Theta, Time Reversal, and Temperature

Abstract

$SU(N)$ gauge theory is time reversal invariant at $\theta =0$ and $\theta =\pi$. We show that at $\theta =\pi$ there is a discrete 't Hooft anomaly involving time reversal and the center symmetry. This anomaly leads to constraints on the vacua of the theory. It follows that at $\theta =\pi$ the vacuum cannot be a trivial non-degenerate gapped state. (By contrast, the vacuum at $\theta =0$ is gapped, non-degenerate, and trivial.) Due to the anomaly, the theory admits nontrivial domain walls supporting lower-dimensional theories. Depending on the nature of the vacuum at $\theta =\pi$, several phase diagrams are possible. Assuming area law for space-like loops, one arrives at an inequality involving the temperatures at which CP and the center symmetry are restored. We also analyze alternative scenarios for $SU(2)$ gauge theory. The underlying symmetry at $\theta =\pi$ is the dihedral group of 8 elements. If deconfined loops are allowed, one can have two $O(2)$-symmetric fixed points. It may also be that the four-dimensional theory around $\theta =\pi$ is gapless, e.g. a Coulomb phase could match the underlying anomalies.

62 pages, 9 figures, important typos fixed, added comments

References in corpus (13)

• Generalized Global Symmetries
• Gravitational Anomaly and Transport
• Level/rank Duality and Chern-Simons-Matter Theories
• Cardy Formulae for SUSY Theories in d=4 and d=6
• Comments on Global Symmetries, Anomalies, and Duality in (2+1)d
• Anomalies of discrete symmetries in three dimensions and group cohomology
• Effective String Theory Revisited
• Hidden supersymmetry in quantum bosonic systems
• Anomalies of discrete symmetries in various dimensions and group cohomology
• Closed flux tubes and their string description in D=3+1 SU(N) gauge theories
• TASI 2003 Lectures on Anomalies
• Topological Model for Domain Walls in (Super-)Yang-Mills Theories
• $\theta$ and the ${\eta }^{\prime }$ in Large $N$ Supersymmetric QCD

Cited by in corpus (298)

• Topological Defect Lines and Renormalization Group Flows in Two Dimensions
• Generalized Symmetries in Condensed Matter
• On finite symmetries and their gauging in two dimensions
• On gauging finite subgroups
• Kramers-Wannier-like duality defects in (3+1)d gauge theories
• Symmetries and Strings of Adjoint QCD${}_2$
• Orbifold groupoids
• Comments on One-Form Global Symmetries and Their Gauging in 3d and 4d
• Anomalies in the Space of Coupling Constants and Their Dynamical Applications II
• Anomalies in the Space of Coupling Constants and Their Dynamical Applications I
• 50 Years of Quantum Chromodynamics
• Time-Reversal Breaking in QCD${}_4$, Walls, and Dualities in 2+1 Dimensions
• Dai-Freed anomalies in particle physics
• Comments on Abelian Higgs Models and Persistent Order
• Intrinsic and emergent anomalies at deconfined critical points
• Non-Invertible Symmetries from Holography and Branes
• Walls, Anomalies, and (De)Confinement in Quantum Anti-Ferromagnets
• Non-Invertible Symmetries of $\mathcal{N}=4$ SYM and Twisted Compactification
• Global Symmetries, Counterterms, and Duality in Chern-Simons Matter Theories with Orthogonal Gauge Groups
• Anomaly constraints on deconfinement and chiral phase transition
• Anomaly and global inconsistency matching: $\theta$-angles, $SU(3)/U(1{)}^2$ nonlinear sigma model, $SU(3)$ chains and its generalizations
• Non-invertible Time-reversal Symmetry
• The Global Form of Flavor Symmetries and 2-Group Symmetries in 5d SCFTs
• An exact quantization of Jackiw-Teitelboim gravity
• A Modified Villain Formulation of Fractons and Other Exotic Theories
• Modified instanton sum in QCD and higher-groups
• Anomaly matching for phase diagram of massless ${\mathbb{Z}}_N$-QCD
• Phases Of Adjoint QCD${}_3$ And Dualities
• A Symmetry Breaking Scenario for QCD${}_3$
• Non-invertible self-duality defects of Cardy-Rabinovici model and mixed gravitational anomaly
• Vacuum structure of bifundamental gauge theories at finite topological angles
• Circle compactification and 't Hooft anomaly
• Generalized global symmetries and holography
• Abelian gauge theories on the lattice: $\theta$-terms and compact gauge theory with(out) monopoles
• Adjoint QCD${}_4$, Deconfined Critical Phenomena, Symmetry-Enriched Topological Quantum Field Theory, and Higher Symmetry-Extension
• Semi-Abelian gauge theories, non-invertible symmetries, and string tensions beyond $N$-ality
• Anomaly constraint on massless QCD and the role of Skyrmions in chiral symmetry breaking
• Higher-form symmetries and 3-group in axion electrodynamics
• Goldstone modes and photonization for higher form symmetries
• Higher-Form Symmetries and Their Anomalies in M-/F-Theory Duality
• Discrete Theta Angles, Symmetries and Anomalies
• Symmetry-Enriched Quantum Spin Liquids in $(3+1)d$
• Global 3-group symmetry and 't Hooft anomalies in axion electrodynamics
• An Adventure in Topological Phase Transitions in 3 + 1-D: Non-abelian Deconfined Quantum Criticalities and a Possible Duality
• Symmetric Mass Generation
• Two-flavor adjoint QCD
• Time-Reversal Symmetry, Anomalies, and Dualities in (2+1)$d$
• Time Reversal, SU(N) Yang-Mills and Cobordisms: Interacting Topological Superconductors/Insulators and Quantum Spin Liquids in 3+1D
• Higher Symmetries of 5d Orbifold SCFTs
• Thermal Order in Conformal Theories
• String Universality and Non-Simply-Connected Gauge Groups in 8d
• Continuous Generalized Symmetries in Three Dimensions
• QCD-Collapsed Domain Walls: QCD Phase Transition and Gravitational Wave Spectroscopy
• Beyond Standard Models and Grand Unifications: Anomalies, Topological Terms, and Dynamical Constraints via Cobordisms
• Anomalies and Symmetry Fractionalization
• From spinon band topology to the symmetry quantum numbers of monopoles in Dirac spin liquids
• Anomaly Inflow for Subsystem Symmetries
• From 4d Yang-Mills to 2d ${\mathbb{CP}}^{N-1}$ model: IR problem and confinement at weak coupling
• Anomalies of Duality Groups and Extended Conformal Manifolds
• Anomaly matching, (axial) Schwinger models, and high-T super Yang-Mills domain walls
• Anomaly Obstructions to Symmetry Preserving Gapped Phases
• An Introduction to Higher-Form Symmetries
• QCD Axion on Hilltop by a Phase Shift of $\pi$
• Candidate Phases for SU(2) Adjoint QCD${}_4$ with Two Flavors from $\mathcal{N}=2$ Supersymmetric Yang-Mills Theory
• Generalized homology and Atiyah-Hirzebruch spectral sequence in crystalline symmetry protected topological phenomena
• Higher Form Symmetries TFT in 6d
• From gauge to higher gauge models of topological phases
• Anomalies, a mod 2 index, and dynamics of 2d adjoint QCD
• Anomaly matching and symmetry-protected critical phases in $SU(N)$ spin systems in 1+1 dimensions
• Quantum 4d Yang-Mills Theory and Time-Reversal Symmetric 5d Higher-Gauge Topological Field Theory
• C-P-T anomaly matching in bosonic quantum field theory and spin chains
• Universal Features of Higher-Form Symmetries at Phase Transitions
• Subsystem Non-Invertible Symmetry Operators and Defects
• Instantons, Symmetries and Anomalies in Five Dimensions
• Anomaly Matching in the Symmetry Broken Phase: Domain Walls, CPT, and the Smith Isomorphism
• Higher-form symmetry breaking at Ising transitions
• Domain walls in high-T SU(N) super Yang-Mills theory and QCD(adj)
• Lieb-Schultz-Mattis anomalies as obstructions to gauging (non-on-site) symmetries
• Anomalous Symmetries End at the Boundary
• Matrix ensembles with global symmetries and 't Hooft anomalies from 2d gauge theory
• On the baryon-color-flavor (BCF) anomaly in vector-like theories
• Fractional $\theta$ angle, 't Hooft anomaly, and quantum instantons in charge-$q$ multi-flavor Schwinger model
• The Bi-Fundamental Gauge Theory in 3+1 Dimensions: The Vacuum Structure and a Cascade
• Spontaneous $CP$ breaking in QCD and the axion potential: an effective Lagrangian approach
• Center vortex and confinement in Yang-Mills theory and QCD with anomaly-preserving compactifications
• Curious Aspects of Three-Dimensional $\mathcal{N}=1$ SCFTs
• Exact emergent higher-form symmetries in bosonic lattice models
• Cosmic strings from pure Yang-Mills theory
• Topological critical slowing down: variations on a toy model
• New Higher Anomalies, SU(N) Yang-Mills Gauge Theory and ${\mathbb{CP}}^{\mathrm{N}-1}$ Sigma Model
• Symmetries and Anomalies of (1+1)d Theories: 2-groups and Symmetry Fractionalization
• QCD domain walls, Chern-Simons theories and holography
• Dual simulation of the 2d U(1) gauge Higgs model at topological angle $\theta =\pi$: Critical endpoint behavior
• On 2-form gauge models of topological phases
• 6d SCFTs, Center-Flavor Symmetries, and Stiefel--Whitney Compactifications
• Axion Mass from Magnetic Monopole Loops
• Quantum Distillation of Hilbert Spaces, Semi-classics and Anomaly Matching
• Lifetimes of (near) eternal false vacua
• Tensor renormalization group study of two-dimensional U(1) lattice gauge theory with a $\theta$ term
• Anomaly matching in QCD thermal phase transition
• Quantization of Axion-Gauge Couplings and Non-Invertible Higher Symmetries
• Exact-WKB, complete resurgent structure, and mixed anomaly in quantum mechanics on $S^1$
• Higgs-confinement phase transitions with fundamental representation matter
• The holography of duality in $\mathcal{N}=4$ Super-Yang-Mills theory
• Non-invertible and higher-form symmetries in 2+1d lattice gauge theories
• Sigma Models on Flags
• Relating 't Hooft Anomalies of 4d Pure Yang-Mills and 2d ${\mathbb{CP}}^{N-1}$ Model
• Topological order in a color-flavor locked phase of $(3+1)$-dimensional $U(N)$ gauge-Higgs system
• Skyrmions, Quantum Hall Droplets, and one current to rule them all
• Nonperturbative effects in the Standard Model with gauged 1-form symmetry
• Mixed global anomalies and boundary conformal field theories
• Living on the walls of super-QCD
• Some Exact Results in Chiral Gauge Theories
• Higher Anomalies, Higher Symmetries, and Cobordisms III: QCD Matter Phases Anew
• Line Operators of Gauge Theories on Non-Spin Manifolds
• Vacuum Structure of Charge k Two-Dimensional QED and Dynamics of an Anti D-String Near an O1- plane
• $\theta =\pi$ in $SU(N)/{\mathbb{Z}}_N$ gauge theories
• TQFT, Symmetry Breaking, and Finite Gauge Theory in 3+1D
• Flow of (higher) Berry curvature and bulk-boundary correspondence in parametrized quantum systems
• Phase Diagram of the Two-Flavor Schwinger Model at Zero Temperature
• Parity and the modular bootstrap
• Remarks on Geometric Engineering, Symmetry TFTs and Anomalies
• Exploring the Universe with Dark Light Scalars
• Notes on Confinement on ${\mathbf{R}}^3\times {\mathbf{S}}^1$: From Yang-Mills, super-Yang-Mills, and QCD(adj) to QCD(F)
• Generalized 't Hooft anomalies on non-spin manifolds
• More Exact Results on Chiral Gauge Theories: the Case of the Symmetric Tensor
• Aspects of Non-Relativistic Quantum Field Theories
• Flag manifold sigma models: spin chains and integrable theories
• Lieb-Schultz-Mattis anomalies and web of dualities induced by gauging in quantum spin chains
• 't Hooft anomaly matching condition and chiral symmetry breaking without bilinear condensate
• Scrambling and decoding the charged quantum information
• Probing the dynamics of chiral SU(N) gauge theories via generalized anomalies
• Novel 3d bosonic dualities from bosonization and holography
• Gauging 1-form center symmetries in simple $SU(N)$ gauge theories
• Landau ordering phase transitions beyond the Landau paradigm
• Emergent generalized symmetries in ordered phases and applications to quantum disordering
• Topological aspects of $4$D Abelian lattice gauge theories with the $\theta$ parameter
• New Anomalies, TQFTs, and Confinement in Bosonic Chiral Gauge Theories
• Deconfinement on axion domain walls
• A study of center and chiral symmetry realization in thermal $\mathcal{N}=1$ super Yang-Mills theory using the gradient flow
• Semiclassics with 't Hooft flux background for QCD with $2$-index quarks
• Effective Field Theory of Force-Free Electrodynamics
• Baryons as Quantum Hall Droplets
• Phase structure of the twisted $SU(3)/U(1{)}^2$ flag sigma model on $\mathbb{R}\times S^1$
• Aspects of ${\mathbb{Z}}_N$ rank-2 gauge theory in $(2+1)$ dimensions: construction schemes, holonomies, and sublattice one-form symmetries
• Unifying Monopole and Center Vortex as the Semiclassical Confinement Mechanism
• A possible phase for adjoint QCD
• Deconfinement and CP-breaking at $\theta =\pi$ in Yang-Mills theories and a novel phase for SU(2)
• Vector dominance, one flavored baryons, and QCD domain walls from the "hidden" Wess-Zumino term
• String tensions in deformed Yang-Mills theory
• The gaugino condensate from asymmetric four-torus with twists
• Ultra Unification
• Anomaly and Cobordism Constraints Beyond the Standard Model: Topological Force
• Covariant fracton gauge theory with boundary
• Oblique Confinement at $\theta \ne 0$ in weakly coupled gauge theories with deformations
• Exploring the CP-violating Dashen phase in the Schwinger model with tensor networks
• First-principle simulations of 1+1d quantum field theories at $\theta =\pi$ and spin-chains
• Vacuum structure of Yang-Mills theory as a function of $\theta$
• Effective gauge theories of superfluidity with topological order
• Gauge Enhanced Quantum Criticality and Time Reversal Domain Wall: SU(2) Yang-Mills Dynamics with Topological Terms
• Theory of oblique topological insulators
• Non-Perturbative Defects in Tensor Models from Melonic Trees
• Domain Walls and the $CP$ Anomaly in Softly Broken Supersymmetric QCD
• Higher-form Symmetries under Weak Measurement
• Higher Berry Phase of Fermions and Index Theorem
• DMRG study of the higher-charge Schwinger model and its 't Hooft anomaly
• Lattice gauge theory for Haldane conjecture and central-branch Wilson fermion
• Time reversal and $\mathbf{CP}$ invariance in Calabi-Yau compactifications
• High-temperature domain walls of QCD with imaginary chemical potentials
• Dihedral symmetry in $SU(N)$ Yang-Mills theory
• Self-conjugate QCD
• Universality between vector-like and chiral quiver gauge theories: Anomalies and domain walls
• Condensates and anomaly cascade in vector-like theories
• Proton Stability: From the Standard Model to Beyond Grand Unification
• High quality Nelson-Barr solution to the strong CP problem with $\theta =\pi$
• Outer Automorphism Anomalies
• Tensor Modes in Pure Natural Inflation
• Semiclassical analysis of the bifundamental QCD on ${\mathbb{R}}^2\times T^2$ with 't Hooft flux
• Topology of $SU(N)$ lattice gauge theories coupled with ${\mathbb{Z}}_N$ $2$-form gauge fields
• Anomalies and phases of strongly-coupled chiral gauge theories: recent developments
• Vacuum structure of large $N$ QCD${}_3$ from holography
• Four-fermion deformations of the massless Schwinger model and confinement
• Domain walls and deconfinement: a semiclassical picture of discrete anomaly inflow
• Vector mesons on the wall
• Is $N=2$ Large?
• Linking Entanglement and Discrete Anomaly
• Note on Generalized Symmetries, Gapless Excitations, Generalized Symmetry Protected Topological states, and Anomaly
• Interaction-induced topological phase transition at finite temperature
• The Hitchhiker's Guide to 4d $\mathcal{N}=2$ Superconformal Field Theories
• Toric 2-group anomalies via cobordism
• Gauge theory geography: charting a path between semiclassical islands
• On thermal transition in QCD
• Varieties and properties of central-branch Wilson fermions
• Composite Dark Matter with Forbidden Annihilation
• Higher-group structure in $2n$-dimensional axion-electrodynamics
• Charting the phase diagram of QCD${}_3$
• Anomaly-induced transport phenomena from the imaginary-time formalism
• If the Weak Were Strong and the Strong Were Weak
• Dynamical Abelianization and anomalies in chiral gauge theories
• Bootstrapping Lieb-Schultz-Mattis anomalies
• Higgs-Confinement Transitions in QCD from Symmetry Protected Topological Phases
• Violation of Eigenstate Thermalization Hypothesis in Quantum Field Theories with Higher-Form Symmetry
• Quantum distillation in QCD
• The Fate of Discrete 1-Form Symmetries in 6d
• Higher-group structure in lattice Abelian gauge theory under instanton-sum modification
• Global Symmetries and Partial Confinement
• Supersymmetric Tools in Yang-Mills Theories at Strong Coupling: the Beginning of a Long Journey
• Magnetic operators in 2D compact scalar field theories on the lattice
• Anomalies for anomalous symmetries
• On Topology of the Moduli Space of Gapped Hamiltonians for Topological Phases
• Average-exact mixed anomalies and compatible phases
• Witten effect, anomaly inflow, and charge teleportation
• Strongly coupled QFT dynamics via TQFT coupling
• Symmetry Transmutation and Anomaly Matching
• Symmetry fractionalization, mixed-anomalies and dualities in quantum spin models with generalized symmetries
• $\theta$-dependence in the small-$N$ limit of $2d$ $C{P}^{N-1}$ models
• Finite-Density Massless Two-Color QCD at Isospin Roberge-Weiss Point and 't Hooft Anomaly
• Study of gapped phases of 4d gauge theories using temporal gauging of the ${\mathbb{Z}}_N$ 1-form symmetry
• Anomaly and Cobordism Constraints Beyond Grand Unification: Energy Hierarchy
• Delineating chiral separation effect in two-color dense QCD
• Generalized symmetries in singularity-free nonlinear $\sigma$ models and their disordered phases
• Global anomaly matching in higher-dimensional ${\mathbb{CP}}^{N-1}$ model
• Complex instantons in sigma models with chemical potential
• Anomaly Matching Across Dimensions and Supersymmetric Cardy Formulae
• Exotic Invertible Phases with Higher-Group Symmetries
• Two-dimensional Anomaly, Orbifolding, and Boundary States
• Lectures on 4d N=1 dynamics and related topics
• An Index for Quantum Integrability
• Geometric Approach to 3D Interfaces at Strong Coupling
• BCF anomaly and higher-group structure in the low energy effective theories of mesons
• Symmetry fractionalization and duality defects in Maxwell theory
• Spontaneous symmetry breaking in pure 2D Yang-Mills theory
• Lattice regularizations of $\theta$ vacua: Anomalies and qubit models
• $3d$ One-form Mixed Anomaly and Entanglement Entropy
• Large-$N$ ${\mathbb{CP}}^{N-1}$ sigma model on a Euclidean torus: uniqueness and stability of the vacuum
• Higher Gauge Theory
• Phase structure of linear quiver gauge theories from anomaly matching
• Natural inflation, strong dynamics, and the role of generalized anomalies
• A Long Exact Sequence in Symmetry Breaking: order parameter constraints, defect anomaly-matching, and higher Berry phases
• Cluster expansion and resurgence in Polyakov model
• Simulating Quantum Mechanics with a $\theta$-term and an 't Hooft Anomaly on a Synthetic Dimension
• $\theta$ dependence of $T_c$ in 4d SU(3) Yang-Mills theory with histogram method and the Lee-Yang zeros in the large $N$ limit
• Domain Walls in 4d N=1 Supersymmetric Yang-Mills
• Role of anomalous symmetry in 0-$\pi$ qubits
• Aspects of the QCD $\theta$-vacuum
• Musings on the Current Status of HEP
• Monopole Breaking of Chern-Weil Symmetries
• Monopole-vortex continuity of $\mathcal{N}=1$ super Yang-Mills theory on ${\mathbb{R}}^2\times S^1\times S^1$ with 't Hooft twist
• Evidence of a CP broken deconfined phase in 4D SU(2) Yang-Mills theory at $\theta =\pi$ from imaginary $\theta$ simulations
• Consistency of $\chi$SB in chiral Yang-Mills theory with adiabatic continuity
• Domain wall moduli in softly-broken SQCD at $\bar{\theta }=\pi$
• Confinement on ${\mathbb{R}}^3\times {\mathbb{S}}^1$ and Double-String Collapse
• Winding theta and destructive interference of instantons
• Topology and index theorem with a generalized Villain lattice action -- a test in 2d
• The 3d $A$-model and generalised symmetries, Part I: bosonic Chern-Simons theories
• Global Aspects of Spaces of Vacua
• Axial Anomaly in the $SU(N)$ Gauge Matrix Model
• The Ultraviolet Structure of Quantum Field Theories. Part 2: What is Quantum Field Theory?
• A Multicritical Point with Infinite Fractal Symmetries
• C-P-T Fractionalization
• Anomaly of $(2+1)$-Dimensional Symmetry-Enriched Topological Order from $(3+1)$-Dimensional Topological Quantum Field Theory
• Topological terms and anomaly matching in effective field theories on ${\mathbb{R}}^3\times S^1$: I. Abelian symmetries and intermediate scales
• Remarks on effects of projective phase on eigenstate thermalization hypothesis
• Chiral condensate and Dirac spectrum of one- and two-flavor QCD at nonzero $\theta$-angle
• Note on lattice description of generalized symmetries in $SU(N)/{\mathbb{Z}}_N$ gauge theories
• 5d SCFTs and their non-supersymmetric cousins
• One-form symmetries and the 3d $\mathcal{N}=2$ $A$-model: Topologically twisted indices and CS theories
• Higher-form symmetry breaking and holographic flavour
• WZW terms without anomalies: generalised symmetries in chiral Lagrangians
• Center-vortex semiclassics with non-minimal 't Hooft fluxes on ${\mathbb{R}}^2\times T^2$ and center stabilization at large $N$
• Dark Matter from 't Hooft Anomaly Matching
• Topological observables and $\theta$ dependence in high temperature QCD from lattice simulations
• Non-Invertible Gauss Law and Axions
• Graded Hilbert spaces, quantum distillation and connecting SQCD to QCD
• Generalized symmetry constraints on deformed 4d (S)CFTs
• Sigma-model analysis of $SU(3)$ antiferromagnetic spins on the triangular lattice
• Charge pumps, pivot Hamiltonians and symmetry-protected topological phases
• Strong CP from a Hidden Chiral Condensate
• Phase diagram of $q$-deformed Yang-Mills theory on $S^2$ at non-zero $\theta$-angle
• Mildly Flavoring Domain Walls in Sp(N) SQCD
• Negative string tension of higher-charge Schwinger model via digital quantum simulation
• Interface Junctions in QCD${}_4$
• Holographic quantum singularity
• Perturbative and global anomalies in bosonic analogues of integer quantum Hall and topological insulator phases
• First-order CP phase transition in two-flavor QCD at $\theta =\pi$ under electromagnetic scale anomaly via a Nambu-Jona-Lasinio description
• Anomaly-induced edge currents in hydrodynamics with parity anomaly
• A landscape of 4d N=1 SCFTs with a=c
• The mixed $0$-form/$1$-form anomaly in Hilbert space: pouring new wine into old bottles
• Symmetry-protected gates of Majorana qubits in a high-$T_c$ superconductor platform
• Chiral vortex-line liquid of three-dimensional interacting Bose systems with moat dispersion
• On the perturbative aspects of deformed Yang-Mills theory
• Topological Manipulations On $\mathbb{R}$ Symmetries Of Abelian Gauge Theory
• Fermionic Matrix Models and Bosonization
• Topology of Calorons Re-examined
• Higher-form entanglement asymmetry and topological order
• Quantum Hall liquids in high-density QCD
• Large $\theta$ angle in two-dimensional large $N$ ${\mathbb{CP}}^{N-1}$ model
• Generalized symmetry enriched criticality in (3+1)d