Table Of ContentQuantum hexatic order in two-dimensional dipolar and charged fluids
Georg M. Bruun
Department of Physics and Astronomy, University of Aarhus, Ny Munkegade, DK-8000 Aarhus C, Denmark
David R. Nelson
Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
Recent advances in cold atom experimentation suggest that studies of quantum two-dimensional
melting of dipolar molecules, with dipoles aligned perpendicular to ordering plane, may be on the
horizon. An intriguing aspect of this problem is that two-dimensional classical aligned dipoles
4
(already studied in great detail in soft matter experiments on magnetic colloids) are known to
1
melt via a two-stage process, with an intermediate hexatic phase separating the usual crystal and
0
isotropic fluid phases. We estimate here the effect of quantum fluctuations on this hexatic phase,
2
for both dipolar systems and charged Wigner crystals. Our approximate phase diagrams rely on
r a pair of Lindemann criteria, suitably adapted to deal with effects of thermal fluctuations in two
a
M dimensions. As part of our analysis, we determine the phonon spectra of quantum particles on
a triangular lattice interacting with repulsive 1/r3 and 1/r potentials. A large softening of the
transverseandlongitudinalphononfrequencies,duetobothlatticeeffectsandquantumfluctuations,
1
plays a significant role in our analysis. The hexatic phase is predicted to survive down to very low
3
temperatures.
]
s
a The melting of crystals is a fundamental topic in con- to the 2d plane exhibits a quantum phase transition di-
g
densed matter physics that has been studied for more rectly to a hexagonal crystal phase at zero temperature
-
t thanacentury. Nevertheless,onelacksaquantitativeun- T = 0.11,12 However, the effects of quantum zero point
n
derstandingofthemeltingformanymaterials. Thisdefi- motion on the hexatic phase, accessible now for the first
a
u ciencyispartlyduetoimpreciseknowledgeoftheparticle time via cold dipolar gases, have not been discussed.
q interactionsthatcontroltherelevantphasetransitionson
In this paper, we explore this question by analysing
. a microscopic scale. In two dimensions (2d), however, a
t the stability of crystal and hexatic order of a 2d system
a defect-mediated theory of melting is available. Building
of dipoles including both thermal and quantum effects.
m
onpioneeringworkproposinganentropically-drivenpro-
First, we calculate the classical elastic coefficients of the
- liferation of dislocations by Kosterlitz and Thouless and
d crystal from the phonon spectrum. We then show how
by Berezinski,1 and on unpublished work of Feynman,2
n quantum effects soften the crystal by decreasing these
o a defect mediated theory was worked out.3 The detailed coefficients. Using Lindemann criteria suitably modified
c theory actually invokes a sequential unbinding of dislo- to treat both 2d thermal fluctuations and quantum ef-
[ cations and disclinations, with the usual latent heat of
fects, we study the successive loss of translational and
a first order melting transition spread out over an in-
2 orientational order that lead to the melting of the crys-
v termediate hexatic phase. The hexatic phase is charac- tal and hexatic phases. The relevant Lindemann num-
7 terised by extended bond orientational order at interme- bers are extracted in the classical regime as well as for
3 diate densities or temperatures. It has been observed in
T = 0 by comparing with Monte-Carlo and experimen-
2 a series of impressive experiments using colloidal parti-
tal results. Throughout the paper, we construct a use-
2
cles with a diameter of 4.5µm and an induced magnetic
ful comparison between the 2d dipolar system and a 2d
.
1 moment, with dynamics confined to an air-water inter- fluid consisting of negatively charged particles immersed
0 face.4,5Althoughthissystem,characterisedbylongrange
inauniformbackgroundofpositivecharges–theJellium
4 1/r3 dipole-dipoleinteractions,isthoroughlyunderstood
model. An experimental realisation of charged systems
1
in the classical regime, it is presently unknown whether
: in the classical limit with a Wigner crystal phase at low
v thehexaticphaseexistswhenquantumfluctuationsplay
temperatures has been known for quite some time, in
i a major role.
X the form of a 2d electron gas trapped by a positively
The advancing field of ultra-cold gases consisting of chargedcapacitorplatetothesurfaceofliquidhelium.13
r
a heteronuclear molecules with an electric dipole moment Forcomputersimulationevidencethatthissystemmelts
promises to change this situation. The dipolar gases are via a dislocation mechanism and may possess an inter-
expected to exhibit qualitatively new physics, with sev- mediate hexatic phase, see Ref. 14 and 15 respectively.
eral experimental groups reporting impressive progress We obtain essentially the same Lindemann numbers for
towards achieving quantum degeneracy in these sys- thedipolarandthechargedsystems,whichsuggeststhat
tems.6–10 Using dipolar gases, one should finally be able the melting of these phases is a geometric phenomenon,
toprobeexperimentallytheroleofquantumfluctuations insensitive to the detailed form of the interaction po-
on the hexatic phase and study its stability at very low tential. Similar conclusions resulted from Monte-Carlo
temperatures. Recent Monte-Carlo calculations predict simulationsofquantumhardspheresystems,16 andfrom
that a 2d dipolar gas with the moments perpendicular a meta-analysis of Monte-Carlo results for the freezing
2
of two-dimensional systems.17 We show that quantum the criterion (2) using crystalline phonon spectra to es-
effects initially increase the temperature range where timate the extent of the hexatic phase seems reasonable,
the hexatic phase is stable when the coupling strength provided local orientational order remains robust after
is decreased from the strong coupling classical regime, longrangetranslationalorderislost,asisthecaseforthe
provided the temperature dependence of the Lindemann dislocation-disclination theory of classical 2d melting,23
numbers can be neglected. A tentative phase diagram is and in situations where a weakly first order transition
then provided showing that quantum effects are impor- leads to a hexatic phase.24 We also note that our use of
tant even for very large interaction strengths where one phonondisplacementsfromanunderlyingreferencecrys-
wouldnaivelyexpectthesystemtobedeepintheclassi- tal implicitly treats quantum particles as distinguishable
cal regime. Finally, we discuss the possible experimental (a similar approximation is used in the Debye theory of
observation of the hexatic phase in the quantum regime thespecificheatofcrystals25), soweareeffectivelylook-
using cold dipolar gases. ing at the melting of quantum particles with Boltzman-
We conclude this introduction with a few observations nian statistics.26 We assume that the exchange interac-
about the Lindemann criterion for melting of quantum tions that distinguish bosons from fermions play only a
and classical systems in 2d and the nature of quan- minorroleindeterminingthelocationsofquantummelt-
tum hexatics. As originally proposed by Lindemann,18 ing transitions.
one first calculates the mean-square displacement W =
(cid:104)|u(r)|2(cid:105) of a single particle away from its equilibrium These Lindemann inequalities are criteria, and of
lattice position, where (cid:104)...(cid:105) represents an ensemble av- course do not themselves constitute a theory of quan-
erage. Melting as a function of, say, the temperature or tum or classical melting. We are not aware of a reli-
densitythenoccurswhentherootmeansquaredisplace- ablemicroscopictheoryof2dquantummelting,andeven
mentexceedsafixedfractionofthelatticespacinga,i.e. the defect-mediated melting of classical particles in two
√
for W ≥c a,wheretheLindemannnumberistypically dimensions could be preempted by a direct first order
L
in the range c ≈ 0.1−0.3.19 This rough criterion fails, transitionfromacrystaltoanisotropicliquid.3 Itisalso
L
however, in 2d classical solids, because W diverges loga- worthnotingtheratherdifferentnatureofclassicalasop-
rithmically with system size. Here we use an alternative posedtoquantummeltingintwodimensions. Thisdiffer-
formulation that focuses on the stretching of a nearest enceisparticularlyevidentintheFeynmanpathintegral
neighbor distance,20 namely formulation of nonrelativistic quantum statistical me-
chanics,27whereclassicalparticlesarereplacedbyconfig-
∆(r)=(cid:112)(cid:104)|u(r+b)−u(r)|2(cid:105)≥γ a (1) urationsofparticleworldlinesinimaginarytime,seeFig.
m
1. Weallow,forsimplicity,onlytheidentitypermutation
with periodic boundary conditions across an imaginary
where b connects nearest neighbor lattice sites (|b|=a)
time slab of thickness β(cid:126). In the absence of interactions,
and γ is an alternative Lindemann number describing
m
these trajectories when projected down the imaginary
this new measure of the loss of translational order. The
timeaxisbehaveliketwodimensionalrandomwalksasa
quantity ∆(r) remains finite in the thermodynamic limit
functionoftheimaginarytimevariable,withasizegiven
evenin2dand,asweshowhere,canalsobecomputedin
bythethermaldeBrogliewavelengthΛ =(cid:112)2π(cid:126)2β/m.
thequantumregimeallthewaydowntoT =0forsimple T
pairpotentials. Moreover,arelatedcriterionallowsusto In the classical limit Λ (cid:28) n−1/2, where n is the areal
T 0 0
estimate where the order associated with the rotational particle density, the particle world lines are short and
broken symmetry of a 2d crystal is lost due to thermal nearly straight; hence, the usual Lindemann picture for
or quantum fluctuations, namely melting of point-like particles applies when interactions
are turned on. However, in the highly quantum limit
∆ (r)=(cid:112)(cid:104)θ2(r)(cid:105)= 1(cid:114)(cid:68)|∂ u (r)−∂ u (r)|2(cid:69)≥γ ΛT (cid:29) n−01/2, quantum and thermal fluctuations act on
θ 2 x y y x i a crystal of long wiggling lines: as illustrated in Fig. 1,
(2) a particle world line r (τ) that makes a large excursion
j
where γ is a Lindemann number for the loss of bond withinitsconfiningcageatimaginarytimeτ isconnected
i
orientational order. Here, θ(r) = [∂xuy(r)−∂yux(r)]/2 bythekineticenergyinteractionm(cid:82)0(cid:126)β|drj(τ)/dτ|2/2to
is the local phonon-induced twist of the crystallo- time slices above and below, and hence can more easily
graphic axes,21 a quantity whose fluctuations are known recover and return to its equilibrium position when the
to remain finite even in the limit of infinite system slabthicknessislarge. Itishardertomeltarraysoflines
size for a classical 2d crystal.22 Note that ∆(r) ≈ in 2+1 dimensions than point-like particles in two di-
(cid:112)
(cid:104)[(b·∇)u(r)]2(cid:105) has a similar gradient structure to mensions with the same pair potential. Thus, we should
∆ (r). These two different Lindemann numbers γ and not be surprised if the Lindemann numbers γ and γ
θ m m i
γ allow for two distinct melting temperatures, charac- dependsomewhatonn Λ2,withlargerLindemannnum-
i 0 T
terized by the successive loss of first translational and bers required to produce melting when n Λ2 (cid:29) 1. This
0 T
thenorientationalorder,23 ascenarioweknow occursfor is indeed what we find fitting to experiments on colloids
classicalcolloidalparticlesinteractingwithrepulsivelong and quantum simulations of power law potentials, with,
range 1/r3 dipole-dipole interaction.4 Our evaluation of e.g.,γ rangingfrom∼0.1intheclassicallimitto∼0.3
m
3
FIG. 1. The quantum partition function (12) is obtained
by integrating over particle world lines in imaginary time τ.
Left: The high temperature regime limit n Λ2 (cid:28) 1 where
0 T
one recovers the classical 2d system. Right: The low tem-
perature quantum regime where the problem becomes 2+1
dimensional.
when quantum effects predominate. Phonon nonlinear- FIG. 2. X-ray diffraction pattern as a function of scatter-
ities can give rise to a weak temperature-dependence of ingangleφfromasampleofalignedDNA-moleculesexhibit-
ingthesix-foldsymmetrycharacteristicofthehexaticphase.
thelongwavelengthelasticconstants,evenintheabsence
Reprinted with permission from Ref. 34.
of quantum fluctuations.14 We neglect such effects here
forsimplicity. Withtheseunderstandings,webelievethe
criteriasketchedabovecanprovidearoughmapofwhere
externalmagneticfield,whereκ−1istheLondonpenetra-
to look for quantum melting in the new arena provided
tiondepth. Whenthelinesareverylongcomparedtothe
by cold quantum gases.
vortex line spacing, and pinning is negligible, the classi-
What would a quantum hexatic look like, if next
cal statistical mechanics of these three dimensional lines
generation experiments were to discover such a thing?
at finite temperatures can be mapped via the transfer
Roughly speaking, it would be a quantum liquid crys-
matrix method onto the quantum statistical mechanics
tal,acousinofthelong-soughtsupersolidphaseof4Heif
of 2d bosons at T =0 interacting with the pair potential
the particles were bosons.28,29 A fermionic analog would
Eq. (3).32 Here, the temperature T of the 3d supercon-
be the quantum nematic studied by Oganesyan et. al.
ductor plays the role of (cid:126) and the thickness of the bulk
in electronic systems,30 which exhibits a two-fold rather
superconductor plays the role of (cid:126)β in the equivalent 2d
than six-fold anisotropy. These authors also considered
quantum system. A dislocation loop unbinding model
the possibility of an electronic quantum hexatic. A
then leads directly to an entangled liquid of vortex lines,
hexatic quantum fluid would display a fuzzy, six-fold-
withlongrangesix-foldbondorientationalorder,equiva-
symmetric diffraction pattern, indicating extended ori-
lentviathepathintegralmappingtoazerotemperature
entational correlations, somewhat like a poorly averaged
quantum hexatic.33 Although it has not yet been possi-
powder diffraction pattern. However, unlike a classical
ble to check for hexatic order in melted vortex liquids in
polycrystal,aquantumhexaticwouldbeafluidwithzero
TypeIIsuperconductors(thesignalfromneutrondiffrac-
shear modulus! If composed of bosons, it could develop
tionisquiteweak),somethingverylikeanentangledline
a nonzero superfluid density and support supercurrents
hexatic has been seen in X-ray diffraction experiments
at sufficiently low temperatures.31 The nature of hexatic
off partially ordered arrays of aligned DNA molecules.34
orderinrealspace,withextendedcorrelationsintheori-
When these charged, linear polymers are aligned by an
entationsofdistantsix-foldparticleclusters, isdiscussed
external field, a screened, Debye-Hu¨ckel interaction arise
in Ref. 3.
in the perpendicular direction which has precisely the
Inthecrudephasediagramsconstructedhere,wehave
form (3). The characteristic line hexatic diffraction pat-
taken a conservative approach, and assumed the inter-
tern from this work is reproduced in Fig. 2. The 2d
mediate hexatic phase is squeezed out as T →0, leaving
structure function (cid:104)|n(q)|2(cid:105) for particles in a quantum
behind a transition from a quantum solid directly to an
hexatic at low temperatures should look very similar.
isotropic quantum liquid. But this need not be the case:
considerparticlesinteractingwithascreened,2dYukawa
potential,
I. LAME´ COEFFICIENTS
V(r)=(cid:15) K (κr) (3)
0 0
Inthissection,wecalculatetheelasticcoefficientsfrom
where K (x) is the modified Bessel function of the sec- thephononmodesofa2dhexagonalcrystalconsistingof
0
ond kind, K (x)∼−logx for x(cid:28)1, K (x)∼exp(−κx) charged particles with a neutralising background charge
0 0
for x (cid:29) 1, and κ−1 is a screening length. Such a poten- density, orparticleswithadipolemomentperpendicular
tialdescribesinteractionsbetweenvortexlineswithweak to the 2d plane. The crystal lattice with lattice con-
√
thermal fluctuations in Type II superconductors with an stant a is spanned by the vectors a =a( 3/2,1/2) and
1
4
these results confirming that the Ewald summation has
√
converged. Notethatthelongitudinalmodescalesas q
forsmallmomentareflectingthelongrangenatureofthe
Coulomb interaction.
TheLam´ecoefficientsaredefinedbywritingtheelastic
energy of the crystal as38
F = 1 (cid:88)(cid:8)µ(k)|u (k)|2+[2µ(k)+λ(k)]|u (k)|2(cid:9)k2
FIG. 3. Left: The hexagonal lattice in reciprocal space and el 2L2 t l
k
the irreducible first Brillouin zone. Middle: The two phonon
(7)
modes along b for dipoles. Right: The two phonon modes
1 √ with|u(k)|2 =u(k)u(−k)andL2 theareaofthesystem.
alongb forchargedparticles. Notethatω(q)∼ qforsmall
1 l The longitudinal component of the displacement field is
q,indicatinganinfinitebulkmodulusinthelongwavelength
u ,andu isthetransversecomponent. Notethat”trans-
limit. The dashed lines are from Eqs. (5)-(6). l t
verse”and”longitudinal”simplyreferstothelowestand
highestphononmodeforagivenk,sincetheeigenvectors
√
a =a(− 3/2,1/2) corresponding to the reciprocal vec- arenotingeneralparallelorperpendiculartokwhenlat-
2 √ √
tors b = 2πa−1(1/ 3,1) and b = 2πa−1(−1/ 3,1). tice effects are taken into account. The relation between
1 2
The reciprocal lattice with the irreducible Brillouin zone the Lam´e coefficients and the phonon modes is then as
is shown in Fig. 3(left). The interaction between two usual
particles separated by a distance r in the plane is (cid:115) (cid:115)
µ(k) 2µ(k)+λ(k)
ω (k)= k and ω (k)= k
D2 Q2 t ρ l ρ
U(r)= dipoles U(r)= charges (4)
r3 r (8)
√
where D2 =d2/4π(cid:15) for electric dipoles with dipole mo- whereρ=m2/ 3a2 isthemassarealdensity. Thenatu-
0
ment d, and Q2 = q2/4π(cid:15) for particles with charge q. ralscalefortheLam´ecoefficientsisD2/a5fordipolesand
0
We set (cid:126)=kB =1 in the following. Q2/a3 for charged particles, and they√are k dependent
due to lattice effects. Since ω (k) ∝ k for k → 0, the
l √
Lam´e coefficient λ(k) diverges as 1/ k for the charged
A. Classical elasticity of point dipoles and point particles.
charges In Figs. 4-5, we plot the classical Lam´e coefficients
along b for the dipoles and the charged particles. The
1
We find the phonon modes of the potential energy in elasticparametersdisplayasignificantk-dependentsoft-
the harmonic approximation in the usual way. To ac- ening due to the discrete lattice symmetry. We also plot
celerate the convergence of the sums, we use the Ewald in Fig. 4 the k = 0 Lam´e coefficients corresponding to
summationtechniqueasdetailedinAppendixA.InFig.3 Eq. (5), i.e.
weplottheresultingtwophononbranchesalongthevec-
D2 D2
tor b for the dipoles (middle) and the charged particles µ(0)(cid:39)2.4 and 2µ(0)+λ(0)(cid:39)26 . (9)
1 a5 a5
(right). Thelowenergymodeispurelytransverseandthe
high energy mode is purely longitudinal for long wave- These values agree very well those reported in Ref. 36.
lengths where the hexagonal crystal is equivalent to an Likewise, we plot in Fig. 5 the k =0 value for the trans-
isotropiccontinuumsystem.21 Thecharacteristicphonon verse mode corresponding to Eq. (6), i.e.37
(cid:112)
energy for the dipolar crystal is ω = D2/ma5, and a
D 23/2Q2
crystal of Coulomb charges, it is ω =(cid:112)Q2/ma3 where µ(0)=η (10)
C 33/4a3
m is the particle mass. For the dipoles, we find for long
which is recovered by our numerics.
wave lengths the isotropic modes
Using Eq. (7), it is straightforward to calculate the
ω (q)(cid:39)4.8ω qa and ω (q)(cid:39)1.4ω qa. (5) meandisplacementoftheparticlesfromtheirequilibrium
l D t D
positions at a given temperature T, and we obtain
These sound velocities differ somewhat from what was
T
reported recently,35 but as we shall demonstrate shortly, (cid:104)u (k)u (k(cid:48))(cid:105)=δ L2
they accurately recover well established values for the l l k,−k(cid:48) [2µ(k)+λ(k)]k2
k = 0 Lam´e coefficients36 which gives us confidence in T
(cid:104)u (k)u (k(cid:48))(cid:105)=δ L2 . (11)
ournumericalcalculations. Forthechargedparticles, we t t k,−k(cid:48) µ(k)k2
have plotted the long wave length formulas,
√
2 π √ 21/4η1/2
ω (q)= ω qa and ω (q)= ω qa (6) B. Quantum softening of the Lam´e coefficients
l 31/4 C t 31/8 C
for the longitudinal and transverse mode respectively We now include quantum effects on the Lam´e coeffi-
with η = 0.25.37 We see that the numerics reproduce cients by quantizing the phonons. This can be done in
5
including quantum effects,
L2β (cid:88) 1
(cid:104)u (k,τ)u (−k,τ)(cid:105)=
σ σ ρ ω2 +ω (k)2
n n σ
L2
= coth[βω (k)/2)] (13)
2ρω (k) σ
σ
wherethesumisoverMatsubarafrequenciesω =2nπT
n
and phonon modes σ = t,l. Recasting this result in the
FIG.4. Left: TheLam´ecoefficientforthelowest(transverse)
form of Eq. (11) defines the quantum Lam´e coefficients
mode of the dipoles along b . The solid and dotted lines in-
1 µ (k) and λ (k) as
cludequantumeffectsfortwodifferenttemperatures,andthe Q Q
dash-dottedlinegivestheclassicallimit. Right: Thesamefor √
2T ρµ (cid:18)(cid:114)µ k (cid:19)
the highest (longitudinal) mode of the dipolar system. The
µ (k)= tanh
dashed lines are the q = 0 results from Eq. (9). Note the Q k ρ2T
significantdownwardrenormalizationduetoquantumfluctu- (cid:112) (cid:32)(cid:115) (cid:33)
ations. 2T ρ(2µ+λ) 2µ+λ k
2µ (k)+λ (k)= tanh ,
Q Q k ρ 2T
(14)
where we have suppressed the k-dependence of the clas-
sical Lam´e coefficients µ(k) and λ(k) for notational
simplicity. Equation (14) reveals that the magnitude
of quantum effects on the Lam´e coefficients is de-
(cid:112)
termined by the parameters µ/ρk/T = ω /T and
t
(cid:112)
(2µ+λ)/ρk/T = ω /T. For ω /T → 0 we recover
l σ
the classical results given by Eq. (8), whereas the elas-
ticcoefficientsaredecreasedduetoquantumfluctuations
FIG.5. Left: TheLam´ecoefficientforthelowest(transverse)
modeofthechargesparticlesalongb1. Thesolidanddotted whenever ωσ/T (cid:38)1.
lines include quantum effects for two different temperatures, In Figs. 4-5, we plot the quantum Lam´e coefficients
andthedash-dottedlinegivestheclassicallimit. Right: The along b for T/ω = T/ω = 1 and T/ω = T/ω =
1 D C D C
same for the highest (longitudinal) mode of the charged par- 0.2. We see that quantum effects significantly soften the
ticles. The dashed line is obtained from Eq. (10). crystalfordecreasingtemperature,andthatitisthehigh
energy fluctuations which are reduced the most. Quan-
tum softening is therefore greater for the longitudinal
mode, reducing [2µ (k) + λ (k)]/[2µ(k) + λ(k)] more
several ways. Here, we use the path integral approach Q Q
than µ (k)/µ(k).
since it allows us to describe the quantum effects on the Q
crystal melting in terms of a simple geometrical picture
of wiggling particle trajectories described in the intro-
duction. For completeness, we also present a canonical II. MODIFIED LINDEMANN CRITERIA FOR
quantization approach in Appendix B. CRYSTAL AND HEXATIC MELTING
ThepartitionfunctionZ forthecrystalcanbewritten
asanintegraloverallpossiblepathsu(r,τ)oftheparticle Sincethereisalgebraic,asopposedtolongrangetrans-
displacements in imaginary time τ as27 lational order in a 2d crystal when T > 0,39 it is not
possibletoestimatethemeltingtemperatureofthecrys-
(cid:90) talfromausualLindemanncriterion,asdiscussedinthe
Z = D[u(k,τ)]e−(cid:82)0βdτ[Fkin(τ)+Fel(τ)] (12) introduction. We will therefore use a modified Linde-
manncriteriumforthemelting. Ourbasicassumptionis
that the melting of the crystal occurs in two steps with
where F (τ)=ρ(cid:80) |∂ u(k,τ)|2/2L2 is the kinetic en- increasing temperature.3,23,40 First, the crystal melts at
kin k τ
ergy, ρ is the 2d mass density, and the elastic energy a temperature T into a hexatic phase, characterised by
m
F (τ) is given by Eq. (7) with the replacement k → longrangebondangleorderbutshortrangetranslational
el
(k,τ). Wedonotincludepermutationsoftheparticlepo- order. Then, at a higher temperature T the hexatic
i
sitionsattheboundaryτ =β oftheimaginerytimeslab, phasemeltsintoanisotropicliquid. Theexistenceofthe
sotheboundaryconditioninEq.(12)isu(r,β)=u(r,0). hexatic phase is well established for classical systems4,
At this level of approximation, we therefore cannot dis- but our knowledge concerning its stability against quan-
tinguishbetweenbosonicandfermionicparticles. Wecan tum fluctuations is limited. A Monte-Carlo study sup-
from Eq. (12) calculate the fluctuations of the particles portstheexistenceofsuchaphaseinthequantumregime
6
in the case of distinguishable particles with Coulomb in- limit of Eq. (14) in Eq. (15), we obtain
teractions.26 We therefore focus on how quantum fluctu-
ations affect on the stability of the hexatic phase. 1 (cid:88)
(cid:104)|u (r)−u (0)|2(cid:105)= (1−cosk·r)
i i ρL2
k
(cid:20)(cid:15) (k)2 1−(cid:15) (k)2(cid:21)
A. Melting of the crystal phase × l,i + l,i (18)
ω (k) ω (k)
l t
To calculate the temperature Tm where the crystal for T = 0. Equation (18) predicts as expected that the
melts into the hexatic phase, we will use the modified zero point motion of the particles scale as the typical
Lindemann criterion given by Eq. (1). It states that the harmonic oscillator length for the phonons, i.e. δu2 ∼
crystalmeltswhentherelative fluctuationsoftheparticle 1/mω for dipoles and δu2 ∼ 1/mω for charged parti-
D C
positions of two nearest neighbours are larger than the cles. Using Eq. (18), we can determine the Lindemann
lattice constant a. Using the quantum Lam´e coefficients number for T = 0 at the critical coupling strength for
in Eq. (11) yields the quantum melting transition, obtaining
2 (cid:88) γ =0.31 dipoles γ =0.31 charges. (19)
δu2 =(cid:104)|u (r)−u (0)|2(cid:105)=T (1−cosk·r) m,0 m,0
i i i L2
k
The Lindemann numbers are again the same for the
(cid:20) (cid:15) (k)2 1−(cid:15) (k)2(cid:21)
× l,i + l,i (15) dipoles and the charges at this level of accuracy, indi-
[2µQ(k)+λQ(k)]k2 µQ(k)k2 cating that the melting of the crystal phase is primarily
determined by geometry also at T =0. The T =0 value
for the fluctuations along a specific direction i = x or of γ at the critical point is consistent with what was
m
i = y. Here, (cid:15)l,i(k) is i’th component of the eigenvector obtained using perturbation theory.42
of the highest mode. Comparing Eqs. (16) and (19) shows that the Linde-
The melting temperature for the crystal phase in the mann numbers are significantly larger at T = 0 than in
classical limit was reported to be Tm (cid:39)0.0907D2/a3 for the classical regime. As discussed in the introduction,
dipoles,36,41 and Tm (cid:39) 0.0136Q2/a for charged parti- the path integral approach provides a simple geometri-
cles.15 From these results, we can determine the Linde- cal interpretation of this result. Indeed, from Eq. (12)
mannnumberintheclassicalregime. UsingEqs.(1)and it is clear that the quantum problem corresponds to the
(15) with the classical Lam´e coefficients to calculate the melting of a crystal of lines in a 3d slab of thickness β,
particlefluctuationsattheclassicalmeltingtemperature see Fig. 1. Only when β → 0 does one recover the clas-
yields sical problem of the melting of a 2d crystal. Since the
lines can wiggle significantly along the β-direction with-
γm,cl =0.14 dipoles γm,cl =0.15 charges. (16) out melting the crystal, it is natural to expect that the
Lindemann number is larger in the quantum regime as
We see that the classical Lindemann numbers are essen- compared to the classical regime.
tially the same for the dipoles and the charged particles.
This result suggests that melting is mainly determined
by the geometry of the crystal, depending very weakly
B. Melting of the hexatic phase
on the detailed form of the interaction potential so that
the Lindemann number is almost universal.
In the two-step melting scenario, the system is in a
As discussed in the introduction, we expect that the
hexatic phase characterised by extended bond angle or-
Lindemann numbers in general depend on temperature
der for temperatures T < T < T . We therefore use
due to quantum effects. We can determine the Linde- m i
theLindemanncriterionbasedonthefluctuationsinthe
mannnumberγ atT =0usingrecentquantumMonte-
m bond angle θ given by Eq. (2). Fourier transforming Eq.
Carlo results suggesting a T = 0 quantum phase tran-
(11)usingthequantumLam´ecoefficientsgivesaftersome
sition between the crystal phase and the liquid phase:
algebra
For dipoles, two independent calculations give the value
triDcle(cid:39)s o1n8e±o4bt1a1i,n12s,tahnedvfaolruedisrting(cid:39)uis1h2a7blfeorchtahregecdritpiacar-l (cid:104)θ2(cid:105)= T (cid:88)(cid:20) (k×(cid:15)l(k))2 + (k×(cid:15)t(k))2(cid:21)
C 4L2 [2µ (k)+λ (k)]k2 µ (k)k2
value of this quantum phase transition.26 Here, the r- k Q Q Q
parameters (20)
where (cid:15) (k) is the eigenvector of the lowest mode. For
t
r = mD2 dipoles r =mQ2a charges (17) an isotropic medium where (cid:15)l (cid:107) k and (cid:15)t ⊥ k, Eq. (20)
D a C reduces to
are the ratios between the nearest neighbour interaction δθ2 = T (cid:88) 1 , (21)
energyandthequantumkineticenergy. UsingtheT →0 4L2 µQ(k)
k
7
i.e. the bond angle fluctuations are determined by the phaseismorerobusttowardsquantumfluctuationscom-
transversemodeonly. Sinceahexagonalcrystalisequiv- pared to the crystal phase, and as a result the hexatic
alenttoanisotropicmediumforlongwavelengths,21 and region in the phase diagram should increase with de-
sinceitistheselowenergymodeswhichcontributemost creasing potential/quantum-kinetic energy ratio r. This
to the fluctuations, Eq. (21) turns out to be a very good conclusiondependsofcourseontheassumptionthatthe
approximation to Eq. (20). Lindemann numbers are independent of temperature, so
To determine the Lindemann numbers for the hexatic we expect it to be valid only when the quantum correc-
phase, we again use results for the melting temperatures tionsaresmall. Whenthetemperaturedependenceofthe
reported in the literature. The melting temperature of Lindemann numbers is significant we cannot determine
the hexatic phase in the classical limit was found to be the fate of the hexatic without further information.
T (cid:39) 0.0968U(a) with U(a) = D2/a3 for dipoles36, and
i
T (cid:39) 0.0159U(a) with U(a) = Q2/a for charged parti-
i
cles.15 Using these temperatures and the classical Lam´e
III. PHASE DIAGRAMS
coefficients in Eq. (20) yields
γ =0.12 dipoles γ =0.13 charges (22) In this section, we provide approximate phase dia-
i,cl i,cl
grams, using the fact that the melting of the crystal and
for the Lindemann numbers determining the melting of hexatic phases are insensitive to the detailed form of the
thehexaticphaseintheclassicalregime. Asforthecrys- interaction potential. However, by comparing the values
tal phase, the Lindemann numbers are essentially the of the Lindemann numbers in the classical regime and
same for the dipoles and the charged particles, suggest- at T = 0 given by Eqs. (16) and (19) we see that they
ingagainthatmeltingofthehexaticphaseisageometric depend on temperature. To provide a tentative phase
phenomenon, largely independent of the precise form of diagram, we therefore write the Lindemann number de-
the interaction. termining the crystal melting on the phenomenological
The T = 0 Monte-Carlo calculations for the dipoles form
did not examine the quantum hexatic phase,11,12 so it is
presently not known whether it exists all the way down (cid:18) T (cid:19)n
γ (T)=γ +(γ −γ ) , (25)
to T = 0. In the case of distinguishable ”Boltzman- m m,0 m,cl m,0 T
m
nian” charged particles, it was found that the hexatic
phase persists to quite low temperatures where quantum whichinterpolatesbetweentheT =0valueandtheclas-
effects are significant, disappearing in a tricritical point sical value for T = T . We write γ (T) in the same
m i
at T (cid:39)0.04U(a).26 Since our analysis indicates that, for phenomenological form. To determine n, we compare
a given value of n0Λ2T, the melting is insensitive to the the phase diagram produced by these phenomenologi-
detailed form of the interaction potential, this Monte- cal forms with the phase diagram obtained by Monte-
Carlo result suggests that the hexatic phase persist deep Carlo calculations for distinguishable charged particles
into the low temperature regime both for dipoles and for in Ref. 26. It turns out that n = 6 yields a reasonable
charged particles. It is therefore interesting to evaluate goodfitasisshowninFig.6. Itmustbeemphasisedthat
the Lindemann numbers at T = 0 for the bond angle we have not performed a systematic fit to determine the
fluctuations. Using the T =0 limit of Eq. (20), optimal value of n, since this is not relevant at this level
of approximation, where our goal is simply to provide a
δθ2 = 1 (cid:88)(cid:20)(k×(cid:15)l(k))2 + (k×(cid:15)t(k))2(cid:21), (23) qualitatively reliable phase diagram. The detailed form
8ρL2 ωl(k) ωt(k) of the actual phase diagram could be different. For in-
k
stance, our analysis does not reproduce the tri-critical
yields
point at T (cid:39)0.004U(a) found in the Monte-Carlo calcu-
lations. Instead, we have chosen to let the hexatic and
γ =0.23 dipoles γ =0.24 charges (24)
i,0 i,0
crystal phases continue down to r (cid:39) 95 for T = 0 cor-
C
at the quantum quantum transition points r (cid:39) 18 and responding to the Lindemann numbers γ = 0.25 and
D i,0
r (cid:39) 127 for dipoles and charged particles respectively. γ = 0.33. Moreover, as stressed in the introduction,
C m,0
Again, the angle fluctuations differ very little between for the case of a Yukawa potential, a quantum hexatic
the dipolar and the charged systems. The precise values phase could exist for a finite parameter range even at
of the Lindemann numbers may, of course, differ at the T = 0. Note that there is no significant increase in the
exact boundaries of the hexatic phase in the quantum temperature range for which the hexatic phase is sta-
regime, which are presently unknown. ble with decreasing r in Fig. 6. This is because the
C
Aswediscussed,themeltingofthehexaticphaseisde- phenomenological form for the temperature dependence
termined almost exclusively by the lowest phonon mode of the Lindemann numbers given by Eq. (25) with n=6
in the sense that Eq. (21) is an excellent approxima- essentiallycancelsthiseffectforthetemperaturesshown.
tion to Eq. (20). In addition, the lowest phonon mode In Fig. 6, we also plot as dashed lines the critical tem-
is less affected by quantum fluctuations than the high- peraturesintheclassicallimit. Weseethatthequantum
est phonon mode. It follows from this that the hexatic suppressionofthecriticaltemperaturesissignificanteven
8
0.016
Hexatic
0.09
0.014 Hexatic
0.08
0.012
0.07
0.01
2Q/a]0.008 23D/a]00..0056
T [0.006 T [0.04 Crystal
0.03
0.004
0.02
Crystal
0.002
0.01
0 0
0 100 200 300 400 500 600 700 800 0 100 200 300 400 500 600
r r
C D
FIG. 6. Approximate phase diagram of the charged particles FIG. 7. Approximate phase diagram of the dipoles as a
as a function of the rescaled temperature T/(Q2/a) and the function of the rescaled temperature T/(D2/a3) and the
potential/quantum-kinetic energy ratio r , using tempera- potential/quantum-kinetic energy ratio r , using tempera-
C D
turedependentLindemannconstantsdeterminedbyEq.(25) turedependentLindemannconstantsdeterminedbyEq.(25)
with n = 6. The dashed lines are the classical limits for with n = 6. The dashed lines are the classical limits for
r (cid:29)1,andthe×’sand+’sarenumericalMonte-Carlodata r (cid:29)1.
C D
from Ref. 26.
forr ∼O(100),whereonewouldnaivelyexpectthesys-
C
temtobewellwithintheclassicalregime. Theenhanced air-water interface.4 On the other hand, the quantum
importance of quantum fluctuations arises because the analogue of the hexatic phase is yet to be explored. The
classical melting temperatures of the crystal and hex- cold assemblies of dipolar molecules seem well suited to
atic phases are so low with Tm/U(a) = 0.0136 (cid:28) 1 study both the classical and the quantum regimes of 2d
and Ti/U(a) = 0.0159 (cid:28) 1. Since quantum soften- melting. Thetypicaldipolemomentofthesemoleculesis
ing of the√Lam´e coefficients sets in for T/ωC (cid:46) 1, and of the order of one Debye. Taking as an example the re-
T/ωC = rCT/U(a) with Tm/U(a) (cid:28) 1 in the hexatic centlytrapped23Na40Kmoleculewhichhasapermanent
and crystal phases, the classical melting temperature is dipole moment of d = 2.7 Debye,6,43 one gets r (cid:39) 24
D
only recovered for rC (cid:29)1. for an average inter particle spacing of 300nm, which
In Fig. 7, with applications to cold dipolar gases in is well inside the quantum regime as can be seen from
mind, we plot our tentative phase diagram for dipo- Fig. 7. The critical temperature for the hexatic phase
lar charges obtained from the Lindemann criteria us- in the classical regime is T (cid:39) 0.2µK for this set of pa-
ingtemperaturedependentLindemannnumbersgivenby rameters. Even though the critical temperature will be
Eq. (25) with n = 6. As for the charged case, quantum lower in the quantum regime we estimate that the quan-
effects on the melting are significant even for rD (cid:29) 1 tum hexatic phase should be within experimental reach
since the classical critical temperatures are so low with once the cooling techniques for the dipolar gases have
Tm/U(a) (cid:28) 1 and Ti/U(a) (cid:28) 1. As for the case of beenoptimised. Onecanfurthermorereachmuchhigher
charged particles, there is no significant increase in the critical temperatures using molecules with larger dipole
temperaturerangewherethehexaticphaseisstablewith moments such as SrO with d=8.9 Debye. The presence
decreasingrD duetothechosentemperaturedependence of the hexatic phase can be detected by measuring the
oftheLindemannnumbers. Again,thisphasediagramis static structure factor via Bragg spectroscopy which is a
onlyqualitativelyreliablebutitsuggeststhatthehexatic well proven experimental probe for quantum gases.44–46
phase extends well into the quantum regime. In the hexatic phase, the structure factor will exhibit a
sixfoldsymmetrywithnosharppeakssimilartowhatit
is shown in Fig. 2. Recent impressive experiments have
IV. EXPERIMENTAL CONSIDERATIONS reported single atom resolution in optical lattices,47,48
and if one is able to achieve the same resolution with
The hexatic phase and other aspects of melting for dipolar systems, the hexatic phase can be seen directly
aligned2ddipolarsystemshavebeenobservedwithgreat bythecharacteristicpresenceoflatticedefectsconsisting
precisionintheclassicalregimeinexperimentsusingcol- of tightly bound disclination pairs, i.e., particles with 5
loidal particles with a magnetic moment, confined to an and 7 neighbors respectively.3
9
V. CONCLUSIONS and E =D2 for the dipoles. We can write D(k) as
0
∂2 (cid:88) 1
In this paper, we analysed the stability of the crys- D (k)=E lim (1−e−ik·R).
tal and hexatic phases of 2d systems consisting of ei- ij 0u→0∂ui∂uj |R+u|n
R(cid:54)=0
ther dipoles or charges. The classical elastic coefficients (A3)
were calculated from the phonon spectra of the trian- Usingr−n =(n+1)π−1/2(cid:82)∞dyyn−1e−r2y2 withn=1,3
0
gular crystal, and we then demonstrated how quantum and splitting the integral into a short range and a long
effects decrease these coefficients thereby softening the range part, i.e. (cid:82)∞dy... = (cid:82)y0dy...+(cid:82)∞dy..., the
crystal. Using Lindemann criteria suitably adapted to 0 0 y0
sum in Eq. (A3) is split into a short range and a long
deal with the large fluctuations in 2d systems, we calcu- range part, D(k)=D<(k)+D>(k). Upon defining the
latedapproximatephasediagramsfortheexistenceofthe
function
hexatic and crystal phases, predicting that the hexatic
phase is stable to very low temperatures. The relevant ϕ (x)= √2 (cid:90) ∞dttne−tx2 (A4)
Lindemann numbers were extracted from experiments in n π
1
the classical regime, and from Monte-Carlo calculations
we get
for T =0. The Lindemann numbers depend strongly on
temperature,buttheyturnouttobeessentiallythesame n+1 (cid:88)
forthechargedandthedipolarsystemforthesamevalue D>ij(k)=E0 2 y0n+2 (1−e−ik·R)
of n Λ2. This suggests that the two-step melting of the R(cid:54)=0
0 T
crystalphasewith anintermediatehexatic phaseisage- ×(cid:2)2y2R R ϕ (y R)−δ ϕ (y R)(cid:3). (A5)
0 i j n/2+1 0 ij n/2 0
ometric phenomenon, insensitive to the detailed form of
the particle interaction. Finally, we discussed the excit- The short range part of the sum is evaluated by Fourier
ing prospect of finally being able to probe the existence transforming. Writing (cid:80) (1 − e−ik·R)exp(−|R +
R(cid:54)=0
hexatic phase in the quantum regime using ultra-cold u|2y2)=F (u,y)−exp(k·u)F (u,y) and Fourier trans-
1 2
dipolar gases. forming the functions F (u,y) which have the same peri-
i
odicity as the lattice yields after some algebra
n+1 (cid:88)
D<(k)=E yn−2 [(K+k) (K+k)
VI. ACKNOWLEDGEMENTS ij 0 4πv 0 i j
K
(cid:3)
×ϕ (|K+k|/2y )−K K ϕ (|K|/2y ) (A6)
It is a pleasure to acknowledge the Kavli Institute −n/2 0 i j −n/2 0
for Theoretical Physics, UCSB, where this work began. √
where v = 3a2/2 is the area of the primitive cell of the
DRN would like to acknowledge the hospitality of the
lattice, and K are reciprocal lattice vectors. With Eqs.
Niels Bohr Institute, Copenhagen, and the support of
(A5)-(A6)wehavesplittheexpressionforD(k)intotwo
the National Science Foundation, through grant DMR
fastconvergingsums. Theseexpressionsagreewithwhat
1306367andthroughtheHarvardMaterialResearchSci-
is found in Refs. 49 and 50. We pick y = 1/a for the
ence and Engineering Center via grant DMR-0820484. 0
numerical calculations.
GMBwouldliketoacknowledgethesupportoftheCarls-
berg Foundation via grant 2011010264 and the Villum
Foundation via grant VKR023163.
Appendix B: Canonical quantisation of the phonons
For clarity, we briefly discuss how quantum effects on
Appendix A: Ewald summation
the Lam´e coefficients are included via canonical quanti-
sation. In this approach, we introduce the bosonic anni-
The phonon modes are as usual found by solving the hilation operatorsˆb for the phonons via
kσ
matrix equation25
L
uˆ (k)= (ˆb +ˆb† ) (B1)
mω2(cid:15)(k)=D(k)(cid:15)(k) (A1) σ (cid:112)2ρω (k) kσ −kσ
σ
where D(k) = (cid:80)RD(R)exp(−k·R) is the dynamical where σ = l,t. Using (cid:104)ˆb†kσˆbkσ(cid:105) = [eβ(cid:126)ωσ(k) −1]−1, we
matrix with R the lattice vectors. We have obtain
Dij(R)=E0×(cid:40)(cid:80)nRRδni(cid:54)=+j20(cid:104)−(n(n++2)2n)nRRRnRiR+niRj4+j4−nRδni+j2(cid:105) RR=(cid:54)=00 (cid:104)uˆσ(k)uˆσ(−k)L(cid:105)2= ρωLσ2(k)(eβωσ(1k)−1 + 21)
(A2) = 2ρω (k)coth[βωσ(k)/2)] (B2)
for repulsive power law potentials V(r) = E /rn with σ
0
n=1 and E0 =Q2 for the charged particles, and n=3 which is identical to Eq. (13).
10
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