Topological states in quantum antiferromagnets

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1 Pierre Pujol Laboratoire de Physique Théorique Université Paul Sabatier, Toulouse Topological states in quantum antiferromagnets Thanks to I. Makhfudz, S. Takayoshi and A. Tanaka

2 Quantum AF systems : GS zoology Non frustrated AF: (anti-ferro) magnetic order

3 Quantum AF systems : GS zoology Non frustrated AF: Frustrated AF: (anti-ferro) magnetic order Non magnetic order

4 Quantum AF systems : GS zoology GS of the AKLT type (SPT) Z 2 spin liquid : The Rokhsar-Kivelson model in the triangular lattice (Moessner and Sondhi)

5 Chiral spin liquids Looking for a QHE state in magnetic degrees of freedom (Kalmeyer and. Laughlin, Wen, Wilczek, and Zee, Yang, Warman, and Girvin..)

6 Chiral spin liquids Looking for a QHE state in magnetic degrees of freedom (Kalmeyer and. Laughlin, Wen, Wilczek, and Zee, Yang, Warman, and Girvin..) " The kagome lattice with explicitely broken TRI is a good candidate (ex. Fradkin et al., Moessner et al.): H =J { S x i Sj x + Sy i Sy j + } λsz i Sz j hext S z i, i,j i H ch = h χ ij k ( ) = h S i (S j S k ),

7 The path integral approach Write down a path integral for spins (Haldane) A particular contribution to the action, the Berry phase term

8 The path integral approach Haldane s NLSM for spin chains: S eff [n(τ,x)] = 1 2g Q τx = 1 4π dτdx { ( τ n) 2 +( x n) 2} dτdxn τ n x n Z. +2 isq x

9 Planar and CP 1 representation: d in the next a µ = µ φ/2, ion correspond S eff n pl (cos φ, sin φ, 0), at the appropriate modi The path integral approach Q v = 1 2π S Θ = i Θ 2π And for an open spin S chain, integrate to get a boundary topological term (Ng, 1994) Example : spin 1 AKLT chain S edge = ±is dτa τ { } dτdx( τ x x τ )φ Z. dτdx( τ a x x a τ ) (Θ =2πS),

10 The path integral approach 2-D case (square lattice), monopoles play a role (Haldane): wo-componen Seff 2d = 1 n = z σ 2 z, a µ = iz µ z, 2K = dτd 2 r dτd 2 r(ϵ µνλ ν a λ ) 2 + i πs 2 Qtot mon { 1 2K (ϵ µνλ ν a λ ) 2 + i S 4 ϵ µνλ µ ν a λ } Only even-integer spins admit non-degenerate GS ( Work donne in colaboration with S. Takayoshi and A. Tanaka, arxiv:

11 The path integral approach And for a system with boundary, integrate again to get a boundary topological term" " " S " y-edge = ±i S dτdx( τ a x x a τ )=±i πs 4 2 Q τx " " 0 (mod 4) vs 2 (mod 4) : Only the second is SPT

12 The path integral approach Adapt the path integral approach to the presence of a magnetic field (Tanaka, Totsuka, Hu) The starting point is a quasi-classical configuration with non-zero neat magnetization

13 The path integral approach ( ) } Integrate out short ranged fields to get the effective action for the Goldstone field : S = dxdτ { Kτ 2 ( τ φ) 2 + K x 2 ( xφ) 2 +i ( S m a ) } ( τ φ) Berry phase term

14 Coupling to (Chern) Charge degrees of freedom Then, extend the study to the presence of moving holes

15 Coupling to (Chern) Charge degrees of freedom Doped electrons feel an effective flux of ± π per plaquette Work donne in colaboration with I. Makhfudz, Phys. Rev. B 92, pp , 2015.

16 Coupling to (Chern) Charge degrees of freedom Dirac-like dispersion relation at half filling for the electrons

17 Coupling to (Chern) Charge degrees of freedom Add dimerization to gap the charge degrees of freedom Effective action for the charge in the continuum :

18 Coupling to (Chern) Charge degrees of freedom Integrate out these harmless degrees of freedom and get an effective action for the spin sector

19 Coupling to (Chern) Charge degrees of freedom Effective action for the spin sector : Dual vortex theory New contribution to the Berry phase term :

20 Coupling to (Chern) Charge degrees of freedom The partition function of an anyon gaz here Protection against the spin gap

21 Perspectives: in search of a chiral plateau state Find a microscopic model where the vortex condensation is possible This would realize the equivalent of a QHE state in the spin sector This scenario is expected to reproduce when coupling to a generic Chern insulator Thank you for your attention!