Jan 24, 2006 - A general analysis of Meissner effect and spin susceptibility of a uniform ... ner effect.1 The key quantity describing the Meissner effect.
PHYSICAL REVIEW B 73, 024511 共2006兲
Paramagnetic Meissner effect and finite spin susceptibility in an asymmetric superconductor Lianyi He, Meng Jin, and Pengfei Zhuang Physics Department, Tsinghua University, Beijing 100084, China 共Received 29 September 2005; revised manuscript received 5 December 2005; published 24 January 2006兲 A general analysis of Meissner effect and spin susceptibility of a uniform superconductor in an asymmetric two-component fermion system is presented in nonrelativistic field theory approach. We found that the pairing mechanism dominates the magnetization property of superconductivity and the asymmetry enhances the paramagnetism of the system. At the turning point from BCS to breached pairing superconductivity, the Meissner mass squared and spin susceptibility are divergent at zero temperature. In the breached pairing state induced by chemical potential difference and mass difference between the two kinds of fermions, the system goes from paramagnetism to diamagnetism, when the mass ratio of the two species increases. DOI: 10.1103/PhysRevB.73.024511
PACS number共s兲: 74.20.⫺z, 12.38.Mh
I. INTRODUCTION
It is well known that there are two fundamental features of an electromagnetic superconductor, the zero resistance and the perfect diamagnetism. The latter is also called Meissner effect.1 The key quantity describing the Meissner effect is the Meissner mass or penetration depth. In the language of gauge field theory, the Meissner mass is the mass of the electromagnetic field obtained through the spontaneous breaking of local U共1兲 gauge symmetry, i.e., the AndersonHiggs mechanism.2 Recently, the study on superconductivity is extended to color SU共3兲 gauge field of quantum chromodynamics 共QCD兲 at finite temperature and baryon density.3 In the linear response theory, the Meissner effect is defined in the static and long wave limit → 0, qជ → 0 of the ជ 共qជ 兲. From the microscopic BCS external magnetic potential A theory, the electric current density can be expressed as1
冉
2 2 ជ 共qជ 兲 ne 1 + ប ជj共qជ 兲 = − A mc 32mn
冕
⬁
0
dpp4
冊
f共⑀⌬兲 , ⑀⌬
共1兲
where m, e, p, and n are, respectively, the mass, electric charge, momentum, and density of electrons, ⑀⌬ = 冑共p2 / 2m − 兲2 + ⌬2 is the quasiparticle energy with electric chemical potential and energy gap ⌬, and f共x兲 is the fermion distribution function. Since f共⑀⌬兲 / ⑀⌬ 艋 0, the second term in the bracket on the right hand side is a paramagnetic one and cancels partially the diamagnetism characterized by the first term. However, the total Meissner mass squared keeps positive in the normal superconductor with the BCS pairing mechanism. At zero temperature, due to the limit f共⑀⌬兲 / ⑀⌬ → −␦共⑀⌬兲, the second term in 共1兲 disappears automatically, and there is no paramagnetic part. In addition to the perfect diamagnetism, the Meissner effect includes also the property of magnetic flux expulsion upon cooling through the critical temperature corresponding to the thermodynamic critical field. Another quantity to describe the magnetization property of a superconductor is the spin susceptibility. Since an electron carries a Bohr magneton, a cold free electron gas exhibits Pauli paramagnetism.1 However, at zero temperature, the spin susceptibility of a metallic superconductor is zero, it 1098-0121/2006/73共2兲/024511共12兲/$23.00
does not possess Pauli paramagnetism. The physical picture is clear: The two electrons in a Cooper pair carry opposite spin. From the microscopic BCS theory, the spin susceptibility of a superconductor can be written as1 2 ⑀F = P 32 n
冕 冉 ⬁
dpp2 −
0
冊
f共⑀⌬兲 , ⑀⌬
共2兲
where P is the Pauli susceptibility of a normal electron gas, and ⑀F the electron energy at the Fermi surface. Due to the above mentioned limit of f共⑀⌬兲 / ⑀⌬, the spin susceptibility of the BCS type superconductor is zero at T = 0. At finite temperature, is nonzero because of the thermoexcitation in the superconductor. The above discussed diamagnetic Meissner mass and zero spin susceptibility are only for a normal BCS superconductor, where the two fermions participating in a Cooper pair are symmetric, i.e., they have the same chemical potential, the same mass, and in turn the same Fermi surface. In many physical cases, however, the difference in chemical potentials, or number densities, or masses of the two kinds of fermions results in mismatched Fermi surfaces. Such physical systems can be realized in, for instance, a superconductor in an external magnetic field4 or a strong spin-exchange field,5–7 an electronic gas with two species of electrons from different bands,8 a superconductor with overlapping bands,9,10 a system of trapped ions with dipolar interactions,11 a mixture of two species of fermionic cold atoms with different densities and/or masses,8,12 an isospin asymmetric nuclear matter with proton-neutron pairing,13 and a neutral quark matter in dense QCD.14–24 In the study on superconductivity in an external magnetic field, Sarma4 found an interesting spatial uniform state where there exist gapless modes. However, compared with the fully gapped BCS state, the Sarma state is energetically unfavorable and, therefore, instable. A spatial nonuniform ground state where the order parameter has crystalline structure was also proposed for such types of superconductors by Larkin and Ovchinnikov and Fulde and Ferrell, the so-called LOFF state. In this ground state, the translational and the rotational symmetries of the system are spontaneously broken. Recently, the above spatial uniform ground state prompted new interest due to
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PHYSICAL REVIEW B 73, 024511 共2006兲
HE, JIN, AND ZHUANG
the work of Liu and Wilczek.8 They considered a system of two species of fermions with a large mass difference. The stability of the state has been discussed in many papers.22,23,25–27 It is now accepted that the Sarma instability can be avoided by two possible ways, finite difference in number densities of the two species23,27 or a proper momentum structure of the attractive interaction between fermions.27 In these states, the dispersion relation of one branch of the quasiparticles has two zero points at momenta p1 and p2, and at these two points it needs no energy for quasiparticle excitations. The superfluid Fermi liquid phase in the regions p ⬍ p1 and p ⬎ p2 is breached by a normal Fermi liquid phase in the region p1 ⬍ p ⬍ p2. The temperature behavior of such a breached pairing 共BP兲 state is very different from that of a BCS state, the temperature corresponding to the maximum gap is not zero but finite.13,23,28 Since the dispersion relation controls the Meissner mass and the spin susceptibility, as shown above, it is natural to guess that the change in ⑀⌬ in a breached pairing superconductor will modify the Meissner effect and spin susceptibility significantly. Recently, it is found that the Meissner mass squared of some gluons in two flavor neutral color superconductors are negative,29,30 which indicates that the quark matter in the breached pairing state exhibits a paramagnetic Meissner effect 共PME兲. In condensed matter physics, PME was observed first in high temperature superconductors such as ceramic samples of Bi2Sr2CaCu2O8+␦,31,32 and later in conventional superconductors such as Nb.33–36 It was suggested that the paramagnetic response might be a manifestation of d-wave superconductivity.37 However, it seems that it does not necessarily have anything to do with the d-wave analysis for the PME in conventional superconductors.34 It is now widely accepted that the PME in these materials is most likely due to extrinsic mesoscopic or nanoscale disorder.38 In this paper, we will investigate the Meissner effect and spin susceptibility in an asymmetric two-component fermion system in nonrelativistic case. We will try to prove that the Meissner effect and spin susceptibility of a superconductor is dominated by the pairing mechanism, and the paramagnetism and nonzero spin susceptibility are universal phenomena of superconductors with mismatched Fermi surfaces. We will model the pairing interaction by a four-fermion point coupling, which is appropriate for electronic systems, cold fermionic atom gas, nuclear matter, and dense quark matter. Since our purpose is a general analysis for the Meissner effect and magnetization property, we will neglect the inner structures of fermions like spin, isospin, flavor, and color, which are important and bring much abundance, while they are not central for pairing. The paper is organized as follows. In Sec. II, we review the BCS theory in a symmetric fermion system and show how to calculate the Meissner mass and spin susceptibility in a nonrelativistic field theory approach. In Sec. III, we extend the investigation to an asymmetric two-component fermion system with mismatched Fermi surfaces and derive the universal formula of Meissner mass squared and spin susceptibility. We then consider two kinds of mismatched Fermi surfaces induced by chemical potential difference and mass difference in Secs. IV and V. In Sec. VI, we apply our general discussion to a relativistic system with spin structure
and, as an example, reobtain the eighth gluon Meissner mass in a neutral color superconductor. We summarize in Sec. VII. We use the natural unit of c = ប = 1 through the paper. II. SYMMETRIC FERMION SYSTEM
In this section, we review the BCS theory, the Meissner effect, and the spin susceptibility in a symmetric fermion system in a field theory approach. We start with a system containing two species of fermions represented by a and b, described by the following nonrelativistic Lagrangian density with a four-fermion interaction, L=
兺 ¯i共x兲 i=a,b
冉
−
冊
ⵜ2 + + i共x兲 2m
+ g¯a共x兲¯b共x兲b共x兲a共x兲,
共3兲
where 共x兲, ¯共x兲 are fermion fields for the two species, and the coupling constant g is positive to keep the interaction attractive. For the symmetric system, the two species have the same mass m and chemical potential . The key quantity to describe a thermodynamic system is the partition function which can be defined as Z=
冕
3 关da兴关d¯a兴关db兴关d¯b兴e兰d兰d xL
共4兲
in the imaginary time 共兲 formulism of finite temperature field theory. According to the standard BCS approach, we introduce the order parameter ⌬共x兲 of superconductivity phase transition and its complex conjugate ⌬*共x兲, ⌬共x兲 = g具b共x兲a共x兲典,
⌬*共x兲 = g具¯a共x兲¯b共x兲典,
共5兲
where the symbol 具 典 means ensemble average. Since we focus in this paper on uniform and isotropic superconductors, we take the condensate to be x independent and real in the following. Introducing the Nambu-Gorkov space1 defined as ⌿=
冉冊
a , ¯ b
¯ = 共¯ b 兲, ⌿ a
共6兲
the partition function in mean field approximation can be written as Z MF =
冕
¯ G−1⌿−兩⌬兩2/g兲 ¯ 兴e兰d兰d3x共⌿ 关d⌿兴关d⌿
共7兲
with the inverse of the mean field fermion propagator
G = −1
冢
−
ⵜ2 + + 2m ⌬*
⌬ −
ⵜ2 − − 2m
冣
.
共8兲
Taking the Gaussian integration in path integral 共7兲 and then the Fourier transformation, the thermodynamic potential of the system can be expressed as
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PARAMAGNETIC MEISSNER EFFECT AND FINITE…
⍀=
⌬2 − T兺 g n
冕
d3pជ Tr ln G−1共in,pជ 兲 共2兲3
n
in momentum space, where 兺n is the fermion frequency summation in the imaginary time formulism. The first term is the mean field contribution, and the second term comes from the quasiparticle excitations with the inverse of the propagator G−1共in,pជ 兲 =
冉
in − ⑀ p
⌬
⌬
in + ⑀ p
冊
共10兲
in terms of momentum and frequency n = 共2n + 1兲T, where ⑀ p = p2 / 2m − is the fermion energy. To determine the order parameter, the occupation number of fermions, the Meissner mass, and the spin susceptibility as functions of temperature and chemical potential, we need to know the fermion propagator itself. Using matrix technics, it can be easily evaluated as G共in,pជ 兲 =
冉
G11共in,pជ 兲 G12共in,pជ 兲 G21共in,pជ 兲 G22共in,pជ 兲
冊
,
共11兲
with the elements G11 =
G12 =
in + ⑀ p 共in兲 − 2
−⌬ 共in兲 − 2
, ⑀⌬2
G22 =
, ⑀⌬2
G21 =
in − ⑀ p 共in兲2 − ⑀⌬2 −⌬ 共in兲2 − ⑀⌬2
共12兲
n
nb共pជ 兲 = − T 兺 G22共in,pជ 兲. n
共14兲 After the Matsubara frequency summation, one has na共pជ 兲 = nb共pជ 兲 =
冉 冊
1 ⑀p ⑀p 1− + f共⑀⌬兲. 2 ⑀⌬ ⑀⌬
冉 冊
1 ⑀p 1− . 2 ⑀⌬
共18兲
After the Matsubara frequency summation, the gap equation reads ⌬共1 − gI⌬兲 = 0
共19兲
冕
共20兲
with the function I⌬ =
1 2
d3pជ 1 − 2f共⑀⌬兲 . 共2兲3 ⑀⌬
It is easy to see that there are two solutions of the gap equation 共19兲: One is ⌬ = 0 which describes the symmetry phase, and the other is ⌬ ⫽ 0 determined by 1 − gI⌬ = 0 which characterizes the symmetry breaking phase.
We show now how to calculate the Meissner mass in terms of the thermodynamic potential. Suppose the fermion field carries electric charge e and couples to a magnetic poជ . In mean field approximation, the magnetic potentential A tial is treated as an external and static potential, and the ជ 兲 of the system can be exthermodynamic potential ⍀共A ជ, panded in powers of A
ជ 兲 = ⍀共0兲 + 1 M 2 A A + ¯ , ⍀共A i j 2 ij with the coefficient M 2ij =
共16兲
The gap equation which determines the gap parameter ⌬ as a function of T and self-consistently can be expressed in terms of the nondiagonal elements of the fermion propagator matrix,
冏
ជ兲 2⍀共A Ai A j
冏
ជ =0 A
共21兲
共22兲
.
Since the thermodynamic potential is just the effective potential of the field system, the coefficients M 2ij can be defined as the components of the Meissner mass squared tensor. If the ground state of the system is isotropic, one has M 2ij = 0 for 2 2 2 = M 22 = M 33 , and the Meissner mass squared M 2 i ⫽ j and M 11 39,40 can be defined as
共15兲
At zero temperature, the fermion distribution function f共⑀⌬兲 goes to zero, the occupation numbers are reduced to na共pជ 兲 = nb共pជ 兲 =
⍀ = 0. ⌬
共13兲
It is easy to see that these excitations are all gapped with minimal excitation energy ⌬. The fermion occupation numbers na = 具+a a典, nb = 具+b b典 can be either calculated from the derivative of the thermodynamic potential with respect to the chemical potential, or equivalently, obtained directly from the diagonal elements of the fermion propagator matrix, na共pជ 兲 = T 兺 G11共in,pជ 兲,
共17兲
A. Meissner effect
where ⑀⌬ = 冑⑀2p + ⌬2 is the quasiparticle energy. The excitation spectra ±共p兲 can be read directly from the poles of the fermion propagator,
±共pជ 兲 = ± ⑀⌬ .
d3pជ G12共in,pជ 兲, 共2兲3
which is equivalent to the minimum of the thermodynamic potential,
,
,
冕
⌬ = gT 兺
共9兲
1 M = 3 2
冏兺 冏 3
i=1
ជ兲 2⍀共A Ai Ai
ជ =0 A
.
共23兲
In our model of the four-fermion point interaction 共3兲, the thermodynamic potential in the presence of external and ជ can be expressed as static magnetic potential A 2 ជ 兲 = ⌬ − T兺 ⍀共A g n
冕
d3pជ Tr ln GA−1共in,pជ 兲, 共2兲3
ជ -dependent propagator is defined as where the A
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PHYSICAL REVIEW B 73, 024511 共2006兲
HE, JIN, AND ZHUANG
GA−1共in,pជ 兲 =
冉
in − ⑀+p
⌬ in +
⌬
⑀−p
冊
,
ជ 兲2 / 2m − . To extract with the fermion energies ⑀±p = 共pជ ± eA the Meissner mass squared, we expand the propagator GA−1, 2 2 e ជ−eA , pជ · A 3 m 2m
GA−1 = G−1 −
冉 冊
ns T , =2 1− n Tc
共25兲
共26兲
T → Tc ,
共33兲
where ⌬0 is the order parameter calculated by the gap equation 共19兲 at zero temperature, and Tc the critical temperature determined by 1 − gI0共Tc兲 = 0. It is also necessary to note that the Meissner mass squared 共31兲 satisfies the renormalization condition M 2共⌬ = 0兲 = 0.
and its contribution to the thermodynamic potential,
共34兲
2
Tr ln GA−1 = Tr ln G−1 −
e ជ TrG − e A2Tr共G 兲 pជ · A 3 m 2m
B. Spin susceptibility
2
−
e ជ 兲2Tr共GG兲 + ¯ 共pជ · A 2m2
共27兲
ជ , where G共i , p兲 is the propagator matrix 共11兲 in powers of A n in the absence of magnetic field. After the momentum inteជ vanishes, and the Meissner mass gration, the linear term in A 2 squared M can be read from the coefficient of the quadratic term in A2 of the thermodynamic potential ⍀共A兲,
冕 兺冕
d3pជ 共G11 − G22兲 共2兲3
e 2T M = 兺 m n 2
e 2T + 2 m
n
n=
冕
冕
d3pជ ne2 共G − G 兲 = 11 22 共2兲3 m
d3pជ 关na共pជ 兲 + nb共pជ 兲兴. 共2兲3
共29兲
共30兲
Employing the Matsubara frequency summation, 兺nGG, calculated in the appendix for the second term, we recover the well-known Meissner mass squared shown in text books,1
冉
1 ne2 M2 = 1+ m 32mn with ns defined as 1 ns = n − 3 2m
冕
⬁
0
冕 冉 ⬁
0
冊
f共⑀⌬兲 n se 2 dpp4 = ⑀⌬ m
冊
f共⑀⌬兲 dpp4 − . ⑀⌬
冑
2⌬0 −⌬ /T e 0 , T
T → 0,
冏
,
冏
= −
B=0
2⍀共B兲 B2
冏
.
共35兲
B=0
⌬2 − T兺 g n
冕
d3pជ Tr ln GB−1共in,pជ 兲, 共2兲3
共36兲
where the B-dependent propagator is defined as GB−1共in,pជ 兲 =
冉
in − ⑀↑p
⌬
⌬
in + ⑀↓p
冊
共37兲
with the spin-up and spin-down fermion energies ⑀↑p = p2 / 2m − + 0B and ⑀↓p = p2 / 2m − − 0B, where 0 is some elementary magneton such as the Bohr magneton or the nucleon magneton. To extract the magnetic moment and spin susceptibility from the expansion of ⍀共B兲 in powers of B, we take the similar way used for the discussion of Meissner effect in Sec. II A. We expand the propagator GB−1 = G−1 − 0B,
共38兲
and its contribution to the thermodynamic potential
共31兲
1 Tr ln GB−1 = Tr ln G−1 − 0BTrG − 共0B兲2Tr共GG兲 + ¯ 2 共39兲
共32兲
The effective density ns is positive at any temperature and chemical potential, which means it is a diamagnetic superconductor for any symmetric fermion system. At a low temperature limit and in the approach to the phase transition line of superconductivity, ns behaves as1 ns =1− n
⍀共B兲 B
In our model, the thermodynamic potential ⍀共B兲 in mean field approximation can be expressed as ⍀共B兲 =
From the comparison with the fermion occupation numbers 共14兲, the first term on the right hand side is proportional to the total number density n,
with the definition
冏
M= −
d3pជ p2 共G11G11 + G22G22 + 2G12G21兲. 共2兲3 3 共28兲
e 2T 兺 m n
Assuming the thermodynamic potential in the presence of a constant magnetic field B to be ⍀共B兲, the magnetic moment M and the spin susceptibility of the system are defined as
in powers of B. Substituting them into the thermodynamic potential, we obtain from the linear term in B the magnetic moment M M = − 0T 兺 n
冕
d3pជ 共G11 + G22兲 = 0共nb − na兲 = 0, 共2兲3 共40兲
and from the quadratic term the spin susceptibility1 024511-4
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PARAMAGNETIC MEISSNER EFFECT AND FINITE…
冕
= − 20T 兺 n
=−
20 22
冕
d3pជ 共G11G11 + G22G22 + 2G12G21兲 共2兲3
⬁
dpp2
0
f共⑀⌬兲 , ⑀⌬
na共pជ 兲 = 共41兲
where we have used again the Matsubara frequency summation of 兺nGG shown in the appendix. It is easy to see that = 0 at T = 0 and = P = 320n / 2⑀F at T = Tc.
冉 冊
1 ⑀S 1− 共⑀⌬ − ⑀A兲, 2 ⑀⌬
nb共pជ 兲 = 1 −
冉 冊
1 ⑀S 1+ 共⑀⌬ − ⑀A兲. 2 ⑀⌬
If there is no breached pairing, namely ⑀⌬ ⬎ ⑀A, the two species have the same occupation number
III. ASYMMETRIC FERMION SYSTEM
We discuss now the fermion pairing mechanism and the Meissner effect and spin susceptibility in an asymmetric fermion system using the same approach for the symmetric system in Sec. II. Our asymmetric two-component system with different masses ma, mb and different chemical potentials a, b is defined through the Lagrangian density
兺 ¯i共x兲 i=a,b
L=
冋
−
册
ⵜ2 + + i i共x兲 2mi 共42兲
The thermodynamic potential of the system in mean field approximation is just the same as 共9兲 for the symmetric system, but the matrix elements of the fermion propagator in Nambu-Gorkov space are different, G11 =
G12 =
in − ⑀A + ⑀S 共in − ⑀A兲2 − ⑀⌬2 −⌬ 共in − ⑀A兲 − 2
,
, ⑀⌬2
G22 =
G21 =
in − ⑀A − ⑀S 共in − ⑀A兲2 − ⑀⌬2 −⌬ 共in − ⑀A兲 − 2
,
, ⑀⌬2
±共pជ 兲 = ⑀A ± ⑀⌬ .
共44兲
Without losing generality, we can choose ⑀A ⬎ 0 in the following. Different from the BCS mechanism for the symmetric fermion system, while one branch of the excitations + in the asymmetric system is always gapped, the other one − can cross the momentum axis and become gapless at the momenta p1 and p2, where p1 and p2 satisfy the equation −共p兲 = 0 and pFa ⬍ p1 ⬍ p2 ⬍ pFb with pFa and pFb the Fermi momenta of the two species. The phenomena of gapless excitation is directly related to the breached pairing mechanism. The occupation numbers for fermions a and b defined in 共14兲 become now
冉 冊 冉 冊
冉 冊
1 ⑀S 1− , 2 ⑀⌬
共47兲
which comes back to the result 共16兲 for ma = mb and a = b. However, when the breached pairing happens, the result 共47兲 is valid only in the momentum regions p ⬍ p1 and p ⬎ p2, and in the breached pairing p1 ⬍ p ⬍ p2, we have nb共pជ 兲 = 1.
共48兲
In this case, the pairing between fermions is breached by the region p1 ⬍ p ⬍ p2, the system is in normal Fermi liquid state in the region p1 ⬍ p ⬍ p2 and superconductivity state in the regions p ⬍ p1 and p ⬎ p2. For the asymmetric system, the gap parameter ⌬ is still determined through the self-consistent equation 共19兲, but the function I⌬ is changed to I⌬ =
共43兲
where ⑀S and ⑀A are defined as ⑀S = 共⑀ap + ⑀bp兲 / 2, ⑀A = 共⑀ap − ⑀bp兲 / 2 with the fermion energies ⑀ap = p2 / 2ma − a and ⑀bp = p2 / 2mb − b, and ⑀⌬ = 冑⑀2S + ⌬2 is the quasiparticle energy. The dispersion relations ±共p兲 can be read from the poles of the fermion propagator,
na共pជ 兲 =
na共pជ 兲 = nb共pជ 兲 =
na共pជ 兲 = 0,
+ g¯a共x兲¯b共x兲b共x兲a共x兲.
共46兲
1 2
冕
d3pជ f共−兲 − f共+兲 . 共2兲3 ⑀⌬
共49兲
A. Meissner effect
Suppose the fermions a and b carry electric charges eQa and eQb, respectively, the thermodynamic potential in the ជ in mean field approximapresence of a magnetic potential A tion is still in the form of 共24兲, but the propagator matrix is now a little bit different, GA−1共in,pជ 兲 =
冉
in − ⑀a+ p
⌬
⌬
in + ⑀b− p
ជ− = G−1 − e⌫1pជ · A
冊
e2 ⌫ 2A 2 2
共50兲
ជ 兲2 / 2m − , ⑀b− ជ + eQaA with the fermion energies ⑀a+ a a p = 共p p ជ 兲2 / 2m − , and matrices ⌫ and ⌫ defined as = 共pជ − eQbA b b 1 2
冉 冊 冉 冊
1 1 ⑀S ⑀S 1− f共−兲 + 1 + f共+兲, 2 2 ⑀⌬ ⑀⌬
1 1 ⑀S ⑀S f共−兲 − 1 − f共+兲 + 1. 共45兲 nb共pជ 兲 = − 1 + 2 2 ⑀⌬ ⑀⌬ At zero temperature, they are reduced to 024511-5
⌫1 =
冢 冣 冢 冣 Qa ma
0
0
Qb mb
,
⌫2 =
Q2a ma
0
0
Q2 − b mb
.
共51兲
ជ, Taking again the expansion of Tr ln GA−1 in powers of A
PHYSICAL REVIEW B 73, 024511 共2006兲
HE, JIN, AND ZHUANG 2 ជ Tr共G⌫ 兲 − e A2Tr共G⌫ 兲 Tr ln GA−1 = Tr ln G−1 − epជ · A 1 2 2
−
e2 ជ 兲2Tr共G⌫ G⌫ 兲 + ¯ , 共pជ · A 1 1 2
共52兲
the Meissner mass squared M 2 can be extracted from the ជ of ⍀共A ជ 兲, quadratic term in A 2
M = 2 MD = e 2T 兺 n
M 2P = e2T 兺 n
+
冕
冕
2 MD
+
M 2P ,
冉 冉
冊
共53兲
The diamagnetic part is related to the number densities, 2 = MD
冉
冊
naQ2a nbQ2b 2 + e , ma mb
共54兲
and the paramagnetic term can, with the help of the frequency summations in the appendix, be expressed as M 2P = e2
冕
冉
d3pជ p2 共2兲3 3
冋冉
Qa Qb − ma mb
冊
冊 冉 2
u2pv2p
f共+兲 − f共−兲 ⑀⌬
As shown in Sec. II, the magnetic moment M and spin susceptibility are, respectively, the coefficients of the linear and quadratic terms in the magnetic field B in the thermodynamic potential ⍀共B兲. For the asymmetric system, the fermion energies in the B-dependent propagator matrix 共37兲 are b↓ 2 2 defined as ⑀a↑ p = p / 2ma − a + 0gaB, and ⑀ p = p / 2mb − b − 0gbB, where ga and gb are constants related to the quantum numbers of angular momentum of species a and b. Expanding the propagator GB−1 = G−1 − 0B⌫3 ,
1 Tr ln GB−1 = Tr ln G−1 − 0BTr共G⌫3兲 − 共0B兲2Tr共G⌫3G⌫3兲 2 共59兲
+¯
冊 册
in powers of B with the matrix
共55兲
u2p
冉 冊
1 ⑀S = 1+ , 2 ⑀⌬
v2p
冉 冊
1 ⑀S = 1− = 1 − u2p . 2 ⑀⌬
共58兲
and
Qa 2 Qb 2 Qa 2 Qb 2 + up + u f ⬘共 −兲 , v p f ⬘共 +兲 + v + ma mb ma p mb p with the definitions
共57兲
B. Spin susceptibility
Q2b d3pជ p2 Q2a G G + G22G22 11 11 共2兲3 3 m2a m2b
冊
M 2共⌬ = 0兲 = M 2共T = Tc兲 = 0 at the critical temperature Tc.
Q2b d3pជ Q2a G − G22 , 11 共2兲3 ma mb
2QaQb G12G21 . m am b
= 0 coincide, p1 = p2, the momentum integration of the third term of 共55兲 goes to infinity, since 兰dp␦共−兲 → 兰d−共d− / dp兲−1␦共−兲 → ⬁ due to d− / dp → 0 at p1 = p2, and then the total Meissner mass squared becomes negative infinity at this point. It is easy to check that there is still the renormalization condition for the total Meissner mass squared,
共56兲
The first term in the square bracket of 共55兲 is a new term that resulted fully from the asymmetric property of the system. Since f共+兲 ⬍ f共−兲 at any momentum, it is always negative. We see that the asymmetry between the paired fermions enhances the paramagnetism of the system. The second and third terms are negative due to the property f ⬘共x兲 ⬍ 0 for any x, they together will be reduced to the paramagnetic part of M 2 共31兲, if we come back to the symmetric system. At zero temperature, we have the limit f共x兲 → 共 −x兲 and f ⬘共x兲 → −␦共x兲, the second term vanishes due to −␦共+兲 → 0, and the third term disappears only in normal superconductor but keeps negative in breached pairing state because of the property f ⬘共−兲 → −␦共⑀⌬ − ⑀A兲 ⫽ 0. In Secs. IV and V, we will discuss in detail the paramagnetism of breached pairing superconductors induced by chemical potential difference and mass difference between the two paired fermions. Here we just point out a singularity of the Meissner mass squared at zero temperature in the general case. At the turning point from gapped excitation to gapless excitation where the two roots p1 and p2 of −共p兲
⌫3 =
冉 冊 ga 0
0 gb
共60兲
,
we obtain from the expansion of ⍀共B兲 the magnetic moment M = − 0T 兺 n
冕
d3pជ 共gaG11 + gbG22兲 = 0共gbnb − gana兲, 共2兲3 共61兲
and the spin susceptibility
= − 20T 兺 n
= − 20
冕
冕
d3pជ 2 共g G11G11 + g2bG22G22 + 2gagbG12G21兲 共2兲3 a
冋
d3pជ 2 2 2 f共+兲 − f共−兲 3 共ga − gb兲 u pv p 共2兲 ⑀⌬
册
+ 共gau2p + gbv2p兲f ⬘共+兲 + 共gav2p + gbu2p兲f ⬘共−兲 ,
共62兲
where we have again taken into account the frequency summations listed in the Appendix . Similar to the discussion for the Meissner mass, the first term here is new and fully due to the asymmetry property between the two species, and is divergent at zero temperature at the turning point from gapped to gapless excitations due to the behavior of the term with f ⬘共−兲 in the limit T → 0. We now turn to the details of the Meissner effect and spin susceptibility in breached pairing state induced by the chemi-
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PARAMAGNETIC MEISSNER EFFECT AND FINITE…
cal potential difference and mass difference between the two paired fermions. IV. ONLY CHEMICAL POTENTIAL DIFFERENCE
We first discuss the Meissner effect and spin susceptibility induced by the chemical potential difference only. It is convenient to replace the chemical potentials of the two species ¯ and difference ␦ defined as a and b by their average
a + b , 2
¯=
␦ =
b − a . 2
共63兲
¯ and ␦, Without losing generality, we can set ␦ ⬎ 0. With the dispersion relation of the elementary excitations can be written as
冑冉
±共pជ 兲 = ␦ ±
p2 ¯ − 2m
冊
2
+ ⌬2 .
共64兲
It is easy to see that only under the constraint ⌬ ⬍ ␦ ,
共65兲
there is breached pairing in the momentum region p1 ⬍ p ⬍ p2 with ¯ − 冑␦2 − ⌬2兲, p1 = 冑2m共
¯ + 冑␦2 − ⌬2兲. p2 = 冑2m共 共66兲
A. Meissner effect
For simplicity, we set Qa = Qb = 1. In this case, the diamagnetic and paramagnetic parts of the Meissner mass squared take the form 2 = MD
M 2P =
e2 m2
冕
⬁
dp
0
ne2 , m
p4 关f ⬘共⑀⌬ − ␦兲 + f ⬘共⑀⌬ + ␦兲兴. 共67兲 62
FIG. 1. The schematic occupation numbers na and nb as functions of momentum. paF and pbF are Fermi momenta of species a and b, and p1 and p2 are the two roots of the dispersion equation −共p兲 = 0. The pairing state is breached by the normal fermion liquid state in the region p1 ⬍ p ⬍ p2.
From the definition of u2p and v2p and their relation to the occupation numbers na共p兲 and nb共p兲, the first and second integrations in 共71兲 are, respectively, the upper and lower shadow regions of na共p兲 and nb共p兲 in the three-dimensional case, shown diagrammatically in Fig. 1. Since na共p1兲 = nb共p1兲 ⬎ 1 / 2 and na共p2兲 = nb共p2兲 ⬍ 1 / 2, the contribution of the shadow regions to the total fermion density n is much smaller compared with the term 共p31 + p32兲 / 62 and we have approximately,
⯝ 1,
At zero temperature, the paramagnetic part is evaluated as M 2P = −
e2 m2
冕
⬁
dp
0
p4 ␦共⑀⌬ − ␦兲. 62
共68兲
It is easy to see that only in the breached pairing state, namely, ⌬ ⬍ ␦, M 2P is nonzero. After the momentum integration, the total Meissner mass squared can be expressed as M2 =
冉
冊
␦共␦ − ⌬兲 ne2 1− 冑␦2 − ⌬2 , m
共69兲
冉冕
p31 + p32 1 + 2 − 62
p1
0
dpp2u2p +
⬁
p2
冊
dpp2v2p .
共71兲
共72兲
We take ga = gb = 1 for simplicity. While the magnetic moment M disappears automatically for the normal superconductor, it is no longer zero in the breached pairing superconductor,
共70兲
冕
冊
B. Spin susceptibility
M = 0共nb − na兲 = 0
p3 + p3 = 1 2 2, 6 n
n=
冉
␦共␦ − ⌬兲 ne2 1− 冑␦2 − ⌬2 , m
which means global paramagnetism in the asymmetric fermion system, if the breached pairing happens.
with the parameter defined as
and the total fermion density
M2 ⯝
p32 − p31 . 62
共73兲
The physical picture is clear: while the paired fermions in the region p ⬍ p1 and p ⬎ p2 have no contribution to the magnetic moment, the unpaired fermions in the region p1 ⬍ p ⬍ p2 do contribute to M. The spin susceptibility in this case is reduced from 共62兲 to
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= − 20
冕
⬁
dp
0
p2 关f ⬘共⑀⌬ − ␦兲 + f ⬘共⑀⌬ + ␦兲兴. 22
共74兲
At zero temperature, it is nonzero only in the breached pairing state characterized by ⌬ ⬍ ␦,
=
20m ␦ 共p1 + p2兲. 22 冑␦2 − ⌬2
共75兲
From the comparison with the Pauli susceptibility P, we have the relation
␦ . ⬃ P 冑␦2 − ⌬2
共76兲
V. BOTH CHEMICAL POTENTIAL DIFFERENCE AND MASS DIFFERENCE
FIG. 2. The parameter  as a function of v21. The dashed and solid lines correspond to = 2 and = 10, respectively.
We now consider the magnetization property of the superconductor with both chemical potential difference and mass difference between the paired fermions. To simplify the calculation, we still take Qa = Qb = 1 and ga = gb = 1. From the dispersion relations
±共pជ 兲 =
2
p + ␦ ± 2mA
冑冉
2
p ¯ − 2mS
冊
M2 =
共77兲 −
with the reduced masses mA = 2mamb / 共mb − ma兲 and mS = 2mamb / 共ma + mb兲, the condition for the system to be in breached pairing state is ⌬ ⬍ ⌬c =
兩mbb − maa兩 2 冑m a m b
=
兩b − a兩 2 冑
,
共78兲
p1 = 冑ma关共a + b兲 − 冑共a − b兲2 − 4⌬2兴, p2 = 冑ma关共a + b兲 + 冑共a − b兲2 − 4⌬2兴.
0
p4 u2 v2 dp 2 p p . 6 ⑀⌬
共80兲
In the state with breached pairing, taking into account the relation
冉
冊
冊
dpp2v2p − 2共 − 1兲
+
⬁
dpp2
⑀A − ␦ 2 2 u pv p ⑀⌬
冊
− 3共 + 1兲
冉冕
冊册
冊
冉冕
p1
dpp2u2p
0
p1
dpp2
0
⑀A − ␦ 2 2 u pv p ⑀⌬ 共82兲
with the shorthand notations v21 = v2p and v22 = v2p . 1 2 It is easy to check that for = 1, we recover the result obtained in Sec. IV for the case with only chemical potential difference. For ⬎ 1, we again take the approximation of neglecting the integration terms in 共82兲. Considering the relation 1−
共1 + 共 − 1兲v22兲2 兩1 − 共 + 1兲v22兩
艋0
共83兲
for any , the sign of M 2 depends on the quantity
We now calculate analytically the Meissner mass squared at zero temperature. When there is no breached pairing, we have from the general expressions 共53兲–共55兲,
冕
兩1 − 共 + 1兲v22兩
p2
共79兲
A. Meissner effect
⬁
冕 冕
关1 + 共 − 1兲v22兴2
⬁
p2
where = mb / ma is the mass ratio, and the corresponding region of breached pairing is located at p1 ⬍ p ⬍ p2 in momentum space with
共 + 1兲ne2 共 − 1兲2e2 M2 = − 2ma 2m2a
冉
+ p32 1 −
2
+ ⌬2
冋冉
关1 + 共 − 1兲v21兴2 e2 3 2 p1 − 6ma 兩1 − 共 + 1兲v21兩
u2 v2 −共p兲 p 共u2 − v2p兲 =p p − p = p mb ma ma p
共81兲
for the integrated function with f ⬘共−兲 in 共55兲, the total Meissner mass squared in the breached pairing superconductor can be written as
共, v21兲 = −
共1 + 共 − 1兲v21兲2 兩1 − 共 + 1兲v21兩
.
共84兲
Figure 2 shows  as a function of v21 at = 2 and = 10. Taking into account the condition v21 = na共p1兲 = nb共p1兲 ⬎ 1 / 2, we focus on the behavior of  in the region of v21 ⬎ 1 / 2. At = 2,  is negative in this region and results in negative Meissner mass squared and paramagnetism of the system, which is continued with the conclusion in Sec. IV. However, at = 10,  becomes positive in the interesting region, the Meissner mass squared tends to be positive and the system tends to be diamagnetism. In fact, for very large , v22 becomes extremely small,12 and 1 − 关1 + 共 − 1兲v22兴2 / 兩1 − 共 + 1兲v22兩 ⯝ 0, the system contains approximately the diamagnetic term only.
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In the case with both chemical potential difference and mass difference, the magnetic moment M in breached pairing state takes still the form of 共73兲, but p1 and p2 determined by the dispersion relation −共p兲 = 0 are shown in 共79兲. As for the spin susceptibility , its general expression at finite temperature is still 共74兲. At zero temperature, it is reduced to
=
冉
冊
ma20 p1 p2 . 2 2 + 2 2 2 兩u1 − v1兩 兩u2 − v22兩
共85兲
M 2P = −
2 where in the calculation of M D we have taken again the approximation used in deriving 共72兲. The paramagnetic part is automatically zero in a normal superconductor with ⌬ ⬎ ␦, but negative in breached pairing superconductor with ⌬ ⬍ ␦. Putting the two terms together, the total Meissner mass squared can be expressed as
VI. EXTENSION TO RELATIVISTIC SYSTEMS
冉
¯2 ␦2 − ⌬2 e 2 1 + 32 ¯2
M 2P =
e2 62
冕
⬁
dp
0
L=
¯ 共i␥ − m + ␥ 兲 兺 ␣ ␣ ␣ 0 ␣ ␣=a,b
p4 关f ⬘共⑀⌬ − ␦兲 + f ⬘共⑀⌬ + ␦兲兴 p + m2
⌿= 共86兲
nr =
冕
⬁
0
dp m 关na共p兲 + nb共p兲兴. 3冑 2 共2兲 p + m2
共87兲
In ultra relativistic limit m → 0 and at zero temperature, they can be evaluated as
冉
共¯␣Ci␥51␣兲
兺
␣,=a,b
共¯␣i␥51␣C兲, 共90兲
冢冣
a Cb , b Ca
¯ = 共¯ ¯C ¯ ¯C 兲. ⌿ a b b a
共91兲
Introducing the order parameter
3ជ
2 ⯝ MD
兺
where ␥ = 共␥0 , ␥1 , ␥2 , ␥3兲 = 共 , ␣1 , ␣2 , ␣3兲 and ␥5 = i␥0␥1␥2␥3 are Dirac matrices with  and ␣i being anticommuting matrices, 2 = ␣2i = 1, 兵␣i , ␣ j其 = 0 for i ⫽ j, and 兵␣i , 其 = 0, their covariant counterparts ␥ and ␥5 are defined as ␥ = 共␥0 , −␥1 , −␥2 , −␥3兲 and ␥5 = ␥5, and ¯ are Dirac spinors, C = C¯T and ¯C = TC are charge-conjugate spinors, C = i␥2␥0 is the charge conjugation matrix, the superscript T denotes transposition operation, and 1 is the first Pauli mabb ab ba trix with the elements aa 1 = 1 = 0 and 1 = 1 = 1. For convenience, we define the Nambu-Gorkov spinors
2
with the relativistic fermion density
冊
␦共␦ − ⌬兲 冑␦2 − ⌬2 . 共89兲
We study now a more realistic relativistic model containing two kinds of fermions. The Lagrangian of the system is defined as
␣,=a,b
n re 2 , m
3−
B. With dirac structure
A. Without dirac structure
2 = MD
冊冉
It is negative in the case of ⌬ ⬍ 冑8 / 9␦ ⬍ ␦. Therefore, a relativistic breached pairing superconductor is also paramagnetic.
+g As a naive calculation, we first neglect the antiparticles and take the same Lagrangian density 共42兲. The relativistic effect is only reflected in the fermion energies ⑀ap = 冑p2 + m2a − a, ⑀bp = 冑p2 + m2b − b. The formulas for Meissner mass squared and spin susceptibility in Sec. III still hold if we replace mi 共i = a , b兲 there by 冑p2 + m2i . As an example, we list here the diamagnetic and paramagnetic parts of the Meissner mass squared in the case with only chemical potential difference between the two species,
冊
共88兲
M2 =
We have investigated the Meissner effect and spin susceptibility in an asymmetric system of two kinds of fermions with different chemical potentials, masses, charges, and magnetic moments in a nonrelativistic case. We found that the magnetization property of the breached pairing superconductor is very different from that of the BCS superconductor, and the system tends to be more paramagnetic. What is the situation in the relativistic case? Are these exotic phenomena just a consequence of nonrelativistic kinematics? In this section, we extend our discussion to relativistic systems, which is relevant for the study of color superconductivity at high baryon density.19,22–24,29,30 We will see that there is still paramagnetic Meissner effect in breached pairing state in relativistic superconductors.
冉
¯2 ␦2 − ⌬2 ␦ e 2 1 + 共␦ − ⌬兲, 32 冑␦2 − ⌬2 ¯2
冊
¯2 ␦2 − ⌬2 e 2 1 + , 2 ¯2
⌬ = − 2g
兺
␣,=a,b
具¯␣Ci␥5␣典,
共92兲
and taking it to be real, the thermodynamic potential of the system in mean field approximation is still in the form of 共9兲 with the propagator matrix defined in the four-dimensional Nambu-Gorkov space,
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G−1 =
冢
关G+0 兴−1 a
i ␥ 5⌬
0
0
i ␥ 5⌬
关G−0 兴−1 b
0
0
0
0
关G+0 兴−1 b
i ␥ 5⌬
0
0
i ␥ 5⌬
关G−0 兴−1 a
冣
G21共in,p兲 = ,
共93兲
+
where G0 is the free propagator, 关G±0 兴␣−1 = in␥0 − ␥ជ · pជ − m␣ ± ␣␥0
G33共in,p兲 =
共94兲
with the vector matrix ␥ជ = 共␥1 , ␥2 , ␥3兲. Assuming that the fermion field ␣ with charge e can couple to some U共1兲 gauge potential A, the Meissner mass squared can be extracted from the expansion of the thermodynamic potential in powers of the external magnetic potenជ . By separating the A ជ -dependent propagator 共93兲 into tial A two terms,
ជ GA−1共in,p兲 = G−1 + e⌺␥ជ · A
G44共in,p兲 =
⌺=
冢
1
0
0
0 0
0
0
1
0
0
0
0 −1
冣
G43共in,p兲 =
M 2 = e 2T 兺 n
冕
共in + ␦兲2 − 共⑀⌬+ 兲2
共in + ␦兲2 − 共⑀⌬− 兲2
共in + ␦兲2 − 共⑀⌬+ 兲2
i⌬ ⌳ +␥ 5 共in + ␦兲2 − 共⑀⌬− 兲2 i⌬ ⌳ −␥ 5 共in + ␦兲2 − 共⑀⌬+ 兲2
In the case of ma = mb, it is straightforward to write down explicitly the propagator matrix elements Gij, by employing the method used in Refs. 41 and 42, 共in − ␦兲2 − 共⑀⌬− 兲2 +
G22共in,p兲 =
⌳ +␥ 0
¯, ⑀±p = 冑p2 + m2 ±
共in − ␦兲 −
共in − ␦兲 − 2
+
⌳± =
⌳ −␥ 0 共⑀⌬− 兲2
in + ⑀+p − ␦ 共in − ␦兲2 − 共⑀⌬+ 兲2
⌳ +␥ 0 ,
共99兲
冊
共100兲
冉
1 ␥0␥ជ · pជ 1± 2 冑p + m2 . 2
It is easy to see that in the ultra relativistic limit m → 0, the expression 共97兲 is just the same as the eighth gluon’s Meissner mass squared in the two flavor gapless color superconductor,29,30 namely,
⌳ −␥ 0 , 共⑀⌬+ 兲2
in − ⑀−p − ␦
⑀⌬± = 冑共⑀±p 兲2 + ⌬2 ,
and the energy projectors
in − ⑀+p − ␦ 2
共98兲
with the quasiparticle energies
共97兲
G11共in,p兲 =
⌳ +␥ 0 ,
i⌬ ⌳ +␥ 5 , 共in + ␦兲2 − 共⑀⌬+ 兲2
+
+ G33␥iG33␥i + G44␥iG44␥i − 2G12␥iG21␥i − 2G34␥iG43␥i兴.
in + ⑀−p − ␦
⌳ −␥ 0
in + ⑀+p + ␦
3
dp 兺 Tr关G11␥iG11␥i + G22␥iG22␥i 共2兲3 i=1
⌳ −␥ 0 ,
i⌬ ⌳ −␥ 5 共in + ␦兲2 − 共⑀⌬− 兲2 +
共96兲
,
⌳ +␥ 0
in − ⑀+p + ␦
in − ⑀−p + ␦
共95兲
ជ of ⍀共A ជ 兲, and taking the coefficient of the quadratic term in A the Meissner mass squared can be evaluated as 3ជ
共in + ␦兲2 − 共⑀⌬− 兲2
+
G34共in,p兲 =
i⌬ ⌳ −␥ 5 , 共in − ␦兲2 − 共⑀⌬+ 兲2 in + ⑀−p + ␦
+
with the self-energy matrix ⌺ defined by
0 −1 0
i⌬ ⌳ +␥ 5 共in − ␦兲2 − 共⑀⌬− 兲2
M2 ⬇
冋
册
¯2 2e2 ␦共␦ − ⌬兲 1− 2 冑␦2 − ⌬2 . 3
共101兲
Since we did not consider here the non-Abelian structure of the color superconductivity, the negative Meissner mass squared for the eighth gluon is just a reflection of the breached pairing mechanism. VII. SUMMARY
i⌬ ⌳ −␥ 5 G12共in,p兲 = 共in − ␦兲2 − 共⑀⌬− 兲2 +
i⌬ ⌳ +␥ 5 , 共in − ␦兲2 − 共⑀⌬+ 兲2
We have investigated the relation between the pairing mechanism and magnetization property of superconductivity in an asymmetric two-component fermion system coupled to a magnetic potential. In the frame of field theory approach, we derived the dependence of the Meissner mass squared, 024511-10
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PARAMAGNETIC MEISSNER EFFECT AND FINITE…
magnetic moment, and spin susceptibility of the system on the chemical potential difference, mass difference, charge difference, and magnetic moment difference between the two kinds of fermions. Compared with the superconductor formed in a symmetric system where the Meissner mass squared is globally diamagnetic, there is no magnetic moment, and the spin susceptibility disappears at zero temperature, we found the following new magnetization properties for the asymmetric system with mismatched Fermi surfaces between the paired fermions: 共1兲 The asymmetry leads to a new paramagnetic term in the Meissner mass squared and a new term in the spin susceptibility, and the magnetic moment is no longer zero in asymmetric systems. Note that the new terms and the finite magnetic moment do not depend on the pairing mechanism, they are only the consequence of asymmetry between the two species. 共2兲 At the turning point from BCS to breached pairing state, the Meissner mass squared and spin susceptibility are divergent at zero temperature. 共3兲 In the breached pairing state induced by chemical potential difference between the two paired fermions, the Meissner mass squared is paramagnetic at zero temperature. 共4兲 In the breached pairing state with not only chemical potential difference but also mass difference between the two kinds of fermions, the system at zero temperature is paramagnetic at small mass ratio and tends to be diamagnetic when the ratio is large enough. While the paramagnetic Meissner effect and finite spin susceptibility discussed above are interesting, how to understand them correctly and their reflection on physically observable quantities is not clear. By comparing the BP and LOFF states,43–45 the paramagnetic Meissner effect might be a signal of instability of the BP state.43,44 The thermodyជ 兲 of a LOFF state with a nonzero monamic potential ⍀共Q ជ of the Cooper pair can be obtained by replacing mentum 2Q ជ in the thermodynamic potential the magnetic potential eA ជ ជ . If the uniform BP ⍀共A兲 derived above by the momentum Q state is the ground state, the thermodynamic potential must ជ = 0, namely, ⍀ / Q = 0 and be the minimum at Q i 2 2 ជ = ⍀ / Q ⬎ 0 at Q = 0. Through the obvious relation M 2 = e2, negative Meissner mass squared leads automatically to negative . Therefore, the paramagnetic Meissner effect may be a signal that the LOFF state is more favored than the BP state. However, the above argument is obtained from the study for systems with fixed chemical potentials, it is not clear if it is still true for systems with fixed number densities of the two species. As we know, the existence of the LOFF phase in conventional superconductors has still not been convincingly demonstrated in any material. If the above argument is true, the LOFF state might be observed in trapped atomic fermion systems.46,47 Our research in this direction is in progress. ACKNOWLEDGMENTS
supported in part by Grants Nos. NSFC10428510, 10435080, 10447122, and SRFDP20040003103. APPENDIX: FERMION FREQUENCY SUMMATIONS
We calculate in this appendix the fermion frequency summations T兺nG11G11, T兺nG22G22, and T兺nG12G21 in the Meissner mass squared and spin susceptibility in Secs. III–V. From the decomposition of these summations, T 兺 G11G11 = T 兺 n
关共in − ⑀A兲2 − ⑀⌬2 兴2
= A + 共⑀⌬ + ⑀S兲2B + 2⑀SC, T 兺 G22G22 = T 兺 n
n
共in − ⑀A − ⑀S兲2 关共in − ⑀A兲2 − ⑀⌬2 兴2
= A + 共⑀⌬ − ⑀S兲2B − 2⑀SC, T 兺 G12G21 = T 兺 n
n
⌬2 = ⌬2B, 关共in − ⑀A兲2 − ⑀⌬2 兴2
with the definitions A = T兺 n
1 共in − ⑀A兲 − 2
C = T兺 n
, ⑀⌬2
B = T兺 n
1 关共in − ⑀A兲2 − ⑀⌬2 兴2
,
1 , 共in − ⑀A + ⑀⌬兲2共in − ⑀A − ⑀⌬兲
we need to complete the summations A, B, and C only, A=
B=
C=
f共− −兲 + f共+兲 − 1 , 2⑀⌬
冋
册
1 f共− −兲 + f共+兲 − 1 , 2⑀⌬ ⑀⌬ 2⑀⌬
f共− −兲 + f共+兲 − 1 4⑀⌬2
−
1 f共− −兲 . 2⑀⌬ ⑀⌬
Defining the function f ⬘共x兲 = f共x兲 / x, we can express the summations we need as T 兺 G11G11 = u2pv2p
f共+兲 − f共−兲 + v4p f ⬘共−兲 + u4p f ⬘共+兲, ⑀⌬
T 兺 G22G22 = u2pv2p
f共+兲 − f共−兲 + u4p f ⬘共−兲 + v4p f ⬘共+兲, ⑀⌬
n
n
T 兺 G12G21 = − u2pv2p n
We thank Professor Hai-cang Ren and Dr. Mei Huang for stimulating discussions and comments. The work was
n
共in − ⑀A + ⑀S兲2
f共+兲 − f共−兲 + u2pv2p关f ⬘共−兲 + f ⬘共+兲兴, ⑀⌬
where the energies ⑀A, ⑀S, ⑀⌬, the dispersion relations +, −, and the functions u2p, v2p are defined in Sec. III.
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PHYSICAL REVIEW B 73, 024511 共2006兲
HE, JIN, AND ZHUANG 1 See,
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