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PHYSICAL REVIEW B 83, 144506 (2011)
Frustrated Cooper pairing and f -wave supersolidity in cold-atom optical lattices
Hsiang-Hsuan Hung, Wei-Cheng Lee, and Congjun Wu
Department of Physics, University of California, San Diego, California 92093, USA
(Received 12 October 2010; revised manuscript received 29 November 2010; published 12 April 2011)
Geometric frustration in quantum magnetism refers to the fact that magnetic interactions on different bonds
cannot be simultaneously minimized. The usual Cooper pairing systems favor uniform spatial distributions
of pairing phases among different lattice sites without frustration. In contrast, we propose “frustrated Cooper
pairing” in non-bipartite lattices which leads to supersolid states of Cooper pairs. Not only the amplitudes of
the pairing order parameter but also its signs vary from site to site. This exotic pairing state naturally occurs
in the p-orbital bands in optical lattices with ultracold spinless fermions. In the triangular lattice, it exhibits an
unconventional supersolid state with the f -wave symmetry.
DOI: 10.1103/PhysRevB.83.144506
PACS number(s): 67.80.kb, 03.75.Mn, 03.75.Ss, 75.50.Cc
I. INTRODUCTION
Frustration is one of the fundamental challenges in classic
and quantum magnetism.1 For the antiferromagnetic states
in non-bipartite lattices, such as triangular, Kagome, and
pyrochlore, it is impossible to simultaneously minimize the
magnetic energy of each bond. Consequentially, the ground
state configurations are heavily degenerate. The enhanced
thermal and quantum fluctuations strongly suppress spin
ordering. The ultimate orderings often occur at much lower
temperatures than the energy scale of the antiferromagnetic
coupling through the “order from disorder mechanism.”2,3
Furthermore, frustration provides a promising way to reach
the spin liquid states, which exhibit exotic properties including
topological ordering and fractionalization.4,5
In the usual superfluid states of paired fermions and
bosons, a uniform distribution of the superfluid phase over
the lattice sites is favored in order to maximally facilitate
phase coherence. However, frustration can indeed occur under
certain conditions. It has been found that in the disordered
superconductors near the superconductor-insulator transition,
the fluctuations of the superfluid density can result in frustrated Josephson coupling among superconducting grains.6–8
Recently, striped superconductivity9,10 has been proposed
for the high-Tc compound La2−x Bax CuO4 . The Josephson
coupling between two adjacent superconducting stripes is
like the π junction leading to opposite signs of the pairing
phases across the junction. The mechanism for frustrated
coupling arises from the interplay between superconductivity
and antiferromagnetism in doped Mott insulators. However, an
intuitive picture of the microscopic origin of this exotic phase
is still needed.
On the other hand, cold-atom optical lattices have opened
up a new opportunity to investigate novel features of orbital
physics which are not exhibited in the usual orbital systems of
transition-metal oxides. Bosons have been pumped into the excited p-orbital bands experimentally with a long lifetime.11–13
This metastable excited state of bosons does not obey the
“no-node” theory and exhibits unconventional superfluidity
with complex-valued many-body wave functions breaking
time-reversal symmetry spontaneously,14–17 which has already
been observed.12,13 For orbital fermions, large progress has
been made in the px,y -orbital bands in hexagonal lattices,
whose physics is fundamentally different from that in the
1098-0121/2011/83(14)/144506(7)
pz -orbital system of graphene. The interesting physics includes the flat band structure,18 the consequential nonperturbative strong correlation effects (e.g., the Wigner crystal19 and
ferromagnetism20 ), frustrated orbital exchange interaction,21
the quantum anomalous Hall effect,22 and the unconventional
f -wave Cooper pairing.23
We are interested in bridging the above important research
directions together by introducing frustration into Cooper
pairing as a new feature of orbital physics. In this article, we
propose the “frustrated Cooper pairing” in the px,y band of the
non-bipartite optical lattices with spinless fermions. Due to the
odd parity nature of the px,y orbitals, the Josephson coupling of
the on-site Cooper pairing is frustrated. In the strong-coupling
limit, the superexchange interaction of the pseudospin algebra
composed of the pairing and density operators is described
by the “antiferromagnetic” Heisenberg model with the Ising
anisotropy. It results in the coexistence of charge density waves
and the superfluidity of Cooper pairs with a nonuniform phase
pattern. This supersolid state of Cooper pairs exhibits the
f -wave pairing symmetry in the triangular lattice within a
large range of particle density.
Before we move on, let us explain some conceptual
subtleties. One might wonder how to justify the validity of
“frustration” of Cooper pairs which are usually extended
objects. Indeed, frustration is most commonly defined in
antiferromagnetism of local spin moments. However, frustration does not necessarily mean “on-site” physics even in
the context of antiferromagnetism in non-bipartite lattices.
For example, antiferromagnetic orders can be considered as
pairing between particles and holes in the spin triplet channel
carrying nonzero momentum, i.e., spin density waves. In
the strong-coupling limit, the particle-hole pairs are strongly
bound to be on site, then the physics reduces to local moments
described by the Heisenberg model. However, in the weak
and intermediate coupling regimes with small charge gaps,
the systems are still locally itinerant. The spatial extensions
of the particle-hole bound states are beyond one lattice site. If
the lattice is non-bipartite, we still have frustrated magnetism
with extended particle-hole pairs. For example, this picture
applies to the intermediate-coupling regime of the Hubbard
model at half-filling in the triangular lattice. In our case, we
will consider the Cooper pairing at intermediate and strong
coupling regimes. In the strong-coupling limit, Cooper pairs
144506-1
©2011 American Physical Society
HSIANG-HSUAN HUNG, WEI-CHENG LEE, AND CONGJUN WU
are bound on a single site, whose exchange physics can be
described by the antiferromagnetic pseudospin Heisenberg
model in the charge channel. In the intermediate-coupling
regime, although a Cooper pair is an extended objects covering
several lattice constants, its location can still be defined by
its center of mass. The associated physical quantity is the
pairing order parameter at each lattice site in the mean-field
theory. This physics can be best explained in terms of the
r ; ω), where R is the center
anomalous Green’s function F (R,
of mass coordinate, and r is the relative coordinate. The order
corresponds to F (R,0;
ω = 0), while the
parameter (R)
size of Cooper pairing is determined by the decay length of
r ; ω = 0) with respect to r. In our context, frustration
F (R,
refers to center of mass motion of (R).
This paper is organized as follows. The model Hamiltonian
and the band structure are introduced in Sec. II. The strongcoupling analysis is given in Sec. III. The mean-field theory
analysis is presented in Sec. IV, and the f -wave supersolid
state is presented in Sec. V. Conclusions are given in Sec. VI.
PHYSICAL REVIEW B 83, 144506 (2011)
(a)
(b)
−t
+
_
+
_
+
+
t
_
_
FIG. 1. (Color online) (a) The σ bonding and π bonding of the
p orbitals have opposite signs due to their odd parity nature. This gives
rise to the antiferromagnetic-like exchange in the change channel with
attractive interactions as expressed in Eq. (8), which is frustrated in the
triangular lattice. (b) The first Brillouin zone of the triangular lattice
,0) represent two nonequivalent
is a regular hexagon. K1,2 = (± 4π
3a
vertices, and the other four are equivalent to K1,2 .
= (px (k),p
y (k))
T for the px and py orbitals. The
spinor ψ(k)
Hamiltonian Eq. (1) becomes
αβ (k)
− μδαβ }ψβ (k),
H =
ψα† (k){H
(2)
k
takes the structure of
where the matrix kernel Hαβ (k)
II. THE MODEL HAMILTONIAN
We take the 2D triangular lattice as an example, which
has been constructed experimentally by three coplanar laser
beams.24 The optical potential on each site is approximated
by a 3D anisotropic harmonic potential with frequencies
ωz ωx = ωy . After the lowest s band is fulfilled, the active
orbital bands become px,y . The pz band remains empty and is
neglected. The free part of the px,y -orbital band Hamiltonian
in the triangular lattice filled with spinless fermions reads
†
H0 = t
(pL,r ,i pL,r +a êi ,i + h.c.)
r,i=1∼3
−t⊥
r,i=1∼3
†
(pT ,r ,i pT ,r +a êi ,i + h.c.) − μ
= f (k)
+ g1 (k)τ
1 + g3 (k)τ
3,
H (k)
where τ1,3 are the Pauli matrices defined for the basis of px ,
the expressions of f (k),
g1 (k),
and
py for the spinor of ψ(k);
are
g3 (k)
= (1 − t⊥ )
f (k)
cos k · êi ,
=−
g1 (k)
3
(1 + t⊥ )(cos k · ê2 − cos k · ê3 ),
2
(4)
= − 1 (1 + t⊥ )(cos k · ê2 + cos k · ê3 )
g3 (k)
2
+ (1 + t⊥ ) cos k · ê1 .
r
√
where r runs over all the sites; eˆ1 = êx , ê2,3 = − 12 êx ± 23 êy
are the three unit vectors along bond directions. pL,i ≡
(px êx + py êy ) · êi are the longitudinal projections of the p
orbitals along the êi direction.
More explicitly, pL,1 = px
√
3
1
and pL,2(3) = − 2 px ± 2 py . The transverse projections of the
p orbitals along the bond read pT ,i ≡ (px êx + py êy ) · (ẑ × êi ).
†
†
nr = px px + py py is the particle number operator; μ is
the chemical potential. The σ bonding t and π bonding
t⊥ describe the hoppings between p orbitals along and
perpendicular to the bond direction, respectively, as depicted
in Fig. 1(a). t is positive due to the odd parity nature of
the p orbitals, which is scaled 1 below. t⊥ is usually much
smaller than t because of the anisotropy of the p orbitals. The
first Brillouin zone (BZ) of the triangular lattice is a regular
hexagon as plotted in Fig. 1(b). The edge length of the first BZ
, where a is the lattice constant.
is 4π
3a
The band structure of the noninteracting Hamiltonian
Eq. (1) is displayed in Fig. 2 with the value of t⊥ /t
chosen as 0.2. The bandwidth is around 6t . There is no
particle-hole symmetry with respect to the zero-energy point,
which hints at an asymmetric phase diagram with respect
to half-filling for the interacting Hamiltonian introduced in
Sec. III. In momentum space, we define the two-component
i
√
(1)
nr ,
(3)
ky
2
−2 0
2
0
−2
−4
−2
0
kx
2
4
FIG. 2. (Color online) The band structure of the Hamiltonian
Eq. (1). Two bands touch at K1,2 with the Dirac spectra, and at the
center of the BZ with the quadratic spectra.
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FRUSTRATED COOPER PAIRING AND f -WAVE . . .
PHYSICAL REVIEW B 83, 144506 (2011)
gives rise the dispersions of two
The diagonalization of H (k)
bands as
± (1 + t⊥ )
E± = f (k)
×
cos2 k · êi −
cos k · êa cos k · êb . (5)
i
1a<b3
These two bands touch each other at K1,2 with the Dirac conelike spectra, and at the center of the BZ with the quadratic
spectra. When the Fermi energy is located at the Dirac points,
the other band contributes a large connected branch Fermi
surface, thus its contributions to thermodynamic quantities
dominate over those from the Dirac points.
The topology of the Fermi surfaces varies at different filling
levels. The energy minima of Eq. (1) are threefold degenerate
located at the middle points of the BZ edges. The middle points
of the opposite edges are equivalent up to a reciprocal lattice
vector. Around the band bottom, the Fermi surfaces only cut
the first band and form three disconnected elliptical pockets.
As filling increases, these pockets become connected forming
a large Fermi surface around the center of the BZ. At the
same time, the two Dirac cones contribute two Fermi surfaces
around the K1,2 points, which shrink to two points when the
Fermi energy is right at the Dirac points. As approaching the
band top where a quadratic band touching exists, there are two
Fermi surfaces around the center of the BZ.
III. STRONG-COUPLING ANALYSIS
It is well known that in the Mott-insulating state of the
positive-U Hubbard model at half-filling, its low-energy
physics lie in the magnetic channel, which is captured by
the antiferromagnetic Heisenberg model.25 In non-bipartite
lattices (e.g., the triangular lattice), the antiferromagnetic
exchange cannot be simultaneously minimized for every
bond, which leads to frustration. Similarly, for the negative-U
Hubbard model, in the strong-coupling limit, the low-energy
physics is described by the exchange interaction in the charge
channel. On each site, the low-energy states are the doubly
occupied state and the empty state, which can be considered
the pseudospin “up” and “down” states, respectively.26 The
Josephson coupling between neighboring sites plays the role
of the ferromagnetic coupling in the xy direction of the
pseudospin, which favors a uniform phase distribution. In
other words, the exchange interaction in the charge channel is
unfrustrated.
This situation is fundamentally changed for the Hubbard
model of spinless fermions in px,y orbitals based on the band
structure in Eq. (1). We add the attractive Hubbard interaction
between spinless fermions in the px,y orbital bands:
nr,px nr,py ,
(6)
Hint = −U
Up to a normalization factor, they are the real and imaginary
parts of the pairing operator and the particle density operator,
respectively. The low-energy Hilbert space in each site consists
of the doubly occupied state and the empty state, which are
eigenstates of ηz with eigenvalues ± 12 , respectively.
This superexchange interaction of the pseudospin has
a remarkable feature that a π -phase difference is favored
between pairing order parameters ηx,y on neighboring sites.
As a pair hops to the neighboring site, it gains a π -phase shift,
because the σ -bonding and π -bonding terms are with opposite
signs as depicted in Fig. 1(a). The perturbation theory gives
rise to the anisotropic “antiferromagnetic” Heisenberg model
(AAFHM) in the “external magnetic field,”
Heff =
ij
−h
Jx,y {ηx (i)ηx (j ) + ηy (i)ηy (j )} + Jz ηz (i)ηz (j )
ηz (i),
(8)
i
where h = 2μ is the external magnetic field; the exchange
constants Jx,y and Jz read
Jx,y =
2(t⊥2 + t2 )
4t⊥ t
, Jz =
.
U
U
The Ising anisotropy in Eq. (8) is because Jz Jx,y . Similar
models apply to the pairing problem of spinless fermions in
the px,y orbital of square27 and hexagonal28 bipartite lattices
which are not frustrated because a canonical transformation
can change Jx,y to −Jx,y .
Equation (8) can be interpreted as a hard core boson model
with the frustrated hopping Jx,y and the nearest neighbor
repulsion Jz . It has been studied at the zero external field
in Refs. 29 and 30 which shows a supersolid ordering31,32 with
a three-site unit cell as depicted in Fig. 3. The competition
between charge density wave and supersolid ordering in
optical lattices has also been studied in Refs. 33 and 34.
Site A has no superfluid component, i.e., η ẑ; sites B
and C develop superfluid orders with a π -phase difference.
However, the experimental realization of hard core bosons
with frustrated hopping is difficult. In comparison, our idea
of the frustrated Cooper pairing of fermions is very natural
in the px,y -orbital bands. Furthermore, previous studies29,30
focus on the pseudospin model Eq. (8) completely neglecting
the fermion degree of freedom. In the following, instead of
using Eq. (8), we directly study the Cooper pairing problem
with the fermion Hamiltonian in the entire filling range from
0 to 2.
r
where U is positive. The frustrated nature of Cooper pairing
can be easily explained in the strong-coupling limit of U t .
Similarly to the usual negative U Hubbard model, we construct
the pseudospin algebra denoted25 as
1
i
ηx = (px† py† + py px ), ηy = − (px† py† − py px ),
2
2
1
(7)
ηz = (nr − 1).
2
(9)
C
B
A
FIG. 3. (Color online) The pattern of G|
η|G at h = 0: The unit
cell of three sites exhibiting supersolid ordering.
144506-3
HSIANG-HSUAN HUNG, WEI-CHENG LEE, AND CONGJUN WU
PHYSICAL REVIEW B 83, 144506 (2011)
IV. MEAN-FIELD THEORY AT INTERMEDIATE
COUPLINGS
Below we will focus on the intermediate-coupling regime
and apply self-consistent mean-field theory to the fermionic
Hubbard model of Eq. (1) and Eq. (6). Unlike the positive-U
Hubbard model with doping, in which mean-field theory is
unreliable, in our case of the negative-U Hubbard model, the
mean-field theory gives qualitatively correct results for the
competition between Cooper pairing and charge density wave
(CDW). For a detailed review, please refer to Ref. 26.
To decouple Hint , we assume the pairing and CDW ordering
taking an enlarged unit cell of three sites, and define
I = G|pr∈I,y pr∈I,x |G, NI = 12 G|n̂r∈I |G,
(10)
where I = A,B,C refers to the sublattice index; G| · · · |G
means the average over the mean-field ground state. The meanfield interaction Hamiltonian becomes
mf
Hint
= −U
{∗I pr∈I,y pr∈I,x + h.c.}
r,I =A,B,C
†
†
+NI {pr∈I,x pr∈I,x + pr∈I,y pr∈I,y } .
(11)
Combining Eqs. (1) and (11) and performing a Fourier transformation to momentum space, we can obtain the resulting
mean-field Hamiltonian as
D̂(k)
† Ĥs (k)
ˆ k),
ˆ
H =
(
(12)
(k)
−Ĥs∗ (−k)
D̂ † (k)
FIG. 4. The average fermion number per site n versus the
chemical potential μ at U/t = 6 and t⊥ /t = 0.2.
those at G|ηz |G = ± 13 observed in the study of the classical
ground state of the AAFHM. These two plateaus correspond
to CDW insulating states without superfluidity. As shown in
Fig. 5, the corresponding pseudospin orientation for CDW
insulating states is that all the pseudospins are fully polarized
along the ẑ axis with two of the sublattices along the same
direction and the remaining one along the opposite direction.
Although Fig. 4 resembles the behaviors of the magnetization obtained by the AAFHM,35 two major differences
exist. First, the widths of the two CDW plateaus in Fig. 4
are different while those of the AAFHM are the same. We
k
(a)
(c)
(b)
means the summation only covers half of the
where
k
ˆ k)
is defined as
reduced Brillouin zone; (
ˆ k)
= (φ(k)
T ,φ(−k)
† ),
(
(13)
= [pA,x (k),p
A,y (k),p
B,x (k),p
B,y (k),p
C,x (k),p
C,y
where φ(k)
Hs contains the free Hamiltonian Eq. (1) combined with
(k)];
is the pairing part. The order
the CDW decoupling; D(k)
parameters are obtained self-consistently. The above definition
of order parameters are related to the pseudospin operators
through
(d)
(e)
(f)
G|ηx (r ∈ I )|G = ReI , G|ηy (r ∈ I )|G = ImI ,
G|ηz (r ∈ I )|G = NI − 1/2.
(14)
Different from the ordinary BCS problem, the pairing of
Eq. (1) is not an infinitesimal instability but occurs at a finite
attraction strength. This is because the eigenstates of the two
time-reversal partners with momentum k and −k of the free
Hamiltonian Eq. (1) have the same real polar orbital configuration. This suppresses pairing at weak interactions because
attraction only exits in orthogonal orbitals. With intermediate
and strong interactions, pairing can occur between different
bands. Below we present results for t⊥ /t = 0.2 and an
intermediate coupling and U/t = 6. This corresponds to the
effective AAFHM with the Ising anisotropy of Jz /Jx,y = 2.6.
We discuss our mean-field results in terms of the pseudospin
orientations at the three sublattices A,B,C. Figure 4 shows
the total fermion number per site n = (nA + nB + nC )/3 as a
function of chemical potential μ, which is the counterpart of
the magnetization in the AAFHM. The first prominent feature
is the plateaus occurring at n = 23 and 43 , which correspond to
(g)
(h)
(i)
FIG. 5. The real space configurations of pseudospin η on the
xz plane at various fillings n from (a) to (i). (a) and (i) indicate fully
polarized states. (b) and (h) show three titled vectors, where two of
them have a relative π phase to the third one. (c) and (g) depict the
CDW insulating state. (d) and (f) exhibit an umbrella-like shape with
opposite orientation. Two of them have a π -phase difference and
the third one does not have a superfluid component. (e) denotes an
intermediate configuration between (d) and (f).
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FRUSTRATED COOPER PAIRING AND f -WAVE . . .
PHYSICAL REVIEW B 83, 144506 (2011)
attribute this discrepancy to the different symmetry properties
between the AAFHM and the Hubbard model of Eq. (1) and
Eq. (6) with the asymmetric band structure shown in Fig. 2. The
AAFHM has the symmetry of the rotation of 180◦ around the
x axis, i.e., ηx → ηx , ηy ,ηz → −ηy , − ηz , and h → −h. Such
an operator corresponds to the particle-hole transformation at
†
†
the fermion level as px → ipy and py → ipx , which is not
kept in the triangular lattice. As a result, for the AAFHM, the
magnetization should be an odd function with respect to h
so that the lengths of the plateaus are the same. This kind of
behavior is not expected in Fig. 4. The other difference is that
at h = 0, the ferrimagnetic state is found in the AAFHM, and
our results show the “paramagnetic” behavior; i.e., there is no
jump around n = 1. This is due to the quantum fluctuations
arising from the singly occupied states as discussed below.
Figure 5 plots the pseudospin orientations on three sublattices at a series of filling levels. Except for the CDW insulating
states at n = 23 , 43 , we find that the pseudospins have nonzero
G|ηx |G and G|ηz |G in most parts of the phase diagram,
indicating the frustrated supersolid states with nonuniform
Cooper pairing density and phase. Moreover, the phase
diagram can be well understood by the rotations of pseudospin
orientation under the magnetic field h = 2μ. At n = 0, h
is large along the −ẑ direction so that all the pseudospins
are completely polarized. As n increases, the magnitude of
h decreases so that the pseudospins gradually rotate upward
with one of the pseudospins [
η(A)] having much faster rotating
rate. They become polarized along the z direction as arriving at
the CDW insulating state n = 23 with one pointing up and the
other two pointing down. The magnitudes of ηA,B,C are smaller
than 12 due to quantum fluctuations. As n increases further,
since ηz (A) cannot increase anymore, η(B) and η(C) gradually
turn upward leaving ηA unchanged forming an umbrella
configuration. Such a state is a coexistence of superfluidity
and CDW, and thus is a supersolid state.
After a critical value nc ∼ 0.7, all the pseudospins start
to rotate simultaneously and continuously evolve between
the two umbrella configurations with opposite orientations
depicted in Figs. 5(d) and 5(f), respectively. A detailed process
of evolution is plotted in Fig. 6 from n = 0.68 to n = 1. This
continuous evolution of the ground state is not present in the
AAFHM since its ground state is ferrimagnetic with nonzero
magnetization at zero field, which corresponds to n = 1 in
our model. This deviation is because the AAFHM model is
only justified at the strong-coupling limit. The larger kinetic
energy in this region leads to the less stringent assumption of
the strong coupling. Consequently, the quantum fluctuations
arising from the singly occupied states are enhanced, which
are in disfavor of CDW but in favor of uniform superfluidity.
The continuous evolution also explains why there is no jump
at n = 1 in Fig. 4. Finally, the rest of the phase diagram can be
easily understood by rotating all the pseudospins upward, and
eventually all the pseudospins are fully polarized along the +ẑ
direction at n = 2.
(a)
(d)
(g)
(j)
(b)
(c)
(e)
(f)
(h)
(i)
(k)
(l)
(m)
FIG. 6. The smooth evolution of the two opposite umbrella-like
configurations from n = 0.68 to n = 1.
example, in Fig. 5 for a wide region of n (0.67 n 0.7,
1 n 1.3), we find G|ηx (A)|G = 0 and G|ηx (B)|G =
−
G|ηx (C)|G = . As shown in Fig. 7(a), the signs of the
pairing order parameter are opposite in sublattices A and B.
As a result, a spatial rotation around a site of sublattice C at
60◦ corresponds to flipping the sign of the order parameters,
which indicates the f -wave pairing symmetry.
The f -wave pairing symmetry is also manifest in the
gap function structure in momentum space. We calculate the
intraband pairing functions nn in the momentum space by
projecting the pairing potential in Eq. (11) onto the band
eigenbasis as
6
n (k)ψ
m (−k)
+ h.c.,
∗nm (k)ψ
(15)
k m,n=1
where
= [Û † (k)D(
Û ∗ (−k)]
nm ,
k)
nm (k)
V. THE f -WAVE SUPERSOLID STATE
One remarkable feature of the frustrated Cooper pairing that
we are studying is that it can give rise to an unconventional
type supersolid state exhibiting non-s-wave symmetry. For
(16)
is the unitary matrix such that
Û (k)
144506-5
s (k)
Û (k)
= diag[E1 (k),...,E
Û † (k)H
6 (k)],
(17)
HSIANG-HSUAN HUNG, WEI-CHENG LEE, AND CONGJUN WU
PHYSICAL REVIEW B 83, 144506 (2011)
(b)
(b)
(a)
2
(a)
kx
0
2
2
0
2
2
0 ky
4
FIG. 7. (Color online) The f -wave pairing pattern in (a) real
space and (b) momentum space. The rotation of 60◦ around the center
site O in (a) (denoted by the hollow circle) in real space or around
the center of the reduced BZ in momentum space is equivalent to
reversing the sign of the pairing order parameters.
D(k)
are given in Eq. (12). We have confirmed that
and Hs (k),
all six intraband pairing functions have three nodal lines and
sign changes under 60◦ rotation.
As a specific example, we present the results of the selfconsistent mean-field theory for the filling n = 1.07 and other
parameters t⊥ = 0.2 and U = 6 as before. The system exhibits
the three-site pattern of the CDW order as nA = nC = 1.52
and nB = 0.154, and the pairing order parameters in the real
space are A = −C = 0.345 and B = 0. We plot the band
structure with the above CDW order parameter but set the gap
functions to zero. The reduced BZ is only 13 of the original
BZ, and there are six bands in total as plotted in Fig. 8. The
chemical potential μ is reset to 0, which lies in the gap between
the fourth and fifth bands and has no crossing with the band
spectra. As a result, although the gap functions have node
lines due to the f -wave symmetry, the Bogoliubov excitations
remain fully gapped. We plot the gap function of 44 in Fig.
7(b) for demonstration purposes. The nodal lines are the three
lines connecting the middle points of the opposite edges of BZ.
Thus this is an unconventional supersolid state of frustrated
Cooper pairing with the f -wave pairing symmetry. Another
interesting feature is that the gap function 44 even changes
sign along the radial direction.
It would also be instructive to compare our f -wave
pairing supersolid state of Cooper pairs with the Fulde-FerrellLarkin-Ovchinnikov (FFLO) state.36,37 Both cases exhibit nonuniform distributions of pairing phase in real space. However,
the FFLO state completely breaks rotational symmetry. Its
pairing pattern does not form a well-defined representation of
the lattice point group in momentum space. In our case, it has
a well-defined f -wave symmetry.
We also consider the extreme anisotropy limit of the
vanishing π -bonding strength, i.e., t⊥ = 0. The bond superexchange only results in the Jz term at the second-order
perturbation level in Eq. (8). The leading order of the
hopping of the Cooper pairs occurs through the three-site ring
exchange
H = −
ij k
J [ηx (i)ηx (j ) + ηy (i)ηy (j )]ηz (k),
(18)
2
FIG. 8. (Color online) (a) The reduced Brillouin zone (RBZ)
associated with the enlarged three-site unit cell compared with the
original BZ. The six vertices of the RBZ are located at the centers
of the six regular triangles composed of the center of the BZ and
the vertices of the original BZ. (b) The six bands in the reduced BZ
with the three-site CDW pattern of nA = nC = 1.52, and nB = 0.154
for t⊥ = 0.2 and U = 6. The chemical potential μ is reset to
zero.
t3
where J = 92 U2 . The hopping is frustrated for a plaquette with
only one site occupied, but it is unfrustrated for a plaquette with
two sites occupied. This means that at low fillings the phase
diagram does not change much from the case of nonzero t⊥ ,
while the system finally evolves to a uniform pairing phase at
n close to 2. A more detailed analysis will be presented in a
later publication.
VI. CONCLUSION
In summary, we introduce the concept of “frustrated Cooper
pairing” of spinless fermions in the p-orbital band in optical
lattices. The frustration occurs naturally from the odd parity
of the p orbitals and is a new feature of orbital physics.
Exotic supersolid states of Cooper pairs with nonuniform
distributions of pair density and phase are obtained with
an unconventional f -wave symmetry. This opens up a new
opportunity to study the physics of frustrated magnets by
using the pseudospin algebra of the charge and pair degrees
of freedom of Cooper pairs. This idea can also be applied
to other even more frustrated lattices, such as Kagome and
pyrochlore. In considering the possibility of the existence of
exciting spin liquid states therein, their counterpart in terms of
“frustrated Cooper pairs” is another interesting direction for
further exploration.
Note added. Upon the completion of this manuscript, we
learned of the work of Cai et al.38 in which a similar problem
in the square lattice is investigated.
ACKNOWLEDGMENTS
C.W. thanks J. Hirsch for helpful discussions. C.W., H.H.H.,
and W.C.L. are supported by Grant No. NSF-DMR-0804775
and the AFOSR YIP program.
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