Pairing with Unconventional Symmetry around BCS-BEC Crossover: Fermionic Atoms in 2D Optical Lattices to Correlated Electron Systems

Abstract

We study superfluid properties of fermions on a 2D lattice using a finite-range pairing interaction derivable from an extended Hubbard model. We obtain signatures of unconventional pair-symmetry states, $d_{x^2-y^2}$ and extended-s ($s^*$), in the BCS-BEC crossover region. The fermion momentum distribution function, $v_k^2$, the ratio of the Bogoliubov coefficients, $v_k/u_k$, and the Fourier transform of $v_k^2$ are among the properties that are strikingly different for d- and $s^*$ symmetries in the crossover region. Fermionic atoms in 2D optical lattices may provide a way to observe these signatures. We discuss possible experimental ramifications of our results.

Full text

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arXiv:0710.0023v1 [cond-mat.supr-con] 28 Sep 2007
Pairing with Unconventional Symmetry around BCS-BEC Crossover: Fermionic
Atoms in 2D Optical Lattices to Correlated Electron Systems
Duˇ san Volˇ cko and Khandker F. Quader
Department of Physics, Kent State University, Kent, OH 44242
(Dated: February 2, 2008)
We study superfluid properties of fermions on a 2D lattice using a finite-range pairing interaction
derivable from an extended Hubbard model. We obtain signatures of unconventional pair-symmetry
states, dx2−y2 and extended-s (s∗), in the BCS-BEC crossover region. The fermion momentum
distribution function, v2
of v2
region. Fermionic atoms in 2D optical lattices may provide a way to observe these signatures. We
discuss possible experimental ramifications of our results.
k, the ratio of the Bogoliubov coefficients, vk/uk, and the Fourier transform
kare among the properties that are strikingly different for d- and s∗symmetries in the crossover
PACS numbers: 03.75.Ss,32.80.Pj,71.10.Fd,72.20.Rp,05.30.Fk
Attainment of boson and fermion condensates [1] of
ultracold neutral atoms has presented an unprecedented
opportunity to study properties of quantum many-
particle systems. Fermionic atoms in optical lattices [2, 3]
constitute yet another intriguing set of systems. While
these are by themselves interesting to study, they may
also provide a way to gain useful insight into proper-
ties of correlated electrons in solids.
suggested that atoms in optical lattices, confined to the
lowest Bloch band, can be represented by the Hubbard
model with hopping kinetic energy t between neighboring
sites, and on-site interaction U. Hubbard model calcula-
tions [5, 6, 7] predict that attractive-U Hubbard model
give rise to s-wave superconductivity, while the repulsive-
U model results in an antiferromagnetic or a d-wave
superconducting phase depending on filling (number of
fermions per lattice site). Owing to the continuous tun-
ability of model parameters such as, density, hopping or
interactions, optical lattices can serve as testing grounds
for such models. This has led, for example, to the sug-
gestion [8] that the underlying physics of the high Tc
superconductors may be understood by studying these
systems. Recent work [3, 9, 10] have pointed out possible
role of additional Bloch bands and multi-band couplings
in optical lattices. In solids this would correspond to
having multiple orbitals and near-neighbor interactions.
Duan [9] has shown that on different sides of a broad
Feshbach resonance, the effective Hamiltonian can be re-
duced to a t-J model, familiar in correlated electron sys-
tems, wherein it has been suggested [7] that t-J model
can give rise to d-wave pairing.
Jaksch et al [4]
Fermionic atoms subjected to positive and negative de-
tuning using Feshbach resonance technique provide real-
izations of BEC-BCS crossover behavior. It has been
recently suggested [11] that it should also be possible
to study superfluid properties of fermions in optical lat-
tices around BEC-BCS crossover regime. Starting with
the seminal work of Eagles [12] and Leggett [13], the
BEC-BCS crossover problem received considerable the-
oretical attention [5, 14, 15, 16, 17, 18, 19] due to
the possibility that high Tc superconductors, possess-
ing short coherence lengths, could fall in the BEC-BCS
crossover region. Several authors employed continuum
models [5, 14, 15, 16, 19], focussing mostly on conven-
tional s-wave pair symmetry. Lattice models with on-
site or nearest-neighbor attractions have also been con-
sidered [5, 14, 17, 18, 19]. More recent theory work [20]
are in the context of cold fermions.
Motivated by these issues, in this paper, we study su-
perfluid properties of fermions in a 2D square lattice in
the BEC-BCS crossover regime using a finite-range pair-
ing interaction, obtainable from a multi-band extended
Hubbard model. As representative cases of unconven-
tional pair symmetry, we consider two even-parity repre-
sentations of the cubic group, namely the ℓ = 2 dx2−y2-
wave, and the ℓ = 0 extended s-wave (s∗). There has
been work [18, 19, 21] employing similar pairing inter-
action; however these have focussed on different systems
and issues. We present several new results, including spe-
cific signatures of superfluid states with unconventional
pairing gap symmetry as one goes between the BEC and
BCS regimes. This could provide a way to distinguish be-
tween different gap symmetry states in systems that al-
low for tuning into the BEC-BCS crossover regime, such
as fermionic atoms in 2D optical lattices, and possibly
high Tccuprates. One of our key results is the remark-
able behavior of the fermion distribution function, v2
(related to momentum distribution, nk): For the d-wave
gap function, v2
kchanges abruptly from having a peak at
the Brilloiun zone (BZ) center (0,0) to a vanishing cen-
tral peak accompanied by a redistribution of the weight
around other parts of the BZ ((0,±π),(±π,0)) as the sys-
tem crosses from the weak-coupling BCS to the strong-
coupling BEC regime. By contrast, v2
in the s∗-wave case. Similar signatures are also found in
the ratio of Bogoliubov coefficients vk/uk, related to the
phase of the superfluid wavefunction. The Fourier trans-
form of v2
kin real space exhibits a “checkerboard” type
pattern that could have consequences for experiments.
The extended Hubbard model for two equal species
k,
kchanges smoothly

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population system on a 2D square lattice is given by:
H =
?
<ij>σ
(−tc+
iσcjσ+ H.c.) + U
?
i
niσni−σ
− V
?
<ij>σσ′
niσnjσ′ − µo
?
i
ni, (1)
where t is the kinetic energy hopping, µothe unrenormal-
ized chemical potential, U the on-site repulsion and V the
nearest-neighbor attraction. In the case of cold fermions
on a lattice, V would be related to inter-band coupling.
σ is the “pseudo-spin” index, that could refer to equally
populated hyperfine states in the case of optical lattices.
At the mean-field level, the Hartree self-energy terms
renormalize µosuch that µ = µo+µU(f)+µV(f) where
µU(f) and µV(f) are filling-dependent corrections to µ.
We work with the renormalized µ so as to properly deal
with weak and strong couplings, and take µJi(f) = Jif,
where Ji= U, −V . The filling f = N/2M, with N the
number of particles, M the number of lattice sites, and
the pseudo-spin degeneracy factor 2. On Fourier trans-
forming and retaining interactions between particles with
equal and opposite momentum, as in BCS theory, the re-
duced pairing Hamiltonian assumes the form:
Hpair=
?
k
(ǫk− µ)c+
kck+
?
kk′
Vkk′c+
k′c+
−k′c−kck
(2)
where in the
−2t(coskx+cosky); Vkk′ = V0(cos(kx−k′
which is non-separable. Using the standard BCS vari-
ational ansatz, |ΦBCS >=?
obtain the T = 0 gap equations for the gap functions
∆d,s
k
= ∆o(f)(coskx± cosky) with dx2−y2(-) and s∗(+)
symmetries,
tight-binding approximation,ǫk
=
x)+cos(ky−k′
y)),
k(uk+ vkc†
kc†
−k)|0 >, we
1
Vo
=
1
2M
BZ
?
k
coskx(coskx± cosky)
Ed,s∗
k
, (3)
where Ed,s∗
Bogoliubov coefficients are given by,
k
= ((ǫk−µ)2+∆2
o(coskx±cosky)2)1/2. The
|uk|2; |vk|2=1
2(1 ±ǫk− µ
Ed,s∗
k
). (4)
The ratio vk/uk = −(Ed,s∗
ing Leggett[13], we readjust µ for strong attractions by
supplementing the T=0 gap equation with the number
equation:
k
− (ǫk− µ))/∆d,s∗
k
. Follow-
N =
BZ
?
k
(1 − (ǫk− µ
Ed,s∗
k
);(5)
This determines the self-consistently readjusted µ, which
is no longer fixed at the Fermi level, and makes the
FIG. 1: Chemical potential µ vs.
fillings f for the d-wave case. BEC pairs appear where µ(V )
crosses the µ/2t = −2 line. The inset shows µ(V ) for the s-
(dash-short dashed line), s∗(dashed line), and d-wave (solid
line) at f = 0.2.
coupling V at different
gap equation applicable over the entire range of filling,
thereby the BCS and BEC regimes. To allow for strong
scattering, the sums are performed over the entire BZ.
The natural momentum cut-off afforded by the lattice
avoids any possible ultraviolet divergences.
Remarkable differences in features stem in an essential
way from differences in gap symmetry. The dx2−y2 gap
∆d
kvanishes along the lines ±kx = ±ky in the 2D BZ,
i.e. at four points on the Fermi surface(fs), the location
of which depends upon filling. The s∗gap ∆s∗
with the tight-binding fs at exact 1/2-filling, and is node-
less otherwise. Here, µ ≤ 0, with µ = −4t at the bottom
of the band. Owing to particle-hole symmetry, it is suffi-
cient to consider 0 ≤ f ≤ 1/2. Upon examination of the
gap functions and Eqs. (3-5), the following distinctions
become apparent:
(a) For very low fillings (f → 0,µ → −4t), a thresh-
old coupling is required for pairing in the d-wave case,
while in the s∗case ∆s∗→ 0 as V → 0 due to a weak
singularity at µ = −4t. On the other hand, at 1/2-filling,
due to a weak singularity at µ = 0 in the d-wave case,
∆d→ 0 as V → 0. In the s∗case such a singularity is not
present and as ∆s∗→ 0, V/4t → π2/8, i.e. a minimum
coupling is needed for pairing. In contrast with ∆s∗
∆d
o(V ) changes slope at µ = −4t, and hence not smooth
everywhere (though continuous).
(b) For small k, we have the following limiting be-
havior:(i) ǫk < µ(= −4t);
this is the strong-coupling BEC limit.
vk/uk ∼ ∆k/2|µ| → (k2
ǫk > µ(= −4t);|uk| → 0,|vk| → 1; this is the weak-
kcoincides
o(V ),
|uk| → 1,|vk| → 0;
Here the ratio
y)/2|µ|, i.e. analytic. (ii)
x− k2

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FIG. 2: d-wave gap functions ∆/2t vs. nearest-neighbor cou-
pling V/4t for different chemical potential µ. Inset: Results
for the s∗case. µ = −4t demarcates BEC and BCS regimes.
coupling BCS limit. Here vk/uk→ 1/(kx−ky), i.e. non-
analytic. (iii) ǫk= µ(= −4t); |uk| ?= 0,|vk| ?= 0, when
Ek→ 0. Then vk/uk∼ (kx− ky)/(kx+ ky), i.e. inter-
mediate between (i) and (ii). It may be noted that for
d-wave, the quasiparticle excitations in the BCS limit (ii)
are “gapless” for some values of k, while in the BEC limit
(i), Ek?= 0, even for gaps with nodes [22].
Self-consistent numerical solutions of Eqs.(3-5) bear
out the above features in detail, and also reveal a number
of other features. We scale µ, V , ∆ by hopping parame-
ter, t. At a given filling f, both ∆d
increasing V . While for d-wave it is easier to pair elec-
trons at higher fillings, this is not necessarily the case for
s∗-wave for the weaker couplings V/4t ≤ 1.5 and small
gaps ∆s∗/2t ≤ 0.5.
In Fig.1 we show µ(V ) for different fillings f. At a fixed
f, in both the d- and s∗-wave cases, µ decreases with
increasing coupling V , changing less rapidly for progres-
sively larger f. However in the s∗case, µ(V ) exhibits a
small “bump” for weaker couplings V/4t ≤ 1.5. The drop
in µ with increasing attraction is significantly more rapid
in the uniform s-wave case; see inset in Fig. 1. Crossover
to the BEC regime here is signalled by µ(V ) going below
the bottom of the band, i.e. crossing the µ = −4t line.
As Fig. 1 shows, for the d-wave case, this develops at
both low and high fillings at some minimum value Vb/4t
of the coupling. It is interesting to note that as f → 0,
Vb/4t → 1.8. At exactly 1/2-filling this coupling tends
to infinitely large values. For couplings V > Vb, the sys-
tem is conducive to BEC pairing; for V < Vb, the system
exhibits BCS-like features.
Fig. 2 shows the behavior of the d-wave gaps as a
function of coupling V for different values of the chemi-
kand ∆s∗
kincrease with
FIG. 3: (a), (b): 3D plots of d-wave electron distribution
functions v2
“jump” in v2
In BCS regime (a), µ = −3t, ∆d= 5.2t,
V = 18.7t, and in BE regime (b), µ = −6t, ∆d= 0.6t,
V = 5.2t. (c), (d): 3D plots of d-wave vk/uk vs kx − ky
for the same parameters as in ((a),(b)) respectively. In BCS
regime (c) it can be seen to be non-analytic; in BEC regime
(d) it is analytic. (e), (f): The same as in (a), (b), but for
s∗-wave; the behavior is smooth.
kvs. kx − ky at filling f = 0.1, showing abrupt
k.
cal potential µ. The µ = −4t curve represents the locus
of Vb/4t for different fillings (see Fig. 1), and demarcates
BEC and BCS -pair regimes. To the left is the µ > −4t
region wherein finite gaps of the BCS or intermediate
BCS-BEC types exist. On a given constant-µ curve it
may not be possible to have solutions for any arbitrary
filling, but only those that satisfy Eqs. (3) and (4) self-
consistently. The inset in Fig. 2 shows the corresponding
∆s∗(V ) curves for the s∗case. There are interesting dif-
ferences with the d-wave results in that the boundary
(µ = −4t) separating BEC/BCS regimes is not as clear-
cut for the weaker couplings V/4t ≤ 1.5 and the smaller
gaps ∆/2t ≤ 0.5, however the µ < −4t region lies to the
right of the µ = −4t curve as in the d-wave case.
Differences in the gap symmetry manifest in a strik-
ing manner in the momentum distribution function, v2
and the ratio vk/uk. For d-wave, for a given filling, in
the weak-coupling BCS regime (V < Vb(f), µ > −4t),
v2
kexhibits a peak centered around the zone center (0,0),
that becomes progressively narrower with decreasing fill-
ing. Then at the crossover point at Vb(f) (µ = −4t), v2
k,
k

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FIG. 4: (a) Fourier transform ρv(x,y) of a typical d-wave
electron distribution function, v2
−4.2t (strong-coupling regime), gap ∆ = .76t. (b) Projection
of (a) to show contrast ratio of ρv(x,y).
k. Here, filling f = 0.01, µ =
abruptly goes to zero around (0,0), and shows a drastic re-
distribution in a different region of the BZ, namely, along
(0,±π), and (±π,0). The abruptness is evident from the
“jump” in v2
kas the chemical potential goes from just
above the bottom of the band (µ > −4t) to just below
(µ < −4t), i.e. from BCS to BEC regime. A representa-
tive case is shown in Figs. 3a,3b. In marked contrast, in
the the s∗-wave case (Figs 3e,3f), the zone center peak in
v2
kdecreases smoothly as one goes from the BCS regime
to the BEC regime; only a slight redistribution occurs at
(±π,±π). We find this behavior to be replicated at all
fillings f. As observed above in the limiting cases, the nu-
merical calculations show (Fig 3c,3d) that for d-wave, in
the weak-coupling BCS regime, vk/ukis non-analytic at
±kx= ±ky; in the strong-coupling BEC regime, vk/ukis
analytic, vanishing along the zone diagonals and peaking
about (±π,0), (0,±π). In the s∗case (not shown), vk/uk
is analytic in both regimes. Similar behavior in nk has
also been reported in other work [14, 16, 18].
Our findings suggest that experiments, that may be
able to directly or indirectly probe v2
ukand vk, could reveal novel aspects of the paired states.
For example, it may be possible to decipher the OP sym-
metry (e.g. d- or s∗- wave) by measuring v2
of filling (especially at low-fillings), and/or for different
interaction strengths, both of which can be controlled in
optical lattices. At the BCS-BEC crossover, we expect
the behavior to be quite different depending on whether
the OP is d- or s∗wave. Also, in the case of d-wave pairs,
kor combinations of
kas a function
quantities sensitive to v2
ferent depending on whether the paired state is BEC or
BCS like. A possible probe may be ARPES. Information
may also be obtained from experiments that sample the
quasiparticle energy Ek= (∆2
the quasiparticle density of states, or coherence factors,
ukvk+ u′
attenuation[23], or quasiparticle tunneling at low fillings
are possible experiments.
The Fourier transform of v2
may provide yet another interesting way to test our
results.In the d-wave case, in marked contrast with
its behavior in the BCS regime, ρv(x,y) is oscilla-
tory in the BEC regime, and exhibits an inhomogenous
“checkerboard-type” pattern as shown in Fig 4(a,b). For
the chosen parameters of Fig. 3, the contrast ratio of the
lowest density to the peak is roughly 50%, being most
sensitive to the location of µ(V ). The length scale is of
the order of fractions of lattice spacing. ρv(x,y) is fairly
uniform in the s∗case in both regimes. Highly sensitive
STM may be able to pick up such distinctions[24].
Much of the phenomena we have discussed are away
from exact 1/2-filling, and at relatively strong coupling,
where possible effects of spin density wave (SDW) and
charge density wave (CDW) instabilities are expected to
be suppressed. Addition of a next-near-neighbor hop-
ping would also stabilize the paired state, as well as
lower the minimum near-neighbor interaction necessary
for a bound-state; we have checked this[14]. We have
not explored here the issues of collective modes or phase
separation[25]. It may be interesting to extend this work
to, for example, finite-T, or to explore whether the inho-
mogenous density that we find bear relationship to the
range/strength of the interaction, or to possible phase
separation.
We thank E. Abrahams, S. Davis, and H. Neuberger
for discussions and comments. The work is supported in
part by ICAM.
kor to (uk,vk) should be very dif-
k+ǫ2
k)1/2(related to uk,vk),
kv′
k. Angle-dependent or transverse ultrasound
k(kx,ky), namely, ρv(x,y)
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