We derive a selection rule among the (1+1)-dimensional SU(2) Wess-Zumino-Witten theories, based on the global anomaly of the discrete Z_{2} symmetry found by Gepner and Witten. In the presence of both the SU(2) and Z_{2} symmetries, a renormalization-group flow is possible between level-k and level-k^{'} Wess-Zumino-Witten theories only if k≡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-invariant quantum spin chains.
We derive a selection rule among the (1+1)-dimensional SU(2) Wess-Zumino-Witten theories, based on the global anomaly of the discrete Z_{2} symmetry found by Gepner and Witten. In the presence of both the SU(2) and Z_{2} symmetries, a renormalization-group flow is possible between level-k and level-k^{'} Wess-Zumino-Witten theories only if k≡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-invariant quantum spin chains.
We study (1+1)-dimensional SU(N) spin systems in the presence of global SU(N) rotation and lattice translation symmetries. Knowing the mixed anomaly of the two symmetries at low energy, we identify, by the anomaly matching argument, a topological index for the spin model-the total number of Young-tableau boxes of spins per unit cell modulo N-characterizing the "ingappability" of the system. A nontrivial index implies either a ground-state degeneracy in a gapped phase, which can be thought of as a field-theory version of the Lieb-Schultz-Mattis theorem, or a restriction of the possible universality classes in a critical phase, regarded as the symmetry-protected critical phases. As an example of the latter case, we show that only a class of SU(N) Wess-Zumino-Witten theories can be realized in the low-energy limit of the given lattice model in the presence of the symmetries. Similar constraints also apply when a higher global symmetry emerges in the model with a lower symmetry. Our results agree with several examples known in previous studies of SU(N) models, and predict a general constraint on the structure factor which is measurable in experiments.
We study the quantum critical phase of an SU(2) symmetric spin-2 chain obtained from spin-2 bosons in a one-dimensional lattice. We obtain the scaling of the finite-size energies and entanglement entropy by exact diagonalization and density-matrix renormalization group methods. From the numerical results of the energy spectra, central charge, and scaling dimension we identify the conformal field theory describing the whole critical phase to be the SU(3)_{1} Wess-Zumino-Witten model. We find that, while the Hamiltonian is only SU(2) invariant, in this critical phase there is an emergent SU(3) symmetry in the thermodynamic limit.
We study (1 + 1)-dimensional SU(N ) spin systems in the presence of global SU(N ) rotation and lattice translation symmetries.Knowing the mixed anomaly of the two symmetries at low energy, we identify, by anomaly matching argument, a topological index for spin model-the total number of Young-tableaux boxes of spins per unit cell modulo N -characterizing the "ingappability" of the system.A nontrivial index implies either a ground-state degeneracy in a gapped phase, which can be thought of as a field-theory version of the Lieb-Schultz-Mattis (LSM) theorem, or a restriction of the possible universality classes in a critical phase, regarded as the symmetry-protected critical phases.As an example of the latter case, we show only a class of SU(N ) Wess-Zumino-Witten theories can be realized in the low-energy limit of the given lattice model in the presence of the symmetries.Similar constraints also apply when a higher global symmetry emerges in the model with a lower symmetry.We also predict a general constraint on the structure factor which is measurable in experiments.To generalize to higher dimensions, we introduce a new boundary condition which renders the flux-insertion argument for the LSM-type theorems in two or higher dimensional lattices free from the specific choice of system sizes.It also enables a formulation of the LSM-type theorems in arbitrary dimensions in terms of the anomaly in field theories of bulk correspondences.Moreover, by our new boundary condition, we also show that the higher-form U(1) LSM theorem and U(1) and SU(N ) LSM theorems in arbitrary higher dimensions have unified form in the bulk viewpoint.At last, we also discuss the discrete-symmetry LSM-type ingappabilities in the lattice language and we find that projective symmetry representations per unit cell implies that the system cannot be gapped with a unique translation-symmetric ground state.I would like to express my sincere gratitude to all those having helped me during my Master and Doctoral courses at the University of Tokyo.First of all, I would like to thank my parents who have supported me financially and encouraged me to do whatever I am keen on in the past 28 years.Also, I am grateful to every one that has taught me great knowledges on life and study at various steps.More specifically, I have to appreciate the guidance and kind supports by my supervisor, Professor Masaki Oshikawa who has given me patient guidance on study and kind financial supports.Moreover, I would thank Prof. Masahiro Sato for his cooperation in our paper, "Theory of electron spin resonance in one-dimensional topological insulators with spinorbit couplings: Detection of edge states".I am also grateful for Dr. Chang-Tse Hsieh for his collaboration on our work "Anomaly Matching and Symmetry-Protected Critical Phases in SU (N ) Spin Systems in 1+1 Dimensions", on which this Thesis is largely based, and Dr. Yoshiki Fukusumi on the work "Bosonization in a background U (1) gauge field".I would also thank Prof. Yuji Tachikawa and Prof. David Tong for patient and kind help on resolving my confusion during study.I would also like to acknowledge with much appreciation the help from our group members.
We suggest the possibility that the two-dimensional SU(2)k Wess-Zumino-Witten (WZW) theory, which has global SO(4) symmetry, can be continued to $2+\ensuremath{\epsilon}$ dimensions by enlarging the symmetry to $\mathrm{SO}(4+\ensuremath{\epsilon})$. This is motivated by the three-dimensional sigma model with SO(5) symmetry and a WZW term, which is relevant to deconfined criticality. If such a continuation exists, the structure of the renormalization group flows at small $\ensuremath{\epsilon}$ may be fixed by assuming analyticity in $\ensuremath{\epsilon}$. This leads to the conjecture that the WZW fixed point annihilates with a new, unstable fixed point at a critical dimensionality dc>2. We suggest that dc<3 for all k, and we compute dc in the limit of large k. The flows support the conjecture that the deconfined phase transition in SU(2) magnets is a ``pseudocritical'' point with approximate SO(5), controlled by a fixed point slightly outside the physical parameter space.
We study a relationship between conformally invariant boundary conditions and anomalies of conformal field theories (CFTs) in 1+1 dimensions. For a given CFT with a global symmetry, we consider symmetric ``gapping potentials'', which are relevant perturbations to the CFT. If a gapping potential is introduced only in a subregion of the system, it provides a certain boundary condition to the CFT. From this equivalence, if there exists a Cardy boundary state, which is invariant under a symmetry, then the CFT can be gapped with a unique ground state by adding the corresponding gapping potential. This means that the symmetry of the CFT is anomaly free. Using this approach, we systematically deduce the anomaly-free conditions for various types of CFTs with several different symmetries. They include the free compact boson theory, Wess-Zumino-Witten models, and unitary minimal models. When the symmetry of the CFT is anomalous, it implies a Lieb-Schultz-Mattis type ingappability of the system. Our results are consistent with, where available, known results in the literature. Moreover, we extend the discussion to other symmetries including spin groups and generalized time-reversal symmetries.
In this paper, we propose using the nonlinear sigma model (NLSM) with the Wess-Zumino-Witten (WZW) term as a general description of deconfined quantum critical points that separate two spontaneously symmetry-breaking (SSB) phases in arbitrary dimensions. In particular, we discuss the suitable choice of the target space of the NLSM, which is in general the homogeneous space G/K, where G G is the UV symmetry and K K is generated by {\mathfrak k}={\mathfrak h}_1\cap {\mathfrak h}_2 𝔨 = 𝔥 1 ∩ 𝔥 2 , and {\mathfrak h}_i 𝔥 i is the Lie algebra of the unbroken symmetry in each SSB phase. With this specific target space, the symmetry defects in both SSB phases are on equal footing, and their intertwinement is captured by the WZW term. The DQCP transition is then tuned by proliferating the symmetry defects. By coupling the G/K G / K NLSM with the WZW term to the background gauge field, the ’t Hooft anomaly of this theory can be determined. The bulk symmetry-protected topological (SPT) phase that cancels the anomaly is described by the relative Chern-Simons term in odd spacetime dimensions or mixed \theta θ term in even dimensions. We construct and discuss a series of models with Grassmannian symmetry defects in 3+1d. We also provide the fermionic model that reproduces the G/K G / K NLSM with the WZW term.