Constraints on the symmetric mass generation paradigm for lattice chiral gauge theories

TL;DR

The paper investigates whether symmetric mass generation (SMG) can yield a chiral massless fermion spectrum on the lattice without spontaneous symmetry breaking. It develops a generalized Nielsen-Ninomiya no-go framework based on a one-particle effective Hamiltonian $H_{ m eff}({\vec p})$ and the momentum-space correlator ${\cal R}({\vec p})$, and argues that propagator zeros in SMG phases are typically kinematical remnants that can be removed by including composite interpolating fields that realize opposite-chirality bound states. Under precise assumptions—local reduced model, continuum limit of relativistic massless fermions with no massless bosons, and a complete interpolating-field set yielding a zero-free ${\cal R}({\vec p})$—the theorem implies any massless sector must be vector-like, challenging SMG constructions for chiral theories in four dimensions. The work also analyzes strong-coupling decoupling, highlights key differences between four- and two-dimensional models, and outlines open questions, using the ZZWY (3450) model as a testbed for bound-state formation and interpolating-field completeness. Overall, it provides a rigorous framework to evaluate SMG scenarios and identify when they can or cannot realize genuine lattice chiral gauge theories.

Abstract

Within the symmetric mass generation (SMG) approach to the construction of lattice chiral gauge theories, one attempts to use interactions to render mirror fermions massive without symmetry breaking, thus obtaining the desired chiral massless spectrum. If successful, the gauge field can be turned on, and thus a chiral gauge theory can be constructed in the phase in which SMG takes place. In this paper we argue that the zeros that often replace the mirror poles of fermion two-point functions in an SMG phase should be ``kinematical'' singularities. We conjecture that the SMG interactions generate opposite-chirality bound states, which combine with the gapped elementary mirror states to form massive Dirac fermions. The propagator zeros can then be avoided by choosing an appropriate set of interpolating fields that contains both elementary and composite fields. This allows us to apply general constraints on the existence of a chiral fermion spectrum which are valid in the presence of (non-gauge) interactions of arbitrary strength, including in any SMG phase. Using a suitably constructed one-particle lattice hamiltonian describing the fermion spectrum, we formulate a generalized no-go theorem which establishes the conditions for the applicability of the Nielsen-Ninomiya theorem to this hamiltonian. If these conditions are satisfied, the massless fermion spectrum must be vector-like. We add some general observations on the strong coupling limit of SMG models. We also elaborate on the qualitative differences between four-dimensional and two-dimensional theories that limit the lessons that can be drawn from two-dimensional models. Finally, we compile a list of open questions which must be addressed in any SMG model in order to determine whether or not it is subject to the generalized no-go theorem.

Figures (4)

• Figure 1: The lattice of the 3450 model of Ref. ZZWY. The lattice shown in Fig. 1(a) of that paper is here rotated by $90^0$ clockwise, so that the physical space direction is horizontal. The lattice consists of two inter-connected one-dimensional chains. Edge A (on the left in Fig. 1(c) of Ref. ZZWY) is here the upper chain, while edge B is the lower chain. The directional links have complex coupling $\tau_1 = t_1 e^{i\frac{\pi}{4}}$, while the undirectional links have real coupling $t_2$ (solid line) or $-t_2$ (dashed line). The unit cell of this lattice is a $2\times 1$ rectangle (in lattice units), and the numbers inside the circles represent different sublattices. To avoid confusion, note that the sublattice index is different from the species index $I$ in Eq. (\ref{['H0']}). For more details, see App. \ref{['HZZWY']}.
• Figure 2: Spectrum of the bilinear hamiltonian $\hat{H}$, reproducing Fig. 1(b) of Ref. ZZWY. The horizontal axis runs from 0 to $\pi$. The green (blue) branch corresponds to the LH (RH) chiral mode supported mainly on edge A (edge B). The plot shows that the spectrum has the correct mod $\pi$ periodicity. Also seen is that the eigenvalues spectrum forms a single smooth curve that winds around the Brillouin zone four times.
• Figure 3: Eigenvalues of ${\cal H}^{-1}(p)$. Left: eigenvalues of the full ${\cal H}^{-1}(p)$. Colors match those in Fig. \ref{['3450spec']}. In particular, the poles of the LH (RH) chiral modes are shown in green (blue). Right: eigenvalues of the upper-left 2-by-2 block of ${\cal H}^{-1}(p)$. While the LH pole (green) supported near edge A is reproduced, the RH pole got replaced by a propagator zero.
• Figure 4: Spectrum of the "left-over" hamiltonian---the upper-left $2\times 2$ block of the hamiltonian matrix (\ref{['Hp']}).