We investigate 't Hooft anomalies in the $\mathbb{CP}^{N-1}$ model in spacetime dimensions higher than two and identify two types of anomalies: One is a mixed anomaly between the $\mathrm{PSU}(N)$ flavor-rotation and magnetic symmetries, and the other is between the reflection and magnetic symmetries. The latter indicates that even in the absence of the flavor symmetry, the model cannot have a unique gapped ground state as long as the reflection and magnetic symmetries are respected. We also clarified the condition for the 't Hooft anomalies to survive under monopole deformations, which explicitly break the magnetic symmetry down to its discrete subgroup. Besides, we explicitly show how the identified 't Hooft anomalies match in the low-energy effective description of symmetry broken phases---the Néel, $\mathrm{U}(1)$ spin liquid, and the valence bond solid phases. An application to the finite-temperature phase diagram of the four-dimensional $\mathbb{CP}^{N-1}$ model is also discussed.
We describe how Goldstone bosons of spontaneous symmetry breaking $G \to H$ can reproduce anomalies of UV theories under the symmetry group $G$ at the nonperturbative level. This is done by giving a general definition of Wess-Zumino-Witten terms in terms of the invertible field theories in $d+1$ dimensions which describe the anomalies of $d$-dimensional UV theories. The hidden local symmetry $\widehat H$, which is used to describe Goldstone bosons in coset construction $G/H$, plays an important role. Our definition also naturally leads to generalized $θ$-angles of the hidden local gauge group …
We discuss the $SU(3)/[U(1)\times U(1)]$ nonlinear sigma model in 1+1D and, more broadly, its linearized counterparts. Such theories can be expressed as $U(1)\times U(1)$ gauge theories and therefore allow for two topological $θ$-angles. These models provide a field theoretic description of the $SU(3)$ chains. We show that, for particular values of $θ$-angles, a global symmetry group of such systems has a 't Hooft anomaly, which manifests itself as an inability to gauge the global symmetry group. By applying anomaly matching, the ground-state properties can be severely constrained. The anomaly…
We elucidate that the phase diagram of massless $N$-flavor QCD under $\mathbb{Z}_N$ flavor-twisted boundary condition (massless $\mathbb{Z}_N$-QCD) is constrained by an 't Hooft anomaly involving two-form gauge fields. As a consequence, massless $\mathbb{Z}_N$-QCD turns out to realize persistent order at any temperatures and quark chemical potentials, namely, the symmetric and gapped phase is strictly forbidden. This is the first result on the finite-$(T,μ)$ phase diagram in QCD-type theories based on anomaly matching related to center and discrete axial symmetries.
We consider the $O(3)$ nonlinear sigma model with the $θ$-term and its linear counterpart in 1+1D. The model has discrete time-reflection and space-reflection symmetries at any $θ$, and enjoys the periodicity in $θ\rightarrow θ+2π$. At $θ=0,π$ it also has a charge-conjugation $C$-symmetry. Gauging the discrete space-time reflection symmetries is interpreted as putting the theory on the nonorientable $\mathbb RP^2$ manifold, after which the $2π$ periodicity of $θ$ and the $C$ symmetry at $θ=π$ are lost. We interpret this observation as a mixed 't Hooft anomaly among charge-conjugation $C$, pari…
Symmetries in Quantum Field Theory may have 't Hooft anomalies. If the symmetry is unbroken in the vacuum, the anomaly implies a nontrivial low-energy limit, such as gapless modes or a topological field theory. If the symmetry is spontaneously broken, for the continuous case, the anomaly implies low-energy theorems about certain couplings of the Goldstone modes. Here we study the case of spontaneously broken discrete symmetries, such as Z/2 and T. Symmetry breaking leads to domain walls, and the physics of the domain walls is constrained by the anomaly. We investigate how the physics of the do…
We calculate the magnetoconductivity of the Weyl semimetal with $\mathbb{Z}_2$ symmetry and chiral anomaly utilizing the recently developed hydrodynamic theory. The system in question will be influenced by magnetic fields connected with ordinary Maxwell and the second $U(1)$-gauge field, which is responsible for anomalous topological charge. The presence of chiral anomaly and $\mathbb{Z}_2$ anomalous charge endow the system with new transport coefficients. We start with the linear perturbations of the hydrodynamic equations and calculate the magnetoconductivity of this system. The holographic …
Following the ideas developed by Komargodski and Schwimmer in arXiv:1107.3987 and arXiv:1112.4538, we argue that the IR dilaton effective action on the cone computes the entanglement entropy of an even dimensional non-conformal field theory interpolating between a UV and an IR fixed point. We restrict our attention to theories which flow to trivial IR fixed point. We get exact non-perturbative results for the coefficients of the logarithmically divergent term of the entanglement entropy in these field theories in arbitrary even dimensions. The results match precisely with the weak coupling res…
Difference between pair creation and anomaly matching: Nontrivial aspects of invertible symmetries and defects in $SU(N)$ models
National Center of Theoretical Sciences, National Taiwan University（国立台湾大学理论物理研究中心）；Institute for Solid State Physics, University of Tokyo（东京大学固体物理研究所）
We study the relation between the quantum spin Hall effect(QSHE) and the $\mathbb{Z}_2$ global gauge anomaly and discover that there exists an one-to-one correspondence between them. By constructing a two dimensional non-abelian gauge theory whose non-abelian gauge field is the Berry connection induced by the Bloch wave function of the quantum spin Hall system, we prove that if the quantum spin Hall system is topologically nontrivial, the corresponding 2D gauge theory has a global gauge anomaly. We further generalize our discussion of the zero modes of the Dirac operator which play a central r…
We study $(1+1)$-dimensional $SU(N)$ spin systems in the presence of the global $SU(N)$ rotation and lattice translation symmetries. By matching the mixed anomaly of the $PSU(N)\times\mathbb{Z}$ symmetry in the continuum limit, we identify a topological index for spin model evaluated as the total number of Young-tableaux boxes of spins per unit cell modulo $N$, which characterizes the "ingappability" of the system. A nontrivial index implies either a ground-state degeneracy in a gapped phase, which can be regarded as a field-theory version of the Lieb-Schultz-Mattis theorem, or a restriction o…
Some time-reversal symmetric topological orders are anomalous in that they cannot be realized in strictly two-dimensions without breaking time reversal symmetry; instead, they can only be realized on the surface of certain three-dimensional systems. We propose two quantities, which we call {\it anomaly indicators}, that can detect if a time-reversal symmetric topological order is anomalous in this sense. Both anomaly indicators are expressed in terms of the quantum dimensions, topological spins, and time-reversal properties of the anyons in the given topological order. The first indicator, $η_…
In this paper we study gapless fermionic and bosonic systems in $d$-dimensional continuum space with $U(1)$ particle-number conservation and $\mathbb{R}^d$ translation symmetry. We write down low energy effective field theories for several gapless phases where $U(1)\times \mathbb{R}^d$ is viewed as internal symmetry. The $U(1)\times \mathbb{R}^d$ symmetry, when viewed as an internal symmetry, has a mixed anomaly, and the different effective field theories for different phases must have the same mixed anomaly. Such a mixed anomaly is proportional to the particle number density, and can be measu…
Recently it was shown that the topological properties of $2D$ and $3D$ topological insulators are captured by a $\mathbb{Z}_2$ chiral anomaly in the boundary field theory. It remained, however, unclear whether the anomaly survives electron-electron interactions. We show that this is indeed the case, thereby providing an alternative formalism for treating topological insulators in the interacting regime. We apply this formalism to fractional topological insulators (FTI) via projective/parton constructions and use it to test the robustness of all fractional topological insulators which can be de…
Anomaly matching constrains low-energy physics of strongly-coupled field theories, but it is not useful at finite temperature due to contamination from high-energy states. The known exception is an 't Hooft anomaly involving one-form symmetries as in pure $SU(N)$ Yang-Mills theory at $θ=π$. Recent development about large-$N$ volume independence, however, gives us a circumstantial evidence that 't Hooft anomalies can also remain under circle compactifications in some theories without one-form symmetries. We develop a systematic procedure for deriving an 't Hooft anomaly of the circle-compactifi…
We study properties of self-duality symmetry in the Cardy-Rabinovici model. The Cardy-Rabinovici model is the $4$d $U(1)$ gauge theory with electric and magnetic matters, and it enjoys the $SL(2,\mathbb{Z})$ self-duality at low-energies. $SL(2,\mathbb{Z})$ self-duality does not realize in a naive way, but we notice that the $ST^{p}$ duality transformation becomes the legitimate duality operation by performing the gauging of $\mathbb{Z}_N$ $1$-form symmetry with including the level-$p$ discrete topological term. Due to such complications in its realization, the fusion rule of duality defects be…
We derive a selection rule among the $(1+1)$-dimensional SU(2) Wess-Zumino-Witten theories, based on the global anomaly of the discrete $\mathbb{Z}_2$ symmetry found by Gepner and Witten. In the presence of both the SU(2) and $\mathbb{Z}_2$ symmetries, a renormalization-group flow is possible between level-$k$ and level-$k'$ Wess-Zumino-Witten theories only if $k\equiv k' \mod{2}$. This classifies the Lorentz-invariant, SU(2)-symmetric critical behavior into two "symmetry-protected" categories corresponding to even and odd levels,restricting possible gapless critical behavior of translation-in…
In this paper, we discuss relativistic hydrodynamics for a massless Dirac fermion in $(2+1)$ dimensions, which has the parity anomaly -- a global 't Hooft anomaly between $\mathrm{U}(1)$ and parity symmetries. We investigate how hydrodynamics implements the party anomaly, particularly focusing on the transport phenomena at the boundary. Based on the parity anomaly matching and the second law of local thermodynamics, we find $\mathrm{U}(1)$ and entropy currents localized at the boundary as well as the bulk anomalous current with vanishing divergence. These edge currents are similar to the $(1+1…
A non-perturbative lattice regularization of chiral fermions and bosons with anomaly-free symmetry $G$ in 1+1D spacetime is proposed. More precisely, we ask "whether there is a local short-range quantum Hamiltonian with a finite Hilbert space for a finite system realizing onsite symmetry $G$ defined on a 1D spatial lattice, such that its low energy physics produces a 1+1D anomaly-free chiral matter theory of symmetry $G$?" In particular, we propose that the 3$_L$-5$_R$-4$_L$-0$_R$ U(1) chiral fermion theory, with two left-moving fermions of charge-3 and 4, and two right-moving fermions of char…
We present microscopic Hamiltonians that gap the Dirac fermions on the surface of topological crystalline insulators (TCIs) protected by reflection symmetry and create symmetry-preserving states with Abelian topological orders. For TCIs with $n = 2$ copies of surface Dirac fermions, we find a fermionic $\mathbb{Z}_4$ topological order with charge-spin self duality and with a novel symmetry action on the anyons, whereby reflections with respect to two parallel planes act differently. We also analyze TCIs with $n = 1$ and $4$ copies of Dirac fermions, which taken together covers every nontrivial…
Quasiparticle excitations in $3+1$ dimensions can be either bosons or fermions. In this work, we introduce the notion of fermionic loop excitations in $3+1$ dimensional topological phases. Specifically, we construct a new many-body lattice invariant of gapped Hamiltonians, the loop self-statistics, that distinguishes two bosonic topological orders that both superficially resemble $3+1$ d ${\mathbb{Z}}_2$ gauge theory coupled to fermionic charged matter. The first has fermionic charges and bosonic ${\mathbb{Z}}_2$ gauge flux loops (FcBl) and is just the ordinary fermionic toric code. The second…
We study the phase diagram of two-flavor massless two-color QCD (QC$_2$D) under the presence of quark chemical potentials and imaginary isospin chemical potentials. At the special point of the imaginary isospin chemical potential, called the isospin Roberge--Weiss (RW) point, two-flavor QC$_2$D enjoys the $\mathbb{Z}_2$ center symmetry that acts on both quark flavors and the Polyakov loop. We find a $\mathbb{Z}_2$ 't Hooft anomaly of this system, which involves the $\mathbb{Z}_2$ center symmetry, the baryon-number symmetry, and the isospin chiral symmetry. Anomaly matching, therefore, constrai…
We consider the hydrodynamical model of topological Dirac semi-metal possessing two Dirac nodes separated in momentum space along a rotation axis. It has been argued that the system in question, except the chiral anomaly, is endowed with the other one $\mathbb{Z}_2$. In order to model such a system we introduce two $U(1)$-gauge fields. The presence of the additional $\mathbb{Z}_2$ anomaly leads to the non-trivial modifications of hydrodynamical equations and to the appearance of new kinetic coefficients bounded with the vorticity and the magnetic parts of Maxwell and auxiliary $U(1)$-gauge fie…
Certain patterns of symmetry fractionalization in topologically ordered phases of matter are anomalous, in the sense that they can only occur at the surface of a higher dimensional symmetry-protected topological (SPT) state. An important question is to determine how to compute this anomaly, which means determining which SPT hosts a given symmetry-enriched topological order at its surface. While special cases are known, a general method to compute the anomaly has so far been lacking. In this paper we propose a general method to compute relative anomalies between different symmetry fractionaliza…
The charge-$q$ Schwinger model is the $(1+1)$-dimensional quantum electrodynamics (QED) with a charge-$q$ Dirac fermion. It has the $\mathbb{Z}_q$ $1$-form symmetry and also enjoys the $\mathbb{Z}_q$ chiral symmetry in the chiral limit, and there is a mixed 't Hooft anomaly between those symmetries. We numerically study the charge-$q$ Schwinger model in the lattice Hamiltonian formulation using the density-matrix renormalization group (DMRG). When applying DMRG, we map the Schwinger model to a spin chain with nonlocal interaction via Jordan-Wigner transformation, and we take the open boundary …
We study bosonic systems on a spacetime lattice defined by path integrals of commuting fields. We introduce branch-independent bosonic (BIB) systems, whose path integral is independent of the branch structure of the spacetime simplicial complex, even for a spacetime with boundaries. In contrast, a generic lattice bosonic (GLB) system's path integral may depend on the branch structure. We find the invertible topological order characterized by the Stiefel-Whitney cocycle (e.g., 4+1d w$_2$w$_3$) to be nontrivial for BIB systems, but this topological order and a trivial gapped tensor product state…
Understanding Non-Split 2-Group Symmetry: (3+1)D SymTFT, Anomaly and Bordism
School of Physics, Peking University（北京大学物理学院）；Center for High Energy Physics, Peking University（北京大学高能物理中心）；Kavli IPMU (WPI), UTIAS, The University of Tokyo（东京大学 Kavli宇宙起源与粒子物理研究所（WPI）、UTIAS）
