Circulating Current States in Bilayer Fermionic and Bosonic Systems
Circulating Current States in Bilayer Fermionic and Bosonic Systems
A. K. Kolezhuk∗
Institut für Theoretische Physik C, RWTH Aachen, 52056 Aachen, Germany and
Institut für Theoretische Physik, Universität Hannover, 30167 Hannover, Germany
It is shown that fermionic polar molecules or atoms in a bilayer optical lattice can undergo the transition to
a state with circulating currents, which spontaneously breaks the time reversal symmetry. Estimates of relevant
temperature scales are given and experimental signatures of the circulating current phase are identified. Related
phenomena in bosonic and spin systems with ring exchange are discussed.
PACS numbers: 05.30.Fk, 05.30.Jp, 42.50.Fx, 75.10.Jm
Introduction.– The technique of ultracold gases loaded
into optical lattices [1, 2] allows a direct experimental study of
paradigmatic models of strongly correlated systems. The pos-
sibility of unprecedented control over the model parameters
has opened wide perspectives for the study of quantum phase
transitions. Detection of the Mott insulator to superfluid tran-
sition in bosonic atomic gases [3, 4, 5], of superfluidity [6, 7]
and Fermi liquid [8] in cold Fermi gases, realization of Fermi
systems with low dimensionality [9, 10] mark some of the re-
cent achievements in this rapidly developing field [11]. While
the atomic interactions can be treated as contact ones for most
purposes, polar molecules [12, 13, 14] could provide further
opportunities of controlling longer-range interactions.
In this Letter, I propose several models on a bilayer op-
tical lattice which exhibit a phase transition into an exotic
circulating current state with spontaneously broken time re-
versal symmetry. Those states are closely related to the “or-
bital antiferromagnetic states” proposed first by Halperin and
Rice nearly 40 years ago [15], rediscovered two decades later
[16, 17, 18] and recently found in numerical studies in ex-
tended t-J model on a ladder [19] and on a two-dimensional
bilayer [20]. Our goal is to show how such states can be real-
ized and detected in a relatively simple optical lattice setup.
Model of fermions on a bilayer optical lattice.– Consider
spin-polarized fermions in a bilayer optical lattice shown in
Fig. 1. The system is described by the Hamiltonian
H = V
n1,rn2,r +
〈rr′〉
V ′σσ′nσ,rnσ′,r′ (1)
1,ra2,r + h.c.)− t
〈rr′〉
(a†σ,raσ,r′ + h.c.)
where r labels the vertical dimers arranged in a two-
dimensional (2d) square lattice, σ = 1, 2 labels two layers,
and 〈rr′〉 denotes a sum over nearest neighbors. Amplitudes t
and t′ describe hopping between the layers and within a layer,
respectively. A strong “on-dimer” nearest-neighbor repulsion
V ≫ t, t′ > 0 is assumed, and there is an interaction between
the nearest-neighbor dimers V ′σσ′ which can be of either sign.
This seemingly exotic setup can be realized by using po-
lar molecules [13, 14], or atoms with a large dipolar magnetic
moment such as 53Cr [12], and adjusting the direction of the
dipoles with respect to the bilayer plane. Let θ, ϕ be the polar
and azimuthal angles of the dipolar moment (the coordinate
axes are along the basis vectors of the lattice, z axis is perpen-
dicular to the bilayer plane). Setting ϕ = ±π
,± 3π
ensures
the dipole-dipole interaction is the same along the x and y di-
rections. The nearest neighbor interaction parameters in (1)
take the following values: V = (d20/ℓ
⊥)(1 − 3 cos
2 θ), and
V ′12 = V
21 = (d
3){1−3R−2(ℓ‖ cos θ+ℓ⊥ sin θ cosϕ)
V ′11 = V
22 = V
12(ℓ‖ = 0), where d0 is the dipole moment
of the particle, ℓ⊥ and ℓ‖ are the lattice spacings in the direc-
tions perpendicular and parallel to the layers, respectively, and
R2 = ℓ2
+ ℓ2⊥. The strength and the sign of interactions V ,
Ṽ ′ can be controlled by tuning the angles θ, ϕ and the lattice
constants ℓ⊥, ℓ‖. Below we will see that the physics of the
problem depends on the difference
Ṽ ′ = V ′11 − V
12, (2)
with the most interesting regime corresponding to Ṽ ′ < 0.
Consider the model at half-filling. Since V ≫ t, t′, we may
restrict ourselves to the reduced Hilbert space containing only
states with one fermion per dimer. Two states of each dimer
can be identified with pseudospin- 1
states |↑〉 and |↓〉. Second-
order perturbation theory in t′ yields the effective Hamiltonian
〈rr′〉
J(SxrS
r′ + S
) + JzS
Sxr ,
J = 4(t′)2/V, Jz ≡ J∆ = J + Ṽ
′, H = 2t, (3)
describing a 2d anisotropic Heisenberg antiferromagnet in
a magnetic field perpendicular to the anisotropy axis. The
twofold degenerate ground state has the Néel antiferromag-
netic (AF) order transverse to the field, with spins canted to-
wards the field direction. The AF order is along the y axis for
∆ < 1 (i.e., Ṽ ′ < 0), and along the z axis for∆ > 1 (Ṽ ′ > 0).
t , V2
FIG. 1: Bilayer lattice model described by the Hamiltonian (1). The
arrows denote particle flow in the circulating current phase.
http://arxiv.org/abs/0704.1644v3
The angle α between the spins and the field is classically given
by cosα = H/(2ZJS), where S is the spin value and Z = 4
is the lattice coordination number. This classical ground state
is exact at the special point H = 2SJ
2(1 + ∆) [21]. The
transversal AF order vanishes above a certain critical field Hc;
classically Hc = 2ZJS, and the same result follows from the
spin-wave analysis of (3) (one starts with the fully polarized
spin state at large H and looks when the magnon gap van-
ishes). This expression becomes exact at the isotropic point
∆ = 1 and is a good approximation for ∆ close to 1.
The long-range AF order along the y direction translates
in the original fermionic language into the staggered arrange-
ment of currents flowing from one layer to the other:
Ny = (−)
r〈Syr 〉 7→ (−)
1,ra2,r − a
2,ra1,r)〉. (4)
In terms of the original model (1), the condition H < Hc for
the existence of such a staggered current order becomes
t < 8(t′)2/V. (5)
The continuity equation for the current and the lattice symme-
try dictate the current pattern shown in Fig. 1. This circulating
current (CC) state has a spontaneously broken time reversal
symmetry, and is realized only for attractive inter-dimer inter-
action Ṽ ′ < 0 (i.e., the easy-plane anisotropy ∆ < 1) [22].
If ∆ = 1, the direction of the AF order in the xy plane is
arbitrary, so there is no long-range order at any finite temper-
ature. For ∆ > 1 (i.e., Ṽ ′ > 0) the AF order along the z axis
corresponds to the density wave (DW) phase with in-layer oc-
cupation numbers having a finite staggered component.
The phase diagram in the temperature-anisotropy plane is
sketched in Fig. 2. At the critical temperature T = Tc the dis-
crete Z2 symmetry gets spontaneously broken, so the corre-
sponding thermal phase transition belongs to the 2d Ising uni-
versality class (except the two lines ∆ = 1 and H = 0 where
the symmetry is enlarged to U(1) and the transition becomes
the Kosterlitz-Thouless one). Away from the phase bound-
aries the critical temperature Tc ∼ J , but at the isotropic point
∆ = 0, H = 0 it vanishes due to divergent thermal fluctua-
tions: for 1−∆ ≪ 1 and H ≪ J , it can be estimated as
Tc ∼ J/ ln[min(|1−∆|
−1, J2/H2)]. (6)
<sz><s >y
current
circulating density wave
FIG. 2: Schematic phase diagram of the model (1), (3) at fixed H =
2t. The line ∆ = 1 corresponds to the Kosterlitz-Thouless phase,
with the transition temperature TKT ∝ J/ ln(J/H) at small H .
FIG. 3: Noise correlation function G(r, r′) from time-of-flight im-
ages in the circulating current (CC) phase, shown as the function of
the relative distance Q(r) − Q(r′), with Q(r) = mr/(~t) ex-
pressed in 1/ℓ‖ units: (a) Q(r) = (0, 0); (b) Q(r) =
(1, 1).
Changing the initial point Q(r) leads to the change of relative weight
of the two systems of dips, which is the fingerprint of the CC phase.
The quantum phase transition at T = 0, H = Hc is of the 3d
Ising type (except at the U(1)-symmetric point ∆ = 1 where
the universality class is that of the 2d dilute Bose gas [23]), so
in its vicinity the CC order parameter Ny ∝ (Hc −H)
β with
β ≃ 0.313 [24], and Tc ∝ JN
y ∝ J(Hc −H)
2β . At T > Tc
or H > Hc the only order parameter is 〈S
x〉, corresponding
to the Mott phase with one particle per dimer.
Bilayer lattice design and hierarchy of scales.– The bi-
layer can be realized, e.g., by employing three pairs of mutu-
ally perpendicular counter-propagating laser beams with the
same polarization and adding another pair of beams with
an orthogonal polarization and additional phase shift δ, so
that the resulting field intensity has the form
E⊥(cos kx +
cos ky)+Ez cos kz
+Ẽ2z cos
2(kz+δ). Taking δ = π
(1+ζ)
and Ẽ2z > Ez(2E⊥+ ζEz), with ζ = ±1 for blue and red de-
tuning, respectively, one obtains a three-dimensional stack of
bilayers, separated by large potential barriers U3d. Eq. (5) im-
plies V ≫ t′ ≫ t, |Ṽ ′|, which can be achieved by making the
z-direction potential barrier U⊥ inside the bilayer sufficiently
larger than the in-plane barrier U‖, so that the condition t ≪ t
will be met; e.g., Ẽz/E⊥ ≈ 20, Ez/E⊥ ≈ 15 yields the bar-
rier ratio U3d : U⊥ : U‖ of approximately 16 : 8 : 1, and the
lattice constants ℓ⊥ ≈ 0.45λ, ℓ‖ = λ, where λ = 2π/k is the
laser wave length. The parameter Ṽ ′ has a zero as a function
of the angle θ, so it can be made as small as needed. Taking
λ = 400 nm, one obtains an estimate of Tc = (0.1÷ 0.3) µK
for cyanide molecules ClCN and HCN with the dipolar mo-
ment d0 ≈ 3 Debye, while the Fermi temperature for the same
parameters is Tf ≈ (0.6÷1.3) µK. This estimate corresponds
to the maximum value of Tc ∼ J reached when Ṽ
′ ∼ −J
and t . J . The hopping t′ was estimated assuming the in-
plane potential barrier U‖ is roughly equal to the recoil energy
Er = (~k)
2/2m, where m is the particle mass.
Experimental signatures. – Signatures of the ordered
phases can be observed [25, 26] in time-of-flight experi-
ments by measuring the density noise correlator G(r, r′) =
〈n(r)n(r′)〉 − 〈n(r)〉〈n(r′)〉. If the imaging axis is perpen-
dicular to the bilayer, n(r) =
σ,raσ,r〉 is the local net
density of two layers. For large flight times t it is proportional
to the momentum distribution nQ(r), where Q(r) = mr/~t.
In the Mott phase the response shows fermionic “Bragg dips”
at reciprocal lattice vectors g = (2πh/ℓ‖, 2πk/ℓ‖),
GM(r, r
′) ∝ f0(r, r
′) = −2〈Sx〉2
Q(r)−Q(r′)−g
In the CC and DW phases the noise correlator contains an
additional system of dips shifted by QB = (π/ℓ‖, π/ℓ‖):
GCC,DW(r, r
′) ∝ f0(r, r
′)− 2
〈Sz〉2 + 〈Sy〉2
Qx(r)ℓ‖
+ cos
Qy(r)ℓ‖
))2]}
Q(r)−Q(r′)−QB − g
In the DW phase 〈Sz〉 6= 0, 〈Sy〉 = 0, and so the density cor-
relator depends only on r − r′. In the CC phase 〈Sz〉 = 0,
〈Sy〉 6= 0, and the relative strength of the two systems of dips
varies periodically when one changes the initial point r, see
Fig. 3. This Q-dependent contribution stems from the intra-
layer currents 〈a†σ,raσ,r′〉 = (−)
σ(−)rδ〈rr′〉i〈S
y〉/4, where
comes from the fact that the inter-layer current splits into
four equivalent intra-layer ones (see Fig. 1), δ〈rr′〉 means r
and r′ must be nearest neighbors, and (−)r ≡ eiQB ·r de-
notes an oscillating factor. If the correlator is averaged over
the particle positions, the CC and DW phases become indis-
tinguishable. A direct way to observe the CC phase could be
to use the laser-induced fluorescence spectroscopy to detect
the Doppler line splitting proportional to the current.
Bosonic models.– Consider the bosonic version of the
model (1), with the additional on-site repulsion U . The ef-
fective Hamiltonian has the form (3) with J = −4(t′)2/V
and Jz = Ṽ
′ + 4(t′)2(1/V − 1/U). Due to ferromag-
netic (FM) transverse exchange, instead of spontaneous cur-
rent one obtains the usual Mott phase. CC states can be in-
duced by artificial gauge fields [27]: The vector potential
A(x) = π
(x + 1/2) makes hopping along the x axis imagi-
nary, t′ 7→ it′. The unitary transformation Sx,yr 7→ (−)
rSx,yr
maps the system onto a set of FM chains along the x axis,
AF-coupled in the y direction and subject to a staggered field
H = 2t along the x axis in the easy (xy) plane. In the ground
state net chain moments are arranged in a staggered way along
the y axis, so a current pattern similar to that of Fig. 1 emerges,
now staggered along only one of the two in-plane directions.
A different type of CC states, with orbital currents localized
at lattice sites, can be achieved with p-band bosons [28].
Yet another way to create a CC state in a bosonic bilayer is
to introduce the ring exchange on vertical plaquettes:
Hring =
〈rr′〉
2,r′b2,rb1,r′ + h.c.). (8)
In pseudospin language, the ring interaction modifies the
transverse exchange, J 7→ J + K , so for K > 0 one can
achieve the conditions J > 0, J > |Jz| necessary for the CC
phase to exist. However, engineering a sizeable ring exchange
in bosonic systems is difficult (see [29] for recent proposals).
Spin- 1
bilayer with four-spin ring exchange.– Consider
the Hubbard model for spinful fermions on a bilayer shown
in Fig. 1, with the on-site repulsion U and inter- and intra-
layer hoppings t and t′, respectively. At half filling (i.e., two
fermions per dimer), one can effectively describe the system
in terms of spin degrees of freedom represented by the opera-
tors S = 1
a†ασαβaβ . The leading term in t/U yields the AF
Heisenberg model with the nearest-neighbor exchange con-
stants J⊥ = 4t
2/U (inter-layer) and J‖ = 4(t
′)2/U (intra-
layer), while the next term, with the interaction strength J4 ≃
10t2(t′)2/U3, corresponds to the ring exchange [30, 31]:
H4 = 2J4
(S1 · S2)(S1′ · S2′)
+ (S1 · S1′)(S2 · S2′)− (S1 · S2′)(S2 · S1′)
, (9)
where the sum is over vertical plaquettes only (the interaction
for intra-layer plaquettes is of the order of (t′)4/U3 and is ne-
glected), and the sites (1, 2, 2′, 1′) form a plaquette (traversed
counter-clockwise). In the same order of the perturbation the-
ory, the nearest-neighbor exchange constants get corrections,
J⊥ 7→ JR = J⊥ + J4, J‖ 7→ JL = J‖ + J4/2,
and the interaction JD =
J4 along the diagonals of verti-
cal plaquettes is generated. Generalization for any 2d bipar-
tite lattice built of vertically arranged dimers is trivial. Since
J⊥ ≫ J‖, J4, we can treat the system as a set of weakly cou-
pled spin dimers. The dynamics can be described with the
help of the effective field theory [32] which is a continuum
version of the bond boson approach [33] and is based on dimer
coherent states |u,v〉 = (1−u2−v2)|s〉+
j(uj + ivj)|tj〉.
Here |s〉 and |tj〉, j = (x, y, z) are the singlet and triplet
states, and u, v are real vectors related to the staggered mag-
netization 〈S1 − S2〉 = 2u(1 − u
2 − v2)1/2 and vector chi-
rality 〈S1×S2〉 = v(1−u
2−v2)1/2 of the dimer. Using the
ansatz u(r) = (−)rϕ(r), v(r) = (−)rχ(r), passing to the
continuum in the coherent states path integral, and retaining
up to quartic terms in u, v, one obtains the Euclidean action
dτd2r
~(ϕ · ∂τχ− χ · ∂τϕ) (10)
+ (ϕ2 + χ2)(JR − 3ZJ4/2)− Z[J‖ϕ
2 + J4χ
+ (Z/2)[J‖(∂kϕ)
2 + J4(∂kχ)
2] + ZU4(ϕ,χ)
where the quartic potential U4 has the form
U4 = (ϕ
2 + χ2)[J‖ϕ
2 + J4χ
+ J4(ϕ
2 + χ2)2 + (J‖ + J4)(ϕ× χ)
2. (11)
Interdimer interactions J‖ and J4 favor two competing types
of order: while J‖ tends to establish the AF order (ϕ 6= 0,
χ = 0), strong ring exchange J4 favors another solution with
ϕ = 0, χ 6= 0, describing the state with a staggered vector
chirality. It wins over the AF one for J4 > J‖, J4 >
which for the square lattice (Z = 4) translates into
J4 > max(J‖, J⊥/9). (12)
On the line J4 = J‖ the symmetry is enhanced from SU(2)
to SU(2)× U(1), and the AF and chiral orders can coexist: a
rotation (ϕ+ iχ) 7→ (ϕ+ iχ)eiα leaves the action invariant.
The chiral state may be viewed as an analog of the cir-
culating current state considered above: in terms of the
original fermions of the Hubbard model, the z-component
of the chirality (S1 × S2)z =
1↓a2↓)(a
2↑a1↑) −
2↓a1↓)(a
1↑a2↑)
corresponds to the spin current (particles
with up and down spins moving in opposite directions).
Summary.– I have considered fermionic and bosonic
models on a bilayer optical lattice which exhibit a phase tran-
sition into a circulating current state with spontaneously bro-
ken time reversal symmetry. The simplest of those models
includes just nearest-neighbor interactions and hoppings, and
can possibly be realized with the help of polar molecules.
Acknowledgments.– I sincerely thank U. Schollwöck,
T. Vekua, and S. Wessel for fruitful discussions. Support by
Deutsche Forschungsgemeinschaft (the Heisenberg Program,
KO 2335/1-1) is gratefully acknowledged.
∗ On leave from: Institute of Magnetism, National Academy of
Sciences and Ministry of Education, 03142 Kiev, Ukraine.
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