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 spin liquids and topological quantum spin liquids (QSLs): chiral and Z2 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 term. For the transition to a chiral spin liquid, it is the Gross-Neveu interaction (QED3-GN), while for the transitions to Z2 QSLs, it is a superconducting pairing term with general spin and valley structure (generalized QED3-Z2GN). Using the state-operator correspondence, we obtain monopole scaling dimensions to subleading 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 Z2 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 QED3 is also obtained, and we find a subleading contribution of -0.038138(5), which is slightly different from the value -0.0383 first obtained in the literature. The scaling dimension of a QED3-GN monopole with minimal charge is very close to the scaling dimensions of other operators predicted to be equal by a conjectured duality between QED3-GN with 2N=2 flavors and the CP1 model. Additionally, nonminimally charged monopoles with equal charges on both sides of the duality have similar scaling dimensions. By studying the large-q asymptotics of the scaling dimensions in QED3, QED3-GN, and QED3-Z2GN, we verify that the constant O(q0) coefficient precisely matches the universal nonperturbative prediction for CFTs with a global U(1) symmetry. Finally, we identify numerous open questions regarding the fate of monopoles and their hierarchies at transitions to spin liquids and ordered phases.