Wigner Crystallization in Rapidly Rotating 2D Dipolar Fermi Gases

0.14 MB·English

by  M. A. Baranov

Wigner Crystallization in Rapidly Rotating 2D Dipolar Fermi Gases
M.A. Baranov
Van der Waals-Zeeman Instituut, Unversiteit van Amsterdam,
Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands and
C Kurchatov Institute, Kurchatov sq. 1. 115582 Moscow, Russian Federation
8
0 H. Fehrmann
0 Institut fu¨r Theoretische Physik, Leibniz Universit¨at Hannover, Appelstraße 2, 30167 Hannover, Germany
2

n M. Lewenstein
a ICREA and ICFO–Institut de Ci`encies Fot`oniques, Castelldefels 08860, Spain
J

4 We study the competition between the Wigner crystal and the Laughlin liquid states in an ul-
tracold quasi two-dimensional rapidly rotating polarized fermionic dipolar gas, and find that the
] Wignercrystalhasalowerenergybelowacriticalfillingfactor. Weexaminethequantumcrystalto
l
l liquidtransitionfordifferentconfinementsinthethirddirection. Ouranalysisofthephononspectra
a
of the Wigner crystal with the account of phonon-phonon interactions also shows the stability of
h
- theWigner crystal for sufficiently low filling factors (ν <1/7).
s
e
m
Recently a remarkable progress has been made in ex- tal”, and ”bubble crystal” lattices [9, 10]. On the other
. perimental studies of strongly correlated systems of ul- hand, Rezayi et al. [11] have recently shown that a
t
a
tra cold gases. A seminal breakthrough was the obser- small amount of dipole-dipole interactions stabilizes the
m
vation of Mott insulator-superfluid transitions in Bose so called bosonic Rezayi-Read strongly correlated state
- gasesin opticallattices [1], followedby studies of lowdi- at ν = 3/2 whose excitations are both fractional, and
d
n mensionalgases,fermionic superfluidity or ultracold dis- non-Abelian.
o ordered gases (for recent reviews of this very rapidly de-
The dipolar interactions could also lead to a crystal
c velopingareasee [2, 3]). Aparticularlyfascinatingroute
[ groundstateinaquasi2Dsystemsimilartothe electron
toward creation of strongly correlated states is the one
Wignercrystal. Inthecaseofanelectrongas,bothinthe
2 thatusesrapidlyrotatinggases. Therotationisformally
presence and in the absence of a magnetic field, Wigner
v equivalent to a magnetic field (in the rotational frame),
2 crystalbecomesenergeticallyfavorableandstableforlow
whichreorganizesfree-particlestatesintodiscrete highly
9 filling factors: this is due to the fact that at low densi-
degenerate Landau levels. As a result, the properties
5 ties Coulomb interactions ( 1/R, where R is a typical
2 of the system become very sensitive to interparticle in- length scale) win over the k∼inetic energy ( 1/R2) (cf.
1 teractions, in complete analogy to the Fractional Quan- ∼
[12]). For non-rotating dipolar gases (where interactions
6 tum Hall Effect (FQHE) for electrons [4], that exhibits
0 scale as 1/R3), it was argued [8] that the ground state
a variety of strongly correlated states, among which the
/ has a crystal order at high densities, as shown in Refs.
t Laughlin liquid [5], and the Wigner crystal [6] are prob-
a [13]. Remarkably, in the case of a rotating dipolar gas,
m ably the most famous. This analogy has been pointed theWignercrystalphaseisexpectedatlow densities [8].
out in the context of ultracold atomic gases with short
- Therefore,thedipolargases,beingtosomeextendsimilar
d range interactions in Refs. [7], but an experimental ob-
toelectrons,presentneverthelessanovelclassofphysical
n servation of the FQHE in this case is difficult, partially
systems exhibiting quantum liquid to crystal transition.
o
duetosmallnessofthe energygapseparatingaLaughlin
c This opens anunprecedented possibility for the observa-
state from the so called quasi-hole excitations. It was
: tion and detailed studies of this transition in clean and
v recentlyargued[8]thattheseproblemsmaybeovercome
i well-controlled conditions. It is especially appealing in
X in rotating quasi-2D polarized dipolar gases, in which
viewofthe recentexperimentalrealizationofthedipolar
the interparticle interaction has a long-range part that
r Bose condensate of Chromium [14, 15], and progress in
a puts dipolar systems somewhat in between atomic sys-
trapping and cooling of dipolar molecules [16].
tems with short-range interactions and electron systems
with Coulomb interactions. As a result, the Laughlin In this paper we investigate the existence of a Wigner
state for the filling factor ν = 1/3 [32] was predicted to crystalphase in a rapidly rotating gas of polarizeddipo-
be incompressible with gapped excitations [8]. lar fermions by comparing it with the competing liquid
states. WethenexaminethestabilityoftheWignercrys-
Several other unexpected predictions have been al- tal by incorporating phonon-phonon interactions, and
ready made for rotating dipolar bosonic gases. The ro- identify the border of stability with the appearance of
tating dipolar condensates in the mean-field regime ex- purely imaginary phonon frequencies. It appears that
hibit novel forms of vortex lattices: square, ”stripe crys- the Wigner crystal is the stable ground state of the sys-
2
tems for low filling factors, typically for ν <1/7. the magnetic length, and the antisymmetrization op-
A
We consider a gas of fermionic polarized dipolar par- erator, which can be omitted for sufficiently low fill-
ticles in a rotating cylindrical trap, tightly confined and ing factors ν = 2πl2n. One can check that similar to
0
polarized along the axis of rotation. For a sufficiently the case of classical dipoles, the energy is minimal for
strong confinement in the axial direction, all particles a triangular lattice with particles centered at positions
are in the ground state of the axial motion and we can R = l b +l b , where l are integers, b = a(0,1),
i 1 1 2 2 1,2 1
write the many-body wave function in the form b = a √3,1 /2, and a is the lattice constant deter-
2
mined by the density n of the gas, a2 = 2/√3n =
ψ3D({ri,ξi})=ψ2D({ri}) l√π −N/2exp(− iξi2/2l2), 4πl02/√3(cid:0)ν. The(cid:1)energy of the crystal state equals
where ri =(x,y) and ξi ar(cid:0)e rad(cid:1)ial and axialXcoordinates UC = ΨC VD ΨC d2n3/2(0.2823+0.2146β
h | | i≈
of the particles, respectively, l is the extension in the
+0.3388β2+0.7456β3+2.0676β4+...),
axial direction, N is the total number of particles, and
ψ2D is the wave function of the system in the xy-plane. where β = πn(2l2 l2) and the first term corresponds
The confinement in the axial direction also modifies the 0 −
to the energy of classical point-like dipoles, E =
cl
interparticleinteractioninthe2D-plane. Fortwodipoles d2/l b +l b 3 =5.513d2/a3.
separated by a distance r in the 2D-plane, the effective i<j |1 1 2 2|
The energy of the Laughlin state (for a small number
interaction potential is P
of particles interacting via dipolar forces, recent compu-
tations show a remarkable overlap of the exact ground
d2 ∞ ξ r2
v (r)= dξ exp( ξ ) . state with the Laughlin state [18]) characterized by an
2D l3√2π Z0 s(ξ+1)3 − 2l2 odd integer M =1/ν is
For r l the potential is v2D d2/r3, while for r < l ν ∞
≫ ≈ U = rg(r)v (r)dr,
the effective potential increases logarithmically, v2D L 2 2D
(d2/l3) 2/πln(l/r). The wave function ψ obeys th≈e Z0
2D
Schr¨odinger equation with the effective 2D Hamiltonian whereg(r)isthepaircorrelationfunction. Wecalculated
p
that in the rotational frame reads this functions and the energies for odd M from 1 to 19
for a gas of 512 particle using a Monte Carlo method.
~2 m
H = ∆ + ω2r2 ΩL +V , (1) We now compare the energies of the Wigner crystal
− 2m i 2 ⊥ i − z D
i (cid:18) (cid:19) and of the Laughlin liquid for different filling factors ν
X
and extensions l in the axial direction. The results are
where Ω is the rotational frequency, ω⊥ is the radial presented in Fig. 1. They show that below some critical
trap frequency, V = v (r r ) describes the
D i<j 2D | i− j| value νc, which depends on l, the Wigner crystal has a
dipole-dipole interparticle interaction, and L is the z-
P z lowerenergy than the Laughlin liquid and, therefore, for
component of the total angular momentum. The Hamil-
ν <ν thegroundstateisexpectedtobeacrystal. Note
c
tonian (1) can be rewritten in the form
thatinobtainingthecriticalfillingfactorwecomparethe
1 energies of two simplest trail wave functions for liquid
H = ( i~ i Ai)2 (Ω ω⊥)Lz +VD (2) and crystal states. A better estimate could follow from
2m − ∇ − − −
Xi considering the wave functions of quantum Hall liquids
with Ai = mω⊥ez ri and, therefore, it formally de- of composite fermions [19] with the filling factors closer
× to the critical one than primary Laughins 1/M, and of
scribes a system of charged particles in a constant mag-
the correlatedWignercrystals[20]. We, however,do not
netic field with the cyclotron frequency ωc = 2ω⊥. For
expect a significant change of our result because for a
non-interacting particles (V = 0) in the regime of the
D
dilute system the two considered wave functions already
criticalrotation(ω⊥ =Ω),thepropertiesoftheHamilto-
catch the most important features of the interparticle
nian (2) are well known; it’s spectrum consists of highly
degenerate levels E = ~ω (n+1/2) called Landau lev- interaction.
n c
els. In the following, we restrict ourself to a system of Letusnowapproachtheliquid-crystaltransitionstart-
particles occupying only the lowest Landau level. ing from the crystalphase. For this purpose we consider
The wave function of a non-correlated Wigner crystal the phonon spectrum with taking into account phonon-
on the lowest Landau level is [17]: phonon interactions (anharmonicity effects). The ap-
pearanceofpurelyimaginaryphononfrequencieswillin-
Ψ ( z ) exp −1(z R 2+z R∗ z∗R ) , dicate the instability of the crystal and, therefore, the
C { i} ∼A 4l2 | i− i| i i − i i transition to a liquid state.
i (cid:20) 0 (cid:21)
Y In the harmonic approximation the phonon eigenfre-
where z = x +iy is the complex representation of the quencies can be obtained from the dynamic equations
i i i
2Dvectorri,Ri the(complex)latticesite,l0 = ~/mωc for displacements uαl of particles along the α-axis from
p
3
ω
0.4 ω/ω8
L
3 17 d
E l0 0.38 1/ν crystal
ν3/2 2 13 6
d
0.36 9
5 liquid 4
0.34
1
0 0.2 0.4 0.6 0.8 1 1.2 1.4
l/l0
0.32 2 ω
T
0.3
−0.5
2 4 6 8 10 12 14 16 18
1/ν
0
aky/2π0.5 0.6 0.4 0.2 0 −0.2 −0.4 −0.6
ak /2π
x
Figure 1: Energy per particle for the Wigner crystal (dotted
line) and for the Laughlin state (solid line) as a function of
thefillingfactorforl=0. Theinsertshowsthecriticalfilling
factor as a function of the extension in thez-direction.
Figure 2: Energy of the transverse ωT (lower surface) and
longitudinal ωL phonons.
their equilibrium positions Rl in the lattice [21]
G G
mu¨αl = βl′Φ(α2l),βl′uβl′ +ωcεαβu˙βl′, (3) Σ =Φ(3)
+
X
where Φ(α2l),βl′ = ∂2UC/∂Rαl∂Rβl′ is the dynamical ma- G Φ(3) Φ(4)
trix, α,β = x,y and ε is the antisymmetric tensor,
αβ
ε = 1. The last term in Eq. (3) corresponds to the
xy
Coriolis (Lorentz) force due to the rotation (magnetic Figure 3: One-loop diagrams for thephonon self-energy.
field).
Without rotation, ω = 0, Eq. 3 in the quasimomen-
c G (ω,k) of the phonon Green function G (t,l l′)=
tum representation reads: αβ αβ −
i[θ(t) uαl(t)uβl′(0) +θ( t) uβl′(0)uαl(t) ]withθ(t)be-
h i − h i
ω2u˜ (k)= F (k)u˜ (k), ing the step function (for more details on the usage
α αβ β
β of Green functions for phonons in crystals see e.g.
X
where mFαα′(k) = lexp[−ik(Rl−Rl′)]Φ(α2l),βl′ is the [m23a]t)i.on TishGe(G0)−re1e(nω,fku)nc=tion in the ha(sr)m[ωo2n(ikc)appωr2o]xi-
Fourier transform of Φ(2) , and the eigenvalues ω2(k), αβ { s=T,LMαβ s − −
s = T,L of Fαα′(kP) dαelt,eβrl′mines the frequencies ofs the iεαβωcω}m/~, where M(αsβ) =Pe(αs)e(βs) is the projector to
transversal (T) and longitudinal (L) phonons. The theeigenmodeswiththepolarizatione(s),andthepoles
phonon frequencies for l = 0 are shown in Fig. 2. They are at ω =ω±(k). In the presence of phonon-phononin-
are linear in k for k a−1: ω (3/√8)ω ak and teractions,the Greenfunction obeysthe Dysonequation
T d
≪ ≈
ω √11ω , where ω = d2/ma5 sets the typical
L T d
≈
value for the frequency of phopnons. G−1(ω,k)=G(0)−1(ω,k) Σ (ω,k),
For a rotating crystal, ωc = 0, the phonon spectrum αβ αβ − aβ
6
becomes [21, 22] wherethe phononself-energyfunctionΣ incorporatesall
the effects of phonon-phonon interactions.
ω2 +ω2 +ω2 1
ω2 = L T c (ω2 +ω2 +ω2)2 4ω2ω2. We calculate the phonon self-energy in the one-loop
± 2 ± 2 L T c − L T
approximation that has proven its reliability in the case
q
In the case ωc ωs, one has ω+ ωc and ω− of an electron Wigner crystal in a strong magnetic field
≫ ≈ ≈
ω ω /ω . [24,25,26]. Thiscorrespondstotakingintoaccountonly
L T c
Higher orders (anharmonic) terms in the expansion of the fourth Φ(4) ∂4U /∂R4 and the square of the third
C
∼
the energy of the crystal with respect to the displace- Φ(3) ∂3U /∂R3 order anharmonic terms, see Fig 3.
C
∼
ments of particles from their equilibrium positions re- We then solve the Dyson equation numerically by suc-
sult in phonon-phonon interactions and, therefore, in cessive iterations until we either obtain a self-consistent
the renormalization of the phonon frequencies. At solution or the iteration breaks down due to the appear-
a given quasimomentum k, the renormalized frequen- ance of purely imaginary phonon frequencies. The lat-
cies correspond to the poles of the Fourier transform ter indicates the instability of the lattice. The results
4
of our calculations can be summarized as follows. At We wish to acknowledge N. Barber`an, N.R. Cooper,
very low filling factors the phonon-phonon interactions J.M.F. Gunn, E. Jeckelmann, A.H. MacDonald, K. Os-
do not play any significant role and the system behaves terloh, and R. Wimmer for helpful discussions. This
harmonically. However, the effects of anharmonicity be- work was supported by the Nederlandse Stichting voor
come progressively important with increasing the filling Fundamenteel Onderzoek der Materie (FOM), the Rus-
factor, and, finally, at some critical filling factor ν we sianFoundationforFundamentalResearch,theDeutsche
c
findpurely imaginaryfrequenciesand,hence,the crystal Forschungsgemeinschaft (SPP1116, SFB 407, GK 282),
ceases to exist. We always observe this phonon instabil- ESF PESC ”QUDEDIS”, EU IP ”SCALA”, Span-
ity for k 0. This indicates the breakdown of the crys- ish MEC (FIS 2005-04627 and Consolider-Ingenio 2010
→
talline order and rules out any possible structural phase ”QOIT”).
transition. Therefore, the value ν sets the upper bound
c
for the stability of the crystal. The critical filling factor
ν depends on the confinement in axial direction as well
c
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