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Physical review. B, Condensed matter

Lagrange-multiplier method in correlated-electron systems: Exact diagonalization study

Takashi Yanagisawa

Published Mar 15, 1998• 3 citations
Publisher

Abstract

We investigate the Lagrange-multiplier method which is frequently used in strongly correlated-electron systems to impose the constraints. We assume that the constraint operators commute with the Hamiltonian. It may be implicitly believed that the Lagrange multipliers are determined uniquely due to the variational conditions. We discuss whether this is true or not by using the exact diagonalization method. For the mean-field Hamiltonian which does not commute with the constraint operators, the Lagrange multipliers may be determined uniquely.

DOI
10.1103/physrevb.57.6208

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Comment on “Lagrange-multiplier method in correlated-electron systems: Exact diagonalization study”

Shun-Qing Shen

We discuss the Lagrange-multiplier method in a many-body system, and how to apply the Lagrange multiplier correctly to realize quantum constraint in a Hamiltonian for a quantum system. It is pointed out that the discussion in Yanagisawa's paper [Phys. Rev. B 57, 6208 (1998)] lacks a basic knowledge of mathematics, and that his conclusion is generally wrong.

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Reply to “Comment on ‘Lagrange-multiplier method in correlated-electron systems: Exact diagonalization study’ ”

Takashi Yanagisawa

We reply to the preceding Comment by Sten. We discuss a role of the Lagrange multipliers using an exact diagonalization.Received 12 November 1998DOI:https://doi.org/10.1103/PhysRevB.59.12692©1999 American Physical Society

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Journal of Mathematical Physics•Oct 1, 1980•13 citations
We investigate the Lagrange-multiplier method which is frequently used in strongly correlated-electron systems to impose the constraints. We assume that the constraint operators commute with the Hamiltonian. It may be implicitly believed that the Lagrange multipliers are determined uniquely due to the variational conditions. We discuss whether this is true or not by using the exact diagonalization method. For the mean-field Hamiltonian which does not commute with the constraint operators, the Lagrange multipliers may be determined uniquely.
We propose a method of determining states for correlated fermions in Fock space, which we supplement with the Lagrange-Euler equation for the single-particle wave functions wi(r){{w}_{i}(\mathbf{r})}wi​(r) contained in the microscopic parameters. The equation is derived by treating the ground state energy of the interacting system as a functional of wi(r).{{w}_{i}(\mathbf{r})}.wi​(r). The method is applied to a linear chain up to N=10N=10N=10 atoms with all pair interactions and all hopping integrals included. Renormalized Wannier functions, the parameters, the distribution function, and the dimerization energy, are all discussed as a function of the lattice parameter.
A new approach for calculating spectral density functions of strongly correlated electron systems is proposed within the exact diagonalization method of dynamical mean-field theory (DMFT). This approach is based on the analytic continuation of spectral density function obtained by conventional exact diagonalization method of DMFT and its results are more reasonable in shape. As an example of its application, the Mott transition in a strongly correlated electron system is investigated using this new approach.
The quantization rules obtained by Le Guillou and Ragoucy in the slave particle theories of correlated systems have been applied to study the superconductivity in extended Hubbard and $t\ensuremath{-}J$ models. The quantization scheme has the virtue of treating the local constraint exactly. We compare our numerical results with those obtained in the mean-field approximation for usual slave boson technique. We also discuss the relation of the present approach with method of equation of motion for Hubbard operators. Where possible the obtained results are also compared with experimental data on high-temperature superconductors, as in these materials the essential physics is believed to depend on the presence of strong correlations.
For many years we have advocated a form of quantum mechanics based on the application of sum rule methods (completeness) to the equations of motion and to the commutation relations, i.e., to Heisenberg matrix mechanics. Sporadically we have discussed or alluded to a variational foundation for this method. In this paper we present a series of variational principles applicable to a range of systems from one-dimensional quantum mechanics to quantum fields. The common thread is that the stationary quantity is the trace of the Hamiltonian over Hilbert space (or over a subspace of interest in an approximation) expressed as a functional of matrix elements of the elementary operators of the theory. These parameters are constrained by the kinematical relations of the theory introduced by the method of Lagrange multipliers. For the field theories, variational principles in which matrix elements of the density operators are chosen as fundamental are also developed. A qualitative discussion of applications is presented.