Effect of Disorder on the Superfluid Transition in Two-Dimensional Systems

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by  Karén G. Balabanyan

Effect of Disorder on the Superfluid Transition in Two-Dimensional Systems
Kar´en G. Balabanyan
Department of Physics, University of Massachusetts, Amherst, Massachusetts 01003, USA
(Dated: February 5, 2008)
Inrecentexperimentsonthin4HefilmsabsorbedtoroughsurfacesLuhmanandHallock(Ref.1)
attempted to observe KT features of the superfluid–normal transition of this strongly disordered
7
2D bosonic system. It came as a surprise that while peak of dissipation was measured for a wide
0
rangeofsurfaceroughnesstherewerenoindicationsofthetheoretically expecteduniversaljumpof
0
the areal superfluid density for the strongly disordered samples. We test the hypothesis that this
2
unusualbehaviorisamanifestationoffinite-sizeeffectsbynumericalstudyofthecorresponding2D

n bosonic model with strong diagonal disorder. We demonstrate that the discontinuous features of
a the underlying KT transition are severely smoothed out for finite system sizes (or finite frequency
J measurements). Weresolve theuniversaldiscontinuityof theareal superfluiddensitybyfittingour

1 datatotheKTrenormalizationgroupequationsforfinitesystems. Inanalogytooursimulations,we
1 suggestthatinexperimentsonstronglydisordered2DbosonicsystemstheveryexistenceoftheKT
scenario canand shouldberevealed onlyfrom aproperfinite-sizescaling of thedata(for 4Hefilms

] finite-size scaling can be effectively controlled by the scaling of finite frequency of measurements).
r
e We also show relevance of our conclusions for a wider class of systems, such as superconducting
h granular films, Josephson junction arrays, and ultracold atomic gases, where similar difficulties
t appear in experiments designed to verify KT transition (especially in disordered cases).
o
.
t PACSnumbers: 68.15.+e,67.70.+n,64.70.Ja,68.35.Ct,67.40.Pm
a
m
- I. INTRODUCTION ments with the superfluid helium films the sample size
d
can simply set finite length scales, e.g., it can be the
n
o For decades 4He films played a prominent role in size of a confining channel,7 or the radius of spherical
c the research of superfluidity of two-dimensional (2D) grains R in the case of helium films absorbed in packed
[ bosonic systems. The Kosterlitz-Thouless (KT) theory2 powders,8 or the pore radius R in the case of helium
3 describes the finite temperature superfluid phase transi- films absorbed on porous glass9 (in the latter two cases
v tion for the 4He films absorbed to the flat substrates.3 crossover from the effective 2D to 3D behavior should
5 One of its main predictions is the universal jump from be kept in mind10). Finite length scales can also be
0 zero of the areal superfluid density σ at the transition related to finite frequency of measurements, when one
4 temperature T given by the relationS4F is bounded by the frequency-dependent vortex diffusion
8 cr
length r = 14D/ω, where D is the vortex diffusivity
0 ω
2 ~2 and ω is thepexperimental frequency.11,12 Finite-size ef-
6 = σ , (1)
0 π m2k T · SFcr fects in experiments smear the universal jump (1), and
B cr
/ disorderisexpectedtoincreasethisfinite-sizebroadening
t
a where m is the mass of a 4He atom. Theoretically, ac- oftheKTtransitionevenmore. Wethinkthisscenariois
m cording to Harris criterion, disorder is irrelevant at the applicable to the finite frequency measurements on 4He
- KT transition, and should not alter the discontinuous films absorbed to the rough surfaces in Ref. 1. The pur-
d KT scenario.5 Recently, Luhman and Hallock (Ref. 1) pose of this study is to show that in these experiments
n performedextensivequartz crystalmicrobalance(QCM) disorder of the substrate can produce such dramatic ad-
o
experiments of the superfluid transition of the 4He films ditionalbroadeningofthe superfluidtransitionontopof
c
: absorbed to rough surfaces. Surprisingly, their obser- finite-sizeeffectssothatitcouldbeindeeddifficulttore-
v vations were not directly agreeingwith the expected KT solvethegrowthofthesuperfluidfractiononthevarying
i
X scenarioofthe2Dsuperfluidtransition. Theyfoundthat normal fluid background.
r it was increasingly difficult to identify signatures of the In Sec. II we start with the description of 4He films
a universaljump(1)asthesurfaceroughnesswasincreased absorbed to rough surfaces studied in Ref. 1 by a 2D
(see also Ref. 6). For strong enough disorder the pres- bosonic model with diagonaldisorder. We want to be as
enceofthe KTtransitionwasseenonlyindirectlybythe close in our simulations to the experimental situation of
observationof a peak in the dissipation. Could their ob- Ref. 1 as possible in order to guarantee that all physics
servations be an indication of some strong-randomness that might be important for explanation of surprising
scenarioof the phase transition, or is there a way to rec- outcome of Ref. 1 will be preserved. We represent the
oncile those experiments with the standard KT theory? microscopically disordered rough surface by a random
Note that for ideal (disorder free) 2D bosonic systems potential (i.e., diagonal disorder). We keep temperature
discontinuityofthesuperfluiddensityatthecriticalpoint fixed,andtunethesuperfluidtransitionbyadjustingthe
is a characteristic of the phase transition in the thermo- chemicalpotential(theaverageheliumfilmthicknessisa
dynamical limit of infinite system sizes. In the experi- monotonicincreasingfunctionofthechemicalpotential).
2
In the ideal case, we observe the anticipated finite-size
broadeningoftheuniversaljumpofthesuperfluiddensity
(1) near the critical point of the KT transition. Inclu-
e
sionofthestrongdisorderleadstosuchsevereadditional r
u
broadeningthat,comparedtotheidealcase,itmayseem at
r Normal Superfluid
impossible to make any definite conclusions about the e
p
underlying transition type. Our analysis utilizes Monte m
e
Carlo simulations of different system sizes, or, equiva- T
n
lently, ability to set r by carrying out measurements at QM
ω
different finite frequencies. We resolve the asymptotic
universal jump of the superfluid density (1) by showing
Boson density
that data do obey the KT renormalization group (RG)
equation. Finally, once we have established the KT sce-
FIG. 1: Generic phase diagram in the density-temperature
nariointhe stronglydisorderedcase,inSec.III wecarry plane for 2D bosonic system with strong diagonal disorder
out a qualitative comparisonof our simulations with the (nQMisthethresholdbosondensity). Westudythesuperfluid
aforementioned Ref. 1 QCM experiments with 4He films transition along thedotted line path.
absorbed to rough surfaces. In the context of the QCM
technique, we also discuss how the dissipation peak is
affected by disorder.
model:
WithrespecttotheKTtransition,weexpectthecom-
bined effect of diagonal disorder and finite-size broaden-
U
itnemg st,oabnedqutoalibteatiinvedleypesinmdielnatr foofrtdhieffepraerntticpuhlayrsictyaplesyos-f H =−thXi,ji(cid:16)aˆ†iaˆj +aˆiaˆ†j(cid:17)+ 2 Xi nˆ2i −Xi (µ+vi)nˆi,
disorderrealizationandchoiceofacontrolparameter. In (2)
Sec. IV we discuss how our results for the 4He films can where aˆ† and aˆ are the creation and annihilation oper-
i i
be related to superconducting granular films, Josephson ators, nˆ = aˆ†aˆ is the number operator on site i, and
i i i
junction arrays,and ultracold atomic gases.
i,j denotes summationoverpairsofnearestneighbors,
h i
each pair counted once. The first term describes nearest
neighbor hopping, and the second term corresponds to
soft-core on-site repulsion. The chemical potential µ de-
II. SIMULATIONS
finestheaveragenumberofbosonspersiten(µ),whereas
v represents random on-site potential. The value of v
i i
InFig.1weshowaphasediagraminthetemperature- is distributed uniformly between ∆ and ∆.
density plane for a 2D bosonic model (2) with strong −
We do not carry out quantum Monte Carlo simula-
diagonal disorder. The transition from the normal to
tions of the Hubbard model (2) in continuous time.15.
superfluid phase happens according to the KT scenario
Instead, we map 2D quantum bosonic model (2) on the
atthe phaseboundary. Atsomefinite temperatureT we
equivalent (2+1)D classical J-current model.16 For this
drivethesystemthroughthetransitionbyincreasingthe
classical (2+1)D counterpart of quantum 2D model (2)
boson density. At the critical point, the superfluid areal
we perform classicalMonte Carlo simulations with high-
density experiences the universal jump σ according
SFcr performancewormalgorithm.17,18 Actionoftheso-called
to (1).
J-currentmodeliswrittenintermsofintegerlink-current
For a system of a given finite size, the superfluid den- variablesJ=(Jx,Jy,Jτ)definedonaN N N space
τ
sity is expected to be a monotonic function of the boson and imaginary time lattice (N δτ = ~/k×T, w×here δτ is
τ B
density,andwestudyitssharpnesswithrespecttodisor-
an imaginary time interval):
der. Thestrengthofdisorderalsodeterminestheposition
of the phase boundary.
Note that the zero temperature quantum phase tran- S = t˜ Jnx 2+ Jny 2 +U˜ Jnτ 2 (µ˜+v˜r)Jnτ ,
sition is in a different universality class.13 In the case of 2Xn h(cid:0) (cid:1) (cid:0) (cid:1) i 2 Xn h(cid:0) (cid:1) − i
4He films on disordered substrates, the threshold boson (3)
density nQM . 1 per lattice site corresponds to up to where n = (r,τ) are discrete space and imaginary time
several atomic layers.14 coordinates. J has to be conserved and therefore sat-
isfies the local zero-divergence constraint, Jn = 0.
∇ ·
SpatialcurrentsJx andJy representhopping,andimag-
inary time current component Jτ represents occupation
A. Model
number. We choose to consider only positive occupation
numbers,Jτ 0. Thisisanaturalchoiceforatomicsys-
≥
We start with the description of bosonic systems with tems, and it cannot affect the universalproperties of the
diagonal disorder by a general lattice bosonic Hubbard transition.18Correspondenceofparametersofthemodels
3
(2) and (3) are given by
t˜=−2ln(n0t· δ~τ), U˜ =U · δ~τ, µ˜= Uµ, v˜r = vUr, n
(4)
2.5
wheren issomeoffsetbosondensityofmodel(2).19Here ~
0
the value of v˜r is uniformly distributed between ∆˜ and
∆˜, where ∆˜ =∆/U. − 2.0
The total boson density σ (proportional to the boson
number per site n) and the superfluid density σSF are 1.5
determined from Monte Carlo estimators of the winding
~
numbers Wi =1/Ni nJni, and are given by16,20 1.0
P
m˜ W ~
τ
σ = n , n= h i, (5a) 0.5
a2 · N2
m˜2k T W2 + W2
σ = B K , K = x y , (5b) 0.0
SF ~2 · (cid:10) (cid:11) 2 (cid:10) (cid:11) -0.5 0.0 0.5 1.0 1.5
where m˜ = ~2/2ta2 is an effective boson mass on a lat-
tice with spacing a. In 2D,the statistics ofspatialwind-
FIG.2: Averagenumberofbosonsperlatticesitenasafunc-
ing numbers are essentially discrete,thus anappropriate
tion of chemical potential µ˜. Three sets of curves represent
way to find K is through fitting collected winding num-
thecases of nodisorder ∆˜ =0,and disorderstrengths∆˜ =3
ber histograms with a discrete gaussian P(Wx,Wy) ∝ and∆˜ =6. Statisticalerrorsaresmallerthanthesymbolsize.
exp(−(Wx2+Wy2)/2K). Data in each set for different system sizes 24×24, 48×48,
Fordefiniteness,wechoosenumericalvaluesofparam- 96×96 and 192×192 collapse on one and the same curve,
eters to be close to those characteristic for experiments i.e., n is independentof system size.
with 4He films at temperatures of about 1K, i.e., hop-
ping, interaction and temperature are considered to be
of the same order t U k T, and n n 1 to Foragivendisorderstrengththeabsenceofsaturation
B 0 QM
∼ ∼ ∼ ∼
correlatewith atransitionofseverallayersofa 4He film. of the reduced superfluid density K(µ˜,N) for different
Corresponding effective parameters of the (2+1)D clas- system sizes below the critical value K and collapse of
cr
sical model (3) are t˜, and U˜. Our choice of finite Lτ is a the dataaboveKcr,see Fig.3,is aqualitativeindication
compromise between the efficiency of discrete imaginary of the KT discontinuity at infinite system sizes. To per-
time codes and staying very close to the original model. formquantitativeanalysisweusetheintegralformofthe
WehavevariedN ,whilet˜andU˜ canbe estimatedfrom KT RG equations (8a,8b) (see Ref. 21)
τ
(4), so that the boson number density in the vicinity of
the superfluid transition in the ideal system would stay K(µ˜,N2)/Kcr dt N
2
about 1 per lattice site. We found t˜= 4, U˜ = 0.2 and Z t2(lnt ξ(µ˜))+t =−4ln(cid:18)N (cid:19). (6)
N =6 to be appropriate values. K(µ˜,N1)/Kcr − 1
τ
Here ξ(µ˜) is the size-independent microscopic param-
eter characterizing the vortex fugacity, and, in con-
B. Results
trast to the thermodynamical reduced superfluid den-
sity K(µ˜) = K(µ˜,N ), ξ(µ˜) is an analytic function
→∞
We model systems of finite spatial size N N to be of µ˜ (i.e., ξ(µ˜) may be expanded into the Taylor series
×
24 24,48 48,96 96and192 192. Threesituationsare ξ(µ˜) 1+ξ′(0) (µ˜ µ˜ )+... in the vicinity of the
con×sidered×: no dis×order ∆˜ =0×,and disorderof strengths critica≈l point µ )·. Th−osecrtwo properties of ξ we use to
cr
∆˜ = 3 and ∆˜ = 6. In the later cases, we average data check the consistency of our data with the KT scenario
over about 500 realizations of disorder. Temperature is of the transition. For different pairs of K(µ˜,N) we solve
kept fixed throughoutall simulations, and we study how (6) for ξ(µ˜) and observe that its values collapse on the
the average boson number per site n and the superfluid line near µ˜ , see Fig. 4. Equation (6) indirectly repre-
cr
density σSF behaves as we pass over the transition point sentsKTRGflowofK(µ˜,N)forincreasinglengthscales
increasing chemical potential µ˜, see Fig. 2 and 3. Note N <N <N <N <.... In the limit N it gives
1 2 3 4
→∞
that it is more informative to study the dimensionless equation
reduced superfluid density K rather than σ itself. Ac-
SF
cordingto(5b),thecoefficientofproportionalityrelating Kcr K(µ˜)
+ln =ξ(µ˜) (7)
K andσSF is~2/m˜2kBT. Thus,theuniversaldiscontinu- K(µ˜) (cid:18) Kcr (cid:19)
ity ofthe superfluid density (1) at the fixed temperature
KT transition is equivalent to the jump of the reduced for the thermodynamical value of K(µ˜). Finally, using
superfluid density K from 0 to K =2/π. determined function ξ(µ˜), we solve (7) for K(µ˜), see
cr
4

K ~ ~ ~
0.7 1.010 ~ ~ ~
2/
0.6 1.005
0.5
1.000
0.4 24x24
0.995
0.3
48x48
0.2 96x96 0.990
(a)
0.1
192x192 4 4
10 x10 0.985
0.0
-0.5 0.0 0.5 1.0 1.5
-0.5 0.0 0.5 1.0 1.5
FIG. 4: RG parameter ξ as a function of µ˜. Three sets rep-
resent thecases of no disorder ∆˜ =0, and disorder strengths
∆˜ =3and∆˜ =6. Ineachset,valuesofξ arededucedbyRG
K
~ ~ ~ analysis (6) of different pairs of K corresponding to pairs of
0.75 systems192×192and48×48,96×96and24×24,192×192
and 24×24 at Fig. 3b. The solid lines are linear fits for all
data points at a given disorder strength.
0.70

0.65 continuity of the superfluid density σSF K. For exam-
∝
2/ ple, consider system of size 24 24. Compared to the
ideal case (disorder strength ∆˜ ×= 0), there is no clearly
0.60
visiblecharacteristicbendinginthedatatrendforthere-
(b) duced superfluid density K at a strong disorder (∆˜ =6)
0.55 near the anticipated discontinuous KT transition. And,
yet, the self-consistent RG analysis of the data for expo-
-0.5 0.0 0.5 1.0 1.5
nentially different system sizes will and is revealing the

KT transition.
FIG. 3: Reduced superfluid density K =~2σSF/m2kBT as a
function of chemical potential µ˜. Three sets of curves repre-
sent the case of no disorder ∆˜ = 0, and disorder strengths III. COMPARISON WITH EXPERIMENTS
∆˜ =3 and ∆˜ =6. In each set, four curves from the bottom
tothetopcorrespondtofinitesystemsizes192×192, 96×96, Our simulations are directly applicable to explain the
48×48 and 24×24. Statistical errors are smaller than the role of disorder in the superfluid transition in 4He film
symbolsize. (a)TheuppergraphshowsK inthewiderange absorbed to microscopically rough surfaces, see Ref. 1.
of µ˜, and (b) the lower graph depicts proximity to the KT
In these experiments, sample surfaces are prepared by
transition, where the solid curves are the KT extrapolation
coating QCMs with CaF . The height distribution of a
to the infinite system size. Horizontal lines are equal to the 2
CaF filmis closeto aGaussianfunction, andits disper-
value of the universal jump Kcr =2/π at the KT transition. 2
sionincreaseswiththenominaldepositionthickness(the
Thedottedcurveontheuppergraph(a)isanexampleofthe
extrapolation of the datato lager system size 104×104. systemofzerodisorderisanuncoatedplainQCMcrystal
with typical gold electrodes). The crystals are mounted
in a cell. The QCM crystals are operated in the third
harmonic f 15MHz of their fundamental resonance
0
Fig. 3b (solid curves). Note that, due to the logarith- frequency. A≈t the constant temperature 1.672 K, 4He
mic form of the finite-size corrections to K near the KT
is incrementally added to the cell, correspondingly, the
transition,11 K(N) Kcr 1/lnN for µ˜ &µ˜cr, data for thickness of the 4He film absorbedto the sample surface
− ∝
systemsofexponentiallydifferentsizesisrequiredforthe
grows. Theresonancefrequencyofoscillationsf =ω/2π
KT RG analysis. oftheQCMcrystalwithanabsorbed4Hefilmisdirectly
The critical observation, see Fig. 3a, is that disorder related to the viscously clamped normal fraction of the
dramaticallyincreasesfinite-sizesmearingoftheKTdis- 4He film. Whenthe filmundergoesa transitionfromthe
5
normal to superfluid state, one expects to see a sharp This broadens the sharp KT transition and leads to a
step in the resonance frequency f because of the decou- finite-sizeroundingoftheKTjumpinσ ,seeFig.3. In-
SF
pling of the zero-viscosity superfluid fraction of the 4He terpretation of experiments at finite frequencies requires
film from the substrate.22 The absorbed film shifts the the theory of Ref. 11 that incorporates the dynamic re-
resonance frequency from the value f by ∆f which sponseofthevortexplasmatoanexternallyappliedoscil-
0
−
is proportional to the difference of total and superfluid lating field. It was shown that vortex pairs of size larger
areal densities of 4He, ∆f σ σ (see insert of than the vortex diffusion length r = 14D/ω cannot
SF ω
− ∝ −
Fig. 5). At a finite (angular) frequency ω, one probes equilibratetotheoscillatingfieldofthepfrequencyω,and
lengthscalesoforderofadiffusionlengthr = 14D/ω, do not contribute to renormalization, so the iterations
ω
where ω ω0 = 2πf0, and D is diffusivity copnstant.11 are truncated at rω. Thus, results of our simulations for
∼
This sets finite-size scaling for σSF and smooths out the σSF for finite-size systems are in one-to-one correspon-
zero frequency sharp step in ∆f. It was found in Ref. 1 dencetofinite-frequencyexperiments,providedthatω is
thatwithhigherdisordernoindicationofthis stepisob- adjustedsothatrω isequaltothe simulatedsystemsize.
servable. Another signature of the superfluid transition
in4Hefilms isapeakinthe dissipation,anditindirectly
manifested the presence of the KT transition in these Let’s specify particular values of length scales for 4He
experiments. films absorbed to rough CaF surfaces studied in Ref. 1.
2
Averageheliumfilmthicknessinthevicinityofthesuper-
fluid transition in those experiments was reported to be
A. Important length scales about4layers 15˚A on the flat substrate (on the disor-
∼
deredCaF substratetheamountofheliumabsorbedwas
2
The superfluid transition in 2D bosonic systems is larger). Temperature was in the range 1 2K, thus the
−
driven by unbinding of thermally activated vortex- vortex core size is expected to be on the order of atomic
antivortex pairs. Kosterlitz and Thouless used the dimensions, a 10˚A. Because a is comparable to the
0 0
∼
renormalization-group technique to consider the atten- heliumfilmthickness no3Drelatedvortexeffects areas-
uating effect of smaller pairs on the interaction between sumedto exist. The diffusivity constantD wasnotmea-
the respective members of larger pairs. They derived suredandmightbeaffectedbydisorder,butfromdimen-
recursion relations for a scale-dependent superfluid den- sional analysis D ~/m 2 10−4cm2/sec.11 (A cau-
≈ ∼ ×
sity σSF(l) (or, corresponding reduced superfluid den- tionaryremark: fordifferentsubstratesthenumericalco-
sity K(l)=σSF(l)~2/m2kBT)andvortex-pairexcitation efficient before ~/m in the formula for D was reported12
probability y2(l) tobe inthe wide rangefrom0.1to20.) Dynamicalmea-
surements of σ were carried out in the course of op-
dK−1(l)/dl = 4π3y2(l)+O[y4(l)], (8a) SF
eration of the QCM crystals at ω 2π 15MHz, and
dy2(l)/dl = 2[2 πK(l)]y2(l)+O[y4(l)], (8b) this gives r 500˚A. QCMs were≈of cen×timeter dimen-
− ω
∼
sions,thustheirwidthisanirrelevantlengthscale. CaF
where l = ln(r/a ), and contribution of vortex- 2
0
surfaces coating the QCMs consisted of peak-type struc-
antivortex pairs of sizes less than r has been already
tures with an average separation R . 100˚A. Because
taken into account. The vortex core size a is of the or- s
0
the roughness scale is less than the diffusivity length,
derofthehealinglengthofthesuperfluid,i.e.,thelength
R r , disorder is considered to be microscopic. In
over which the superfluid density can change from zero s ω
≪
Ref. 1, by varying deposition time, surfaces with aver-
to its mean value. Here, we explicitly note that KT RG
equations are only valid for y2 1 (for y2 1 the vor- age height of the CaF2 structures from 0 to 50˚A were
∼
≪ ≫ prepared and used (0 stands for uncoated (plain) QCM
texdensity issohighthatitisambiguousto groupthem
crystals).
intosomehierarchyofvortex-antivortexpairsofdifferent
sizes and then to carry out renormalization).
For an idealized experiment carried out at zero fre-
quencyoninfinitesizesystem,therecursionrelationsare With respect to the lattice model (2) used in our sim-
iterated from a to infinite scale. In the normal phase ulations in Sec. II, the vortex core size a stands for the
0 0
y2 diverges at infinite scales, and σ is renormolized to latticespacinga,andtheratior /a 50defineslattice
SF ω 0
∼
zero. In contrast, the superfluid phase is characterized spatial size to be N N 50 50. We consider height
by vanishing y2 and some finite σ . Phase transition variations of the Ca×F p∼rofile×to act as a random po-
SF 2
from the superfluid to the normal phase corresponds to tential. We chose on-site potential disorder to represent
unbinding of vortex-antivortex pairs as they get excited the closely spaced, raised structures of CaF . Particular
2
to infinitely large scales of their mutual separation, and, choiceofthedisorderrepresentationcannotaffectuniver-
in addition, the superfluid transition is accompanied by sal properties of the second order KT transition. How-
the universal jump (1) of σ . ever,tobeinlinewithexperimentsofRef.1wemustsat-
SFcr
In finite systems the maximum vortex separation is isfy criterion of the microscopic disorder r R . And
ω s
≫
bounded by the system size, and the KT iterative renor- indeedthisistrueforourlatticemodel,becauseitcorre-
malization process is cut off at a corresponding scale. sponds to R =a a and system size r /a N 1.
s 0 ω 0
∼ ∼ ≫
6
B. Broadening of the KT transitions

The experiments of Ref. 1 with 4He films absorbed to
n- K
rough CaF surfaces can be compared to our model in
2
the following way (see (5a,5b)) 2.0 ~
∆f σ σ n γK, γ =m˜a2k T/~2. (9) 192x192
SF B 96x96
− ∝ − ∝ − 1.5 48x48
Forourchoiceofparametersγ 1(wechooseγ =0.75). 24x24
∼
Our simulations, specifically Fig. 5, qualitatively resem-
f
1.0
ble the experimental situation. Actually, Fig. 5 contains
~
excessive information relative to those experiments of
Ref.1. There,QCMmeasurementswerecarriedoutata 0.5 ~
fixed frequency, i.e., r was fixed, while CaF substrates
ω 2
cr
of different roughness were used. In other words, at a
given disorder strength (i.e., on a given coated QCM) 0.0
-0.5 0.0 0.5 1.0 1.5
only a single curve for mass decoupling (9) as a function
ofthechemicalpotentialwasobserved,forexample,let’s
say that these were upper curves in each set on Fig. 5
FIG. 5: Frequency shift theoretically defined −∆f ∝ n−
for which r corresponded to 192 192 lattice sites in
ω × γK as a function of chemical potential µ˜. The three sets of
oursimulations. Infullagreementwithexperimentalob- curvesrepresentthecasesof nodisorder ∆˜ =0,and disorder
servations,we see that, as disorderstrengthis increased, strengths ∆˜ = 3 and ∆˜ = 6. In each set, the four curves
finite-size behavior completely masks the step in ∆f re- fromthebottomtothetopcorrespondstofinitesystemsizes
lated to the KT universaljump of the superfluid density 24×24, 48×48,96×96and192×192. Statisticalerrors are
σ . Thisisexactlytheobservationthatdidnotgivefirm smaller than the symbol size. The insert shows an idealized
SF
evidence to unambiguously claim the KT mechanism of sketch of −∆f(µ) in the vicinity of the superfluid transition
the superfluid transition in Ref. 1. If, at each disorder atµcr;thestepcorrespondstothejumpofsuperfluiddensity
strength, QCM measurements were carried out at a se- σSF in thesystem of infinite size, see (1).
ries of frequencies, then similar to our results, Fig. 5,
one could have had enough data to perform the KT RG
analysis. For further discussion see Sec. V. to deduce from (6) the size-independent microscopic pa-
An essential condition of the KT transition is an ob- rameter ξ. Too deep in the normal phase y2 & 1, and
servation of the dissipation peak during dynamic finite the KT recursion relations (8a,8b) break down. Thus,
frequency measurements. It signifies change in the dy- we had to extrapolate our data to higher system sizes,
namical response from the one determined by the free so that y2 . 1 and the dissipation peak is positioned in
vortices moving diffusively at the normalside to the one the samerangeofµ˜. Foreachdisorderstrength,weused
produced by the vortex pairs of size greater r failing to linear fits for ξ(µ˜) from Fig. 4 to recalculate the finite-
ω
equilibrate to external oscillating field at the superfluid size smeared reduced superfluid density K(µ˜,N = 192)
side. In Ref. 1 even for the strongly disordered samples ofthesystemofsize192 192intoK(µ˜,N =104).24 The
×
the peak in dissipation was present and was the basis to value of y2(µ˜,N) for known K(µ˜,N) and ξ(µ˜) is given
claim the KT scenario of the superfluid transition. Our by
simulations are statistical mechanical and do not simu-
1 K K
latepersethedissipationpeakofdynamicalorigin. How- y2 = cr +ln ξ . (11)
ever, using the results of the linear dynamical theory for 2π2 (cid:20) K (cid:18)Kcr(cid:19)− (cid:21)
real and imaginary parts of effective dielectric constant
ofvortexplasma,11 wecanpredicttheshapeofthedissi- Ourobservationsfor∆(1/Q),see Fig.6,arethatforone
and the same r with an increase of disorder strength
pation peak from our numerical data. The dissipation is ω
thedissipationpeakwillbebroadenedandwillbeshifted
measuredinterms ofthe changeinqualityfactorQ that
is given by12,23 deeper in the normal phase away from the critical point
of the superfluid transition. A high level of noise in the
1 π2 (Ky)2 data for 4He films coating rough CaF2 surfaces, Ref. 1,
∆ . (10)
(cid:18)Q(cid:19)∝ 2 · 1+(π4Ky2)2 did not allow a validation of this statement (note that
dissipation peaks were clearly seen for all values of dis-
The dissipation peak is positioned close to the point of order). However, in those experiments there is room left
theinflectionofthegrowingσ asonegoesfromthenor- to raise accuracy at least by the order of magnitude.25
SF
mal to the superfluid phase while increasing the control We have also studied finite-size scaling of the width of
parameter,chemicalpotentialµ. ButKT RGworksin a the dissipation peak for a given disorder strength. Its
narrowwindowofthetransition,e.g.,wehadtouseonly broadening one-to-one resembles the finite-size broaden-
aportionofthesimulatedsuperfluiddensitydataFig.3b ing of the superfluid density jump (correspondingly, po-
7
temperatures (in Ref. 1 T =1.672K) or,equivalently, at
high average helium film thicknesses, see the phase di-

agram Fig. 1. For T & 1K we expect that the surface
tensionpredominates,andmicroscopicpuddlesofhelium
2/
0.6 arecreatedinthevalleysofthedisorderedsurfaceprofile
K
whiletheheliumfilmthinsclosetotheridgesofthepeak-
0.5 (1/Q) typestructuresoftheCaF2 substrate. Astudysimilarto
Ref. 1 of the superfluid transition of 4He films absorbed
0.4
tothesameroughCaF substrateswascarriedoutatlow
2
temperatures T . 1K. It turned out that in this case
0.3
broadeningoftheKTtransitionwasalmostindependent
~ ~ ~
of the strength of disorder.28 This allows us to speculate
0.2
that a different regime from the one considered in this
0.1 paper was achieved for T . 1K, when van der Waals’
forceswinoverthe surfacetensionandthe heliumfilmis
0.0 uniformly flattened over the rough surface, and, hence,
-0.4 -0.3 -0.2 -0.1 0.0
the surface potential is uniform.

cr
IV. RELATED PHYSICAL SYSTEMS
FIG. 6: Dotted lines are the reduced superfluid density
K ∝ σSF and solid lines are the dissipation peak ∆(1/Q)
as a function of chemical potential µ˜ for the systems of size Ourresearchiscenteredaroundthe KT-type2Dfinite
104×104. Thethreepairsofcurvesrepresentthecasesofno temperature superfluid transitions in 4He films. Phase
disorder ∆˜ = 0, and disorder strengths ∆˜ = 3 and ∆˜ = 6. transitions in thin-film superconductors,29 Josephson
Chemical potential is shifted by the critical value µ˜cr of the junction arrays,30 weakly interacting2D Bosegases,21,31
KT transitions in thecorresponding systems. 2D gas of spin-polorized atomic hydrogen on liquid-
helium surface,32 and Bose-Enstein condensates loaded
on a 2D optical lattice33 belong to the same universal-
sition of the peak shifts to higher chemical potential as ity class. Our analysis of the role of diagonaldisorder in
rω is increased). 4He films can be qualitatively extended to some of these
cases.
Microscopic details are irrelevant for the study of uni-
C. Disorder description in detail versal properties of the second order KT phase tran-
sition characterized by the diverging correlation length
Our simulations explain broadening of the KT tran- scales.13,16 For example, universal jump (1) of the areal
sition in microscopically disordered 2D bosonic systems superfluid density is innate for both discrete lattice type
suchasthe4Hefilms absorbedtoroughCaF substrates and continuous geometry of experimental realizations of
2
in Ref. 1. In our case, R r , the roughness scale 2Dbosonicsystemswithdiagonaldisorderandcontrolled
s ω
R is much less compared wi≪th the dimension r of the value of the chemical potential. Instead of 4He film ab-
s ω
region effectively studied at finite frequency. Different sorbedto roughsurface with the chemicalpotential con-
case of macroscopic disorder, R r , can be achieved, trolledbythe pressureinthe samplecell,1 wecouldhave
s ω
≫
forexample,inexperimentswith4Hefilmsabsorbedona consideredaswellanatomicgasinanopticallatticewith
Mylarsubstratewhenthetorsionaloscillatortechniqueis a givennumber of atoms per site anda varying trapping
implemented.23,26Inthesecases,typicalω 2π 1500Hz potential, or a Josephson junction array (or a supercon-
≈ ×
leadstor 5µm. Mylarcontainsbubbles andflakesof ducting granular film) with an external voltage applied
ω
∼
sizes1 100µmandaverageseparationof10 150µm,i.e., to the ground plane and random gate voltages induced
R −100µm. Macroscopic variations of t−he substrate by trapped charge in the substrate.34 Critical behavior
s
∼
potential result in the inhomogeneous broadening of the in all mentioned cases falls within the framework of the
KTtypesuperfluidtransition. Similartooursuggestion, lattice bosonic Hubbard model (2). The same finite-size
frequency dependence of the inhomogeneous broadening analysisas inSec.IIB canbe usedtoresolvethe univer-
of the jump of the superfluid density and the dissipa- salKTjump(1). However,microscopicdetailsdetermine
tion peak was successfully studied by a two-torsional- howstrongwillbefinite-sizeeffects. Thus,similartoour
oscillators technique.27 study targeting 4He films absorbed to rough surfaces in
The fact that in our simulations we relate rough sur- Ref. 1, separate simulations are requiredfor the particu-
face of the CaF substrate of Ref. 1 with the spatially larexperimentalsituationtoreveal(ordisproof)primary
2
varying potential implies that helium film thickness is importanceoffinite-sizeeffectsasasourceofexperimen-
modulated by the microscopic surface roughness. This tal smearing of the discontinuous KT transition.
is true for the case of the superfluid transition at high Along with diagonal disorder, off-diagonal disorder
8
can take place in 2D systems that exhibit finite tem- dered substrates. We speculate that operation of QCM
perature KT transition. Off-diagonal disorder can be crystals at different harmonics of their fundamental res-
present in the form of site or bond dilution disorder in onance will be required. Note that QCM measurements
Josephson junction arrays35, or as nonmagnetic disor- of the superfluid transition of 4He films on a flat sub-
der in planar symmetry spin models36, or random gauge strates at two different harmonics of fundamental reso-
disorder37 (suchas disorderinthe phaseof the complex- nancefrequencywerecarriedout. Finite-sizebroadening
number tunnelling amplitudes for bosonic systems), etc. that increasedat higher frequencies was observed.42 An-
Our study of non-universal corrections to the finite-size other study of the ω dependance of the KT transition in
broadening of the KT transition produced by diagonal 2D 4He films used anultrasonic technique.43 For a given
disorder cannot be directly related to the case of off- sample the superfluid transition was relatively broad at
diagonal disorder. 2D XY models are used for descrip- highfrequencieswiththeuniversaljumpofthesuperfluid
tion of aforementioned disordered cases. But an XY densitysmearedtowardthenormalphase,butsharpened
model simply does not admit conventional diagonal dis- asthe frequencywasdecreased(correspondingly,the po-
order. Formally within the bosonic Hubbard hamilto- sition of the dissipation peak was shifting). Our simu-
nian(2)theXY limitisobtainedbywritingthelasttwo lations (Sec. IIIB) predict the same scenario for a given
terms as U/2(nˆi (µ vi)/U)2 and setting simultane- disorder strength as rω is increased.
− −
ouslyU ,andµ . Inthis limitaveragenumber TheexperimentsofRef.1wereouroriginalmotivation.
→∞ →∞
of particles per site is fixed, n = µ/U, and there are no Due to the noise in the measured data we sought only a
local density fluctuations at all, i.e., any finite diagonal qualitative description of those experiments. Thus, we
disorderv inµisignoredwithrespecttotheXY models. used a lattice Hubbard model with on-site potential dis-
i
On a passing note, finite-size effects and disorder will order to describe helium films and discrete space and
playanimportantroleintheobservationofthesupercon- imaginary time worm algorithm to carry out classical
ductingtransitionin2D(theKTscenarioisdebatednow Monte Carlo simulations (Sec. II). Universal properties
for low-T granular films,38 high-T films,39 and Joseph- of the superfluid transition were preserved, but one-to-
c c
son junction arrays.40) However, we do not continue our onematchingoftheexperimentsandthesimulationscan-
discussion of both neutral superfluid and superconduct- not be achieved in this kind of simulations setup. Exact
ing [chargedsuperfluid] systemsdue to differences in the ab initio numerical simulations of 4He film with realistic
correspondingmeasurementtechniques,andduetoaddi- substratepotentialsandgeometryispossiblewithexten-
tionalmagnetic fieldscreeningspecific for superconduct- sion of the used worm algorithm17 for continuous-space
ing systems.41 path Monte Carlo simulations.44 Another aspect is that,
withregardtotheexperimentsofRef.1on4Hefilms,we
have considered the case of a 2D bosonic system driven
V. DISCUSSION AND CONCLUSIONS to the point of the superfluid transition by the increase
of boson density (i.e., chemical potential was the con-
We have studied a 2D bosonic system with diagonal trol parameter). We expect qualitatively similar results
disorder in the vicinity of the finite temperature super- for broadening of the KT transition if, instead, at a con-
fluidtransition. Wefoundthatthedisorderdramatically stantbosondensitythesystemwillbebroughtacrossthe
increasesfinite-sizesmearingoftheuniversaljumpofthe superfluid–normal phase boundary by an increase of the
superfluid density (1). And, an analysis of its finite-size temperature (see Fig. 1).
scalingisrequiredtoresolvefaithfullytheunderlyingKT
scenario of the superfluid transition.
Our simulations consistently explained difficulties of Acknowledgments
identification of the KT transition by QCM frequency
shift in the experiments of Ref. 1 with 4He absorbed to AuthorisverygratefulR.B.HallockandD.R.Luhman
microscopically rough CaF surfaces. In those experi- for illuminating discussions and for explanation of de-
2
ments QCM measurements were carried out at a fixed tails of experiments carried out with 4He films absorbed
frequency ω. Thus, for a given disorder strength, the to rough surfaces. Thanks are also due to N. Prokofev
effective finite length scale set by the vortex diffusivity and B. Svistunov for a large number of valuable sugges-
length r 1/√ω was fixed. Without finite-size scal- tions and discussions. The research was supported by
ω
∝
ing it was not possible to extract the universal jump of the NationalScience FoundationunderGrantNo. PHY-
the superfluid density (1) in the case of strongly disor- 0426881,and, in part, under Grant No. PHY99-07949.
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