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arXiv:cond-mat/9503089 · Physical review. B, Condensed matter

Motion of heavy particles coupled to fermionic and bosonic environments in one dimension

A. O. Caldeira, A. H. Castro Neto

Published Aug 1, 1995• 16 citations
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Abstract

Making use of a simple unitary transformation we change the Hamiltonian of a particle coupled to a one-dimensional gas of bosons or fermions to a new form from which the many-body degrees of freedom can be easily eliminated by taking the trace. The effective dynamics of the particle allows us to compute its damping constant in the semiclassical approximation in terms of the reflection coefficient of the interaction potential and the occupation number of the environmental particles. We apply our results to a delta-function repulsive potential.

DOI
10.1103/physrevb.52.4198

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Making use of a simple unitary transformation we change the Hamiltonian of a particle coupled to a one-dimensional gas of bosons or fermions to a new form from which the many-body degrees of freedom can be easily eliminated by taking the trace. The effective dynamics of the particle allows us to compute its damping constant in the semiclassical approximation in terms of the reflection coefficient of the interaction potential and the occupation number of the environmental particles. We apply our results to a delta-function repulsive potential.
An energy spectrum is exactly examined for a one-dimensional many boson system where bosons are interacting to each other through a delta-functional repulsive potential. After a unitary transformation, the Hamiltonian is diagonalized to be the following compact form ∑ p ( p2 /2 m ) np + (π\hbar/2 mL ) ×∑ p , q | p - q | npnq + (1/6 m )(π\hbar/ L ) 2 [(∑ pnp ) 3 - ∑ pnp )] at the infinitely large limit of the coupling constant. This form is non-linear with respect to the numbers np of quasi-particles. The non-linear terms are Galilean-invariant and produce a phonon-like spectrum. In a case of a finite coupling contant g , the total energy is expanded to power series of (1/ g ).
The motion of a particle subject to an external force and coupled to a large number of harmonic oscillators is reducible to a nonlinear integro-differential equation for the position operator. This is achieved by eliminating the degrees of freedom corresponding to the motion of the oscillators. Upon this elimination the commutator of the position and the mechanical momentum of the particle becomes time dependent and a c number for a quadratic potential, but in general a q number for other fields of force. For a harmonically bound particle interacting with the heat bath the time evolution of the position and momentum operators are given in terms of the solutions of an integro-differential equation. Certain forms of the coupling of the motion of the particle to the heat bath allow for further reduction of these equations to quartic differential equations with constant coefficients. For these cases the time dependence of the position-momentum commutator, the energy of the particle as well as the heat bath are studied.
The main work of this thesis involves the calculation, using the Bethe ansatz, of two of the signature quantities of the one-dimensional delta-function Bose gas. These are the density matrix and concomitantly its Fourier transform the occupation numbers, and the correlation function and concomitantly its Fourier transform the structure factor. The coefficient of the delta-function is called the coupling constant; these quantities are calculated in the finite-coupling regime, both expansions around zero coupling and infinite coupling are considered. Further to this, the density matrix in the infinite coupling limit, and its first order correction, is recast into Toeplitz determinant form. From this the occupation numbers are calculated up to 36 particles for the ground state and up to 26 particles for the first and second excited states. This data is used to fit the coefficients of an ansatz for the occupation numbers. The correlation function in the infinite coupling limit, and its first order correction, is recast into a form which is easy to calculate for any N, and is determined explicitly in the thermodynamic limit.
The interplay between fermionic and bosonic degrees of freedom, in the context of many-body systems and with reference to applications in nuclear and elementary particle physics, is revisited. Starting from the formulation due to Geyer, Hahne, and co-workers, bifermion operators which obey bosonic transformation rules are introduced. Fermion-boson coupling terms are included in the original Hamiltonian. Exact solutions to this Hamiltonian are compared to approximate ones. The occurrence of fermion and boson condensates is discussed.
A unitary operator, which transforms free states into the exact eigenstates for a one-dimensional many boson system with repulsive delta-function potential, is explicitly constructed. By the unitary operator, original bare creation and annihilation operators are transformed into new operators which are shown to create or annihilate the exactly dressed bosons with the interaction cloud. The total Hamiltonian, total number and total momentum in the system are expressed in the diagonalized form in terms of the dressed operators. It is shown that the exact ground state is the condensed state of all exactly dressed bosons with zero momentum.