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Thermal Boson Expansion

Zoheir Aouissat, Jochen Wambach

Published Mar 16, 2011• 0 citations
OpenAlex

Abstract

Based on recent progress in the application of thermo field dynamics (TFD) [1] to thermal many-body systems, several authors [2, 3, 4] have considered a consistent thermal boson expansion (TBE). In this regard two points of view have been taken [2]. The first (path I) consists of a bosonization of the original degrees of freedom of the system, substituting for these ideal boson images. The thermalization is then achieved by doubling those newly introduced bosons as according to TDF prescription. The second possibility (path II) proceeds on the other hand via a thermalization of the system by doubling the original degrees of freedom and a subsequent bosonization of the entire new system. The two paths do not lead to the same results. Moreover, when applied to the Lipkin model, a closer look shows that the choice of path I implies that

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Based on recent progress in the application of thermo field dynamics (TFD) [1] to thermal many-body systems, several authors [2, 3, 4] have considered a consistent thermal boson expansion (TBE). In this regard two points of view have been taken [2]. The first (path I) consists of a bosonization of the original degrees of freedom of the system, substituting for these ideal boson images. The thermalization is then achieved by doubling those newly introduced bosons as according to TDF prescription. The second possibility (path II) proceeds on the other hand via a thermalization of the system by doubling the original degrees of freedom and a subsequent bosonization of the entire new system. The two paths do not lead to the same results. Moreover, when applied to the Lipkin model, a closer look shows that the choice of path I implies that
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.
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),
The algebraic structure of Thermo Field Dynamics for bosons can be fully incorporated in the q-deformation of the Weyl-Heisenberg algebra hq (1) . The doubling of the degrees of freedom, the set of the tilde-conjugation rules, the Bogoliubov transformation and its generator have a direct and simple interpretation in terms of operators and of properties of hq (1) . The notion of “thermal degree of freedom” introduced by Umezawa also finds a more specific formalization since the corresponding “thermal conjugate momentum” can be formally introduced, thus providing us with a set of canonical “thermal” variables.
The thermal Bogoliubov transformation in thermo field dynamics is generalized in two respects. First, a generalization of the alpha\\alphaalpha--degree of freedom to tilde non--conserving representations is considered. Secondly, the usual 2times22\\times22times2 Bogoliubov matrix is extended to a 4times44\\times44times4 matrix including mixing of modes with non--trivial multiparticle correlations. The analysis is carried out for both bosons and fermions.
Abstract We develop a wave‐function‐based method for the simulation of quantum dynamics of systems with many degrees of freedom at finite temperature. The method is inspired by the ideas of Thermo Field Dynamics (TFD). As TFD, our method is based on the doubling of the system's degrees of freedom and thermal Bogoliubov transformation. As distinct from TFD, our method implements the doubling of thermalized degrees of freedom only, and relies upon the explicitly constructed generalized thermal Bogoliubov transformation, which is not restricted to fermionic and bosonic degrees of freedom. This renders the present approach computationally efficient and applicable to a large variety of systems.