Symmetric Mass Generation

TL;DR

SMG shows that fermions can acquire a mass through strong interactions without breaking anomaly-free global or gauge symmetries, provided the anomaly index vanishes in the relevant cobordism class and the fermion representation lacks a trivial antisymmetric product. The review unifies SMG through fluctuating bilinear mass pictures, fermion fractionalization, and symmetry-extension constructions, tying SMG to SPT/ANOMALY inflow, and to lattice approaches for chiral fermions. Numerical evidence across dimensions supports the existence of SMG phases and reveals diverse transition behaviors, including direct, continuous (often DPQCP-like) and occasionally intermediate-SMB scenarios. The work highlights the potential of SMG to resolve longstanding lattice regularization issues for chiral theories and to inform beyond-Standard-Model constructions, while signaling significant theoretical challenges in obtaining controlled analytic descriptions of the critical behavior.

Abstract

The most well-known mechanism for fermions to acquire a mass is the Nambu-Goldstone-Anderson-Higgs mechanism, i.e. after a spontaneous symmetry breaking, a bosonic field that couples to the fermion mass term condenses, which grants a mass gap for the fermionic excitation. In the last few years, it was gradually understood that there is a new mechanism of mass generation for fermions without involving any symmetry breaking within an anomaly-free symmetry group. This new mechanism is generally referred to as the "Symmetric Mass Generation (SMG)." It is realized that the SMG has deep connections with interacting topological insulator/superconductors, symmetry-protected topological states, perturbative local and non-perturbative global anomaly cancellations, and deconfined quantum criticality. It has strong implications for the lattice regularization of chiral gauge theories. This article defines the SMG, summarizes current numerical results, introduces a novel unifying theoretical framework (including the parton-Higgs and the s-confinement mechanisms, as well as the symmetry-extension construction), and overviews various features and applications of SMG.

Figures (7)

• Figure 1: Phase diagram of the interacting Majorana chain model described by Eq. \ref{['eq:H_maj_chain']}
• Figure 2: (a) Honeycomb lattice. (b) Graphene band structure (inset defines high symmetry points in the Brillouin zone).
• Figure 3: Typical features of SMG observed in numerics. Continuous gap opening in (a) (1+1)D SMG driven by four-fermion interactions and (b) higher dimensional SMG in general. (c) Vanishing fermion bilinear expectation $\langle\bar{\psi} \Gamma \psi\rangle$ with its source field $m$ in the SMG phase. (d) Diverging Yukawa field susceptibility $\chi$ at the SMG transition.
• Figure 4: Classification of mass generation mechanisms.
• Figure 6: Schematic phase diagram between a trivial phase and a (fake) fermionic SPT phase that can be trivialized by interaction. Massless bulk fermions along the critical line in the weak-coupling limit undergoes the SMG as the interaction $g$ exceeds a critical value $g_c$. Although the two phases (technically one phase) are smoothly connected through the strong-coupling regime, the topological number $n$ must still jump across the missing "phase boundary", along which the Green's function must have a zero.