Superfluidity in 2D: Kosterlitz Thouless Transition

Martina Dombrowski ([email protected]) 04. Juni 2013

In 1966 Mermin and Wagner proved the absence of long-range order in two-dimensional systems with a continuous symmetry (e.g. XY-model) for finite T [1]. Though, Kosterlitz and Thouless showed the existence of a in 1972. A phase transition is produced by unbinding -antivortex pairs at a critical temper- ature TKT [2].

πJ 1 XY-Model Indentify critical temperatur TKT = 2kB Two dimensional square lattice with planar rotors of unit length. 2 Phase Transition

2.1 Mermin Wagner Theorem Looking at the mean magnetisation hSi Mermin and Wagner showed that there is no long-range order to be found in 2D sys- tems (e.g XY-model) for finite tempera- ture. hSi goes to zero for finite tempera- ture. Fluctuations at even low temperature will destroy the long-range order. Figure 1: Schematic XY model 2.2 Find Hamiltonian H and interaction term J: Looking at the correlation function hS(r)S(0)i, you will find quasi-long-range X X H = −J Si · Sj = −J cos(θi − θj) order: hi,ji hi,ji (1) – Low temperature:

For low temperature with ground state en-  r −T/2πJ hS(r)S(0)i ' (4) ergy E0 = 2JN: L J Z H = E + d2r(∇θ)2 (2) → quasi-long-range order. 0 2 – High temperature Free energy F of a single vortex with sys- − r tem size L and lattice spacing a: hS(r)S(0)i = e ξ (5)

F = E0 + (πJ − 2kBT ) ln(L/a) (3) → exponential decay.

Hauptseminar Physik der kalten Gase“, SS 2023, Univesit¨at Stuttgart ”

1 VOLUME 40, NUMBER 26 PHYSICAL REVIEW LETTERS 26 JvNE 1978

eters of the static theory, three new param- the quartz-microbalance experiments of Chester eters appear in this fit of the dynamical theory: and Yang. " These experiments, which are very a dimensionless parameter In(2D/&ua') related to similar in concept to the present , were per- the vortex diffusion constant D, the vortex-core formed in the MHz frequency range and showed, radius a, and the frequency ~ of the oscillator; as would now be expected on the basis of the dy- the coefficient of the free vortex dissipation; and namic theory, ' considerable broadening of the a background term, referred to above, which transition region. plays almost no role in the transition region it- These older experiments can now be analyzed self. In the ratio A/M, A is the area of the Mylar in terms of our present understanding to obtain substrate and M is the effective of the pen- estimates of the Kosterlitz-Thouless jump in the 2.3 Superfluid Densitydulum bob when the pendulum is treated as aUniversalsuperfluid formass per allunit area systemsat the two-dimen- you get linear oscillator. This ratio is obtained from Ra −sional phase transition, Although it is not possi- knowledge of the area of the substrate and a ρmea-s (T ble )to/TfollowKTa =third-sound 2/π. signal through the surement of the sensitivity of the oscillator peri- TK Superfluid density ρs measures the energy transition region, it can be followed to the point od to changes in the mass per unit area of the where the dissipation begins to rise rapidly. If change caused byadsorbed a twist.. Therefore it the third-sound signal disappears at this point, In the analysis of our data taken for different then as can be seen in Fig. 2 the value of the describes the effectcoverages of thermallyof adsorbed helium, activatedwe allow the 3value Kosterlitz-Thoulesssuperfluid mass is still up on the shoulder Phaseof the vortex pairs. of the Kosterlitz-Thouless jump in p, to be a free curve and a reasonably good estimate for the parameter to be determined by an optimization ofTransitionstatic value of the Kosterlitz-Thouless jump can the fit through a nonlinear least-squares routine. " be obtained. Recently, Rudnick" has rear-:xlyzed ( R − As an example the value for p, (T,1/)2obtained for his third-sound data us1ng his latest estimate of R ρs (TKT )[1the +fit const.shown in (Fig.TKT2 is −0.96Ttimes) the] exactIncreasingthe van der theWaals temperatureconstant, and has obtained activates vor- ρs = theoretical value, 8~k B(m/k)'T„given by Koster- values for p, (T, ) which are in good agreement 0 litz and Thouless. tices.with FortheTvalue TobservationsKT . are made, the adsorbed helium 3.1is Experimenttained by Rudnick" and additional values provided entirely locked to the substrate and contributes by Mochel and Hallock from their third-sound its entire moment of inertia to the pendulum bob. This serves to calibrate the torsion oscillator. The surface area is taken to be the geometric 14 area of the Mylar film. Tc=1 Tc=5 Tc=10 ρ R(T −) 12 Since 1968 there has beens c evidence that the super fluid density in two-dimensional helium 10 films might be nonzero at the superfluid onset. E 5 R s

ρ This was shown most clearly in the third-sound 8 experiments of Rudnick et al. "and also by per- 4 sistent-current measurements on helium films, " Q 6 3 which indicated that the superfluid critical veloc- I

Superfluid Density I— while den- 4 ity was becoming zero the superfluid M2 sity was still finite. However, neither third-

2 sound nor persistent-current measurements can

be used to pursue the question of the superfluid 0 I & I 0 .2 4 .6 .8 t,0 t,2 IA t.6 t, 2. 2,2 0 density behavior in the transition region, since e 0 0 2 4 6 8 10 c~ the third-soundTemperature T signals become heavily damped FIG. 3. Results of all of our data, in addition to and do not and persistent currents de- propagate, previous third-sound results for the discontinuous cay away. superfluid density jump p, {T, ) as a function of temper- The first experiments to show that the dynamicFigureature. 3:The Experimentsolid line is the Kosterlitz-Thouless with superfluid(beefs. he- Figure 2: Superfluidsuper fluid density densitygoes continuously over T. to zero were 3 and 4)liumstatic theory. films. 1729

References

[1] N.D. Mermin and H. Wagner, Phys. Rev. Lett. 22 1133 (1966).

[2] J.M. Kosterlitz and D.J. Thouless, J. Phys. C 6, 1181 (1973).

[3] P.M. Chaikin and T.C. Lubensky, Principles of condensed physics, Cam- bridge University Press (1995).

[4] H. J. Jensen, The Kosterlitz-Thouless Transition, Department of Mathematics, Im- perial College (2003).

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