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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. [1], 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 [2].

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 [3]. 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 [7]

H = −

?

i

?

txb†

ix,iybix+1,iy+ tyb†

ix,iybix,iy+1+ h.c.

?

+U

2

?

i

ni(ni− 1) − µ

?

i

ni,(1)

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

[2]. 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 [1].

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 [8]. 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 [9]. 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-

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2

0 0.10.20.30.4

0

0.1

0.2

0.3

tx/U

ty/U

2D SF

1D MI

2D MI

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 [10]. 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

tc∝ L

−2+

x

1

2Kx

.(3)

Since it is known that Kx> 2 here [9], 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 [1]. 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,

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3

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) [13], 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

coherence [13].

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 [14]. 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,

The profile

y?/β, where W2

yis the so-

A third observable

0.010.1

1

10

tyU-1Lx

1.75

0.0001

0.01

1

100

expectation values

6*6

8*8

10*10

12*12

14*14

40∆

2

N

10Γy(L/2)

ρs/(tyU-1)

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.

x

/U, for different sys-

∆N2=

∆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 [9].

(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

[15], and if the number of particles is held fixed as in the

?

iyvar??

ixnix,iy

?/Ly.It is expected that

x

/U. The exponent

x

/U.

x

. The three

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4

0.11

tyU-1Lx

1.75

0

1

2

ρs,XY/[1+1/(2lnLy-1)]

0.11

tyU-1Lx

1.75

0.4

0.8

1/(2Ky)

6*6

6*10

10*10

6*14

10*14

14*14

6*18

10*18

6*24

1/2K=0.25

a)

b)

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.

x

/U. The hori-

present study, it is known that the transition considered

here is a KT transition [8].

Characteristic for such a transition is a discontinuous

jump of the superfluid density at the critical point [14]. 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. [4], 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

EC∝ L−1

this model is in turn mapped onto a 2D XY-model if

one identifies βXY =

Ly,XY = Ly [15], 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 [14]

x

and hopping energy EJ∝ tyL1−1/2Kx

?EJ/EC, Lx,XY = β√EJEC, and

x

. Next,

ρs,XY ∝ ?W2

y?Ly/βtyL1−1/2Kx

x

. (4)

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

[16], 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

x

.

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 [9].

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.

x

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