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Physical Review C

Extended Holstein-Primakoff mapping for the next-to-leading order of the 1∕N expansion at finite temperature

Alan A. Dzhioev, Zoheir Aouissat, Aleksej Nikolaevich Storozhenko, A.I. Vdovin, Jochen Wambach

Published Jan 30, 2004• 15 citations
Publisher

Abstract

An extended Holstein-Primakoff mapping which incorporates both single- and double-fermion mappings is used in the context of thermofield dynamics to study the next-to-leading order of the $1∕N$ expansion at finite temperature. For the Lipkin-Meshkov-Glick model it is shown that the extended mapping naturally leads to the correct Fermi statistics both in leading and next-to-leading order.

DOI
10.1103/physrevc.69.014318

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An extended Holstein-Primakoff mapping which incorporates both single- and double-fermion mappings is used in the context of thermofield dynamics to study the next-to-leading order of the 1∕N1∕N1∕N expansion at finite temperature. For the Lipkin-Meshkov-Glick model it is shown that the extended mapping naturally leads to the correct Fermi statistics both in leading and next-to-leading order.
Within the formalism of thermo field dynamics, the boson-fermion Holstein-Primakoff transformation is constructed for the case of nonzero temperatures. For the example of the Lipkin model, the transformation in question is used to construct a thermal Hamiltonian in the form of an expansion in the parameter $${1 \mathord{\left/ {\vphantom {1 {\sqrt N }}} \right. \kern-\nulldelimiterspace} {\sqrt N }}$$ , where N is the number of particles in the system. The temperature dependence of quasiparticle (fermion) and collective-excitation (boson) energies is calculated to terms of order 1/N.
For the last decades Boson expansion theory has played an effective role in the development of approximate solutions to the many-body problem in nuclear physics [1]. These techniques also provide promising tools to address solutions for quantum field theories [2]. In a recent work, we have succeeded in combining an extended form of the Holstein-Primakoff mapping (EHPM) [3] with thermofield-dynamics [4] to generate a consistent thermal boson expansion for quantum mechanical systems such as the bosonic O(N) anharmonic oscillator as well as the fermionic Lipkin model [7]. The analogue of the first of these two models in four space-time dimensions can be used to address important questions such as the chiral transition for two flavors QCD or the scattering of pions at zero and finite temperature. It is also found that the EHPM removes number of ambiguities when computing anharmonicities in the next-to-leading order of the 1/N expansion [7], which are very much relevant to the ππ scattering problem, for instance. Furthermore, the EHPM allows to extend the use of the boson expansion approach to quantum systems with internal SU(N) symmetry where one is confronted with theories having tensor couplings LI = fijklφiφjφkφl + gijkφiφjφk +..., (1) and which calls for a simultaneous bosonization of both pair and single modes. This opens, indeed, the possibility to ultimately address the question of the chiral transition in SU(3) flavor QCD in four space-time dimensions. This point, although crucial for the understanding of the QCD phase diagram, has not been adequately addressed before due to the fact that the 1/Nf expansion cannot be correctly sorted out with known techniques [6]. The approach highlighted above and which fundamentally relies on the bosonization of the generators of the Sp2⊗N1 semidirect product group, (aa)I = √ 2+4(n + m)A, (a + a)I =2n+ m, (a)I = √ 2+4(n + m)Γ(m)α +2α + A Γ(m),
In the context of the boson expansion theory, it is usually the case that the bosonization of single-boson (fermion) states is ignored. Although this is tolerable to some extent in cold systems, it causes serious difficulties in the treatment of thermal ensembles where the single-boson (fermion) density of states plays an important role. In the framework of the thermo-field dynamics it is shown how extended forms of the Holstein-Primakoff mapping for both bosons and fermions can lead to consistent thermal-boson expansions. Applications to the O(N)O(N)O(N) anharmonic oscillator and the Lipkin model are presented.
When the Holstein-Primakoff (Hermitian)-type boson expansion theory is applied for nuclear structure calculations, truncation at the fourth-order expansion of the boson Hamiltonian is often used. The Dyson-type boson mapping has no difficulty concerning boson expansion, since the Dyson boson Hamiltonian is written in up to sixth-order boson operators, and we do not have to make any truncation of the boson expansion. In the present work, we propose a new clear-cut method to obtain the normal-ordered Holstein-Primakoff boson expansion. Using this method, we test whether or not the truncation of the boson expansion is good approximation in various cases. We show that truncation at the fourth-order expansion of the normal-ordered Holstein-Primakoff boson Hamiltonian is a rather good approximation, but it is not sufficient for detailed analyses.