Spin-charge separation in doped 2D frustrated quantum magnets p.

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1 0.5 setgray0 0.5 setgray1 Spin-charge separation in doped 2D frustrated quantum magnets Didier Poilblanc Laboratoire de Physique Théorique, UMR5152-CNRS, Toulouse, France Spin-charge separation in doped 2D frustrated quantum magnets p.

2 Outline 1. Disordered state in frustrated magnets 2. Lanczos methods & ARPES 3. Spin-charge separation in single hole dynamics? 4. New results for the J 1 -J 2 -J 3 model (Exotic superconductivity in VBS host) Spin-charge separation in doped 2D frustrated quantum magnets p.

3 Collaborations and references Single hole dynamics A. Läuchli & DP, PRL 92, (2004) On the J 1 -J 2 -J 3 model: ongoing work with A. Läuchli, M. Mambrini & F. Mila Other work: doped Shastry-Sutherland lattice P W. Leung, PRB 69, (2004) Pairing in VBS host DP, PRL 93, (2004) Spin-charge separation in doped 2D frustrated quantum magnets p.

4 2D Frustrated magnets Lattices with AF frustrating interactions Melzi et al., PRB 85, 1318 (2000) J 1 J2 frustrated square lattice (S=1/2): Li 2 VOSiO 4 Kagome lattice like SrCr 9 x Ga 3+x O 19 (S=3/2) Ramirez et al., PRL 64 ( 90) Broholm et al., PRL 65 ( 90) Spin-charge separation in doped 2D frustrated quantum magnets p. Uemura et al., PRL 73 ( 94)

5 3D Frustrated magnets pyrochlores and spinels Transition metal oxides - ZnCr 2 O 4 spinel - A 2 Ti 2 O 7 titanates Ramirez et al., PRL 89, (2002) -no ordering at low temperatures -spin gap formation Spin-charge separation in doped 2D frustrated quantum magnets p.

6 Exotic disordered groundstates Low-spin (S=1/2) strong quantum fluctuations nature of disordered phases? many studies (and controversies!) Schulz, Ziman & DP, "Magnetic systems with competing interactions", p120, Ed. H.T. Diep, W.-S.(1994) Misguich & Lhuillier, "Frustrated spin systems", Ed. H.T. Diep, World-Scientific (2003) Spin-charge separation in doped 2D frustrated quantum magnets p.

7 Confinement vs deconfinement Idea: use doping (or ARPES) to probe nature of the ground state (a) string potential (b) deconfined spinon Spin-charge separation in doped 2D frustrated quantum magnets p.

8 Checkerboard lattice: a Valence Bond Solid Finite spin gap 0.6J 2D array of corner-sharing tetrahedra: 2D pyrochlore VBS phase (plaquette) Fouet et al., PRB (2003) Translation symmetry breaking 1. E singlet (Q = (π, π)) E 0 0 when N 2. Spin-charge Plaq l Plaq l finite when l separation in doped 2D frustrated quantum magnets p. l

9 Kagome: paradigm of a spin liquid (?) Magnetically disordered Leung & Elser PRB 47, 5459 (1993) Small spin gap 0.05J Lecheminant et al., PRB 56, 2521 (1997) No symmetry breaking (neither SU(2) nor lattice symmetries) Large number of low energy singlets Waldtmann et al., EPJB 2, 501 (1998) Mila et al., PRL 81, 2356 (1998) Spin-charge separation in doped 2D frustrated quantum magnets p.

10 framework Γ Σ M (a) (b) (c) Γ M (d) H = t i,j,σ P ( ) c i,σ c j,σ + h.c. P + J S i S j 1 4 n in j i,j Spin-charge separation in doped 2D frustrated quantum magnets p. 1

11 Frustrated hole motion J 0 limit singlet triplet t < 0 E = t E = 2 t t > 0 E = 2t E = t Spin-charge separation in doped 2D frustrated quantum magnets p. 1

12 Single particle Green-function "time-ordered" Green s function "electron" (ω < 0) and "hole" (ω > 0) parts G(q, ω) = N e c 1 q,σ c q,σ N e ω iɛ + H E Ne N e c q,σ c ω + iɛ H + E q,σ N e Ne +1 half-filling: N e = N, system size "hole" and "electron" parts related: t t Spectral fct Im G(q, ω) IPES & ARPES Spin-charge separation in doped 2D frustrated quantum magnets p. 1

13 Dynamics within Lanczos ED - Continued-fraction: z = ω + iɛ, A = c q,σ or A = c q,σ G(z) = z + E 0 ẽ 1 Ψ 0 AA Ψ 0 z + E 0 ẽ 2 b2 2 b2 3 z + E 0 ẽ Physical meaning: I(ω) = m Ψ m A Ψ 0 2 δ(ω E m + E 0 ) 1. poles and weights dynamics of A 2. symmetry of A well defined quantum numbers & selection rules 3.! calculation of eigen-states/vectors not required! Spin-charge separation in doped 2D frustrated quantum magnets p. 1

14 Static hole (checkerboard) - Switch off t first already some insight! Spin-charge separation in doped 2D frustrated quantum magnets p. 1

15 Static hole (checkerboard) - Switch off t first already some insight! A static (ω) weight=90.1% ω/j Overlap Z = Φ 1h c i, Φ 0h 2 finite on checkerboard lattice Spin-charge separation in doped 2D frustrated quantum magnets p. 1

16 Static hole (checkerboard) - Switch off t first already some insight! A static (ω) weight=90.1% ω/j Overlap Z = Φ 1h c i, Φ 0h 2 finite on checkerboard lattice Similarity with 1D spin-peierls g.s. confining potential between "holon" and "spinon" Spin-charge separation in doped 2D frustrated quantum magnets p. 1

17 Static hole (Kagome) - Static correlations: Dommange et al., PRB 68, (2003) - Dynamic correlations: Z = Φ 1h c i, Φ 0h 2 0 Spin-charge separation in doped 2D frustrated quantum magnets p. 1

18 Static hole (Kagome) - Static correlations: Dommange et al., PRB 68, (2003) - Dynamic correlations: Z = Φ 1h c i, Φ 0h A static (ω) ω/j Spin-charge separation in doped 2D frustrated quantum magnets p. 1

19 Static hole (Kagome) - Static correlations: Dommange et al., PRB 68, (2003) - Dynamic correlations: Z = Φ 1h c i, Φ 0h 2 0 A static (ω) ω/j Incoherent spectrum Weights distributed on many poles even at low energies Spin-charge separation in doped 2D frustrated quantum magnets p. 1

20 A(k, ) [a.u.] ω Hole dynamics in the VB Solid Γ x 5 Checkerboard t>0 t<0 Γ x 5 0.2% x % x 1.5 Σ 6.7% x % x 1.5 Σ 7.1% 0.04% / 2% x 1.5 M 13.5% 4.2% x 1.5 M ω/t ω/ t Spin-charge separation in doped 2D frustrated quantum magnets p. 1

21 A(k,ω) [a.u.] Single hole doped in a spin liquid Γ kagome t>0 t<0 Γ x 0.5 x 0.5 Μ Μ ω/t ω/ t Spin-charge separation in doped 2D frustrated quantum magnets p. 1

22 spin & charge repel! (a) (b) Hole-spin correlations t>0 t<0 Dimer correlations in "Holon" wavefct (c) (d) Spin-charge separation in doped 2D frustrated quantum magnets p. 1

23 spin & charge repel! (a) (b) Hole-spin correlations t>0 t<0 Dimer correlations in "Holon" wavefct (c) Spin & charge separation: holon benefits from large dimer (d) correlations in neighboring triangle (like static case: see Spin-charge separation in doped 2D frustrated quantum magnets p. 1 e.g. Dommange et al, PRB)

24 Hole localisation in the VBS host J/ t =0.3, 0.4 & 0.6 (a) t<0 k=k X =( π /2, π/2) (c) 0,1 Z k 0,05 A(k=k X, ) [a.u.] ω (b) t>0 square lattice t<0 t>0 (d) 0 0,4 W / t 0,2 ΓΜ ω/ t Electron-hole asymmetry 0 0,2 0,4 0,6 0,8 0 For t > 0: destructive interference effects single hole almost localized Spin-charge separation in doped 2D frustrated quantum magnets p. 1 singlet corr. robust & no Nagaoka F J/ t

25 J 1 -J 2 -J 3 square lattice Heisenberg J 3 /J B Z imp <0.84 (q,q) Classical phase diagram (Moreo et al., PRB 90) A ( q,π) collinear vs spiral (π,π) 0.5 (0,π) J /J 2 1 Quantum case VBS vs spin liquid Spin-charge separation in doped 2D frustrated quantum magnets p. 2

26 J 1 -J 2 -J 3 square lattice Heisenberg J 3 /J B A Z imp <0.84 (q,q) ( q,π) Classical phase diagram (Moreo et al., PRB 90) collinear vs spiral (π,π) 0.5 (0,π) J /J 2 1 columnar dimer: Leung & Lam, PPB 96 Quantum case VBS vs spin liquid spin liquid: Capriotti, Scalapino & White, PRL 2004 = Z imp = Φ 1h Φ bare 1h 2 with Φ bare 1h = ci, Φ 0h (t = 0) Spin-charge separation in doped 2D frustrated quantum magnets p. 2

27 Spin distribution <S i z > S z i at distance r = ri r O from defect on a = 32-site square cluster A bare wf ground state r i r O r i r O B S z i spin-spin bare correlation in host S z i location of gs spinon Typically, ξ conf > ξ AF ξ conf finite when N? Spin-charge separation in doped 2D frustrated quantum magnets p. 2

28 Discussion & Conclusions Spin-charge separation in a spin-liquid Generic? Finite density of holes? Spinon-holon bound-state in translational symmetry breaking VBS Frustration of hopping electron-hole asymmetry Progress on frustrated square lattice AF help from dimer basis (Mambrini et al.) Pairing mechanism based on kinetic energy (another time!) Spin-charge separation in doped 2D frustrated quantum magnets p. 2

29 Metallic frustrated systems? spinel oxide LiTi 2 O 4 Sun et al., PRB 70, (2004) 5d transition-metal pyrochlores as Cd 2 Re or KOs 2 O 6 Hanawa et al., PRL 87, (2001) Hiroi et al., JPSJ 73, 1651 (2004) CoO triangular layer based compound Takada et al., Nature 422, 53 (2003) All superconducting with T c up to 13.7 K! Spin-charge separation in doped 2D frustrated quantum magnets p. 2

30 Dynamics within Lanczos ED - A is applied to GS: Φ 1 = 1 ( Ψ 0 AA Ψ 0 ) 1/2A Ψ 0 C(z) = Ψ 0 AA Ψ 0 Φ 1 (z H) 1 Φ 1 z H = - Lanczos procedure: z ẽ 1 b b b M (1) 0... b M z ẽ M Spin-charge separation in doped 2D frustrated quantum magnets p. 2

31 Non-magnetic dopant in spin-peierls chain Experiment Doping in CuGeO 3 : Cu 2+ Zn 2+ or Mg 2+ Hase et al., PRL 71, 4059 (1993) Theory (numerics) Augier et al., PRB 60, 1075 (1999) Spin-charge separation in doped 2D frustrated quantum magnets p. 2

32 Pairing energy Binding on 4 4 & site clusters binding = E 2holes + E Heis 2E 1hole Energy 0-0,2-0,4 4x4 h-h continuum 4x4 E B mag >0 square lattice d-wave t<0 d-wave t>0 s-wave t>0 s-wave t<0 d xy t>0 g-wave t>0 0 0,1 0,2 0,3 0,4 0,5 0,6 J E B kin <0 Feynman-Hellmann: magnetic energy: E mag B = J de B dj kinetic energy: E kin B = E B E mag B < 0 gain!! s-wave and d-wave symmetries favored Spin-charge separation in doped 2D frustrated quantum magnets p. 2

33 Hole-hole correlations 0.05 C hh (r) 0.04 t>0 (s-wave) & J=0.3 t<0 (d-wave) & J=0.3 J=0.6 t>0 (s-wave) t<0 (d-wave) 2.6 R hh (a) (b) r J/ t - No hole-hole repulsion for t > 0 - Pair size 3 lattice spacings Spin-charge separation in doped 2D frustrated quantum magnets p. 2

34 Correlated pair hopping - Analogy with fully frustrated TB model: interaction-induced delocalized 2-particle BS Vidal & Douçot, PRB 65, (2002) Spin-charge separation in doped 2D frustrated quantum magnets p. 2