Formation of topological vortices during superfluid transition in a rotating vessel
FFormation of topological vortices during superfluid transition in a rotating vessel
Shreyansh S. Dave
1, 2, ∗ and Ajit M. Srivastava
1, 2, † Institute of Physics, Bhubaneswar 751005, India Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai 400085, India
Formation of topological defects during symmetry breaking phase transitions via the
Kibble mech-anism is extensively used in systems ranging from condensed matter physics to the early stages ofthe universe. Kibble mechanism uses topological arguments and predicts equal probabilities for theformation of defects and anti-defects. Certain situations, however, require a net bias in the produc-tion of defects (or antidefects) during the transition, for example, superfluid transition in a rotatingvessel, or flux tubes formation in a superconducting transition in the presence of external magneticfield. In this paper we present a modified Kibble mechanism for a specific system, He superfluidtransition in a rotating vessel, which can produce the required bias of vortices over antivortices.Our results make distinctive predictions which can be tested in superfluid He experiments. Theseresults also have important implications for superfluid phase transitions in rotating neutron starsand also for any superfluid phases of QCD arising in the non-central low energy heavy-ion collisionexperiment due to an overall rotation.
Topological defects arise in a wide range of systemsranging from condensed matter physics to the earlystages of the universe. Formation of these defects dur-ing symmetry breaking transitions has been a very activearea of research, especially in last few decades, bringingout important interconnections between condensed mat-ter physics and particle physics. Indeed, the first detailedtheory of formation of topological defects via a domainstructure arising during a phase transition was proposedby Kibble [1] in the context of early universe. It wasproposed by Zurek that certain aspects of Kibble mech-anism can be tested in superfluid helium systems [2]. Itis now well recognized that the basic physical picture ofthe Kibble mechanism applies equally well to any sym-metry breaking transition [3, 4] thereby providing thepossibility of testing the predictions of Kibble mechanismin various condensed matter systems, see refs. [5–9]. Itis particularly important to note that the basic mecha-nism has many universal predictions making it possible touse condensed matter experiments to carry out rigorousexperimental tests of these predictions made for cosmicdefects [8, 9]. Defect formation in continuous transitionsraises important issues due to critical slowing down. TheKibble-Zurek mechanism incorporates these aspects andleads to specific predictions of the dependence of defectdensities on the rate of transition etc. [2, 3].Basic physics of Kibble mechanism lies in the forma-tion of a domain structure during a phase transitionwhere order parameter field varies randomly from onedomain to another. Individual domains represent corre-lation regions where order parameter field is taken to beuniform. Another important physical input in the Kib-ble mechanism is the assumption of smallest variation ofthe order parameter field in between the two adjacent do-mains (the so called geodesic rule). With these two phys-ical inputs, a geometrical picture emerges for the physicalregion undergoing phase transition, and straightforwardtopological arguments can be used to calculate the prob- ability of formation of defects and anti-defects. It is im-portant to note that the probability of defect formationin the Kibble mechanism is calculated per correlation do-main and it is a universal prediction. Indeed, utilizingthis universality, defect formation probability for Kib-ble mechanism was experimentally tested in liquid crys-tal experiments [7] for a first order transition case wherecorrelation domains could be directly identified as bub-bles of the nematic phase nucleating in the background ofisotropic phase. However, for a continuous transition, di-rect identification of correlation domains is not possible.Further, here effects of critical slowing down introducedependence of relevant correlation length on the rate oftransition [3]. The Kibble-Zurek mechanism incorporatesthese non-trivial aspects of phase transition dynamics forthe case of continuous phase transitions in prediction ofdefect density [2, 3]. We now note that for the casesunder consideration, these topological calculations giveequal probability for the formation of defects and anti-defects. Of course this is on the average, and there can beexcess of defects or antidefects in a given event of phasetransition. Kibble mechanism leads to important predic-tions about the typical value of this excess which, for thecase of U(1) vortices in 2 space dimensions is found tobe proportional to N / where N is the total number ofdefects plus antidefects [9].There are many physical situations which require a netexcess of defects or anti-defects (i.e. a non-zero value ofthe average net defect number) in a phase transition dueto external conditions. For example, formation of fluxtubes in type II superconductors in the presence of ex-ternal magnetic field will lead to a net excess of flux tubesoriented along the direction of external field. Similarly, a He system undergoing a superfluid transition in a rotat-ing vessel will lead to a net excess of vortices. Along withthese excess defects (or anti-defects), there will also be arandom network of defects/antidefects resulting from do-main structure via the conventional Kibble mechanism. a r X i v : . [ c ond - m a t . o t h e r] J un Normally, the net defect formation (e.g. superfluid vortexformation in a rotating vessel) is studied using argumentsof energetics [10, 11]. But the formation of superfluidvortices in a rotating vessel during the superfluid tran-sition also includes contribution from a non-equilibriumdefect production process (via the Kibble mechanism)due to which number of formed vortices during the tran-sition can deviate from the vortex model prediction. (Adeviation from the vortex model prediction was indeedobserved by Hess and Fairbank in their experiment [13],and in view of the above discussion, Kibble vortices maybe able to account for this). As we elaborate below,in the presence of external influence (rotation of initialfluid here, or external field for superconductor) the basicphysics of Kibble mechanism needs to be modified.Two most important ingredients of Kibble mechanismare, existence of correlation domains inside which theorder parameter is taken to be uniform, while the or-der parameter varies randomly from one domain to an-other, and the geodesic rule which says that the orderparameter variation in between two domains is along theshortest path in the order parameter space. (We men-tion that the geodesic rule becomes ambiguous for thecase of superconductors as discussed in [12]. This makesour considerations of the present paper non-trivial forsuperconductors, we will present it in a follow up work.)We will show below that to get a net excess of defectsor antidefects in the presence of external influence (e.g.rotating vessel) both of these aspects of Kibble mecha-nism need to be modified; a given domain can no longerrepresent uniform value of the order parameter, rathereach domain will have certain systematic variation of theorder parameter field originating from the external influ-ence. Further, the same external influence also affectsthe geodesic rule. In certain situations, the variation oforder parameter in between two adjacent domains maytrace a longer path on the vacuum manifold in apparentviolation of the geodesic rule. We will show that thismodified Kibble mechanism leads to reasonable predic-tions of a net excess of defects, along with a random net-work of defects/antidefects. Interestingly, it shows verysystematic deviations for the random component of theexcess of defects or antidefects from the Kibble predic-tion of N / . We find that this excess becomes largerwith larger external bias. This is an important predic-tion of the biased Kibble mechanism proposed here, andcan be tested in experiments. This fluctuation in the netexcess of defects resulting from the phase transition, ontop of the average net defect number arising from the ro-tation may account for the experimental results of Hessand Fairbank [13] for superfluid transition in a rotatingvessel where deviations from the energetics based net vor-tex number (at times even negative vortex number) werefound.Superfluid component is characterized by a multi-particle condensate wave function, Ψ = Ψ e iθ , where Ψ gives number density of superfluid component. The su-perfluid velocity is given by (cid:126)v s = (cid:126) m (cid:126) ∇ θ , where m is themass of He atom. We use the expression for the free en-ergy of the superfluid system in the presence of rotation[10, 14] as F (cid:48) = F − (cid:126)L.(cid:126) Ω, where F is the free energy for su-perfluid without rotation and (cid:126)L = ρ s (cid:82) ( (cid:126)r × (cid:126)v s ) d x is theangular momentum of the superfluid just after the phasetransition generated due to external rotation ( ρ s = m Ψ is the mass density), (cid:126) Ω being the angular velocity of thevessel containing superfluid. Here we are assuming thatpart of normal component which undergoes superfluidcondensation carries same angular momentum as beforethe transition. (Though, it may be possible that only afraction of the momentum of the normal fluid part whichis condensing is carried over to the superfluid momentum.Effects of this possibility on our analysis requires a fur-ther study. One can determine the value of this fractionexperimentally using a rotating annulus of the kind sug-gested in ref.[2].) In two spatial dimensions, free energydensity is given by, f (cid:48) = f − ρ s ( (cid:126)r × (cid:126)v s ) .(cid:126) Ω , (1)where f is the free energy density of superfluid withoutany rotation. We thus get [3], f (cid:48) = α | Ψ | + β | Ψ | + (cid:126) m Ψ | (cid:126) ∇ θ | − Ω ρ s r (cid:126) m | (cid:126) ∇ θ | , (2)where α and β are phenomenological coefficients. Fortemperatures less than the superfluid transition temper-ature, α < by minimizing the free energy neglectingthe rotation. (One can discuss the effect of rotation onΨ , even far away from vortices, especially in presenceof boundaries. We keep analysis of this issue for futurediscussions.) With constant superfluid density Ψ , weminimize this free energy density with respect to | (cid:126) ∇ θ | and get, | (cid:126) ∇ θ | bias = m Ω r (cid:126) . (3)This shows that the equilibrium configuration of Ψ re-quires a non-zero value of | (cid:126) ∇ θ | in the presence of rotation.(Note, for the non-rotating case, we get θ = constant, asis assumed inside a domain in the conventional Kibblemechanism.) Note that | (cid:126) ∇ θ | bias is proportional to thedistance from the origin, this will play an important rolefor the biasing in the production of vortices over antivor-tices as we will see below.One of the main ingredients of Kibble mechanism isthe randomness of the condensate phase θ from one cor-related domain to other. As we have discussed, for su-perfluid phase transition in the presence of rotation, or-der parameter θ cannot be uniform inside any domain,it must vary systematically inside each domain. In thismodified domain picture we still use the fact that all do-mains are independent from each other and have com-pletely random θ value at the center of domain. (Thistype of picture was invoked in an earlier work by some ofus where biased Skyrmion production due to non-zerobaryon chemical potential was studied via a modifiedKibble mechanism for a toy model in 1+1 dimensions[15].) Further, the order parameter variation inside do-main has to be such that it preserve the curl free motionof superfluid. As we have mentioned, here we are as-suming that part of normal components which under-goes superfluid condensation carries the same angularmomentum as before the transition, and we know thatnormal components follow rigid-body rotation with ve-locity given by (cid:126)v n = Ω r ˆ θ which has non-zero curl. Withtransition to the superfluid phase, we model the domainstructure in the presence of initial rotation such that curlfree property of superfluid does not get violated inside adomain. We assume that only on the circular arc withina given domain, drawn using the center of the vessel andpassing through the center of that domain has super-fluid velocity as that was of normal component before thetransition. This will give the gradient of θ on that arcto be the same as given by Eq.(3). We can see this byrelating velocity of superfluid components with normalcomponents on the circular arc, i.e., v s = v n , which gives | (cid:126) ∇ θ | bias = m Ω r (cid:126) , which is the same as earlier obtainedby minimizing the free energy density. It means thatlarger r domain will have more variation in θ than thedomains with smaller r . As we will see, this is preciselythe feature that will cause the biasing in the formationof vortices over antivortices.Now as there is no initial radial flow, we don’t expectany radial superflow inside a domain also. This meansthat θ will be uniform in the radial direction inside eachdomain. With these considerations, we obtain well de-fined values of θ at every point of a domain. We notethat inside a given domain, gradient of θ decreases withincrease in r , this domain structure provides curl free mo-tion of superfluid. So with this, for the rotation of theinitial normal component whose velocity increases with r ,after becoming superfluid, the velocity becomes 1 /r de-pendent inside a given domain. This can be viewed as theeffect of superfluid transition on the velocity profile in-side a given correlation domain. Since with all this, outerdomains have stronger variation of θ (see Eq. 3), there-fore, for the anti-clockwise rotation of vessel, we shouldget more number of vortices than anti-vortices. This biaswill depend upon Ω, system size ( r dependence) and alsocorrelation length ξ (large values of ξ will give larger θ variation inside a domain). Below we will see that bias-ing will also depend on the inter-domain separation dueto modified geodesic rule.We now consider the effect of the bias on the geodesicrule, the way phase θ interpolates in between two adja-cent domains. Conventional Kibble mechanism assumes the geodesic rule which states that θ in between two ad-jacent domains traces the shortest path on the vacuummanifold. Physical motivation for this rule comes fromminimizing the free energy in the inter-domain region.(As we mentioned, for gauged case, as for a supercon-ductor, phase variation between two different points is agauge degree of freedom and has no physical significancelike gradient energy. Hence assumption of geodesic rulefor gauge case raises conceptual issues, see ref.[12].) Oneshould note that this conventional geodesic rule does notrequire specification of how large the inter-domain regionactually is. However, we will see that for the biased case,the physical extent of the inter-domain region becomesan important parameter. We will still follow the physi-cal consideration of minimizing the net free energy in theinter-domain region.For the inter-domain region also we assume that at thecenter of this region, the superfluid velocity is the sameas the velocity of the initial normal fluid component. Forgeodesic rule only the gradient terms of free energy den-sity are important, so by ignoring | Ψ | terms from freeenergy density we have, f (cid:48) = a | (cid:126) ∇ θ | − b | (cid:126) ∇ θ | , (4)where a = (cid:126) m Ψ and b = Ω ρ s r (cid:126) m . We are interestedin gradient in the direction of shortest distance betweenboundaries of two successive domains. So in this direc-tion gradient can be written as | (cid:126) ∇ θ | = ( θ − θ ) /d , where θ and θ are the order parameter values at the bound-ary of 1st and 2nd domain respectively when we traverse,in the physical space, from right to left (anti-clockwisepath) and d is the shortest distance between two succes-sive domains. Now we have to determine path for whichfree energy density gets minimized. There are two possi-ble paths on the order parameter space. If θ > θ , foranti-clockwise path free energy density, f (cid:48) = a ( θ − θ ) /d − b ( θ − θ ) /d (5)and for clockwise path, f (cid:48) = a ( θ − θ − π ) /d − b ( θ − θ − π ) /d. (6)Out of these two paths, one of the path will have lowerfree energy density. Clockwise path will be preferable ifcondition, f (cid:48) − f (cid:48) < θ − θ >bd/ (2 a ) + π . Putting values of a and b , we get,( θ − θ ) > d | (cid:126) ∇ θ | bias + π, (7)which is more restrictive condition to have clockwise pathon order parameter space than the case when there is norotation.Now, if θ < θ , free energy density f (cid:48) given by Eq.(5)will be for clockwise path. For anti-clockwise path freeenergy density will be, f (cid:48) = a ( θ − θ + 2 π ) /d − b ( θ − θ + 2 π ) /d. (8)Now in this case, condition f (cid:48) − f (cid:48) < θ − θ < d | (cid:126) ∇ θ | bias − π, (9)which is more supportive condition to have anti-clockwisevariation of θ than without any rotation. Thus, in boththe cases, rotation of vessel supports anti-clockwise vari-ation of θ on the order parameter space over clockwisevariation even though the path is longer. This showsthat rotation generates biasing in the geodesic rule also.These modified geodesic rules (Eq.(7) and Eq.(9)) willalso contribute in the biasing of vortices formation overantivortices, along with modified domain structure. Notethat for Eq.(7) and Eq.(9), we have considered that thevariation of θ is along the direction of initial flow. If θ variation is considered along a different direction, thensuitable projection of | (cid:126) ∇ θ | bias should be taken.We consider a cylindrical vessel of radius R = 40 µm ,and study the formation of vortices in an essentially twodimensions system. We have taken such a small vesselbecause of computational limitations. Note that effec-tive two dimensions requires that the height of the cylin-der should be small (i.e. not too large compared to thecorrelation length). This will avoid string bending andformation of string loops which has to be handled in afull three-dimensional simulation. Certainly, it will bevery interesting to see the effects of rotating cylinder inthe formation of strings (including string loops) in a fullthree-dimensional simulations and we plan to investigateit in future. Further, we consider correlation length ξ (which determines the domain size) equal to 140 ˚ A as anexample. This corresponds to a temperature which is justbelow the Ginzburg temperature T G (as the domain pic-ture is well defined only below T G , so defect productionis essentially determined just below T G , see ref. [1]). ForHe II system, the critical temperature T c = 2 . K andGinzburg temperature T G = 2 . K (ref. [14]). (We men-tion values of T c and T G here ignoring effect of rotation.)We take inter-domain distance d = 5 ˚ A (as a sample value,we will discuss the effect varying d on our results). Wehave considered anti-clockwise rotational of the vesselwith angular velocity Ωˆ z . Critical angular velocity forthis system for the production of vortices using energet-ics argument, will be Ω cr = (cid:126) mR log( R/ξ ) ∼ = 78 rad s − (note that radius of the vessel is very small here).For our two-dimensional simulation, we take a squarelattice with the correlated domains centered at the lat-tice points. Domains are assumed to be circular with radius ξ so that lattice constant is ( ξ + d ) with d be-ing the inter-domain separation as mentioned above. Wehave performed simulation only in first quadrant of thevessel. So the numbers we get should be multiplied by 4to get the total number of vortices for the whole vessel.Our focus will be on the probability of vortices per do-main. (Note that even for the whole system, the centerof the vessel is within a domain so cannot accommodatea vortex at that point.) We take the lattice to start fromnon-zero coordinates (excluding the x and y axes). Forwinding number calculations (to locate vortices) we haveexcluded domains which touch the boundary of the ves-sel.The essential physics of the Kibble mechanism is imple-mented by taking random θ value at each lattice points(i.e. at the center of domains). We know from the Eq.(3)the gradient of θ at the circular arc, passing through thecenter of the domain. By knowing the value of θ at thecenter of the domain, and gradient of θ on this arc, wecan determine θ at each point on the arc. With this, byusing the fact that there is no flow in the radial direction,so θ is uniform in this direction, we obtain phase valueat the domain boundaries which lie on the side of lattice.We also use modified geodesic rule Eq.(7) and Eq.(9) forvariation of θ in the inter-domain region. To implementthis rule, as we mentioned, we assume that at the center-point of inter-domain region (which is the middle pointof a link) superfluid has same velocity as was of normalcomponents before the transition (given by Eq.(3)). Weproject this velocity along the direction of lattice side toget (cid:126) ∇ θ along the lattice side. With this, and knowingthe values of θ at domain boundaries, we implement themodified geodesic rule Eq.(7) and Eq.(9) to know θ vari-ation in that region. With all this, we calculate windingin each plaquette. Depending upon the winding, at thecenter of plaquette we obtain vortices or antivortices.Now we present the results of our simulation. Weconsider different values of the angular velocity Ω, andfor each Ω we generate 5000 events for defect formationto get good statistics of vortex-anti-vortex production.Fig.1 shows the distribution of net defect number ∆ n (= defect number − anti-defect number) for 5000 events.Upper plot shows the distribution without any rotationof vessel (Ω = 0), we get standard distribution as pre-dicted by the Kibble mechanism. This distribution fol-lows Gaussian distribution f (∆ n ) = ae − (∆ n − ∆ n )22 σ . Byfitting the distribution, we obtain the parameters of thisGaussian as: a = 656 . , ∆ n ∼ = 0 , σ = 30 .
46 (we havetaken bin width 10 with error bars on the plot taken as[ f (∆ n )] / for each bin value). Important point to noteis that center of Gaussian ∆ n has zero value which isthe standard prediction of Kibble mechanism; no bias-ing in the formation of vortices and antivortices (on theaverage). We obtained average total number of defectsfrom the simulation to be N = 1857948. Kibble mecha-nism makes an important prediction of relation between σ and N Ref. [9], σ = CN ν , where value of C for squaredomains is 0.71. The exponent ν is universal and its the-oretical value is ν = 1 / σ and N with simulation, we derivevalue of ν = 0 . .
25 and matches well with the experimentalvalue of ν = 0 . ± .
11 obtained for liquid crystal case,see ref.[9].The lower plot in Fig.1 gives the distribution of ∆ n forthe case of vortex formation during superfluid transitionin a rotating vessel with angular velocity 10 rad s − .We see that in this case also we get a Gaussian distri-bution but shifted with the mean value ∆ n = 25, whichclearly shows that there is a biasing in the formation ofvortices over antivortices. For the whole cylinder, wethus expect to get on an average more than 100 vor-tices over antivortices in the vessel. This bias in the netvalue of ∆ n occurs here because of the modification inthe domain structure and geodesic rule in the presence ofrotation. Thus our proposed modification of the Kibblemechanism, with modified domain structure along withthe modified geodesic rule, is able to accommodate theexpected bias in the net value of ∆ n due to the rotationof the vessel.Table I shows the obtained values of ∆ n , σ , and N from simulations at different Ω values; for each Ω we haveperformed 5000 events. Values of ν is obtained from therelation σ = CN ν , C = 0 .
71. It is very clear that withΩ all the other quantities are increasing.
TABLE I: Effect of rotation on the formation of vorticesΩ ∆ n σ N ν ± ± ± ± ± ± × ± ± ± ± ± ± Fig.2 shows the dependence of ∆ n on Ω (axes are inlog-log scale). This plot clearly shows that ∆ n linearlyincreases with Ω with slope 0 . n = 0 . n (cid:39) . n = 0 . .
0. For full vessel thiswould mean ∆ n (cid:39) cr ( (cid:39)
78 rad s − as mentioned earlier) we willhave on an average net 12 vortices (for the whole vessel).Note when number of vortices is calculated using only f ( D n ) D n W =0 rad s −1 f( D n)=a e −( D n− −− D n) /2 s a=656.40 −− D n=0 s =30.46 data D nfitted D n 0 100 200 300 400 500 600 700−200 −150 −100 −50 0 50 100 150 200 f ( D n ) D n W =10 rad s −1 f( D n)=a e −( D n− −− D n) /2 s a=657.79±8.19 −− D n=25.29±0.44 s =30.41±0.44 data D nfitted D n FIG. 1: Distribution of vortices − antivortices. Upper plotshows the case without any rotation of the vessel (Ω = 0) giv-ing the mean value of Gaussian distribution, ∆ n = 0. Lowerplot corresponds to the case with angular velocity of the vesselΩ = 10 rad s − showing that ∆ n gets shifted from zero tovalue 25, showing a net biasing in formation of vortices overantivortices. As we simulate only a quadrant, the full vesselwill give value of net ∆ n of about 100. energetics arguments in the vortex model, we expect asingle vortex at the critical angular velocity. However,just after the superfluid transition, number of vorticesalso gets contributions from the Kibble mechanism (suit-ably modified as proposed here) whose contributions havea Gaussian spread with σ as given in Table 1. Thus thefinal value of ∆ n will be expected to deviate from thevortex model prediction in general. It is still interestingto ask that with proper incorporation of the Kibble vor-tices, what is the new critical angular velocity at whichone expects to get ∆ n = 1. With our results, angu-lar velocity of the vessel will be smaller than a differentcritical velocity, say, Ω Kibble , which also depends on sys-tem parameters system size, the inter-domain separation d , etc. It is very interesting to study the behavior ofΩ Kibble in comparison to Ω cr and we plan to study thisin future. Especially interesting will be to investigate thedependence of our results on the parameter d . For a firstorder transition, with a simple situation of nucleation ofa large density of critical bubbles (almost at close pack-ing) the value of d will be given by 2 × the bubble wallthickness (while ξ corresponds to the bubble diameter).By considering different experimental situations, the ra-tio d/ξ can be varied and its effects on various results,especially on Ω Kibble can be studied. For a second or-der transition such a study will be more complicated. Inview of these issues, it is clear that a proper interpre-tation of Hess and Fairbank experiment [13] requires amore detailed analysis. Measurement of average numberof vortices in experiment with sufficiently large numberof events for superfluid transition with angular velocityjust below Ω cr may give a good test for the model herewe propose. A non-zero value of angular momentum ofsuperfluid below Ω cr will give a solid support for thismodel. It will also show that there is a critical angularvelocity Ω kibble which is different from Ω cr for the phasetransition in the presence of rotation.As mentioned above, the best fit line for results in Ta-ble I gives ∆ n = 0 .
1Ω (ignoring the intercept, hence forlarge Ω). This matches very well with the vortex modelprediction which gives n (cid:39) πR m Ω /h (cid:39) .
1Ω (Ref.[14]). This is expected as for very large Ω, number ofvortices should be dominated by the effects of rotation.We again mention that our results depend on variousparameters, such as ξ, d etc. Thus one needs to studywhether this agreement with the vortex model prediction(for large Ω) is valid in general.We emphasize that the free energy of individual de-fects plays no role in the Kibble mechanism (even withthe modifications we propose). Still, with our incorpora-tion of initial rotation of the normal fluid (and its somefraction getting transferred to the superfluid flow afterthe transition) at least some part, if not all, of the ”ro-tation induced vortices” have been included in this pro-posed modified Kibble mechanism. This point will beparticularly important for small rotations where very fewvortices are expected from energetics arguments. Thismodified Kibble mechanism gives defect density right af-ter the transition which will evolve in time, and approachthe density expected using equilibrium free energy argu-ments. Thus, if the (modified) Kibble mechanism giveslesser number of net produced vortices then with time,more number of vortices will get produced and ultimatelyin the equilibrium, system will have n number of vorticesas predicted by the vortex model using energetics argu-ments. It is also interesting to study the distribution ofvortices and antivortices as a function of distance fromcenter in our model. The equilibrium distribution is uni-form but as mentioned above, the distribution right afterthe transition may be different due to non-equilibriumcontributions from the (modified) Kibble mechanism. Anon-uniform initial distribution will have very importantimplications for the case of neutron stars where migrationof vortices to achieve uniform (equilibrium) distributionwill lead to change in moment of inertia of the neutronstar (as in the model discussed in [16]). This requireslarge statistics and this study is underway.
10 100 1000 10000 100000 1e+06 100 1000 10000 100000 1e+06 1e+07 — ∆ — n Ω (rad s -1 ) slope = 0.024datafitted FIG. 2: Variation of ∆ n with Ω in log-log scale. This plotshows that ∆ n linearly depends on Ω with slope 0.024. Slopewill be about 0.1 if simulation perform in full cylinder. Table I also shows that the width of the Gaussian σ increases with Ω (slowly initially but strongly for largevalues of Ω). σ represent randomness in the formationof vortices and anti-vortices. If formation of vorticesand antivortices is completely uncorrelated then valueof σ goes like ∼ N / ; width of Binomial distribution.But there is correlation between production of defectand anti-defects in Kibble mechanism (Ref.[9]) causingsuppression in randomness and hence σ ∼ N / . Bywriting σ ∼ N ν we see from the Table I, that ν in-creases with Ω showing that correlation between pro-duction of vortices and antivortices is getting suppressedwith Ω. We also fit the dependence of σ on Ω. Areasonable fit for σ as a function of Ω is obtained by σ = a Ω p + b where fitted values of parameters are foundto be a = 0 . ± . , p = 0 . ± . , b = 30 . ± . a is entirely dominated by error,this fit does suggest a systematic variation of σ with Ωwith exponent p (cid:39) .
5. We plan to carry out a systematicstudy of this result and increase of ν with Ω in future.Fig.3 presents results for a single event for the num-ber of defects per domain, i.e., probability of formationof defects. Fig.3 shows probability of formation of sin-gle winding defects and anti-defects as a function of Ω.We note that both probabilities increase with Ω, withwinding +1 defect probability increasing faster than theprobability for winding − > × rads − and changes differently with Ω, again reflecting bias-ing in the formation of defects over anti-defects. It is wellknown fact that winding number two defects are unsta-ble in superfluid systems and split into two single winding P r ob a b ilit y Ω (rad s -1 ) winding +1 defectswinding -1 defects FIG. 3: Plot shows probability of formation of single windingdefects and anti-defects as a function of Ω. Probabilities forboth the cases changes differently with Ω and causing biasingin the formation of defects over anti-defects. defects eventually enhancing single winding defects for-mation probabilities. We have also checked the effects ofvarying the inter-domain separation d on our results. ForΩ = 10 , increase of d by a factor of 20 (from d = 1 ˚ A to d = 20 ˚ A ) increases probabilities for winding one defect aswell as antidefect by about 15 %. Change in winding twodefect probabilities is very small and dominated by fluc-tuations. For smaller Ω = 10 the change in probabilitiesis very small and dominated by fluctuations. The effectof d on various probabilities is a complex issue and weplan to study it systematically in future. We note thatwhile increase of vortex formation probability is expectedas a function of increasing angular velocity, it may appearpuzzling why anti-defect probability also increases withthe rotation. The explanation for this may lie in the cor-relation of defects and antidefects which is an importantand non-trivial prediction of the Kibble mechanism. Aswe see from Table I, the defect-antidefect correlation ex-ponent ν , while increasing slightly with angular velocityto a value of about 0.28, still remains far below the valueof 0.5 for uncorrelated case. Thus, while vortex probabil-ity increases naturally with the rotation, the underlyingdomain structure forces larger probability of formationof anti-vortices close to vortices for winding number 1 aswell as for winding number 2 case. (Basically from thefact that positive winding across two domains appears asanti-winding for the neighboring region.)Experimental tests of our predictions based on thismodified Kibble mechanism will lend strong support tothe whole underlying picture of the Kibble mechanismwhich is adaptable for varying experimental conditionssuch as biased formation of flux tubes in superconduc-tors in the presence of external field etc. We mentionhere an important aspect of vortex formation in super-fluids via the Kibble mechanism which is not present forother types of topological defects (as emphasized in ourearlier work [17]). We mentioned above that we assume that part of normal component which undergoes super-fluid condensation carries the same angular momentumas it had before the transition (along an arc at the centerof the domain). This just reflects the local conservationof linear momentum during the superfluid transition onthat arc. However, even if there was no initial motion ofthe fluid, still during phase transition, spontaneous gen-eration of flow of the superfluid will arise simply fromthe spatial variation of the condensate phase. Indeed,it is this (random) phase variation from one domain toanother which leads to formation of vortex network andhence spontaneous generation of superflow. What hap-pens then to local linear momentum conservation? Ba-sically, some fraction of ( He) atoms form the superfluidcondensate during the transition and develop momentumdue to the non-zero gradient of the phase of the conden-sate. The only possibility is that the remaining frac-tion of atoms (which form the normal component of fluidin the two-fluid picture) develop opposite linear momen-tum so that the momentum is locally conserved. (Herewe avoid conceptual question of an ideal instantaneousquench to almost zero temperature where there is no nor-mal component left). This means that there is no netmomentum flow anywhere right after the transition. Forsuperfluid transition in a rotating vessel, same consider-ation will apply to the normal component in a domainin regions away from the central arc as in those regionssuperflow will in general not match with the initial flowdue to rotation implying generation of extra counterbal-ancing normal flow component. Note, this argument isquite different from the conventional argument of net an-gular momentum conservation for Kibble superfluid vor-tices where one knows that spontaneous generation of netrotation of the superfluid has to be counter balanced bythe opposite rotation of the vessel containing the super-fluid [2]. Here, we are arguing for local linear momentumconservation which implies generation of complex flowpattern for normal component depending on the gener-ation of spontaneous part of the superflow during thetransition. The final picture is then that, the original ro-tation of the normal fluid (before the transition) is sim-ply transferred to the rotation of the superfluid which,via our modified Kibble mechanism, accounts for the netbias of vortices over anti-vortices. At the same time gen-eration of extra vortices and anti-vortices via the randomdomain formation (via the Kibble mechanism) leads toextra local superfluid circulation in the system which willbe accompanied by opposite circulation being generatedin the normal component of the fluid (to balance the mo-mentum conservation). To incorporate both these con-tributions accurately, one must carry out simulations ofthe transition with a two fluid picture in a rotating ves-sel. These consideration must be incorporated for anyexperimental test of the Kibble mechanism (either theconventional one, or the modified one presented here). Itis possible that a due consideration of this spontaneouslygenerated counterbalancing flow of the normal fluid mayimprove agreement of the results of various superfluid he-lium experiments with the Kibble mechanism. We planto carry out a detailed investigation of this issue in afuture work.In conclusion, we have proposed a modification of theconventional Kibble mechanism for the situation of pro-duction of topological defects when physical situation re-quires excess of windings of one sign over the oppositeones. We have considered the case of formation of vor-tices for superfluid He system when the transition iscarried out in a rotating vessel. As our results show,this biased formation of defects can strongly affect theestimates of net defect density. Also, these studies maybe crucial in discussing the predictions relating to defect-anti-defect correlations. The modified Kibble mechanismwe presented here has very specific predictions about netdefect number which shows a clear pattern of larger fluc-tuations (about mean value governed by the net rotation)compared to the conventional Kibble prediction. Thiscan be easily tested in experiments. Further, even theaverage net defect number deviates from the number ob-tained from energetics considerations, especially for lowvalues of Ω. This implies that exactly at the time oftransition, a different net defect number will be formedon the average, which will slowly evolve to a value ob-tained from energetic considerations. These considera-tions can be extended for the case of flux tube formationin superconductors (with appropriate modifications forthe gauged case), and we hope to present it in a futurework. Such a modified Kibble mechanism is also neededto study formation of baryons at finite chemical potentialin the framework of chiral sigma model where baryonsappear as Skyrmions which are topological solitons (ex-tending our earlier work on 1+1 D Skyrmion formationto 3+1 D [15]). Our results will have implications for su-perfluid transition in rotating neutron stars (where phasetransition induced density fluctuations could be detectedby observing pulsar signal changes, as proposed by someof us [16]). In an earlier work [17], we considered the pos-sibility of superfluid phases of QCD, e.g. neutron super-fluid and color-flavor-locked phase, in low energy heavy-ion collisions and showed that this will lead to productionof few vortices via the (conventional) Kibble mechanismwhich can strongly affect the hydrodynamical evolutionof the system and can be detected by measuring flow fluc-tuations. 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