Anomalous Dimensions of Monopole Operators at the Transitions between Dirac and Topological Spin Liquids

Abstract

Monopole operators are studied in a large family of quantum critical points between Dirac and topological quantum spin liquids (QSLs): chiral and Z$_{2}$ QSLs. These quantum phase transitions are described by conformal field theories (CFTs): quantum electrodynamics in 2+1 dimensions with 2N flavors of two-component massless Dirac fermions and a four-fermion interaction. For the transition to a chiral spin liquid, it is the Gross-Neveu interaction (QED$_{3}$-GN), while for the transitions to Z$_{2}$ QSLs it is a superconducting pairing term with general spin/valley structure (generalized QED$_{3}$-Z$_{2}$GN). Using the state-operator correspondence, we obtain monopole scaling dimensions to sub-leading order in 1/N. For monopoles with a minimal topological charge q=1/2, the scaling dimension is 2N*0.26510 at leading-order, with the quantum correction being 0.118911(7) for the chiral spin liquid, and 0.102846(9) for the simplest Z$_{2}$ case (the expression is also given for a general pairing term). Although these two anomalous dimensions are nearly equal, the underlying quantum fluctuations possess distinct origins. The analogous result in QED$_{3}$ is also obtained and we find a sub-leading contribution of -0.038138(5), which differs slightly from the value first obtained in the literature. The scaling dimension of a QED$_{3}$-GN monopole with minimal charge is very close to the scaling dimensions of other operators predicted to be equal by a conjectured duality between QED$_{3}$-GN with 2N=2 flavors and the CP$^{1}$ model. Additionally, non-minimally charged monopoles on both sides of the duality have similar scaling dimensions. By studying the large-q asymptotics of the scaling dimensions in QED$_{3}$, QED$_{3}$-GN, and QED$_{3}$-Z$_{2}$GN we verify that the constant O(q$^{0}$) coefficient precisely matches the universal non-perturbative prediction for CFTs with a global U(1) symmetry.