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NORDITA-2002-60 HE, HIP 2002-39/TH The inverted XY universality of the superconductivity phase transition T. Neuhausa,A.Rajantieb and K. Rummukainenc
aFinkenweg 15, D-33824 Werther, Germany bDAMTP, CMS, University of Cambridge, Cambridge CB3 0WA, United Kingdom cNORDITA, Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark and Department of Physics, P.O.Box 64, 00014 University of Helsinki, Finland
It has been conjectured that the phase transition in the Ginzburg-Landau theory is dual to the XY model transition. We study numerically a particular limit of the GL theory where this duality becomes exact, clarifying some of the problems encountered in standard GL theory simulations. This may also explain the failure of the superconductor experiments to observe the XY model scaling.
1. INTRODUCTION One reason for the confusing results both in ex- periments and numerical simulations is that the The (3-d) U(1) gauge + Higgs (Ginzburg- duality is expected to be valid only in a very nar- Landau, GL) theory is an effective theory for the row temperature range around the critical tem- superconductor-insulator phase transition. De- perature, requiring extreme precision and large spite the formal simplicity of the GL theory, and volumes in the measurements. Moreover, the du- numerous analytical and numerical studies, the ality relates “simple” observables (like the field universal properties of the transition have not expectation value in the XY model) to non-local yet been fully resolved. Theoretical duality argu- observables in the dual theory (vortex network in ments [1] suggest that the phase transition is in the GL theory). This makes it very difficult to the 3d XY model universality class, but with an know whether the problems are caused by insuffi- inverted temperature axis. Thus, the symmetric cient resolution near the critical temperature, or and broken phases in the XY model correspond by the difficult nature of the observables them- to the insulator and superconducting phases of selves. the superconductor. In order to gain insight into this problem we The duality gives concrete predictions for the study a special limit of the GL theory, the frozen behaviour of several critical observables in super- superconductor (FZS) (an integer-valued gauge conductors. For example, the Abrikosov vortex theory), which is exactly dual to the XY-Villain tension should scale with the XY model expo- model at all temperatures. Thus, the transition T 1 nent νXY =0.6723 . Experimentally, it is easier in FZS is bound to be in the XY model universal- to access the penetration depth λ (or the inverse ity class. Studying the critical quantities of FZS photon mass), which is also argued to scale with can shed light on the problems faced in both the the XY exponent ν0 = νXY. GL theory simulations and superconductor exper- However, the XY universality has not been iments. Detailed results are published in [6]. unambiguously observed. Two different high-Tc Our starting point is the lattice GL model in YBa2Cu3O7 δ experiments [3] report ν0 in the the London limit: range 0.3−...0.5. Monte Carlo simulations of ≈ 2 1 2 the GL model favour ν0 0.3[4]. = F +κ s (θ θ qA ) .(1) ∼ LGL 2 ~x,ij ~x+i − ~x − ~x,i Xi Figure 1. Vortex tension in FZS (left) and the scalar mass in the XY model (right), plotted against β =1/κXY. The continuous lines show power law fits, and, for comparison, the dashed lines show the fits transferred from the other plot. Here Ax,i is a real-valued gauge field, θx is the 2. CRITICAL OBSERVABLES spin angle variable, and F~x,ij is the non-compact plaquette. We use the Villain form hopping term The XY-Villain model has a symmetry break- ing transition at κ = κ 0.333068(7) [6]. Be- c ≈ ∞ 1 cause of the exact duality, FZS must have a tran- s(x)= ln exp (x 2πk)2 . (2) − − 2 − sition at βc =1/κc, which is of XY model univer- k=X −∞ sality, and the phases are related as follows: We shall study the following 2 limiting cases of XY model: FZS: the GL model: ↔ symmetric κ<κc superconducting β>βc i) Let and define =4 2 2.Now ↔ κ β π /q broken κ>κc Coulomb β<βc the GL model→∞ becomes the Frozen Superconduc- ↔ tor (FZS): Vortex tension The duality implies that the XY model scalar β 2 correlation function equals the FZS Abrikosov- ZFZS(β)= exp 2~x,ij . (3) − 2 Nielsen-Olesen “vortex operator.” In particular, XI~x,i ~x,i>jX { } the XY model scalar mass m equals the ten- Here 2~x,ij = I~x,i + I~x+i,j I~x+j,i I~x,j,andthe sion of the vortex line between a monopole- − − T link variables I~x,i take integer values. antimonopole pair in FZS. Thus, in the symmet- ii) Let q 0, and the GL model becomes the ric/superconducting phase of the XY/FZS model XY model with→ the Villain action we should find m(κ)= (1/κ) κ κ νXY , (6) ZXY(κ)= Dθ exp κ s(θ~x+i θ~x) . (4) T ∝| − c| Z − X − ~x,i and in the broken/Coulomb phases m = =0. T The frozen superconductor and the XY-Villain This is indeed the case: in Fig. 1 we show the model are exactly dual to each other with the results for and m. As predicted by duality, T identification β =1/κ, i.e. = m within the statistical errors, and the ten- sionT critical exponent is 0.672(9), compatible with ZXY(κ)=ZFZS(β =1/κ). (5) νXY. This relation is valid at infinite volume; on a finite Photon mass volume there are corrections proportional to the The most natural observable for the FZS model surface area of the volume. For a proof of this (and the one usually measured in high-Tc su- relation see [6]. perconductor experiments) is the photon mass 3 3. CONCLUSIONS Because of the exact duality between the frozen superconductor (FZS) and the XY model, the critical observables in FZS must behave accord- ing to the corresponding XY model critical expo- nents. We have measured several observables in the frozen superconductor, and we duly find the behaviour predicted by the duality (for quantities other than reported here, see [6]). The sole ex- ception is the photon mass mγ, which appears to scale with an exponent substantially smaller than the XY model prediction. Exactly analogous be- Figure 2. Photon mass measured from the FZS haviour has been observed superconductor exper- model in close proximity to the critical point. The iments and in GL theory simulations. continuous line is a power law fit, and the dashed Since the observed behaviour of m differs from line shows the curve 2 . γ ×T predictions even when the duality is exact, the similar discrepancies observed in other studies do not mean that the duality hypothesis is not valid mγ =1/λ, the inverse of the penetration depth. The duality relates λ to the correlation length of in real superconductors. The apparent incorrect the Noether current operator in the XY-Villain scaling behaviour may be due to the large anoma- model. Parametrizing the critical behaviour of lous dimension ηA of the photon propagator. ν0 mγ with the exponent ν0,wehavemγ β βc in the superconducting phase. The∝| theoretical− | REFERENCES prediction is ν0 = νXY [7]. Fig. 2 shows the mγ values measured from the 1. M. E. Peskin, Annals Phys. 113 (1978) 122; plaquette correlation functions in FZS. A power C. Dasgupta and B.I. Halperin, Phys. 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