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Physical review. B, Condensed matter

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

Shun-Qing Shen

Published May 15, 1999• 0 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.59.12689

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

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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.
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 show that Shi's conclusion about the singlet pairing in the Hubbard model at strong-coupling limit [Phys. Rev. B 51, 16 619 (1995)] is wrong. The singlet pairing is related to many-particle correlated pairings. There generally is no exact identity among the singlet pairing functions. We give an intuitive argument that the nearest-neighbor ${\mathit{d}}_{{\mathit{x}}^{2}\mathrm{\ensuremath{-}}{\mathit{y}}^{2}}$-wave pairing is the predominant one in the doped Hubbard model. \textcopyright{} 1996 The American Physical Society.
Minimum-time quantum control protocols can be obtained from the quantum brachistochrone formalism [Carlini, Hosoya, Koike, and Okudaira, Phys. Rev. Lett. 96, 06053, (2006)]. We point out that the original treatment implicitly applied the variational calculus with fixed boundary conditions. We argue that the genuine quantum brachistochrone problem involves a variational problem with a movable endpoint, contrary to the classical brachistochrone problem. This formulation not only simplifies the derivation of the quantum brachistochrone equation but introduces an additional constraint at the endpoint due to the boundary effect. We present the general solution to the full quantum brachistochrone equation and discuss its main features. Using it, we prove that the speed of evolution under constraints is reduced with respect to the unrestricted case. In addition, we find that solving the quantum brachistochrone equation is closely connected to solving the dynamics of the Lagrange multipliers, which is in general governed by nonlinear differential equations. Their numerical integration allows generating time-extremal trajectories. Furthermore, when the restricted operators form a closed subalgebra, the Lagrange multipliers become constant and the optimal Hamiltonian takes a concise form. The new class of analytically solvable models for the quantum brachistochrone problem opens up the possibility of applying it to many-body quantum systems, exploring notions related to geometry such as quantum speed limits, and advancing significantly the quantum state and gate preparation for quantum information processing.