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Bose-Einstein Condensation in an Ultrahigh Magnetic Field

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Bose-Einstein Condensation in an Ultrahigh Magnetic Field View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Loughborough University Institutional Repository PHYSICAL REVIEW B VOLUME 54, NUMBER 21 1 DECEMBER 1996-I Theory of the charged Bose gas: Bose-Einstein condensation in an ultrahigh magnetic field A. S. Alexandrov Interdisciplinary Research Center in Superconductivity, University of Cambridge, Madingley Road, Cambridge CB3 OHE, United Kingdom and Department of Physics, Loughborough University, Loughborough LE11 3TU, United Kingdom W. H. Beere Interdisciplinary Research Center in Superconductivity, University of Cambridge, Madingley Road, Cambridge CB3 OHE, United Kingdom V. V. Kabanov Interdisciplinary Research Center in Superconductivity, University of Cambridge, Madingley Road, Cambridge CB3 OHE, United Kingdom and Frank Laboratory of Neutron Physics, Joint Institute for Nuclear Research, Dubna, Russia ~Received 25 March 1996; revised manuscript received 8 July 1996! The Bogoliubov–de Gennes equations and the Ginzburg-Landau-Abrikosov-Gor’kov-type theory are for- mulated for the charged Bose gas ~CBG!. The theory of the Bose-Einstein condensation of the CBG in a magnetic field is extended to ultralow temperatures and ultrahigh magnetic fields. A low-temperature depen- dence of the upper critical field Hc2(T) is obtained both for the particle-impurity and particle-particle scatter- ing. The normal-state collective plasmon mode in ultrahigh magnetic fields is studied. @S0163-1829~96!08845-5# I. INTRODUCTION comparable with or even less than the unit cell volume. This favors a charged 2e Bose liquid as a plausible microscopic A charged Coulomb Bose gas ~CBG! recently became of scenario for the ground state.1 particular interest motivated by the bipolaron theory of high- 1 2 Our objective is the theory of the CBG in a magnetic temperature superconductivity. A long time ago Schafroth field. In this paper we first extend to finite temperatures the demonstrated that an ideal gas of charged bosons exhibits the BdG-type equations derived earlier15 for T50. Then we ana- Meissner-Ochsenfeld effect below the ideal Bose-gas con- lyze the linearized Ginzburg-Landau-type equation for the densation temperature. Later on, the one-particle excitation order parameter and formulate the condition of the Bose- spectrum at T50 was calculated by Foldy,3 who worked at Einstein condensation in a homogeneous magnetic field. By zero temperature using the Bogoliubov4 approach. The Bo- the use of the sum rule we calculate the upper critical field goliubov method leads to the result that the ground state of Hc2(T) both for a short-range particle-impurity and long- the system has a negative correlation energy, whose magni- range particle-particle Coulomb interactions at low tempera- tude increases with the density of bosons. Perhaps more in- tures. The plasmon dispersion of CBG in the ultrahigh mag- teresting is the fact that the elementary excitations of the netic field is analyzed as well. system have, for small momenta, energies characteristic of plasma oscillations which pass over smoothly for large mo- II. BOGOLIUBOV–de GENNES EQUATIONS menta to the energies characteristic of single-particle excita- FOR THE CBG AT FINITE TEMPERATURES tions. Further investigations have been carried out at or near The superfluid properties of charged bosons as well as Tc , the transition temperature for the gas. These works have their excitation spectrum and the response function can be been concerned with the critical exponents5 and the change studied by the use of the Bogoliubov–de Gennes- ~BdG-! in the transition temperature from that of the ideal gas.5,6 The type equations, fully taking into account the interaction of random phase approximation ~RPA! dielectric response quasiparticles with the condensate.15 The Hamiltonian of function and screening in the CBG have been studied in the charged bosons on an oppositely charged background ~to en- high-density limit,7 including a low-dimensional @two- sure charge neutrality! in an external field with the vector dimensional ~2D!# CBG.8,9 The theory of the CBG beyond potential A(r,t) is given by the lowest-order Bogoliubov approximation was discussed 2 10 11 ~¹2ie*A! by Lee and Feenberg and by Brueckner. They obtained H5 drc†~r! 2 2m c~r! the next-order correction to the ground-state energy. Woo E F 2m G 12 and Ma calculated numerically the correction to the Bogo- 1 liubov excitation spectrum. Alexandrov13 found the critical 1 dr dr8V~r2r8!c†~r!c~r!c†~r8!c~r8!. ~1! 2E E magnetic field Hc2(T), at which the CBG is condensed. The predicted temperature dependence of Hc2(T) was observed For 3D charged bosons of mass m the Fourier component of 14 2 2 both in low-Tc and high-Tc cuprates, where the coherence the interaction potential V(r)isV(k)54pe*/k with volume estimated from the heat capacity measurements is bosonic charge e*. For a 2D system with a three- 0163-1829/96/54~21!/15363~9!/$10.0054 15 363 © 1996 The American Physical Society 15 364 A. S. ALEXANDROV, W. H. BEERE, AND V. V. KABANOV 54 dimensional interaction V(k)52pe*2/k. Respecting elec- ] ~¹2ie*A!2 i c~r,t!5@H,c~r,t!#5 2 2m c~r,t! troneutrality one takes V(k[0)50, which is a consequence ]t F 2m G of the compensation of the boson-boson repulsion by the attraction due to a spatially homogeneous charged back- † ground. In this paper we will treat the purely academic prob- 1E dr8V~r2r8!c ~r8,t!c~r8,t!c~r,t!. ~2! lem of the charged Bose gas; however, we state that the results obtained are also applicable to the case of preformed If the interaction is weak, one can expect that the occupation electron pairs, which we postulate to exist in the high-Tc numbers of one-particle states are not very much different cuprates. In the more realistic case of preformed pairs their from those in the ideal Bose gas. In particular the state with hard core nature needs to be considered. This introduces an zero momentum k50 remains macroscopically occupied and additional term missing from the equations given in this pa- the corresponding Fourier component of the field operator per as we no longer have V(k[0)50 and instead have c(r) has an anomalously large matrix element between the V(k[0)5const. When considering the self-energy, as we ground states of the system containing N11 and N bosons. shall do later in the paper, it can be seen that this zero- It is convenient to consider a grand canonical ensemble, in- momenta interaction term will give a constant contribution to troducing a chemical potential m. In this case the quantum the self-energy, which amounts to a renormalization of the state is a superposition of states uN& with slightly different chemical potential. The hard core nature of the pairs also total numbers of bosons. The weight of each state is a gives a constant contribution to the Fourier component of the smooth function of N which is practically constant near the interaction potential at all momenta. The most significant average number N¯ on the scale 6AN¯. Because c changes part of the interaction is Coulomb interaction as can be seen the number of particles only by 1 its diagonal matrix ele- from Eq. ~15!, where, if we include the hard core interaction ment coincides with the off diagonal, calculated for the states as well as the Coulomb interaction, the main result—that the ¯ ¯ 4 excitation spectrum contains a plasma gap—is unchanged. with fixed N5N11 and N5N. Following Bogoliubov one The consequences of including only the hard core interaction can separate the large diagonal matrix element c0 from c by ˜ have been discussed earlier by Alexandrov et al.16 One final treating the rest c as a small fluctuation: point worth mentioning on the choice of the interaction po- tential deals with the screening within the system. Simply c~r,t!5c ~r,t!1˜c~r,t!. ~3! taking a screened Coulomb potential as our starting potential 0 can lead to an erroneous double-counting result as discussed 15 by Alexandrov and Beere. In this paper we start with the The anomalous average c0(r,t)5^c(r,t)& is equal to bare interaction potential and derive a self-consistent form of An0 in a homogeneous system, where n0 is the condensate the self-energies, carefully taking into account the self- density. screening of the interaction by the bosons in the high-density Substituting the Bogoliubov displacement transformation, limit, rs!1 ~see below!. If we have in mind a metal with Eq. ~3!, into the equation of motion and collecting preformed pairs, to avoid their overlap, the density is also c-number terms of c0 and supracondensate boson opera- restricted in the upper limit. Here and further \5c51. tors ˜c, we obtain a set of the BdG-type equations. The mac- The equation of motion for the field operator c is derived roscopic condensate wave function, which plays the role of using this Hamiltonian, the order parameter, obeys the equation ] ~¹2ie*A!2 i c ~r,t!5 2 2m c ~r,t!1 dr8V~r2r8!n~r8,t!c ~r,t!1 dr8V~r2r8!@^˜c†~r8,t!˜c~r,t!&c ~r8,t! ]t 0 F 2m G 0 E 0 E 0 ˜ ˜ 1^c~r8,t!c~r,t!&c0*~r8,t!#. ~4! Taking explicitly into account the interaction of supracondensate bosons with the condensate and applying the Hartree-Fock approximation for the interaction between supracondensate particles one obtains ] ~¹2ie*A!2 i ˜c~r,t!5 2 2m ˜c~r,t!1 dr8V~r2r8!n~r8,t!˜c~r,t!1 dr8V~r2r8!@c*~r8,t8!c ~r,t! ]t F 2m G E E 0 0 ˜† ˜ ˜ ˜ ˜ ˜† 1^c ~r8,t!c~r,t!&#c~r8,t!1E dr8V~r2r8!@c0~r8,t!c0~r,t!1^c~r8,t!c~r,t!&#c ~r8,t!1E dr8V~r2r8! ˜† ˜ ˜† ˜ ˜† ˜ ˜† ˜ 3@c ~r8,t!c~r8,t!2^c ~r8,t!c~r8,t!&#c0~r,t!1E dr8V~r2r8!@c ~r8,t!c~r,t!2^c ~r8,t!c~r,t!&#c0~r8,t! ˜ ˜ ˜ ˜ * 1E dr8V~r2r8!@c~r8,t!c~r,t!2^c~r8,t!c~r,t!&#c0 ~r8,t!.
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