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Parity anomaly from LSM: exact valley symmetries on the lattice

Salvatore D. Pace, Minho Luke Kim, Arkya Chatterjee, Shu-Heng Shao

TL;DR

This work provides an exact lattice realization of the ${2+1}$D parity anomaly by identifying an Onsager-algebra-based valley symmetry in the honeycomb tight-binding model, whose continuum limit reproduces the SU(2) valley symmetry of two massless Dirac fermions. The central construction is a set of conserved, integer-quantized charges $Q_oldsymbol{x}$ that generate the non-Abelian ${ m Ons}_2$ algebra and map to the valley symmetry in the IR, thereby linking lattice symmetries to continuum anomalies. An LSM anomaly is shown for the lattice symmetries, matching the IR parity anomaly, which forbids a symmetric, unique gapped ground state on the lattice; the authors also analyze a simple symmetric model, its phase diagram under deformations, and the effect of stacking two copies. Overall, the paper establishes a concrete anomaly-matching framework that connects exact lattice symmetries to IR 't Hooft anomalies, enriching our understanding of how lattice models realize and constrain topological and symmetry-protected phenomena.

Abstract

We show that the honeycomb tight-binding model hosts an exact microscopic avatar of its low-energy SU(2) valley symmetry and parity anomaly. Specifically, the SU(2) valley symmetry arises from a collection of conserved, integer quantized charge operators that obey the Onsager algebra. Along with lattice reflection and time-reversal symmetries, this Onsager symmetry has a Lieb-Schultz-Mattis (LSM) anomaly that matches the parity anomaly in the IR. Indeed, we show that any local Hamiltonian commuting with these symmetries cannot have a trivial unique gapped ground state. We study the phase diagram of the simplest symmetric model and survey various deformations, including Haldane's mass term, which preserves only the Onsager symmetry. Our results place the parity anomaly in ${2+1}$D alongside Schwinger's anomaly in ${1+1}$D and Witten's SU(2) anomaly in ${3+1}$D as 't Hooft anomalies that can arise from the Onsager symmetry on the lattice.

Parity anomaly from LSM: exact valley symmetries on the lattice

TL;DR

This work provides an exact lattice realization of the D parity anomaly by identifying an Onsager-algebra-based valley symmetry in the honeycomb tight-binding model, whose continuum limit reproduces the SU(2) valley symmetry of two massless Dirac fermions. The central construction is a set of conserved, integer-quantized charges that generate the non-Abelian algebra and map to the valley symmetry in the IR, thereby linking lattice symmetries to continuum anomalies. An LSM anomaly is shown for the lattice symmetries, matching the IR parity anomaly, which forbids a symmetric, unique gapped ground state on the lattice; the authors also analyze a simple symmetric model, its phase diagram under deformations, and the effect of stacking two copies. Overall, the paper establishes a concrete anomaly-matching framework that connects exact lattice symmetries to IR 't Hooft anomalies, enriching our understanding of how lattice models realize and constrain topological and symmetry-protected phenomena.

Abstract

We show that the honeycomb tight-binding model hosts an exact microscopic avatar of its low-energy SU(2) valley symmetry and parity anomaly. Specifically, the SU(2) valley symmetry arises from a collection of conserved, integer quantized charge operators that obey the Onsager algebra. Along with lattice reflection and time-reversal symmetries, this Onsager symmetry has a Lieb-Schultz-Mattis (LSM) anomaly that matches the parity anomaly in the IR. Indeed, we show that any local Hamiltonian commuting with these symmetries cannot have a trivial unique gapped ground state. We study the phase diagram of the simplest symmetric model and survey various deformations, including Haldane's mass term, which preserves only the Onsager symmetry. Our results place the parity anomaly in D alongside Schwinger's anomaly in D and Witten's SU(2) anomaly in D as 't Hooft anomalies that can arise from the Onsager symmetry on the lattice.
Paper Structure (20 sections, 70 equations, 4 figures, 2 tables)

This paper contains 20 sections, 70 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Exact lattice valley symmetries
  3. Symmetries and anomalies in the IR
  4. Parity anomaly from Onsager symmetries
  5. LSM anomaly of Onsager symmetries
  6. Outlook
  7. Crystalline symmetries of the honeycomb lattice
  8. G-Onsager algebras in a general tight-binding model
  9. Square lattice
  10. Majorana honeycomb model
  11. The Hamiltonian
  12. Lattice zero modes
  13. Continuum zero modes
  14. Matching the continuum and lattice symmetries
  15. More on the complex honeycomb model
  16. ...and 5 more sections

Figures (4)

  • Figure 1: The honeycomb lattice is a triangular lattice with a two-site basis. The primitive lattice vectors of the triangular lattice formed by the white (W) sites are denoted as $\mathbf{a}_1$ and $\mathbf{a}_2$. The black (B) sites form another triangular lattice displaced from the white lattice by the vector $\bm{\delta}$. The yellow dashed arrows indicate the next-to-nearest neighbor term in Eq. \ref{['HaldaneTerm']}.
  • Figure 2: The continuum limits of the on-site symmetry charge $Q$ and the valley charges $Q_\mathbf{x}$ generate an SU(2) internal global symmetry parametrized by a two-sphere.
  • Figure 3: The phase diagram of \ref{['HNNNN']}, which preserves both the Onsager symmetry and spacetime reflections. The number of gapless modes changes from 2 to 8 at ${t" = \frac{1}{3}}$ where there is a Lifshitz transition, and there is an additional point ${t"=\frac{1}{2}}$ where there are two gapless modes with quadratic dispersions.
  • Figure 4: Locations of the gapless modes in the Brillouin zone of the Hamiltonian \ref{['HNNNN']} as $t"$ varies. For ${0 \leq t"<1/3}$, all gapless modes reside at the $\mathbf{K}/\mathbf{K}'$ points. Additional modes appear when ${t"=1/3}$ and moves throughout the Brillouin zone as $t"$ increases.