(Inverse) Magnetic Catalysis in Bose-Einstein Condensation of Neutral Bound Pairs

# (Inverse) Magnetic Catalysis in Bose-Einstein Condensation of Neutral Bound Pairs

## Abstract

The Bose-Einstein condensation of bound pairs made of oppositely charged fermions in a magnetic field is investigated. We find that the condensation temperature shows the magnetic catalysis effect in weak coupling and the inverse magnetic catalysis effect in strong coupling. The different responses to the magnetic field can be attributed to the competition between the dimensional reduction by Landau orbitals in pairing dynamics and the anisotropy of the kinetic spectrum of fluctuations (bound pairs in the normal phase).

###### pacs:
74.20.Fg,03.75.Nt,11.10.Wx,12.38.-t

## I Introduction

The behavior of a system consisting of charged fermions in a magnetic field had attracted considerable interests in recent years especially in strongly interacting matter, where fundamental constituent-quarks exhibit a host of interesting phenomenalectnotes (), such as Chiral Magnetic effect and Magnetic Catalysis of chiral symmetry breaking. The latter one, which will be the main motivation to the present work, involves the dimensional reduction by the Landau orbitals of charged fermions under a magnetic field. We shall investigate another (nonrelativistic) system that shares the same physics, the Bose-Einstein Condensation (BEC) of composite bosons—-neutral bound pairs made of two oppositely charged fermions in the presence of an external magnetic field.

The underlying theory of strong interaction-Quantum Chromodynamics(QCD) possesses chiral symmetry for massless quarks, which is spontaneously broken by a long range order because of the condensation of bound pairs formed by quark and antiquark. As the density of states , with respect to the single quark energy , vanishes at the Dirac point (analog of the Fermi surface in a metal), a threshold coupling has to be attained for pairing. The terminology ”Magnetic Catalysis” refers to the fact that chiral symmetry is always spontaneously broken at finite magnetic field regardless of the coupling strengthKlimenko (); Miransky (). The physical reason of this effect is the dimension reduction in the dynamics of fermion pairing in a magnetic field. The motion of charged particle would be squeezed to a discrete set of Landau orbitals and is one-dimensional within each orbital. The system would thus become dimension when the magnetic field is sufficiently strong than the mass and energy of the fermions, which would be restricted entirely in the lowest Landau level (LLL) only. Consequently, the density of states at the Dirac point becomes a nonzero constant proportional to the magnetic field . Such an enhancement would make the chiral condensate happen regardless of the interaction strength, the magnetic field thus plays a role as the catalysis. This is quite similar to the Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity, where a nonzero density of states at the Fermi surface supports Cooper pairing with an arbitrarily weak attraction.

It would be natural to expect a higher transition temperature from the chiral broken phase to the chiral symmetric phase due to magnetic catalysis effect. This is indeed the case within mean-field approximations of effective model studies, it was found that the chiral phase transition is significantly delayed by a nonzero magnetic field even including the meson contributionFukushima (); Chernodub (); Skoko (). The pseudo-critical temperature of chiral restoration was also found to increase linearly with the magnetic field in a quark-meson model using the functional renormalization group equationKanazawab (). The recent lattice calculationslatticetemp (), however, provide surprising results that the pseudo-critical temperature of chiral restoration drops considerably for increasing magnetic field. On the other hand, the chiral condensate increase with increasing magnetic field at low temperature consistent with magnetic catalysis while it turns out to be monotonously decreasing at high temperaturelatticegap (), which is in apparent conflict with the magnetic catalysis and termed as inverse magnetic catalysis evoking an extensive studiesSchmitt (); Fukushimapawlowski (); Fukushimahidaka (); Kojo (); Bruckmann (); Mei (); Efrain (); Pinto ().

While mean field approximation gives sensible results in certain circumstances, fluctuations can break it down, especially in strong coupling domain or in lower dimensions. As was shown in CMWH () in the absence of magnetic field, a long range order cannot survive at a nonzero temperature in the spatial dimensionality two or less because of the fluctuation of its phase. A long wavelength component of the fluctuation variance goes like with the momentum, which gives rise to infrared divergence of the momentum integration in two and lower dimensions. The anisotropy introduced by a magnetic field renders the long wavelength fluctuation , with a positive constant between zero and one. Such a distortion of the bosonic spectrum towards dimensionality one (), as a consequence of the dimension reduction of the pairing fermions, would enhance the phase fluctuation. A preliminary study of the Ginzburg-Landau theory of the chiral phase transitionGinzburgwindow () reveals the same effect and the Ginzburg critical window gets widened in the presence of magnetic field, indicating the enhancement of the long wavelength fluctuations.

The BEC of bound pairs made of oppositely charged fermions in a magnetic field provides another platform to explore the competition between the enhanced Cooper pairing by Landau orbitals and the enhanced phase fluctuation by the distortion of the bosonic spectrum. Our system corresponds to the BEC limit of the BCS/BEC crossover, which has been studied extensively in the absence of magnetic field for nonrelativistic fermions Nozieres (); Sademelo () and relativistic ones abuki (); Dengjian (); Efraincrossover (); Zhuang (). We follow the functional integral formulation developed in Sademelo () and calculate the leading (Gaussian) correction to the effective action. A technical simplification in the BEC limit is that all summations over Landau orbitals involved can be carried out analytically, resulting in an explicit formula of the critical temperature under an aribitrary magnetic field. We found that the critical temperature for the BEC was dramatically affected by the magnetic field exhibiting magnetic catalysis or inverse magnetic catalysis depending on the coupling strength. In the weak coupling domain, where no bound pairs(composite bosons) exist at zero magnetic field, the magnetic catalysis induces bound pairs and thereby a BEC. The critical temperature increases with increasing magnetic field. In the strong coupling domain, where bound pairs exist without magnetic field, an inverse magnetic catalysis was found. The critical temperature decreases as increasing magnetic field, signaling the enhanced fluctuation in a magnetic field.

The rest of the paper is organized as follows: in Section II we lay out the general formulation and present the mean field approximation. The fluctuations beyond the mean field theory, which is necessary for BEC, is calculated under the Gaussain approximation in Section III. The magnetic field dependence of the BEC temperature is investigated in Section IV. Section V is devoted to the conclusions and outlooks. Some calculation details and useful formulas are presented in the Appendices A, B and C. Throughout the paper, we will work Euclidean signature with the four vector represented by with the Matsubara frequency for bosons and for fermions .

## Ii General Formulation and Mean Field Theory

We consider a system consisting of nonrelativistic fermions of mass and chemical potential with opposite charge interacting through a short ranged instantaneous attractive interaction. The Hamiltonian density reads

 H[ψ,ψ†]= ∑σ=±ψ†σ(x)[(−i∇+σeA)22m−μ]ψσ(x) −gψ†+(x)ψ†−(x)ψ−(x)ψ+(x). (1)

where is the charge magnitude carried by each fermion, , and is the vector potential underlying an external magnetic field, . To avoid the Meissner effect, only fermions with opposite charges can pair. For the sake of simplicity, we ignore the spin degrees of freedom. The thermodynamic potential density of the system reads

 Ω=−1βVlnZ (2)

where and is the volume of the system. The path integral representation of the partition function reads

 Z=∫Dψ†σ(x)Dψσ(x)exp[S]. (3)

with the action given by

 S=∫dτd3x(−∑σψ†σ(x)∂∂τψσ(x)−H[ψ,ψ†]). (4)

where the Grassmann variables and are antiperiodic in and independent of each other. The number density of fermions is given by

 n=−(∂Ω∂μ)T,B. (5)

Introducing the standard Hubbard-Stratonovich field coupled to , the partition function is converted to

 Z= ∫Dψ†σ(x)Dψσ(x)DΔ∗(x)DΔ(x)exp{∫dτd3x(−ψ†σ(x)∂∂τψσ(x)−ψ†σ(x)(−i∇+σeA)22mψσ(x) +μψ†σ(x)ψσ(x)+Δ(x)ψ†+(x)ψ†−(x)+Δ∗(x)ψ−(x)ψ+(x)−|Δ(x)|2g)}. (6)

and becomes bilinear in fermion fields. In terms of the Nambu-Gorkov(NG) spinors

 Ψ(x)=(ψ+(x)ψ†−(x)),Ψ†(x)=(ψ†+(x),ψ−(x)). (7)

the partition function becomes

 Z= N∫DΨ†(x)DΨ(x)DΔ∗(x)DΔ(x)exp∫dτd3x [∫dτ′d3x′Ψ†(x)G−1(x,x′)Ψ(x′)−|Δ(x)|2g]. (8)

with

 G−1= ⎡⎢ ⎢⎣−∂∂τ−(−i∇+eA)22m+μΔ(x)Δ∗(x)−∂∂τ+(−i∇+eA)22m−μ⎤⎥ ⎥⎦ ×δ4(x−x′). (9)

where is a constant. Integrating out the fermionic NG fields, we obtain the parition function

 Z=N∫DΔ∗(x)DΔ(x)exp(S[Δ(x)]), (10)

with the action given by

 S[Δ]=−∫dτd3x|Δ(x)|2g+TrlnG−1(x,x′), (11)

where the trace in (11) is over space, imaginary time and NG indices.

For a uniform magnetic field considered in this work, we choose the Landau gauge, in which the vector potential is and the magnetic field is thus along direction and the system is translationally invariant. To explore the long range order of the system, we make a Fourier expansion

 Δ(x)=√1βV∑ωnk,ke−iωnkτ+ik⋅xΔ(iωnk,k)=Δ0+Δ′(x), (12)

where we have singled out the zero energy-momentum component of the expansion. Carrying out the path integral over , we end up with

 Z=N∫DΔ∗0DΔ0exp[−βVΞ(|Δ0|)], (13)

and the thermodynamic potential density in the infinite volume limit equals to the value of the function at its saddle point determined by

 (∂Ξ∂|Δ0|2)T,μ,B=0. (14)

A nontrivial saddle point, , corresponds to a long range order and the superfluidity phase of the system. drops to zero at the transition to the normal phase. Expanding the function in a power series in ,

 Ξ(|Δ0|2)=Ξ(0)+α(T,μ,B)|Δ0|2+..., (15)

where , and the coefficients of higher order terms of (15) include the contribution from the fluctuation field defined in (12). A negative value of the coefficient signals the instability of the normal phase, , and the critical temperature , and the chemical potential for the instability satisfy the condition

 α(Tc,¯μ,B)=0. (16)

The critical temperature at a given density is obtained by solving both eqs. (16) and (5) simultaneously.

The mean field approximation ignores and the eigenvalues of the inverse propagator (9) with can be easily found. We obtain

 Ξ(|Δ0|2)= 1g|Δ0|2−1βV∑n∑ky,kz;l (17) ln[(iωn)2−(εkz+lωB−χ)2−|Δ0|2].

where are the Landau levels and . We have also defined with the cyclotron frequency. The symbol is the abbreviation of . The coefficient under the mean field approximation can be readily extracted from the Taylor expansion of RHS of (17) and the condition (16) becomes

 1g=12V∑ky,kz;l1εkz+lωB−¯χtanhεkz+lωB−χ2Tc. (18)

In BCS limit, this equation would be solved to yield the critical temperature with the chemical potential given by that of an ideal Fermi gas at a given density (the limit of eq.(5) with at , and ). In BEC limit, however, the role is reversedSademelo (). Eq.(18) determines the chemical potential. In the latter case, the fluctuation contribution to has to be restored to determine the critical temperature at a given density through (5).

For negative with , the hyperbolic tangent function in (18) may be approximated by one and we end up with

 −m4πas=12V⎡⎣∑ky,kz;l1εkz+lωB−¯χ−∑k12εk⎤⎦ (19)

where we have introduced a renormalized coupling constant according to

 1gR≡1g−1V∑k12εk≡−m4πas (20)

with the -wave scattering length extracted from the low energy limit of the two-body scattering in vacuum and in the absence of a magnetic field so that the RHS is free from UV divergence. Carrying out the summation explicitly (for details, see appendix A), we find that

 −m4πas=√ωBm3/24√2πζ(12,|¯χ|ωB), (21)

In obtaining this equation, the contributions from all Landau levels have been taken into account and this summation gives rise to the Hurwitz zeta function, which was defined by

 ζ(s,a)=∞∑n=01(n+a)s. (22)

for and can be continuated to the entire -plane with a pole at in terms of its integral representation.

The eq.(21) sets the chemical potential at the energy of a bound pair of zero center-of-mass momentum in vacuum and this is the condition for the BEC of an ideal Bose gas. The contributions of the bound pairs of nonzero momentum, however, is ignored here. Therefore the mean field approximation is not sufficient and the contribution from the bound pairs with nonzero momenta to the density equation (5) has to be restored to determine the transition temperature (the density will be set low enough to justify the approximation .).

In the absence of magnetic field, the RHS of (21) becomes and we have a solution only for , which defines the strong coupling domain. The weak coupling domain, , however, entirely resides on the BCS side of the BCS/BEC crossover. When the magnetic field is turned on, the RHS of (21) can take both signs and a solution emerges in the weak coupling domain. This is caused by the dimensional reduction of the Landau orbitals, i.e., magnetic catalysis and the BEC limit can be approached in both strong and weak coupling domains.

## Iii Gaussian Fluctuation

The Guassian approximation of the fluctuation effect maintains to the quadratic order in the path integral (10), while including to all orders. To locate the pairing instability starting from the normal phase, where , the Gauss approximation amounts to replace of (11) by its expansion to the quadratic order in the entire boson field .

 S[Δ] ≃Seff.[Δ]=S[0]−∫dτd3x|Δ(x)|2g −∫dτdτ′d3xd3x′[G+(x,x′)Δ(x′)G−(x′,x)Δ∗(x)], (23)

with

 G±(x,x′)=[−∂τ∓((−i∇+eA)22m−μ)]−1δ4(x−x′). (24)

In terms of the Fourier transformation (12),

 Seff[Δ]=S[0]−∑ωnp,pΓ−1(iωnp,p)|Δ(iωnp,p)|2 (25)

where the dependence of the coefficient on , and has been suppressed and the thermodynamic potential density reads

 Ω=Ω0−1βV∑ωnp,plnΓ(iωnp,p). (26)

where is the thermodynamic potential of an ideal Fermi gas. It follows that

 α(T,μ,B)=Γ−1(0,0), (27)

and

 n=n0+1βV∂∂μ∑ωnp,plnΓ(iωnp,p), (28)

with the fermionic contribution to the density. Continuating to an arbitrary real frequency according to the prescription in BaymMermin () and introducing a phase shift defined by , the number equation can also be written asSademelo ()

 n=n0+1V∑p∫∞−∞dωπnB(ω)∂δ∂μ(ω,p). (29)

with the Bose-Einstein distribution function. The pair of equations, (18) and (29), at zero magnetic field are widely employed in the context of BCS/BEC crossover in the literature.

To calculate , we write of (24) in terms of the eigenvalues and eigenfunctions of ,

 G±(x,x′)=∑KψK(τ,x)ψ∗K(τ′,x′)iωnk∓(εkz+lωB−χ), (30)

with abbreviation and the notation . The eigenfunction in Landau gauge reads

 ψK(τ,x)=1√LyLze−iωnτ+i(kyy+kzz)ul(x−kyeB), (31)

where are the normalization lengths along and axes and the -function is the wavefunction of a harmonic oscillator given by

 un(z)=(eB)14π1/4√2n⋅n!e−eBz22Hn(√eBz). (32)

with the Hermite polynomial. The -functions satisfy the orthonormality relation .

In terms of the Fourier components of , the trace term in (23) becomes

 tr[G+(x,y)Δ(y)G−(y,x)Δ∗(x)] = ∑K,l′∑ωnp,p∑p′x[∫dx′eipxx′ul(x′−kyeB)ul′(x′−qyeB)][∫dxe−ip′xxul(x−kyeB)ul′(x−qyeB)] ×Δ(iωnp,p)iωnk−(εkz+lωB−χ)Δ∗(iωnp,p′x,py,pz)iωnq+(εqz+l′ωB−χ). (33)

where , and . Upon shifting the integration to , the last two -functions will no longer be coordinate dependent and the first two u-functions depend only on the relative coordinates. The translational invariance becomes explicit then. It would be convenient to introduce the center of mass coordinate and the relative one and we obtain that

 tr[G+(x,y)Δ(y)G−(y,x)Δ∗(x)]= ∑ωnk,l,kz;l′∑ωnp,p∫dreipxr[∫dky2πul(r−kyeB)ul′(r−qyeB)ul(−kyeB)u−l′(qyeB)] ×1iωnk−(εkz+lωB−χ)1iωnq+(εqz+l′ωB−χ)|Δ(iωnp,p)|2. (34)

Making the variable transformation and , we have

 tr[G+(x,y)Δ(y)G−(y,x)Δ∗(x)]=eB2π∑ωnk,l,kz;l′∑ωnp,p ×|Δ(iωnp,p)|2|Ill′(px,py)|2[iωnk−(εkz+lωB−χ)][iωnq+(εqz+l′ωB−χ)], (35)

with

 Ill′(px,py)=∫∞−∞dξeipxξul(ξ)e−pyeBddξul′(ξ), (36)

where the identity

with an arbitrary function is employed. As is shown in Appendix B, the integral (36) can be calculated explicitly with the aid of the raising and lowering operators pertaining to the harmonic oscillator wave function ,

 ξ=1√2eB(a+a†), (38) ddξ=√eB2(a−a†). (39)

and we obtain that

 |Ill′|=√le−p2⊥4eB(p2⊥2eB)|l−l′|2L|l−l′|l<(p2⊥2eB). (40)

where , , and is the generalized Laguerre polynomial. Combining (23), (35) and (40) and carrying out the summation over the Matsubara frequency in (35), we end up with

 Γ−1(iωnp,p)=eB2πe−p2⊥2eB∑l,l′,kz⎧⎪⎨⎪⎩l(p2⊥2eB)|l−l′|[L|l−l′|l<(p2⊥2eB)]2nF(εkz+lωB−χ)−nF(−εqz−l′ωB+χ)−iωnp+(εkz+εqz)+(l+l′)ωB−2χ}+1g. (41)

with the Fermi-Dirac distribution function. The isotropy perpendicular to the magnetic becomes evident in . Setting and , we verify the relation with given by the mean field theory of the previous section and vanishes at and according to (18).

For a negative with , the case considered in this work, the numerator on RHS of (41) may be approximated by -1 and the integration over can be carried out analytically. We have

 Γ−1(iωnp,p) (42) = −m12eB4πe−p2⊥2eB∑l,l′l(p2⊥2eB)|l−l′|[L|l−l′|l<(p2⊥2eB)]2√p2z4m−2χ+(l+l′)ωB+iωnp+1g.

The singularity structure of , with continuated to the entire complext plane, reflects the two-fermion spectrum. There will be an isolated real pole representing the two-body bound pair and a branch cut along the real axis representing the continuum of two-fermion excitations. For sufficiently large , the contribution to the density is dominated by the bound pair pole. We henceforth consider the expansion of (III) around this pole, which is determined by with the solution to the mean field equation (21) and , to the second order in terms of and first order in terms of . We obtain that

 Γ−1≃a1[−ω−2(μ−¯μ)+p2z4m]+a2p2⊥4m. (43)

with

 a1=m3/216π√2ωB∞∑l=0(l+|¯χ|ωB)−32=m3/216π√2ωBζ(32,|¯χ|ωB), (44)

and

 a2= m3/2π√2ωB∞∑l=0⎛⎜ ⎜ ⎜ ⎜⎝l+12√l+|¯χ|ωB−l2√l−12+|¯χ|ωB −l+12√l+12+|¯χ|ωB⎞⎟ ⎟ ⎟ ⎟⎠ = m3/2π√2ωB{ζ(−12,|¯χ|ωB)−ζ(−12,12+|¯χ|ωB) +(12−|¯χ|ωB)[ζ(12,|¯χ|ωB)−ζ(12,12+|¯χ|ωB)]}. (45)

where the frequency is the continuation of the Matsubara frequency to the neighborhood of the pole. Obviously, the kinetic term becomes anisotropic with respect to the directions along and perpendicular to the magnetic field because of the rotational symmetry breaking by the magnetic field.

The partition function (10) under the Gaussian approximation of fluctuations may be written as

 Z=N∫Dϕ∗Dϕexp{∑ωnp,pϕ∗p(ω−ωb+2μ)ϕp}. (46)

where is the rescaled field of the fluctuation and is the bosonic dispersion relation with the binding energy that is measured from the lowest Landau level. We have also the explicit expression of the anisotropy factor

 κ≡a2/a1=16ζ(−12,|¯χ|ωB)−ζ(−12,12+|¯χ|ωB)+(12−|¯χ|ωB)[ζ(12,|¯χ|ωB)−ζ(12,12+|¯χ|ωB)]ζ(32,|¯χ|ωB). (47)

As is shown in the Appendix C, for an arbitrary value of the ratio and is a monotonically increasing function of this ratio.

The partition function (46) is nothing but an ideal Bose gas with anisotropy in kinetic term and is proportional to the boson propagator. The condensation temperature is determined by setting the chemical potential in (29) at the solution of the mean field equation (21), i.e. , and the phase shift there reads

 δ(ω,p)=πθ(ω−ωb+2¯μ). (48)

where is the Heaviside step function with and otherwise zero. It follows then that

 n=2∫d3p(2π)3[exp(p2z+κp2⊥4mTc)−1]−1, (49)

where the term of eq.(29) is ignored with . Consequently, the BEC temperature is given by

 Tc=κ23T0c, (50)

where

 T0c=[n2ζ(3/2)]2/3πm. (51)

is the condensation temperature of an ideal Bose gas of the same density at zero magnetic field.

Beyond the Gaussian approximation, we have also calculated the quartic term, , of the effective action (11) in the limit of low energy and momentum of and obtained a term

 −3m32ω−32B64√2πζ(52,¯χωB)∑ωnp,p|Δ(iωnp,p)|4. (52)

to be added to eq.(25). This term gives rise to a repulsive interaction between the bound pairs.

## Iv Bose-Einstein Condensation in a Magnetic Field

In this section, we shall explore the magnetic field dependence of the BEC temperature (50) for both strong coupling, and weak coupling, .

As the mean-field equation (21) and the formula (47) depend on the ratio through the Hurwitz zeta function, we shall begin with an examination of the two asymptotic behaviors and of the Hurwitz zeta function .

The large expansion follows from the Hermite formula

 ζ(s,r)=r−s2+r1−ss−1+2∫∞0(r2+y2)−s/2sinsθe2πy−1dy, (53)

 ζ(s,r)≃ r−s+1s−1+r−s2+sr−s−112−s(s+1)(s+2)r−s−3720 +O(r−s−5). (54)

The negative value of in this limit leads us to the strong coupling domain via the mean-field equation (21)

 1as≃√2m|¯χ|−12√m2|¯χ|ωB>0, (55)

If follows that the approximate binding energy

 |¯χ|≃12ma2s(1+eBa2s), (56)

with . The anisotropy factor (47) reads

 κ≃1−116(ωB|¯χ|)2≃1−14(eB)2a4s. (57)

and gives rise to a slight suppression of the condensation temperature according to (50), corresponding to an inverse magnetic catalysis.

The small behavior follows from the relation

 ζ(s,r)=r−s+ζ(s,1+r)≃r−s+ζ(s), (58)

which, for , is dominated by the first term on RHS and corresponds to the lowest Landau level approximation in our problem. The positivity of in this case, i.e. , together with the mean-field equation (21) implies a negative and thereby the weak coupling domain, i.e.

 1as≃−√m2|¯χ|ωB<0. (59)

It follows that the binding energy

 |¯χ|≃12mω2Ba2s, (60)

is entirely induced by the magnetic field, as a consequence of the magnetic catalysis. In terms of the solution (60), the inequality implies . The anisotropy factor

 κ≃8|¯χ|ωB≃4eBa2s<<1. (61)

in this case and maximizes the suppression of the condensation temperature.

Since is a monotonically decreasing function of and is negative (positive) for a large (small) , its zero, , serves a demarcation between the strong coupling domain, where and , and the weak coupling domain, where and