Numerical evidence of a critical point in the (2+1)D SO(5) nonlinear sigma model with Wess-Zumino-Witten term

Abstract

We develop an optimized continuous-field quantum Monte Carlo (QMC) algorithm to investigate the SO(5) nonlinear sigma model with a Wess-Zumino-Witten term, which describes half-filled Dirac fermions in 2+1 space-time dimensions akin to graphene and Yukawa coupled to a quintuplet of compatible mass terms. To regularize the theory, we project onto the lowest Landau level for both spherical and torus geometries. Our algorithm reduces the computational complexity to $O(\beta N_{\mathbf{q}} N_\phi^2)$, yielding a speedup of a factor of $N_\phi$ (the number of magnetic fluxes, i.e., system size) relative to prior works [1-3]. This advance enables us to simulate system sizes up to $N_\phi=140$ on torus and $N_\phi=49$ on sphere, far exceeding the maximum sizes accessed, and to map out the universal phase diagram of the model on both geometries. Most notably, we identify and characterize a critical point that separates an SO(5)-broken ordered phase at small coupling from an SO(5)-symmetric disordered phase at large coupling. The critical point becomes multicritical upon the inclusion of terms that break the SO(5) symmetry down to $\mathrm{U}(1) \times \mathrm{SU}(2)$, relevant for the deconfined phase transition between N\'eel antiferromagnetic and valence-bond-solid orders in quantum magnets. While the precise nature of the disordered phase in the thermodynamic limit remains to be determined, we argue that it is neither conformal nor trivially gapped, akin to a chiral quantum spin liquid with a small gap. Our finding of a multicritical point in the phase diagram of the SO(5) nonlinear sigma model with Wess-Zumino-Witten term resolves the long-standing open question of its global structure, and our QMC framework opens a new avenue for systematic studies of projected Hamiltonians, ranging from correlated flat bands to fractional quantum (anomalous) Hall systems.