Boson-fermion pairing in a boson-fermion environment

Boson-fermion pairing in a boson-fermion environment A Storozhenko,1,2 P Schuck,1,3 T Suzuki,4 H Yabu,4 and J Dukelsky5 1Institut de Physique Nucléaire, IN2P3-CNRS, Université Paris-Sud, F-91406 Orsay Cédex, France

ElectroweakMonopole Productionat theLHC- a Snowmass WhitePaper

represents the Cho-Maison dyon, where Z = A − B and we have chosen f(0) = 1 and A(∞) = M W/2 but not the Dirac’s monopole It has the electric charge 4π/e, not 2π/e [2] Second, this monople naturally ad-mits a non-trivialdressing of weak bosons With the non-trivial dressing, the monopole becomes the Cho-Maison dyon

Finite Energy Electroweak Dyon - CERN

have the Cho-Maison monopole with A = B = 0 In general, with A0 6= 0, we nd the Cho-Maison dyon solution shown in Fig 1 [6] The solution looks very much like the well-known Prasad-Sommer eld solution of the Julia-Zee dyon But there is a crucial di erence The Cho-Maison dyon now has a non-trivial B − A, which represents the non-vanishing

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Structural and dynamic origin of the boson peak in a Cu-Zr

2Laboratoire de Physique et Modelisation des Milieux Condens´ ´es, UMR-CNRS 5493, Maison des Magist `eres, BP 166 CNRS, 38042 Grenoble-Cedex 09, France

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A. Storozhenko,

1,2

P. Schuck,

1,3

T. Suzuki,

H. Yabu,

4 and J. Dukelsky 5 1

Institut de Physique Nucléaire, IN2P3-CNRS, Université Paris-Sud, F-91406 Orsay Cédex, France

Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Russia

Laboratoire de Physique et Modélisation des Milieux Condensés, CNRS & Université Joseph Fourier, Maison des Magistères,

Boîte Postale 166, 38042 Grenoble Cedex 9, France 4

Department of Physics, Tokyo Metropolitan University, 1-1 Minami-Ohsawa, Hachioji, Tokyo 192-0397, Japan

5 Instituto de Estructura de la Materia, Serrano 123, Madrid 28006, Spain ?Received 29 April 2004; published 30 June 2005

Propagation of a Boson-Fermion?BF?pair in a BF environment is considered. The possibility of formation

of stable strongly correlated BF pairs, embedded in the continuum, is pointed out. The Fermi gas of correlated

BF pairs shows a strongly modified Fermi surface. The interaction between like particles is neglected in this

exploratory study. Various physical situations where our pairing mechanism could be of importance are

invoked.

DOI: 10.1103/PhysRevA.71.063617 PACS number?s?: 03.75.Hh, 05.30.FkThe physics of ultracold atomic gases is making progress

at a rapid pace, which has led to a realization of Boson- fermion mixtures of atomic gases?1Ð4?. Boson-fermion?BF? mixtures may exhibit the richest variety of phenomena of all. They may show very different behavior from pure fermion or pure Bose gases?5,6?. Especially interesting is a possible instability of the mixture when there is an attraction between bosons and fermions?5,7Ð9?, as a recent experiment in fact suggests a collapse of the mixture?10?. In the present work we propose and study quite a different scenario for an attractively interacting boson-fermion mix- ture. To simplify the problem in a first survey we shall con- sider the situation where there is no interaction between at- oms of the same kind. As we will discuss at the end of the paper, this is not a severe approximation to cases where the interaction between like atoms is repulsive. More precisely we want to address the question of what happens to a mix- ture of free fermions and bosons when a?tunable?attraction is switched on between fermions and bosons. We imagine that correlated BF pairs will be created. These BF pairs are composite fermions and as such these BF pairs should form a Fermi gas of composites. Besides in ultracold atomic gases such a situation can exist in other branches of physics. For example, in nuclear systems?e.g., neutron stars?of high- densityK- mesons and nucleons may form a gas of?'s and the?'s may then form a Fermi gas of their own?11?.Orin a quark-gluon plasma additional quarks may bind to pre- formed diquarks or color Cooper pairs?the ÒbosonsÓ??12?to form a gas of nucleons in the so-called hadronization transi- tion. Further examples may be added to this list.

For a numerical example, we take a mixture of40

K?fer-

mion?and 41

K?boson?atoms throughout the paper. They are

known as candidates for a realization of this kind of quantum systems. While their scattering lengths are not well fixed at present, and different values have been reported experimen- tally?13?, it is not crucial at the moment because our study will be mostly academic, elaborating on the basic phenom- enon. Applications to realistic systems will be left for the future.

Let us consider a single BF pair propagating in the back-ground of a homogeneous gas of free one-component fermi-

ons and spinless bosons. We will formulate our approach for a situation at finite temperatureT, though later on in our application we will concentrate on theT=0 case. We have in mind an analogous study Cooper performed a long time ago ?14?for the propagation of two fermions?spin up or down? in the background of a homogeneous gas of two-component free fermions. In other words we consider a situation where in the original Cooper problem one fermion type?let us say spin down?is replaced by spinless bosons. The BF propaga- tor at finite temperatureTand finite center-of-mass momen- tumPof the pair that is added to the system with momenta

P/2+p?fermion?andP/2-p?boson?is

Gp,p t-t? ?P?=-i??t-t?????b P/2-p c P/2+p t ,?c P/2+p b P/2-p t? where?,?is the anticommutator andc andb are fermion and boson creation operators, respectively. In the ladder ap- proximation the integral equation forGp,p ?P,E?reads?15? G p,p ?P,E?=G p0 ?P,E???p-p?? dp 1 ?2?? 3 G p0 ?P,E?V?p,p 1 ?G p 1,p ?P,E?. ?1? In graphical form this equation is represented in Fig. 1.

In Eq.?1?V?p,p

1 ?is the BF interaction andG p0 ?P,E?is the free retarded BF propagator in the BF background: FIG. 1. Graphical representation of Eq.?1?. Dashed line stands for the boson, straight line for the fermion. Dotted vertical line is the interaction.PHYSICAL REVIEW A71, 063617?2005?

G p0 ?P,E?=1-f?P/2+p?+g?P/2-p? E-e f ?P/2+p?-e b ?P/2-p?+i? +?2 3 n 0 E-P 2 /2m+i?? P 2-p ?.?2? Heref?p?andg?p?are the Fermi-Dirac and Bose-Einstein distributions with chemical potentials f and? b , respec- tively, and the term with the condensate fractionn 0 of bosons only appears forT?T cr whereT cr is the critical temperature for Bose condensation. We further havee f ?p?=e b ?p? =p 2 /2mwhich are the kinetic energies of fermions and bosons which we suppose of equal mass:m b =m f =m. For simplicity we disregard mass shifts from self-energy correc- tions which may drive the masses of fermions and bosons apart, even if in free space they are equal. Had we considered fermion-fermion?FF?propagation in a two-component Fermi gas?spin up or down?, as Cooper did in his original work, then in Eq.?2?the bosonic distribution +g?P/2-p? would have to be replaced by -f?P/2-p?with, of course, n 0 =0. As in Cooper's work, Eq.?1?only treats the propaga- tion of one pair and neglects the inßuence of the other pairs on the pair under consideration. We therefore only can study situations with a very low density of BF pairs. For the BF case we will make the schematic ansatz of separability of the force: V?p,p ??=-?v?p?v?p??,??0,?3? with a Yukawa type of form factor v?p?=1 m?p 2 2 where, in principle, the two parameters?and ?may be related to the scattering length and the effective range param- eters of the low-energy BF scattering in free space?16?. However, in this exploratory study we will consider?and as free parameters especially in view of the fact that the interaction strength can be shifted using the Feshbach reso- nance phenomenon, whose application to K atoms has been discussed in?13,17?. The integral equation can then easily be solved with only a quadrature to be done numerically. The result is G p,p ?P,E?=G p0 ?P,E???p-p?? 1 ?2?? 3 ?G p0 ?P,E?v?p?v?p??G p 0 ?P,E? 1+?J 0 ?E,P??4? where J 0 ?E,P?= dp ?2?? 3 G p0 ?P,E?v 2 ?p?.?5? Without loss of generality we can consider the simpler propagator integrated over relative momentumG?P,E?= dp? ?2?? 3 dpv?p?v?p??G p,p ?P,E? J 0 ?E,P? 1+?J 0 ?E,P?.?6?

We will be interested in theTmatrix?18?

T EP ?q,q??=-? v?q?v?q?? 1+?J 0 ?E,P??7? and want to study the pole structure of this function, first at T=0, as a function ofP. To this purpose we show in Fig. 2 the imaginary part ofJ 0 as a function ofEfor different values of the center-of-mass momentumP. Calculations have been done for a 40
Ku 41

K system with equal masses:

m=m B =mquotesdbs_dbs5.pdfusesText_10

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