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

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

3. ZnCu3(OH)6Cl2 (S=1/2) Kagome Magnet
Helton et al. 98, 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, (1956) ; Zinkin et al PRB 56, (1997);
Hermele et al. PRB 69, (2004); Isakov et al PRL 93, (2004);
Helney PRB 71, (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, (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, (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. Re-write spin as a fermion bilinear: Solve the mean-field theory with
MIT Work on Kagome HAF
Ran et al. PRL 98, (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 1 2 3
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, (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: v1.
Acknowledgment: Dung-Hai Lee, Ying Ran, Ashvin Vishwanath at UCB

19. 1 3 4 2

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

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