Magnetic three states of matter: A quantum Monte Carlo study of spin liquids

Abstract

We present thermodynamic phase diagrams showing magnetic analog of "three states of matter," namely, spin liquid, paramagnetic, and magnetically ordered phases, obtained by unbiased quantum Monte Carlo simulations. Our simulations are carried out for Kitaev's toric codes in two and three dimensions, i.e., the simplest realizations of gapped topological $Z_2$ spin liquids, extended by a nearest-neighbor ferromagnetic Ising coupling. We find that the ordered phase borders on the spin liquid by a discontinuous transition line in three dimensions, while it grows continuously from the quantum critical point in two dimensions. In both cases, our results elucidate peculiar proximity effects to the nearby spin liquids in the high-temperature paramagnetic phase, even when the ground state is magnetically ordered. The thorough study of magnetic three states of matter is achieved by introducing the "fictitious vertex" method into the directed loop algorithm. This provides a generic prescription to simulate models with off-diagonal multispin interactions, in which the conventional scheme may suffer from intrinsic ergodicity breakdown as in the present case.