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Bulletin of the American Physical Society

Assessing approximations to spin dynamics using a geometric representation of spin correlations

Rick Mukherjee, Kenneth Wang, Tony Mirasola, Kaden R. A. Hazzard

Published Mar 17, 2017• 0 citations
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arXiv:cond-mat/0605133•Physical Review B•May 21, 2007•20 citations
We compare the continuous and discrete truncated Wigner approximations - TWA and dTWA, respectively - of various spin models' dynamics to exact analytical and numerical solutions. We account for all components of spin-spin correlations on equal footing, facilitated by a recently introduced geometric correlation matrix visualization (CMV) technique. We find that the two approximations capture different aspects of the dynamics of correlations: dTWA captures periodic partial revivals of spin-spin correlations missed in the TWA; but at least for some conditions, TWA better captures the directions in which spins have correlations. We also find that at modestly short times, the dominant error in both approximations is to substantially suppress spin correlations along one direction. In addition to assessing the accuracy of approximations, our comparisons illustrate the utility of CMVs as an intuitive visual tool to understand the full complexity of spin correlations.
By revisiting the path-integral formulation of the Hubbard model, we propose a theoretical approach based on a semiclassical approximation employing an unconventional coherent-state representation. Within this framework, a subset of the dynamical variables is treated as static, yielding a nonperturbative scheme that is applicable at finite temperature, incorporates intersite correlations, and can be naturally extended to multiorbital systems. We assess the validity of the approximation by comparing its results with exact solutions for one- and two-site systems, focusing in particular on the particle number, double occupancy, hopping amplitude, and spin correlations, and find that the present approach qualitatively reproduces the exact behavior. Quantitatively, deviations arise, which is associated with the continuum (non-discretized) character of the underlying density of states. Furthermore, we derive the exact transformation associated with the coherent-state construction, thereby providing additional insight into the representation of the Hubbard model.
Quantum Monte Carlo and semiclassical methods are used to solve two- and four-site cluster dynamical mean-field approximations to the square-lattice Hubbard model at half filling and strong coupling. The dyanmical cluster approximation, cluster dynamical mean-field theory, and fictive-impurity approaches are compared. The energy, spin correlation function, phase boundary, and electron spectral function are computed and compared to available exact results. The comparision permits a quantitative assessment of the ability of the different methods to capture the effects of intersite spin correlations. Two real-space methods and one momentum-space representation are investigated. One of the two real-space methods is found to be significantly worse: in it, convergence to the correct results is found to be slow and, for the spectral function, nonuniform in frequency, with unphysical midgap states appearing. Analytical arguments are presented showing that the discrepancy arises because the method does not respect the pole structure of the self-energy of the insulator. Of the other two methods, the momentum-space representation is found to provide the better approximation to the intersite terms in the energy but neither approximation is particularly acccurate and the convergence of the momentum-space method is not uniform. A few remarks on numerical methods are made.