C-P-T anomaly matching in bosonic quantum field theory and spin chains

Abstract

We consider the $O(3)$ nonlinear sigma model with the $\theta$-term and its linear counterpart in 1+1D. The model has discrete time-reflection and space-reflection symmetries at any $\theta$, and enjoys the periodicity in $\theta \to \theta +2\pi$. At $\theta =0,\pi$ it also has a charge-conjugation $C$-symmetry. Gauging the discrete space-time reflection symmetries is interpreted as putting the theory on the nonorientable $\mathbb{R}{P}^2$ manifold, after which the $2\pi$ periodicity of $\theta$ and the $C$ symmetry at $\theta =\pi$ are lost. We interpret this observation as a mixed 't Hooft anomaly among charge-conjugation $C$, parity $P$, and time-reversal $T$ symmetries when $\theta =\pi$. Anomaly matching implies that in this case the ground state cannot be trivially gapped, as long as $C$, $P$ and $T$ are all good symmetries of the theory. We make several consistency checks with various semi-classical regimes, and with the exactly solvable XYZ model. We interpret this anomaly as an anomaly of the corresponding spin-half chains with translational symmetry, parity and time reversal (but not involving the $SO(3)$-spin symmetry), requiring that the ground state is never trivially gapped, even if $SO(3)$ spin symmetry is explicitly and completely broken. We also consider generalizations to $\mathbb{C}{P}^{N-1}$ models and show that the $C$-$P$-$T$ anomaly exists for even $N$.

arXiv comment14 pages, 5 figures; (v2) title changed, refs added, some appendices added

References in corpus (35)

• Classification of Gapped Symmetric Phases in 1D Spin Systems
• Lieb-Schultz-Mattis in Higher Dimensions
• Physics of three dimensional bosonic topological insulators: Surface Deconfined Criticality and Quantized Magnetoelectric Effect
• Theta, Time Reversal, and Temperature
• Commensurability, excitation gap and topology in quantum many-particle systems on a periodic lattice
• Classifying gauge anomalies through SPT orders and classifying gravitational anomalies through topological orders
• The "Parity" Anomaly On An Unorientable Manifold
• Time-Reversal Breaking in QCD${}_4$, Walls, and Dualities in 2+1 Dimensions
• Symmetry Protected Topological Phases, Anomalies, and Cobordisms: Beyond Group Cohomology
• Gauging spatial symmetries and the classification of topological crystalline phases
• Comments on Abelian Higgs Models and Persistent Order
• Intrinsic and emergent anomalies at deconfined critical points
• Anomalies of discrete symmetries in three dimensions and group cohomology
• Symmetric-Gapped Surface States of Fractional Topological Insulators
• Anomaly Manifestation of Lieb-Schultz-Mattis Theorem and Topological Phases
• Walls, Anomalies, and (De)Confinement in Quantum Anti-Ferromagnets
• Anomaly constraints on deconfinement and chiral phase transition
• Anomaly matching for phase diagram of massless ${\mathbb{Z}}_N$-QCD
• Anomalies of discrete symmetries in various dimensions and group cohomology
• Lieb-Schultz-Mattis Theorem and its generalizations from the Perspective of the Symmetry Protected Topological phase
• Vacuum structure of bifundamental gauge theories at finite topological angles
• Circle compactification and 't Hooft anomaly
• Global anomalies on the surface of fermionic symmetry-protected topological phases in (3+1) dimensions
• Confinement in the bulk, deconfinement on the wall: infrared equivalence between compactified QCD and quantum magnets
• Global symmetries, volume independence and continuity
• Time Reversal, SU(N) Yang-Mills and Cobordisms: Interacting Topological Superconductors/Insulators and Quantum Spin Liquids in 3+1D
• On time-reversal anomaly of 2+1d topological phases
• Bosonic Topological Insulators and Paramagnets: a view from cobordisms
• Conflicting Symmetries in Topologically Ordered Surface States of Three-dimensional Bosonic Symmetry Protected Topological Phases
• More on time-reversal anomaly of 2+1d topological phases
• Topological Phases Protected By Reflection Symmetry and Cross-cap States
• Tunneling Topological Vacua via Extended Operators: (Spin-)TQFT Spectra and Boundary Deconfinement in Various Dimensions
• $S$-duality of $u(1)$ gauge theory with $\theta =\pi$ on non-orientable manifolds: Applications to topological insulators and superconductors
• Effective field theory for one-dimensional valence-bond-solid phases and their symmetry protection
• Critical behavior of gauge theories and Coulomb gases in three and four dimensions

Cited by in corpus (42)

• Anomaly and global inconsistency matching: $\theta$-angles, $SU(3)/U(1{)}^2$ nonlinear sigma model, $SU(3)$ chains and its generalizations
• Modified instanton sum in QCD and higher-groups
• Abelian gauge theories on the lattice: $\theta$-terms and compact gauge theory with(out) monopoles
• Ising ferromagnet to valence bond solid transition in a one-dimensional spin chain: Analogies to deconfined quantum critical points
• Anomaly constraint on massless QCD and the role of Skyrmions in chiral symmetry breaking
• Topological theory of Lieb-Schultz-Mattis theorems in quantum spin systems
• Anomaly matching, (axial) Schwinger models, and high-T super Yang-Mills domain walls
• Lieb-Schultz-Mattis type theorem with higher-form symmetry and the quantum dimer models
• Anomalies, a mod 2 index, and dynamics of 2d adjoint QCD
• Anomaly Matching in the Symmetry Broken Phase: Domain Walls, CPT, and the Smith Isomorphism
• Domain walls in high-T SU(N) super Yang-Mills theory and QCD(adj)
• Symmetry enhancement and closing of knots in 3d/3d correspondence
• Fractional $\theta$ angle, 't Hooft anomaly, and quantum instantons in charge-$q$ multi-flavor Schwinger model
• New Higher Anomalies, SU(N) Yang-Mills Gauge Theory and ${\mathbb{CP}}^{\mathrm{N}-1}$ Sigma Model
• Dual simulation of the 2d U(1) gauge Higgs model at topological angle $\theta =\pi$: Critical endpoint behavior
• Quantum Distillation of Hilbert Spaces, Semi-classics and Anomaly Matching
• Generalized 't Hooft anomalies on non-spin manifolds
• Lieb-Schultz-Mattis type constraints on Fractonic Matter
• New Anomalies, TQFTs, and Confinement in Bosonic Chiral Gauge Theories
• Phase structure of the twisted $SU(3)/U(1{)}^2$ flag sigma model on $\mathbb{R}\times S^1$
• Deconfinement and CP-breaking at $\theta =\pi$ in Yang-Mills theories and a novel phase for SU(2)
• Effective gauge theories of superfluidity with topological order
• First-principle simulations of 1+1d quantum field theories at $\theta =\pi$ and spin-chains
• Lattice gauge theory for Haldane conjecture and central-branch Wilson fermion
• Universality between vector-like and chiral quiver gauge theories: Anomalies and domain walls
• Condensates and anomaly cascade in vector-like theories
• High-temperature domain walls of QCD with imaginary chemical potentials
• Self-conjugate QCD
• Varieties and properties of central-branch Wilson fermions
• Anomaly-induced transport phenomena from the imaginary-time formalism
• Topological susceptibility of $2d{\mathrm{CP}}^1$ or $\mathrm{O}(3)$ non-linear $\sigma$-model: is it divergent or not?
• Finite-Density Massless Two-Color QCD at Isospin Roberge-Weiss Point and 't Hooft Anomaly
• $\theta$-dependence in the small-$N$ limit of $2d$ $C{P}^{N-1}$ models
• Global anomaly matching in higher-dimensional ${\mathbb{CP}}^{N-1}$ model
• Complex instantons in sigma models with chemical potential
• Lieb-Schultz-Mattis Theorem for 1D Quantum Magnets with Antiunitary Translation and Inversion Symmetries
• LSM and CPT
• Topology and index theorem with a generalized Villain lattice action -- a test in 2d
• Anomaly constraint on chiral central charge of (2+1)d topological order
• Sigma-model analysis of $SU(3)$ antiferromagnetic spins on the triangular lattice
• Anomaly-induced edge currents in hydrodynamics with parity anomaly
• The mixed $0$-form/$1$-form anomaly in Hilbert space: pouring new wine into old bottles