We consider the $O(3)$ nonlinear sigma model with the $\ensuremath{\theta}$ term and its linear counterpart in 1+1D. The model has discrete time-reflection and space-reflection symmetries at any $\ensuremath{\theta}$, and enjoys the periodicity in $\ensuremath{\theta}\ensuremath{\rightarrow}\ensuremath{\theta}+2\ensuremath{\pi}$. At $\ensuremath{\theta}=0,\ensuremath{\pi}$ 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 $2\ensuremath{\pi}$ periodicity of $\ensuremath{\theta}$ and the $C$ symmetry at $\ensuremath{\theta}=\ensuremath{\pi}$ are lost. We interpret this observation as a mixed 't Hooft anomaly among charge-conjugation $C$, parity $P$, and time-reversal $T$ symmetries when $\ensuremath{\theta}=\ensuremath{\pi}$. 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 semiclassical 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 $\mathbb{C}{P}^{N\ensuremath{-}1}$ models and show that the $C\text{\ensuremath{-}}P\text{\ensuremath{-}}T$ anomaly exists for even $N$.
We consider the O(3) nonlinear sigma model with the \ensuremathθ term and its linear counterpart in 1+1D. The model has discrete time-reflection and space-reflection symmetries at any \ensuremathθ, and enjoys the periodicity in \ensuremathθ\ensuremath→\ensuremathθ+2\ensuremathπ. At \ensuremathθ=0,\ensuremathπ it also has a charge-conjugation C symmetry. Gauging the discrete space-time reflection symmetries is interpreted as putting the theory on the nonorientable RP2 manifold, after which the 2\ensuremathπ periodicity of \ensuremathθ and the C symmetry at \ensuremathθ=\ensuremathπ are lost. We interpret this observation as a mixed 't Hooft anomaly among charge-conjugation C, parity P, and time-reversal T symmetries when \ensuremathθ=\ensuremathπ. 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 semiclassical 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 CPN\ensuremath−1 models and show that the C\ensuremath-P\ensuremath-T anomaly exists for even N.
C-P-T anomaly in bosonic systems
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→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 RP2 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, parity P, and time-reversal T symmetries when θ=π. 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 CPN−1 models and show that the C-P-T anomaly exists for even N.
We discuss the $SU(3)/[U(1)\ifmmode\times\else\texttimes\fi{}U(1)]$ nonlinear sigma model in 1+1D and, more broadly, its linearized counterparts. Such theories can be expressed as $U(1)\ifmmode\times\else\texttimes\fi{}U(1)$ gauge theories and therefore allow for two topological $\ensuremath{\theta}$ angles. These models provide a field theoretic description of the SU(3) chains. We show that, for particular values of $\ensuremath{\theta}$ 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 matching is an avatar of the Lieb-Schultz-Mattis (LSM) theorem for the spin chain from which the field theory descends, and it forbids a trivially gapped ground state for particular $\ensuremath{\theta}$ angles. We generalize the statement of the LSM theorem and show that 't Hooft anomalies persist even under perturbations which break the spin-symmetry down to the discrete subgroup ${\mathbb{Z}}_{3}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{3}\ensuremath{\subset}SU(3)/{\mathbb{Z}}_{3}$. In addition, the model can further be constrained by applying global inconsistency matching, which indicates the presence of a phase transition between different regions of $\ensuremath{\theta}$ angles. We use these constraints to give possible scenarios of the phase diagram. We also argue that at the special points of the phase diagram the anomalies are matched by the SU(3) Wess-Zumino-Witten model. We generalize the discussion to the $SU(N)/U{(1)}^{N\ensuremath{-}1}$ nonlinear sigma models as well as the 't Hooft anomaly of the SU(N) Wess-Zumino-Witten model, and show that they match. Finally, the (2+1)-dimensional extension is considered briefly, and we show that it has various 't Hooft anomalies leading to nontrivial consequences.
We consider two sets of related models: initially, these are SU(2) antiferromagnetic spin chains with N sites of spin S, and the O(3) nonlinear sigma model in two dimensions with topological coefficient Θ a multiple of π (and later, the extensions of these with any semisimple Lie group symmetry). It is known that, in a continuum description, the low-energy behavior of the spin chain is given by the sigma model with Θ=2πS. We study these models with N odd and with antiperiodic (A) boundary condition (b.c.), respectively, which correspond. The A b.c. in the sigma model involves the Z2 inversion symmetry n→−n, and amounts to a flux of a Z2 gauge field through a spacetime torus; summing over the two b.c.s for each direction would amount to gauging the Z2 inversion symmetry. We show directly that, if and only if (−1)Θ/π=−1, the gauging cannot be carried out; there is an 't Hooft anomaly. The partition function for the A b.c. exists, but is not gauge invariant; consequently, the sum over b.c.s cannot be made modular invariant. The gauged model would be a sigma model with target space RP2≅S2/Z2, and hence this model does not exist for Θ=π (mod 2π). A related result is that, using semiclassical quantization, in the spin chain we obtain the known values of the ground-state crystal momentum, which at leading order depend only on N modulo 4 and 2S modulo 2. For a large class of spin chains and associated sigma models we find similar results, but now (−1)Θ/π is replaced by the value ±1 of the square of the time-reversal operator acting on a single spin, which is still determined by the coefficients of the topological terms, in a way that depends on the symmetry group.
We investigate 't Hooft anomalies in the ${\mathbb{C}\mathbb{P}}^{N\ensuremath{-}1}$ model in space-time dimensions higher than two and identify two types of anomalies: one is a mixed anomaly between the 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 N\'eel, U(1) spin liquid, and the valence bond solid phases. An application to the finite-temperature phase diagram of the four-dimensional ${\mathbb{C}\mathbb{P}}^{N\ensuremath{-}1}$ model is also discussed.
A bstract We study a number of 1+1d lattice models with anti-unitary symmetries that simultaneously reflect space and reverse time. Some of these symmetries are anomalous, leading to Lieb-Schultz-Mattis-type constraints, thus excluding a trivially gapped phase. Examples include a mod 8 anomaly in the Majorana chain and various mod 2 anomalies in the spin chain. In some cases, there is an exact, non-anomalous lattice symmetry that flows in the continuum to CPT. In some other cases, the CPT symmetry of the continuum theory is emergent or absent. Depending on the model, the anomaly of the lattice model is matched in the continuum in different ways. In particular, it can be mapped to an emergent anomaly of an emanant symmetry.
Symmetry protected topological phases protected by crystalline symmetries and internal symmetries are shown to enjoy fascinating one-to-one correspondence in classification. Here we investigate the physics content behind the abstract correspondence in three or higher-dimensional systems. We show correspondence between anomalous boundary states, which provides a new way to explore the quantum anomaly of symmetry from its crystalline equivalent counterpart. We show such a correspondence directly in two scenarios, including the anomalous symmetry-enriched topological orders (SET) and critical boundary states. (1) First of all, for the surface SET correspondence, we demonstrate it by considering examples involving time-reversal symmetry and mirror symmetry. We show that one 2D topological order can carry the time reversal anomaly as long as it can carry the mirror anomaly and vice versa, by directly establishing the mapping of the time reversal anomaly indicators and mirror anomaly indicators. Besides, we also consider other cases involving continuous symmetry, which leads us to introduce some new anomaly indicators for symmetry from its counterpart. (2) Furthermore, we also build up direct correspondence for (near) critical boundaries. In this perspective, we first consider the edge-corner correspondence between edge theory as 1+1D conformal field theory of internal fermionic SPT and the 0+1D corner modes of (higher-order) crystalline fermionic SPT. By viewing the corner modes on 1D boundary as perturbed CFT is crucial insight for the correspondence, but also help to discover the boundary theory of some intrinsically interacting fermionic SPT, whcih are challenging. Next, we discuss this correspondence bosonic systems, again taking topological phases protected by time reversal and mirror symmetry as examples, the direct correspondence of their (near) critical boundaries can be built up by coupled chain construction that was first proposed by Senthil and Fisher. The examples of critical boundary correspondence we consider in this paper can be understood in a unified framework that is related to hierarchy structure of topological O(n) O ( n ) nonlinear sigma model (NL \sigma σ M), that generalizes the Haldane’s derivation of O(3) O ( 3 ) sigma model from spin one-half system. To our best of knowledge, this hierarchy structure of the O(n) O ( n ) NL \sigma σ M is first exposed, which inspires us to explore more interesting connection between different theories in the light of the boundary correspondence.