Theory of the Charged Bose Gas: Bose-Einstein Condensation in an Ultrahigh Magnetic Field

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Theory of the charged Bose gas: Bose-Einstein condensation in an ultrahigh magnetic field

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Citation: ALEXANDROV, BEERE and KABANOV, 1996. Theory of the charged Bose gas: Bose-Einstein condensation in an ultrahigh magnetic field. Physical Review B, 54(21), pp 1536315371

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• This article was published in the journal, Physical Review B[ c American Physical Society]. It is also available at: http://link.aps.org/abstract/PRB/v54/p15363.

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Publisher: c American Physical Society

Please cite the published version. 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 1996͒ The Bogoliubov–de Gennes equations and the Ginzburg-Landau-Abrikosov-Gor’kov-type theory are for- mulated for the charged Bose gas ͑CBG͒. 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 dependence of the upper critical field Hc2(T) is obtained both for the particle-impurity and particle-particle scattering. The normal-state collective plasmon mode in ultrahigh magnetic fields is studied. ͓S0163-1829͑96͒08845-5͔

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 ͑CBG͒ 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 Tϭ0. 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 Tϭ0 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 ͑RPA͒ 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 neutrality͒ in an external field with the vector dimensional ͑2D͔͒ CBG.8,9 The theory of the CBG beyond potential A(r,t) is given by the lowest-order Bogoliubov approximation was discussed 2 Ϫie*Aٌ͒͑ 11 10 by Lee and Feenberg and by Brueckner. They obtained Hϭ dr␺†͑r͒ Ϫ Ϫ␮ ␺͑r͒ the next-order correction to the ground-state energy. Woo ͵ ͫ 2m ͬ 12 and Ma calculated numerically the correction to the Bogo- 1 liubov excitation spectrum. Alexandrov13 found the critical ϩ dr drЈV͑rϪrЈ͒␺†͑r͒␺͑r͒␺†͑rЈ͒␺͑rЈ͒. ͑1͒ 2͵ ͵ 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)ϭ4␲e*/k with volume estimated from the heat capacity measurements is bosonic charge e*. For a 2D system with a three-

Ϫie*A͒2ٌ͑ ץ -dimensional interaction V(k)ϭ2␲e*2/k. Respecting elec i ␺͑r,t͒ϭ͓H,␺͑r,t͔͒ϭ Ϫ Ϫ␮ ␺͑r,t͒ ͬ t ͫ 2mץ troneutrality one takes V(kϵ0)ϭ0, which is a consequence 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- ϩ͵ drЈV͑rϪrЈ͒␺ ͑rЈ,t͒␺͑rЈ,t͒␺͑r,t͒. ͑2͒ 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 kϭ0 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(kϵ0)ϭ0 and instead have ␺(r) has an anomalously large matrix element between the V(kϵ0)ϭconst. When considering the self-energy, as we ground states of the system containing Nϩ1 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 ␮. In this case the quantum the self-energy, which amounts to a renormalization of the state is a superposition of states ͉N͘ 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 ϮͱN¯. Because ␺ 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. ͑15͒, 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 NϭNϩ1 and NϭN. Following Bogoliubov one The consequences of including only the hard core interaction can separate the large diagonal matrix element ␺0 from ␺ by ˜ have been discussed earlier by Alexandrov et al.16 One final treating the rest ␺ as a small fluctuation: point worth mentioning on the choice of the interaction potential deals with the screening within the system. Simply ␺͑r,t͒ϭ␺ ͑r,t͒ϩ˜␺͑r,t͒. ͑3͒ 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 ␺0(r,t)ϭ͗␺(r,t)͘ is equal to bare interaction potential and derive a self-consistent form of ͱn0 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, rsӶ1 ͑see below͒. If we have in mind a metal with Eq. ͑3͒, into the equation of motion and collecting preformed pairs, to avoid their overlap, the density is also c-number terms of ␺0 and supracondensate boson opera- restricted in the upper limit. Here and further បϭcϭ1. tors ˜␺, we obtain a set of the BdG-type equations. The mac- The equation of motion for the field operator ␺ is derived roscopic condensate wave function, which plays the role of using this Hamiltonian, the order parameter, obeys the equation

Ϫie*A͒2ٌ͑ ץ i ␺ ͑r,t͒ϭ Ϫ Ϫ␮ ␺ ͑r,t͒ϩ drЈV͑rϪrЈ͒n͑rЈ,t͒␺ ͑r,t͒ϩ drЈV͑rϪrЈ͓͒͗˜␺†͑rЈ,t͒˜␺͑r,t͒͘␺ ͑rЈ,t͒ t 0 ͫ 2m ͬ 0 ͵ 0 ͵ 0ץ ˜ ˜ ϩ͗␺͑rЈ,t͒␺͑r,t͒͘␺0*͑rЈ,t͔͒. ͑4͒ 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

Ϫie*A͒2ٌ͑ ץ i ˜␺͑r,t͒ϭ Ϫ Ϫ␮ ˜␺͑r,t͒ϩ drЈV͑rϪrЈ͒n͑rЈ,t͒˜␺͑r,t͒ϩ drЈV͑rϪrЈ͓͒␺*͑rЈ,tЈ͒␺ ͑r,t͒ t ͫ 2m ͬ ͵ ͵ 0 0ץ

˜† ˜ ˜ ˜ ˜ ˜† ϩ͗␺ ͑rЈ,t͒␺͑r,t͔͒͘␺͑rЈ,t͒ϩ͵ drЈV͑rϪrЈ͓͒␺0͑rЈ,t͒␺0͑r,t͒ϩ͗␺͑rЈ,t͒␺͑r,t͔͒͘␺ ͑rЈ,t͒ϩ͵ drЈV͑rϪrЈ͒

˜† ˜ ˜† ˜ ˜† ˜ ˜† ˜ ϫ͓␺ ͑rЈ,t͒␺͑rЈ,t͒Ϫ͗␺ ͑rЈ,t͒␺͑rЈ,t͔͒͘␺0͑r,t͒ϩ͵ drЈV͑rϪrЈ͓͒␺ ͑rЈ,t͒␺͑r,t͒Ϫ͗␺ ͑rЈ,t͒␺͑r,t͔͒͘␺0͑rЈ,t͒

˜ ˜ ˜ ˜ * ϩ͵ drЈV͑rϪrЈ͓͒␺͑rЈ,t͒␺͑r,t͒Ϫ͗␺͑rЈ,t͒␺͑r,t͔͒͘␺0 ͑rЈ,t͒. ͑5͒ 54 THEORY OF THE CHARGED BOSE GAS: BOSE- . . . 15 365

ik•rϪi⑀kt Here uk͑r,t͒ϭuke , ͑12͒ 2 ˜† ˜ n͑r,t͒ϭ͉␺0͑r,t͉͒ ϩ͗␺ ͑r,t͒␺͑r,t͒͘ ͑6͒ ik•rϪi⑀kt vk͑r,t͒ϭvke , ͑13͒ is the boson density, and so the general sum rule and the condensate wave function is (r,t) independent, ␺0ϭͱn0. Thus the solution to Eq. ͑4͒ is ͵ drn͑r,t͒ϭnB ͑7͒ ␮ϭV͑kϵ0͒ϭ0 ͑14͒ is satisﬁed. Here nB is the number of bosons in the normalized volume, which is taken to be unity. in the leading order in rs , when the last term in Eq. ͑4͒ can In the high-density limit r ϭ *2 1/3Ӷ be neglected. The excitation spectrum is that obtained by s me /(4␲nB/3) 1 and 3 for the temperature close to zero the number of bosons Foldy, pushed up from the condensate by the repulsion is small. 4 2 Therefore the contribution of terms nonlinear in ˜␺ is negli- k k V͑k͒n0 ⑀kϭ 2 ϩ , ͑15͒ gible. Applying a linear Bogoliubov transformation for ˜␺, ͱ4m m

2 with a gap ␻ p0ϭͱ4␲e n0 /m, which is the classical plasma ˜ * † ␺͑r,t͒ϭ͚ un͑r,t͒␣nϩvn ͑r,t͒␣n , ͑8͒ frequency for a plasma of density n0. n At finite temperatures and ͑or͒ in a strong magnetic field † we are left with the extremely complicated integro- where ␣n and ␣n are bosonic quasiparticle operators for the one-particle quantum state n, and omitting nonlinear terms, differential nonlinear equations ͑4͒–͑7͒. we obtain two coupled Schro¨dinger equations for the wave functions u(r,t) and v(r,t):15 III. UPPER CRITICAL FIELD OF CBG: GENERAL FORMULATION Ϫie*A͒2ٌ͑ ץ i u͑r,t͒ϭ Ϫ Ϫ␮ u͑r,t͒ The situation, however, is not hopeless in the region of t 2mץ ͫ ͬ the second-order phase transition near the upper critical field Hc2(T). In this region one can apply the expansion in pow- ϩ drЈV͑rϪrЈ͓͉͒␺ ͑rЈ,t͉͒2u͑r,t͒ ͵ 0 ers of the order parameter ␺0(r,t) to obtain the equation similar to that of the Ginzburg-Landau-Abrikosov-Gor’kov 19,20 ϩ␺0*͑rЈ,t͒␺0͑r,t͒u͑rЈ,t͔͒ ͑GLAG͒ theory. In particular, the linearized equation ͑4͒ takes the form

ϩ drЈV͑rϪrЈ͒␺0͑rЈ,t͒␺0͑r,t͒v͑rЈ,t͒ Ϫie*A͒2ٌ͑ ץ ͵ i ␺ ͑r,t͒ϭ Ϫ Ϫ␮ ␺ ͑r,t͒ t 0 ͫ 2m ͬ 0ץ 9͒͑ and ϩ͵ drЈV͑rϪrЈ͒n͑rЈ,t͒␺0͑r,t͒ ϩie*A͒2ٌ͑ ץ Ϫi v͑r,t͒ϭ Ϫ Ϫ␮ v͑r,t͒ ˜ †˜ ͬ t ͫ 2mץ ϩ͵ drЈV͑rϪrЈ͒͗␺ ͑rЈ,t͒␺͑r,t͒͘␺0͑rЈ,t͒. 2 ϩ͵ drЈV͑rϪrЈ͓͉͒␺0͑rЈ,t͉͒ v͑r,t͒ ͑16͒ ϩ␺ ͑rЈ,t͒␺*͑r,t͔͒v͑rЈ,t͒ The last two terms are the Hartree-Fock corrections to the 0 0 normal-state single-boson energy, respectively. This can be reduced to the renormalization of the normal-state single- * * ϩ͵ drЈV͑rϪrЈ͒␺0 ͑rЈ,t͒␺0 ͑r,t͒u͑rЈ,t͒. boson energy spectrum (m) and of the chemical potential, as discussed in the last section. One of the solutions to Eq. ͑16͒ ͑10͒ is a trivial ␺0ϭ0. If this solution is compatible with the sum There is also another sum rule rule, Eq. ͑7͒, we have the normal state. On the other hand, if ␺0Þ0, the linearized equation ͑16͒ can be satisfied with the appropriate choice of the chemical potential ͑in a stationary ͓un͑r,t͒un*͑rЈ,t͒Ϫvn͑r,t͒vn*͑rЈ,t͔͒ϭ␦͑rϪrЈ͒, ͑11͒ homogeneous magnetic field͒. Therefore, the linearized ͚n equation for the order parameter does not determine the up- which retains the Bose commutation relations for all opera- per critical field at all. The upper critical field is determined tors. by the sum rule, Eq. ͑7͒, as the lowest field at which this rule 13 Unfortunately, the last set of BdG equations ͑9͒–͑11͒ is cannot be satisfied with ␺0ϭ0, rather than by the linear- applied only for low temperatures and small magnetic fields, ized GL equation of the GLAG theory, where ␮ϳTcϪT is when the depletion of the condensate is small. As an ex- fixed. ample for the homogeneous case and Aϭ0 the excitation Introducing the one-particle normal-state Green’s func- wave functions are plane waves tion 15 366 A. S. ALEXANDROV, W. H. BEERE, AND V. V. KABANOV 54

and Ecϭ␻H/2. Substituting this DOS into the sum rule we obtain the familiar Schafroth result2

2 4␲ nB Hc2͑T͒ϭ ϱ Ϫ3/2 ϵ0. ͑22͒ e*ͱ2mT͐0 d⑀⑀ The condensation of ideal charged bosons in the magnetic field is impossible which is a consequence of the one- dimensional motion in the lowest Landau level. To obtain a finite value of Hc2 one should go beyond the Hartree-Fock approximation, taking into account the broadening of the Landau levels. If this broadening is to have an effect on Hc2, then it must act to make the density of states converge to zero at Ec , ensuring that the integral in Eq. ͑19͒ will be finite.

FIG. 1. Contour transformation of the sum rule. IV. CONDENSATION OF BOSONS SCATTERED BY IMPURITIES 1 13 ␯͑i⍀n͒ϭ , ͑17͒ Here we extend the earlier study of the condensation of G i⍀nϪ⑀␯ϩ␮Ϫ⌺␯͑i⍀n͒ charged bosons scattered by impurities to low temperatures we can write the sum rule in the Matsubara representation as and ultrahigh magnetic ﬁelds, where the Born approximation for a single-impurity scattering fails. ϩ In the lowest order in impurity concentration the self- ei⍀n0 ϪT ϭn , ͑18͒ energy ⌺ in a magnetic ﬁeld is expressed in terms of a ͚ B 17 n,␯ i⍀nϪ⑀␯ϩ␮Ϫ⌺␯͑i⍀n͒ single-impurity t matrix

2 ͑0͒ Ϫ1 Ϫ1 where ⑀␯ϭ␻H(Nϩ1/2)ϩkz /2m is the free-particle spectrum ⌺ ⑀͒ϭ Gˆ ⑀͒ Ϫ Gˆ ⑀͒ , ͑23͒ ␯,␯Ј͑ ͓ ͑ ͔␯,␯Ј ͓ ͑ ͔␯,␯Ј in a homogeneous magnetic field H, ␻Hϭe*H/m, kz is the z component of the momentum, Nϭ0,1,2,...,⍀ ϭ2␲Tn, n ⌺␯,␯Ј͑⑀͒ϭnimt␯,␯Ј͑⑀͒, ͑24͒ nϭ0,Ϯ1,Ϯ2,...,andtheself-energy ⌺␯(⍀n) takes into account the interaction terms. The complete set of the quantum where nim is the concentration of impurities. This expression corresponds to the summation of all noncrossing diagrams numbers is ␯ϭ(N,kz ,kx) with the energy degenerate for the ͑ladder approximation͒.18,19,21–23 The t matrix in the mag- quantum number kx . It is convenient to replace the summa- netic field is derived by the use of the general formalism17 tion for the contour integral over C1, C2, and C3 as shown in Fig. 1. Then shifting the contour to the real axis and by the fully taking into account the multiple scattering: analytical continuation of the Green’s function to the upper R A 2␲ f ␾␣*͑r͒␾␤͑r͒ (G ) and lower (G ) half-planes the sum rule is given by t ϭ dr , ͑25͒ ␣,␤ m ͵ 2␲ f * ϱ d⑀␳͑⑀,H͒ 1ϩ ͚ ␾␯ ͑r͒␾␯Ј͑r͒G␯,␯Ј͑⑀͒ m ␯,␯ ϭnB , ͑19͒ Ј ͵␮ϩ0exp͓͑⑀Ϫ␮͒/T͔Ϫ1 where ␾␯(r) is the one-particle wave function in the mag- A,R A,R Ϫ1 netic field. In the Landau representation with the magnetic where G␯ (⑀)ϭ͓⑀Ϫ⑀␯Ϫ⌺␯ (⑀)͔ , and field along the z axis ␾␯(r) is given by

1 A R ␳͑⑀,H͒ϭ ͚ ͓G␯ ͑⑀͒ϪG␯ ͑⑀͔͒ ͑20͒ 1 2␲i ␯ ␾ ͑r͒ϭ exp i͑k xϩk z͒ ␯ 1/4 1/2 N ͫ x z ␲ aH ͱ2 N! is the one-particle density of states ͑DOS͒ in the magnetic 2 field, with the edge of the spectrum, E , determined by the 1 yϪy 0 yϪy 0 c Ϫ H . ͑26͒ condition ␳(E ,H)ϭ0. The integral over C is compensated 2 ͩ a ͪ ͬ Nͩ a ͪ c 2 H H for by the opposite sign contribution of C˜ and C˜ ͑Fig. 1͒. 1 3 H (y) is a Hermite polynomial, y ϭk a2 is the center of This and the condition ␮рE determine the lower limit in N 0 x H c the cyclotron motion, and a ϭ1/ e*H is the magnetic Eq. ͑19͒ as ␮ϩ0. It is apparent now that the upper critical H ͱ length. G␯,␯ (⑀) is the one-particle Green’s function aver- field is determined by Eq. ͑19͒ with HϭHc2 and ␮ϭEc . Ј Hence, the problem of the Bose-Einstein condensation in the aged over the position of the impurities, and f is the scatter- magnetic field is reduced to the calculation of the normal- ing amplitude of a particle with zero energy in zero magnetic field. This equation is similar to that discussed by Skobov22 state DOS, Eq. ͑20͒. In particular, for free particles we have 23 A,R and Magarill and Savvinykh. ⌺␯ (⑀)ϭϮi␦,␦ ϩ0, so that → It should be pointed out that the self-energy ⌺␯,␯Ј(⑀) has diagonal (␯ϭ␯Ј) and nondiagonal (␯Þ␯Ј) components. ͱ2m3/2␻ ϱ H Ϫ1/2 Both diagonal and nondiagonal components contribute to the ␳͑⑀,H͒ϭ 2 Re ͚ ͓⑀Ϫ␻H͑Nϩ1/2͔͒ , ͑21͒ 4␲ Nϭ0 density of states, N(⑀)ϭTr ImG␯,␯(⑀)/␲. 54 THEORY OF THE CHARGED BOSE GAS: BOSE- . . . 15 367

There are two dimensionless parameters in this problem. 3/2 m ␻H The first is the ratio of the scattering amplitude f to the N͑⑀͒ӍNcol͑⑀͒ϭ ͱ⑀ϪEc, ͑33͒ ͱ621/3⌫ magnetic length aH . At low temperatures, where Hc2 is 0 large, this parameter is not small. Therefore, at low tempera- 2 2 2/3 where the parameter ⌫0ϭ(4␲f nim /aH) /2m is the colli- tures, one has to go beyond the Born approximation for a sion broadening of the Landau levels. In this limit tempera- single-impurity scattering by the use of Eq. ͑25͒. The second ture dependence of the upper critical field has been found by parameter is the ratio of the mean-free path, lϭv␶,tothe Alexandrov:13,14 magnetic length. This is the same as ␻H␶ for particles at the Ϫ1 lowest Landau level where vϷ(maH) . Here 1/␶ is the ⌽0 H Ϸ ͑T /T͒9/2, ͑34͒ order of the collision broadening of the Landau levels, which c2 2␲␰2 c0 depends on the magnetic field as H2/3 ͑see below͒.Asa with ⌽ ϭ␲/e the flux quantum, T 3.3n2/3/m, and the result in the low-temperature limit, when Hc2 is large, the 0 c0Ӎ B 3/4 2 nondiagonal components of the Green’s function G␯,␯Ј are ‘‘coherence’’ length ␰ϭ␲ͱ(e*/2e)/3 f nim is proportional small when compared with the diagonal components as to the mean-free path in zero magnetic field. 1/(␻H␶)Ӷ1. It should be pointed out that the t matrix, Eq. When the temperature decreases the low-energy excita- ͑25͒, is essentially nondiagonal, but the contribution of the tions become more important. Then the multiple scattering nondiagonal elements of the t matrix to the density of states gives the leading contribution. In that case (␥ϽϽ1) broad- is small, as 1/(␻H␶). ening of the Landau levels is determined by the parameter The equation for the diagonal part of the self-energy, ⑀0, and so ⌺␣(⑀), has the following form: 2 ⑀cϭ͑␥/2͒ , ͑35͒ 2␲ fnim ␾␣*͑r͒␾␣͑r͒ ⌺ ͑⑀͒ϭ dr . 3/2 ␣ m ͵ 1ϩ͑2␲ f /m͚͒ ␾*͑r͒␾ ͑r͒G ͑⑀͒ m ␯ ␯ ␯ ␯ N͑⑀͒ӍN ͑⑀͒ϭ ͱ⑀ϪE . ͑36͒ ͑27͒ scat 2 6 2 c 4␲ ͱ2aHnim

By the use of the fact that in a homogeneous system G␯ is It should be pointed out that the ratio Ncol(⑀)/Nscat(⑀) 2 kx independent one can perform the summation over kx in ϰ(⌫0 /⑀0) . The temperature dependence of the upper criti- the denominator. Then integrating over r we obtain the fol- cal field is different in that case and is given by lowing equation in the ultraquantum limit (Nϭ0): ⌽0 H Ϸ ͑T /T͒1/2 ͑37͒ 1 c2 2␲␰2 c0 ␴͑˜⑀͒ϭ␥ , ͑28͒ 1ϩi/ͱ˜⑀Ϫ␴͑˜⑀͒ Ϫ1/3 where ␰ϭͱe*/2e(2nim) . With the temperature lowering the upper critical field di- where ␥ϭ2␲ fnim /⑀0m, ␴ϭ⌺/⑀0, ˜⑀ϭ⑀/⑀0, and ⑀ ϭ f 2/2ma4 . verges. Therefore in the ultralow-temperature limit we have 0 H Ϫ4.5 The density of states is given by ⑀0ӷ⌫0. As a result the T divergence in Eq. ͑34͒ trans- forms into TϪ0.5 in Eq. ͑37͒ when T goes to zero. In this 3/2 limit we also expect that the localization of bosons, due to ͱ2m ␻H 1 N͑⑀͒ϭ Re . ͑29͒ the high magnetic field, in the random potential might be 4 2 ␲ ͱ⑀Ϫ⌺͑⑀͒ important. Solving the cubic equation ͑28͒ we find the edge, where a nonzero density of states appears, as V. NORMAL-STATE PLASMON IN THE ULTRAQUANTUM LIMIT E 1/9Ϫ3␥ c 1/3 As discussed above the condensation temperature in a ⑀cϵ ϭA ϩ 1/3 ϩ␥Ϫ2/3, ͑30͒ ⑀0 A magnetic field is zero without scattering because the density with Aϭ1ϩ(3/2)3͕ͱx͓␥ϩ(2/3)3͔3Ϫ(10/27Ϫ␥)2͖. Ex- of states diverges at low energy. In a clean system the self- panding the imaginary part of the self-energy near the edge energy arising from the Coulomb scattering is the main we arrive with the following expression for N(⑀): source of the level broadening, resulting in a vanishing density of states at low energy and thus in a finite condensation 3/2 2 temperature Tc . We can expect that at low temperatures den- ͱ2m ␻H R ͑Rϩ1͒ N͑⑀͒ϭ 2 ͱ⑀ϪEcͱ , ͑31͒ sity fluctuations of the CBG, i.e., plasmons, play the role of 4␲ ⑀0 ⑀c͑3Rϩ1͒Ϫ␥ impurities. Plasmons are defined as poles of the renormalized Cou- where RϭϪ(6⑀cϪ2Ϫ6␥)/(8⑀cϩ␥). One can distinguish two limiting cases. The first one cor- lomb potential. We calculate the renormalized Coulomb po- responds to the collision broadening of the Landau tential as a summation series of polarization loops ͑Fig. 2͒, levels21–24 when the multiple scattering by a single impurity given by is negligible. This is ␥ӷ1, and so V͑q͒ 2/3 ͑q,i⍀n͒ϭ , ͑38͒ ⑀cϭ␥Ϫ3/͑␥/2͒ ͑32͒ D 1ϪV͑q͒⌸͑q,i⍀n͒ and where 15 368 A. S. ALEXANDROV, W. H. BEERE, AND V. V. KABANOV 54

FIG. 2. Plasmon propagator.

␯Ј 2 ⌸͑q,i⍀n͒ϭT ͚ ͚ ͉I␯ ͑q͉͒ ␯͑i⍀m͒ ␯Ј͑i⍀mϩi⍀n͒ i⍀m ␯,␯Ј G G ͑39͒ is the polarization loop in the Matsubara representation, and ␯Ј I␯ (q) is the matrix element defined as FIG. 3. Plasma frequency as a function of momentum. 2 The factor 2␲aH arises from the summation over kx , I␯Ј q ϭ dreiq–r * r r . 40 ␯ ͑ ͒ ͵ ␾␯ ͑ ͒␾␯Ј͑ ͒ ͑ ͒ 1 In the Landau representation, Eq. ͑26͒, one obtains ͚ ϭ 2 . ͑46͒ kx 2␲aH NЈ! 1/2 This summation is finite because it is limited by the condi- I␯Ј͑q͒ϭ␦ ␯ kxϩqx ,kЈ␦k ϩq ,kЈ tion that the Landau eigenfunction lies within the normalized x z z z ͩ N! ͪ volume (ϵ1) being considered. This condition places a NϪNЈ 2 ͑Ϫqxϩiqy͒aH boundary on allowed values of kx as y 0ϭkxaH is the center ϫ LNϪNЈ͑a2 q2 ͒ ͫ ͱ2 ͬ NЈ H Ќ of the orbit. Using the approximation in Eq. ͑45͒, the single-particle a2 polarization loop H 2 2 qx ϫexp Ϫ q ϩia qy kxϩ ͑41͒ ͫ 4 Ќ H ͩ 2 ͪͬ 2 2 2 nB qz exp͑ϪaHqЌ/2͒ ⌸͑q,i⍀ ͒ϭ ͑47͒ 2 2 NϪNЈ n 2 4 2 for N N, where q ϭ q ϩq and L (x) is a Laguerre m ͑i⍀n͒ Ϫqz /4m Јр Ќ ͱ x y NЈ polynomial. takes on the same form as for zero temperature.15 This gives Now considering the case of a high magnetic field and the plasmon propagator as low temperature, ␻HӷT, all the bosons are in the lowest ϭ ͑q,i⍀n͒ Landau level, i.e., N NЈϭ0. Also at the condensation D point, as has been previously noted, the chemical potential 4␲e*2 ␮ is at the edge of the spectrum, Ec , which for free bosons ϭ q2 is Ecϭ␻H /2. Taking the free particle Matsubara Green’s functions, as a first-order approximation for the polarization ͑i⍀ ͒2Ϫq4/4m2 loop, we obtain n z ϫ 2 4 2 2 2 2 2 2 , ͑i⍀n͒ Ϫqz /4m Ϫ4␲e nBqz exp͑ϪaHqЌ/2͒/mq n k ϩq 2/2m Ϫn k2/2m ͓͑ z z͒ ͔ ͑ z ͒ ͑48͒ ⌸͑q,i⍀n͒ϭ ͚ 2 kz ,kx i⍀nϪkzqz /mϪkz /2m 2 2 2 where q ϭqz ϩqЌ . 2 2 The frequency of plasma excitations is given by the poles ϫexp͑ϪaHqЌ/2͒, ͑42͒ of the retarded plasmon propagator, which is found from the where the Bose distribution function is analytical continuation of the Matsubara plasmon propagator to the real axis, 1 n͑⑀͒ϭ ⑀/T . ͑43͒ DR͑q,⍀͒ϭ ͑q,i⍀ ⍀ϩi␦͒. ͑49͒ e Ϫ1 D n→ The poles of the retarded plasmon propagator, and thus the We note that the sum rule, Eq. ͑19͒, for free particles, plasma excitations, occur at after summation over Matsubara frequencies is 2 2 2 1/2 qz qz k2 ⍀ϵ␻ ϭ ϩ␻2 exp͑Ϫ 1 a2 q2 ͒ . ͑50͒ z q ͫͩ2mͪ p q2 2 H Ќ ͬ ͚ n ϭnB . ͑44͒ k ,k ͩ 2mͪ z x For qЌϭ0 this is the same as the nonmagnetic case, For ultralow temperatures we can approximate the Bose dis- ⍀ϳ␻ for q 0. However, for q Þ0 the plasmon is now p z→ Ќ tribution function to a ␦ function of weight nB at zero mo- gapless and sound like for low momenta, ͑Fig. 3͒. The effect mentum, of the magnetic field is to confine the bosons to their Landau

2 orbitals along the z axis. This allows freedom of movement kz along the z axis, but restricts movement perpendicular to the n Ϸ␦ 2␲a2 n . ͑45͒ ͩ 2mͪ kz,0 H B z axis, making the system stiffer to perturbations in that di- 54 THEORY OF THE CHARGED BOSE GAS: BOSE- . . . 15 369

All terms containing the Bose-distribution factor n(⑀) are removed, leaving the zero-temperature result as this is the most signiﬁcant part. The retarded self-energy is easily derived from the Mat- subara self-energy,

FIG. 4. Coulomb self-energy. ⌺R͑⑀͒ϭ⌺ ͑i⍀ ⑀ϩi␦͒. ͑55͒ ␯ ␯ → rection. So the polarization perpendicular to the z axis is The result is similar to the case of scattering from the impu- lowered in a magnetic field, resulting in soundlike, rather rities if we take the plasma frequency in the denominator, than plasmalike, excitations. ⑀Ϫ⑀ЈϪ␻qϩi␦, as zero. This is an elastic scattering approxi- The approximation used in Eq. ͑45͒ produces a result mation, which is reasonable if the characteristic plasma fre- similar to the zero-temperature superfluid Bogoliubov ͑3D͒ quency is small compared with the characteristic broadening spectrum.3 The validity of this approximation arises from the of the Landau level. We will show this to be true for the condition T Ӷ␻ . ultrahigh magnetic field, which lowers the condensation tem- c0 H perature. This ensures that all of the bosons are at very low The spectral function in terms of the retarded self-energy energies even in the normal state, hence validating Eq. ͑45͒. is

1 VI. CONDENSATION OF CBG DUE A␯͑⑀͒ϭϪ2Im 2 R . ͑56͒ TO THE COULOMB SCATTERING ⑀Ϫkz/2mϪ⌺␯ ͑⑀͒ϩ␮

The self-energy due to boson-boson scattering in the Mat- We now have a self-consistent equation for the self- subara representation is energy,

2 2 2 ␯ A ͑⑀Ј͒ Ј 2 qz exp͑ϪaHqЌ͒ kzϩqz ⌺␯͑i⍀n͒ϭϪT ͚ ͉I␯ ͑q͉͒ ͑q,i⍀m͒ ␯Ј͑i⍀nϩi⍀m͒. R 2 2 D G ⌺k ͑⑀͒ϭe* ␻ p d⑀Ј 2 2 2 . q,␯Ј,i⍀m z ͚ ͵ Ϫ ϩ q ͑qz ϩqЌ͒ ␻q ͑⑀ ⑀Ј i␦͒ ͑51͒ ͑57͒ This expression includes all ‘‘bubble’’ diagrams which is a Extending the summation over q to an integral, fair approximation for the long-range Coulomb potential ͑Fig. 4͒. 1 As the density of states, Eq. ͑20͒, is obtained from the ϭ dqz dqЌqЌ , ͑58͒ ͚ ͑2␲͒2 ͵ ͵ retarded Green’s functions, we will transform this Matsubara q representation using we can begin to build an approximate solution. For high magnetic fields aH is small and for low qz the integral dqЌ 1 A␯͑⑀Ј͒ can be approximated as ͑i⍀ ͒ϭ d⑀Ј , ͑52͒ ␯ n ͵ Ϫ G 2␲ i⍀n ⑀Ј 2 2 2 ϪaHqЌ qЌqze 1 where A␯(⑀) is the spectral function defined as dq . 59 ͵ Ќ 2 2 2 Ϸ ͑ ͒ ͑qzϩqЌ͒ ␻q ␻p A ͑⑀͒ϭ͓GAϪGR͔ϭϪ2ImGR. ͑53͒ ␯ ␯ ␯ ␯ To eliminate the chemical potential from the equation we A R* choose that ⑀ is zero at the mobility edge, i.e., Ecϭ0, and Note that G␯ ϭG␯ . Summing over the Matsubara frequencies the resulting thus ␮ is also zero at the condensation temperature. As in the equation is case of impurity scattering of bosons we expect the self- energy to be momentum independent. The integration of 2 A (⑀Ј) over dq gives the density of states, which for a 1 4␲e* kzϩqz z ⌺ ͑i⍀ ͒ϭ ͉I␯Ј͑q͉͒2 d⑀Ј A ͑⑀Ј͒ ␯ n ͚ ␯ 2 ͵ q2 ␯Ј momentum-independent self-energy is q,␯Ј ␲ 2 2 2 2 4 i ␻ pqz exp͑ϪaHqЌ/2͒ n͑␻q͒ ␲ ϫ dqzAk ϩq ͑⑀Ј͒ϭͱ2m Im R 1/2 . ͑60͒ ͫ 2q2␻ ͩ i⍀ Ϫ⑀Јϩ␻ ͵ z z ͓⑀ЈϪ⌺ ͑⑀Ј͔͒ q n q Then the imaginary part of ⌺ is obtained by integrating over n͑␻q͒ϩ1 ϩ the energy in Eq. ͑57͒ as i⍀ Ϫ⑀ЈϪ␻ ͪ n q 2 4 2 ͱ2 i ͓͑⑀ЈϪi⍀n͒ Ϫqz/4m ͔n͑⑀Ј͒ R 3/4 3/2 ϩ , ͑54͒ Im⌺ ͑⑀͒ϭϪ 1/4 rs ␻ p Im R 1/2 . ͑61͒ ͑⑀ЈϪi⍀ Ϫ␻ ͒͑⑀ЈϪi⍀ ϩ␻ ͒ͬ 3 ͓⑀Ϫ⌺ ͑⑀͔͒ n q n q where ␻q is the plasma frequency for momentum Despite the appearance of the minus sign in Eq. ͑61͒ the qϭ͕qz ,qЌ͖ given by Eq. ͑50͒. This can be simplified for the imaginary self-energy is positive. This is not obvious from case of ultralow temperatures when the only remaining sig- Eq. ͑61͒, but comes from the relationship of Im⌺R to the nificant term is from the spontaneous emission of plasmons. density of states, to be given later. As an exercise we can use 15 370 A. S. ALEXANDROV, W. H. BEERE, AND V. V. KABANOV 54

e* ␰Ϸ0.92nϪ1/3rϪ1/2 ͑68͒ B s ͱ2e is the coherence length due to the Coulomb scattering. As was stated earlier, the elastic scattering approximation is only valid if the plasma frequency associated with the scattering is less than the width of the Landau level broadening, i.e., Re⌺R. This gives the condition

R 1/2 ␻qӶRe⌺ Ϸ␻ prs . ͑69͒ We can estimate the plasma frequency, from Eq. ͑50͒,as

qz ␻qϷ ␻p, ͑70͒ qx FIG. 5. Density of states in a magnetic ﬁeld without ͑a͒ and with with the momenta associated with the scattering process as Ϫ1 1/2 ͑b͒ the Coulomb interaction. qxϷaH and qzϷͱm␻ prs . The estimation for qz comes as the momentum associated with the broadening, i.e., R 2 R a ﬁrst-order approximation for the self-energy, ⌺ ϭϩi␦. qz /2mϷRe⌺ . Thus the condition for using the elastic scat- Substituting this into Eq. ͑61͒, tering approximation is

Ϫ1/2 i 1 ␻Hӷ␻ prs ϷTc . ͑71͒ Im ϭϪ , ͑62͒ 0 ͓⑀Ϫ⌺R͑⑀͔͒1/2 ͱ⑀ In short the effect of including the interaction is to restore where the minus sign arises from taking the square root cut the density of states to its 3D form, and it is the energy line on the positive real axis. dependence of the DOS which determines the temperature Expanding the cubic equation ͑61͒, and taking the limit of dependence of Hc2. The broadening of the Landau levels is R small Im⌺ and small ⑀, we ﬁnd that for Eq. ͑61͒ to be proportional to the interaction strength, i.e., rs . The lower satisﬁed in the lowest order, Im⌺Rϭ⑀ϭ0, the interaction, the less the damping of the Landau levels and thus the lower Hc2 is. 1/2 ␻ prs Re⌺R͑⑀͒ϭ . ͑63͒ ͑2ͱ3͒1/3 The next order in Im⌺R and ⑀ gives VII. CONCLUSION

12 1/2 In contrast to the Fermi liquid, in which the long-range R 1/2 1/4 Coulomb interaction is screened and high-energy plasmons Im⌺ ͑⑀͒ϭ 1/3 ␻ p rs ͱ⑀. ͑64͒ ͫ 5͑2ͱ3͒ ͬ are irrelevant for low-frequency kinetics, allowance for the Coulomb interaction at finite temperatures in the CBG is a The real part of ⌺R can also be derived from the principal more complicated matter because plasmons and one-particle part of Eq. ͑57͒, which for low ⑀ gives a constant of the same excitations are essentially the same in the long-wave limit. In order as Eq. ͑63͒. this paper we derived the Bogoliubov–de Gennes-type equa- The density of states is directly related to the imaginary tions for the CBG at finite temperatures and studied the part of the self-energy, Bose-Einstein condensation in an ultrahigh magnetic field. In 1 contrast with the canonical GLAG theory the upper critical ␻H R ␳͑⑀,H͒ϭ 2 nB Im⌺ ͑⑀͒. ͑65͒ field of CBG is determined by the sum rule beyond the ␲ ␻ p mean-field approximation. The damping of the Landau levels For high energies the density of states decreases as 1/ͱ⑀, the due to the scattering is a key feature allowing Bose-Einstein condensation in a magnetic field. We have extended the ear- same as the undamped result, Eq. ͑21͒. Using this to plot the 13,14 density of states ͑Fig. 5͒, and remembering the sum rule lier results on the impurity scattering in CBG to very low temperatures, when the critical magnetic field is very high, 1 so that the multiple impurity scattering becomes important. n ϭϪ dq d A n , 66 Ϫ4.5 Ϫ0.5 B 3 2 ͵ z ͵ ⑀ ␯͑⑀͒ ͑⑀͒ ͑ ͒ As a result the crossover from the T to the T expo- ͑2␲͒ aH nent is found in the divergent Hc2(T) when the temperature we can see that the low-energy solution, Eq. ͑64͒, is applied is lowered. The plasmon dispersion as well as the Bose- R for the temperature range TӶRe⌺ . The value of Hc2 for Einstein condensation due to Coulomb scattering is studied which the sum rule can no longer be satisfied is then in an ultrahigh magnetic field as well. The plasmon is gapless. The Coulomb scattering results in the TϪ1.5 exponent of ⌽0 Hc2 at low temperatures. In general, the CBG has an unusual H ϭ ͑T /T͒3/2, ͑67͒ c2 2␲␰2 c0 phase transition from the normal to the condensed phase in the magnetic field with remarkable positive curvature. At Ϫ␯ where now low temperatures we find Hc2ϳT . Depending on the tem- 54 THEORY OF THE CHARGED BOSE GAS: BOSE- . . . 15 371 perature interval and impurity concentration the critical ex- ACKNOWLEDGMENTS ponent ␯ varies from 0.5 up to 4.5. We believe that the CBG is a relevant model of the ground state of high-Tc cuprates We acknowledge helpful discussions with our colleagues with a very small coherence volume. If it is so, our predic- at IRC, Cambridge University, and at the Department of tions can be verified by the resistive and magnetic measure- Physics, Loughborough University, and the Royal Society ments in high magnetic fields. financial support for one of us ͑V.V.K.͒.

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