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Register a new userA quantum Monte Carlo method on asymptotic Lefschetz thimbles for quantum spin systems: An application to the Kitaev model in a magnetic field
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The quantum Monte Carlo method on asymptotic Lefschetz thimbles is a numerical algorithm devised specifically for alleviation of the sign problem appearing in the simulations of quantum many-body systems. In this method, the sign problem is alleviated by shifting the integration domain for the auxiliary fields, appearing for example in the conventional determinant quantum Monte Carlo method, from real space to an appropriate manifold in complex space. Here we extend this method to quantum spin models with generic two-spin interactions, by using the Hubbard-Stratonovich transformation to decouple the exchange interactions and the Popov-Fedotov transformation to map the quantum spins to complex fermions. As a demonstration, we apply the method to the Kitaev model in a magnetic field whose ground state is predicted to deliver a topological quantum spin liquid with non-Abelian anyonic excitations. To illustrate how the sign problem is alleviated in this method, we visualize the asymptotic Lefschetz thimbles in complex space, together with the saddle points and the zeros of the fermion determinant. We benchmark our method in the low-temperature region in a magnetic field and show that the sign of the action is recovered considerably and unbiased numerical results are obtained with sufficient precision.
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Frustrated spin systems generically suffer from the negative sign problem inherent to 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 anti-unitary operator that commutes with the one body Hamiltonian coupled to the auxiliary field, then the phase of the action is pinned to $0$ and $pi$. 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 thermodynamical and dynamical properties of the Kitaev-Heisenberg model down to temperatures corresponding to half of the exchange coupling constant. Our dynamical data reveals finite temperature properties of ordered and spin-liquid phases inherent to this model.
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We investigate the impact of two types of disorder, bond randomness and site dilution, on the spin dynamics in the Kitaev model on a honeycomb lattice. The ground state of this model is a canonical quantum spin liquid with spin fractionalization into two types of quasiparticles, itinerant Majorana fermions and localized fluxes. Using unbiased quantum Monte Carlo simulations, we calculate the temperature evolution of the dynamical spin structure factor, the magnetic susceptibility, and the NMR relaxation rate. In the dynamical spin structure factor, we find that the two types of disorder affect seriously the low-energy peak dominantly originating from the flux excitations, rather than the high-energy continuum from the Majorana excitations, in a different way: The bond randomness softens the peak to the lower energy with broadening, whereas the site dilution smears the peak and in addition develops the other sharp peaks inside the spin gap including the zero energy. We show that the zero-energy spin excitations, which originate from the Majorana zero modes induced around the site vacancies, survive up to the temperature comparable to the energy scale of the Kitaev interaction. For the bond randomness, the low-temperature susceptibility does not show any qualitative change against the weak disorder, but it changes to divergent behavior while increasing the strength of disorder. Similar distinct behaviors for the weak and strong disorder are observed also in the NMR relaxation rate; an exponential decay changes into a power-law decay. In contrast, for the site dilution, we find no such crossover; divergent behavior in the susceptibility and a power-law decay in the NMR relaxation rate appear immediately with the introduction of the site dilution. We discuss the relevance of our results to experiments for the Kitaev candidate materials with disorders.
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