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2007 Luttinger Liquid in the Core of a Screw Dislocation in Helium-4 M Boninsegni

Kuklov

Pollet

Nikolai Prokof 'ev University of Massachusetts - Amherst, [email protected]

Boris Svistunov University of Massachusetts - Amherst, [email protected]

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Recommended Citation Boninsegni, M; Kuklov; Pollet; Prokof'ev, Nikolai; Svistunov, Boris; and Troyer, "Luttinger Liquid in the Core of a Screw Dislocation in Helium-4" (2007). Physics Review Letters. 1177. Retrieved from https://scholarworks.umass.edu/physics_faculty_pubs/1177

This Article is brought to you for free and open access by the Physics at ScholarWorks@UMass Amherst. It has been accepted for inclusion in Physics Department Faculty Publication Series by an authorized administrator of ScholarWorks@UMass Amherst. For more information, please contact [email protected]. Authors M Boninsegni, Kuklov, Pollet, Nikolai Prokof'ev, Boris Svistunov, and Troyer

This article is available at ScholarWorks@UMass Amherst: https://scholarworks.umass.edu/physics_faculty_pubs/1177 arXiv:0705.2967v2 [cond-mat.other] 27 Jul 2007 aefrhrlmtdpsil ehnssfrauniform perfect a A for mechanisms state: supersolid possible limited further have fietai solid in inertia of ecniee sawl frrl spaced can rarely boundary of wall of a type as this considered because be angles tilt boundaries small grain con- insulating with of 18] however, observation [17, our theories, model with previous The trasts the conclusive. and of quantitative was None cores. location solid Re- of [8]. simulations ex- vacancies Carlo however, zero-point Monte such past, cent find the not sta- In did of concentration periments vacancies. finite delocalized a and to ground leading ble the crystal, in a vanishes of vacancies state standard the creating if the for develops gap state to review energy supersolid According (see uniform the matter [7], therein). of paradigm references state and supersolid the [6] in interest of al nieadsoaincr 1] rmt Fsealso see SF of promote role on [18], [19] core particu- dislocation Ref. [17], a deformations their inside model crystal larly along phenomenological where a SF recently, proposed, More was host 16]. might [15, cores order, translational turing superglass. as su- considered regular be a rather should of They grain example an superfluid persolid. constitute the not order, do to boundaries translational due discrete Thus, or- of two superfluid. lack and by translational characterized discrete symmetry is regular – translational ders supersolid – cases regular exist both broken, may in is supersolid While of glassy. types [14]. and boundaries two grain general, of presence In samples the to non-uniform it attributed in recent and a superflow while observed [13], experiment superfluid generically are boundaries polycrystalline scenario. persolid eg hs eaain[] hspoie togevidence strong a provides This that are [9]. perfect separation vacancies a phase When dergo into hand interstitials. by and for creation introduced one vacancy larger toward gap even energy an large a with tor h eakbeosrain fanncasclmoment non-classical a of observations remarkable The n a ojcueta srih)dsoain,fea- dislocations, (straight) that conjecture may One h iuto sqiedffrn o eetrde,or defect-ridden, for different quite is situation The 4 ecytlde o ofr oaysadr su- standard any to conform not does crystal He .Boninsegni, M. ASnmes 51.m 53.p 74.h 74.25.Dw 67.40.Kh, 05.30.Jp, 75.10.Jm, numbers: Luttinge PACS H a solid of in cases dislocation cleanest screw the of of th type one – and supersolid orders, quasi-one-dimensional superfluid regular a of example h eaoa xso an of axis hexagonal the ntebsso rtpicpeMneCrosmltosw fin we simulations Carlo Monte first-principle of basis the On hcp utne iudi h oeo ce ilcto nHelium- in Dislocation Screw of Core the in Liquid Luttinger 4 2 e[,2 ,4 ]hv pre wave a sparked have 5] 4, 3, 2, [1, He 4 eateto niern cec n hsc,CN,State CUNY, Physics, and Science Engineering of Department 4 3 e iuain hwdta grain that showed Simulations He. eateto hsc,Uiest fMsahsts Amher Massachusetts, of University Physics, of Department ei nuigsprudt ndis- in superfluidity inducing in He 1 eateto hsc,Uiest fAbra dotn Al Edmonton, Alberta, of University Physics, of Department 5 usa eerhCne KrhtvIsiue,138 Mos 123182 Institute”, “Kurchatov Center Research Russian 1 ..Kuklov, A.B. 3 hcp hoeicePyi,EHZuih H89 Z¨urich, Switze Z¨urich, CH-8093 ETH Physik, Theoretische 4 ecytli ninsula- an is crystal He hcp hcp 4 e[,1,1,12] 11, 10, [9, He rsa,te un- they crystal, 2 edge 4 ecytlfaue uefli (at superfluid a features crystal He .Pollet, L. dislocations 2 Dtd eray1 2008) 1, February (Dated: sinsulating. is 3 ..Prokof’ev, N.V. a raglrlyr eaae yhl fthe the identical of along two half center by superimpose its separated first, layers in we, triangular dislocation Specifically, screw axis. a with tal inladsprudodr xs in exist orders transla- superfluid by and characterised remains: tional defects question were supersolid the they Thus, extended contrary, Do list. the the on from properties; excluded screw often SF in that their dilocations suggestion of edge any over terms of advantage aware any have not dislocations are We 20]. [13, ieysekn,awnigccemprpeet h local Qualita- the response. represents spa- map superfluid component. winding-cycle that the a superfluid visualize so speaking, the tively to [22], of one distribution response allows tial wind- superfluid map The winding-cycle the the define cycle). cycles (winding ing with number worldline the winding of of non-zero (bead) position element spatial selected instant randomly av- measuring a i.e., points maps, of winding-cycle set spa- the eraged For employ we [21]. imaging, function tial Green the for Matsubara space Monte single-particle configuration Integral the Path on a defined technique i.e., Carlo algorithm, worm the on based of h oainlsmer fa of symmetry rotational the well. as planes 3 basal with of samples pairs simulated the 4 have in and we periodic effects, is finite-size sample out The 2. and z 1 Figs. in lustrated fasrwdsoainaindwt the with core aligned question, the dislocation inside this screw SF to a of of evidence answer theoretical affirmative provide we an i.e., give we Letter, eut ihsalrsamples. smaller with the results in cell instead non-periodic opt a we using restore but dislocation, xy could pair, single One dislocation a a for dislocation. simulating single by a it of presence the nthe in drcin ih8piso aa lns nodrt rule to order In planes. basal of pairs 8 with -direction, z apegeometry. Sample are simulations Carlo Monte grand-canonical Our elproiiyi the in periodicity Cell u iuainsml ossso nideal an of consists sample simulation Our hcp = ln,a hsalw sto us allows this as plane, p hs etrn ohtasainland translational both featuring phase e 4 xy 8 He. ,5 4, -iudsse.I otat h same the contrast, In system. r-liquid / -plane, 3 a httesrwdsoainalong dislocation screw the that d ..Svistunov, B.V. ln the along T a t A003 USA 01003, MA st, → sad Y10314 NY Island, n en h atc eidaogthe along period lattice the being et 6 2J1 T6G berta )cr.Ti stefirst the is This core. 0) h emtyo u apei il- is sample our of geometry The o,Russia cow, c ai n hfe y( by shifted and -axis rland xy ,5 4, hcp ln sicmail with incompatible is plane a n .Troyer M. and iuaesmlswith samples simulate ) crystal 4 ecytl?I this In crystals? He 4 c − b banrobust obtain ) ymtyaxis symmetry c ai period -axis a/ 3 hcp 2 a/ , crys- √ x 3) c - 2

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FIG. 1: (Color online. For the best perception, look from a distance.) Columnar winding-cycle map (blue dots) in the 4 T . core of the screw dislocation in solid He at = 0 25 K and FIG. 2: (Color online.) Columnar winding-circle map in the ˚−3 density 0.0287 A . View is along the hexagonal axis, with core of screw dislocation. View is along the x-axis in the the core parallel to it. Shown with red dots (in the lower basal plane—perpendicular to the core. Shown with red dots half of the plot only) are the atomic positions in the initial (in the lower half of the plot only) are the atomic positions in ∼ a configuration. At distances 3 from the core, the atomic initial configuration. The unit of length is 1 A.˚ positions averaged over the imaginary time are only slightly shifted against those of the original configuration.The unit of length is 1 A.˚ the x-direction is then a = 3.666 A.˚ The convenience of working at the melting point is that we know the chemical potential, µ =0.02 K, from the simulations of the liquid axis. The crystal, then, has been cut along the xz-half- phase at its freezing density [21]. Our simulations span plane, whose edge (located at the center of one of the a temperature range from 0.2 to 1 K. hexagons formed when all basal layers are projected to In Figs. 1 and 2 we present the projected winding- a single xy-plane) is the dislocation core. Atoms have cycle maps on the xy- and yz-planes, together with the been displaced so that, upon completing a full revolu- initial configuration of the atoms. We see that the super- tion around the core, each atom advances by a lattice fluid density in the dislocation core is strongly correlated period az. This procedure creates an ideal (classical) with the insulating environment, respecting the hexag- screw dislocation in essentially infinite medium. Then, onal symmetry modified by the presence of the screw all particles located outside a (pencil shape) cylinder, dislocation. The superfluid density is most robust along with the dislocation core being its axis of symmetry, the bonds of the ideal structure. (and inside the rectangular simulation cell) have been Quantitatively, our data can be analyzed using the con- pinned. These particles are not moved in our Monte cepts of Luttinger liquid theory. Luttinger liquids are Carlo simulation, but they do interact with the rest of characterized by two parameters, the superfluid stiffness (1D) (1D) the system via the helium pair potential. The purpose of Λs = ns /m (where ns is the 1D superfluid den- these “frozen” particles is that of confining our sample sity and m is the helium atom mass) and the compress- to the inner (cylindrical) part of the cell, while, practi- ibility κ. Both can be extracted from the distributions cally, completely eliminating the effects of open boundary PW (W ), PN (N) of the winding W (along the core) and conditions. Our rectangular simulation cell is twice the particle number N fluctuations directly obtained in the size of the physical sample inside the cylinder. The ini- simulation: tial numbers of (physical) atoms used in simulations are LW 2 Nini = 384, 768, 1912. PW (W ) exp , Superfluid response. The simulations were performed ∼ −2βΛs  at the number density 0.0287 A˚−3, that is, in the close βN 2 P (N) exp , (1) vicinity of the melting point. The lattice period along N ∼ − 2Lκ  3 where N is taken relative to its mean N [23], L is temperatures 0.2K. To rule out such a scenario, we the sample length along the c-axis and βh standsi for in- first relaxed the∼ initial configuration by treating atoms as verse temperature. We fitted the collected histograms classical particles with a repulsive potential 1/r6, and ∼ PW (W ), PN (N) with Gaussians and have obtained the minimizing the energy of the configuration. [In the case superfluid stiffness and the compressibility. The Lut- of edge dislocations, this protocol leads to an insulating −1 √ tinger parameter is given as KL = π Λκ. Our data (for groundstate.] Our observation was that in both cases— the different temperatures, and different system sizes in with or without classical relaxation—the Monte Carlo the z-direction) yield KL = 0.205(20), where the finite- process converges to one and the same result. Another size (along the c-axis) and statistical errors are combined. indication that our superfluid signal is not an artifact The initial number of (updatable) particles in the state of poor thermalization comes from the perfect regularity shown in Figs. 1 and 2 was N = 768 corresponding to a of the superfluid density map, in combination with its relatively tight confinement in the xy-plane. We clearly strong correlation to the insulating environment. This see superfluidity in samples with much larger number of outcome rules out concerns based on a lack of thermal- particles in the xy-plane (1912 updatable particles in the ization in the glassy state. Analogous simulations have sample with 3 pairs of basal planes). In particular, the been performed for the hcp para-hydrogen crystal, using superfluid stiffness was found to be the same within the the same sample preparation procedure, at a density of error bars (of about 20%). However, the accuracy of data 0.0261 A˚−3, corresponding to the T 0 equilibrium den- for κ was not sufficient for resolving finite-size effects in sity. In this case, the dislocation core→ is found to be an KL due to the xy-confinement. insulator doped with vacancies (forming a commensurate For the value of the chemical potential corresponding density wave). to the melting point, the average equilibrium number of Discussion. The observation of a superfluid dislocation particles in the 8 basal-pair sample was found to be about in a 4He crystal is an interesting example of a regular 770. By varying chemical potential and, in particular, quasi-one-dimensional supersolid. However, the question lowering it below the point where N = 768, we saw no is whether this observation is relevant to the effect of qualitative difference in the superfluid response, which non-classical inertia [1]. Superfluid dislocations can be a is fully consistent with the fact that KL < 1/2 (in the part of the disordered superfluid network, together with Luttinger liquid, an external periodic potential is rele- superfluid grain boundaries, ridges, and liquid (or glassy, vant for opening the insulating gap only at KL > 1/2 at higher pressure) pockets, etc. Considering a network [24, 25, 26, 27]). A sample half the size in z-direction of dislocations only, the effective three-dimensional su- yielded the same qualitative conclusion. In the 3 basal- perfluid fraction of such a network is limited by the vol- pair sample with much larger number of particles in the ume fraction of the atoms located at the cores of dislo- xy-plane, we observe that the core dopes itself with va- cations. This fraction is likely to be significantly smaller cancies (of the order of 3-4), rather than with interstitial than 1 %. The dislocation density in solid 4He typ- atoms, apparently due to lower effective chemical poten- ically∼ varies over a range as wide as 106 1010 cm−2 tial in the core. We do not attribute special significance [28, 29, 30, 31, 32]. It is, however, also possible÷ to grow to this fact because by increasing the chemical potential crystal without screw dislocations at all [33]. Taking the by 4 K, far smaller than the interstitial energy in the (1D) − 1D superfluid density of n 1 A˚ 1 observed in our hcp solid, the system can be made to have the original s simulations, we estimate the required≈ order-of-magnitude (classical solid) particle number without changing the su- 2 12 −2 dislocation density nd 1/l 3 10 cm in order perfluid nature of the state. (3D) ∼ (1D∼) × (1D) to account for ns /n = (ns nd)/n 1% of the su- From our data for Λs we deduce that ns is about perfluid fraction observed in [1]. Here∼l is the typical ˚ 1 particle per 1 A, which is equivalent to saying that su- distance between dislocations. perfluid phase in the dislocation core involves nearly all atoms in a “tube” of diameter 6 A.˚ More quantitatively, The Luttinger liquid parameter is all we need to char- one can assign the superfluid response to partial con- acterize superfluid properties of the dislocation network consisting of segments of length l a. As it has been tributions from the coordination shells. Nearly 90 % of ≫ points in the winding-cycle map in Fig. 1 belong to the pointed out by Shevchenko [34], in this system one has to first three coordination shells with the following division: distinguish between static and dynamic responses. Below 35 % in the first shell, 29 % in the second one, and 26 % the thermodynamic transition temperature Tc T∗a/l (3D) ∼ 1/2 in the third one. (which can be also written as Tc T∗ [ns /n] , in ∼ (3D) Special care should be taken to make sure that the terms of the zero-point superfluid fraction ns /n) the observed superfluid response is not an artifact of poor network features a non-zero static superfluid response thermalization of the worldline configurations in the dis- [34]. Here, T∗ is the characteristic temperature of the location core. This concern stems from our previous ex- superfluid helium liquid. In our simulations we find that perience with edge dislocations, where we saw that a the core has robust phase coherence properties at all tem- superfluid glassy region in the core—created either by peratures below 1 K, which allows us to set T∗ 1 K (the hand, or as a result of quenched relaxation of the initial continuation of the λ-line in liquid helium to∼ the region configuration—remains essentially metastable, at least at of solid densities gives T 1.5 K [13]). This estimate is λ ≈ 4 in agreement with the small value of KL. fore, must also be superfluid with an additional feature In the temperature range Tc

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