Non-abelian bosonization in two and three spatial dimensions and applications Yen-Ta Huanga,b, Dung-Hai Leea,b aDepartment of Physics, University of California, Berkeley, California 94720, USA bMaterials Sciences Division, Lawrence Berkeley National Laboratories, Berkeley, California 94720, USA Abstract In this paper, we generalize Witten's non-abelian bosonization in (1+1)-D to two and three spatial dimensions. Our theory applies to fermions with relativistic dispersion. The bosonized theories are non-linear sigma models with level-1 Wess-Zumino-Witten terms. We apply the bosonization results to the SU(2) gauge theory of the π-flux phase, critical spin liquids in 1,2,3 spatial dimensions, and twisted bilayer graphene. Contents Introduction 6 I Bosonization 7 1 The idea 7 2 Emergent symmetries of the massless fermion theory 8 2.1 Complex class . .8 2.2 Real class . .9 3 Mass terms and mass manifolds 10 3.1 Complex class . 11 3.2 Real class . 12 4 The symmetry anomalies of the fermionic theories 13 4.1 The 't Hooft anomaly of continuous symmetry . 13 arXiv:2104.07015v2 [cond-mat.str-el] 23 Apr 2021 4.2 A heuristic way to determine the 't Hooft anomaly . 15 4.3 Discrete global symmetry anomaly . 16 Email addresses: [email protected] (Yen-Ta Huang), [email protected] (Dung-Hai Lee) Preprint submitted to Elsevier April 27, 2021 5 Breaking the emergent symmetry by the mass terms 19 5.1 Complex class . 19 5.2 Real class . 19 6 Restoring the emergent symmetries 21 7 The conditions for the effective theory being bosonic 22 8 Fermion integration 22 8.1 Complex class . 22 8.2 Real class . 25 9 Non-linear sigma models in (2 + 1)-D and (3 + 1)-D 25 9.1 Complex class in (2 + 1)-D . 26 9.2 Real class in (2 + 1)-D . 27 9.3 Complex class in (3 + 1)-D . 28 9.4 Real class in (3 + 1)-D . 28 9.5 The value of the stiffness constant and the phases of non-linear sigma models . 29 10 Non-linear sigma models as the effective theories of interacting fermion models 30 11 Global symmetries of the non-linear sigma models 32 12 The symmetry anomalies of the nonlinear sigma models 33 12.1 Gauging the non-linear sigma models and the 't Hooft anomalies 34 12.2 Discrete symmetry anomalies . 35 13 Soliton of the non-linear sigma models and the Wess-Zumino- Witten terms 37 13.1 Soliton classification . 37 13.2 Soliton charge and the conserved U(1) charge Q ......... 38 13.3 Statistics of soliton . 39 14 A summary of part I 39 II Applications 40 15 The SU(2) gauge theory of the π-flux phase of the half-filled Hubbard model 40 15.1 The \spinon" representation of the spin operator . 40 15.2 The π-flux phase mean-field theory and the SU(2) gauge fluctu- ations . 42 15.3 Antiferromagnet, Valence bond solid, and the “deconfined” quan- tum crtical point . 46 2 16 The critical spin liquid of \bipartite Mott insulators" in D = 1 + 1; 2 + 1 and 3 + 1. 48 16.1 (1+1)-D . 49 16.1.1 The analog of the π-flux phase . 49 16.1.2 The charge-SU(2) confinement . 50 16.2 (2 + 1)-D . 51 16.2.1 The analog of the π-flux phase . 51 16.2.2 The charge-SU(2) confinement . 53 16.3 (3 + 1)-D . 54 16.3.1 The analog of the π-flux phase . 54 16.3.2 The charge-SU(2) confinement . 57 16.3.3 Gapping out the U(1) mode . 58 17 Twisted bi-layer graphene 60 17.1 Charge neutral point ν =0 ..................... 61 17.2 ν = ±1; ±2; ±3 ............................ 63 Conclusions 66 Acknowledgement 67 Appendices 67 A The emergent symmetries for (2 + 1) and (3 + 1)-D 67 A.1 Complex class in (2 + 1)-D . 67 A.2 Real class in (2 + 1)-D . 68 A.3 Complex class in (3 + 1)-D . 68 A.4 Real class in (3 + 1)-D . 69 B The mass manifolds, homotopy groups and symmetry transfor- mations 70 B.1 Complex class in (2 + 1)-D . 70 B.2 Real class in (2 + 1)-D . 71 B.3 Complex class in (3 + 1)-D . 73 B.4 Real class in (3 + 1)-D . 74 C The anomalies of the fermion theories 76 C.1 The choice of discrete symmetry generators . 76 C.2 Complex class in (1 + 1)-D . 76 C.3 Real class in (1 + 1)-D . 77 C.4 Complex class in (2 + 1)-D . 78 C.5 Real class in (2 + 1)-D . 79 C.6 Complex class in (3 + 1)-D . 79 C.7 Real class in (3 + 1)-D . 80 3 D Fermion integration 80 D.1 Integrating out real versus complex fermions . 81 D.2 Integrating out complex fermions . 82 D.2.1 The stiffness term . 84 D.2.2 The WZW (topological) term . 85 D.3 The fermion integration results for sufficiently large n so that the WZW term is stabilized . 86 D.3.1 Complex class in (1 + 1)-D . 86 D.3.2 Real class in (1 + 1)-D . 88 D.3.3 Complex class in (2 + 1)-D . 89 D.3.4 Real class in (2 + 1)-D . 90 D.3.5 Complex class in (3 + 1)-D . 91 D.3.6 Real class in (3 + 1)-D . 93 D.4 The less relevant real terms originate from Eq.(118) and Eq.(119) 94 D.4.1 (1 + 1)-D . 94 D.4.2 (2 + 1)-D . 95 D.4.3 (3 + 1)-D . 96 E Emergent symmetries of the nonlinear sigma models 100 E.1 Complex class in (2 + 1)-D . 100 E.2 Complex class in (3 + 1)-D . 102 F Anomalies of the nonlinear sigma models 104 F.1 The ('t Hooft) anomalies associated with continuous symmetries 104 F.1.1 Complex class in (1 + 1)-D . 104 F.1.2 Real class in (1 + 1)-D . 105 F.1.3 Complex class in (2 + 1)-D . 106 F.1.4 Real class in (2 + 1)-D . 109 F.1.5 Complex class in (3 + 1)-D . 110 F.1.6 Real class in (3 + 1)-D . 110 F.2 Anomalies with respect to the discrete groups . 112 G Soliton's statistics 112 G.1 Complex class in (2 + 1)-D . 113 G.2 Real class in (2 + 1)-D . 114 G.2.1 How to test whether a proposed Z2 soliton is trivial or not 115 G.2.2 The Berry phase of a self-rotating Z2 soliton . 119 G.3 Complex class in (3 + 1)-D . 119 G.4 Real class in (3 + 1)-D . 121 H Bosonization for small flavor number 122 H.1 Complex class in 2 + 1 D with n =2................ 122 H.2 Mass manifold enlargement . 123 H.3 The derivation of the Hopf term . 123 H.4 Gauging small n non-linear sigma models . 125 4 I Massless fermions as the boundary of bulk topological insula- tors/superconductors 126 I.1 The Z classification . 126 I.2 Construction of the bulk SPT . 127 J The decoupling of the charge-SU(2) gauge field from the low energy non-linear sigma model after confinement 128 K The WZW term in the (3 + 1)-D real class non-linear sigma model 129 5 Introduction Bosonization in (1 + 1)-D has been a very useful theoretical tool. It allows one to map a theory, where the fundamental degrees of freedom are fermionic, to a theory with bosonic degrees of freedom. Often, things that can be seen easily in one picture are difficult to see in the other. The best-known bosonization is the abelian bosonization [1, 2, 3, 4], where fermions are solitons in the Bose field. A shortcoming of the abelian bosonization, when fermions have flavor (e.g., spin) degrees of freedom, is that the flavor symmetries are hidden. This problem was solved by Witten's non-abelian bosonization [5]. In this paper we generalize Witten's non-abelian bosonization to (2 + 1) and (3 + 1) space-time dimensions. The limitation of our theory is that it only applies to fermions with relativistic dispersion. (However, we do not restrict the Fermi velocity to be the speed of light.) In the absence of a mass gap, such theories have Dirac-like dispersion rela- tion. In one space dimension, massless fermions are generically relativistic at low energies. In two and three space dimensions, relativistic massless fermions have been discovered in many experimental condensed matter systems. Examples in- clude graphene and twisted bilayer graphene, Dirac and Weyl semi-metal,...etc. Moreover, relativistic massless fermions can appear in the mean-field theory of strongly correlated systems. Such theory serves as the starting point of a more rigorous treatment. For example, the \spinon π-flux phase" mean-field theory sets the stage for a gauge theory description of the Mott insulating state of the half-filled Hubbard model. Another important area where relativistic massless fermions appear is at the boundary of topological insulators or superconductors, which are simple ex- amples of symmetry-protected topological (SPT) phases. The classification of topological insulator/superconductor[6, 7] can be viewed as asking how many copies of the massless fermion theories on the boundary are required to couple together before a symmetry-allowed mass term emerges. The paper contains two major parts: I. bosonization and II. applications. Each of them contains several sections, namely, 14 sections in part I and 3 sections in part II.