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.

arXiv comment37 pages

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