Frustrated spin systems generically suffer from the negative sign problem inherent in Monte Carlo methods. Since the severity of this problem is formulation dependent, optimization strategies can be put forward. We introduce a phase pinning approach in the realm of the auxiliary field quantum Monte Carlo algorithm. If we can find an antiunitary operator that commutes with the one-body Hamiltonian coupled to the auxiliary field, then the phase of the action is pinned to 0 and $\ensuremath{\pi}$. For generalized Kitaev models, we can successfully apply this strategy and observe a remarkable improvement of the average sign. We use this method to study the thermodynamical and dynamical properties of the Kitaev-Heisenberg model down to temperatures corresponding to half of the exchange coupling constant. Our dynamical data reveal finite temperature properties of ordered and spin-liquid phases inherent in this model.
The long-sought quantum spin liquid is a quantum-entangled magnetic state leading to the fractionalization of spin degrees of freedom. Quasiparticles emergent from the fractionalization affect not only the ground-state properties but also thermodynamic behavior in a peculiar manner. Here we investigate how the spin dynamics evolves from the high-temperature paramagnet to the quantum spin-liquid ground state, for the Kitaev spin model describing the fractionalization into itinerant matter fermions and localized
Z2 fluxes. Beyond the previous studies [J. Yoshitake, J. Nasu, and Y. Motome, Phys. Rev. Lett. 117, 157203 (2016); J. Yoshitake, J. Nasu, Y. Kato, and Y. Motome, Phys. Rev. B 96, 024438 (2017)], in which the mean-field nature of the cluster dynamical mean-field theory prevented us from studying low-temperature properties, we develop a numerical technique by applying the continuous-time quantum Monte Carlo (CTQMC) method to statistical samples generated by the quantum Monte Carlo (QMC) method in a Majorana fermion representation. This QMC+CTQMC method is fully unbiased and enables us to investigate the low-temperature spin dynamics dominated by thermally excited
Z2 fluxes, including the unconventional phase transition caused by
Z2 flux loops in three dimensions, which was unreachable by the previous methods. We apply this technique to the Kitaev model in both two and three dimensions. Our results clearly distinguish two cases: while the dynamics changes smoothly through the crossover in the two-dimensional honeycomb case, it exhibits singular behaviors at the phase transition in the three-dimensional hyperhoneycomb case. We show that the low-temperature spin dynamics is a sensitive probe for thermally fluctuating
Z2 fluxes that behave very differently between two and three dimensions.
Determinant quantum Monte Carlo (DQMC), formulated in complex-fermion representation, has played a key role in studying strongly-correlated fermion systems. However, its applicability is limited due to the requirement of particle-number conservation after Hubbard-Stratonovich transformation. In going beyond the conventional DQMC, one encouraging development occurred when Majorana fermions were introduced for QMC [1,2]. But in previous Majorana-based QMC, Boltzmann weight is determined often with a sign ambiguity. Here we successfully resolved this ambiguity by deriving a closed-form Pfaffian formula for the weight, enabling efficient calculation of the weight with its sign in polynomial time. We call it ''Pfaffian quantum Monte Carlo'' (PfQMC), which can be applied to generic interacting fermion models. We have successfully employed PfQMC to explore how robust Majorana edge modes in Kitaev chain are against strong interactions. By offering greater flexibility, PfQMC can potentially enhance existing sign-mitigating and approximation methods and help address challenging issues such as the ground-state properties of the doped Hubbard model.