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arXiv:cond-mat/0605526 · Physical Review B

Quantum Monte Carlo study for multiorbital systems with preserved spin and orbital rotational symmetries

志朗 酒井, Ryotaro Arita, Karsten Held, Hideo Aoki

Published Oct 4, 2006• 41 citations
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Abstract

We propose to combine the Trotter decomposition and a series expansion of the partition function for Hund's exchange coupling in a quantum Monte Carlo (QMC) algorithm for multiorbital systems that preserves spin and orbital rotational symmetries. This enables us to treat the Hund's (spin-flip and pair-hopping) terms, which is difficult in the conventional QMC method. To demonstrate this, we first apply the algorithm to study ferromagnetism in the two-orbital Hubbard model within the dynamical mean-field theory (DMFT). The result reveals that the preservation of the SU(2) symmetry in Hund's exchange is important, where the Curie temperature is grossly overestimated when the symmetry is degraded, as is often done, to Ising $({\mathrm{Z}}_{2})$. We then calculate the ${t}_{2g}$ spectral functions of ${\mathrm{Sr}}_{2}{\mathrm{RuO}}_{4}$ by a three-band DMFT calculation with tight-binding parameters taken from the local density approximation with proper rotational symmetry.

DOI
10.1103/physrevb.74.155102

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We propose to combine the Trotter decomposition and a series expansion of the partition function for Hund's exchange coupling in a quantum Monte Carlo (QMC) algorithm for multiorbital systems that preserves spin and orbital rotational symmetries. This enables us to treat the Hund's (spin-flip and pair-hopping) terms, which is difficult in the conventional QMC method. To demonstrate this, we first apply the algorithm to study ferromagnetism in the two-orbital Hubbard model within the dynamical mean-field theory (DMFT). The result reveals that the preservation of the SU(2) symmetry in Hund's exchange is important, where the Curie temperature is grossly overestimated when the symmetry is degraded, as is often done, to Ising (Z2)({\mathrm{Z}}_{2})(Z2​). We then calculate the t2g{t}_{2g}t2g​ spectral functions of Sr2RuO4{\mathrm{Sr}}_{2}{\mathrm{RuO}}_{4}Sr2​RuO4​ by a three-band DMFT calculation with tight-binding parameters taken from the local density approximation with proper rotational symmetry.
We have studied numerically exact auxiliary field quantum Monte Carlo scheme for the spin and orbital rotational invariant Hamiltonian by combining series expansion and Trotter decomposition. We have used algorithm to solve the dynamical field theory equations. We have employed Boltzmann operator with respect to the total interaction. The method significantly relaxes the sign problem and allows in principle for more than two orbitals. The computational effort increases with the perturbation orders appearing in the Mote Carlo samples. The order become very large in multiorbital systems. To remove this difficulty we have used the Hirsch-Fye quantum Monte Carlo and parturbation series expansion quantum Monte Carlo methods. This algorithm enabled us not only to handle three or more orbitals but also to reach much stronger coupling than in Hirsch-Fye quantum Monte Carlo calculations, even for the two orbital Hubbard model.
In order to numerically study electron correlation effects in multiorbital systems, we propose a new type of discrete transformation for the exchange (Hund's coupling) and pair-hopping interactions to be used in the dynamical mean field theory+quantum\text{theory}+\text{quantum}theory+quantum Monte Carlo method. The transformation, which is real and exact, turns out to suppress the sign problem in a wide parameter region including non-half-filled bands. This enables us to obtain the dominant pairing symmetry in the double-orbital Hubbard model, which shows that the spin-triplet, orbital-antisymmetric pairing that exploits Hund's coupling is stable in a wide region of the band filling.
We implement a multiorbital cluster dynamical mean-field theory (DMFT) by improving a sample update algorithm in the continuous-time quantum Monte Carlo method based on the interaction expansion. The proposed sampling scheme for the spin-flip and pair-hopping interactions in the two-orbital systems mitigates the sign problem, giving an efficient way to deal with these interactions. In particular, in the single-site DMFT, we see that the negative signs vanish. We apply the method to the two-dimensional two-orbital Hubbard model at half-filling, where we take into account the short-range spatial correlation effects within a four-site cluster. We show that, compared to the single-site DMFT results, the critical interaction value for the metal-insulator transition decreases and that the effects of the spin-flip and pair-hopping terms are less significant in the parameter region we have studied. The present method provides a firm starting point for the study of intersite correlations in multiorbital systems. It also has a wide applicable scope in terms of realistic calculations in conjunction with density functional theory.
We present a modification of the continuous-time hybridization-expansion quantum Monte Carlo method, which allows one to take into account spin rotational symmetry. At present, studies of real compounds by means of dynamical mean-field theory employ an approximate form of the Coulomb interaction which breaks the rotational symmetry, leading, in particular, to considerable overestimation of the Curie-Weiss temperature. The obtained temperature dependence of magnetic susceptibility and Curie-Weiss temperature for the three-band model are in good agreement with the exact solution. Owing to its computational efficiency, the presented method is suitable for investigation of magnetic properties of real compounds with strong electronic correlations.
A recently introduced one-dimensional three-orbital Hubbard model displays orbital-selective Mott phases with exotic spin arrangements such as spin block states [J. Rincón et al., Phys. Rev. Lett. 112, 106405 (2014)PRLTAO0031-900710.1103/PhysRevLett.112.106405]. In this publication we show that the constrained-path quantum Monte Carlo (CPQMC) technique can accurately reproduce the phase diagram of this multiorbital one-dimensional model, paving the way to future CPQMC studies in systems with more challenging geometries, such as ladders and planes. The success of this approach relies on using the Hartree-Fock technique to prepare the trial states needed in CPQMC. We also study a simplified version of the model where the pair-hopping term is neglected and the Hund coupling is restricted to its Ising component. The corresponding phase diagrams are shown to be only mildly affected by the absence of these technically difficult-to-implement terms. This is confirmed by additional density matrix renormalization group and determinant quantum Monte Carlo calculations carried out for the same simplified model, with the latter displaying only mild fermion sign problems. We conclude that these methods are able to capture quantitatively the rich physics of the several orbital-selective Mott phases (OSMP) displayed by this model, thus enabling computational studies of the OSMP regime in higher dimensions, beyond static or dynamic mean-field approximations.
We present a strategy to alleviate the sign problem in continuous-time quantum Monte Carlo (CTQMC) simulations of the dynamical-mean-field-theory (DMFT) equations for the spin-orbit-coupled multiorbital Hubbard model. We first identify the combinations of rotationally invariant Hund coupling terms present in the relativistic basis which lead to a severe sign problem. Exploiting the fact that the average sign in CTQMC depends on the choice of single-particle basis, we propose a bonding-antibonding basis Vj3/2BA{V}_{j3/2\mathrm{BA}}Vj3/2BA​ which shows an improved average sign compared to the widely used relativistic basis for most parameter sets investigated. We then generalize this procedure by introducing a stochastic optimization algorithm that exploits the space of single-particle bases and show that Vj3/2BA{V}_{j3/2\mathrm{BA}}Vj3/2BA​ is very close to optimal within the parameter space investigated. Our findings enable more efficient DMFT simulations of materials with strong spin-orbit coupling.