Phases of 2d Gauge Theories and Symmetric Mass Generation (2509.12305v1)

Abstract: We study the dynamics and phase structure of Abelian gauge theories in $d=1+1$ dimensions. These include $U(1)$ gauge theory coupled to a scalar and a fermion, as well as the two-flavour Schwinger model with different charges. Both theories exhibit a surprisingly rich phase diagram as masses are varied, with both $c=1$ and $c=1/2$ critical lines or points. We build up to the study of 2d chiral gauge theories, which hold particular interest because they provide a mechanism for symmetric mass generation, a phenomenon in which fermions become gapped without breaking chiral symmetries.

• The paper reveals that symmetric mass generation enables fermions in 2D gauge theories to acquire mass without breaking non-anomalous chiral symmetries.
• It employs techniques including bosonisation, anomaly matching, and partition function analysis to map out confining, Higgs, and gapless CFT phases.
• The findings connect theoretical insights to lattice realizations and dualities, setting the stage for studies on chiral gauge dynamics in higher dimensions.

Phases of 2d Gauge Theories and Symmetric Mass Generation

Overview and Motivation

This paper presents a comprehensive analysis of the phase structure and dynamics of Abelian gauge theories in $d=1+1$ dimensions, focusing on the interplay between scalars, fermions, and gauge fields. The central theme is the phenomenon of symmetric mass generation (SMG), where fermions acquire a mass gap without breaking non-anomalous chiral symmetries. The work systematically explores several models: QED with scalars and fermions, the two-flavour Schwinger model with arbitrary charges, and chiral gauge theories, culminating in a detailed paper of SMG mechanisms.

Abelian Higgs Model and Schwinger Model: Phase Structure

The analysis begins with the Abelian Higgs model in two dimensions, described by a $U(1)$ gauge field coupled to a complex scalar. The phase structure is controlled by the scalar mass $m_s^2$:

• Confining Phase ($m_s^2 \gg e^2$): The scalar is heavy and can be integrated out, leaving pure Maxwell theory. The system exhibits confinement, with Wilson loops obeying an area law. The ground state structure depends on the $\theta$ angle: for generic $\theta$, there is a unique ground state, but at $\theta = \pi$, a two-fold degeneracy emerges due to a mixed anomaly between time reversal and large gauge transformations.
• Higgs Phase ($m_s^2 \ll -e^2$): The scalar condenses, leading to a Higgs phase. Quantum effects, notably instanton contributions from vortices, restore confinement, but with a reduced string tension.

When a massless Dirac fermion is added, the phase diagram becomes richer. For $N_f=1$, the theory is gapped except along critical lines where gapless modes emerge, protected by global symmetries. The phase transitions are characterized by $c=1$ and $c=1/2$ conformal field theories (CFTs), with the $c=1$ line terminating at a point where it splits into two $c=1/2$ Ising lines.

Two-Flavour Schwinger Model: Charge Dependence and Bosonisation Subtleties

The two-flavour Schwinger model, $U(1)$ gauge theory coupled to two Dirac fermions of charges $p$ and $q$, exhibits a phase diagram sensitive to the parity and coprimality of the charges:

• Bosonic vs. Fermionic Theories: If both $p$ and $q$ are odd, the theory is bosonic; if one is even, it is fermionic. The IR fixed point is a compact boson CFT with radius $R^2 = (p^2 + q^2)/2$, but the partition function and operator content depend crucially on the $\mathbb{Z}_2$ gauging inherent in bosonisation.
• Ground State Degeneracy: The number of ground states is determined by the one-form symmetry associated with the gauge group and the global symmetry structure. Careful treatment of $\mathbb{Z}_2$ twists in the partition function is essential for correctly counting vacua, especially in the fermionic case.

The phase diagram, as a function of the fermion masses, features regions of spontaneously broken charge conjugation (with degenerate ground states) and lines of Ising transitions. The structure is robust under variations of the charges and persists across the moduli space of the theory.

Chiral Gauge Theories and Symmetric Mass Generation

The paper advances to chiral gauge theories, where left- and right-moving fermions carry different charges. The prototypical example is the 3450 model, with anomaly-free charge assignments. Anomaly matching dictates that the IR theory is a single left-moving Dirac fermion, with no accompanying TQFT unless the UV theory possesses a one-form symmetry.

The mechanism of symmetric mass generation is elucidated by considering chiral gauge theories coupled to scalars. In the Higgs phase, the global chiral symmetry is preserved, and the theory flows to a gapless CFT with no relevant singlet operators, ensuring stability. As the scalar VEVs are tuned, operator dimensions change, and the system can transition to a gapped confining phase without breaking the global symmetry. The transition is controlled by the emergence of relevant operators in the CFT, and the phase diagram features a $c=2$ region (gapless), $c=1$ regions (partially gapped), and a fully gapped phase.

Bosonisation, Partition Functions, and Operator Content

A significant technical contribution is the careful treatment of bosonisation in two dimensions, especially the role of $\mathbb{Z}_2$ gauging. The equivalence between fermionic and bosonic theories is established only after summing over twisted sectors in the partition function. The analysis employs modular invariance, T-duality, and the structure of the $c=1$ and $c=2$ conformal manifolds to classify IR fixed points and operator spectra.

The stability of gapless phases is determined by the absence of relevant singlet operators under the global symmetry. The transition to gapped phases is triggered when such operators become relevant as a function of the scalar VEVs, leading to RG flows away from the CFT.

Implications and Future Directions

The results have several implications:

• Lattice Realizations: The field-theoretic understanding of SMG provides guidance for constructing lattice models of chiral gauge theories, with potential applications in condensed matter and high-energy physics.
• Dualities and Anomalies: The interplay between global symmetries, anomalies, and dualities in 2d gauge theories is clarified, with explicit constructions of partition functions and operator algebras.
• Generalization to Higher Dimensions: While the mechanisms are specific to two dimensions, the conceptual framework may inform studies of SMG and chiral symmetry protection in higher-dimensional theories.

Future work may focus on non-Abelian generalizations, the role of topological phases, and numerical studies of phase transitions in lattice implementations.

Conclusion

This paper provides a detailed and rigorous analysis of the phase structure of 2d Abelian gauge theories, with a particular focus on symmetric mass generation. The combination of anomaly matching, bosonisation, and careful partition function analysis yields a precise characterization of gapless and gapped phases, the role of global symmetries, and the conditions for SMG. The results deepen the understanding of chiral gauge dynamics in two dimensions and offer a foundation for further theoretical and computational exploration.