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Progress of Theoretical Physics

On Quasi-Particles at Finite Temperature

Yuji Nakawaki, Azuma Tanaka, K. Ozaki

Published Feb 1, 1989• 4 citations
Publisher

Abstract

To investigate validity of non-scattering of quasi-particles ( S = 1) at finite temperature, which Narnhofer et al. claimed, we assume asymptotic completeness at zero temperature and consider whether out-going particles are well-defined or not at finite temperature in Fock space of in-coming particles. It is found in nonlinear Schrödinger model that contrary to the anticipation of Narnhofer et al., not only one particle operator but also continuous background operator contribute to construct in-coming and out-going particles, which are not identical. However Fock spaces of them become inequivalent so that non-scattering of quasi-particles at finite temperature remains not to be denied. It is also noted that LSZ reduction formula for S -matrix does not apply at finite temperature.

DOI
10.1143/ptp.81.498

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To investigate validity of non-scattering of quasi-particles ( S = 1) at finite temperature, which Narnhofer et al. claimed, we assume asymptotic completeness at zero temperature and consider whether out-going particles are well-defined or not at finite temperature in Fock space of in-coming particles. It is found in nonlinear Schrödinger model that contrary to the anticipation of Narnhofer et al., not only one particle operator but also continuous background operator contribute to construct in-coming and out-going particles, which are not identical. However Fock spaces of them become inequivalent so that non-scattering of quasi-particles at finite temperature remains not to be denied. It is also noted that LSZ reduction formula for S -matrix does not apply at finite temperature.
Creation and annihilation operators of quasi-particles are introduced into the description of a finite temperature system of degenerate bosons with the aid of a canonical transformation which generalizes the one due to Bogoliubov. A variational calculation taking into account the interaction between non-zero momentum particles shows that the quasi-particles are phonons in the long wave-length limit. The state of the system, which is approximated by the grand canonical ensemble of the quasi-particles, approaches the Hartree-Fock state when the condensate is depleted completely at the critical temperature.
We consider a mean-field model to describe the dynamics of [Formula: see text] bosons of species one and [Formula: see text] bosons of species two in the limit as [Formula: see text] and [Formula: see text] go to infinity. We embed this model into Fock space and use it to describe the time evolution of coherent states which represent two-component condensates. Following this approach, we obtain a microscopic quantum description for the dynamics of such systems, determined by the Schrödinger equation. Associated to the solution to the Schrödinger equation, we have a reduced density operator for one particle in the first component of the condensate and one particle in the second component. In this paper, we estimate the difference between this operator and the projection onto the tensor product of two functions that are solutions of a system of equations of Hartree type. Our results show that this difference goes to zero as [Formula: see text] and [Formula: see text] go to infinity.
We study the interacting homogeneous Bose gas in two spatial dimensions in the thermodynamic limit at fixed density. We shall be concerned with some mathematical aspects of this complicated problem in many-body quantum mechanics. More specifically, we consider the dilute limit where the scattering length of the interaction potential, which is a measure for the effective range of the potential, is small compared to the average distance between the particles. We are interested in a setting with positive (i.e., non-zero) temperature. After giving a survey of the relevant literature in the field, we provide some facts and examples to set expectations for the two-dimensional system. The crucial difference to the three-dimensional system is that there is no Bose–Einstein condensate at positive temperature due to the Hohenberg–Mermin–Wagner theorem. However, it turns out that an asymptotic formula for the free energy holds similarly to the three-dimensional case. We motivate this formula by considering a toy model with δ interaction potential. By restricting this model Hamiltonian to certain trial states with a quasi-condensate we obtain an upper bound for the free energy that still has the quasi-condensate fraction as a free parameter. When minimizing over the quasi-condensate fraction, we obtain the Berezinskii–Kosterlitz–Thouless critical temperature for superfluidity, which plays an important role in our rigorous contribution. The mathematically rigorous result that we prove concerns the specific free energy in the dilute limit. We give upper and lower bounds on the free energy in terms of the free energy of the non-interacting system and a correction term coming from the interaction. Both bounds match and thus we obtain the leading term of an asymptotic approximation in the dilute limit, provided the thermal wavelength of the particles is of the same order (or larger) than the average distance between the particles. The remarkable feature of this result is its generality: the correction term depends on the interaction potential only through its scattering length and it holds for all nonnegative interaction potentials with finite scattering length that are measurable. In particular, this allows to model an interaction of hard disks.
We study the interacting homogeneous Bose gas in two spatial dimensions in the thermodynamic limit at fixed density. We shall be concerned with some mathematical aspects of this complicated problem in many-body quantum mechanics. More specifically, we consider the dilute limit where the scattering length of the interaction potential, which is a measure for the effective range of the potential, is small compared to the average distance between the particles. We are interested in a setting with positive (i.e., non-zero) temperature. After giving a survey of the relevant literature in the field, we provide some facts and examples to set expectations for the two-dimensional system. The crucial difference to the three-dimensional system is that there is no Bose–Einstein condensate at positive temperature due to the Hohenberg–Mermin–Wagner theorem. However, it turns out that an asymptotic formula for the free energy holds similarly to the three-dimensional case. We motivate this formula by considering a toy model with δ interaction potential. By restricting this model Hamiltonian to certain trial states with a quasi-condensate we obtain an upper bound for the free energy that still has the quasi-condensate fraction as a free parameter. When minimizing over the quasi-condensate fraction, we obtain the Berezinskii–Kosterlitz–Thouless critical temperature for superfluidity, which plays an important role in our rigorous contribution. The mathematically rigorous result that we prove concerns the specific free energy in the dilute limit. We give upper and lower bounds on the free energy in terms of the free energy of the non-interacting system and a correction term coming from the interaction. Both bounds match and thus we obtain the leading term of an asymptotic approximation in the dilute limit, provided the thermal wavelength of the particles is of the same order (or larger) than the average distance between the particles. The remarkable feature of this result is its generality: the correction term depends on the interaction potential only through its scattering length and it holds for all nonnegative interaction potentials with finite scattering length that are measurable. In particular, this allows to model an interaction of hard disks.