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Phase Transitions in the Two-Dimensional Random Gauge XY Model Petter Holme∗ Department of Physics, Ume˚a University, 901 87 Ume˚a, Sweden Beom Jun Kim† 3 Department of Molecular Science and Technology, Ajou University, Suwon 442-749, Korea 0 0 2 Petter Minnhagen‡ NORDITA, Blegdamsvej 17, DK-2100, Copenhagen, Denmark and n Department of Physics, Ume˚a University, 901 87 Ume˚a, Sweden a J The two-dimensional random gauge XY model, where the quenched random variables are mag- 6 netic bond angles uniformly distributed within [−rπ,rπ] (0 ≤ r ≤ 1), is studied via Monte Carlo 1 simulations. We investigate the phase diagram in the plane of the temperature T and the disorder strengthr,andinfer, incontrasttoaprevailingconclusion inmanyearlier studies,thatthesystem ] n issuperconductingatanydisorderstrengthrforsufficientlylowT. Itisalsoarguedthatthesuper- o conducting to normal transition has different nature at weak disorder and strong disorder: termed c Kosterlitz-Thouless (KT) type and non-KT type, respectively. The results are compared to earlier - works. r p u PACSnumbers: 64.60.Cn,64.70.Pf,74.60.Ge,74.76.Bz s . at I. INTRODUCTION m 0.89 N - The XY gaugeglassmodel1 has attractedmuchinter- d estinconnectionto the vortexglassphase ofhigh-T su- n c perconductors2. Inthreedimensions (3D)thereis agen- o c eral consensus that the XY gauge glass model exhibits T [ a finite-temperature glass transition1,3,4,5. However, in 2D there exist conflicting evidences: On the one hand, ~0.2 1 equilibrium studies of defect energy3,5,6,7, Monte Carlo SI SII v 9 (MC) simulations of the root-mean-squarecurrent8, and 0 7 resistance calculation9 have suggested that no finite-T 0 ~0.4 r 1 2 ordering exists. The glass order parameter has further- 1 morebeenanalyticallyshowntovanishatanyfiniteT10. FIG.1: Sketchofthephasediagram in thedisorderstrength 0 On the other hand, the MC studies of the glass suscep- r and thetemperatureT (inunitsofJ/kB with thecoupling 3 tibility11, as well as dynamical simulations of the resis- strengthJ)plane. Thereexisttwosuperconductingphasesat at/0 tcaantecde1a2,1p3osasnibdilnitoyno-efqauifilinbirtieu-mT trrealnaxsiattioionn.14A,lshoavtheeinMdCi- ltTohhweedtheismigtihpnectrteamptuhpraeesrsea:t“tuShrIeeI”lnoofwor-rmTsamlKapTlhlaprsheaanissdemmlaarargrkekeerdd,abrseys“p“NeSc”It”.ivTaenhlyde. m simulations of the helicity modulus in Ref.15 were inter- solid phase boundary represents the boundary of the low-T - preted as being compatible with a finite-T transition. KT phase whereas the dashed represents the normal to su- d In this paper we study a generalization of the 2D XY perconductingtransition for larger r. n gaugeglassmodel—therandomgaugeXY model—where o both the temperature T and disorder strength r can be c varied. Whenr hasthemaximumvalue1,itcorresponds : v totheusualXY gaugeglassmodel,whileintheopposite tions to study the phase transition of the random gauge Xi limit of r = 0, the standard XY model without disor- XY model. It is found that the system is superconduct- der is recovered15,16,17. It has been proposed that there ingatanyrandthatthetransitionfromnormaltosuper- r a is a Kosterlitz-Thouless (KT) like transition at a finite conductingphaseisconsistentwithaKosterlitz-Thouless temperatureT whenthe disorderstrengthissufficiently (KT)typeatweakdisorderandanon-KTtypeatstrong c small, and that as r is increased T becomes smaller un- disorder. In addition to this we suggest that there ex- c til it vanishes as r reaches the critical disorder strength ist two different superconducting phases at sufficiently r (< 1) and T = 0 for r > r 16,17. However, even if low temperatures separated by a non-KT phase transi- c c c the glass order parameter is zero and even if there is no tion (see Fig. 1). For the special case of r = 1, which finite-T KTtransitionforr>r ,theexistenceofafinite- corresponds to the 2D XY gauge glass model, we also c T transitionwithadifferentcharactercannota priori be compute the root-mean-square current in the same way ruled out. as in Ref.8 and, in contrast to Ref.8, again consistently Inthe presentpaperweperformextensiveMCsimula- find a finite-temperature transition. 2 II. MODEL AND SIMULATIONS 0.7 0.6 T =0:05 The Hamiltonian of the 2D random gauge XY model 0.5 on an L×L square lattice under the fluctuating twist 0.4 boundary condition (FTBC)19 is given by T =0:15 (cid:7)℄ 0.3 [ 1 Hˆ =−J cos φ ≡θ −θ − r ·∆−A , (1) ij i j L ij ij 0.2 T =0:25 Xhiji (cid:18) (cid:19) (a) where J is the coupling strength (set to unity from now on), the sum is over nearest neighbor pairs, rij ≡ xˆ T =0:35 (yˆ) if j = i+xˆ(yˆ). The phase angle θi at the lattice 0.1 4 5 6 7 8 910 20 30 40 point i satisfies the periodicity θ = θ = θ , and i+Lxˆ i+Lyˆ i L A ∈ r[−π,π] is a uniform quenched random variable ij with the disorder strength 0≤r ≤1. The twist variable 1.6 (cid:23) =1:1 4(cid:2)4 ∆=(∆x,∆y) corresponds to the globaltwist acrossthe 1.4 b=0:4 8(cid:2)8 system, i.e., the summation of the gauge invariantphase T =0:19 16(cid:2)16 difference φ along the x (y) direction equals ∆ (∆ ). 1.2 24(cid:2)24 ij x y Foragivendisorderrealization,wefirstcomputethedis- b L 1.0 (T(cid:0)T )L1=(cid:23) tribution P(∆), which is related to the free energy F by (cid:7)℄ 0.8 1.4 [ m∂Fin/i∂m∆ize=sF−T(o(r∂mlnaPx/im∂∆ize)s.PT)heistdweitsetrvmairniaedblfero∆m0Pwhainchd 0.6 1.2 b(cid:7)℄L then fixed when the helicity modulus Υ≡∂2F/∂∆2 and 0.4 1.0 [ the 4th order modulus Υ4 ≡∂4F/∂∆4 are computed. 0.2 0.8 To ensure that the cooling is slow enough, we simul- 0.6 (b) 0.0 -1 0 1 taneously cool two replicas (α and β) of the system and 0.0 0.1 0.2 0.3 0.4 measure ∆α,β. For the first cooling at a new temper- 0 T ature we use 120000 update sweeps (for spin and twist variables respectively). Then we check that FIG. 2: (a) The size dependence of the helicity modulus [Υ] at different temperatures T for the fully disordered case |∆α −∆β |<δ and |∆α −∆β |<δ (2) (r = 1)—the gauge glass model. At high T, [Υ] is shown to x,0 x,0 y,0 y,0 vanishasLisincreased,whereasitsaturatestoafinitevalue atlowT (qualitativelygivenbythedashedlinesarelinearex- where δ sets the precision of the cooling. The idea with trapolations from thesmallest sizes and correspond to power theannealingconditionEq.(2)istokeepthesystemclose law behavior)(b)Finite-size scaling of [Υ]of thedatain (a). tothelowest-energystateataparticulartemperature(so From the standard finite-size scaling form, the critical tem- thesystemdoesnotfreezeintoalocalminimumthatbias perature Tc ≈ 0.19 and the critical exponent ν ≈ 1.1 are [Υ] and [Υ4]). Since the system moves more swiftly over determined. the configurationspace the higher the temperature is we canchoose δ increasing with temperature. A choice that proves good in practice is compute the root-mean-square current defined by8 0.02π for T <0.3 δ = 0.15π for 0.3≤T <0.6 (3)   ∞ for T ≥0.6 2 1/2 1 I ≡ sinφ , (4) IftheannealingconditionEq.(2)failwerepeatthecool- rms *L ij+  ing with three times as many update sweeps; if it fails hXijix   again we increase the number of cooling sweeps a fac- tor three again, and so on until the condition is fulfilled. for comparisons with earlier works. When∆ ischosen,beforecooling,weletthesystemrun 0 until Eq. (2) is fulfilled with ∆ replaced by ∆. 0 ForI ,foreachdisorder,weuse50000updatesweeps rms We repeat the above calculations for more than 500 forthermalizationsand500000sweepsformeasurements differentdisorderrealizations(thedisorderaverageisde- wheretheactualmeasurementwasperformedeverytenth noted by [···] throughout this paper). sweep. Weusedthiscomputertimesavingstrategysince Forthecaseofr =1,wealsousetheperiodicboundary we found that performing the measurement every sweep condition PBC, corresponding to ∆ = 0 in Eq. (1), and gave closely the same result. 3 where [Υ]∼L−b at the critical temperature T , and the c 1.4 4(cid:2)4 Ref. 8 critical exponent ν is related to the divergence of the (a) 8(cid:2)8 Ref. 8 coherence length. At T , the scaling function f has the 1.2 16(cid:2)16 Ref. 8 same value irrespectivecof L, implying that Lb[Υ] versus 4(cid:2)4 =(cid:23) 1.0 6(cid:2)6 T should have a unique crossing point at Tc for various 1 L 8(cid:2)8 sizes. InFig.2(b),itisclearlyshownthat[Υ]hasascale- ms 0.8 10(cid:2)10 invariant behavior [Υ] ∝ L−0.4 at a unique T, signaling Ir 0.6 16(cid:2)16 a phase transition. The inset of Fig. 2(b) furthermore 12(cid:2)12 confirmsthat this scalingbehavioris consistentwith the 0.4 standard form (5) with ν ≈1.1. Simple dimensional analysis for the non-disordered 0.2 case, r = 0, gives for [Υ] the exponent b = 0 and from 0.0 such a dimensional analysis one would likewise conclude 0.0 0.5 1.0 1.5 2.0 2.5 thatb=0alsoforthedisorderedcase. Thusif[Υ]scales 1=(cid:23) T L withb6=0forthedisorderedcasethisisequivalenttothe 1.8 appearanceofananomalousdimensionnotaccountedfor 4(cid:2)4 bysimpledimensionalanalysis. Oursuggestion,basedon 1.6 (b) 6(cid:2)6 8(cid:2)8 the simulation results, is consequently that the disorder 1.4 10(cid:2)10 introduces such an anomalous dimension. Irms 11..02 (T(cid:0)T )L1=(cid:23) 1126(cid:2)(cid:2)1126 theFigPuBrCe.3Wsheowfisrstthenortoeoti-nmeFaing.-sq3u(aa)rethcuartretnhte(fi4)niftoer- b L 0.8 1.5 ms size-scaling form for Tc = 0 used in Ref.8, Irms = Ir L−1/νf(TL1/ν) with ν = 2.2, shows systematic devia- 0.6 bL 1.0 tions from the data collapse to a single scaling curve at 0.4 lowertemperatures,incontrasttowhatwasconcludedin 0.2 0.5 Ref.8,whenmoreandbetterconvergeddataareincluded. -1 0 1 Furthermore a T = 0-collapse cannot be achieved with 0.0 c 0.0 0.1 0.2 0.3 0.4 any value of ν. On the other handif we, in analogywith T the finding for [Υ] above, use the scaling form sFiIzGe-.sc3a:linTgheplroototu-smedeainn-sRqueaf.r8efcourrrTecnt=Ir0mss.ho(wa)s Tsyhsetefimnaitteic- Irms =L−bf L1/ν(T −Tc) , (6) deviations in the low-temperature region (ν = 2.2 in Ref.8 (cid:16) (cid:17) is used). We include the data in Ref.8 for comparisons. (b) whichallows for the same anomalousdimension, one ob- Finite-sizescalingforminEq.(6)yieldsTc ≈0.2andb≈0.5. tains the scaling plot in Fig. 3(b) with b ≈ 0.5 and Inset: Alldatapointsinthemainpanelcollapsetoasmooth T ≈0.2.18 From Figs. 2 and 3 we conclude that the 2D c curve with ν = 1.1, the same value as found for the helicity XY gauge glass (r = 1) exhibits a non-KT type super- modulus in Fig. 2. conductingtonormaltransitionatT ≈0.2characterized c bythe existenceofthe anomalousdimensionb≈0.5and ν ≈1.1. III. RESULTS FROM SIMULATIONS Next we investigate the phase transition with decreas- ing r. For r = 0.9,0.8,···,0.5, we obtain the finite-size scaling plots of the same quality as in Fig. 2(b), and de- We first investigate the standard 2D XY gauge glass termine the phase boundary (the dashed line in Fig. 1). model,correspondingtothefullydisorderedcase(r =1), As r is changed from 0.5 to 0.4 the exponents b and ν andshowin Fig.2(a) the helicity modulus [Υ]asa func- exhibit quite rapid changes, b from 0.27 to 0.06 and ν tion of the system size L for various temperatures. It from 1.1 to 2.0. This, in our interpretation, reflects that is clearly shown that at high T, [Υ] goes to zero as the near r = 0.4 the nature of the superconducting-normal system size L is increased. The crucial point is that, at transition changes from a non-KT type to a KT type. low enough temperatures (T <∼0.2), [Υ] changes its cur- Forr=0the phasetransitionisofthe KTnatureand vature in terms of L, and appears to saturate to a finite it has been suggested that this character should persist value as L is increased. This behavior suggests a phase along the phase boundary up to some r < 116,17. The c transition with a scale-invariant power law dependence KT transition is characterized by that [Υ] jumps from atthecriticaltemperatureandadivergingcharacteristic a finite value [Υ] = 2T/π to zero as the phase line is length. Such a behavior can often be described by the crossed from the small (T,r) region. For r = 0 we find finite-size scaling form that the KT transition is also characterized by the increase of the 4th order modulus |[Υ ]| ∝ Lc at T with 4 c a positive exponent c, and [Υ ] stays at a constantvalue 4 [Υ]=L−bf L1/ν(T −Tc) , (5) below Tc as L is increased20. Figure 4 (a) shows [Υ4] (cid:16) (cid:17) 4 0 concluded that such a phases line does indeed exist but separates a superconducting phase from a normal phase allthewaydowntoT =0. Oursuggestedphasediagram -5 r is thus consistentwith earlierworkasto the existence of 0.8 (a) a phase line ending at (T = 0, r ≈ 0.4). For tempera- c ℄4 tures largerthan the mergingwith the secondphase line (cid:7) 0.6 [ -10 (above dashed line in Fig. 1) the transition is consistent 4(cid:2)4 (cid:7)℄ with a KT transition although a different character can- 0.4 [ 6(cid:2)6 not be ruled out. However, below the merging with this -15 0.2 8(cid:2)8 second line the transition is not a KT transition. It may 16(cid:2)16 be that there still is a jump in [Υ] at this transition, but 0.0 this is then between two non-vanishing values. 0.0 0.5 1.0 -20 0.0 0.1 0.2 0.3 0.4 Basedonthenumericalevidenceswesuggestthestruc- r tureofthephasediagramissketchedinFig.1. Onestrik- ing feature is the finite-T transition line between normal 0 andsuperconductingphases,startingfromtheboundary -2 ofthe low-T KT-phase(solidline)andchanginginto the r (dashed) line which ends at Tc ≈ 0.2 at r = 1. The lat- -4 (b) ter line (dashed line) is characterized by the appearance ℄4 0.8 of an anomalous dimension b. Since the KT transition (cid:7) [ -6 0.6 4(cid:2)4 does not have such an anomalous dimension it follows (cid:7)℄ that, if the transition along the boundary to the low-T -8 0.4 [ 6(cid:2)6 KTphaseboundaryhasKTcharacter,thenbshouldap- 8(cid:2)8 proach b = 0 when the two transition line merge. Thus 0.2 -10 the rapid drop from b≈0.27 to b≈0.06 as r is changed 16(cid:2)16 0.0 from r = 0.5 to 0.4 is consistent with a change over to 0.0 0.5 1.0 -12 a KT transition. Another interesting feature is that the 0.0 0.1 0.2 0.3 0.4 phaseline(solidlineinFig.1),associatedwiththediver- r gence of [Υ ], continues even inside the superconducting 4 regionandendsatr ≈0.4forT =0. Althoughthetran- FIG. 4: The fourth order modulus and helicity modulus (in- sitionseparatingtheKTphaseandthenormalphase(SI set) as a functions of r for temperatures (a) 0.6 and (b) 0.1. and N, respectively, in Fig. 1), may well be of the true Thesolid linein theinset representstheuniversal-jump con- KT type, manifested by a universal jump in the helicity dition for a KT transition [Υ]=2T/π. modulus, the phase line separating SI and SII in Fig. 1 is not a KT transition and the helicity modulus has a non-zero value on both sides. for T =0.6 as a function of r. The KT condition in the inset gives r ≈ 0.4. This means that if the transition is It is interesting to note that earlier works have found of KT type then it would occur aroundr ≈0.4. The on- evidencesforonlytwophasesseparatedbya single phase set of size dependence for [Υ ] in Fig. 4 (a) is consistent line;either,inthemoreprevailingview,aphaselineend- 4 with a transition at around r <∼0.4. Thus the transition ing at a point T = 0 for a finite rc < 1, or, in the less is compatible with a KT-character, although a different prevailing view, a phase line ending at a point T >0 for charactercannotbe ruledout. The situationfor T =0.1 r =1. ¿From our numerical simulations we instead sug- in Fig. 4 (b) is quite different: the inset shows that the gest three distinct phases separated by two phase lines, KT jump condition is not fulfilled and a KT transition which combines the two earlier proposed scenarios and can be ruled out. Yet there is a marked structure in [Υ] provides a unified picture: On the one hand, in Ref.5, around r ≈0.4. This structure corresponds to the onset the end point of the phase boundary to the low-T KT- ofstrongsizedependencein[Υ ]. Weinterpretthisonset phase has been obtained to be r ≈ 0.37 and T = 0, 4 c as the reflection of a true divergence in [Υ ] consistent whichisconsistentwithourcruderestimate r ≈0.4. On 4 with a phase transition. Thus we suggestthat the whole the other hand, the phase transition point at T ≈ 0.22 boundary to the low-T KT phase (solid line in Fig. 1) is with ν =1.1, found in Refs.11 and13 for r=1, is in very reflected in a divergence of [Υ ] and that this line ends goodagreementwith the end point of our finite-T phase 4 at (T = 0,r ≈0.4). A phase line which ends at (T =0, line (T ≈0.2, ν ≈1.1). However,the difference with the c r ≈ 0.4) has been found in many earlier investigations previouswork5isthataccordingtoourinterpretationthe c (see e.g. Ref.16 and references therein)21. The differ- phase line below T ≈ 0.2 ending at T = 0 and r ≈ 0.4 c ence with earlier work is that in our case such a phase separatestwodistinct superconductingphases,while the line is for lower temperatures between the two different whole phase line in Ref.5 is for the superconducting to superconducting phases SI and SII (as demonstrated by normal transition. A very hand-waving picture of the our direct calculation of [Υ]), whereas earlier work have scenario of our phase diagramin terms of vortex motion 5 is sketched in Fig. 1: In SI the vortex motion is sup- is the T =0 calculations of the size scaling of the defect pressed by vortex pair binding, in N pair unbinding has energy3,5,6,7. However, as discussed in Refs.7 and23, the occurred and free vortices exists which are not entirely local vorticity conservation must be properly taken into pinned by the disorder, whereas in SII vortex pair un- account when calculating the energy barriers. Thus the binding has occurred but the vortices are pinned by the energy barrier for vortex dissipation increases with sys- disorder. temsizewhentakingthelocalvorticityconservationinto account23. This growing of the energy barrier for vortex dissipation with system size supports the possibility of a IV. SUMMARY finite-T transition23. The appearance of a new superconducting phase for One main conclusion from this phase diagram is that theXY randomgaugemodelisintriguing. Inparticular, the XY gauge glass model (r = 1) has a finite-T tran- since it is neither a low T KT phase nor a phase with sition. How is this possible in view of earlier conflicting a finite glass order parameter. The true nature of this evidences? In Ref.8 it was concluded, on the basis of phaseandtheexistenceofsimilarphasesinotherrelated an analysis of data for the root-mean-square current for models are open questions. standard periodic boundary conditions22, that no finite- Support from the Swedish Natural Research Council T transitionexistsin2D.Wehavefoundthat,takinginto through Contract No. F 5102-659/2001 is gratefully ac- account the possibility of an anomalous scaling dimen- knowledged. B.J.K.wassupportedby the KoreaScience sion, I displays a transition at a finite-temperature and Engineering Foundation through Grant No. R14- rms (seeFig.3)20. Anotherpuzzlingevidencetothecontrary 2002-062-01000-0. ∗ Electronic address: [email protected] 13 B. J. Kim, Phys.Rev.B 62, 644 (2000). † Electronic address: [email protected] 14 B. J. Kim et al.,Phys. Rev.B 56, 6007 (1997). ‡ Electronic address: [email protected] 15 G. S. Jeon, S. Kim, and M. Y. Choi, Phys. Rev. B 51, 1 D.A.HuseandH.S.Seung,Phys.Rev.B42,1059(1990). 16211 (1995). 2 M. P. A. Fisher, Phys. Rev. Lett. 62, 1415 (1989); 16 J. Maucourt and D. R. Grempel, Phys. Rev. B 56, 2572 D. S. Fisher, M. P. A. Fisher, and D. A. Huse, Phys. Rev (1997). B 43, 130 (1991). 17 M.Rubinstein,B.Shraiman,andD.R.Nelson,Phys.Rev. 3 M. Cieplak, J. R. Banavar, and A. Khurana, J. Phys. A B 27 1800 (1983); T. Natterman et al., J. Phys. (France) 3, L145 (1991); M. Cieplak et al., Phys. Rev. B 45, 786 I 5, 565 (1995); M.-C. Cha and H. A. 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