VOLUME 75, NUMBER 7 PH YSICAL REVIEW LETTERS 14 AUGUsT 1995

Vortex Entanglement and Broken Symmetry

Andreas Schonenberger, ' Vadim Geshkenbein, ' and Gianni Blatter'I Theoretische Physik, ETH-Hong gerberg, CH-8093 Zii rich, SwitzerlandL D L. an. dau Institute for Theoretical Physics, 117940 Moscow, Russia

(Received 6 January 1995)

Based on the London approximation, we investigate numerically the stability of the elementaryconfigurations of entanglement, the twisted pair and the twisted triplet, in the vortex-lattice and -liquidphases. We find that, except for the dilute limit, the twisted pair is unstable and hence irrelevant in thediscussion of entanglement. In the lattice phase the twisted triplet constitutes a metastable, confinedconfiguration of high energy. Loss of lattice symmetry upon melting leads to deconfinement, and thetwisted triplet turns into a low-energy helical configuration.

PACS numbers: 74.60.Ec, 74.60.Ge

The combination of the soft elastic moduli and the largetemperatures attainable in the vortex system of the high-T, superconductors boosts the importance of fluctuationsand leads to interesting phenomena such as vortex-latticemelting and the appearance of vortex-liquid phases [1]. Inthis context, topological excitations in the vortex systemleading to entanglement of the Aux lines play an importantrole, with respect to both statistical mechanics as wellas dynamical properties of the vortex-solid and -liquidphases. In this Letter, we present a detailed analysis of thestability and recombination properties of the elementaryentangled configurations, the twisted pair and the twistedtriplet (see Fig. 1), both for the vortex solid and for amodel vortex-liquid phase.

Topological excitations of the vortex lattice in the formof edge and screw dislocations are long-known objects[2]. Recently, interest has concentrated on more exoticconfigurations such as interstitials and vacancies [3]. Thelatter are relevant in the discussion of a novel supersolidphase in layered high-T, superconductors [3,4] and canbe viewed as bound pairs of oppositely "charged" edgedislocations. Similarly, vortex entanglement is consideredto be at the basis of yet another new vortex phase, namely,the vortex liquid [5], where the entanglement loops can beviewed as bound pairs of concentric screw dislocations ofopposite sign. The presence or absence of entanglementloops determines the nature of the vortex-liquid phase:The entangled liquid is characterized by a finite density ofloops at all scales and is equivalent to the normal metallicphase, whereas a disentangled liquid constitutes a newthermodynamic phase showing superconducting responsealong the field direction [6].

Vortex entanglement is equally relevant for the dynami-cal properties of the vortex liquid: the prohibitively longrelaxation times via reptation [7] make the barriers forvortex cutting and reconnection the limiting factor in thevortex dynamics. These barriers then determine the innerviscosity of the liquid and hence its pinning, creep, andflow properties [1].

The determination of the barriers for vortex cutting andreconnection is a subtle issue. An unambiguous definition

can be given if a metastable entangled state can be shownto exist —in this case a force-free saddle-point configu-ration separating the metastable entangled state from thestable rectilinear configuration defines the two energy bar-riers for entanglement and for reconnection, which neednot be the same in general. As we will show below, theexistence of a metastable entangled configuration dependsstrongly on the number of twisting vortices as well ason the surrounding vortex system exerting a pressure onthe twist. In particular, no metastable state exists for thetwisted-pair configuration embedded in a vortex solid, andthe precise physical meaning of previously obtained cross-

Twisted tripletme tastable

FIG. 1. Twisted-triplet configuration embedded within a vor-tex lattice. The nine nearest neighbors are allowed to relaxwhen twisting the inner vortex triplet. A confined (LTr =2.0ao), high-energy rnetastable state is found, with an excitationenergy ETT —Ep = 2.06epap and stabilized against reconnec-tion by the small barrier E, —E» = 0.23epap.

1380 0031-9007/95/75(7) /1380(4) $06.00 1995 The American Physical Society

VOLUME 75, NUMBER 7 PH YS ICAL REVIEW LETTERS 14 AUGUsT 1995

ing barriers for an entangled vortex pair [8—11] may haveto be reconsidered (a notable exception is the dilute limitwith B ~ H„). A metastable twisted-pair configurationdoes exist in a model vortex liquid, where the pressure ofthe surrounding vortices acting on the pair is modeled by acircular potential. This situation has been investigated byCarraro and Fisher [12], who calculated the crossing bar-rier for the limiting case of an infinitely extended twist us-ing quite ingenious symmetry arguments. Below we willargue that in a realistic description of the vortex fluid thepressure exerted on a vortex pair by its surroundings de-stroys the metastable twist in the same way as in a vortexlattice and hence entanglements involving three or morevortices have to be considered. Similar observations re-garding the absence of metastable twisted configurationshave been reported by Dodgson and Moore [11] basedon work formulated within the framework of the lowestLandau level approximation valid close to H„.

In the following, we present a detailed stability analysisof the vortex entanglement problem: we determine theexcitation energies and the reconnection barriers forthe elementary metastable entangled configurations, thetwisted pair (TP) and the twisted triplet (TT) in boththe vortex-solid and -liquid phases. The analysis is donefor an isotropic superconductor (penetration depth A,coherence length $) within the London approximation;generalization of the results to the anisotropic situationinvolves simple rescaling [1], at least for B » H„.We start out with the vortex lattice and show that nometastable twisted-pair configuration exists for fields B )H„. The TP state can be rendered metastable either byartificially enhancing the vortex core energy or by goingover to the dilute limit B ~ H„, where the interactionbetween vortices becomes short range. Next, we considerthe twisted triplet in the vortex solid and find a metastablestate. The twist is restricted to a finite length alongthe field axis, and we call this a "confined" excitation.The confinement is a consequence of the discrete latticesymmetry and leads to a high excitation energy whencompared to the rectilinear ground state. In comparison,the barrier stabilizing the TT state against reconnectionis small, and one concludes that the lattice phase showsonly a little entanglement. In the model vortex liquidthe situation remains essentially unchanged as regards thetwisted pair —the pressure of the neighboring vorticesdestroys the metastable state in the same manner asin the vortex-solid phase. For the twisted triplet theassumption of a circular effective potential mimicking theliquid environment is realistic. The restoration of planarrotational symmetry upon melting leads to deconfinement,and the TT turns into a low-energy helical configurationstabilized by a high barrier against reconnection. As aconsequence, one expects the liquid to become entangled;however, due to the large cutting barrier, a nonentangledsystem only slowly transforms into an entangled state.Whether a thermodynamic vortex-fluid phase becomes

entangled immediately upon melting is a complicatedstatistical mechanics problem [6], and we will not gointo this discussion here. In the following, we give abrief description of the (numerical) technique on whichour analysis is based and then present the results forthe twisted-pair and twisted-triplet configurations in thevortex lattice and the model vortex liquid.

We base our analysis on the London approximationvalid for the important low and intermediate field rangeB ( 0.2H„. Choosing a set of n vortices (labeled byp„v = 1, . . . , n) involved in the excitation, the free energyfunctional takes the form [see Ref. [1],r~ = (R~, z)]

with

—Qr, +$2/AdI'p ' dI p r„+$2

+ ep (dr~(c,

+,„[r„]= 2ap dr~ z V(R~),

where r~ and r~ (r» = r„—r~i) refer to separatepoints on the same line. Here, we treat the constant cdescribing the vortex core energy as a parameter; withinthe London model its physical value is cp = 0.38. Tak-ing all nearest neighbors into account (see Fig. 1), wechoose n = 10 and n = 12 for the TP and the TT,respectively. Twisted metastable configurations are ob-tained from a topologically correct initial state with sub-sequent application of a conventional conjugate gradientmethod to minimize the energy. In case a metastablestate exists, the cutting barrier is found by imposing aconstraint dragging the configuration from the metastableminimum over the saddle towards the rectilinear groundstate. Within a constrained configuration the vortex pair/triplet is forced to have a prescribed distance dp inthe z = 0 symmetry plane. With this constraint im-posed, the metastable minimum is force free; however,the saddle configuration in general remains forced, andhence we obtain only an upper estimate for the en-ergy of the true (force-free) saddle configuration. Wepoint out that neglecting the contribution of the core tothe self-energy +„&r produces severe instabilities lead-ing to unphysical fluctuations upon relaxation. The ori-gin of these short-range single-vortex fluctuations can betraced back to the dispersive nature of the line tension

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VOLUME 75, NUMBER 7 PH YS ICAL REVIEW LETTERS 14 AU|-UsT 1995

gf(k, ) = ~p ln(1/k, se). The functional containing a fi-nite core energy (cp = 0.38) is not only physically cor-rect but also stable with respect to such pathologicalfluctuations.

Before turning to the specific discussion we brieflymention the natural scales in the problem: For a confinedexcitation the natural length scale along the field axis isthe lattice constant ap, whereas the scale for the excitationenergy is soap. For the (uniaxially) anisotropic situationthese scales change to sap for the length and eepapfor the energy, where we have introduced the effectivemass ratio s = m/M « 1 II]. The energy scale isconveniently expressed through the melting temperature

I I], eRpap = T /2 7cl, =. 6 T, where we have used aLindemann number cL = 0.25 in the last equation.

Vortex lattice. —For the interaction with the surround-

ing vortices we choose the lattice potential

I (R) = g &o(IR R I/~) P &o(IR R I/~),m p

with R denoting the equilibrium lattice sites and Kpthe zero-order modified Bessel function. For intermedi-ate fields H„& B & H„we can use the limit A ~ ~in V, leading to a simple and rapidly convergent seriesupon resummation in Fourier space. We then proceedalong the lines described above and first search for ametastable twisted-pair configuration. For fields B ) H„and using the correct core energy with cp = 0.38 no suchmetastable state exists; in fact, although such a state doesexist for the isolated pair, it is squeezed away by the pres-sure of the surrounding vortices as the lattice potentialV is switched on. Thereby the twisted vortices tend toalign antiparallel in the crossing region, and the resultingattractive force between the segments leads to their col-lapse and mutual annihilation. This tendency of antipar-allel alignment can be suppressed by artificially increas-ing the vortex core energy: for large enough c such thatc ) c, = 4.9 —In(ap/s) the metastable TP is recov-ered. Extrapolating this result (obtained for ap/se ( 20)to smaller fields, a metastable TP configuration is pre-dicted for ap/$ ~ exp(4. 9 —cp) = 90 even in the limitA ~ (x, a result relevant for He and marginally rele-vant for the high-T, superconductors with ~ = 50—100.A (more physical) alternative to stabilize the twisted-pairconfiguration is to decrease the surrounding pressure viareducing the vortex density. For fields B ~ H„ the in-teraction becomes short range, and we have to return to afinite screening length A ( ~ in V. Indeed, in the dilutelimit we find a metastable TP for fields B ( 1.59H„, i.e.,

ap ) 1.851, where ~ = 10 has been chosen.Next we discuss the twisted-triplet state in the vortex

lattice (results for ap/s = 10 are quoted in the text;see Table I for a summary). Following the schemedescribed above we find a metastable TT configurationwell in the London regime, i.e., with a minimal separationd;„= 0.76ap &) s between the vortices. Because of

TABLE I. Numerical results obtained for the triplet configu-rations in the vortex-solid (s) and -liquid (I) phases.

ap/g

5 (s)5 (l)

10 (s)10 (I)20 (s)20 (I)

ZT~ ~/ap

1.6

2.0

2.5

(~TT,h &o)/&pap

1.600

2.060

2.380

(E. +TT,h)/soap

0.021.80.232.6

—1.13.5

Twisted —triplet a /q=tO

e

0 I I I

0.2 0.4 0.6 0.8d, la]

1.0

FIG. 2. Excitation energy ETT(dp) —Fp versus distance dpfor the clamped twisted-triplet configuration in a vortex lat-tice. Inset shows a top view of the clamped configurationsmarked by the arrows, particularly the metastable state and thereconnection geometry close to the saddle. Note the small re-laxation amplitude of the surrounding vortices. Upon melting,the confined high-energy twisted configuration turns into a low-energy helical state, whereas the saddle-point configuration for(re)connection stays confined and remains at high energy (seeTable I).

the hexagonal symmetry of the surrounding lattice, thetwist is constrained within a length LIT = 2.0ap and itsenergy is high as compared to the rectilinear ground state,ETT —Ep = 2.06epap', see Fig. 2. On the other hand,the barrier stabilizing the metastable state is small, E, —ETT = 0.23epap. The saddle still is in the London regimewith d;, ~ 0.28ap. The reconnection goes through afirst collapse, leading to the creation of a small transverseloop that subsequently shrinks to zero, leaving behindthree rectilinear vortices in their ground state; see theinset of Fig. 2. Note the sharp drop in energy whencrossing the saddle, which is a consequence of the suddenreconnection and an indication that the present constraintdoes not produce a force-free configuration at the saddlepoint. We defer the detailed discussion of the varioustypes of (re)connection saddles (loop creation, hystereticeffects) to a forthcoming paper.

Model vortex liquid. Here we reduce the descriptionto those vortices directly involved in the entanglement, i.e.,

VOLUME 75, NUMBER 7 PH YS ICAL REVIEW LETTERS 14 AUGUs~ 1995

n = 2 and n = 3 for the twisted pair and twisted triplet,respectively. We choose the surrounding potential mim-icking the vortex liquid to be of the form

V(R) = (x +sy),2

with f = 2/ao (f = 3/a o) for the TP (TT) configuration;s denotes an asymmetry parameter. In their analysis ofthe TP problem, Carraro and Fisher [12] chose a circularsymmetry with s = 1, where the metastable twisted statetakes a helical shape (with a pitch L„) and is degeneratewith the rectilinear ground state in the limit Lz ~ ~.In this limit, the cutting barrier is characterized by ahigh degree of symmetry, and using ao/$ = 10 we findE~ Fh = 0.87epap, in rough agreement with previouswork [12]. However, in a liquid phase one does notexpect the potential acting on a vortex pair to have acircular symmetry; rather the pressure of the surroundingvortices produces an asymmetric potential with s ) 1 (oneexpects s = 2.7 on simple geometrical grounds, assuminga similar pressure to act in the liquid phase as in thesolid one). Increasing s beyond the critical value s, =1.17 the entangled configuration becomes unstable and nometastable TP configuration is expected to exist in theliquid phase either. One could argue that in a vortex liquidthe asymmetric potential can accommodate itself and rotatealong with the pair; however, such a configuration involvesthe twist of four lines at least. The approximation ofa circular effective potential mimicking the pressure ofthe surrounding liquid is already quite reasonable forthe triplet (and further improves for the configurationsinvolving larger loops with 6 and more vortices). In fact,the main reason for studying the triplet as the elementaryentangled configuration is the existence of a metastabletwisted configuration in the solid, implying the existence ofa similar metastable state in the liquid, which still exhibitslattice order on short spatial and temporal scales; in a(viscous) liquid, the entanglement will first go through aconfined high-energy twist, similar to the one in the lattice,which subsequently relaxes into a low-energy helical statedue to the absence of shear forces in the liquid. Theexcitation energy of this relaxed state depends on thepitch L~ of the helix, and we have verified that forLp )& ao the tiltenergy Fh —Eo = (2~ /9) [In(ao/s) +0 5]aoao/L~ prov.ides a good approximation within theLondon regime (note that here the vortices twist by theangle 27r/3 on the length L~). The saddle-point energyfor (re)connection remains high, however; in the limit

Lp ~ ~ an analysis based on symmetry arguments similarto those used by Carraro and Fisher [12] provides theresult F., —Fh = 2.6aoao, where ao/s = 10 and A ~ ~have been chosen. Results for other values of ao/s aresummarized in Table I.

In conclusion, we have investigated the elementary en-tangled configurations, the twisted pair and the twistedtriplet in the vortex-lattice and -liquid phases. Away fromthe dilute limit, we find that the twisted pair is unstableand hence irrelevant for the discussion of entanglement.The basic loop of entanglement is the twisted triplet,which is metastable both in the vortex-lattice and -liquidphases. In the vortex lattice, the TT is a high-energy statestabilized by a comparatively small barrier, and hence thevortex lattice does not entangle. The high energy of theTT excitation is a consequence of the confinement pro-duced by the lattice symmetry of the surrounding poten-tial. Upon melting, the shear modulus disappears and thehexagonal lattice symmetry changes to a rotational one inthe liquid. The high-energy twisted triplet dissolves intoa low-energy helix with a pitch L~ determined by the mu-tual entanglement in the liquid. It is the phenomenon ofdeconfinement and its associated drop in excitation en-ergy that tends to bind the two (originally unrelated) tran-sitions of melting (loss of translational lattice symmetry)and entanglement (loss of longitudinal superconductivity)together. Whereas melting immediately triggers entangle-ment in a system with short-range interactions, it remainsto be shown whether a disentangled liquid state can bestabilized in a system characterized by an interaction withlong range.

We thank M. Feigel'man, E. Heeb, A. Larkin,M. Moore, H. Nordborg, and A. van Otterlo for interest-ing and useful discussions. Financial support from theFonds National Suisse and the International Soros Foun-dation (Grant No. M6MOOO) is gratefully acknowledged.

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[2] R. Labusch, Phys. Lett. 22, 9 (1966).[3] E. Frey, D. Nelson, and D. Fisher, Phys. Rev. B 49, 9723

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(Amsterdam) 167C, 177 (1990).[5] D. R. Nelson, Phys. Rev. Lett. 60, 1973 (1988).[6] The existence of this phase has been proposed by

M. Feigel'man [Physica (Amsterdam) 168A, 319 (1990);see also Ref. [1]].

[7] S.P. Obukhov and M. Rubinstein, Phys. Rev. Lett. 65,1279 (1990).

[8] P. Wagenleithner, J. Low Temp. Phys. 48, 25 (1982).[9] A. Sudbp and E.H. Brandt, Phys. Rev. Lett. 67, 3176

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