Fully Frustrated Cold Atoms

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by  Marco Polini

Fully Frustrated Cold Atoms
Marco Polini,1,∗ Rosario Fazio,1 A.H. MacDonald,2 and M.P. Tosi1
1NEST-INFM and Classe di Scienze, Scuola Normale Superiore, I-56126 Pisa, Italy
2Physics Department, The University of Texas at Austin, Austin, Texas 78712
(Dated: February 2, 2008)
5 Fully frustrated Josephson Junction arrays (FF-JJA’s) exhibit a subtle compound phase tran-
0 sition in which an Ising transition associated with discrete broken translational symmetry and a
0
Berezinskii-Kosterlitz-Thouless (BKT) transition associated with quasi-long-range phasecoherence
2
occur nearly simultaneously. In this Letter we discuss a cold atom realization of the FF-JJA sys-

n tem. Wedemonstratethatbothorderscanbestudiedbystandardmomentum-distribution-function
a measurementsandpresentnumericalresults,basedonasuccessfulself-consistentspin-waveapprox-
J imation, that illustrate theexpected behavior of observables.

7
1 PACSnumbers: 03.75.Lm

]
r
The preparation of cold atomic gases trapped in an of choosing the optimal phase difference for each bond.
e
h optical lattice has opened up attractive new possibil- In a cold atom optical lattice system, frustration can be
t
o ities for the experimental study of strongly correlated introduced by altering the phase factors for atom hop-
.
t many-particle systems [1] and has inspired much the- ping between optical potential minima more explicitly,
a
m oretical activity (see e.g. Ref. 2 for a review). In forexamplebyfollowingproceduressimilartothosepro-
- particular, the experimental observation by Greiner et posedrecentlybyJakschandZoller[9],Mueller[10],and
d
al. [1] of a Superfluid-Mott Insulator (SI) transition in a Sørensen et al. [11]. The laser configurations suggested
n
o three-dimensional (3D) optical lattice explicitly demon- inthesepapersalsoenablespatiallyperiodicmodulation
c
strated the possibility of realizing strongly-correlated ofthe magnitudeofbosonhoppingamplitudes, afeature
[
cold bosons. The SI transition in an optical lattice was that is important to the proposal outlined below.
1
predicted in Ref. 3 and can be described by the Bose- In a FF square-lattice JJA the sum of the optimal
v
7 Hubbard model [4], which has also been employed to phase differences for individual bonds around every pla-
8
model 2D granular superconductors [5] and JJA’s [6]. quette is π, fully incompatible with the integer multiple
3
1 This success has motivated many new proposals [7] for of 2π phase winding constraint imposed by the single-
0
cold-atom simulations of strongly correlated boson phe- valuedcondensatewavefunction. ForsquarelatticeJJA’s
5
0 nomena. fullfrustrationcanbeintroducedbyapplyinganexternal
/ In this Letter we propose that cold atoms be used to magneticfieldthatgeneratesonehalfofasuperconduct-
t
a
studytheincompletelyunderstoodphasetransitionsthat ingfluxquantumthrougheachplaquetteofthearray. In
m
occur in FF-JJA’s [6, 8]. The boson Hubbard model for theLandaugaugethefrustrationisimposedbychanging
-
d JJA’s accounts for Cooper pair hopping between small the signofeverysecondverticalhopping parameter. For
n
superconducting particles and for Coulomb interactions a FF-JJA, the Gross-Pitaevskii mean-field equation of
o
c which can be dominantly intra-particle. For supercon- thecorrespondingbosonHubbardmodelhastwodistinct
:
v ductingparticlesthemodelapplieswhenthethermalen- degenerate solutions, illustrated schematically in Fig. 1,
i
X ergy kBT is much smaller than the bulk energy gap, i.e. which break the discrete translational symmetry of the
r when the underlying fermionic character of electrons is lattice,andforeachsolutionafreeoverallphasefactorin
a
suppressed. Cold atoms in optical lattice potentials pro- thecondensatewavefunctionwhichbreaksgaugesymme-
vide, in some senses at least, a closer realization[1, 3] of try. The surprising property of FF square lattice JJA’s,
the boson Hubbard model because other degrees of free- andbyextensionofFFsquarelattice coldatoms,is that
dom are more completely suppressed and because the the Ising order and the quasi-long-rangephase order ap-
interactionsaremoredominantlyon-site. Frustration[8] peartovanishnearlysimultaneouslyandcontinuouslyat
can be introduced into JJA’s by introducing an external a common critical temperature. When quantum fluctu-
magneticfieldtochangetheenergeticallypreferredphase ations are included, similar phase changes are expected
relationship between boson amplitudes on neighboring to occur at zero temperature as the on-site interaction
sites. Frustration in this case refers to the impossibility strength is increased. If these orders do in fact disap-
2
θD θA is weakened,asdescribedbelow,andthe Ising transition
D A D temperature is driven to zero. In this Letter we point
out that these subtle phase changes can be studied by
measuring the momentum distribution function (MDF)
×
of a FF cold atom cloud, and report on theoretical esti-
C B C
mates for the MDF based on a self-consistent harmonic
θC θB
approximation(SCHA) [18].
We assume that atom hopping between sites on the
×
optical lattice is weak enough to justify a single-band
Wannier basis [3] with Wannier function w(x). The lat-
D A D
tice Hamiltonian we study is
U
JFJIAG..1D:oGurboleunvde-rsttiactaeldliengeesnesrtaatnedsofolurtimonosdufolrataedcla“sasnictaifleFrrFo-- Hˆf = 2 nˆ2xi − ExJi,δcos(φˆxi −φˆxi+δ) (1)
magnetic” bonds (−αEJ) while single vertical and horizon- Xxi xXi,δ
tallinesstandfor unmodulated“ferromagnetic bonds”(EJ). where x = d(n,m) with n,m [ , ) is on a 2D
i
The configuration shown corresponds to α = 0.5 for which ∈ −N N
θA = −θB = π/12 and θD = −θC = π/4. The distinct squarelatticewithlatticeconstantd,δ isthevectorcon-
configuration with equal energy is obtained by (θA,θD) → necting a lattice site to its neighbours, and the Joseph-
−(θA,θD) or equivalently by vertical translation by one lat- son energy or atom hopping energies EJ are identi-
tice constant. xi,δ
cal (equal to E ) on all bonds except the vertical bonds
J
on every second column. These modulated frustrating
pear simultaneously, the phase transition would have to bonds have the value αEJ with α > 0 [18]. In Eq. (1)
−
be in a new universality class and could not have a nat- the phase operator φˆx has been introduced by approx-
i
uraldescriptionin termsof the condensate wavefunction imating the atom annihilation operator on site xi by
order-parameter, a situation reminiscent of the decon- ˆbx √n¯exp(iφˆx ), allowed when the mean occupa-
i ≃ i
fined quantum critical behavior discussed recently [12] tion n¯ on each lattice site is large. The density nˆx
i
by Senthil et al. and phase φˆx operators are canonically conjugate on
i
each site. The negative hopping parameters introduce
The compound phase change in a frustrated JJA is
frustration, which can be energetically weakened [18] by
closelyrelatedtothephasechangesthatoccurinthevor-
choosing α<1.
texlatticesofthemixedstateofsuperconductors,andin
When quantum fluctuations are neglected, the T = 0
rotating4He andcoldatomsystems[13, 14]. The vortex
condensatephasepattern[8]isdeterminedbyminimizing
lattice ground state has broken translational symmetry
the classical energy with respect to the phase difference
instigated by frustrating order-parameter-phase depen-
χ across positive E links; the single-valued condition
dent terms in the Hamiltonian. The key difference be- J
requiresthatthemagnitudeofthephasedifferenceacross
tween vortex lattices and frustrated JJA’s is that the
negative E links χ′ = 3χ, implying [8] that sin(χ) =
broken translational symmetry is discrete rather than J −
αsin(3χ) and hence that
continuous in the latter case. Thermal fluctuations of
a vortex lattice imply [15] that quasi-long-range phase
χ= arcsin( [(3α 1)/α]/2) (2)
order cannot exist at any finite temperature in 2D sys- ± p −
tems. For superconductors it has been argued [16] that for α > 1/3, while χ = 0 for α < 1/3. For α < 1/3, the
given the absence of phase coherence, broken transla- energy penalty of frustration is paid completely on the
tional symmetry will not occur either. For the FF-JJA negative E link and the classical ground ground state
J
case,the opposite conclusionhas been reachedin a care- condensate phase is spatially constant. As α increases
ful Monte Carlo study by Olsson [17]; he finds that that beyond this value, the energy penalty of frustration is
vortex position fluctuations suppress the phase stiffness increasinglyshiftedtothe positiveE links. Theground
J
and instigate a BKT transition as the Ising phase tran- state configuration in this regime is doubly degenerate
sition temperature is approachedfrom below. If correct, with currents circulating in opposite directions around
thisconclusionwouldhavetobealteredwhenfrustration alternating plaquettes, as illustrated in Fig. 1. Thermal
3
and quantum fluctuations will degrade both Ising and by that of an effective harmonic model defined by mean
phase coherence orders. condensate phases on each site and harmonic coupling
Phase coherence of cold atoms in an optical lattice constants K on eachnearestneighbour link. Minimizing
can be directly detected by observinga multiple matter- the variational free-energy with respect to mean phases
wave interference pattern after ballistic expansion with enforcesaveragecurrentconservationateachnodeofthe
all trapping potentials switched off. As time evolves, lattice. Minimization with respect to the harmonic cou-
phase-coherentmatter waves that are emitted from each pling constants sets them equal to the self-consistently
latticesiteoverlapandinterferewitheachother. Narrow determinedmeancurvatureoftheJosephsoninteraction.
peaks appear in the MDF due a combination of lattice Thephasechangesacrosstheverticalandhorizontalpos-
periodicity and long-range phase coherence [19, 20, 21]. itive E links, θ and θ , are unequal in this approxi-
J h v
The vortex superlattice of the α > 1/3 mean-field state mation, as are the harmonic coupling constants K and
h
resultsintheappearanceofadditionalpeaksintheMDF; K and (of course) the coupling constant on frustrated
v
n (k) = e Ψˆ†(k)Ψˆ(k) /A where A is the system area, links K . For U 0 and T 0, the θ = θ χ,
f α h v
ℜ h i → → →
and Ψˆ(k) is the 2D Fourier transform of the field opera- K =K E cosχ and K αE cos(3χ).
h v J α J
→ →−
tor, Ψˆ(x)= xiw(x−xi)ˆbxi. It follows that The SCHA phase correlation function C(xi,xj) =
P Cµ,νCµ,ν(X ,X ) is the product of a long-range factor
nf(k)= n¯|wA(k)|2ℜexXi,xjeik·(xi−xj)C(xi,xj) (3) CsyNNµm,FFνm,deQterpyenudnieitntcjeolnl,lyanodnpaoGsiatiuosnsiwanithfaincttohreC2Qµ×,ν2(Xbrio,kXenj)-
whichcapturesthepower-lawdecayofphasecorrelations
where we have defined a Wannier function form factor
in 2D superfluids (here X is a lattice vectorof the large
w(k)= d2xe−ik·xw(x) and the phase-phase correlator i
unit cell so that sites are labelled by µ and i). We find
C(xi,xjR) ≡ hexp[i(φˆxi −φˆxj)]i. In the broken transla- that Cµ,ν is given by
tionsymmetrystaten (k)is non-zeroatsuperlatticere- NF
f
ciprocallattice vectorsGn,m =π(n,m)/d; for the classi- 1 eiθv ei(θv+θh) e−iθh
cal(i.e. U =0)groundstateatzerotemperaturewefind Cµ,ν = e−iθv 1 eiθh e−i(θh+θv)  ,
that n (G) = (N2/A)n¯ w(G)2S (G) where N = 4 2 NF e−i(θv+θh) e−iθh 1 e−i(θv+2θh)
is the tfotal numbesr of la|ttice |site0s, and the supserlattNice  eiθh ei(θh+θv) ei(θv+2θh) 1 
(5)
structure factors are
and that
S0(G0,0) = [cos(χ)cos(χ/2)]2 Cµ,ν(X ,X ) = exp U Fkµ,,σν(Xi−Xj)
S0(G1,0) = [sin(χ)sin(χ/2)]2 Q i j (cid:26)−Ns2 Xσ kX∈BZ′ ξk,σ
S (G ) = [sin(χ)cos(χ/2)]2
0 0,1
S (G ) = [cos(χ)sin(χ/2)]2 (4) × [1+2NBE(ξk,σ/kBT)](cid:27). (6)
0 1,1
InEq.(6)thesumisoverthefourBogoliuboveigenmodes
with S (G ) = S (G ) for any integers n,
0 n+2k,m+2k 0 n,m
oftheharmonicJosephsontermateachwavevectorinthe
m, and k. Phase coherence in a lattice leads to con-
2 2 super cell’s Brillouin zone. Because we have cho-
densation peaks in nf(k) at all reciprocal lattice vec- ×
sen strictly on-site interactions, the quantum harmonic
tors G . Coherence and Ising broken translational
2n,2m
problem can be solved by first diagonalizing the Joseph-
symmetry leads to additional peaks (satellites) with the
son interaction term, as in the classical case, and then
characteristicpatternofstructurefactorssummarizedby
performing independent Bogoliubov transformations on
Eqs. (4) at the 2 2 superlattice reciprocal lattice vec-
× eachmode. ThecontributionofagivenBogoliubovmode
tors. MDF measurements therefore probe both types of
to the mean square phase difference between sites (µ,i)
order.
and (ν,j) in Eq. (6) is therefore characterized by the
These results will be altered by both quantum and
quantity [18]
thermal fluctuations. At low temperature (k T
B
≪
EJ) and well inside the superfluid regime (U EJ), µ,ν(X X ) = vσ(k)2+ vσ(k)2 2 e [vσ(k)]⋆
the phase correlation functions are given reliab≪ly by a Fk,σ i− j | µ | | ν | − ℜ n µ
SCHA [18] in which the density matrix is approximated × vνσ(k)eik·(Xi−Xj+bµν)o (7)
4
2π 10 3 1.2
−4 1 GGGG0011,,,,0101
π G01 G11
0.8
3
0.6
kyd 0 G00 G10 2 0.4
0.2
1
π 0
− 0 0.05 0.1 0.15 0.2
0 U/EJ
2π 2π FIG.3: Condensation andIsingpeaksofthestructurefactor
− −π 0 π S(G)asafunctionofU/EJ. ThevalueofS(G)forG1,0,G0,1
kxd andG1,1hasbeenmultipliedbyafactorof10forclarity. The
vertical dashed line indicates thevalue of UIcS.
FIG.2: ThestructurefactorS(k)forFFcoldbosonsina2D
array with α = 0.5 as a function of the continuous variable
kd ∈ [−2π,2π]×[−2π,2π]. Here T = 0.242EJ/kB [22], and symmetries.
U =0.1EJ.
This work was partially supported by EC-RTN
“Nanoscale Dynamics” and by an Advanced Research
where bµν is the site separation for i = j, NBE(x) is Initiative of S.N.S. A.H.M. acknowledges support from
a Bose-Einstein thermal factor, ξk2,σ = Uλk,σ, λk,σ and the Welch Foundation and from the National Science
vµσ(k) being the eigenvalues and the µ-th component of Foundation under grant DMR-0115947. We gratefully
the eigenvectors of the harmonic Josephson interaction. acknowledgetheearlycontributionstothisworkbyJairo
We have evaluated S(k) = n (k)A/(n¯N2 w(k)2) in Sinova. We wish to thank R. Asgari, P. Capuzzi, B.
f s| |
the presence of both quantum and thermal fluctuations Davoudi, M. Gattobigio, M. Greiner, and K. Madison
by summing over a finite lattice with N = 1296 sites for useful discussions.
s
in Eq. (3) and applying periodic boundary conditions to
make the wavevectors in Eq. (6) discrete. A typical re-
sult is reported in Fig. 2. The presence of non-zero Ising
satellites at k = G ,G and G is evident. These ∗ Electronic address: [email protected]
1,0 0,1 1,1
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