RAPID COMMUNICATIONS
PHYSICAL REVIEW B 81, 020508共R兲 共2010兲
Mixed triplet and singlet pairing in ultracold multicomponent fermion systems
with dipolar interactions
Congjun Wu and J. E. Hirsch
Department of Physics, University of California, San Diego, California 92093, USA
共Received 27 October 2009; revised manuscript received 4 January 2010; published 26 January 2010兲
The symmetry properties of the Cooper pairing problem for multicomponent ultracold dipolar molecular
systems are investigated. The dipolar anisotropy provides a natural and robust mechanism for both triplet and
singlet Cooper pairing to first order in the interaction strength. With a purely dipolar interaction, the triplet
pz-like polar pairing is the most dominant. A short-range attractive interaction can enhance the singlet pairing
to be nearly degenerate with the triplet pairing. We point out that these two pairing channels can mix by
␲
developing a relative phase of ⫾ 2 , thus spontaneously breaking time-reversal symmetry. We also suggest the
possibility of such mixing of triplet and singlet pairing in other systems.
DOI: 10.1103/PhysRevB.81.020508
PACS number共s兲: 74.20.Rp, 67.30.H⫺, 03.75.Ss, 05.30.Fk
The study of ultracold dipolar molecules has recently become a research focus of cold atom physics.1–4 The prominent feature of the dipolar interaction is its dr2−3z2-type anisotropy when the dipolar moments are aligned by an
external electric field. Considerable progress has been made
in studying anisotropic condensation of dipolar bosons.5–8
Furthermore, dipolar fermionic systems provide an exciting
opportunity to study exotic anisotropic many-body physics
of fermions. Experimentally, a near quantum-degenerate gas
of the dipolar fermion 40K- 87Rb has been achieved.1 A number of theoretical works have been done for the anisotropic
Fermi-liquid properties of dipolar Fermi gases,9–13 including
both singlet particle and collective excitations.
The dipolar interaction also has important effects in the
Cooper pairing symmetry as studied in Refs. 14–18. In the
single component case, the only possible pairing channels
are of odd parity. Assuming dipole moments along the z axis,
the pairing symmetry is mainly of pz with slight hybridization with other odd partial wave components. Dipolar molecules can have an internal degree of freedom arising from
the hyperfine configurations of the constituent atoms. The
electric dipolar interaction is independent of these internal
components which will be denoted as spin below. The intercomponent interaction opens up the possibility of both spin
singlet and triplet pairings for the simplest two-component
case. It would be interesting to study even richer Cooper
pairing patterns and the competition among them.
In this Rapid Communication, we show that the dipolar
interaction favors Cooper pairing in the triplet channel over
the singlet channel. This is an effect directly arising from the
anisotropy of the dipolar interaction, and it occurs to first
order in the interaction strength. In contrast, it does not appear in the usual condensed-matter triplet pairing systems
such as superfluid 3He 共Refs. 19–21兲: the spin-fluctuation
mechanism based on the strong ferromagnetic tendency in
3
He arises from the repulsive part of the 3He-3He interaction
at second order. For a two-component dipolar fermion system, we find the dominate pairing in the spin triplet pz-like
channel with the purely dipolar interaction. It can mix with
the singlet s + dr2−3z2 pairing whose pairing strength is tunable
through the short-range nondipolar s-wave scattering. The
mixing occurs with a relative phase of ⫾ ␲2 which breaks
1098-0121/2010/81共2兲/020508共4兲
time-reversal 共TR兲 symmetry spontaneously. Pairing in dipolar Fermi gases with more than two components is also
discussed.
Samokhin et al. studied nonuniform mixed parity superfluid states in the presence of dipolar interactions.22 They
considered coupling between singlet and triplet channels
with zero relative phase only. Kabanov23 also considered recently mixture of singlet and triplet pairing with zero relative
phase. While this work was being completed, a study of the
competition between triplet and singlet pairing in dipolar fermionic systems appeared that analyzed some of the cases
considered here.24
We begin with the two-component dipolar fermionic system with the electric dipole moments aligned along the z
axis. The dipolar interaction reads V3D共rជ1 − rជ2兲 = −2d2 / 兩rជ1
− rជ2兩3 P2共cos ␪rជ1−rជ2兲, where ␪rជ1−rជ2 is the angle between
ជ ; d is the electric dipole mo共rជ1 − rជ2兲 and the electric field E
ment. The anisotropy is manifested in the angular dependence of the V3D with the form of the second Legendre polynomial. The Fourier transform of the three-dimensional 共3D兲
2
dipolar interaction, V共kជ 兲 = 8␲3d P2共cos ␪k兲, only depends on
the polar angle of kជ . The Hamiltonian is written as
H = 兺 关⑀共kជ 兲 − ␮兴c␣† 共kជ 兲c␣共kជ 兲 +
k,␣
⫻
† ជ
P␤␣
共k ;qជ 兲P␣␤共kជ ⬘ ;qជ 兲,
1
兺 V共kជ − kជ⬘兲
2V k,k ,q
⬘
共1兲
where ⑀共kជ 兲 = បk2 / 共2m兲; ␮ is the chemical potential;
P␣␤共kជ ; qជ 兲 = c␣共−kជ + qជ 兲c␤共kជ + qជ 兲 is the pairing operator; ␣ , ␤ refer to ↑ and ↓. Please note that V共kជ − kជ ⬘兲 depends on the polar
angle of the vector kជ − kជ ⬘, not the relative angle between kជ
and kជ ⬘. We define a dimensionless parameter to describe the
interaction strength as the ratio between the characteristic
interaction energy and the Fermi energy: ␭ ⬅ Eint / EF
d2mk
= 32 ␲2ប2f .
We only consider uniform pairing states at the mean-field
level, thus set qជ = 0 in the pairing interaction in Eq. 共1兲. We
define P␣␤共kជ 兲 = P␣␤共kជ ; qជ = 0兲 which satisfies P␣␤共kជ 兲 = −P␤␣
共−kជ 兲. The pairing operators can be decomposed into the spin
020508-1
©2010 The American Physical Society
RAPID COMMUNICATIONS
PHYSICAL REVIEW B 81, 020508共R兲 共2010兲
CONGJUN WU AND J. E. HIRSCH
x,y,z
singlet Psi and triplet channels Ptri
: Psi共kជ 兲 = 冑12 关P↑↓共kជ 兲
z ជ
x ជ
− P↓↑共kជ 兲兴, Ptri
共k兲 = 冑12 关P↑↓共kជ 兲 + P↓↑共kជ 兲兴, Ptri
共k兲 = − 冑12 关P↑↑共kជ 兲
i
y
␮ ជ
共k兲
− P↓↓共kជ 兲兴, and Ptri共kជ 兲 = − 冑2 关P↑↑共kជ 兲 + P↓↓共kជ 兲兴. Psi共kជ 兲 and Ptri
␮
are even and odd functions of kជ , respectively; Ptri describes
the triplet pairing operators whose total spin is the eigenstate
of ê␮ · Sជ pair with zero eigenvalue. Using these operators, the
pairing interaction of Eq. 共1兲 with qជ = 0 can be rewritten as
H pair =
1
兺 兵Vtri共kជ ;kជ⬘兲
2V k,k ,
⬘
冋兺
␮=x,y,z
†,␮ ជ
␮ ជ
Ptri
共k兲Ptri
共k⬘兲
册
+ Vsi共kជ ;kជ ⬘兲Psi† 共kជ 兲Psi共kជ ⬘兲其,
共2兲
where Vtri,si共kជ ; kជ ⬘兲 = 21 兵V共kជ − kជ ⬘兲 ⫿ V共kជ + kជ ⬘兲其. Vsi共kជ ; kជ ⬘兲 is an
even function of both arguments kជ and kជ ⬘, while Vtri共kជ ; kជ ⬘兲 is
an odd function of both.
The decoupled mean-field Hamiltonian reads
冉
⬘
冊
␰共kជ 兲I ⌬␣␤共kជ 兲
⌿共kជ 兲,
Hmf = 兺 ⌿†共kជ 兲 ⴱ
⌬␤␣共kជ 兲 − ␰共kជ 兲I
ជk
共3兲
where 兺k⬘ means summation over half of momentum space;
␰共kជ 兲 = ⑀共kជ 兲 − ␮; ⌿共kជ 兲 = 关c↑共kជ 兲 , c↓共kជ 兲 , c†↑共−kជ 兲 , c†↓共−kជ 兲兴T. The
mean-field gap function is defined as ⌬␣␤共kជ 兲 = V1 兺kជ ⬘V
共kជ − kជ ⬘兲具兩P␣␤共kជ ⬘兲兩典. ⌬␣␤ can be decomposed into singlet and
y
+ ⌬tri,␮共kជ 兲共i␴␮␴y兲␣␤.
triplet channels as ⌬␣␤共kជ 兲 = ⌬si共kជ 兲i␴␣␤
The Bogoliubov quasiparticle spectra become E1,2共kជ 兲
2 ជ
2 ជ
共k兲, where ␭1,2
共k兲 are the eigenvalues of the
= 冑␰2k + ␭1,2
positive-definite Hermitian matrix ⌬†共kជ 兲⌬共kជ 兲. Its trace satisfies ␭21共kជ 兲 + ␭22共kជ 兲 = 兩⌬si共kជ 兲兩2 + 兺␮兩⌬tri,␮共kជ 兲兩2. The free energy
becomes
F=−
冋
册
␤Ekជ ,i
2
1
ln 2 cosh
−
兺
␤ k,i=1,2
2
2V kជ ,kជ
兺
⬘,a=si,共tri,␮兲
ជ ជ
ជ
⫻ 关⌬ⴱa共kជ 兲V−1
a 共k ;k⬘兲⌬a共k兲兴,
共4兲
−1 ជ ជ
共k ; k⬘兲 is the inverse of the interaction matrix
where Vsi,tri
−1 ជ ជ
共k⬘ , k⬙兲 = ␦kជ ,kជ ⬙.
defined as V1 兺k⬘Vsi,tri共kជ , kជ ⬘兲Vsi,tri
The gap equations are expressed as
⌬tri,␮共kជ 兲 = −
⌬si共kជ 兲 = −
冕
冕
d 3k ⬘
Vtri共kជ ;kជ ⬘兲K共kជ ⬘兲⌬tri,␮共kជ ⬘兲,
共2␲兲3
d 3k ⬘
Vsi共kជ ;kជ ⬘兲K共kជ ⬘兲⌬si共kជ ⬘兲,
共2␲兲3
共5兲
where K共kជ ⬘兲 = tanh关 ␤2 Ei共kជ ⬘兲兴 / 关2Ei共kជ ⬘兲兴. Equation 共5兲 formally
diverges. It can be regularized following the standard procedure explained in Ref. 14 and 25 by replacing the bare interaction Vtri,si with the renormalized zero energy vertex functions ⌫tri,si. At the level of the Born approximation, this
regularization is equivalent to just introducing an energy cut¯ for ␰k⬘ជ in Eq. 共5兲, where ␻
¯ is at the scale of the
off of ⫾␻
Fermi energy.
To analyze the dominant pairing instability around Tc, Eq.
共5兲 is linearized. Considering that the strongest pairing occurs at the Fermi surface and following the standard proce-
j
dure in Ref. 14, we define the eigengap functions ␾tri,si
共kជ 兲
satisfying
N0
冕
d⍀k⬘
4␲
j
j
j
Vtri,si共kជ ;kជ ⬘兲␾tri,si
共kជ ⬘兲 = wtri,si
␾tri,si
共kជ 兲,
共6兲
where N0 = mk f / ␲2ប2 is the density of state at the Fermi surj
are dimensionless eigenvalues; kជ , kជ ⬘ are at the
face; wtri,si
Fermi surface. We neglect the effect of Fermi surface distortion on pairing which is a higher order effect in the interaction strength ␭. Then Eq. 共5兲 is linearized into a set of decoupled equations
j
¯ 兲/共␲kBT兲兴其 = 0.
兵1 + wtri,si ln关共2e␥␻
␾tri,si
共7兲
The spherical harmonics decomposition of Vtri,si共kជ ; kជ ⬘兲 reads
N0
ⴱ
Vtri,si共kជ ;kជ ⬘兲 = 兺 Vll⬘;mY lm
共⍀k兲Y l⬘m共⍀kជ ⬘兲,
4␲
l,l ;m
共8兲
⬘
where Vll⬘;m remains diagonal for m but couples partial wave
channels with l⬘ = l , l ⫾ 2. Vtri,si only have matrix elements in
odd and even partial wave channels, respectively. Vll;m has
the same expressions of the Landau parameters of the dipolar
Fermi gases given by Fregoso et al.13 except for an overall
minus sign due to pairing and a trivial overall numerical
factor difference. For l = l⬘ = m = 0, V00;0 = 0 because the average value of the dipolar interaction is zero.
We diagonalize the matrix Vll⬘;m to find the dominant
negative eigenvalues which determine the dominant pairing
channels. Two eigenvectors are found with eigenvalues much
more negative than other channels. One lies in the triplet
odd-parity sector with dominant pz-wave character with
slight hybridization with other odd-parity channels as same
as in the single component case:14 ␾z共⍀k兲 with the most
z
= −3.820␭, whose eigenvector is
negative eigenvalue wtri
z
␾ 共⍀k兲 ⬇ 0.993Y 10 − 0.120Y 30. The other one lies in the evenparity spin singlet channel. For the purely dipolar interaction,
its eigenvalue is wsis+d = −1.935␭ and the eigenvector lies in
the mixed channel of s + dk2−3k2 as ␾s+d共⍀k兲
z
⬇ 0.6Y 00 – 0.8Y 20 with nodes. However, this channel is sensitive to the strength of the short-range s-wave scattering,
which contributes only to the matrix element of V00;0 as depicted in Fig. 1. Experimentally, this scattering can be tuned
to the scale of the Fermi energy through Feshbach resonance,
i.e., V00;0 can become of order 1. As for ␭, as estimated in
Ref. 13, it could reach 0.1⬃ 0.2. Thus the competition between ␾s+d and ␾s can be studied experimentally in the future. When they become degenerate at V00;0 / ␭ ⬇ −3.15, ␾s+d
becomes mostly of s-wave character as ␾s+d共⍀k兲
⬇ 0.901Y 00 − 0.434Y 20.
We first consider the case of the dominant triplet pairing,
whose critical temperature is determined from Eq. 共7兲 as
z
¯ / ␲兲exp共−1 / 兩wtri
兩兲. Its order-parameter configuraT pz ⬇ 共2e␥␻
tion is ⌬tri,␮共kជ 兲 = ⌬tri,␾z共⍀k兲d̂␮. d̂ is a spin space unit vector.
Without losing generality, it is taken along the z axis as d̂
= êz. ⌬tri,␾z共⍀k兲 = ⌬triei␥␾z共⍀k兲, where ␥ is the pairing phase.
This phase has line nodes on the equator. It breaks the Uc共1兲
gauge, and the spin SUs共2兲 symmetries but maintains TR,
020508-2
RAPID COMMUNICATIONS
PHYSICAL REVIEW B 81, 020508共R兲 共2010兲
MIXED TRIPLET AND SINGLET PAIRING IN…
-1
F = ␣z共T兲 兺 兩⌬␮z 兩2 + ␣s+d共T兲兩⌬s+d兩2 + ␤z兩⌬␮z 兩4 + ␤s+d兩⌬s+d兩4
s+d
Eigenvalues / λ
wsi
␮
+ ␥1 兺 兩⌬␮z 兩2兩⌬s+d兩2 + ␥2 兺 兵⌬␮zⴱ⌬␮zⴱ⌬s+d⌬s+d + c.c.其,
␮
-2
␮
共9兲
-3
wtriz
-4
-3
-2
-1
0
V00;0 / λ
1
2
3
z
FIG. 1. 共Color online兲 The eigenvalues wtri
for the spin triplet ␾z
channel and wsis+d for the singlet ␾s+d channel. The latter depends on
the short-range s-wave interaction V00;0.
parity, and a Z2 symmetry of the combined operation of d̂␮
→ −d̂␮ and ␥ → ␥ + ␲.26,27 Its Goldstone manifold is G
= 关Uc共1兲 丢 SUs共2兲兴 / 关Us共1兲 丢 Z2兴 = 关Uc共1兲 丢 Ss2兴 / Z2. The corresponding low-energy excitations include the phonon and the
spin-wave modes. It supports two different classes of vortices: the usual integer vortex of superfluidity and the halfinteger quantum vortex of superfluidity combined with a ␲
disclination of the d vector.
Next we consider the coexistence of the singlet ␾s+d pairing and the triplet ␾z pairing when they become nearly degenerate. The order parameter can be chosen as ⌬si共⍀k兲
= ⌬0␾s+d共⍀k兲 and ⌬tri,␮共⍀k兲 = ⌬z␾z共⍀k兲␦␮,z. An important observation is that a relative ⫾ ␲2 -phase difference between ⌬si
and ⌬tri is favored, thus spontaneously breaking TR symmetry. This can be proved as follows: the quasiparticle spectra
2
with
␭1,2
= 兩⌬0␾s+d共⍀k兲兩2
reads
Ei = 冑␰2 + ␭2i ,
ⴱ
z
2
s+d
z
+ 兩⌬z␾ 共⍀k兲兩 ⫾ 2 Re共⌬z ⌬0兲␾ 共⍀k兲␾ 共⍀k兲. The last term
vanishes for relative phase ⫾ ␲2 between ⌬0 and ⌬z. In other
words, ␾z + i␾s+d is unitary pairing, i.e., ⌬†共kជ 兲⌬共kជ 兲 is an identity matrix up to a factor, and ␭21 = ␭22 = 兩⌬0␾s+d共⍀k兲兩2
+ 兩⌬z␾z共⍀k兲 兩2兩. All other phase differences give ␭1 ⫽ ␭2, thus
are nonunitary pairing. To show that unitary pairing is optimal, we follow the method presented in Ref. 28 to define the
function
f共x兲 = − ␤2 ln关2 cosh ␤2 冑␰2k + x兴, which satisfies
␭2+␭2
d2 / dx2 f共x兲 ⬎ 0, thus f共␭21兲 + f共␭22兲 ⱖ 2f共 1 2 2 兲. Then the first
term in Eq. 共4兲 is minimized by the unitary pairing, and the
second term is degenerate for unitary and nonunitary pairings. Therefore, ␾z + i␾s+d is favored. This is an exotic fully
gapped TR breaking pairing state because the nodes of ␾s+d
and ␾s do not coincide. It also breaks parity but is invariant
under the combined parity and TR operation. Its Goldstone
manifold for the continuous symmetry breaking is the same
as in the purely triplet-␾z pairing phase.
The above analysis can be recaptured in the GinzburgLandau 共GL兲 framework. The bulk pairing order parameters
and
⌬␮s+d
are
defined
as
⌬␮z = 兺k␾z共kជ 兲⌬tri,␮共kជ 兲
s+d ជ
= 兺k␾ 共k兲⌬si共kជ 兲. The GL free energy is constructed as
where ␣z = N0 ln共T / T pz兲, ␣s+d = N0 ln共T / Ts+d兲, and Ts+d is de¯ / ␲兲exp共−1 / 兩wsis+d兩兲; ␤z and ␤s+d terms
fined as Ts+d ⬇ 共2e␥␻
are the standard quartic terms for the triplet ␾z and singlet
␾s+d channels, respectively; ␥1,2 describe the coupling between the ␾z and ␾s+d channels. Following the analysis on a
similar problem in Ref. 29, we consider the situation where
the two channels are nearly degenerate and T pz is slightly
larger than Ts+d, then the triplet pairing develops first at Tc
= T pz. Defining r = 共␥1 − 2兩␥2兩兲 / 共2␤z兲, the condition for the
second instability to occur is that there exists a lower temperature T⬘ below which 兩␣s+d共T⬘兲兩 ⬎ r兩␣ pz共T⬘兲兩. This can be
satisfied for r ⬍ 1, which results in T⬘ / Tc = 共Ts+d / Tc兲1/共1−r兲.
The ⫾ ␲2 -phase difference between the triplet and singlet
channel pairing requires that ␥2 ⬎ 0. In Refs. 22 and 23, coupling between the singlet and triplet pairings through a linear
spatial derivative is considered, which leads to spatially nonuniform states. Due to spin conservation, such a term is not
allowed here.
It is natural to further consider competing pairings for
even larger number of components represented by the internal hyperfine spin degrees of freedom, which is an even
number. The dipolar interaction is independent of them, thus
the system has an SU共2N兲 symmetry. The 2N ⫻ 2N pairing
matrix ⌬␣␤共kជ 兲 can be classified as N共2N + 1兲-component
symmetric 共odd-parity兲 pairing and N共2N − 1兲-component antisymmetric 共even-parity兲 pairing, which are generalizations
of the triplet and singlet channel pairings, respectively. They
can be explicitly constructed as follows. We define the
charge conjugation matrix R as Rij = 共−兲i␦i,2N−i+1 and the
time-reversal operators T = RC, where C is complex conjugation. For 2N = 2, R reduces to the familiar −i␴y. R satisfies
RT = −R and R2 = −1. On the other hand, any 2N ⫻ 2N Hermitian matrix can be expanded in the basis of the identity
matrix and 4N2 − 1 generators of the SU共2N兲 group. They can
be classified as even and odd under TR transformation.
N共2N + 1兲 of them are TR odd which can be constructed as
spin-tensor matrices P␮ with odd rank numbers 共e.g., spin,
spin octupoles, etc.兲, and N共2N − 1兲 of them are TR even
which can be constructed as spin-tensor matrices Q␯ with
even rank numbers 共e.g., the identity matrix, spin quadrupole, etc.兲. Using TP␮T−1 = −P␮, TQ␯T−1 = Q␯, and RT = −R, it
can be shown that the matrices of P␮R are symmetric and
Q␯R are antisymmetric, respectively. Thus we can decompose
⌬␣␤共⍀k兲 = ⌬asy,␮共⍀k兲共Q␮R兲␣␤ + ⌬sym,␯共⍀k兲共P␯R兲␣␤,
where ⌬asy,␮共⍀k兲关␮ = 1 ⬃ N共2N − 1兲兴 and ⌬sym,␯共⍀k兲关␯ = 1
⬃ N共2N + 1兲兴 are even and odd functions of ⍀k, respectively.
The eigenvalue analysis for competing pairing channels is
the same as in Eq. 共8兲 by replacing the triplet 共singlet兲 pairing with the spin-symmetric 共antisymmetric兲 pairing, respectively. We next consider the unitary pairing in both spinsymmetric and spin asymmetric channels, respectively. A
convenient choice for the matrix kernels is that Pz = ␴z 丢 IN
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PHYSICAL REVIEW B 81, 020508共R兲 共2010兲
CONGJUN WU AND J. E. HIRSCH
for the spin-symmetric channel, and Q = I2N where IN and I2N
are the identity matrices with N and 2N dimensions. The first
†
共−kជ 兲,
one corresponds to the pairing of 兺i=1⬃2Nc†i 共kជ 兲c2N−i
†
†
共−kជ 兲.
while the second corresponds to 兺i=1⬃2N共−兲i−1ci 共kជ 兲c2N−i
In the ␾z-channel pairing states, the spin-SU共2N兲 symmetry
is broken down into SU共N兲 ⫻ SU共N兲 ⫻ U共1兲, thus it has 2N2
branches of spin-wave Goldstone modes. The vortex configuration is similar to the N = 1 case including the usual
integer vortex and the half-quantum vortex combined with a
␲ disclination of spin texture. Again for the mixing between
pairing in the ␾s+d 共spin antisymmetric兲 and ␾z 共spin symmetric兲 channels, a relative phase ⫾ ␲2 is needed to maintain
the unitary pairing.
In summary, we have investigated the competing pairing
symmetries in ultracold multicomponent dipolar molecular
systems, which provides a wonderful opportunity to investigated exotic pairings. We predict that the anisotropy of the
dipolar interaction provides a well-defined pairing mechanism to the spin triplet or, more generally, the spinsymmetric channel Cooper pairing. The spin singlet evenparity channel pairing in the ␾s+d channel is tunable by the
short-range s-wave scattering. It mixes with the spin triplet
odd-parity channel pairing by developing a relative phase
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⫾ ␲2 to maintain the unitary pairing. This is another type of
unconventional Cooper pairing breaking TR symmetry.
We point out that our mechanism of TR breaking mixing
between triplet and singlet pairings is very general. For example, in superfluid 3He it was originally proposed that pairing would occur in the singlet d-wave channel,30 induced by
the attractive part of the van der Waals interaction. Later,
attention focused exclusively on pairing in the triplet p-wave
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It is natural to expect that both channels could contribute to
pairing at sufficiently low temperatures, leading to coupled
Balian-Werthamer 共BW兲 triplet pairing and singlet channel
pairing with a relative phase ⫾ ␲2 . In metallic superconductors, coupling of an isotropic s-wave state with a BW triplet
state will lead to a single isotropic gap only for the particular
case where the relative phase is ⫾ ␲2 . These possibilities will
be discussed separately.
C.W. thanks helpful discussions with E. Fradkin and S.
Kivelson. C.W. was supported by NSF under Grant No.
DMR-0804775 and Sloan Research Foundation. This work
was also supported in part by NSF under Grant No. PHY0551164 at the KITP.
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