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The Effect of Columnar Disorder on the Superconducting Transition of a Type-II Superconduct

The E?ect of Columnar Disorder on the Superconducting Transition of a Type-II Superconductor in Zero Applied Magnetic

arXiv:cond-mat/0404179v1 [cond-mat.supr-con] 7 Apr 2004

Anders Vestergren and Mats Wallin Condensed Matter Theory, Department of Physics, KTH, SE-106 91 Stockholm, Sweden S. Teitel Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627 Hans Weber Division of Physics, Lule? University of Technology, SE-971 87 Lule? Sweden a a,
(Dated: February 2, 2008)

We investigate the e?ect of random columnar disorder on the superconducting phase transition of a type-II superconductor in zero applied magnetic ?eld using numerical simulations of three dimensional XY and vortex loop models. We consider both an unscreened model, in which the bare magnetic penetration length is approximated as in?nite, and a strongly screened model, in which the magnetic penetration length is of order the vortex core radius. We consider both equilibrium and dynamic critical exponents. We show that, as in the disorder free case, the equilibrium transitions of the unscreened and strongly screened models lie in the same universality class, however scaling is now anisotropic. We ?nd for the correlation length exponent ν = 1.2 ± 0.1, and for the anisotropy exponent ζ = 1.3 ± 0.1. We ?nd di?erent dynamic critical exponents for the unscreened and strongly screened models.
PACS numbers:




The discovery of high temperature superconductors, in which thermal ?uctuations are important and mean ?eld theory can no longer be applied, has led to a resurgence of interest in phase transitions and critical phenomena in type-II superconductors.1 Of particular interest has been understanding the e?ects of random quenched disorder on the nature of the ordered phases and the universality of the phase transitions. For the high temperature superconductors, this quenched disorder can take many forms: random point defects due to oxygen vacancies, planar twin grain boundaries, and columnar defects introduced by ion irradiation. Most of the work in this area has focused on the case of a ?nite applied magnetic ?eld, where one seeks to understand how the randomness distorts or destroys the Abrikosov lattice of magnetic ?eld induced vortex lines that forms in a pure system. Columnar defects2,3,4 have received considerable attention, as they are particularly e?ective in pinning vortex lines and reducing ?ux ?ow resistance. In contrast, in this paper we will focus on the e?ect of columnar defects on the superconducting phase transition in zero applied magnetic ?eld. We expect this case to be interesting for the following reason. A generalized Harris criterion5,6,7 argues that disorder will be a relevant perturbation, and change the universality class of a phase transition, whenever 2 ? d? ν > 0, where d? is the number of dimensions in which the system is disordered, and ν is the usual correlation length critical exponent. For a disorder free superconductor, the transition in zero applied magnetic ?eld is in the universality class of the three dimensional (3D) XY model,8 for which ν 2/3. For random point disorder, d? = 3, so 2?d? ν < 0, and the generalized Harris criterion argues that the universality of the transition remains unchanged. For columnar disorder, however, d? = 2, and so 2 ? d? ν > 0. Columnar defects should therefore cross the zero ?eld transition over to a new universality class. Stability6 of this new disordered ?xed point with respect to the generalized Harris criterion implies that it should have a new value ν > 1. In our equilibrium simulations we indeed ?nd behavior consistent with this, and we obtain a value for the correlation length exponent ν = 1.2 ± 0.1. Moreover, we ?nd scaling is now anistropic and we ?nd the value of the anisotropy exponent to be ζ = 1.3 ± 0.1. Experimental measurement of these exponents would therefore provide a precision test of the theoretical model. The model we study also has application to the T = 0 superconductor to insulator 2

quantum phase transition in two dimensional thin ?lms with random substrate disorder,5,9,10 and to the Mott transition for bosons in 2D optical lattices with the addition of random scattered laser intensity.11 In these cases, the two dimensional quantum problem can be mapped onto a corresponding three dimensional classical problem with the same symmetries as the one we study here.12 To study the e?ect of columnar disorder on the zero ?eld transition of a type-II superconductor, we will consider two di?erent limits. The ?rst is the limit of an “unscreened” superconductor4,13 in which magnetic ?eld ?uctuations are frozen out, corresponding to the approximation of an in?nite bare magnetic penetration length, λ0 → ∞. Here, vortex line segments have long ranged Coulombic-like interactions. For the extreme type-II high temperature superconductors, for which κ ≡ λ0 /ξ0 ? 100, where ξ0 is the bare coherence length that also sets the radius of a vortex line core, the unscreened model should give a good description except in an extremely narrow temperature window about the transition.14 The second limit is that of a strongly screened superconductor,3,15 corresponding to the case λ0 ? ξ0 . In this case, vortex line segments have short range interactions. This description should also become valid extremely close to the transition when the diverging correlation length ξ becomes comparable to the renormalized magnetic penetration length λ, λ ξ, and magnetic ?eld ?uctuations on such large length scales must be included in determining the true critical behavior. This region near Tc may, however, be too small to observe in practice.14 As in the disorder free case, a duality transformation8,16,17 establishes that these two limits lie in the same universality class as regards equilibrium critical behavior. They may be di?erent, however, for dynamic critical behavior.15 In this work we carry out detailed Monte Carlo (MC) simulations of the XY model for the unscreened superconductor to determine the equilibrium critical exponents, and we demonstrate the presence of anisotropic scaling; by duality, these exponents also apply to the strongly screened case. Then, using simple local Monte Carlo dynamics, we compute the dynamic critical exponent for both the unscreened XY model, and for the strongly screened vortex loop model. We ?nd that the dynamic exponent is di?erent for these two limits. The remainder of this paper is organzied as follows. In Section II we describe the XY model and the loop model for the unscreened and strongly screened limits, respectively. The duality transformation between the two is given in Appendix A. In Section III we discuss 3

the equilibrium critical behavior of the XY model, presenting our ?nite size scaling analysis, de?ning the observables we measure, and giving the numerical results of our simulations. In Section IV we discuss the dynamic critical behavior of the XY and loop models, within a simple local Monte Carlo dynamics. We de?ne the observables we measure and give our numerical results. In Section V we give our discussion and conclusions.



To model the e?ects of thermal ?uctuations in a type-II superconductor, we start with the commonly used 3D XY model.13 This models the phase ?uctuations of the superconducting order parameter in the “unscreened” limit where magnetic ?eld ?uctuations are frozen out, corresponding to the approximation of an in?nite bare magnetic penetration length, λ0 → ∞. For zero applied magnetic ?eld we have, HXY [θi ] = ? Ji? cos(θi ? θi+? ) . ? (1)


Here θi represents the phase angle of the complex superconducting order parameter on node i of a periodic cubic grid of N = L × L × Lz sites, with periodic boundary conditions in all directions. The sum is over all nearest neighbor bonds (i, ?) of the grid, with ? = x, y , z , ? ? ? ? and the cosine term represents the kinetic energy of ?uctuating supercurrents. The short length cuto? of the discrete grid models the bare vortex core size ξ0 . In a pure system, the couplings Ji? are all equal, except for a possible variation with bond direction ?. Here, we take the Ji? randomly distributed in order to model quenched random columnar defects. For the work reported on here, with columnar defects aligned parallel to the z axis, we have chosen the following distribution: in the z direction, we take ? ? all Jiz = 1; in the xy plane, we take Ji? , ? = x, y, distributed equally likely with the two values 0.1 and 1.9, keeping the Ji? translationally invariant along the z axis so as to model ? columnar disorder. Note that the random Ji? introduce no frustration into the system; in the ground state all the θi are equal. The variations in the Ji? result in spatially random pinning energies for vortex loop excitations of the phase angles θi . We have chosen the above bimodal distribution for Ji? to give strong pinning energies (for ?xed average Ji? ), so as to be able to approach the asymptotic scaling limit with reasonable size systems. 4

Although we will simulate the Hamiltonian of Eq.(1) using periodic boundary conditions on the phase angles θi , it is useful to consider a more general ?xed twist boundary condition, θi+L? ? = θi + ?? , ? (2)

where ?? is a ?xed (non-?uctuating) total twist in the phase angle applied across the system in direction ?. Periodic boundary conditions correspond to the twist ?? = 0. Transforming ? to new variables,
′ θi = θi ? (?? /L? )ri · ? , ?


the Hamiltonian of Eq.(1) becomes,
′ HXY [θi ; ?? ] = ? ′ ′ Ji? cos(θi ? θi+? ? ?? /L? ) , ?



′ ′ ′ where the θi obey periodic boundary conditions, θi+L? ? = θi . Using the fact that the cosine ? ′ is periodic in 2π, the partition function integrals over θi can be taken over the interval ′ θi ∈ [0, 2π), as were the integrals over the original phase angles θi . Considering how the free

energy varies with the twist ?? will be useful later for discussing phase coherence in the model.


Loop Model

Although we carry out our equilibrium simulations directly in terms of the XY model of Eq.(1), we also consider a di?erent formulation of the model. If instead of the cosine interaction of Eq.(1), one uses the periodic Gaussian interaction of Villain,18 then a standard duality transformation8,16,17 (see Appendix A) maps the XY model, HXY /T , onto a model ? of sterically interacting loops, Hloop /T , where, Hloop = 1 2 gi? n2 . i?


The ni? are integer valued variables on the bonds (i, ?) and satisfy a divergenceless constraint,

[ni? ? ni??,? ] = 0 . ?


The ni? thus form connected paths through the system that must eventually close upon themselves. The couplings gi? of Eq.(5) are related to the couplings of the XY model by, ? gi? /T = T /Ji? , 5 (7)

? where the temperature scale of the loop model, T , is inverted with respect to the temperature scale of the XY model, T . While the loop model of Eq.(5) is exactly dual to the XY model of an unscreened super? conductor, taking it on its own with T as the physical temperature, we can give Hloop the as the vortex loops of a strongly screened superconductor with λ0 ? ξ0 . The short ranged vortex line interaction of this case is then modeled by the simple onsite repulsion of Hloop . Further details of this analogy may be found in Ref.15. If we regard each site of our numerical grid as representing a columnar pin, the random gi? in the xy plane can be thought of as modeling the random distances between such pins, and hence giving the random energies associated with a vortex loop segment hopping from one pin to another. This duality between Hloop and HXY thus implies that the unscreened and the screened superconductor models fall in the same equilibrium universality class, just as is the case for the disorder free model.8

following di?erent physical interpretation.3,15 We can regard the divergenceless variables ni?



In this section we report on our equilibrium XY model simulations. To extract critical exponents, we use the method of ?nite size scaling. We ?rst, therefore, discuss this method.


Finite Size Scaling

Because the columnar disorder singles out the special direction z , we must allow for the ? possibility that scaling will be anisotropic. If ξ denotes the correlation length in the xy plane, then anisotropic scaling assumes that, as T → Tc and ξ diverges, the correlation length along the z axis diverges as, ? ξz ? ξ ζ , where ζ is the anisotropy exponent. Consider now an observable O whose scaling dimension is zero. As a function of reduced temperature t ≡ (T ? Tc )/Tc and system size L × L × Lz , we expect the scaling relationship, ? O(T, L, Lz ) = O(tb1/ν , L/b, Lz /bζ ) , 6 (9) (8)

? where b is an arbitrary length rescaling factor, O is the scaling function, and ν is the usual correlation length exponent, ξ ? t?ν . Taking b = L in Eq,(9) then gives, ? O(T, L, Lz ) = O(tL1/ν , 1, Lz /Lζ ) . (11) (10)

For the case of isotropic scaling, with ζ = 1, choosing a constant aspect ratio Lz = γL reduces the right hand side of Eq.(11) to a function of the single scaling variable tL1/ν . Measuring O vs. T for systems with varying L but ?xed Lz /L is then su?cient to determine the exponent ν. However when ζ = 1, and its value is unknown, it becomes necessary to consider systems with varying aspect ratio Lz /L, greatly increasing the complexity of the computation. To deal with this case we take the following approach, originally used to study the phase transition in the quantum Ising spin glass.19 Assume that the observable O(T, L, Lz ) when viewed as a function of Lz , for ?xed T and L, has a maximum at a particular value Lz max . Because of the scaling law Eq.(11), this value Lz max must occur when Lz max /Lζ = γ (tL1/ν ) , ? where γ is a scaling function of the single variable tL1/ν . We then de?ne, ? ? ? ? Omax (T, L) ≡ O(T, L, Lz max ) = O(tL1/ν , 1, γ (tL1/ν )) ≡ Omax (tL1/ν ) . (13) (12)

Plotting Omax (T, L) vs. T for di?erent values of L, the curves will intersect at the common point T = Tc (i.e. t = 0). The slopes of these curves at Tc then determine the exponent ν. In practice, we will determine the values of Tc and the exponent ν by the following approach.20 ? Close to Tc (i.e for small t) we can expand the scaling function Omax as a polynomial for

small values of its argument tL1/ν ,

? Omax (tL1/ν ) ? a0 + a1 [(T ? Tc )/Tc ]L1/ν + a2 [(T ? Tc )/Tc ]L1/ν

+ ...


We then ?t the data for Omax (T, L) to the above form using Tc , ν, a0 , a1 , a2 . . . as free ?tting parameters. Varying the system sizes L and temperature window |T ?Tc | of the data used in the ?t, as well as varying the order of the above polynomial expansion, will give con?dence on the signi?cance of the ?t. 7

Having obtained the value of Tc , plotting Lz max (Tc ) vs. L determines the anisotropy exponent ζ by Eq.(12), Lz max (Tc ) ? Lζ . collapse the respective data to a single scaling curve. (15)

Knowing Tc , ν and ζ, plotting Omax (T, L) vs. tL1/ν and O(Tc , L, Lz ) vs. Lz /Lζ should



To carry out the scaling analysis outlined in the previous section, we now have to determine appropriate observables to measure. For the 3D XY model of Eq.(1), we expect below Tc a non-vanishing order parameter, ψ = (1/N)

eiθi . We de?ne the real part of ψ as, M= 1 N cos θi ,


and construct its Binder ratio21 g(T, L, Lz ) ≡ 1 ? M4 3 M2

= g (tL1/ν , Lz /Lζ ) . ?


Because the scaling dimension of M cancels in taking the ratio above, the Binder ratio g has scaling dimension zero, and so has the scaling form of Eq.(11). In the above, . . . denotes the usual thermal average, while [. . .] denotes the average over di?erent realizations of the columnar disorder. In the denominator of Eq.(17), the square of the expectation value is evaluated using two replicas with identical disorder, indexed by a and b, M 2 (M a )2 (M b )2 , in order to avoid bias.22

Another observable we have measured is obtained by considering the dependence of the total free energy on the total applied twist across the system.23 Sensitivity to boundary conditions, in this case speci?ed by the twist ?? in Eq.(2), is one of the signatures of an ordered phase. The XY model is therefore phase coherent when the total free energy F
′ varies with twist ?? . F is computed from a partition function sum over the θi using the

Hamiltonian HXY [?? ] of Eq.(4). A convenient measure of the dependence of F on ?? is obtained by looking at the curvature of F (?? ) at its minimum. In Appendix A we show that this minimum always occurs at ?? = 0. We therefore consider, ?2F ??2 ? =
?? =0

? 2 HXY ??2 ? 8


1 T

?HXY ???




where HXY is that of Eq.(4), and the averages on the right hand side are taken in the ensemble with ?? = 0. then ? 2 F /??2 has scaling dimension zero. These derivatives are usually de?ned in terms ? Since the total free energy F and the total twist ?? are both scale invariant quantities,

of the helicity modulus13 Υ? , which is the derivative of the free energy density with respect to the twist per length. We have in three dimensions, ?2F ??2 z ?2F ??2 x =
?? =0

L2 Υz , Lz

(19) (20)

= Lz Υx .
?? =0

Averaging over the disorder, we ?nd that, for ?xed T and L, (L2 /Lz )[Υz ] decreases monotonically as Lz increases, while Lz [Υx ] increases monotonically as Lz increases. In order to have an observable which has a maximum as a function of Lz , we therefore consider the product, L2 [Υx Υz ] = u(tL1/ν , Lz /Lζ ) , ? which has the same scaling form as Eq.(11). (21)


Monte Carlo Methods and Error Estimation

In order to achieve accurate results, averaging over many disorder realizations for many di?erent aspect ratios, Lz /L, it is essential to have an e?cient simulation algorithm. For equilibrium simulation of the 3D XY model, the lack of frustration allows us to use the Wol?24 cluster algorithm to avoid critical slowing down. We typically use 100 Wol? sweeps to approach equilibrium, followed by 200 Wol? sweeps to compute averages; one Wol? sweep is de?ned as building clusters until each phase angle θi has been updated once on average. Between 3000 and 5000 di?erent realizations of the random disorder are averaged over near the critical point, with fewer realizations used away from the critical point. A test of the equilibration of our simulations is shown in Fig. 1, where we see that the above simulation lengths are su?cient. To estimate the statistical error in our results we use the following method. For our raw data, our average is just the average over the individual values obtained in Nd independent realizations of the random disorder. Our estimated error is determined from the standard 9

FIG. 1: Binder ratio maximum for the 3D XY model, gmax at T = 2.05 ? Tc , computed with Neq equilibration sweeps followed by Neq sweeps to compute averages. Neq = 100 is su?cient for good equilibration for all sizes L.

√ deviation σ of these Nd independent values, error = σ/ Nd . To estimate the statistical error in the ?tting parameters of our ?nite size scaling analysis, we take the following approach. From our original data set we construct many (typically 1000) ?ctitious data sets by adding to each data point a random Gaussian variable with zero mean, and standard deviation equal to the estimated statistical error of the data point. We then ?t each of the ?ctitious data sets. The standard deviation of the values of the resulting ?tting parameters then gives our estimate of the statistical error in the ?tting parameter. Harder to estimate are the possible systematic errors in our results. Here we rely on varying parameters of our analysis, such as the order of a polynomial ?t, or the system sizes L used in the ?t, in order to get a feeling for the likely accuracy of our results.



We now present our results from simulations of the XY model of Eq.(1). In Fig. 2 we plot our results for the Binder ratio of Eq.(17), g(T, L, Lz ) vs. Lz , for sizes L = 8, 12, 20, at the ?xed temperature T = 2.05. We see that for each L, g(T, L, Lz ) has a clear maximum


FIG. 2: Binder ratio, g(T, L, Lz ) vs. Lz for several values L at the ?xed T = 2.05. Solid curves are cubic polynomial ?ts in ln Lz .

at a particular Lz max . Note that the maximum values of these curves appear to be equal for the di?erent values of L. From Eq.(13) we therefore infer that the temperature T = 2.05 is approximately the critical temperature Tc . To determine the precise values of Lz max and the maximum values gmax (T, L) = g(T, L, Lz max ), we ?t the data for each L to a cubic polynomial in ln Lz (these are the solid curves in Fig. 2). The Lz max obtained this way are not, in general, integer values. We have also done such ?ts using a quadratic polynomial in ln Lz ; the di?erence in values obtained from the cubic vs. the quadratic ?t provides our estimate of the systematic error of this procedure. We ?nd that for gmax (T, L) this systematic error is always smaller than the estimated statistical error of the cubic ?ts; for Lz max the systematic error is bigger. This re?ects the simple fact that g(T, L, Lz ), being a maximum with zero slope, varies only quadratically with deviations from the true Lz max , and hence may be determined more accurately. Henceforth, the error bars we use for gmax are the above estimated statistical errors, while the error bars we use for Lz max are the above de?ned systematic errors. Proceeding in this way at other temperatures, we plot in Fig. 3 the values of gmax (T, L) vs. T for L = 8, 10, 12, 16 and 20. The di?erent curves all intersect at a common point, determining Tc ? 2.05. To determine the correlation length exponent ν, and get a more 11

FIG. 3: Binder ratio maximum gmax (T, L) vs. T for various system sizes L. The common intersection determines Tc ? 2.05. Solid lines are linear ?ts to the data.

precise estimate of Tc , we now ?t the data of Fig. 3 to a polynomial expansion as in Eq.(14). In Table I we show the results from both quadratic and cubic polynomial ?ts, using di?erent system sizes L; we systematically drop the smallest sizes since scaling holds only in the asymptotic large L limit. Our results give a consistent value of Tc ? 2.05. The values of ν that we obtain are consistent within the estimated statistical errors, however we see a small systematic increase in the value of ν when we restrict the data to larger system sizes. We therefore estimate ν = 1.2 ±0.1. Note that our value ν > 1, satis?es the Chayes lower bound condition,6 as generalized5 for correlated disorder, ν > 2/d?, where d? = 2 is the number of dimensions in which the system is disordered. In Fig. 4 we replot the data of Fig. 3 in a scaled form, gmax (T, L) vs. ((T ? Tc )/Tc )L1/ν . We use the value of Tc and ν from Table I for the cubic ?t to sizes L = 12 ? 20. The solid line is the ?tted cubic polynomial. As is seen, the data collapse is excellent. Having found the value of Tc , we next determine the anisotropy exponent ζ. In Fig. 5 we show a log-log plot of our data for Lz max vs. L, at the temperature T = 2.05 ? Tc . Fitting to Eq.(15), Lz max ? Lζ , we get the results summarized in Table II for di?erent ranges of ζ = 1.3 ± 0.1. To check the consistency of our value for ζ, in Fig. 6 we plot g(Tc , L, Lz ) vs. 12 system sizes L. The results are consistent within the estimated statistical error and we ?nd

L 8 ? 20 10 ? 20 12 ? 20

order quadratic cubic quadratic cubic quadratic cubic

Tc 2.052 ± 0.001 2.052 ± 0.001 2.052 ± 0.001 2.052 ± 0.001 2.051 ± 0.001 2.051 ± 0.001

ν 1.07 ± 0.03 1.04 ± 0.06 1.10 ± 0.04 1.10 ± 0.06 1.18 ± 0.04 1.18 ± 0.05

TABLE I: Fitting parameters Tc and ν from quadratic and cubic polynomial scaling ?ts to the data of Fig. 3. Results for di?erent ranges of system sizes L are shown.

FIG. 4: Scaling collapse of gmax (T, L) vs. ((T ?Tc )/Tc )L1/ν . The values of Tc = 2.051 and ν = 1.18 from the last row of Table I are used. The solid curve is the ?tted cubic polynomial.

Lz /Lζ , using our data at T = 2.05 and the above determined value of ζ = 1.3. As expected from Eq.(11), the data for the di?erent values of L and Lz show a very good collapse to a single scaling curve. We have also tried a similar scaling analysis for the helicity moduli product, L2 [Υx Υz ], of Eq.(21). However here we have found less satisfactory results. We ?nd that for a given system size L, the Lz max where L2 [Υx Υz ] has its maximum occurs at a smaller value of Lz 13

FIG. 5: Log-log plot of Lz max vs. L, at T = 2.05 ? Tc . The solid straight line is the best power law ?t using the data for L = 10 ? 20 and yields the value ζ = 1.329 ± 0.08 (see Table II). Lmin ζ 8 1.29 ± 0.05 10 1.33 ± 0.08 12 1.37 ± 0.12

TABLE II: Anisotropy exponent ζ from power law ?ts, Lz max ? Lζ to system sizes L = Lmin ? 20.

than was the case for the Binder ratio gmax . Such smaller system sizes presumably have larger corrections to scaling. We have also found the statistical error in L2 [Υx Υz ] to be larger than we found for gmax , possibly because the Binder ratio g involves a ratio between ?uctuating quantities and so has smaller sample to sample ?uctuations.25 As a consequence of these two e?ects, we could not arrive at a convincing determination of Tc and ν from the L2 [Υx Υz ] data alone. However, to illustrate our results we can make use of the values of Tc ? 2.05 and ζ ? 1.3 found in our analysis of gmax . In Fig. 7 we therefore show a T = 2.05 ? Tc and the above value of ζ. We see clearly in Fig. 7 the above e?ects: error bars are considerably larger than in Fig. 6, and the peak is at a smaller value of Lz /Lζ . The scaling collapse is not bad for the bigger systems sizes, corresponding to larger values of Lz /Lζ . However it is rather scattered near scaling collapse similar to that of Fig. 6, plotting L2 [Υx Υz ] vs. Lz /Lζ , using our data at


FIG. 6: Scaling collapse of g(Tc , L, Lz ) vs. Lz /Lζ , for data at T = 2.05 ? Tc , using ζ = 1.3. The solid line is a guide to the eye only.

the peak and below it. We conclude that it would be necessary to average over many more disorder realizations to reduce the errors, and perhaps also go to larger system sizes, in order to get a convincing scaling analysis from the helicity product L2 [Υx Υz ] on its own.



As one approaches the critical temperature Tc , we expect relaxation times to diverge as τ ? ξ z , de?ning the dynamic critical exponent z. To compute equilibrium critical exponents, it is su?cient that the simulation dynamics satis?es detailed balance; the details of the dynamics are otherwise irrelevant. Thus the exact duality between HXY and Hloop implies that the unscreened and the strongly screened superconductors have the same equilibrium critical behavior. For the dynamic critical behavior, however, the value of z will in general depend on the details of the dynamics,26 and some works suggest that it may even vary for di?erent types of relaxational dynamics or di?erent boundary conditions.27,28 There is thus no reason, a priori, to expect the same dynamic critical behavior for the XY model, expressed in terms of a dynamical rule for the phase variables θi , as compared to the loop model, expressed in terms of a dynamical rule for the vortex line variables ni? . In this


FIG. 7: Attempted scaling collapse of L2 [Υx Υz ] vs. Lz /Lζ . Data is for T = 2.05 ? Tc , using ζ = 1.3.

section, therefore, we will present results from explicit simulations of the loop model as well as the XY model. Because the true dynamics of a superconductor is local, it is not physically meaningful to compute the dynamic critical exponent within accelerated global algorithms such as the Wol? algorithm, which we used to compute equilibrium properties. We therefore will use a local Monte Carlo dynamics for both HXY and Hloop . Even within such local algorithms, it is not obvious how universal the dynamical critical behaviors may be. Thus it is unclear that our results will correspond to what is seen in experiments. Nevertheless it will be interesting to see if the two models give similar or di?erent values of z. The relative loss of e?ciency that results from using such local algorithms means that we will be unable to do as extensive an exploration of the parameter space as we did for our equilibrium analysis. But this is not necessary. We can make use of our already obtained equilibrium results, and simulate only at the value of T = Tc , using system aspect ratios Lz = γLζ . For our simulation of Hloop , we will simulate the loop model which is exactly dual (see Appendix A, Eq.(41)) to the cosine XY model that we have used in our equilibrium simulations, so as to make use of these known values of Tc and ζ.



Monte Carlo Methods and Scaling

For the XY model of an unscreened superconductor we use a standard single spin heat bath algorithm, with ?xed periodic boundary conditions on the θi . In this algorithm, a phase
′ angle θi is selected at random and replaced with a new randomly chosen θi . This update

attempt is then accepted with probability 1/[1 + exp(?E/T )] where ?E is the change in energy. One sweep, consisting of N = L2 Lz update attempts, is taken as one time step, ?t = 1. We average over 300 ? 700 disorder realizations depending on system size L. For the loop model of a strongly screened superconductor, we again use a heat bath algorithm in which the attempted excitation consists of an elementary vortex loop circulating about a randomly chosen plaquette of the grid. Adding only such closed loop excitations corresponds to the ensemble in which the average internal magnetic ?eld is constrained to B = 0 (see Appendix A). One sweep, consisting of 3N such update attempts, is taken as one time step, ?t = 1. We average over 1000 ? 2000 disorder realizations depending on system size L. In general, we expect the relaxation time τ to obey the scaling equation, τ (T, L, Lz ) = bz τ (tb1/ν , L/b, Lz /bζ ) , ? (22)

where b is an arbitrary length rescaling factor. For b = L, T = Tc , and Lz = γLζ , this reduces to the simple, τ ? Lz . (23)

For both the XY model and the loop model, we simulate with values of Lz = γLζ as determined by the ?t shown in Fig. 5. For the XY model, to approximate non-integer values of Lz we use linear interpolation of simulation data for the two closest integer values of Lz . For the loop model we simply use results from the closest integer value of Lz .

B. 1.

Observables XY Model

For the XY model we have tried two independent methods of determining z, analogous to the two quantities g and L2 [Υx Υz ] used in our equilibrium simulations. The ?rst is to look 17

at the decay of correlations in the order parameter M of Eq.(16), de?ning the relaxation time τ by,

τ =1+2

M(t)M(0) M2

? Lz ,


where t0 is chosen large enough so that τ is independent of t0 . The ratio in the above ensures that the quantity being summed over has scaling dimension zero, and hence the sum scales as τ ? Lz . The second method is to look at correlations of the supercurrent I? , de?ned by, I? = ?HXY ??? =
?? =0

1 L?


Ji? sin(θi+? ? θi ) . ?


In terms of I? one can de?ne the conductance in the ? direction by the Kubo formula,27 ? 1 G? = 2T


?t [ I? (t)I? (0) ] ? Lz ,

(26) Since I? =

where again t0 is chosen large enough that G? is independent of t0 .

(?HXY /??? )|?? =0 , and HXY and ?? are scale invariant, then I? , and hence the correlation summed over in in the de?nition of G? , has scaling dimension zero. Therefore, the sum which de?nes G? scales as τ ? Lz .
2. Loop Model

For the loop model we consider the total resistance, de?ned as follows.3,5 Let Q? (t) be the total projected loop area with normal in direction ? at simulation time t. Each time an ? oriented elementary vortex loop with normal in direction ±? is accepted, Q? changes by ±1. ? Let ?Q? (t) ≡ Q? (t) ? Q? (t ? 1) be the total change in this area after one sweep through the entire system; each sweep represents ?t = 1. In one such sweep, the total average phase angle change across the length of the system (in the dual screened XY superconductor model) in direction ? is just 2π?Q? /Lν Lσ , where ?, ν, σ are a cyclic permutation of x, y, z. By ? the Josephson relation, the total voltage drop across the system in direction ? will then be ? V? (t) = 2e 2π Lν Lσ ?Q? ?t = h 2e 1 Lν Lσ ?Q? ?t . (27)


Lmin zXY

6 2.72 ± 0.04

8 2.69 ± 0.05

10 2.63 ± 0.07

12 2.60 ± 0.03

TABLE III: Dynamic exponent zXY from power law ?ts, τ ? Lz , to system sizes L = Lmin ? 20.

Henceforth we de?ne our units of voltage such that h/2e ≡ 1. We then de?ne the total resistance in direction ? by the Kubo formula,29 ? R? = 1 2T

?t[ V? (t)V? (0) ] ,


where again t0 is chosen large enough so that R? is independent of t0 . Since the total voltage drop V? is the time rate of change of the total phase angle di?erence across the system, and since the total phase angle di?erence is a scale invariant quantity, we have the scaling V? ? 1/τ . Thus the resistance above scales as, R? ? 1/τ ? L?z .
C. 1. Results XY Model


In Fig. 8 we show a log-log plot of our results for the order parameter relaxation time τ of Eq.(24) vs. system size L, for T = 2.05 ? Tc and Lz ? Lζ . Our results are obtained using 5 × 105 MC sweeps to equilibrate, followed by 106 sweeps to compute averages. Fitting to

the power law, τ ? Lz , we get the results summarized in Table III, for di?erent ranges of systems size L. The results are consistent within the estimated statistical error, with a small systematic tendency to lower values as we restrict the ?tted data to larger system sizes. We ?nd zXY = 2.63 ± 0.07. As another check on our above determination of zXY , we consider the following. In principal, τ is de?ned by taking t0 in Eq.(24) su?ciently large so that τ is independent of t0 ; our data in Fig. 8 satis?es this condition. How big t0 must be for this to happen is set by the time scale τ itself. Therefore, we expect that if we compute τ for arbitrary t0 , then τ (t0 ) should scale as, τ (t0 ) ? Lz τ (t0 /τ ) ? Lz τ (t0 /Lz ) . ? ? 19


FIG. 8: Log-log plot of order parameter relaxation time τ of Eq.(24) vs. system size L, for T = 2.05 ? Tc and Lz ? Lζ . Solid line is the best power law ?t for sizes L = 10 ? 20, and determines z = 2.63 ± 0.07 (see Table III).

In Fig. 9 we show a log-log plot of τ (t0 )/Lz vs. t0 /Lz for various sizes L (again using T = 2.05 ? Tc and Lz ? Lζ ). Choosing the value zXY = 2.63 obtained from the ?t in Fig. 8 we ?nd an excellent collapse of all the data. For large t0 /Lz we see that the curve does indeed saturate to a ?nite constant as expected, however the collapse holds for the entire range of t0 . and Lz ? Lζ . Our results are for 2×105 MC sweeps to equilibrate, followed by 4×105 sweeps to compute averages. Fitting to the power law, G? ? Lz? , we get the results summarized in Table IV, for di?erent ranges of systems size L. For Gx the results z ? 2.66 ± 0.04 are consistent, within errors, with that obtained from our analysis of the order parameter relaxation time τ . For Gz , we get values for z that are somewhat larger. However if one compares the data points for Gx and Gz directly, one sees that the values are all roughly equal within the estimated error, except for the smallest size L = 4 (probably too small to be in the scaling limit) and for the largest size L = 20. Our ?t for z from the Gz data is skewed by this one L = 20 data point. If we restrict our ?t to sizes L = 8 ? 16, we then ?nd zz = 2.82 ± 0.03. This is still somewhat larger than what we get from Gx , but within two 20 Finally, we plot in Fig. 10 the conductances of Eq.(26), Gx and Gz vs. L, for T = 2.05 ? Tc

FIG. 9: Log-log scaling plot of order parameter relaxation time τ (t0 )/Lz vs. t0 /Lz for T = 2.05 ? Tc , Lz ? Lζ , and various values of L. Using zXY = 2.63 gives an excellent collapse for the entire range of t0 . Lmin zx zz 6 2.71 ± 0.02 2.77 ± 0.02 8 2.75 ± 0.03 2.87 ± 0.03 10 2.66 ± 0.04 2.91 ± 0.04 12 2.69 ± 0.06 3.06 ± 0.07

TABLE IV: Dynamic exponent zXY from power law ?ts, G? ? Lz? , to system sizes L = Lmin ? 20.

standard deviations of zx for the same range of sizes L = 8 ? 16.
2. Loop Model

For our loop simulations we use the interaction of Eq.(41), exactly dual to our XY model. This interaction is computed using the same distribution of Ji? as we used for the XY model, and we simulate at the same value of T = 2.05 as gives the critical point of the XY model. We also use the same values of Lz = γLζ as we used for the XY model, as determined from Fig. 5. In Fig. 11 we give our results for the resistance of the loop model, Eq.(28), as a log-log plot of Rx and Rz vs. system size L. Our results are from 12 × 104 MC sweeps 21

FIG. 10: Log-log plot of conductances Gx and Gz vs. L, for T = 2.05 ? Tc and Lz ? Lζ . The ?tted straight lines determine zx = 2.66 and zz = 2.91.

of Eq.(29), Rx ? L?z , we get the results summarized in Table V, for di?erent ranges of systems size L. The results are consistent within the estimated statistical error, and we ?nd zloop = 3.4 ± 0.1. For the case of Rz , parallel to the columnar defects, our simulations were not su?ciently long to observe the necessary saturation of Rz (t0 ) with increasing t0 , except for the smallest system sizes L ≤ 12. We do not believe that any estimate of zloop based on such small system sizes would be meaningful. We can, however, perform the following consistency check. Similar to our discussion concerning τ (t0 ) (see Eq.(30)), we can compute R? of Eq.(28) for ?nite times t0 , and we expect R? (t0 )Lz to scale with the variable t0 /Lz . In Fig. 12 we make such a log-log scaling plot using the value of z = 3.4 found for Rx in Fig. 11. For Rx we see that the collapse is excellent for all times t0 , and the scaling curve saturates to a constant at large t0 /Lz as expected. For Rz , we ?nd a good collapse for all but the largest times. We see that Rz (t0 ) saturates only for the smallest systems, and it is only here that the collapse appears to be breaking down. We conclude that these small system sizes are not large enough to expect scaling for Rz to hold. We can also try to independently determine the dynamic exponent z by ?tting to a data

to equilibrate, followed by 24 × 104 sweeps to compute averages. Fitting to the power law


Lmin zloop

6 3.23 ± 0.05

8 3.33 ± 0.07

10 3.39 ± 0.11

12 3.38 ± 0.14

TABLE V: Dynamic exponent zloop from power law ?ts, Rx ? L?z , to system sizes L = Lmin ? 20.

collapse as in Fig. 12 for all times t0 , rather than just the asymptotic large time limit. The inset to Fig. 12 shows the resulting χ2 of such ?ts as the ?tting parameter z is varied. For Rx , the χ2 shows a sharp minimum at z = 3.45, in good agreement with our earlier value of z = 3.4 from Fig. 11. For Rz , the χ2 has a minimum at the somewhat higher value of z = 3.7, however the minimum is very shallow, indicating a relative insensitivity of the data to variations in z. We conclude that both the data for Rx and Rz are consistent with a dynamic exponent zloop = 3.4 ± 0.1.

FIG. 11: Log-log plot of resistance Rx and Rz of the loop model vs. L. The solid line is the best power law ?t, Rx ? L?z , for sizes L = 10 ? 20, and determines the value zloop = 3.4 ± 0.1 (see Table V).


FIG. 12: Log-log scaling plot of time dependent resistance Rx (t0 )Lz and Rz (t0 )Lz of the loop model vs. t0 /Lz . The value z = 3.4 obtained from the ?t in Fig. 11 is used. The inset gives the χ2 error of the data collapse, as the exponent z is varied. V. DISCUSSION AND CONCLUSIONS

We have studied the equilibrium and dynamic critical behavior of the zero magnetic ?eld superconducting phase transition for a type-II superconductor with quenched columnar disorder. We have considered both the “unscreened” XY model in which λ0 → ∞, and the “strongly screened” loop model in which λ0 ? ξ0 . A duality transformation establishes that these two models are in the same equilibrium universality class. Using numerical simulations of the XY model, we ?nd, in agreement with a generalized Harris criterion, that the universality class of the transition is di?erent from the pure model, and we ?nd that scaling is anisotropic. We ?nd the value for the correlation length exponent, ν = 1.2 ± 0.1, and for the anisotropy exponent, ζ = 1.3 ± 0.1. Using the value of the critical temperature and the anisotropic scaling determined from the equilibrium analysis, we carry out simulations at the critical point to determine the dynamic critical exponent z of both the XY and loop models for local Monte Carlo dynamic rules. For the “unscreened” XY model, with a single spin heat bath dynamics, we ?nd zXY = 2.6 ± 0.1. For the “strongly screened” loop model, with a heat bath dynamics applied 24

to elementary loop excitations, we ?nd zloop = 3.4 ± 0.1.

A similar random 3D XY model has been studied by Cha and Girvin9 in the context of

the quantum phase transition in the two dimensional boson Hubbard model. In their model disorder was introduced as uniformly distributed random bonds in the z (imaginary time) ? direction, Jiz , so as to model bosons with random charging energy. They found equilibrium critical exponents ν = 1.0 ± 0.3 and ζ = 1.07 ± 0.03 (our anisotropy exponent ζ for the classical 3D model is equivalent to their “quantum dynamic exponent” z for the 2D quantum problem). However their analysis for such a system with anisotropic scaling, ζ > 1, was based on a more ad hoc approach of (i) trying various values of ζ and seeing which appeared to give the best data collapse for systems of di?erent size L, and (ii) measuring real space correlations in a system of a ?xed size and ?tting to assumed power law decays. Their largest system size, 162 × 15 is also smaller than ours and they do not use the Wol? algorithm to accelerate their equilibration. While it is possible that introducing the randomness di?erently (along z rather than in the xy plane) might e?ect the universality class, we believe it is more likely ? that this is not the case, and that our results are more systematic and hence more accurate than those of Cha and Girvin. Prokof’ev and Svistunov10 have simulated the loop model of Eq.(5) in the context of the same two dimensional disordered boson problem as Cha and Girvin. For their “o?-diagonal” disorder case they put the disorder into the bonds along the z direction, making their model ? dual to that of Cha and Girvin. They report an anisotropy exponent ζ = 1.5 ± 0.2, which agrees with ours within the estimated errors. They were unable to determine the correlation length exponent ν. We note that while they use an accelerated “worm” algorithm and have good statistics for quite large system sizes, they determine their exponents by ?tting to real space correlation functions for their biggest size system, as did Cha and Girvin, rather than doing any systematic ?nite size scaling that takes into account the anisotropic scaling present in the model. Experimental investigation of the zero ?eld transition with colummnar disorder has been undertaken by K¨tzler and co-workers30 for YBaCuO thin ?lms. Measuring the frequency o dependent conductivity transverse to the columnar disorder, which is expected to scale as14 σ⊥ (ω, T ) = t(ζ?z)ν σ (ωt?zν ) (where t = (T ? Tc )/Tc ), they ?nd31 the combinations νζ = 1.7 ? and z/ζ = 5.53. This compares with our values νζ = 1.56, and z/ζ = 2 for the unscreened XY model, and z/ζ = 2.6 for the strongly screened loop model. Our value of νζ is conceivably 25

consistent with the experimental value, within possible errors. However both of our values of z/ζ seem too small. It may be that our simple local Monte Carlo dynamics does not adequately capture the true dynamics of a real superconductor. On the other hand, if we use our value of ζ = 1.3, then K¨tzler’s results imply a dynamic exponent of z = 7.2, which o seems extraordinarily large. We may also compare our dynamic exponents with those obtained from the disorder free model. For the the strongly screened limit of the loop model, Lidmar et al.15 ?nd the value zloop ? 2.7; moreover they ?nd this value to be insensitive to the presence of uncorrelated point disorder. For relaxational dynamics of the phase angle variable in the XY model, a et al.27 using a method similar to our scaling of conductance, Eq.(26). The result zloop > zXY thus seems common for both the pure and columnar disordered cases. In our work we have considered only simple relaxational dynamics for the phase angles of the unscreened XY model. Two other possible dynamics might be considered. One would be to do a loop dynamics, similar to what we have done here for the strongly screened loop model, only now as applied to the strongly interacting loops of the unscreened XY model. The other would be to use resistively shunted junction (RSJ) dynamics for the phase angles of the XY model. Both such approaches have been previously used for the disorder free case. For both loop dynamics15,32 and RSJ dynamics27,33 the dynamic exponent z ? 1.5 was found, smaller than the value obtained by simple phase angle relaxational dynamics. Investigating these other dynamics for the case of columnar disorder remains for future work. We only note here that if the above trend remains true for columnar disorder, and that these other dynamics reduce z from that of relaxational dynamics, then it becomes even harder to explain the large value of zζ observed experimentally in Ref. 30. Finally, we note that similar equilibrium exponents to those found in this work were also found for the case of an unscreened superconductor with columnar defects in a ?nite applied magnetic ?eld. For that case the values4 ν = 1.0 ± 0.1 and ζ = 1.25 ± 0.1 were found. Although these are close to the values we ?nd here for zero applied ?eld, there is no apparent reason that the zero and ?nite ?eld cases should be in the same universality class. We also note that once a ?nite ?eld is applied, the duality between the unscreened and strongly screened superconductor models, that exists for zero ?eld, breaks down. value of z ≈ 2 is expected,26 and this is what was found in numerical simulations by Jensen



We wish to thank U. C. T¨uber for originally suggesting this problem. We thank T. a J. Bullard, H. J. Jensen, U. C. T¨uber and M. Zamora for contributions at early stages a of this work. The work of A. V. and M. W. has been supported by the Swedish Research Council, PDC, NSC, and the G¨ran Gustafsson foundation. S. T. acknowledges support o from DOE grant DE-FG02-89ER14017, and travel support from NSF INT-9901379. H. W. acknowledges support from Swedish Research Council contract 621-2001-2545.

Appendix A

In this section we review the duality transformation8,16,17 from HXY of Eq.(1) to Hloop of Eq.(5). Consider ?rst a general 2π periodic interaction Vi? (φ) instead of the ?(Ji? /T ) cos(φ) of Eq.(1). For the generalized ?xed twist boundary condition and the corresponding Hamiltonian of Eq.(4), we can write the partition function as,

i 0

′ dθi 2π



′ ′ Vj? (θj ?θj+? ??? /L? ) ?



′ ? where the θi obey periodic boundary conditions. De?ning the Fourier transform Vj? by, ∞


?Vj? (φ)

e?Vj? (nj? ) einj? φ ,
nj? =?∞



and substituting into Eq.(31) gives,

Z =
{nj? } i 0

′ dθi 2π



? Vj? (nj? )+i


′ ′ nj? (θj ?θj+? ??? /L? ) ?


{nj? }



? [Vj? (nj? )+inj? ?? /L? ] i 0

′ dθi 2π



′ ′ nj? (θj ?θj+? ) ?



′ One is now free to do the integrals over the θj . The result is a product of Kronecker deltas

constraining the variables nj? to be divergenceless, as in Eq.(6). De?ning the “winding numbers” W? by, W? ≡ we get, Z=
′ {nj? }

1 L?

ni? ,




? Vj? (nj? )?i


W ? ??




where the prime on the summation denotes the divergenceless constraint of Eq.(6). A common choice for Vi? (φ) is the Villain interaction,18


?Vj? (φ)

e? 2T





In this case one has for its transform, T 2 ? Vi? (n) = n . 2Ji? (38)

The partition function of Eq,(36), with periodic boundary conditions ?? = 0, then becomes, Z= with ? gi? /T = T /Ji? . (40)
′ {nj? }
? e? 2T 1 j?

gj? n2 j?



The above is just a model of short ranged interacting loops with onsite repulsion ? n2 and ? inverted temperature scale T ? 1/T . For our simulatons, with Vi? (φ) = ?(Ji? /T ) cos(φ), one has17 e?Vi? (n) = In (Ji? /T ) ,


where In (x) is the modi?ed Bessel function of the ?rst kind. Since In (x) is an increasing function of |n| for ?xed x, the above similarly gives a short ranged loop model with onsite repulsion. It is this interaction of Eq.(41) that we use in our dynamic simulations of the loop model in Section IV. We can now demonstrate several interesting results concerning phase coherence in the XY model, by considering the behavior as a function of the twist ?? . The XY model is phase coherent when the total free energy F varies with ?? . Using F (?? ) = ?T ln Z(?? ), and Eq.(36) above, we ?nd, 1 ?F T ??? where . . .

= i W?
?? =0




indicates an average in the ensemble with ?? = 0. Now since ?F /??? must be must similarly be real (as may be seen by considering Hloop ),

a real quantity (as may be seen by considering its evaluation in the original XY model HXY of Eq.(4)), and since W?

the only way for Eq.(42) to hold is if ?F /??? |?? =0 = W? 28

= 0. This then demonstrates

that ?? = 0, i.e. periodic boundary conditions on the θi , is the twist that minimizes the free energy. Finally, returning to Eq.(36), we note that in the ?uctuating twist ensemble34 for the XY model, in which ?? is averaged over as a thermally ?uctuating degree of freedom, the corresponding loop model obeys the additional constraint of zero winding, W? = 0, in each individual con?guration. When viewing Hloop as the Hamiltonian of vortex loops in a strongly screened superconductor, this corresponds to the ensemble in which the average internal magnetic ?eld is constrained to vanish, B? = 0, in each con?guration.


For reviews see, G. Blatter, M. V. Feigel’man, V. B. Geshkenbein, A. I. Larkin and V. M. Vinokur, Rev. Mod. Phys. 66, 1125 (1994); E. H. Brandt, Rep. Prog. Phys. 58, 1465 (1995); T. Nattermann and S. Scheidl, Adv. Phys. 49, 607 (2000).


D. R. Nelson and V. M. Vinokur, Phys. Rev. Lett. 68, 2398 (1992); Phys. Rev. B 48, 13060 (1993); ibid 61, 5917 (2000).

3 4 5 6

J. Lidmar and M. Wallin, Europhys. Lett. 47, 494 (1999). A. Vestergren, J. Lidmar and M. Wallin, Phys. Rev. B 67, 92501 (2003). M. Wallin, E. S. S?renson, S. M. Girvin and A. P. Young, Phys. Rev. B 49, 12115 (1994). J. T. Chayes, L. Chayes, D. S. Fisher and T. Spencer, Phys. Rev. Lett. 57, 2999 (1986); Commun. Math. Phys. 120 501 (1989).

7 8 9 10 11

A. B. Harris, J. Phys. C 7, 1671 (1974). C. Dasgupta and B. I. Halperin, Phys. Rev. Lett. 47, 1556 (1981). M.-C. Cha and S. M. Girvin, Phys. Rev. B 49, 9749 (1994). N. Prokof’ev and B. Svistunov, Phys. Rev. Lett. 92, 15703 (2004). D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner and P. Zoller, Phys. Rev. Lett. 81, 3108 (1998); B. Damski, J. Zakrzewski, L. Santos, P. Zoller and M. Lewenstein, Phys. Rev. Lett. 91, 080403 (2003).

12 13

S. Sachdev, Quantum Phase Transitions, (Cambridge, 1999). Y.-H. Li and S. Teitel, Phys. Rev. Lett. 66, 3301 (1991); Phys. Rev. B 47, 359 (1993); T. Chen and S. Teitel, Phys. Rev. B 55, 15197 (1997).


D. S. Fisher, M. P. A. Fisher and D. A. Huse, Phys. Rev. B 43, 130 (1991).


15 16

J. Lidmar, M. Wallin, C. Wengel, S. M. Girvin and A. P. Young, Phys. Rev. B 58, 2827 (1998). H. Kleinert, Gauge Fields in Condensed Matter, (World Scienti?c, Singapore, 1989), Vol. 1; T. Banks, R. J. Myerson and J. Kogut, Nucl. Phys. B 129, 493 (1977).

17 18 19

R. Savit, Phys. Rev. B 17, 1340 (1978). J. Villain, J. Phys. Paris 36, 581 (1975). H. Rieger and A. P. Young, Phys. Rev. Lett. 72, 4141 (1994); M. Guo, R. N. Bhatt and D. A. Huse, Phys. Rev. Lett. 72, 4137 (1994).

20 21 22 23 24 25 26 27

M. P. Nightingale and H. W. Bl¨te, Phys. Rev. Lett. 60, 1562 (1988). o K. Binder, Phys. Rev. Lett. 47, 693 (1981). T. Olson and A. P. Young, Phys. Rev. B 61, 12467 (2000). J. D. Reger, T. A. Tokuyasu, A. P. Young and M. P. A. Fisher, Phys. Rev. B 44, 7147 (1991). U. Wol?, Phys. Rev. Lett. 62, 361 (1989). D. A. Huse, private communication. P. C. Hohenberg and B. I. Halperin, Rev. Mod. Phys. 49, 435 (1977). L. M. Jensen, B. J. Kim and P. Minnhagen, Phys. Rev. B 61, 15412 (2000) and Europhys. Lett. 49, 644 (2000).

28 29

P. Minnhagen, B. J. Kim and H. Weber, Phys. Rev. Lett. 87, 37002 (2001). A. P. Young, in Random Magnetism, High-Temperature Superconductivity”, eds. W. P. Beyerman, N. L. Huang-Liu and D. E. MacLaughlin (World Scienti?c, Singapore, 1994).


G. Nakielski, A. Rickertsen, T. Steinborn, J. Wiesner, G. Wirth, A. G. M. Jansen and J. K¨tzler, Phys. Rev. Lett. 76, 2567 (1996); J. K¨tzler, in Advances in Solid State Physics 39, ed. o o B. Kramer, (F. Vieweg & Sohn Verlags-GmbH, Braunschweig/Wiesbaden, 1999), p. 371 (also as cond-mat/9904279).

31 32 33 34

Due to di?erences of notation, the exponents ν and z in Ref. 30 correspond to our νζ, and z/ζ. H. Weber and H. J. Jensen, Phys. Rev. Lett. 78, 2620 (1997). K. H. Lee and D. Stroud, Phys. Rev. B 46, 5699 (1992). P. Olsson, Phys. Rev. Lett. 73, 3339 (1994) and Phys. Rev. B 52, 4511 (1995).



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