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Physical review. B, Condensed matter

Autocorrelation functions for Hermitian many-body systems: Necessary conditions

M. Howard Lee

Published Apr 1, 1993• 5 citations
Publisher

Abstract

Brown [Phys. Rev. B 45, 10 809 (1991)] points out an error made in an article on the dynamics of a Heisenberg model by Wysin and Bishop [Phys. Rev. B 42, 810 (1990)]. After correcting the error, he arrives at the conclusion that the dynamic structure in this model cannot be in the form of Lorentzians. We show that this conclusion represents an orthogonality condition of certain basis vectors that span Hilbert space. Hence, it is applicable to all Hermitian many-body models. Also, a simple model is provided to illustrate both the orthogonality and transformation ideas. They can be very useful in the dynamical studies of Hermitian many-particle systems.

DOI
10.1103/physrevb.47.8293

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Brown [Phys. Rev. B 45, 10 809 (1991)] points out an error made in an article on the dynamics of a Heisenberg model by Wysin and Bishop [Phys. Rev. B 42, 810 (1990)]. After correcting the error, he arrives at the conclusion that the dynamic structure in this model cannot be in the form of Lorentzians. We show that this conclusion represents an orthogonality condition of certain basis vectors that span Hilbert space. Hence, it is applicable to all Hermitian many-body models. Also, a simple model is provided to illustrate both the orthogonality and transformation ideas. They can be very useful in the dynamical studies of Hermitian many-particle systems.
The derivation by Wysin and Bishop [Phys. Rev. B 42, 810 (1990)] of a general result for the dynamic structure function, S(q,t), of classical isotropic Heisenberg systems is shown to be incorrect. A correct quantum-mechanical proof of a general theorem, which is valid for all translationally invariant systems and which produces the result of Wysin and Bishop as a special classical case, is provided. It is also pointed out that another condition on S(q,t) for rotationally invariant systems such as the isotropic Heisenberg model does not appear to be satisfied by the results of Wysin and Bishop.
It is proven that a necessary and sufficient condition for an nnn-dimensional Hermitian matrix $\ensuremath{\rho}$ to be positive definite is that it be expressible in the form $\ensuremath{\rho}=OEO\ifmmode\dagger\else\textdagger\fi{}$, where OOO is a complex orthogonal matrix and EEE is a diagonal matrix with positive elements. This accomplishes a parametrization since OOO has ${n}^{2}\ensuremath{-}n$ real parameters and EEE has nnn of them. The proof is constructive, giving OOO and EEE. It is further shown that the limit forms of this expression yield all the non-negative definite matrices. The parametrization for the polarization matrix of a spin-one particle is given explicitly.
We introduce a formalism for time-dependent correlation functions for systems whose evolutions are governed by non-Hermitian Hamiltonians of general type. It turns out that one can define two different types of time correlation functions. Both these definitions seem to be physically consistent while becoming equivalent only in certain cases. Moreover, when autocorrelation functions are considered, one can introduce another function defined as the relative difference between the two definitions. We conjecture that such a function can be used to assess the positive semidefiniteness of the density operator without computing its eigenvalues. We illustrate these points by studying analytically a number of models with two energy levels.
Lee [Phys. Rev. B 47, 8293 (1993)] has shown that the odd-numbered derivatives of the Kubo autocorrelation function vanish at t=0. We show that this condition is based on a more general property of nondiagonal Kubo correlation functions. This general property provides that certain functional forms (e.g., simple exponential decay) are not admissible for any symmetric or antisymmetric Kubo correlation function in a Hermitian many-body system. Lee's result emerges as a special case of this result. Applications to translationally invariant systems and systems with rotational symmetries are also demonstrated.