Chiral Spin States in the Pyrochlore Heisenberg Antiferromagnet

1. ChiralSpin States in the Pyrochlore Heisenberg Antiferromagnet Jung Hoon Han (SungKyunKwan U, Korea) JH Kim, JH Han, PRB(R) 2008

2. Pyrochlore Lattice (3D) Motif=tetrahedron Motif=triangle Kagome Lattice (2D)

3. Kagome Magnet ZnCu3(OH)6Cl2 (S=1/2) Helton et al. 98, 107204 PRL (07) • Heisenberg AFM (J=17meV) on Kagome lattice • Lack of spin gap • Lack of LRO down to T << J • “quantum spin liquid” with “exotic excitation”

4. Pyrochlore Magnet • (spin-ice)Dy2Ti2O7 , Nd2Mo2O7 • (coop. para.) Tb2Ti2O7 • (spin-Peierls) ZnCr2O4 • (Kondo lattice, AHE) Nd2Mo2O7 , Pr2Ir2O7 • (Heavy fermion) LiV2O4 • Some are insulating, others metallic • (We will focus on non-metallic pyrochlore magnet) • Some spins Ising, others Heisenberg + anisotropy • No known examples of insulating, S=1/2 • HAFM with pristine pyrochlore lattice

5. Classical Magnet on Pyrochlore Lattice • Classical Heisenberg AFM spins show • No LRO, No ObyD, extensive GS degeneracy • Reimers PRB 45, 7287 (1992); Moessner, Chalker PRL 80, 2929 (1998) • Ising AFM spins show • No LRO, No ObyD, extensive GS degeneracy, • GS manifold shows dipolar spin-spin correlations • Anderson PR 102, 1008 (1956) ; Zinkin et al PRB 56, 11786 (1997); • Hermele et al. PRB 69, 064404 (2004); Isakov et al PRL 93, 167204 (2004); • Helney PRB 71, 014424 (2005)

6. Constraints • Ground state has for all tetrahedra (up/down) • Infinitely many classical solutions • Quantum case is more uncertain

7. Quantum S=1/2 Magnet on Pyrochlore Lattice • Begin with one-tetrahedron solution of HAFM • Inter-tetrahedra coupling is treated perturbatively (J’/J) • Harris, Berlinsky, Bruder, JAP 69, 5200 (1991) • Canals, Lacroix, PRL 80, 2933 (1998) • Tsunetsugu, JPSJ 70, 640 (2000); PRB 65, 024415 (2001) J’ J

8. A Single Tetrahedron • Three dimer solutions of S=1/2 HAFM • [12][34] , [13][24], [14][23], only two are independent • Two chiral solutions

9. A Lattice of Tetrahedra • When continued to pyrochlore lattice, the quantum GS may be • (i) dimer solid with broken translation symmetry • (ii) chiral spin liquid with broken time symmetry • (iii) mixture of the two • Previous theories based on J’/J expansion found • enhanced dimer correlations, no sign of T-breaking • We find T-breaking chiral spin liquid

10. Spin Chirality of Wen, Wilczek, Zee PRB 39, 11413 (1989) • Notion of chirality proposed as a hidden order of spin liquid state • Need some sort of frustration to realize chirality • (large J’ or geometric frustration)

11. MIT Work on Kagome HAF Ran et al. PRL 98, 117205 (2008) • Re-write spin as a fermion bilinear: • Solve the mean-field theory with • Refine the MF state with Gutzwiller projection of doubly occupied sites

12. Rokhsar’s Rule 3 2 1 Rokhsar PRL 65, 1506 (1990)

13. Connection of MFT to Chirality • If Rokhsar’s Rule was right, maximal chirality should obtain for triangle-based lattices (Kagome, pyrochlore)

14. VMC Ground State is Not Chiral for Kagome Ran et al. PRL 98, 117205 (2008)

15. VMC Ground State is Chiral for Pyrochlore JH Kim, JH Han, PRB(R) 2008

16. VMC Energies • Among a number of different flux configurations, • the chiral states stand out

17. Flux after Projection • Average flux through each triangle can be calculated within VMC • Amount of flux is reduced from mean-field value pi/2 • CSL state with long-range ordered chiralities

18. Summary • Combined fMFT+VMC finds CSL ground state • of Heisenberg model on pyrochlore lattice • Dimer instability found previously contrasts with T-breaking state • found here • Resolving the contending views will require new ideas See also: F. J. Burnell, Shoibal Chakravarty, and S. L.Sondhi, arXiv:0809.0528v1. Acknowledgment: Dung-Hai Lee, Ying Ran, Ashvin Vishwanath at UCB

19. 2 1 3 4 1 3 2

20. 12 16 11 9 10 4 15 13 14 3 1 8 2 5 7 6

21. Effective Theory? • Construction of effective theory requires knowledge of • mean-field band structure