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Gauge Enhanced Quantum Criticality and Time Reversal Domain Wall: SU(2) Yang-Mills Dynamics with Topological Terms

Juven Wang, Yi-Zhuang You, Yunqin Zheng

TL;DR

This work analyzes the low-energy fate of SU(2) Yang-Mills theory at θ = π under four O(3,1) Lorentz symmetry enrichments, driven by a mixed anomaly with the center symmetry. It identifies two principal IR pathways: time-reversal breaking with a nontrivial domain-wall theory, and a deconfined, gapless U(1) phase whose symmetry enrichments mirror those of corresponding U(1) spin liquids; anomaly matching then constrains possible IR phases and their wall theories. The authors construct precise domain-wall theories for all four siblings, determine how Lorentz and unitary symmetry fractionalization manifest on the wall, and connect these to deconfined U(1) spin liquids via dualities and Higgsing in SU(2) QCD4 with odd Nf. This framework leads to Gauge Enhanced Quantum Critical Points (GEQCPs), where SU(2) is enlarged at criticality to mediate direct second-order transitions between U(1) spin liquids and trivial vacua, enriching the landscape of quantum criticality and symmetry-enriched topological phases.

Abstract

We explore the low energy dynamics of the four siblings of Lorentz symmetry enriched SU(2) Yang-Mills theory with a theta term at $θ=π$ in $(3+1)$d. Due to a mixed anomaly between time reversal symmetry and the center symmetry, the low energy dynamics must be nontrivial. We focus on two possible scenarios: 1) time reversal symmetry is spontaneously broken by the two confining vacua, and 2) the low energy theory is described by a U(1) Maxwell gauge theory (e.g. U(1) spin liquid in condensed matter) which is deconfined and gapless while preserving time reversal symmetry. In the first scenario, we first identify the global symmetry on the time reversal domain wall, where time reversal symmetry in the bulk induces a $\mathbb{Z}_2$ unitary symmetry on the domain wall. We discuss how the Lorentz symmetry and the unitary $\mathbb{Z}_2$ symmetry enrich the domain wall theory. In the second scenario, we relate the symmetry enrichments of the SU(2) Yang-Mills to that of the U(1) Maxwell gauge theory. This further opens up the possibility that SU(2) QCD with large and odd flavors of fermions could be a direct second order phase transition between two phases of U(1) gauge theories as well as between a U(1) gauge theory and a trivial vacuum (e.g. a trivial paramagnet), where the gauge group is enhanced to be non-Abelian at and only at the transition. We characterize these transitions, and name them as Gauge Enhanced Quantum Critical Points.

Gauge Enhanced Quantum Criticality and Time Reversal Domain Wall: SU(2) Yang-Mills Dynamics with Topological Terms

TL;DR

This work analyzes the low-energy fate of SU(2) Yang-Mills theory at θ = π under four O(3,1) Lorentz symmetry enrichments, driven by a mixed anomaly with the center symmetry. It identifies two principal IR pathways: time-reversal breaking with a nontrivial domain-wall theory, and a deconfined, gapless U(1) phase whose symmetry enrichments mirror those of corresponding U(1) spin liquids; anomaly matching then constrains possible IR phases and their wall theories. The authors construct precise domain-wall theories for all four siblings, determine how Lorentz and unitary symmetry fractionalization manifest on the wall, and connect these to deconfined U(1) spin liquids via dualities and Higgsing in SU(2) QCD4 with odd Nf. This framework leads to Gauge Enhanced Quantum Critical Points (GEQCPs), where SU(2) is enlarged at criticality to mediate direct second-order transitions between U(1) spin liquids and trivial vacua, enriching the landscape of quantum criticality and symmetry-enriched topological phases.

Abstract

We explore the low energy dynamics of the four siblings of Lorentz symmetry enriched SU(2) Yang-Mills theory with a theta term at in d. Due to a mixed anomaly between time reversal symmetry and the center symmetry, the low energy dynamics must be nontrivial. We focus on two possible scenarios: 1) time reversal symmetry is spontaneously broken by the two confining vacua, and 2) the low energy theory is described by a U(1) Maxwell gauge theory (e.g. U(1) spin liquid in condensed matter) which is deconfined and gapless while preserving time reversal symmetry. In the first scenario, we first identify the global symmetry on the time reversal domain wall, where time reversal symmetry in the bulk induces a unitary symmetry on the domain wall. We discuss how the Lorentz symmetry and the unitary symmetry enrich the domain wall theory. In the second scenario, we relate the symmetry enrichments of the SU(2) Yang-Mills to that of the U(1) Maxwell gauge theory. This further opens up the possibility that SU(2) QCD with large and odd flavors of fermions could be a direct second order phase transition between two phases of U(1) gauge theories as well as between a U(1) gauge theory and a trivial vacuum (e.g. a trivial paramagnet), where the gauge group is enhanced to be non-Abelian at and only at the transition. We characterize these transitions, and name them as Gauge Enhanced Quantum Critical Points.

Paper Structure

This paper contains 21 sections, 79 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Standard Lore of ${\rm SU}(N)$ Yang-Mills
  3. New Aspects: Lorentz Symmetry Enrichments
  4. SU(2) Yang-Mills Theory at $\theta=\pi$
  5. Four Siblings and Anomalies
  6. Low Energy Dynamics: Overview and Questions
  7. Time Reversal Domain Wall
  8. Domain Wall for $(K_1, K_2)=(0,0)$: Semion with ${\mathcal{U}}^2=1$
  9. Domain Wall for $(K_1, K_2)=(1,0)$: Semion with ${\mathcal{U}}^2=-1$
  10. Domain Wall for $(K_1, K_2)=(0,1)$: Anti-Semion with ${\mathcal{U}}^2=1$
  11. Domain Wall for $(K_1, K_2)=(1,1)$: Anti-Semion with ${\mathcal{U}}^2=-1$
  12. Remarks On ${\mathcal{C}}{\mathcal{P}}_{\perp}$ and $\mathcal{T}$, and Summary
  13. Application I: Domain Wall Theory $N_f<N_{CFT}$
  14. Lorentz Symmetry Fractionalization, $K_2=1$
  15. ${\mathcal{U}}$ Unitary Symmetry Fractionalization
  16. ...and 6 more sections

Figures (1)

  • Figure 1: Schematic RG flow diagram around the QCD$_4$ fixed point for odd $N_f$ and $N_f>11$. Possible IR fates are listed for completeness, although some (such as the U(1) SL on the $\theta=0$ side) may be extremely unlikely.