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International Journal of Quantum Chemistry

Symmetric‐group algebraic variational solutions for Heisenberg models at finite temperature

Douglas J. Klein, W. A. Seitz

Published Jan 5, 1992• 6 citations
Publisher

Abstract

Abstract The Heisenberg spin Hamiltonian for a collection of N spin‐1/2 sites is viewed, as favored by Professor Matsen, to be an element of the group algebra of the symmetric group 𝒮N. Several computationally tractable, variational group–algebraic approximations for the finite‐temperature density matrix are made so as to minimize the Gibb's free–energy functional. Relations to previous quite differently motivated approximations are identified, though improvements are noted with the present approach.

DOI
10.1002/qua.560410107

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Abstract The Heisenberg spin Hamiltonian for a collection of N spin‐1/2 sites is viewed, as favored by Professor Matsen, to be an element of the group algebra of the symmetric group 𝒮N. Several computationally tractable, variational group–algebraic approximations for the finite‐temperature density matrix are made so as to minimize the Gibb's free–energy functional. Relations to previous quite differently motivated approximations are identified, though improvements are noted with the present approach.
We show how the partition functions for finite clusters with spin-\textonehalf{} Heisenberg interactions may be computed efficiently and generally to any desired number of powers in reciprocal temperature. As an example, we have expanded the zero-magnetic-field free energy to the twenty-first power for the linear Heisenberg model and for nonzero magnetic field give an expression good through the tenth power. We introduce the concept of the two-point Pad\'e approximant and use it to analyze the energy for the linear Heisenberg model.
We derive a variational cluster approximation for Heisenberg spin systems at finite temperature based on the ideas of the self-energy functional theory by Potthoff for fermionic and bosonic systems with local interactions. Partitioning the real system into a set of clusters, we find an analytical expression for the auxiliary free energy, depending on a set of variational parameters defined on the cluster, whose stationary points provide approximate solutions from which the thermodynamics of spin models can be obtained. We explicitly describe the technical details of how to evaluate the free energy for finite clusters and remark on specific problems and possible limitations of the method. To test the approximation we apply it to the antiferromagnetic spin chain and compare the results for varying cluster sizes and choices of variational parameters with the exact Bethe ansatz solution.
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Within the tensor network framework, the (positive) thermal density operator can be approximated by a double layer of infinite Projected Entangled Pair Operators (iPEPO) coupled via ancilla degrees of freedom.To investigate the thermal properties of the spin-1/2 Heisenberg model on the square lattice, we introduce a family of fully spin-SU(2) and lattice-C 4v symmetric on-site tensors (of bond dimensions D = 4 or D = 7) and a plaquette-based Trotter-Suzuki decomposition of the imaginary-time evolution operator.A variational optimization is performed on the plaquettes, using a full (for D = 4) or simple (for D = 7) environment obtained from the single-site Corner Transfer Matrix Renormalization Group fixed point.The method is benchmarked by a comparison to quantum Monte Carlo in the thermodynamic limit.Although the iPEPO spin correlation length starts to deviate from the exact exponential growth for inverse-temperature β 2, the behavior of various observables turns out to be quite accurate once plotted w.r.t the inverse correlation length.We also find that a direct T = 0 variational energy optimization provides results in full agreement with the β → ∞ limit of finite-temperature data, hence validating the imaginary-time evolution procedure.Extension of the method to frustrated models is described and preliminary results are shown.
Variational methods are of fundamental importance and widely used in theoretical physics, especially for strongly interacting systems. In this work, we present a set of variational equations of state (VES) for pure states of an interacting quantum Hamiltonian. The VES can be expressed in terms of the variation of the density operators or static correlation functions. We derive the algebraic relationship between a known pure state density matrix and its variation, and obtain the VES by applying this relation to the averaged Heisenberg-equations-of-motion for the exact density matrix. Additionally, we provide a direct expression of the VES in terms of correlation functions to make it computable. We present three nontrivial applications of the VES: a perturbation calculation of correlation functions of the transverse field Ising model in arbitrary spatial dimensions, a study of a longitudinal field perturbation to the one-dimensional transverse field Ising model at the critical point and variational calculation of magnetization and ground state energy of the two-dimensional spin-1/2 Heisenberg model on a square lattice. For the second one, our results not only recover the scaling limit, but also indicate the possibility of continuous tuning of the critical exponents by adjusting the longitudinal fields differently from the scaling limit. For the Heisenberg model, we obtained results numerically comparable to established results with simple calculations. The VES approach provides a powerful and versatile tool for studying interacting quantum systems.
Abstract A procedure is developed for obtaining approximate thermodynamic properties of statistical models, based on a well known variational principle for the free energy. The procedure involves the choice of a suitable parametrized trial Hamiltonian which contains some elementary interactions. The free trial Hamiltonian leads to the usual mean‐field approximation. When the trial Hamiltonian includes a certain number of pair interactions only, the results are regained of the constant‐coupling approximation for the Ising model if the number of interacting pairs is conveniently chosen by a comparison with the first few terms of the exact high‐temperature expansion. This procedure is simpler than existing approximations and can be easily generalized. Besides treating the Ising model, some applications are presented to the spin‐1/2 antiferromagnetic Heisenberg model, and to the spin‐1 Heisenberg model with uniaxial anisotropy.