F k B T = 1 2 K. θ q e iq x, (2) q 2 θq. d 2 1

Transcription

1 Physics 7c: Sttisticl Mechnics Kosterlitz-Thouless Trnsition The trnsition to superfluidity in thin films is n exmple of the Kosterlitz-Thouless trnsition, n exotic new type of phse trnsition driven by the unbinding of vortex pirs. Mny of the predictions of the theory hve been verified in experiments on thin films of He 4 on surfce. This type of trnsition occurs in the universlity clss of two dimensionl XY models, where the broken symmetry vrible is n ngle: the phse in superfluidity, the orienttion of the spins in the xy plne for the mgnet, etc. The full tretment of the Kosterlitz-Thouless trnsition is rther dvnced topic, but the clcultion illustrtes mny of the techniues introduced in the first two terms, nd the result is interesting! The novel behvior of the trnsition rises from the long wvelength logrithmic divergence of the phse or orienttion fluctutions. It is therefore sufficient to tke s the free energy F = F k B T = K θ) d x ) where for the superfluid θ = nd K = ρ s /k B T = h/m) ρ s /k B T. From now on we will use the reduced free energy F. Specil Fetures Phse fluctutions diverge This clcultion mirrors the one on the d Heisenberg model in Homework 4 of Ph7b.) Introducing the usul expnsion in Fourier modes θx) = θ e i x, ) the free energy becomes F = K θ, 3) with the re of the system. Euiprtition gives θ = K, 4) nd then the men sure fluctution is θ = π) d K = πk d R 5) where is lrge smll distnce) cutoff, nd the system size R sets the smll cutoff. The integrl diverges logrithmiclly for lrge systems, R. Note tht the free energy ws only expnded up to udrtic order in devitions of θx) or θ. By nlogy with the clcultion for mgnets, this is clled the spin wve pproximtion.

2 Phse correltions decy with power lw We now clculte the decy of correltions of the phse or ngle coming from these smll fluctutions, gin strting from E. ). We wnt to clculte the correltion function Gx) = e iθx) θ) = cosθx) θ) 6) since this gives the ψx)ψ ) correltion function for the superfluid, or the mx) m) correltion function for the mgnet the verge of the corresponding sin is zero since positive nd negtive θx) θ) re eully likely). Expnding in the Fourier modes θx) θ) = θ e i x ). 7) The verge is given by integrting over the Gussin distribution of ech θ given by the Boltzmnn fctor from the free energy E. 3) dθ ) exp ) K θ iθ e i x ) Gx) = dθ ) exp K ), 8) θ where the first term in the sum in the exponentil in the numertor is from the Boltzmnn fctor, the second is from E. 7), nd the products of exponentils hs been written s single sum. The integrls re most relibly done by writing θ in rel nd imginry prts θ = R + ii, 9) where the fct tht θx) is rel mens θ = θ so tht R is even nd I odd in. Then we cn write the sum in the numertor...= K R + I ) ir cos x ) + ii sin x ) where the other terms such s R sin x vnish on summing over nd. Now complete the sures...= K [ R icos x ) K ) + I + i sin x K ) ] + cos x) K, ) where to get the lst term use cos x ) + sin x) = cos x). The integrls over R,I in the numertor of E. 8) cncel the integrls in the denomintor the shift of the center of the Gussins does not chnge the integrl) so tht Gx) = e gx) ) with Using s usul gx) = cos x). 3) K π) d, 4) nd introducing φ the ngle of in the plne mesured from the direction of x gives gx) = πk = πk d π π dφ[ cos x cos φ)] 5) d J x ), 6)

3 with lrge cutoff. Without the Bessel function in the numertor the integrl would diverge logrithmiclly from the smll rnge. Since J x ) for lrge, nd J x ) for / x we hve for lrge x gx) = lnc x ) 7) πk with c some constnt. This gives for the correltion function Gx) x η with η = πk = k BT π ρ s. 8) Thus the order prmeter correltion function decys s power lw with n exponent η tht depends on temperture. The correltions decy more rpidly s T increses, s you might expect. Eution 8) predicts power lw correltions for ll tempertures, wheres we would expect exponentil decy t lrge enough tempertures. We should ctully hve some confidence in the result. The clcultion ws trctble becuse of the udrtic nture of the effective Hmiltonin. But this rises not from n pproximtion tht the devitions of the ngles θx) from some uniform stte re smll which we would doubt), but from the ssumption of smll grdients, i.e. tht the sptil vrition of θx) is slow, nd the difference between neighboring ngles is smll. The power lw correltions t long distnces comes from the smll prt of the behvior, which is where this pproximtion should be good! Wht we hve left out is the possibility of vortex excittions, for which E. ) does not pply everywhere. Eution 8) turn out to be ccurte, until vortex excittions proliferte. How this develops is the next topic. Vortex prolifertion size We hve seen tht the energy of single vortex depends logrithmiclly on the system ) E v αr = πk ln, 9) k B T nd so we might expect no therml excittion of vortex. However the entropy, proportionl to the log of the number of wys we cn put down single vortex, lso depends on the log of the system size. An estimte would be CR ) S v = k B ln, ) with C some numericl constnt. The reduced) free energy F v = E v TS v )/k B T diverges with incresing system size s F v = πk ) ln R +, ) where the denotes unimportnt constnt terms. As first pointed out by Kosterlitz nd Thouless, this diverging free energy switches sign t criticl temperture T KT given by K KT = k BT KT ρ s = π. ) Above this temperture, t lest in the pproximtion of isolted vortices, F v is negtive, nd vortices should proliferte. Notice from E. 8) tht η = /4 tt = T KT so tht the correltions decy s x /4 here. Complete picture Perhps surprisingly, given the simplicity of the ssumptions, the preceding results turn out to be exctly correct, with the one modifiction tht K or ρ s ) is itself temperture dependent. There is phse trnsition t T KT. Above this temperture correltions decy exponentilly. Below this temperture there is no long rnge 3

4 Vortex unbinding Power lw correltion T KT Exponentil correltions T,K - Figure : Schemtic of the Kosterlitz-Thouless trnsition. order consistent with the Mermin-Wgner theorem for ny continuous order prmeter in two dimensions). However there re power-lw correltions with the exponent ηt ) given by E. 8) where the temperture dependent ρ s T ) s would be mesured in n experiment) is to be used. Below T KT the long rnge order is eliminted by the ccumultion of smll phse fluctutions spin wve theory ). The trnsition occurs by the prolifertion of free vortices, the topologicl defect of the broken symmetry. Since t ny temperture there re thermlly excited vortex pirs, with smll seprtions t low tempertures, probbly better wy to think of the trnsition is s vortex pir unbinding. AtT KT the superfluid density jumps discontinuously between nonzero vlue nd zero. The rtio ρ s T KT )/k B T KT is universl nd tkes on the vlue /π. Power lw correltions re ssocited with criticl point; the power lw correltions for ll T < T KT cn be understood in terms of criticl line. The correltions behve s x /4 t T KT, gin universl behvior t the trnsition temperture. The simple clcultions brek down in ignoring the effect of the vortices on the θ modes. In fct the vortices ct to renormlize the stiffness K of these modes. This is wht mkes ρ s for T>T KT, nd is why E. 8) is not correct t ll tempertures. This cn be understood using RNG tretment, s outlined in the next section. RNG Tretment We first wnt to see how the vortices chnge the effective long-distnce stiffness constnt. We lredy know tht vortex pirs reduce the mss flow ρ s v s cf. the discussion of criticl velocities in Lecture 5, where the vortex pir in the tube reduces the totl flow). We cn clculte the effect more completely from the chnge in free energy of n imposed superfluid velocity v. In this chpter I will define v = without fctor of h/m.) The full, renormlized ρ s is then or in terms of K = ρ s /k B T The free energy is ρ R s Fv) F) = lim, 3) v v K R = lim v Fv) F). 4) v [ F ] = ln Tr e H eff v) 5) with the effective Hmiltonin divided by k B T ) H eff = K d xv + v v ). 6) 4

5 Here Tr denotes the integrl over ll configurtions of the vortices, nd v v x) is the superfluid velocity field due to the vortices. The smll ngle fluctutions re supposed lredy included in the bre K.) Expnding out H eff = K d xv + K d xvv + K d x v v v. 7) The first term is n dditive constnt, wht we would hve without vortices; the second plys the role of the unperturbed Hmiltonin H for the vortices no externl v), nd the third is the smll perturbtion. Now expnding the exponentil to second order in v [ ] [ { ln Tr e H eff v) K v ln Tr e H K d x v v v+ }] K d x v v v) + 8) [ F + K v ln + K ) d x v v v F + K v ) K d x v v v, 3) where the verge is with respect to H, nd v v = hs been used. Using isotropy nd homogeneity we hve d x v v v) v i v j d x d x v v,i x)v v,j x 3) = ij = v ] 9) d x v v x) v v. 3) Thus from E. 4) K R = K K d x v v x) v v. 33) This reltes the superfluid density to the velocity-velocity correltion function, result reminiscent liner response theory. The velocity v v is due to the vortices. Now we introduce configurtion of vortices represented by the vorticity density n v x) v v = πn v x)ẑ = πẑ k α δx X α ), 34) α with the αth vortex hving sign k α =±nd position X α the higher chrged vortices hve lrger energy, nd cn be neglected). We wnt to relte the v v correltion function to the vortex density correltion function, which we will then evlute from the sttisticl mechnics of the intercting vortices. The finl result is d x v v x) v v = π d xx n v x)n v. 35) It seems to me I should be ble to do this by integrting E. 34) directly, but the stndrd pproch goes through Fourier spce. Introducing the Fourier representtion nd using v v ) v v δ, d x v v x) v v = d x e i x v v ) v v 36), = d x e i x v v ) v v 36b) = lim v v ) v v. 36c) 5

6 Normlly we would just evlute the result t =, but here we need to keep the limit.) In terms of n v ), the Fourier trnsform of n v x) we cn write lim v v) v v = 4π n v )n v lim. 37) Now we wnt to clculte n v )n v for smll : n v )n v = d x d x n v x)n v x ) ei x x ) 38) d x d x n v x)n v x ) [ i x x ) i j x x ) i x x ) j + ]. The first term is zero since the totl vortex chrge d x n v x) is zero. The second is odd in x x nd integrtes to zero. This leves the lst term, which gives doing the ngulr verge, giving fctor of /) n v )n v 4 d xx n v x)n v. 4) Hence with E. 36) we get E. 35). Thus from Es. 33) nd 35) we hve our finl forml result for the renormliztion of the superfluid density by vortices K R = K + π d xx n v x)n v. 4) K We re left with clculting the correltion function of the vortex density. The vortices interct with logrithmic dependence on seprtion, which in the present nottion is ) x Ē pir = πk ln, 4) where I hve bsorbed constnts inside the logrithm into the smll scle cutoff ). The problem ctully reduces to two dimensionl gs of chrges. Eution 4) is nlogous to the dielectric constnt for the chrged gs: the polriztion of intervening chrge pirs chnges the interction of well seprted chrges. This is nontrivil problem! We cn mke progress ssuming dilute gs. This is resonble if the core energy of the vortex is lrge compred to k B T, or in other words if the fugcity y = exp E c /k B T),is smll. In this limit we cn evlute the correltion function in terms of the Boltzmnn fctor for E pir x [ n v x)n v = 4 y exp πk ln 39) )], 43) where the fctor of 4 is from two fctors of the verge density which we estimte s n v = y, nd the minus sign is becuse the chrges must be opposite to get low energy configurtion. The fctor of is for the two configurtions + t x nd t nd the reverse, but ctully ny numericl prefctor cn be bsorbed into slightly redefined fugcity, since E c is not precisely known. The importnt prt is the dependence on x, which is the exponentil of the interction potentil. It is convenient to rewrite E. 4) in terms of K = k B T/ ρ s. Becuse we re ssuming dilute gs, so tht the second term is smll, this finlly gives K ) R K + π 3 y 6 dr r 3 πk. 44)

7 This expression imgines strting with microscopic y nd K nd clculting the mcroscopic K = K R tking into ccount the reduction in the totl momentum nd energy due to the polriztion relignment nd stretching) of the vortex pirs. Inspecting the integrl reproduces the elementry Kosterlitz-Thouless result: for K >π/ the integrl diverges. We cn understnd this more completely using renormliztion group tretment llowing the coupling constnts K nd y to depend on scle fctor l, nd evluting the integrl over ll scles piece by piece. Suppose we integrte out the smll seprtions between nd = + δl) K ) R K + π 3 y +δl dr r 3 πk + π 3 y dr r 3 πk. 45) The first integrl gives n dditive correction tht cn be bsorbed into new K: K ) = K + π 3 y δl. 46) The second integrl cn be put in the sme form s before, now in terms of the cutoff, by defining new y: ) y = y + δl) 4 πk. 47) These trnsformtions leve the expression for K R unchnged, with K K,y y,, nd the process cn be iterted. y K - Figure : RNG flows for K nd y generted by numericl solutions of Es. 46) nd 47). The eutions 46) nd 47) cn be written s differentil eutions for the evolution of K l) nd yl) dk = π 3 y, 48) dl dy = πk)y. dl 48b) 7

8 These re the RNG flows for the Kosterlitz-Thouless trnsition. Some numericlly generted solutions for the flows re shown in Fig.. We hve derived the results for smll y. The finl step of the RNG is to rescle lengths L L/ + δl), so tht the cutoff returns to its originl vlue. Thus lengths such s the correltion length evolve s dl dl = L i.e. L e l. 49) The line y = isfixed line. ForK <π/the line is stble, for K >π/the line is unstble. The superfluid stte corresponds to initil vlues of K nd y such tht y nd K K <π/sl. The vlue K is the lrge distnce reduced stiffness constnt ρ s /k B T tht would be mesured in experiment. As we increse temperture, the initil y increses, nd K decreses, until we rech temperture T KT which gives initil vlues of y,k such tht the RNG flow termintes t y =, K = π/. For slightly lrger temperture the RNG flows pss ner this point, but then flow wy to lrge y nd K. Presumbly this corresponds to the disordered stte, lthough we cnnot follow the behvior to lrge y. For T = T KT the physicl lrge length scle) superfluid density is ρ s = /π)k B T KT ; for slightly lrger tempertures ρ s =. Thus the phse trnsition is signlled by discontinuous jump in the superfluid density. The rtio of the jump in ρ s t the trnsition to k B times the trnsition temperture is universl. Note tht ρ s is stiffness constnt, not thermodynmic vrible, so this is not first order trnsition. In fct the entropy is continuous, nd the specific het show only very wek singulrity t T KT. This is becuse the energy in the vortices is smll most resides in the elementry excittions or udrtic modes. However the vortices re vitl to the properties such s the superfluid density.ner y =,K = π/ the flow trjectories re hyperbole, nd the scling behvior of other untities such s the correltion length ner the trnsition cn be derived from this. For exmple it is found tht the correltion length diverges pproching T KT from bove s ξ expc/ T T c ), rther thn the usul power lw divergence. Further Reding The originl pper is Ordering, Metstbility, nd Phse Trnsitions in -Dimensionl Systmes, by J.M. Kosterlitz nd D. J. Thouless, J. Phys. C8, 973), vilble here. Kosterlitz introduced the RNG tretment in Criticl Properties Of -Dimensionl XY-Model J. Phys. C7, ) or here, but the stndrd reference on this is Renormliztion, Vortices, nd Symmetry-Breking Perturbtions in -Dimensionl Plnr Model, by J. V. Jose, L. P. Kdnoff, S. Kirkptrick, nd D. R. Nelson, Phys. Rev. B6, 7 977) or online. April, 4 8