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Vector Chiral States in Low-dimensional Quantum Spin Systems
Raoul DillenschneiderDepartment of Physics, University of Augsburg, Germany
Jung Hoon Kim & Jung Hoon HanDepartment of Physics, Sungkyunkwan University, Korea
arXiv : 0705.3993
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Background Information
In Multiferroics :Control of ferroelctricity using magnetism
Magnetic Control of Ferroelectric Polarization (TbMnO3) T. Kimura et al., Nature 426 55, 2003
Magnetic Inversion Symmetry Breaking Ferroelectricity in TbMnO3
Kenzelmann et al., PRL 95, 087206 (2005)
Connection to Magnetism
Spiral Order Ferroelectricity
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Background Information (2)
“Conventional” magnetic order
Spiral magnetic order
Define an order parameter concerned with rotation of spins
Ferromagnetic Antiferromagnetic
+1
-1
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Chirality (ij) can couple to Polarization (Pij)
Microscopic Spin-polarization coupling
Inverse Dzyaloshinskii-Moriya(DM) type:
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Is a (vector) Chiral Phase Possible?
T, frustrationMagnetic
FerroelectricChiral Paramagnetic
T, frustrationSpiralMagnetic
CollinearMagnetic
Paramagnetic
Ferroelectric
Usually,
Possible?
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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
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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)
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Search for Chiral Phases – Previous Works (Hikihara) Hikihara et al. considered a spin chain with nearest
and next-nearest neighbour interactions for S=1Hikihara JPSJ 69, 259 (2000)
DMRG found chiral phase for S=1 when j=J1/J2 is sufficiently large
Define spin chirality operator
No chirality when S=1/2
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Search for Chiral Phases – Previous Works (Zittarz) Meanwhile, Zittartz found exact ground state for the class of anisotro
pic 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
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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)
Questions that arise
What about 2D (classical & quantum) ? How do you construct a spin chiral state? Applicable to AKLT states?
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Search for Chiral Phases– Recent Works (More or Less) A classical model of a spin chiral state in the absence of magnetic o
rder was recently found for 2DJin-Hong Park, Shigeki Onoda,
Naoto Nagaosa, Jung Hoon HanarXiv:0804.4034 (submitted to PRL)
Antiferromagnetic XY model on the triangular lattice with biquadratic exchange interactions
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Search for Chiral Phases– Recent Works (Park et al.)
Order parameters New order parameter
2N degenerate ground states
--
++++
++++++
++++
++
++
-- --
------
-- ----
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JJ22/J/J11
TT• ParamagneticParamagnetic (Non-magnetic)(Non-magnetic)• NonchiralNonchiral
• MagneticMagnetic• ChiralChiral
• Non-magneticNon-magnetic• ChiralChiral• NematicNematic
JJ22/J/J11=9=9
Search for Chiral Phases– Recent Works (Park et al.) With a large biquadratic exchange interaction (J2 ), a non-magnetic c
hiral phase opens up
TT
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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
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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
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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
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Eigenstates are similarly connected:
Connecting Nonchiral & Chiral Hamiltonians
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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
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Given a Hamiltonian with non-chiral eigenstates, a new Hamiltonian with chiral eigenstates will be generated with non- uniform U(1) rotations:
Generating Eigenstates
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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
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Aforementioned U(1) rotations correspond to
Chiral-AKLT ground state is
From AKLT to Chiral AKLT
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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
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Calculate excited state energies in single-mode approximation (SMA) for uniformly chiral AKLT state:
With AKLT:
With chiral-AKLT:
Excitations in Single Mode Approximations
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Excitation energies in SMA
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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)