Standard-Basis Operator Method in the Green's-Function Technique of Many-Body Systems with an Application to Ferromagnetism

Abstract

A new formalism, using standard-basis matrix operators, is presented for the study of the collective excitations and the thermodynamic properties of an ensemble of identical interacting quantum-mechanical systems subsequently called "ions, " each ion having discrete energy levels. A model Hamiltonian is formulated in terms of these operators. The Hamiltonian contains terms which express the interaction of the ion with the crystal field and the external fields as well as terms which arise from the mutual interaction of ions. Using the doubletime temperature-dependent Green's-function technique, the equations of motion of the Green's functions of standard-basis operators are developed in the random-phase decoupling approximation.It is demonstrated that the temperature-dependent correlation functions of standard-basis operators, which are obtained from associated spectral Green's functions, lead to a set of equations that can be solved for the occupation probabilities of the singleion energy levels. Hence, one can calculate the thermal-average expectation value of any quantum-mechanical operator representing a microscopic observable of a single ion, or a pair of ions (correlations). An important feature of the standard-basis matrix-operator formalism is that the single-ion terms, such as crystal field, molecular field, or external fields, are always treated exactly in any Green's-function decoupling scheme. In contrast, Green's-function methods which use angular momentum operators often necessarily treat single-ion terms in a decoupling approximation.As an example, the general standard-basis operator theory is applied to the Heisenberg ferromagnet in the presence of uniaxial singleion crystal-field anisotropy, which has received extensive theoretical attention previously, with widely varying results.

Quantum melting of magnetic long-range order near orbital degeneracy: Classical phases and Gaussian fluctuations

“…The local correlation functions which renormalize the order parameter in Eq. ͑5.1͒ are determined in the standard way, 71 FIG. 12.…”

Dynamics of the two-dimensionalS=12dimer system(C5Hong1, Gvasaliya2, Herringer3 et al. 2011Phys. Rev. B100120View full textAdd to dashboardCite

“…We can treat the weak inter-dimer coupling J ′ at the Random Phase Approximation (RPA) level. 19 This method has been applied to numerous spin systems, In application to 5FAP-CuCl 4 , we can treat two decoupled dimers in a unit cell as the basic spin cluster (tetramer). These tetramers will then form a Bravais lattice.…”

Towards Spin-Wave Like Approach to the J-J' Heisenberg Antiferromagnet in Terms of Composite Spins

“…Therefore, it seems that the described MFA for dimers is a good zeroth order approximation, on top of which a low energy effective theory, analogous to spin-wave expansion, can be constructed by treating the dimers as composite four-level systems (abstract spin- operators are rather cumbersome. A more fruitful avenue in this regard is to resort to the standard basis operator (SBO) formalism [7].…”