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Quasiclassical Hamiltonians for large-spin systems.

Dmitry A. Garanin, K. Kladko, Peter Fulde

Published Jan 1, 2000• 6 citations
OpenAlex

Abstract

We propose a method for obtaining effective classical Hamiltonians \cal H for many-body quantum spin systems with large spins. This method uses the coherent-state representation of the partition function Z and the cumulant expansion in powers of 1/S. For the quantum Hamiltonian \hat H of a Heisenberg form, the 1/S corrections in \cal H have a non-Heisenberg many-spin form. The effective Hamiltonian \cal H can be treated by methods familiar for classical systems. The non-Heisenberg terms in \cal H may be responsible for such effects as spin-Peierls transition and uplifting of the classical degeneracy by quantum fluctuations.

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We propose a method for obtaining effective classical Hamiltonians \cal H for many-body quantum spin systems with large spins. This method uses the coherent-state representation of the partition function Z and the cumulant expansion in powers of 1/S. For the quantum Hamiltonian \hat H of a Heisenberg form, the 1/S corrections in \cal H have a non-Heisenberg many-spin form. The effective Hamiltonian \cal H can be treated by methods familiar for classical systems. The non-Heisenberg terms in \cal H may be responsible for such effects as spin-Peierls transition and uplifting of the classical degeneracy by quantum fluctuations.
A method is proposed for obtaining a systematic expansion of thermodynamic functions of spin systems with large spin S in powers of 1/ S . It uses the cumulant technique and a coherent-state representation of the partition function . The expansion of in terms of cumulants yields an effective classical Hamiltonian with temperature-dependent quantum corrections. For the Heisenberg quantum Hamiltonian, they have a non-Heisenberg form. The effective Hamiltonian can be solved by methods familiar for classical systems.
(received; accepted) PACS. 05.30.-d – Quantum statistical mechanics. PACS. 75.10.Jm – Quantized spin models. PACS. 75.10.Hk – Classical spin models. Abstract. – A method for obtaining a systematic expansion of thermodynamic functions of spin systems with large spin S in powers of 1/S is proposed. This method uses the cumulant technique and a coherent-state representation of the partition function Z. The expansion of Z in terms of cumulants yields an effective classical Hamiltonian with temperature-dependent quantum corrections. For the Heisenberg quantum Hamiltonian, they have a non-Heisenberg form. The effective Hamiltonian can be solved by methods familiar for classical systems. In recent years considerable interest in condensed matter physics has been focusing on the properties of new magnetic materials, such as magnetic molecules with a large effective spin S, or intermetallic compounds containing magnetic layers or chains. For the latter, most of the theoretical studies deal with S = 1/2 systems which, in the one-dimensional case, allow for analytical solutions using special methods such as the Bethe Ansatz or Jordan-Wigner
Generalized Coherent States (GCS) are constructed (and discussed) in order to study quasiclassical behaviour of quantum spin models of the Heisenberg type. Several such models are taken to their semiclassical limits, whose form depends on the spin value as well as the Hamiltonian symmetry. In the continuum approximation, SU(2)/U(1) GCS when applied give rise to the well known Landau - Lifshitz classical phenomenology. For arbitrary spin values one obtains a lattice of coupled nonlinear oscillators. Corresponding classical continuum models are described as well.
Quantum spin systems are shown to be rigorously equivalent to certain Bose systems. As a result the phase transitions occuring in such systems can be regarded as generalized Bose condensation processes. In those processes it is not the number of bosons that is kept constant but a certain function of it. The problem of degeneracy arising in Bosonizing a spin system is solved in a simple fashion. The low-temperature limit of the Bose systems reproduces the magnon gas with correct interactions. The Bloch sum rule becomes exact at all temperatures, when one replaces the magnon number by the number of bosons. The crossover to classical behavior at criticality is discussed and it is shown that at Tc{T}_{c}Tc​ the quantum S=12S=\frac{1}{2}S=21​ spin systems behave like the corresponding classical $S=\ensuremath{\infty}$ systems. A classical effective Hamiltonian whose corresponding partition function is equal to the partition function of the quantum spin systems is derived. Finally, possible application to dynamics are briefly discussed.