice construction for SMG in the 3-4-5-0 model using tangent fermions, successfully avoiding fermion doubling and enabling systematic numerical study at weak coupling. The controlled introduction of a Hubbard-type interaction renders the six-fermion SMG perturbation relevant, manifesting numerically as an excitation gap with a nondegenerate ground state and suppression of Luttinger-liquid singularities. The tangent-fermion framework and the decoupling of gap-opening from scaling-dimension tuning pave the way for further studies of interacting chiral fermions and lattice gauge theories.