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
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
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)
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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
  (S1/2)
• with nonzero chirality in the ground state

Nersesyan PRL 81, 910 (1998)

• Arrows indicate sense of chirality

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Search for Chiral Phases Previous Works
(Nersesyan)

• Nersesyans model equivalent to a single spin
  chain (XXZ model) with both NN and NNN spin-spin
  interactions

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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 S1

Hikihara JPSJ 69, 259 (2000)

• Define spin chirality operator

• DMRG found chiral phase for S1 when jJ1/J2 is
  sufficiently large

No chirality when S1/2
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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
• Zittartzs 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 Hikiharas 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?

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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 order was recently found for
  2D

Jin-Hong Park, Shigeki Onoda, Naoto Nagaosa, Jung
Hoon Han arXiv0804.4034 (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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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/J19
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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 arXiv0705.3993 (Submitted to JKPS)
Include DM interaction

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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

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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

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Connecting Nonchiral Chiral Hamiltonians

• Eigenstates are similarly connected

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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.

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Generating Eigenstates

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

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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

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From AKLT to Chiral AKLT

• Aforementioned U(1) rotations correspond to

• Chiral-AKLT ground state is

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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

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Excitations in Single Mode Approximations

• Calculate excited state energies in single-mode
  approximation
• (SMA) for uniformly chiral AKLT state

• With AKLT

• With chiral-AKLT

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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)