Chern-Simons fermionization approach to two-dimensional quantum magnets: Implications for antiferromagnetic magnons and unconventional quantum phase transitions

Abstract

We develop an approach to describe antiferromagnetic magnons on a bipartite lattice supporting the N\'{e}el state using fractionalized degrees of freedom typically inherent to quantum spin liquids. In particular we consider a long-range magnetically ordered state of interacting two-dimensional quantum spin$-1/2$ models using the Chern-Simons (CS) fermion representation of interacting spins. The interaction leads to Cooper instability and pairing of CS fermions, and to CS superconductivity which spontaneously violates the continuous $\mathrm{U}(1)$ symmetry generating a linearly-dispersing gapless Nambu-Goldstone mode due to phase fluctuations. We evaluate this mode and show that it is in high-precision agreement with magnons of the corresponding N\'{e}el antiferromagnet irrespective to the lattice symmetry. Using the fermion formulation of a system with competing interactions, we show that the frustration gives raise to nontrivial long-range four, six, and higher-leg interaction vertices mediated by the CS gauge field, which are responsible for restoring of the continuous symmetry at sufficiently strong frustration. We identify these new interaction vertices and discuss their implications to unconventional phase transitions. We also apply the proposed theory to a model of anyons that can be tuned continuously from fermions to bosons.