Vector Chiral States in Low-dimensional Quantum Spin Systems

1. Vector Chiral States in Low-dimensional Quantum Spin Systems Raoul Dillenschneider Department of Physics, University of Augsburg, Germany Jung Hoon Kim & Jung Hoon Han Department of Physics, Sungkyunkwan University, Korea arXiv : 0705.3993

2. Spiral Order Ferroelectricity Background Information • In Multiferroics : Control of ferroelctricity using magnetism • Magnetic Control of Ferroelectric Polarization (TbMnO3) T. Kimura et al., Nature 426 55, 2003 Connection to Magnetism • Magnetic Inversion Symmetry Breaking Ferroelectricity in TbMnO3 Kenzelmann et al., PRL 95, 087206 (2005)

3. Antiferromagnetic Ferromagnetic -1 +1 Define an order parameter concerned with rotation of spins Background Information (2) • “Conventional” magnetic order • Spiral magnetic order

4. Microscopic Spin-polarization coupling Inverse Dzyaloshinskii-Moriya(DM) type: Chirality (ij) can couple to Polarization (Pij)

5. T, frustration Paramagnetic Magnetic Chiral Ferroelectric Is a (vector) Chiral Phase Possible? Usually, T, frustration Spiral Magnetic Collinear Magnetic Paramagnetic Ferroelectric Possible?

6. Search for Chiral Phases– Previous Works (Nersesyan) • Nersesyan et al. proposed a spin ladder model (S=1/2) with nonzero chirality in the ground state Nersesyan PRL 81, 910 (1998) • Arrows indicate sense of chirality

7. Search for Chiral Phases – Previous Works (Nersesyan) • Nersesyan’s model equivalent to a single spin chain (XXZ model) with both NN and NNN spin-spin interactions

8. Search for Chiral Phases – Previous Works (Hikihara) • Hikihara et al. considered a spin chain with nearest and next-nearest neighbour interactions for S=1 Hikihara JPSJ 69, 259 (2000) • Define spin chirality operator • DMRG found chiral phase for S=1 when j=J1/J2 is sufficiently large No chirality when S=1/2

9. Search for Chiral Phases – Previous Works (Zittarz) • Meanwhile, Zittartz found exact ground state for the class of anisotropic spin interaction models with NN quadratic & biquadratic interactions Klumper ZPB 87, 281 (1992) • Both the NNN interaction (considered by Nersesyan, Hikihara) and biquadratic interaction (considered by Zittartz) tend to introduce frustration and spiral order • Zittartz’s ground state does not support spin chirality

10. Questions that arise Search for Chiral Phases– Previous Works • All of the works mentioned above are in 1D • Chiral ground state carries long-range order in the chirality correlation of SixSjy-SiySjx • No mention of the structure of the ground state in Hikihara’s paper; only numerical reports • Spin-1 chain has a well-known exactly solvable model established by Affleck-Kennedy-Lieb-Tesaki (AKLT) • What about 2D (classical & quantum) ? • How do you construct a spin chiral state? • Applicable to AKLT states?

11. Search for Chiral Phases– Recent Works (More or Less) • A classical model of a spin chiral state in the absence of magnetic order was recently found for 2D Jin-Hong Park, Shigeki Onoda, Naoto Nagaosa, Jung Hoon Han arXiv:0804.4034 (submitted to PRL) • Antiferromagnetic XY model on the triangular lattice with biquadratic exchange interactions

12. - - - + + + - - - + + + - - - + + + Search for Chiral Phases– Recent Works (Park et al.) Order parameters New order parameter • 2N degenerate ground states

13. T Search for Chiral Phases– Recent Works (Park et al.) • With a large biquadratic exchange interaction (J2 ), a non-magnetic chiral phase opens up T • Paramagnetic • (Non-magnetic) • Nonchiral • Non-magnetic • Chiral • Nematic • Magnetic • Chiral J2/J1 J2/J1=9

14. Search for Chiral Phases– Recent Works (Dillenschneider et al.) • Construction of quantum chiral states • Start with XXZ Hamiltonian Raoul Dillenschneider, Jung Hoon Kim, Jung Hoon Han arXiv:0705.3993 (Submitted to JKPS) Include DM interaction

15. Search for Chiral Phases– Recent Works (Dillenschneider et al.) • Consider “staggered” DM interactions M O M O M O M O M O M O M O M • Staggered oxygen shifts gives rise to “staggered” DM interaction “staggered” phase angle, “staggered” flux • We can consider the most general case of arbitrary phase angles:

16. Connecting Nonchiral & Chiral Hamiltonians • Define the model on a ring with N sites: • Carry out unitary rotations on spins • Choose angles such that • This is possible provided • Hamiltonian is rotated back to XXZ:

17. Connecting Nonchiral & Chiral Hamiltonians • Eigenstates are similarly connected:

18. Connecting Nonchiral & Chiral Hamiltonians • Correlation functions are also connected. In particular, • Since and • It follows that a non-zero spin chirality must exist in • Eigenstates of are generally chiral.

19. Generating Eigenstates • Given a Hamiltonian with non-chiral eigenstates, a new Hamiltonian with chiral eigenstates will be generated with non- uniform U(1) rotations:

20. AKLT States • Well-known Affleck-Kennedy-Lieb-Tasaki (AKLT) ground states and parent Hamiltonians can be generalized in a similar way Arovas, Auerbach, Haldane PRL 60, 531 (1988) • Using Schwinger boson singlet operators • AKLT ground state is

21. From AKLT to Chiral AKLT • Aforementioned U(1) rotations correspond to • Chiral-AKLT ground state is

22. Correlations in chiral AKLT states • Equal-time correlations of chiral-AKLT states easily obtained as chiral rotations of known correlations of AKLT states: • With AKLT: • With chiral-AKLT:

23. Excitations in Single Mode Approximations • Calculate excited state energies in single-mode approximation (SMA) for uniformly chiral AKLT state: • With AKLT: • With chiral-AKLT:

24. Excitation energies in SMA

25. Summary and Outlook • Created method of producing ground states with nonzero vector spin chirality • Well-known AKLT states have been generalized to chiral AKLT states. • Excitation energy for the uniformly chiral AKLT state has been calculated within SMA along with various correlation functions. • Need to search for a quantum spin model with long-range vector spin chirality correlation (without “artificial” DM interactions)