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arXiv:2009.05315 · Journal of Mathematical Analysis and Applications

Classical dynamics generated by long-range interactions for lattice fermions and quantum spins

J.-B. Bru, W. de Siqueira Pedra

Published Jul 23, 2020• 9 citations
arXivHTMLPDF
DOI
10.1016/j.jmaa.2020.124434

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We study the emergence of magnetic order in localized spins that interact solely through their coupling to a Kitaev-type spin liquid. Using three toy models – the Kitaev model, the Yao–Lee model, and a square-lattice generalization of the Kitaev model – we calculate the effective exchange Hamiltonians mediated by the fractionalized excitations of these spin liquids. This setup is analogous to a Kondo lattice model, where conduction electrons are replaced by itinerant Majorana fermions. In the Kitaev model, our results show that the lowest-order perturbation theory generates short-range interactions with modified couplings and extending to sixth order introduces longer-range interactions while preserving the quantum spin-liquid ground state . Models involving more Majorana flavors on honeycomb and square lattices exhibit more complex behavior. The honeycomb Yao–Lee model with three flavors of itinerant Majorana fermions generates long-range RKKY-type interactions, leading to antiferromagnetic order and partial gapping of the Majorana fermion spectrum. In contrast, the square-lattice model produces a combination of anisotropic short- and long-range interactions, which can give rise to either a dimerized quantum paramagnetic state or an Ising antiferromagnet, depending on the parameters. These results illustrate the rich variety of magnetic orders that can be mediated by Kitaev-type spin liquids.
We introduce the fermionized time-dependent Hartree-Fock (fTDHF), a real-time quantum dynamics method for spin-1/2 Hamiltonians following their mapping to fermions via the Jordan-Wigner transformation. fTDHF is formally equivalent to exact dynamics in the case of free fermions and can efficiently handle non-local string operators arising from long-range interactions via transition matrix elements between non-orthogonal Slater determinants. We show that the fTDHF method can be implemented on a classical computer with a cost that scales polynomially with system size, and linearly with the time steps. We benchmark fTDHF against exact dynamics on three separate spin-1/2 models, representing adiabatic preparation of states with long-range correlations, disorder-driven observation of many-body localization, and particle production in the Schwinger model. For each of these systems, fTDHF is shown to reproduce the qualitative dynamics generated by the exact evolutions, while maintaining a simple physical picture due to its mean-field nature.