arXiv:cond-mat/0701616v2 [cond-mat.other] 8 Aug 2007
Transition from a two-dimensional superfluid to a one-dimensional Mott insulator
Sara Bergkvist and Anders Rosengren
Department of Theoretical Physics, Royal Institute of Technology, AlbaNova, SE-106 91 Stockholm, Sweden
Robert Saers, Emil Lundh, Magnus Rehn, and Anders Kastberg
Department of Physics, Ume˚ a University, SE-90187 Ume˚ a, Sweden
A two-dimensional system of atoms in an anisotropic optical lattice is studied theoretically. If
the system is finite in one direction, it is shown to exhibit a transition between a two-dimensional
superfluid and a one-dimensional Mott insulating chain of superfluid tubes. Monte Carlo (MC)
simulations are consistent with the expectation that the phase transition is of Kosterlitz-Thouless
(KT) type. The effect of the transition on experimental time-of-flight images is discussed.
Cold atomic gases in optical lattices provide a means to
study many-body quantum phenomena that offers both
versatility, precision, and control. These techniques al-
lowed for the spectacular realization of the Mott tran-
sition by Greiner et al. , which has inspired numer-
ous proposals for creating and detecting exotic quantum
phases in such systems.
By manipulating the alignment and the polarizations
of the laser beams that build up the optical lattice, the
geometry of the lattice can be changed. The probability
for tunneling can be adjusted by changing the separa-
tion between potential wells, or by simply varying the
irradiance. In conjunction, this brings about the possi-
bility of designing optical lattices where the probability
for tunneling differs significantly between different direc-
tions, and where this difference in tunneling rates can be
a control parameter. By allowing the atoms to tunnel
only along one direction, one essentially has created a
2D lattice of 1D quantum gases. By allowing tunneling
in two directions, an array of 2D systems is created .
The tunability allows to explore the crossover between
different dimensionalities. Thus, with a properly chosen
geometry, it should be possible to start off with high laser
irradiance and thus with deep potential wells. This sys-
tem will be a Mott insulator, where the number of par-
ticles per site is fixed to an integer and there is no phase
coherence. By lowering the irradiance in one direction,
increasing the tunneling probability, one should reach a
point where tunneling becomes probable along just one
direction. This would give us isolated 1D tubes, known
as Luttinger liquids . Decreasing the irradiance in a
second direction, we expect to reach a point where there
is a crossover from this to a system of 2D superfluids,
and eventually one global 3D superfluid [4, 5, 6]. In this
Letter, this dimensional crossover is addressed by quan-
tum MC simulations of an anisotropic bosonic Hubbard
model in two dimensions. We show how the model can be
readily realized in experiment, and predict the outcome
of absorption imaging on both sides of the transition.
System. – Consider a gas of bosonic atoms at zero
temperature in a 2D, anisotropic optical lattice. The gas
is described by the Bose-Hubbard Hamiltonian 
H = −
ni(ni− 1) − µ
where i = (ix,iy); bi, b†
ation operators, ni= b†
the chemical potential, and tx, ty the tunneling matrix
elements in the Cartesian directions. The crossover from
2D to 1D behavior considered in this paper can be re-
alized in a simple cubic optical lattice with a potential
of the form V (r) =?
tial barrier in the z direction Vzis strong enough, there
can be no tunneling in the z direction, and the sample
can be considered as a stack of independent 2D systems
. Vxshould be chosen relatively weak in order to allow
for superfluid flow in the x direction, and Vy is scanned
over the physically interesting range. As will be shown,
if the lattice has an extent of eight sites in the x direc-
tion there will occur a phase transition for tx/U = 0.3
and ty/U ≈ 6 × 10−3. As an example, for a gas of87Rb
atoms in an optical lattice with wavelength λ = 850 nm,
these tunneling matrix elements are obtained by choos-
ing potential strengths Vx = 2.5Erec, Vy = 30Erec, and
Vz= 80Erec, where Erecis the single-photon recoil energy
for a wavelength of 780nm .
Phase diagram. – The Hubbard model exhibits a zero-
temperature quantum phase transition from a superfluid
to a Mott insulating state when the ratio of the tunnel-
ing matrix elements and the coupling strength, tx/U and
ty/U, are decreased below a critical value . In one
dimension (i.e., either tx = 0 or ty = 0), it is known
that the phase transition occurs at t/U = 0.3 for the
system with an average of one particle per site . For
the anisotropic 2D model considered here, a mean-field
argument implies that t should be replaced by the sum
tx+tyin the 1D expression, so the transition occurs when
(tx+ty)/U = 0.3. For finite systems the phase transition
occurs at a smaller value of t/U.
Consider the anisotropic case, where one of the tun-
iare bosonic annihilation and cre-
ibi, U the interaction strength, µ
α=x,y,zVαcos2kα. If the poten-
0 0.10.20.3 0.4
FIG. 1: Phase diagram for the finite anisotropic Hubbard
model at number density n = 1. The 2D superfluid and Mott
insulating phases are denoted by “2D SF” and “2D MI”, re-
spectively. The size of the 1D Mott insulating phase “1D MI”
has been exaggerated in the figure. The vertical double arrow
indicates the interval along which the MC calculations in this
study have been performed.
neling matrix elements, say, tx, is larger than the critical
1D value and the other one, ty, is varied. In an infi-
nite 2D system, it is known that the system remains a
2D superfluid (2D SF) right down to the point where
ty vanishes . In other words, when the coupling in
one direction vanishes, the system can be seen as an ar-
ray of uncoupled Luttinger liquids. However, any finite
tunneling, no matter how small, will induce phase co-
herence between the Luttinger liquid tubes, so that an
anisotropic phase, with superflow only in one direction,
does not exist. However, the situation changes when the
system is not infinite. If each Luttinger liquid has a fi-
nite length, there is an energy barrier against tunneling
into and out of it, and as a result there exists, for small
ty, an anisotropic state with coherence only along the x
direction [4, 5, 6]. Hence, when the weaker tunneling
matrix element ty is small enough, the state of the sys-
tem is a 1D Mott insulator (1D MI), where each site is a
Luttinger liquid tube extending along the x direction. If
the tunneling tyis increased above a critical value tc, the
tubes become phase correlated and the system undergoes
a quantum phase transition to a superfluid state. In a
geometry where the system is infinite in the weakly cou-
pled direction, the state is an 1D SF, i. e., a superfluid
chain of Luttinger liquids, just above the phase transi-
tion, but is expected to undergo a crossover into a 2D SF
as the coherence length becomes smaller than the tube
length Lx, and the picture of separate Luttinger liquid
tubes can no longer be sustained. The crossover between
1D SF and 2D SF presumably happens over a very short
interval, why we in the following shall speak of the 1D
MI - 2D SF transition. In an actual experiment where
the system is finite, there is a crossover between 1D MI
and 2D SF without an intermediate 1D SF state. In the
FIG. 2: Velocity distribution along the weakly coupled lattice
direction. The curves corresponds to a ratio of the tunnel-
ing matrix element to the interaction strength, ty/U = 0.012
(dashed line), 0.0025 (full), and 1.5×10−3(dotted). The inset
shows the full 2D profile of the velocity distribution for the
case ty/U = 1.5 × 10−3. The system size is 12×12 sites.
1D MI phase, each tube along the x direction behaves
as an independent Luttinger liquid, characterized by an
(inverse) exponent K [3, 4, 5]. This so-called Luttinger
parameter determines the algebraic decay of the single-
particle correlation function Γ(j), according to
Γx(j) ≡ ?b†
ix,iybix+j,iy? ∼ j−1/(2Kx).(2)
This form for the asymptotic behavior holds only in the
1D case, i. e., for decoupled Luttinger liquid tubes. Γyis
defined analogously. The critical tunneling was in Refs.
[4, 5, 6] found to depend on the parameters as
Since it is known that Kx> 2 here , the exponent in
Eq. (3) is expected to be slightly less than -1.75. The
constant of proportionality in Eq. (3) depends on the
model used to integrate out the dynamics in the tubes
as well as on the dimensionality. These phase boundaries
have been indicated at arbitrarily chosen positions in the
phase diagram in Fig. 1.
Experimental signal. – Detection of the transition can
be achieved by applying an optical lattice potential, V (r),
to a Bose-Einstein condensate in a magnetic trap. Ramp-
ing up V (r) adiabatically ensures that the atoms are not
heated, and that they are loaded to the lowest energy
band of the lattice . For high lattice barriers, the cor-
relations between wells are lost after some hold time, re-
sulting in a Mott insulating state, whereas in the case of
low barrier heights the sample stays coherent. The state
of the system is detected by standard time of flight ab-
sorption imaging. When the sample is released by ramp-
ing down the lattice and magnetic trap non-adiabatically,
it will expand freely. The expansion reveals the sample’s
velocity distribution n(k), given by the Fourier trans-
form of the trapped sample density distribution in real
space n(r) , n(k) ∝ |w(k)|2S(k), where the first part
is the Fourier transform of the Wannier function for the
first Bloch band w(r), and the second part, the structure
factor S(k), is the Fourier transform of the correlation
function Γ(r) defined in Eq. (2). The structure factor
has the periodicity 4π/λ.
Knowing the structure factor from MC simulations (see
below), and calculating the Wannier functions for a cu-
bic lattice, a theoretical prediction for the actual exper-
imentally observable time-of-flight profiles can thus be
obtained.The result is shown in Fig. 2.
of the cloud in the direction of weak tunneling is deter-
mined by the state of the system. When distinct peaks
are visible at k = 0,±4π/λ, a 2D SF is detected. This is
connected with a slow decay of the real-space correlation
function Γ(r). If the lattice depth is increased, the sam-
ple passes into the 1D MI and the interference fringes are
blurred as Γ(r) becomes exponentially decaying. The sig-
nal would be even more pronounced for a larger system.
However, the peaks will be visible far into the Mott insu-
lator regime, an effect caused by maintained short-range
Method. – To solve for the ground state of the Bose-
Hubbard model, Eq. (1), Quantum MC calculations with
the stochastic series expansion algorithm are used [11,
12]. We simulate a system of size Lx×Lylattice sites with
periodic boundary conditions to minimize edge effects.
The inverse temperature is β = (kBT)−1= 1000U−1;
this is large enough to ensure that the system is in its
ground state. The results presented in the article are
calculated with a fixed tx= 0.3U with ty/U being varied
across the anticipated phase transition as indicated by
the double arrow in Fig. 1. Trial calculations not shown
here, using larger values of tx/U, have yielded similar
results.The calculations have been performed at the
fixed average number of particles per site n = 1.
Transition. – To locate the 2D SF↔1D MI transition,
three different observable quantities are calculated, each
of which can be considered a measure of superfluidity.
The superfluid density, ρs, is readily obtained from the
simulation data as ρs = ?W2
called square winding number in the y direction, defined
as the squared net number of times a particle crosses
the periodic boundary in the calculations . Second,
the off-diagonal correlation function along the y direc-
tion, Γy(j), is, as we have seen, experimentally accessible
through the momentum distribution, which is closely re-
lated to its Fourier transform. We now study the corre-
lation function in real space evaluated at half the system
size, Γy(y = Ly/2) which we expect to behave differently
above and below the transition. Above, Γy(y) decays
algebraically, below exponentially.
is the compressibility, defined as the mean of variances,
y?/β, where W2
yis the so-
A third observable
FIG. 3: From top to bottom, the density fluctuation, the cor-
relation, and the superfluid density are shown as a function of
the size-independent parameter tyL1.75
tem sizes Lx×Ly measured in lattice sites, as indicated in the
legend. The scale is logarithmic on both axes. The correlation
function and the compressibility are multiplied by arbitrarily
chosen numerical factors to separate the three groups of lines.
The superfluid density is divided by the tunneling matrix el-
ement for better visual clarity.
/U, for different sys-
∆N2changes its behavior at the transition: due to the
finite length of the tubes, the fluctuations in the number
of particles per tube will be suppressed once the system
becomes insulating in the weakly coupled y direction.
In Fig. 3, the three quantities, ρs, Γy(Ly/2), and ∆N2,
are shown for lattices 6 ≤ Lx= Ly ≤ 14. The observ-
ables are plotted as functions of tyL1.75
of Lxis given by Eq. (3) and the value of Kx is deter-
mined by a finite-size scaling of the superfluid density
as explained below. For a system with tx/U = 0.4, not
shown here, the best scaling is obtained for tyL1.79
The superfluid density in the y direction is in the super-
fluid phase seen to be approximately proportional to the
coupling constant ty/U, hence the superfluid density is
divided by ty/U. It is clear from Fig. 3 that there is a
change in behavior around ty/U ≈ 0.3L−1.75
quantities in Fig. 3 are behaving as in a 1D MI-SF tran-
sition. The true 1D system can be obtained by putting
Lx = 1 and letting Ly → ∞, and we then recover the
known value for the true 1D transition ty/U = 0.3 .
(The same curves are obtained for Ly > Lx, but these
data are not shown here not to overload the figure). The
finite size effects above the transition point are taken care
of by scaling Lx with the power 1.75. If the curves in-
stead were plotted as a function of ty/U, they would not
tend to coincide.
Since in the critical region, the system can be described
as a 1D chain of Luttinger liquid tubes, the phase transi-
tion is a 1D quantum phase transition. Such a transition
can be mapped onto a classical phase transition in 2D
, and if the number of particles is held fixed as in the
?/Ly.It is expected that
/U. The exponent
. The three
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FIG. 4: a) Superfluid density in the dual XY model, Eq. (4),
scaled according to Weber-Minnhagen scaling. b) Exponent
of the algebraic decay of the correlation Γy, as a function of
the dimensionless coupling parameter tyL1.75
zontal line shows the critical value expected for a 1D Mott
transition at fixed density, 1/(2K) = 0.25. Different curves
correspond to different system sizes; the same symbols are
used in both parts of the figure.
/U. The hori-
present study, it is known that the transition considered
here is a KT transition .
Characteristic for such a transition is a discontinuous
jump of the superfluid density at the critical point . In
Fig. 4 a) a finite size scaling of the data for the superfluid
density is illustrated. In order to do the scaling, a series
of mappings has to be performed: First, according to
Ref. , the original Hubbard model can be rewritten as
a Josephson junction chain model, where each Luttinger
liquid tube is treated as a site, with a charging energy
this model is in turn mapped onto a 2D XY-model if
one identifies βXY =
Ly,XY = Ly , where the quantities referring to the
dual 2D XY-model carry the subscript XY. As a result,
the superfluid density in the 2D XY-model is 
and hopping energy EJ∝ tyL1−1/2Kx
?EJ/EC, Lx,XY = β√EJEC, and
ρs,XY ∝ ?W2
The quantum phase transition occurring at a critical tun-
neling in the underlying Hubbard model corresponds to
the well-known finite-temperature KT transition in the
2D XY-model. According to Weber-Minnhagen scaling
, ρs,XY should be proportional to 1+1/(2lnLy+C)
at the critical point, where C is a fitting constant to be
determined. The quantity ρs,XY divided by this func-
tion should therefore assume the same value for all sys-
tem sizes at the critical point, if C and Kx are chosen
correctly. The best scaling is obtained with C = −1 and
Kx = 2 and is shown in Fig. 4 a). The scaling behav-
ior supports the conclusion that the transition is a KT
transition and the critical point is tc/U ≈ 0.27L−1.75
An independent method for calculating the location of
the transition is provided by fitting the correlation func-
tion in the y-direction, Γy, to an algebraic decay func-
tion according to Eq. (2). The phase transition for a 1D
Hubbard model at a fixed number of atoms per site is
found by determining the point where the exponent in
the power law, 1/(2Ky), becomes equal to 0.25 .
In Fig. 4 b) the exponent obtained from an algebraic
fit is displayed for different system sizes. There is a clear
finite-size effect, the larger Lythe larger the critical value.
The critical point at which 1/(2Ky) = 0.25 for the infi-
nite system is determined to be tc/U ≈ 0.3L−1.75
agreement with the scaling in Fig. 4 a) and the point of
loss of superfluidity observed in Fig. 3.
Summary – We have reported theoretical evidence for
a transition in a 2D Hubbard model between a 2D super-
fluid and a 1D Mott insulating state consisting of isolated
1D tubes. The location of the transition is consistent
with predictions based upon a random phase approxi-
mation made in Refs. [4, 5, 6]. By scaling it is shown
that the system undergoes a KT transition. The transi-
tion can be induced using atoms in an optical lattice and
observed by time-of-flight imaging.
This work was supported by the G¨ oran Gustafsson
foundation, the Swedish Research Council, the Knut and
Alice Wallenberg Foundation, the Carl Trygger founda-
tion, and the Kempe foundation.
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