CHIRAL SYMMETRY BREAKING IN MOLECULES AND SOLIDS

Transcription

1 CHIRAL SYMMETRY BREAKING IN MOLECULES AND SOLIDS T. Nattermann University of Cologne * F.Li, TN and V. Pokrovsky Phys. Rev.Lett. 108, (2012) TN arxiv: Acknowledgments to SFB 608 of DFG.

2 Outline Introduction Hund s Paradox A second paradox Symmetry breaking in chiral magnets Topological defects in chiral magnets Conclusions

3 Chirality handedness: object and mirror image do not agree enantiomers, optical isomers

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6 Examples of chiral substances. (a) Schematic of d and l quartz. (b) Sodium chlorate crystal, and (c) D (upper) and L (lower) alanine. In (b), d and l crystals are differentiated by their color under polarized light.

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8 Hund s Paradox: molecules and solids built by Coulomb interaction Parity transformation: ˆP r = ˆP E = r E ˆP B = B re = rb =0 r E = 1 c Ḃ r B = 1 c j + 1 c Ė m r = ee + e c ṙ B

9 Leipzig (1929 bis 1946) Leipzig

10 Hund s explanation V (x) Ri, Li harmonic oscillator states Ĥ = 1 + x Li Ri ±i = 1 p 2 ( Ri ± Li) x ± =, prepare system at t =0in Ri,ti = 1 p 2 ( +,ti +,ti) = 1 p 2 e i +t +i + i e 2i t spontaneous symmetry breaking

11 3E b ~ e 2 p 2mEb a/~ PF 3 no inversion

12 2nd paradox: why don t we observe the ground state? Coupling to the environment: photons, gas atoms,... 2 /k Li -a Ri a x (t) = A(x, t) =k R L =2 Z t D( R (t) L(t)) 2E 1/2 =2 t dec dec = [sin(ka)]2 2 thermal photons dec a 2 c k BT ~c 0 t/ X i t/ X i dt 0 x(t 0 ) A(x, t 0 ) (t t i ) cos(kx i ) cos i cos(ka) 5

13 Time evolution: description by density matrix = 1 2 (1 + y ˆ y + z ˆ z), y = i RL i LR z = LL RR RL RL e i( R L) RL e h ( R L ) 2 i/2 i =[Ĥ, ] y = dec y z Harris and Stodolsky 1982 z = y 2 z (t) = z (0) e dect + e ( 2 / dec )t dec dec Trost and Hornberger (2009) D 2 S 2 in He: 176 s 1, dec s 1

14 Motivation film plane perpendicular to helical axis 450µm

15 Motivation How do the domain walls look like? Neel wall Bloch wall (Bloch 1932) 15

16 Time reveral symmetry Time reversal transformation: ˆT t = t ˆT E = E ˆT B = B re = rb =0 r E = 1 c Ḃ r B = 1 c j + 1 c Ė m r = ee + e c ṙ B

17 Chiral Magnets: time reversal + spatial inversion symmetry broken simultaneously separately centrosymmetric crystals: space group contains inversion center non-centrosymmetric crystals: space group contains no inversion center

18 Centrosymmetric Crystals Tb,Dy,Ho Ho T<19K TbMnO 3 both right-handed and left-handed helices appear

19 Centrosymmetric Crystals: Model = qa 1 = ±1 chirality = ±1 conicity m = m(#, ') H = J 2 Z d 3 r ' 2? (' x ) 2 (' x + ) '2 xx H = J 2 Z d 3 r ' 2? + 2 [' x A(r)] '2 xx A = A =

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21 Classification of defect structures according to Toulouse & Kleman (1976) Volovik & Mineev (1977) Mermin (1979) Degeneracy space R Consider mapping of a subspace V d of on Ensemble of equivalent mappings: R 3 homotopy groups : R homotopy group d (R) Example: Ising domain wall, 0(R) =Z 2

22 Defects: vortex line perpendicular and parallel to helix z y? 2 J n q ln Lq x + p o 5 16a ln1/2 qa x FeGe, Tokura et al. 08 degeneracy space n w = H C d /2 = 1 R = S 1 S 0 22

23 I d' =2? Very different from Bloch or Neel walls

24 Defects: domain walls Hubert walls perpendicular to helix Vortex domain wall parallel to helix z path in real space z CuCrO 2 x x R = S 1 S 0 path in degeneracy space

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26 Centrosymmetric Crystals: Topological Hall effect* *Ye, Kim, Millis, Shraiman, Majumdar, and Tesanovic Electrons moving adiabatically in exchange field of magnetization experience Berry's magnetic field m 3 =2 : Skyrmeon force on vortex line 26

27 Centrosymmetric Crystals: Topological Hall effect* force on vortex line m 3 =1 : Meron 27

28 Centrosymmetric Crystals: Multiferroics Coupling to electric polarization vortex lines are charged Wall in yz-plane similar to Neel wall: magnetization rotates around axis in the wall

29 Centrosymmetric Crystals: Domain

30 Non-Centrosymmetric Crystals: systems and model FeGe, MnSi, Fe 1-x Co x Si Heisenberg-spins + weak cubic anisotropy Dzyaloshinskii-Moriya interaction gm i M j H ncs = J/a Z d 3 r ((rm) 2 +2q m [r m]+v X m 4 ) v = O(q 4 ) m(r) =e 1 cos (r)+q 1 r e 1 sin (r),

31 R = SO(3)/Z 2 3(R) =Z (non-trivial texture) 2(R) = 0 (no point defects) 1(R) =Z 4 (stable line defects) e 1 e 1 C 0 C 3 C 1 C 2 G.E. Volovik, V.P. Mineev, 1977, "Investigation of singularities in superuid 3He and liquid crystals by homotopic topology methods," Sov.Phys. JETP π#disclina+on- π#disclina+on- #π#disclina+on-

32 Non-Centrosymmetric Crystals: Hubert walls Domain wall bisector to wave vectors of adjacent domains

33 Domain wall is not bisector to the wave vectors of the adjacent domains Numerically calculated vortex domain wall Vortex domain walls much heavier than Hubert walls, may decay in zig-zag structure of vortex-free domain walls

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35 FeGe, Uchida et al. 2008

36 Weak pinning : Basics Pinning by randomly distributed impurities H imp = N X imp i=1 Z d 2 v imp ( i,u( ) x i ) Pinning force due to impurity i: f i = H imp x i Total pinning force on rigid wall N X imp i=1 f i ± f L 2 an imp 1/2 L Total driving force on rigid wall ~ L 2 => no pinning of rigid walls 36

37 Weak pinning : Basics domain wall has to adopt elastically to pinning forces Energy of elastic distortion u E elast L u 2 x x0)] short range elasticity =0 ) 2io. F pin f L 1/(1 On scales L> L pin (J/ f pin ) ) the domain wall decays into individually pinned regions F elast JL u L 37 F pin JL pin L 2 L pin L pin

38 Elasticity of Hubert wall H = J Z d 3 r ' 2? + 2 [' x A(r)] '2 xx ' buckling (no vortices) A = q A = q (u( ), ) =0 x x0)] ) 2io. 0 u( ) x =(y, z) solution of saddle point equation '(r) = Z x 0 dx 0 A(x 0 ) q Z k e ik k x / u k long range elasticity 38 E elast J 3 X k k u k 2 L

39 Weak pinning - the real thing: Functional renormalization group x x0)] ) 2io. develops cusp at 39

40 Bulk pinning of Hubert walls x weak in plane anisotropy vcos(p') (x) 1 (v/ 4 )sinp(qx + ' 0 ) (x) x 40

41 Chiral molecules and solids are the result of spontaneous symmetry breaking and/or decoherence effects. Chiral magnets exhibit broken time and space inversion symmetry. These symmetry breakings can appear sequentially or simultaneously. Topological defects in chiral magnets are vortices, domain walls and skyrmion lattices. Domain walls in helical magnets are always two dimensional textures (vortex lines), in contrast to Bloch or Neel walls, with the exception of special orientations (Hubert walls). Conclusions 41