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

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

Background Information (2)

“Conventional” magnetic order

Spiral magnetic order

Define an order parameter concerned with rotation of spins

Ferromagnetic Antiferromagnetic

+1

-1

Chirality (ij) can couple to Polarization (Pij)

Microscopic Spin-polarization coupling

Inverse Dzyaloshinskii-Moriya(DM) type:

Is a (vector) Chiral Phase Possible?

T, frustrationMagnetic

FerroelectricChiral Paramagnetic

T, frustrationSpiralMagnetic

CollinearMagnetic

Paramagnetic

Ferroelectric

Usually,

Possible?

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

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)

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

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

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?

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

Search for Chiral Phases– Recent Works (Park et al.)

Order parameters New order parameter

2N degenerate ground states

--

++++

++++++

++++

++

++

-- --

------

-- ----

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

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

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

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

Eigenstates are similarly connected:

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.

Connecting Nonchiral & Chiral Hamiltonians

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

Generating Eigenstates

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

Aforementioned U(1) rotations correspond to

Chiral-AKLT ground state is

From AKLT to Chiral AKLT

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

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

With AKLT:

With chiral-AKLT:

Excitations in Single Mode Approximations

Excitation energies in SMA

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)