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Bulletin of Pure & Applied Sciences- Physics

Trotter decomposition and a series expansion of the partition function for hund's exchange coupling in a quantum monte carlo algorithm

Seema Priyadarshini, Rajiv Mishra, Ruchi Kumari, Shashi Shekhar, Sanjay Kumar Singh, Gyandeo Prasad Tiwari

Published Jan 1, 2013• 0 citations
OpenAlex

Abstract

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.

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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.
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.
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.
By precisely writing down the matrix element of the local Boltzmann operator (e(−τhe^(-{\tau}he(−τh), where h is the paired operator formed by two interacting terms conjugate to each other), we have proposed a new path integral formulation for quantum field theory and developed a corresponding Monte Carlo algorithm. Our formula presents remarkable outcomes such that: 1) the Hubbard-Stratonovich transformation is not necessary, and the auxiliary field is no longer needed, which can improve the computational efficiency. 2) The new formula improves the accuracy of the Suzuki-Trotter decomposition. Our method also provides a potential theoretical framework for analyzing and solving the fermion sign problem. As an example, we studied the one-dimensional half-filled Hubbard model at a finite temperature. The obtained results were in excellent agreement with the known solutions. The new formula and Monte Carlo algorithm could be used in various studies.
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.
A numerically exact continuous-time quantum Monte Carlo algorithm for finite fermionic systems with nonlocal interactions is proposed. The scheme is particularly applicable for general multiband time-dependent correlations, since it does not invoke Hubbard-Stratonovich transformation. The present determinantal grand-canonical method is based on a stochastic series expansion for the partition function in the interaction representation. The results for the Green function and for the time-dependent susceptibility of multiorbital supersymmetric impurity model with a spin-flip interaction are presented.
We present a framework of an auxiliary field quantum Monte Carlo (QMC) method for multi-orbital Hubbard models. Our formulation can be applied to a Hamiltonian which includes terms for on-site Coulomb interaction for both intra- and inter-orbitals, intra-site exchange interaction and energy differences between orbitals. Based on our framework, we point out possible ways to investigate various phase transitions such as metal-insulator, magnetic and orbital order-disorder transitions without the minus sign problem. As an application, a two-band model is investigated by the projection QMC method and the ground state properties of this model are presented.