Eur. Phys. J. B (2024) 97: 145
https://doi.org/10.1140/epjb/s10051-024-00793-2

Regular Article - Statistical and Nonlinear Physics

XY-VBS phase boundary for the square-lattice $J_1$-$J_2$XXZ model with the ring exchange

Department of Physics, Faculty of Science, Okayama University, 700-8530, Okayama, Japan

^a nisiyama@science.okayama-u.ac.jp

Received: 28 January 2024
Accepted: 20 September 2024
Published online: 3 October 2024

Abstract

The square-lattice $J_1$-$J_2$XXZ model with the ring-exchange interaction K was investigated numerically. As for the hard-core-boson model with the nearest-neighbor hopping $J_1/2$, namely, the $J_1$-KXY model, it has been reported that the ring exchange leads to a variety of exotic phases such as the valence-bond-solid (VBS) phase. In this paper, we extend the parameter space to investigate the phase boundary between the XY (superfluid) and VBS phases. A notable feature is that the phase boundary terminates at the fully frustrated point, $J_2/J_1\to 0.5^{-}$. As a scaling parameter for the multi-criticality, the distance from the multi-critical point $\delta (\ge 0)$ is introduced. To detect the phase transition, we employed the high-order fidelity susceptibility ${\chi }_F^{(3)}$, which is readily evaluated via the exact-diagonalization scheme. As a demonstration, for a fixed value of $\delta$, the XY-VBS criticality was analyzed by the probe ${\chi }_F^{(3)}$. Thereby, with properly scaling $\delta$, the ${\chi }_F^{(3)}$ data were cast into the crossover-scaling formula to determine the multi-criticality.