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. Background Information In Multiferroics : Control of ferroelctricity using magnetism  Magnetic Control of Ferroelectric Polarization (TbMnO 3 ) T. Kimura et al., Nature 426 55, 2003  Magnetic Inversion Symmetry Breaking Ferroelectricity in TbMnO 3 Kenzelmann et al., PRL 95, 087206 (2005) Connection to Magnetism Spiral Order Ferroelectricity

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

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

5. Is a (vector) Chiral Phase Possible? T, frustration Magnetic Ferroelectric Chiral Paramagnetic T, frustration Spiral Magnetic Collinear Magnetic Paramagnetic Ferroelectric Usually, 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.  Nersesyan’s model equivalent to a single spin chain (XXZ model) with both NN and NNN spin-spin interactions Search for Chiral Phases – Previous Works (Nersesyan)

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)  DMRG found chiral phase for S=1 when j=J 1 /J 2 is sufficiently large  Define spin chirality operator 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. 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 S ix S jy -S iy S jx 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) Questions that arise 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 2 N degenerate ground states - + + ++ + + + + + - - --- ---

13. J 2 /J 1 T Paramagnetic Paramagnetic (Non-magnetic) (Non-magnetic) Nonchiral Nonchiral Magnetic Magnetic Chiral Chiral Non-magnetic Non-magnetic Chiral Chiral Nematic Nematic J 2 /J 1 =9 Search for Chiral Phases – Recent Works (Park et al.)  With a large biquadratic exchange interaction (J 2 ), a non-magnetic chiral phase opens up T

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

15. Search for Chiral Phases – Recent Works (Dillenschneider et al.)  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: M O M O M O M O M O M O M O M  Consider “staggered” DM interactions

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

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

18.  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. Connecting Nonchiral & Chiral Hamiltonians

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

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

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

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

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

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)