Glassy Vortex State in a Two-Dimensional Disordered XY-Model Stefan Scheidl Universität zu Köln, Institut fu¨r Theoretische Physik, Zu¨lpicher Str. 77, D-50937 Köln, Germany (February 1, 2008) The two-dimensional XY-model with random phase-shifts on bonds is studied. The analysis is basedonarenormalization groupforthereplicatedsystem. Themodelisshowntohaveanordered phase with quasi long-range order. This ordered phase consists of a glass-like region at lower temperaturesandofanon-glassyregionathighertemperatures. Thetransitionfromthedisordered phaseintotheorderedphaseisnotreentrantandisofanewuniversalityclassatzerotemperature. Incontrasttopreviousapproachesthedisorderstrengthisfoundtoberenormalizedtolargervalues. Several correlation functions are calculated for the ordered phase. They allow to identify not only 6 the transition into the glassy phase but also an additional crossover line, where the disconnected 9 vortex correlation changes its behavior on large scales non-analytically. The renormalization group 9 approach yields theglassy features without a breakingof replica symmetry. 1 PACS: 64.40; 74.40; 75.10 n a J I. INTRODUCTION cused on the absence of reentrance. The purpose of this 6 article is to examine in more detail the ordered phase, 2 The two-dimensional XY-model with random phase- whichisshowntobecomposedofdifferentregions. They 1 shift, defined in Eq. (1) below, captures the physics are distinguished by the behavior of vortex correlation v of a variety of physical systems. Among them are functions on large length scales, which are calculated 1 XY-magnets with non-magnetic impurities, which are quantitatively. A low-temperature regime is found to ex- 3 coupled to the XY-spins via the Dzyaloshinskii-Moriya hibit glassy behavior in its correlations. 1 1 interaction,1 Josephson-junction arrays with geometric In addition to the previous studies we find that the 0 disorder,2 and crystals with quenched impurities.3 Dis- strength of disorder is increased under renormalization. 6 ordered XY-models are related to even more physical This effect does not destroy the ordered phase but it is 9 systems, like impure superconductors. It is in partic- crucial for the universality type of the transition at low t/ ular the interest in vortex-glass phases in type-II su- temperatures. a perconductors, which motivates an analysis of possi- In the following a generalized self-consistent screening m ble glassy features in the paradigmatic two-dimensional approach is applied to the replicated system. The main - bond-disordered system. improvement compared to earlier approaches1,6,7 on the d n Themostfundamentalquestionis,whetherthismodel same basis is achieved by including all relevant fluctu- o has an ordered phase. In the absence of disorder, there ations and by taking special care when the number of c existsatlowtemperaturesaphasewithquasilong-range replicas is sent to zero. v: order. The famous Kosterlitz-Thouless transition sep- Weusethereplicaformalismmainlyfortworeasons: i) i arates it from the high-temperature phase with short- The calculationof correlationsbecomes very convenient: X ranged order.4,5 The analysis by Cardy and Ostlund6 one may use an expansion technique (small fugacity and r andmoreexplicitlybyRubinstein,ShraimanandNelson1 density of vortices) which fails11–13 in the unreplicated a predicted that the phase transition should be reentrant system. ii)Weshowthatinthepresentmodelthereisno inthepresenceofdisorder. LaterKorshunovsuggested,7 need for replica symmetry breaking, even at lowest tem- thattheorderedphasemightbecompletelydestroyedby peratures. In particular, the reentrance disappears and disorder. Ozeki and Nishimori8 showed for models with the glassiness occurs without a breaking of replica sym- gauge-invariant disorder distributions, that the phase metry. This might astonish, since for various disordered transitioncannotbe reentrant,leavingopenwhetherthe systemsreentrancehasbeenexplainedasanartifactofa orderedphasedoesexistornot. Experiments9onJoseph- replica-symmetric(exactmean-fieldorapproximatevari- sonjunctionarraysandsimulations10wereinfavorofthe ational)approach,whichhasbeenovercomeby breaking existence of an ordered phase without reentrance. the replica symmetry.14 In this controversial situation Nattermann, Scheidl, The paper is organized as follows: In Sec. II we set Korshunov, and Li11 as well as Cha and Fertig12 recon- up our model and mapit onto an effective Coulombgas. sidered only recently the problem and found that the We recapitulate the breakdown of fugacity expansion in orderedphase does exist and its boundary does not have the presence of disorder. In Sec. III we set up the renor- reentrant shape. The earlier observation of reentrance malization group within a self-consistent formulation of was attributed to an overestimation of vortex fluctua- screening. The consistency of this approach in the orig- tions at low temperatures. inal and replicated system is demonstrated. Sec. IV is The present study extends Refs. 11 and 12, which fo- devoted to the evaluation of the self-consistency and the 1 proper performance of the replica limit. The results of is associated with plaquettes (dual sites), which are la- the renormalization group treatment are worked out in beledbyRincontrasttositesr. ThermalvorticesN cou- Sec. V, which are discussed and compared to the results pletoabackgroundofquenchedvorticesQR = 1 A. 2π∇× of previous work in Sec. VI. The last expression means, that the plaquette variable QR is given by the rotation of variable A on the sur- rounding bonds. From this relation one immediately de- II. THE MODEL rives, that the quenched vortices are Gaussian random variableswithvarianceQ−kQk = 4σπ2k2. Asinglephase- The XY-model with random phase-shifts is given by shift A creates a dipole of quenched vortices situ- ated in the plaquettes which have the bond < r,r′ > as the reduced Hamiltonian side. Thereforequenchedvorticesareanti-correlatedand their total vorticity vanishes. = K cos(θr θr′ A). (1) H − − − We recognize from Eq. (3a), that only neutral con- X figurations of vortices will have finite energy. Hence the It refers to a square lattice with unit spacing in two di- partitionsum canbe restrictedto neutral configurations mensions. An XY-spin with angle θr [ π,π[ is at- (i.e. RNR =0) like in the pure case. Everysuchneu- tached to every lattice site (position r)∈. T−he reduced tral configuration can be imagined as a superposition of P spin coupling between nearest neighbors < r,r′ > reads dipolesofavortex(N =1)andanantivortex(N = 1). − K = J/T. The interaction involves quenched random In analogy to electrostatics,vortices of the same sign re- pel each other and vortices of different sign attract each phase-shifts with variance A2 = σ, which are un- other. correlated on different bonds. Thermal fluctuations are Since the interaction is logarithmic, vortices build a weighted according to the partition sum Coulomb gas. The interaction is identical among the π thermalandquenchedcomponent. TheFourierrepresen- Z = θ e−H. (2) tation(3a)ofthevortexHamiltonianmakesevident,that D{ } Z−π thermal vortices try to compensate quenched vortices as wellas possible. A perfect compensationis preventedby ThisXY-modelcanbemappedapproximatelyontoan effectiveCoulombgas. Sincethisprocedureiswellknown the fact, that NR is restricted to integer values, whereas intheabsenceofdisorder15,16 andcanbeperformedsim- QR is a continuous random variable. ilarly in the presence of disorder, we recall only briefly In Eq. (3b) we introduced the vortex core energy the main manipulations leading to the effective model. E = πγK, where γ takes the value γ 1.6 on a square ≈ Inordertosimplifytheanalysisofmodel(2),onecanre- lattice. It emerges from the lattice Greens function [cf. placeapproximatelythecosine-interactionbytheVillain- Eq. (4.13a) of Ref. 16]. Although γ has a unique value interaction.15 Thereby an additionalvariable, the vortex for the lattice model, it will occasionally be considered density N is introduced. The decisive advantage of this asanindependent parameterwhichallowstocontrolthe approximationis,thatthe Hamiltonianbecomesbilinear density of vortices irrespectively of K. In a dilute vor- intheangles. Allconfigurationsoftheoriginalanglescan tex system with NR = 0, 1 only, E is comparable to a ± beexpressedintermsofspin-wavesandvortices. Justas chemical potential. in the pure case, spin-waves and vortices decouple en- In a region of parameters, where vortices are negligi- ergetically. Since spin-waves are simply harmonic, all ble, fluctuations of the original angle variables are given non-trivial physics arises from vortices with an effective only by spin waves. They lead to a decay of the spin Hamiltonian correlationfunction1,6 2π2K Γ(r)= cos[θ(r) θ(0)] r−η (5) Hv = k k2 |Nk−Qk|2 (3a) h − i∼ Z with exponent η = (1/K+σ)/2π. If the correlation de- =E (NR QR)2 (3b) cays algebraically even in the presence of vortices, the − − XR system has quasi long-range order. πKln R R′ (NR QR)(NR′ QR′), As vortices are excited thermally, they give rise to ad- − | − | − − R6=R′ ditional fluctuations in the angles, the correlation will X decayfaster. Thiseffectcanbeexpressedbyarenormal- in Fourier and real space, and the partition sum izedη,whichbecomesscaledependent. Ifvorticesleadto a divergence of the renormalized η on large scales, quasi Z = e−Hv. (4) long-range order is destroyed. However, vortex fluctu- v ations are not only responsible for such a quantitative {XN} effect, they also drive the phase transitioninto the high- Here NR is the integer-valued vortex density, which is temperature phase, where the spin-spin correlation de- subjecttothermalfluctuations. OnthesquarelatticeNR cays exponentially.4 2 Let us estimate, to what extent disorder can favor the It is convenient to introduce the ordinary, disconnected, excitationofvortices. InthepurecaseQ=0,theground and connected correlation, state of (3a) is vortex-free. The density of vortices vanishes for vanishing temperature, since the creation of C(R):= NRN0 , (8a) h i quantized vortices always costs finite energy. Therefore Cdis(R):= NR N0 , (8b) the pure system has a phase with quasilong-rangeorder C (R):=Ch (Ri)h Ci (R). (8c) con dis (5) at low temperatures. − A simple argument shows, that the disordered sys- For a specific realization of disorder, the screened inter- tem has even at zero temperature a finite density of action in principle depends explicitly on the coordinates vortices:11,12 we consider two positions at distance R of both interacting vortices, it is not translation invari- and determine the probability of finding there a vortex- ant. Although we should in principle consider screening antivortex pair. The energy of such a dipole is given by intheparticulardisorderenvironment,weapproximately Udipole(R) 2πJlnR, where we neglect the core energy evaluate only the average screening effect by taking the ≈ for large R. Disorder gives and additional energy contri- disorder average, which restores the translation symme- bution Vdipole(R). At T = 0, the dipole will be present, try of the interaction. ifitstotalenergyUdipole(R)+Vdipole(R)isnegative. The From the large-scale behavior of the screened interac- originalGaussiandistributionoftherandomphase-shifts tion and the variance of the screened potential one can resultsinaGaussiandistributionofVdipole(R)withvari- then identify screened parameters Kscr and σscr as (for ance ∆2(R):= V2 (R) 4πσJ2lnR. Thus the prob- the precise definition see Appendix A) dipole ≈ ability ∞ P(R)= −Udipole(R) dV e−V2/2∆2(R) Kscr =K+2π3K2Z1 dRR3Ccon(R), (9a) 2π∆(R) ∞ Z−∞ σscr(Kscr)2 =σK2 2π3K2 dRR3C (R)+ dis σ Rp−π/2σ (6) − Z1 ≈r2π2lnR +4π4σK3 ∞dRR3C (R). (9b) con of finding a pair at T = 0 is finite. In other words, this Z1 shows that in the presence of disorder the ground state Althoughwedonotyetknowthecorrelationfunctions, is no longer vortex-free and that renormalization effects we may establish from these expressions criteria for the canbeimportantevenatarbitrarilysmalltemperatures. existence of an ordered phase. In this phase, we expect vortices to give rise only to finite screening effects, i.e. the screened parameters must have finite values. This III. SCREENING requires lim R4C (R) < and a similar condi- R→∞ con ∞ tion for the disconnected correlation. The goalof the present work is therefore a calculation In the limit of large core energy, one can calculate the of such renormalization effects in the presence of ran- correlations to leading order by considering only a sin- dom phase-shifts. We use the conceptual most simple gle vortex dipole. In th absence of disorder, one finds approach,aself-consistentlinearresponseapproach. For C (R) = C(R) R−2πK and C (R) = 0. The con- con dis the pure Coulomb gas, this approach is equivalent17 to dition for the orde∼redphase then simply readsK 2/π, the real-space renormalization group of Kosterlitz.5 We in agreement with the condition that the free en≥ergy a havecheckedthatthis equivalencestillholdsinthepres- singlevortexhastobepositive.4 Asimilarargumentcan ence of disorder. beconstructedforthedisorderedsystematT =0,where We now define vortex renormalization effects by one can estimate C(R) p(R) R−π/2σ using Eq. (6). macroscopic properties of our system. As a probe we Then the condition for∼order re∼ads σ π/8. This ar- introduce additional test vortices into the system. They gument already disproves previous pred≤ictions,1,6,7 that experience a screenedinteractionandthe screenedback- infinitesimal σ >0 would destroy order at T =0. ground potential acting on them. Let us denote the Forfinitecoreenergy,screeningeffectshavetobetaken unscreened interaction between vortices Uk = 4π2Kk12. intoaccountquantitativelyforthecalculationofthe cor- Due to quenched disorder vortices Q, thermal vortices relations. For this purpose we develop a self-consistent N are subject to a Gaussian background potential Vk = scheme in analogy to the pure case.4,18 We introduce UkQk with variance V−kVk = 4π2σK2k12. The screened scale-dependentvariablesK K(l)andσ σ(l), which backgroundpotential and interactionare identified from include screening by dipoles≡of radius 1 ≡R < el only. thecontributionstothefreeenergyoftestvortices,which Then Eq. (9) can be cast in differential fo≤rm: are of first and second order in their vorticity18 (see also Appendix A for some intermediate steps) dK−1 = 2π3e4lC (R=el), (10a) con dl − VUkkssccrr ==UUkk(−QkUk−2ChcNonk(ik)),. ((77ba)) ddlσ =−2π3e4lCdis(R=el). (10b) 3 Both equations combine to a simple flow equation of the in analogyto Eq. (10). We introduced the replica corre- spin-spin correlation exponent, dη = π2e4lC(R=el). lation Cab(R):= NaNb , where ... denotes a ther- dl − h R 0in h in Eqs. (10) suggests a simple “two-component picture” mal average in the replicated system. Since the initial illustrating the screeningeffects ofvortices. This picture interactions are replica-symmetric, i.e. the coupling has virtually separates dipoles of thermal vortices N into a the form Kab = Kδab Kˆ, we use the replica symmet- − frozen and a polarizable component. The frozen compo- ric ansatz Cab = C(n)δab +C(n) for correlations. The con dis nent gives rise to the Cdis. It is not polarizable and does relation to the correlations in the unreplicated system is not contribute to screening of the interaction. However, provided by since it is frozen, it behaves like the quenched disorder background and leads to an effective disorder strength C (R)= lim C(n)(R) (14) X X represented by the screened σ. The polarizable compo- n→0 nent is the source of C and leads to the screening of con for all three types X (ordinary, disconnected, and con- K. Both components contribute equally to a renormal- nected)ofcorrelations,ifweidentifyC(n)(R):=Caa(R) izationofthespinexponent. Thisvirtualseparationinto consistently. two components must not be takenliterally. It is impos- Exploiting the replica symmetry of correlations, Eq. sible to assign a particular dipole uniquely to one of the (13) decays into flow equations for K and of Kˆ, which two components. The two-component picture is analo- read in terms of K and σ =Kˆ/(K2 nKKˆ): gous to the two-fluid model of superfluidity, which also − must not be takenliterally in the sense that a particular d atom would be either superfluid or normalfluid. K−1 =4π3y2 , (15a) dl con Inordertotakeadvantagefromthedifferentialscreen- d ingequations,wewillcalculatethecorrelationsatR=el σ =4π3y2 . (15b) in a self-consistent way which includes screening effects dl dis by smaller dipoles. At each differential step we will also In order to approach the usual notation of renormal- perform a simultaneous rescaling R e−dlR. Then we → ization groups, we define effective fugacities can interpret the differential equations as renormalization group flow equations. 1 y2 y2 (l):= e4lC(n)(R=1) (16) For the further investigation of the model, we apply X ≡ X −2 X the standard replica technique19 to the partition function (4). We are going to rewrite screening in the repli- againforallthreetypesofrescaledcorrelations. Theyare cated system in order to expose the consistency between related by y2 = yd2is+yc2on. In these definitions we took screening in the unreplicated and replicated system. already account of rescaling of lengths, which generates The Coulomb gas (3b) is replicated n times and after an additional flow of the core energies, disorder averagingone obtains the effective Hamiltonian d E =πK, (17a) n = EabNRaNRb dl H − d R ab Eˆ =πKˆ. (17b) XX 1 dl −2R6=R′ a,b 2πKabln|R−R′|NRaNRb′ . (11) Eqs. (15)togetherwith(17)constitutethemainrenor- X X malization group flow equations. From relations (14) we We introduce the couplings Kab = Kδab Kˆ, Kˆ = recognize, that the flow equations of K and σ of the un- σK2/(1+nσK) and the core energy Eab =−πγKab = replicated system coincide with those of the replicated Eδab Eˆ, where E = πγK and Eˆ = πγKˆ. On the system in the limit n → 0. In the next section these − equations will be completed by a flow equation of the partition sum the neutrality condition free energy, which however does not feed into the other Na =0 (12) flow equations. R R X is imposed for every replica. IV. SELF-CONSISTENT CLOSURE Inthereplicatedsystem,screeningeffectscanbecalcu- lated in the same linear response scheme as before. The Theflowequations(15)togetherwith(17)cannotyet leading term of the free energy of test replica-vortices is be evaluated since they are not closed. In this section of second order in the infinitesimal test vorticity. From we are going to achieve this closure by expressing the this order, we derive a differential screening effective fugacities in terms of K, σ, E and Eˆ. Inthereplicalanguage,theeffectofdisorderisencoded d Kab =2π3e4l KacCcd(R=el)Kdb (13) in additional interactions between vortices and we first dl have to discuss their nature. An inspection of Eq. (11) cd X 4 shows, that vortices of opposite sign in the same replica E (Na)2, which suppresses large Na , we may re- interact with a potential 2π(K Kˆ)lnR. For n 1 strictRaourRconsideration for large E or|γ t|o states with they always attract each other. T−hese dipoles are st≥able NaP= 0, 1: one vortex-free state and 3n 1 different R ± − against dissociation due to thermal fluctuations only for types of replica-vortices,antivortices included. (K Kˆ) 2/π.4,5Inpreviouswork1,6thestabilityofpre- Because of the neutrality condition (12) the elemen- − ≥ ciselysuchdipoleswasusedascriteriontodeterminethe tary excitations in the replicated system are again vor- boundary of the ordered phase. However, for n = 0 and tex dipoles. In the dilute limit (large γ), some type low temperature or strong disorder, K Kˆ = K σK2 of such replica-vortices will form with their antivortices − − becomesnegative(repulsion!) andthiscriterionbecomes those dipoles, which are most instable against thermal questionable. The increasing instability of these dipoles fluctuation and thus destroy order. To find out this gives rise to a reentrant phase boundary.1 type, we have to discuss the energy of such dipoles. The However, it is not sufficient to consider interactions core energy E of replica-vortex ν = N1,...,Nn de- ν only within replicas. Independently on temperature and pends only on the total vortex number{m := (N} a)2 0 a replica number n, vorticesof the same signbut in differ- and the number m := 1(m + Na) of vortices with entreplicasattracteachotherwithapotential2πKˆ lnR. Na = +1. The sa1me h2olds0for thae coupling KP, which ν P Inthe limitn=0the couplingbetweendifferentreplicas givesthe strengthofthe logarithmicinteractionbetween reads Kˆ = σK2. For low temperature or strong disor- a replica-vortex of type ν and its antivortex: der, the interaction between different replicas becomes stronger to the same extent as the intra-replica cou- Eν =m0E (m0 2m1)2Eˆ, (18a) − − pling becomes weaker and correlations have to be calcu- K =m K (m 2m )2Kˆ. (18b) ν 0 0 1 latedtakingintoaccountthe competitionbetweenintra- − − and inter-replica interactions. The importance of inter- Due to replica symmetry, there is a certain degeneracy replica interactions has been recognized by Korshunov.7 between different types ν, which we include when we The essence of the replica problem is a suitable treat- speak of the class (m ,m ) of replica vortices. 0 1 ment of the inter-replica interaction for integer n 1 The existence of order requires stability of all types of ≥ and to construct an analytic continuation which allows dipoles against thermal dissociation. For the dilute sys- to take then the limit n 0 in a proper way. In the fol- tem this is guaranteed by min K 2π according to lowing this is done in a→way different from Korshunov’s thesimplestabilitycriterionofνK6=0osteνrl≥itzandThouless.4 way, leading to opposite conclusions. On the technical The least stable class with the weakest coupling can be level, this step is the essential progress of the present determined explicitly: it is given by replica-vortices of work. class (m = 1,m = 0,1) with K = K Kˆ in a high 0 1 ν − temperatureregion(n+1)Kˆ K,andbyreplica-vortices ≤ of classes (m = n,m = 0,n) with K = nK n2Kˆ in 0 1 ν A. Physics of n≥1 a low temperature region (n+1)Kˆ K. The−resulting ≥ phase diagram for n=1,2,4,8,16is shown in Fig. 2. Asnecessaryprerequisiteforthe analyticcontinuation The fact, that the class of the least stable replica- n 0, we must discuss vortex fluctuations in the repli- vortices for n > 1 depends on the physical parameters, ca→ted system for integer n 1. This happens in some canbe interpretedasanindicationforaphasetransition detail since our final conclus≥ions contradict Refs. 1,6,7. even for n = 0. We take it as a warning, that we must not focus on a single class of vortices. The origin for the We first check, that the ground state of the replicated Hamiltonian is the state without vortices. For this failure of the earlyreplica approaches1,6 is just the focus on classes (m = 1,m = 0,1) only. We even do not purpose we examine the Hamiltonian in Fourier space, 0 1 = (2π2/k2)KabNa Nb. There we easily rec- restrict ourselves to the two classes which are dominant Hn k ab −k k for n>1, since it turns out a posteriori, that also other ognize, that every creation of vortices costs energy, because KR aPb is positive definite: Kab has one eigenvalue classes contribute to the replica limit n 0 . → κ = K Kˆ = K(1+(n 1)σK)/(1+nσK) > 0 and 1 − − n 1eigenvaluesκ =K >0. Sincethegroundstate 2,...,n ha−s no vortices, we are allowed to use for low tempera- B. Towards the replica limit n→0 tures a low density expansion. This is in contrast to the unreplicated system, where the ground state has a finite Now we calculate the correlationsand the effective fu- density of vortices, as discussed above. gacities taking into account contributions by all types of For the following it is convenient to introduce the no- replica-vortices. For a low density of replica-dipoles (re- tion of a “replica-vortex” (compare Fig. 1): in a state alized for large γ or at low temperature) we can neglect with vorticity N1,...,Nn at position R, we say that the interaction between different replica-dipoles (“inde- R R a “replica-vort{ex” of type ν} N1,...,Nn is at po- pendentdipoleapproximation”). Thismeansthatwecan R R sition R. (Upper indices at≡N{are not exp}onents but calculate approximate correlations in the partition sum replica indices!) Since the core energy contains a term containing only up to one dipole of replica-vortices, 5 1 Z(n) =1+ y2 d2R+d2R− R+ R− −2πKν. anexplicitvariableandwemaynowperformtheanalytic 2 ν ν ν| ν − ν| continuation n 0, νX6=0 Z → (19) d =2 π e4l lnz. (23) dlF F − The fugacity of replica-vortex ν is defined by y := ν e2l−Eν. The factor 1/2 is present, since ν runs over vor- As shown in Appendix B, this yields a well defined free tices and their antivortices and we should count every energy per replica only because we have retained all realizationof a dipole only once. In Eq. (19) integration classesofreplica-vortices. Thismeans,thattypesoffluc- is restrictedto R+ R− 1 foralll 0, since lengths tuations, which seem to be irrelevant for n 1 (in our | ν − ν|≥ ≥ ≥ are rescaled. case: energeticallyexpensivereplica-dipoles)canbecome Before we turn to the calculation of correlations, we important in the replica limit. wishtomakesurethatcalculationsinthisensemblemake Inasimilarwaywecanproceedtodeterminetheeffec- sense in the limit n 0. For this purpose we have to tive fugacities. In the independent dipole approximation → convince ourselves, that Zn 1. This is equivalent to (19) the rescaled correlation functions read (for R 1) → ≥ the condition, that the free energy per replica has a fi- nthiteeflriemeite.neTrghyenpearlsroepthliecar,ewnohrimchalfiozlalotiwosnfgrroomupthfleowcono-f hNRaN0bin =− yν2NνaNνbR−2πKν. (24) ν tributionofdipolesofradius1 R1) y2 , (31a) component picture, we may call y2 “density of dipoles”, ≈ σ πτ e−(π/2σ)l+4l (τ <1) f := e4llnz/2y2 “reduced free energy per dipole”, p := (cid:26) 2π2lsin(πτ) y2 /y2“fractionofpolarizabledipoles”,andq :=1 p= pe−4πKl+4l e8πσK2 1 (τ >2) gydci2voisne/ny2by“fEraqc.ti(o2n7)ofasfrfouznecntidoinpsoloefs”E. aTnhdesEeˆ.quFaonrttihtiee−scoanre- yd2is ≈( 2πσ2lπsτin((1π(cid:16)−ττ))e−(π/−2σ)l(cid:17)+4l (τ <2) , (31b) pe−2πK(1−1/2τ)l+4l (τ >1) vweinthienthcee towfot-hceomrepaodnerenwtepsaurammmeaterirzse: the flow equations yc2on ≈( 2πσ2lsinπ(τπ2τ)e−(π/2σ)l+4l (τ <1) . (31c) d =2 2πfy2, (28a) Althoughpwe were starting from unique expressions, dlF F − the asymptotic behavior differs for high temperature or d weak disorder and for low temperature or strong disor- K−1 =4π3py2, (28b) dl der. Therearethreeregimesofparameters,namelyτ >2 d (called regime IA), 2 > τ > 1 (called regime IB), and σ =4π3qy2, (28c) dl τ < 1 (called regime II). Their physical properties will d be analyzed in the next section. E =πK, (28d) Approachingthe separatricesτ =1 or τ =2 using the dl expressions for the high- and low-temperature regime, d Eˆ =πσK2. (28e) theexpressionsshouldcoincide. Thisistruefortheirex- dl ponents. The comparison of the prefactors is not mean- As initialvaluesforK andσ onehastouse the unrenor- ingful, since in the low-temperature expression the di- malized values, which also enter E =πγK, Eˆ =πγσK2 verging factor 1/sin(πτ) is multiplied with the asymp- and thereby y, f, p, and q. Initially (l = 0) = 0 since totically vanishing factor 1/√l. F no fluctuations are included. Remarkably, for zero temperature (τ = 0) we find y2 = y2 = R4P, which means that the probability P, con Eq. (6), coincides with the ordinary and connected cor- C. Asymptotic approximation relation. For the analysis of the flow equations, we wish to use In the present form the flow equations are not very the quantities of the two-component picture, as intro- convenient since the dependence of the effective fugaci- duced in Eqs. (28). The reduced free energy per dipole, ties and the two-component parameters on E and Eˆ is fraction of polarizable dipoles, and fraction of frozen still quite intricate. Therefore we perform an additional dipoles converge asymptotically to approximation which is valid on large length scales. asFdiirssotrtdheer ianvteergargaelssoinveErqAs.(w(h2i7c)h) aarreetsoplbite ipnetrofoirnmteerd- f :=e4l l2nyz2 ≈ 11/τ ((ττ <>11)) , (32a) (cid:26) vals where one of the contributions to the shell partition y2 1 (τ >1) sum dominates. Then the normalizing denominators in p:= con , (32b) y2 ≈ τ (τ <1) Eqs. (27) can be expanded with respect to the smaller (cid:26) contributions. The resulting infinite series can be inte- q := yd2is 0 (τ >1) . (32c) grated over analytically term by term. From these y2 ≈ 1 τ (τ <1) A (cid:26) − series one then can extract easily the leading terms for large l after approximating E πKl and Eˆ πσK2l. Here we have neglected contributions which vanish on Since dE =πK and dEˆ =πσK≈2,theseappro≈ximations large scales. For example q is not strictly zero but van- dl dl ishes exponentially with l for σ > 0 and τ > 1. areasymptoticallycorrectprovidedK andσ convergeto → ∞ Remarkably, the asymptotic two-component parameters finite values. depend on Kˆ and σ only through the effective tempera- Forconvenienceweintroduce aneffective temperature ture τ. variable In order to eliminate E and Eˆ completely from the 1 asymptotic flow equations, we determine the flow equa- τ := . (29) 2σK tion for the fugacity: The asymptotic approximation yields for the flow of the d ∂y ∂y y2 =2y+πK +πKˆ reduced free energy dl ∂E ∂Eˆ (4 2πK(1 σK))y2 (τ >1) d 2 2πe−2πK(1−1/2τ)l+4l (τ >1) − − . (33) F F − . (30) ≈ 4 π y2 (τ <1) dl ≈( 2F − 2lσsin(ππτ)e−(π/2σ)l+4l (τ <1) (cid:26) − 2σ q To closethi(cid:0)s subse(cid:1)ctionofthe paper,we wish to point For the fugacities we find analogously out, that the flow equations (28) for K and (33) for y 7 withexpression(32b)forthe polarizabilitycoincidewith to disorder the contribution of dipoles is modified by a those of Nattermann et al11 apart from factors smaller factor f =1/τ∗ 1. ≥ than 2. This small deviation is due to a rougher asymp- The increase of σ seems to contradict the tendency of totic approximationin Ref. 11. thermal vortices N to neutralize quenched vortices Q. This neutralization is expected from Eq. (3). How- ever, the screened σ is not defined by the fluctuations V. RESULTS ofQ N . Rather,itwasintroducedviathefluctuation −h i of the screened disorder potential V. The fluctuations of V are proportional to σK2. The expected neutraliza- In the previous sections the technical part of deriving tion shows up as dσK2 0. This holds even though the flow equations has been achieved. The most impor- dl ≤ dσ 0 since the reduction of K dominates the increase tantflowequationsareEqs. (28a-c). Originally,thetwo- dl ≥ of σ, which follows from 2σKp q that is easily proven component parameters f, p, and q have been functions ≥ of E and Eˆ. In the asymptotic approximation E and with the asymptotic expressions. Eˆ have been eliminated, the two-component parameters The most important quantity to analyze is the flow of entropy. From spin-glass models one knows, that a are givenby Eq. (32) as simple functions of τ =1/2σK, replica-symmetric theory might lead to a negative en- and the fugacity flows accordingto Eq. (33). These flow tropy. This unphysical feature then indicates that the equations can be summarized by symmetric approach is insufficient and one has to allow d for a breaking of replica symmetry. We now show that y2 = (4 2πτ∗K(1 στ∗K))y2 (34a) thepresentreplica-symmetrictheorydoesnotsufferfrom dl − − this problem. Therefore we believe that a breaking of d K−1 = 4π3τ∗y2 (34b) replica symmetry is not necessary. dl When the entropy = d T is derived from the d S −dT F σ = 4π3(1 τ∗)y2 (34c) reduced free energy, the unrenormalized values of J and dl − σ have to be kept fixed. According to the flow equa- d 1 = 2 2π y2 (34d) tion for , a non-negative entropy is guaranteed by dlF F − τ∗ d ln(TfyF2) 0. Thedemonstrationofthisinequalityis τ∗ :=min(1,τ) (34e) cdoTmplicatedb≥ythetemperaturedependence,whichisim- plicittothescreenedJ andσforl >0. Ifoneignoresthis Thus we are in position to discuss the physical implica- implicit dependence, one easily verifies ∂ ln(Tfy2) 0, tions of these flow equations. since ∂ Tf 0 and ∂ y2 0 hold for∂τT>1 and τ≥<1 ∂T ≥ ∂T ≥ separately. The implicit temperature dependencies can be taken into account by showing ∂ d ln(Tfy2) 0. A. Qualitative aspects This is verified in the leading ord∂eTr,dlwhere one≥has d ln(Tfy2) = (4 2πK(1 σK))+ (y2) for τ > 1. If dl − − O one would use the same expression for τ <1, one would Tostartwith,weaddressqualitativeaspectsandshow obtain a negative entropy flow. However, for τ < 1 we that the flow equations satisfy some fundamental physi- find d ln(Tfy2) = (4 π )+ (y2). Therefore the en- cal criteria. dl − 2σ O tropyflowisnon-negativeevenatverylowtemperatures. AccordingtoEqs. (16)and(27)allcorrelationsC are X negative, since all fugacities are positive y2 0. This is X ≥ consistent with a screening that weakens the attractive interaction between vortices of the same sign and weak- B. Ordered phase enstherepulsiveinteractionbetweenvorticesofthesame sign. A numerical integration of the flow equations (even As a consequence, the two-component parameters are without asymptotic approximations) shows, that below also non-negative. Our asymptotic expressions show, some transition line L in the (T/J,σ)-plane the renor- c that the fraction of polarizable dipoles p = τ∗ vanishes malization flow indeed does converge for l to fixed linearly at small temperature, whereas the fraction of pointswithK ,σ ,andy =0. Thisistr→ue∞notonlyin ∞ ∞ X frozen dipoles q =1 τ∗ becomes unity. thelimitoflargeγ (smallfugacity),butalsoforarbitrar- − ScreeningreducesthecouplingJ =KT since dK 0, ilysmallγ. Onlythe extentofthe orderedphaseshrinks dl ≤ anditincreasesthestrengthofdisorder, dσ 0. Incom- with increasing γ. This implies in particular, that the dl ≥ parisonto the pure case the screeningof J is reduced by original XY-model described by γ 1.6 has an ordered ≈ the factor 0 p 1, the fraction of polarizable dipoles. phase. Above L all quantities diverge under renormal- c ≤ ≤ Whereas previous theoretical approaches did not yield a ization and the present renormalization group leaves its renormalizationofσ,Eq. (34c)manifeststhatourrenor- range of validity, which is limited to small fugacities al- malizationschemeleadstoanincreaseofσ. Thefreeen- ready due to the independent dipole approximation. ergy density is reduced by screening, see Eq. (34d). Due The properties of the ordered phase are evident in 8 terms of the renormalizedparameters K∞ and σ∞. Due forτ∞ <> 2. The limiting valuesforthe fractionofpolar- to screening they are related to the bare parameters by izable and frozen dipoles are related to the correlations K K and σ σ, where the inequalities become by ∞ ∞ ≤ ≥ equalitiesforvanishingfugacitywithintheorderedphase. C (R) In the ordered phase, the irrelevance of vortices recon p = lim , (37a) quires that the fugacities vanish on large scales. There- ∞ R→∞ C(R) foretheflowequation(34a)offugacitymusthaveanega- C (R) dis q = lim . (37b) tiveeigenvalue. Thisistrueattemperaturesanddisorder ∞ R→∞ C(R) strength below L , which is located at c Naturally the question arises, whether these regimes σ∞ = K1∞ 1− πK2∞ (2/π≤K∞ ≤4/π) (35) aarnedseLpar.atAedsbwyeahtarvueetohbesremrvoeddy,ntahmaitcatsryanmspittiootnicatfoLrmc1 ( π/8(cid:16) (cid:17) (K∞ 4/π) c2 ≥ of the flow equations and of the correlations changes its dependence on σ and K non-analytically at τ = 1 and in terms of the renormalized parameters. It is depicted τ = 2. However, a phase transition in the strict sense inFig. 3asfunctionofthe renormalizedparametersand requiresanon-analyticdependenceoftheintegratedfree of the unrenormalized parameters for γ =1.6. energy on the unrenormalized parameters σ and K. We Themoststrikingfeatureofthephaseboundaryisthe werenotable to provesucha singularityatL andL , absence of reentrance. Reentrance was predicted by ear- c1 c2 whichwethereforeconsiderascrossoverlinesonly. How- lier approaches.1,6 Absence of reentrance without a loss ever, a true transition is expected at L just as in the oftheorderedphasewasfoundonlyrecentlyinimproved c pure case. approaches.11,12 Region II can be distinguished qualitatively from the As a function of the renormalized parameters, the other regions. One can consider q as order parameter phase boundary σ (K ) = π/8 has a horizontal part ∞ ∞ ∞ for the low-temperature regime II. It measures the rela- at large K , i.e. at low temperatures. In terms of ∞ tive strength of the disconnected vortex density correla- bare parameters, the line L can be parametrized by a c tion to the usual correlation on large scales. In the def- function σ (K) or K (σ). Due to the renormalization c c inition of this order parameter the normalization of the of disorder strength, dσ > 0 and due to the reduction dl disconnected correlation by the usual correlation is nec- of the coupling dK < 0 the end points are found at dl essary, since true long-range order is absent. The glassy σ (T = 0) < π/8 and K (σ = 0) > 2/π and σ (T) has c c c orderreflectedbythe Edwards-Andersonlikecorrelation no longer a horizontal part for finite fugacity. can therefore be only quasi long-ranged, too. Neverthe- Concerning the asymptotic behavior of the vortex lessonemaycompareq totheEdwards-Andersonorder ∞ correlation functions, one can distinguish three regions parameterinspin-glassesandweinterpretitasorderpa- within the ordered phase. They are separated by a line rameter for the glassiness of in the vortex system. L , fixed by 2σ K = 1 (equivalent to τ = 1), and c1 ∞ ∞ ∞ The regions IA and IB differ only by the analytic de- a line L , fixed by 4σ K =1 (equivalent to τ =2), c2 ∞ ∞ ∞ pendence of the connected correlation on K and σ . ∞ ∞ see Fig. 3. In the (J/T,σ)-diagram these lines emerge The correlation under consideration do not provide an from the origin. They are straight lines in the limit of order parameter comparabe to q for the line L . c2 zero fugacity and bent upwards for finite fugacity. The correlations are retrieved from the fugacities, Eq. (31), by omitting rescaling and inserting l = lnR. C. Universality Thereby we find (R 1) ≫ The universalityclassofthe pure XY-modelis defined 2R−2πK∞(1−1/2τ∞) C(R) − σ πτ (36a) by the behavior of the critical exponents η = 1/4 and ≈ 2 ∞ ∞ R−π/2σ∞ δ =15atthetransitionandtheexponentialdivergenceof − r2π2lnR sin(πτ∞) the correlationlength when the transition is approached 2R−2πK∞(1−1/2τ∞) from above.5  C (R) − σ πτ2 (36b) Inthe presenceofdisorder,thespincorrelationdecays con ≈ 2 ∞ ∞ R−π/2σ∞ with exponent1 − r2π2lnR sin(πτ∞) η =(1/K +σ )/2π. (38) for τ∞ <> 1 and ∞ ∞ ∞ AlongthelineL theexponenttakesthevalueη =1/4 c ∞ 2R−4πK∞ R4πK∞/τ∞ 1 for σ = 0,5 which decreases with temperature until it Cdis(R)≈−−2 2πσ2∞lnR(cid:16) πτs∞in((1π−τ τ−)∞)(cid:17)R−π/2σ∞ roeradcehreeds tphheasvea,luwehηe∞re =cor1r/e1la6tifoonrsTde=cay0.1e1xpIonnethnetiadlilsy-, r ∞ η = formally. Thus the value of η at the transi- ∞ ∞ ∞  (36c) tionbecomesnon-universal,i.e. itisdisorderdependent. 9 Only for σ = 0 or T = 0 the jump of η is universal essentially Eq. (10a), which they evaluated only to the ∞ (from 1/4or 1/16to respectively)in the sense that it leading order in the fugacity of vortices in the unrepli- ∞ does not depend on the bare coupling J or bare fugacity cated system. Consequently, they neglected the discon- y. For σ > 0 or T > 0 the jump lies inbetween and will nected correlation and could not find a flow of σ. Their depend on microscopic details. flowequationsforK andy2coincidewithourasymptotic In the present work we have not included a magnetic flow equations for τ > 1. However, they used them also field. Therefore we can not determine δ in the presence for τ < 1, which gave rise to reentrance and a negative of disorder. flow of entropy. Wenowattempttoanalyzehowthecorrelationlength The present approach is in principle equivalent to CO ξ diverges when the transition line is approached from andRSN.Theessentialdifferenceliesinthecalculationof + above. Even though the flow equations become invalid the correlations, or in the contributions which are taken forlargefugacity,Kosterlitz5 hasarguedthatξ isgiven into account when the dipoles of smallest distance are + by the length scale where the fugacity becomes of or- integrated out. CO and RSN thereby made too crude der unity. In the pure case, where the flow equations approximations, which treat the disorder effectively as reduce to dK−1 = 4π3y2 and dy2 = (4 2πK)y2 he annealed. This yields correct results only for high tem- dl dl − found ξ exp(b/ K−1 K−1) with a non-universal peratures τ >1 but fails for τ <1. + c ∼ − Technically, they considered only contributions of constant b. In the zero-temperature limit the flow equations read dσ = 4pπ3y2 and dy2 = (4 π )y2. By a replica-vortices of class (m0 =1,m1 =0,1). We pointed dl dl − 2σ out, that already for replica numbers n > 1 this restric- replacement σ 1/4K they can be mapped onto the → tion is insufficient at low temperatures. Therefore we flow equations of the pure case. Therefore we conclude, included all classes of replica-vortices. The additional that the zero temperature transition has vortex classes turned out to be important for τ <1 and ξ exp(b/√σ σ ) (39) remove reentrance and the negative flow of entropy. + c ∼ − From the disappearence of reentrance after the inclu- sion of additional vortex classes we have to conclude, with some other non-universal b. For finite temperature and finite disorder, our flow equations are too compli- that these additional fluctuations in the replicated system (n > 1) reduce the effect of thermal fluctuations in cated for being integrated up analytically. However, we the original unreplicated system (n = 0). This strange naturally expect that the functional form of the diver- featureremindsofthefact,thatinvariationalreplicaap- gence is preserved and that only the argument of the proaches one has to maximize and not to minimize the square-root has to be replaced by the distance to the transition line, say min [(K−1 K−1)2+(σ freeenergyforn=0inordertofindthephysicalstate.20 (σc,Kc)∈Lc − c − When Korshunov7 considered the problem in the σ )2]1/2. c replicarepresentation,heattemptedtoidentifythetran- sitioninthelimitofzerofugacitywiththedissociationof the least stable class of dipoles. The N-complexes in his VI. DISCUSSION AND CONCLUSIONS notation are identical to dipoles formed by replica vor- ticesofclass(m ,m =0,m ),whereonehasto identify 0 1 0 The present theory is based on a selfconsistent linear N m . They indeed drive the transition for n 1. 0 ≡ ≥ screeningapproach,whichissolvedinadifferentialform. From my point of view, there are problems in his way of Thisdifferentialformhasbeenpresentedintherenormal- taking the limit n 0, which cause the conclusion, that → izationgrouplanguage. Infact,theverysamedifferential the ordered phase should not exist at all. One problem equations can be directly derived by a real space renor- is, that it does not make sense to consider replica vor- malization group, generalizing the scheme of Kosterlitz5 tices which occupies m > 1 replicas even for a number 0 to the replicated system. In this scheme the flow equa- of replicas n < 1, since (n m )(m 1) 0 should be 0 0 − − ≥ tionsobtaincontributionsfromintegratingoutallclasses preserved during the analytic continuation. of replica dipoles with the smallest (cut-off) diameter. Inaddition,theconsiderationofsuchrestrictedclasses The controversy among earlier theoretical treatments of dipoles is not sufficient for an analytic continuation of the disordered XY-model requires to put the present n 0. Eventhen, if one takesinto accountallclassesof → work into perspective with these earlier approaches. replica-vortices,itisthemoststableandnottheleaststa- When Cardy and Ostlund (CO) considered the more ble class, which drives the transition in the limit n 0. → complicated problem of the XY-model in the presence This is again a strange feature of replica theory. The of random fields,6 they used a replica-symmetric ap- argumentsforthesestatements aredetailedinAppendix proach to derive renormalization group flow equations, B. whichgeneratedalsorandombonddisorder. Rubinstein, In Appendix C we further show, that L can be de- c Shraiman, and Nelson1 (RSN) focused on the random rived already from considering a single vortex in a finite bond problem, which they analyzed without using the system. After replication its free energy can be calcu- replica formalism. However,the resulting flow equations latedbytakingintoaccountcertaintypesoffluctuations, were a subset of those of CO. Their starting point was which are again parametrized by some m in the range 10

Glassy Vortex State in a Two-Dimensional Disordered XY-Model

Similar Glassy Vortex State in a Two-Dimensional Disordered XY-Model