Workshop on Supersolid August Dislocations and Supersolidity in solid He-4. A. Kuklov CUNY, USA

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1 Workshop on Supersolid August 2008 Dislocations and Supersolidity in solid He-4 A. Kuklov CUNY, USA

2 Dislocations and Supersolidity in 4 He Anatoly Kuklov (CUNY,CSI) supersolid 2008, ICTP, Trieste

3 First principles QMC Massimo Boninsegni (Univ of Alberta) Lode Pollet (ETH) Nikolay Prokof ev (UMASS ) Gunes Soyler (UMASS) Boris Svistunov (UMASS) Matthias Troyer (ETH) Dislocation roughening Darya Aleinikava (CUNY,CSI) Eugene Dedits (CUNY,CSI) David Schmeltzer (CUNY,CCNY) Thanks to: NSF and CUNY grants

4 Outline Motivation: search for superfluid network of defects History: Studies of dislocations in He4 Types of dislocations and their superfluid properties Core splitting and self-pinning Mechanical strain and vacancy gap Approximate phase diagram Quantum roughening Long-range forces between kinks Kosterlitz-Thouless, RG and MC arguments against quantum roughening J. Day & J. Beamish experiment Summary

5 Superfluid network of dislocation (?) in metals: dislocation densities ~ cm -2

6

7 Shevchenko state of quasi-1d SF random netwok S.I. Shevchenko, Sov. J. Low Temp. Phys. 14, 553 (1988). Incompressible vortex fluid by P.W. Anderson, cond-mat/ Coreless vortices a T c in the network T * a/d ~1/D T*~1K, n d ~ (a/d) 2 T c ~1mK for n d ~10 9 cm -2 T c ~10mK for n d ~10 11 cm -2 T c <T<T *, wide range of quasi-superfluid frequency dispersive response < normal state at T>T c ω τ ω > τ superfluid response at T>T c

8 time of phase slip in a single loop: τ T T * 2 K SF 1 1 T * s K SF -Luttinger parameter; T* ~ 1K; K =0.205(20); Tc~0.1K V.A. Kashurnikov, et al, PRB, 53, (1996); Yu. Kagan, et al, PRA,61, (2000). Not much is known about dynamics in the network!

9 Edge and screw dislocations r ϕ bϕ deformation vector u ~ 2 π strain and stress u σ b / r b - Burgers vector

10 Y Dislocation as almost free classical string J.Appl.Phys. 27,583; 789(1956) Y(x,t) X Peierls barrier (still classical) Peierls, Frenkel-Kontorova, in the book A. M. Kosevich, The Crystal Lattice: Phonons, Solitons, Dislocations, Superlattices, Wiley, 2005 Measurements of sound absorption and internal friction in solid He4 R.Wanner, I.Iwasa, S.Wales, Solid State Commun., 18,853(1976) Y.Hiki, F. Tsuruoka, Phys.Lett. 56A, 484; 62A, 50(1977); V.L.Tsymbalenko, JETP 47,787(1978); 49, 859(1979) I.Iwasa,K.Araki, H.Suzuki, J.Phys.Soc.Jpn 46, 1119(1979) J.Beamish, J.P.Frank, PRL 47,1736(1981)...

11 Quantum effects Semiclassical tunneling of kinks: B.V. Petukhov and V.L. Pokrovsky, JETP 36, 336 (1973) Kinks as quantum quasiparticles: A.Andreev, Soviet Uspekhi, 19, 137(1976) Semiclassical description of dislocation dynamics in sound absorption: A. Y.Hiki, F. Tsuruoka, PRB 27, 696(1983) I.Iwasa, N.Saito, H.Suzuki, J.Phys.Soc. Japan 52, 952(1983) A.V.Markelov, JETP 61,118(1985) Models of quantum kinks+ superfluid properties: P.G. de Gennes,C.R. Physique 7, 561(2006) G.Biroli,J.P. Bouchaud, cond-mat see review S. Balibar and F. Caupin, J. Phys.: Condens. Matter 20, (2008) Role of deformations in inducing core SF : V. M. Nabutovskii and V. Ya. Shapiro, JETP 48, 480 (1978) S. I. Shevchenko, Sov. J. Low Temp. Phys. 13, 61; 553 (1987) A. T. Dorsey,P.M. Goldbart, J. Toner PRL 96, (2006) J. Toner, PRL 100, (2008)

12 Main focus of the talk Deformations, structure and core superfluidity ) M. Boninsegni, et al., PRL 99, (2007) L. Pollet, et al, cond-mat Mechanically gapped dislocations and J. Day & J. Beamish experiment Nature 450, 853 (2007)

13 Stable dislocations in hcp He4 z-screw x-screw zy-edge xy-edge xz-edge a b= 8 a 3 b= a b= 8 a b= a b= a SF 3 SF SF strongly insulator splits by I strongly split order split insulator transition

14 Core splitting (partial dislocations) in He4

15 Full edge dislocation Z two atoms in unit cell X b- Burger s vector

16 Full edge dislocation Z two atoms in unit cell Edef ~ b 2 X b- Burger s vector

17 Partial (split) edge dislocation Z two atoms in unit cell E def ( ) 2 2 b b ~2 ~ 2 2 L X fault

18 PIMC simulations (columnar view): split xy-edge dislocation 11 layers 11 layers 10 layers 10 layers

19 Split ZY-edge dislocation Z A B A B A B C b z /2 b z /2 b z X L

20 Y Fault A B X

21 Y Fault: C-layer A B ½ cores X

22 Columnar view of fault + split core: many layers Y X

23 PIMC simulations (columnar view): zy-edge (SF) dislocation ½ cores

24 Strong self-pinning of zy-edge dislocation Z gliding motion is pinned by the barrier ~ L b z /2 b z /2 split zy-edge dislocation L Y b x /2 gliding motion is not pinned b x /2 split xy-edge dislocation X

25 Fault size 2 b μ Efull = ln R 4π 2 b μ Esplit Efull = ln L+ σ L 8π 2 b μ L =, σ fault surface tension 8πσ MC simulations: σ < 0.1K / particle o b = 3.7 A, μ bar L > A o

26 condensate map Z-Screw M. Boninsegni, et al., PRL 99, (2007)

27 condensate map X-screw G. Soyler et al.

28 Fully SF loops z-screw zy-edge + SF splits by I order transition SF strongly split and selfpinned

29 Splitting of z-screw dislocation makes it insulator Z b z /2 b z /2 L split=insulator

30 Z b z /2 b z /2 superfluid L insulator

31 (Preliminary) phase diagram for dislocation SF I order transition superfluid screw dislocations insulating screw dislocations

32 Strain and SF gap F = r u u + int 2 ( T ij ij T ijkl iju kl...) ψ... T = T = r ; T = r T xx yy zz ijkl sym metry of elastic m oduli 2 ψ ~ ( r + T ij u ij + T ijklu iju kl +...) Higher order terms in strain are equally important no proximity to superfuidity of ideal crystal (no perfect supersolid)

33 Density change and vacancy gap V. M. Nabutovskii and V. Ya. Shapiro, JETP 48, 480 (1978) S. I. Shevchenko, Sov. J. Low Temp. Phys. 13, 61; 553 (1987) J. Toner, PRL 100, (2008) L. Pollet, et al, cond-mat δ n u n ii u = u = u xx yy zz No significant quadratic terms!!! δ n/ n 0.13 T + 2T + 2T + 4T 0 zzzz xxyy xxxx xxzz

34 XY-edge dislocation: Insulating b uii, r = a, b= a δ n/ n π r Splitting: b b/2 δ n/ n 0.08 < 0.13 ZY-edge dislocation: marginally SF b u,, 8 ii r = a b= a δ n/ n π r 3 Splitting: b b /2 δ n/ n 013.

35 Anistropic compression uxx + uyy + uzz = 0 L. Pollet, et al, cond-mat No linear term! r = r

36 Anistropic compression 2 2 ( 2 ~ ( /2 /2 2 ) ) uzz ψ r+ Tzzzz + Txxyy + Txxxx Txxzz uzz ~ 1 ( 0.1 2) 2 T + 2T + 2T + 4T 0 zzzz xxyy xxxx xxzz split ZY-edge is marginally SF u u zz zz bz = π a /

37 Shear strain Z u z ~x L x X 2 u zx r T u 1, 2 ( 0.15) 2 2 ψ ~ ( 4 xzxz xz )~ screw dislocation: superfluid if unsplit: insulating if split: 2 2 bz a 8 uzx + uzy = 0.22, r =, bz = a 4π r 3 3 u + u ( 0.15) ( 2 zx zy ) 0.22 ψ ~ 1 = 1+ > 0, < 0 ψ =

38 Quantum (non-)roughening and J.Day & J.Beamish experiment J. Day & J. Beamish,Nature 450, 853 (2007)

39 Y Quantum dislocation Y(x,t) X Quantum rough dislocation Peierls barrier u p is effectively zero at T=0 Zero point wandering ratio K of phonon and kink energies becomes ~1 in He4: K Luttinger parameter

40 Long-range interaction between kinks due to 3d elastic matrix J.P.Hirth, J.Lothe, Theory of Dislocations, McGraw-Hill, 1968 A. M. Kosevich, The Crystal Lattice: Phonons, Solitons, Dislocations, Superlattices, Wiley, 2005 Long-range space-time interaction suppresses quantum roughening U (, ) LR x t x g + t 2 2 dislocation loop L

41 Coulomb gas mapping Kostrelitz-Thouless transition is destroyed for arbitrary small g Pairs are ionized into plasma state Peierls barrier is relevant at T=0!!!

42 RG u p grows at large L for any g(0) and K(0)

43 MC simulations (of dual model) at T=0 Monte Carlo Worm algorithm (Prokof ev&svistunov (1998)): KT

44 Consequences No quantum roughening = no quantum macroscopic wandering Gapped spectrum rather than sound-like (gap << kink energy) ω = q + Δ 2 2 Intrinsic hardening of shear modulus at T< gap (at pinning distances smaller than inverse concentration of thermal kinks)

45 If gap were zero: Thermal activation of He3 linear density of He3 in units a=1: L p pinning length

46 Single Arrhenius: poor fit fit: n d ~ cm -2 n He3 ~ 7. p.p.m actual n He3 ~ 1. p.p.b

47 Finite gap (and L p -1 >> soliton density) Δ( T) Δ0 exp 2π K ( 1 glnl ) p + Δ 0 T

48 Modulus hardening data: courtesy of John Beamish,J. Day & J. Beamish,Nature 450, 853 (2007) For 1p.p.b of He3, L p =10 3 : W = D 65T K 2 from different curves DW self-consistent!

49 Summary Selfpinned SF loops (due to core splitting) Splitting as I order transition for Z-screw - hysteresis and metastability Strain criterion for SF and non-sf dislocations Preliminary phase diagram Self - pinning in the Peierls potential at low T: no quantum macroscopic wandering Bimodality in the modulus hardening due to Peierls gap