Symmetry Protection of Critical Phases and a Global Anomaly in1+1Dimensions

Abstract

We derive a selection rule among the (1 + 1)-dimensional SU(2) Wess-Zumino-Witten theories, based on the global anomaly of the discrete Z2 symmetry found by Gepner and Witten. In the presence of both the SU(2) and Z2 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 Lorentzinvariant, 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.PACS numbers: 75.10. Jm, 75.10.Pq, 03.65.Vf, 11.25.Hf Introduction.-The classification of quantum phases is a central problem in condensed-matter and statistical physics. They are first classified into gapped and gapless phases. There has been a significant progress in further classification of gapped phases, which are relatively easy to be handled theoretically. In particular, symmetry protected topological (SPT) phases [1,2] have become an important concept in the classification. That is, even when two states have no long-range entanglement and are indistinguishable in terms of any local observables, they could still belong to distinct phases separated by a quantum phase transition, in the presence of a certain symmetry. In contrast, classification of gapless quantum phases remains very much open. Symmetries are naturally expected to play an important role also in the classification of gapless quantum phases.The issue of classification of quantum phases is also deeply related to that of quantum field theories. Since a quantum field theory can be regarded as an effective description of universal low-energy behaviors of quantum many-body systems, we may expect that they are essentially the same problem. An intriguing feature of quantum field theory is the anomaly [3][4][5][6]. A particularly interesting anomaly is the global anomaly, which emerges after promotion of a global symmetry to gauge symmetry [6]. As it is the case with many connections between quantum field theory and condensed-matter physics, investigation of the consequences of the anomaly in quantum field theory in a condensed-matter physics context has been quite fruitful, including the discovery of the "Chern insulator" [7]. Nevertheless, the exact correspondence in many concrete cases is not yet understood.An anomaly in a field theory may imply that the field theory cannot be realized in a condensed-matter system or a lattice model in the same dimensions. A renowned case is the impossibility of realization of a chiral fermion with (noninteracting) fermions on a lattice, known as the Nielsen-Ninomiya theorem [8,9], which is deeply related to the chiral anomaly, a representative example of the quantum anomaly. Nevertheless, such a field the-

Field-induced dimer orders in quantum spin chains

“…Quantum spin-1/2 chains do not have a unique gapped ground state in the presence of the time-reversal symmetry unless either the U(1) spin-rotation symmetry or the translation symmetry is broken [1][2][3]. For example, the spin-1/2 Heisenberg antiferromagnetic (HAFM) chain has a unique gapless ground state called the Tomonaga-Luttinger (TL) liquid state [4].…”

Anyonic Chains, Topological Defects, and Conformal Field Theory

“…In this case, the full phase diagram is unknown; however, it was recently argued that in the AF case, no matter how complicated the interactions are, if the microscopic Hamiltonian is translation invariant and the chain is gapless in the thermodynamic limit then it is generically described by an su(2) 1 WZW theory [27].…”

“…Our above discussion also proves the following corollary Corollary 4.6. Consider a unitary CFT, T out , arising from some modular input fusion category via the thermodynamic limit of an anyonic chain (satisfying the conditions discussed below (3.2) 27 ) be isomorphic to C out .…”

Global anomaly matching in the higher-dimensional CPN−1 model

“…quantum system under consideration has an 't Hooft anomaly, it constrains possible lowenergy dynamics due to the 't Hooft anomaly matching [1][2][3]. Recently, new 't Hooft anomalies involving generalized global symmetries such as discrete symmetries [4] and higher-form symmetries [5] are found and applied to a large variety of systems both in condensed matter physics [4,[6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22] and high-energy physics [23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38]. In these applications, the global anomaly induced by a large gauge transformation and a discrete symmetry transformation often plays a central role rather than the perturbative anomaly (such as the chiral anomaly) induced by an infinitesimal transformation (see Refs.…”

“…Quantum field theoretical approach to low-dimensional spin systems provides one representative ground where 't Hooft anomalies play a pivotal role in understanding their possible low-energy behaviors such as properties of their ground states and energy spectra [11,[16][17][18][19][20][21]. One can see this because the 't Hooft anomaly can be regarded as an avatar of the Lieb-Shultz-Mattis (LSM) theorem for the lattice model [43][44][45][46] in the continuum field theory [15,17] (see also Refs.…”