Excitation energy and structure factor of many-boson systems at the absolute zero temperature and at a finite temperature

In this paper we suggest a new ground-state wave function and low-temperature density matrix for a strongly interacting system of bosons. Our basic assumption is that the system can support long-wavelength phonons and that these can propagate independently of any other mode of motion. We therefore write the ground-state function as the product of two factors. One factor arises from the zero-point motion of the phonons, and we show that it has the form ${\ensuremath{\Pi}}_{i<j}f({r}_{\mathrm{ij}})$, where $\mathrm{ln}f(r)$ has an infinite range. The other factor is assumed to have this same form but with $\mathrm{ln}f(r)$ of finite range; it takes into account the short-range correlations arising from the strong repulsive part of the interparticle potential. The function we have chosen to represent the short-range correlations is not new; functions of this kind were first introduced by Bijl and later by Jastrow. At finite temperatures we use a density matrix for an ensemble of excited phonon states. We find that for small wave vectors $k$ , the structure factor $S(k)$ is equal to $\frac{\ensuremath{\hbar}k}{2mc}$, a result that was first derived by Feynman. At a finite temperature $T$ , $S(k)$ tends to the constant value $\frac{{k}_{B}T}{m{c}^{2}}$ , where $m$ is the mass of the particles and $c$ the velocity of propagation of the phonons. The momentum distribution ${n}_{k}$ has a ${k}^{\ensuremath{-}1}$ singularity at absolute zero and a stronger, ${k}^{\ensuremath{-}2}$ singularity at finite temperatures. The small- $k$ behavior of both $S(k)$ and ${n}_{k}$ is completely controlled by the correlations introduced by the phonons. Both the ground-state function and the density matrix imply that there is a finite fraction of particles in the zero-momentum state in three dimensions; this fraction does not seem to be appreciably affected by the infinite-range correlations introduced by the phonons. We find, however, that these correlations imply that a one-dimensional Bose system does not exhibit Bose-Einstein condensation at any temperature, while a two-dimensional system exhibits Bose-Einstein condensation only at absolute zero.In this paper we suggest a new ground-state wave function and low-temperature density matrix for a strongly interacting system of bosons. Our basic assumption is that the system can support long-wavelength phonons and that these can propagate independently of any other mode of motion. We therefore write the ground-state function as the product of two factors. One factor arises from the zero-point motion of the phonons, and we show that it has the form ${\ensuremath{\Pi}}_{i<j}f({r}_{\mathrm{ij}})$, where $\mathrm{ln}f(r)$ has an infinite range. The other factor is assumed to have this same form but with $\mathrm{ln}f(r)$ of finite range; it takes into account the short-range correlations arising from the strong repulsive part of the interparticle potential. The function we have chosen to represent the short-range correlations is not new; functions of this kind were first introduced by Bijl and later by Jastrow. At finite temperatures we use a density matrix for an ensemble of excited phonon states. We find that for small wave vectors $k$, the structure factor $S(k)$ is equal to $\frac{\ensuremath{\hbar}k}{2mc}$, a result that was first derived by Feynman. At a finite temperature $T$, $S(k)$ tends to the constant value $\frac{{k}_{B}T}{m{c}^{2}}$, where $m$ is the mass of the particles and $c$ the velocity of propagation of the phonons. The momentum distribution ${n}_{k}$ has a ${k}^{\ensuremath{-}1}$ singularity at absolute zero and a stronger, ${k}^{\ensuremath{-}2}$ singularity at finite temperatures. The small-$k$ behavior of both $S(k)$ and ${n}_{k}$ is completely controlled by the correlations introduced by the phonons. Both the ground-state function and the density matrix imply that there is a finite fraction of particles in the zero-momentum state in three dimensions; this fraction does not seem to be appreciably affected by the infinite-range correlations introduced by the phonons. We find, however, that these correlations imply that a one-dimensional Bose system does not exhibit Bose-Einstein condensation at any temperature, while a two-dimensional system exhibits Bose-Einstein condensation only at absolute zero.

Physical ReviewMar 5, 1967281 citations

Energy Spectrum in Many-Boson Systems at Finite Temperature

正人大矢

Jun 1, 19790 citations

Elementary excitation spectrum for the multichannel Kondo model at finite temperatures

Sei-ichiro Suga, A. Okiji, N. Kawakami

Spectral density for the elementary excitation of the multichannel Kondo model is investigated at finite temperatures with the use of the Bethe-ansatz method. Our formulation is based on the method of Yang and Yang developed in one-dimensional interacting boson systems. In the orbital singlet case, the narrow peak structure develops in the low-energy region as the temperature is decreased, whereas in the underscreened case, the excitation spectrum exhibits a divergence property at zero excitation energy, irrespective of the temperature. In the overscreened case, in which the non-Fermi-liquid fixed point is stable in the ground state, the peak structure develops in the low-energy region and the weight at zero excitation energy increases as the temperature is decreased. We discuss characteristic properties of the excitation spectrum in connection with bulk properties.Spectral density for the elementary excitation of the multichannel Kondo model is investigated at finite temperatures with the use of the Bethe-ansatz method. Our formulation is based on the method of Yang and Yang developed in one-dimensional interacting boson systems. In the orbital singlet case, the narrow peak structure develops in the low-energy region as the temperature is decreased, whereas in the underscreened case, the excitation spectrum exhibits a divergence property at zero excitation energy, irrespective of the temperature. In the overscreened case, in which the non-Fermi-liquid fixed point is stable in the ground state, the peak structure develops in the low-energy region and the weight at zero excitation energy increases as the temperature is decreased. We discuss characteristic properties of the excitation spectrum in connection with bulk properties.

Physical review. B, Condensed matterNov 1, 19949 citations

Excitation spectrum and structure factor for interacting bosons

Kapil M. Khanna, S. Chauba Singh, O. P. Sinha

Physica B+CAug 1, 19820 citations

Specific Heat and Structure Factor of a Many-Boson System.

H. Scher

Jan 1, 19640 citations

Temperature: The ignored factor in quantum mechanics

Ashkan Shekaari, Mahmoud Jafari

We have developed a theoretical formalism to introduce temperature as a parameter into the framework of non-relativistic quantum mechanics using the laws of classical thermodynamics and the canonical ensemble scheme of statistical mechanics. A self-consistent Hamiltonian has then been constructed for a given quantum many-body system which includes the effect of temperature in the form of correction terms added to the corresponding zero-temperature Hamiltonian of the system. Investigating some quantum mechanical systems with exact zero-temperature solutions including the particle-in-a-box model, the free particle, and the harmonic oscillator within our finite-temperature approach up to the first order of self-consistency has led to temperature-dependent Hamiltonians describing these systems above absolute zero without encountering any physically unacceptable brand of behavior for their wave functions and energy spectra. Results firmly support the view that a quantum mechanical system at a finite temperature behaves as if it is in a zero-temperature excited state.We have developed a theoretical formalism to introduce temperature as a parameter into the framework of non-relativistic quantum mechanics using the laws of classical thermodynamics and the canonical ensemble scheme of statistical mechanics. A self-consistent Hamiltonian has then been constructed for a given quantum many-body system which includes the effect of temperature in the form of correction terms added to the corresponding zero-temperature Hamiltonian of the system. Investigating some quantum mechanical systems with exact zero-temperature solutions including the particle-in-a-box model, the free particle, and the harmonic oscillator within our finite-temperature approach up to the first order of self-consistency has led to temperature-dependent Hamiltonians describing these systems above absolute zero without encountering any physically unacceptable brand of behavior for their wave functions and energy spectra. Results firmly support the view that a quantum mechanical system at a finite temperature behaves as if it is in a zero-temperature excited state.

arXiv:2001.05212Jan 15, 20200 citations

\mathrm{ln}f(r)

has an infinite range. The other factor is assumed to have this same form but with

\mathrm{ln}f(r)

of finite range; it takes into account the short-range correlations arising from the strong repulsive part of the interparticle potential. The function we have chosen to represent the short-range correlations is not new; functions of this kind were first introduced by Bijl and later by Jastrow. At finite temperatures we use a density matrix for an ensemble of excited phonon states. We find that for small wave vectors

k

, the structure factor

S(k)

is equal to $\frac{\ensuremath{\hbar}k}{2mc}$, a result that was first derived by Feynman. At a finite temperature

T

S(k)

tends to the constant value

\frac{{k}_{B}T}{m{c}^{2}}

, where

m

is the mass of the particles and

c

the velocity of propagation of the phonons. The momentum distribution

{n}_{k}

has a ${k}^{\ensuremath{-}1}$ singularity at absolute zero and a stronger, ${k}^{\ensuremath{-}2}$ singularity at finite temperatures. The small-

k

behavior of both

S(k)

and

{n}_{k}

is completely controlled by the correlations introduced by the phonons. Both the ground-state function and the density matrix imply that there is a finite fraction of particles in the zero-momentum state in three dimensions; this fraction does not seem to be appreciably affected by the infinite-range correlations introduced by the phonons. We find, however, that these correlations imply that a one-dimensional Bose system does not exhibit Bose-Einstein condensation at any temperature, while a two-dimensional system exhibits Bose-Einstein condensation only at absolute zero.