Spin Parity Effect in monoaxial chiralferromagnetic chain*
Sohei Kodama1, Akihiro Tanaka2, and Yusuke Kato1* arXiv:2209.04227

1: Department of Basic Science,
The University of Tokyo, Japan
2: International Center for Materials Nanoarchitectonics,
National Institute for Materials Science, Japan
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Background: monoaxial chiral magnetsz

D // ey “Monoaxial”

xy

Period of
Chiral Soliton Lattice～J/D for J » D

Chiral Soliton

Review: Togawa et al., J. Phys. Soc. Jpn. 85, 112001 (2016)
Kishine-Ovchinnikov, Solid State Physics(2015)

Dzyaloshinskii (1965)
Chiral Soliton Lattice.

2Issue:
Understanding of Different behaviour in magnetization process
between half-Integer and Integer Spins (a “Spin Parity Effect”)
Numerical
Diagonalization

J=D

J=D

Under the periodic
boundary condition

Numerical
Diagonalization
Under the periodic
boundary condition

Kodama et al
arXiv:2209.04227

Discontinuous
Magnetization=Level-Crossing

J=D

J=D

Continuous
Magnetization=Level-Repulsion
3Earlier Studies
Semiclassical Approach (Spin coherent state and Berry phase argument)
for➢Nonchiral nanomagnets (Braun-Loss PRB1996) :solitons (domain wall ) are generated by Ising anisotropy➢2D chiral magnets (Takashima-Ishizuka-Balents 2016 PRB):
Quantum skyrmion
Those are valid for large S, ⇔We seek for a theory valid for small S.

Cf：Haldane Gap problem
O(3)Nonlinear Sigma model (large S + Berry phase ) ⇔4
AKLT model (S=1)

Clue: J/D ⇒0 limit
J=D/50J=D/50Discontinuous
Magnetization=Level-Crossing

J=D/50J=D/50Continuous
Magnetization=Level-Repulsion
5Model in the J/D ⇒ 0 limit is a canonical modelto understand for spin parity effect in chiralmagnet➢ Difference between half-odd-Integer S and Integer S is also observed in M-H curves in thelimit J/D ⇒ 0 (i.e. Spin Parity Effect exists: ) cf Previous page
➢ Number of Solitons becomes a conserved quantity. (Next page)6

Conserved Quantity=Soliton Number Operator

Conventional Basis0
Sz=+1/2

1

Sz=-1/2
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Numerical Results for HDH model (under the PBC)
➢ Magnetization/deMagnetization
Processes consisting of
successive escape/penetration of Solitons,
N=0
N=1
N=2

N=3
N=4

k=0

k=0
k=p

k=0

k=p

➢ One-Soliton states (N=1) haveminimum energy at k=p
Two-Soliton states (N=2) haveminimum energy at k=0,
which can be proven for the HDH model.

8Three ingredients for proof➢ There exists a signed basis in which Off-diagonal Matrix Elements of theHamiltonian (HDH) are non-positive.

➢ The relation among sign for each basis, translation, and Soliton
numbers➢ All expansion coefficients for the ground state wavefunction in the signedbasis have a definite sign (Perron-Frobenius Theorem)

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Signed BasisEx.

d(n): sum of the coordinates of the left edge of each soliton

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The relation among sign for each basis, translation,and Soliton numbers

Translation Operator

N=1,3,5 (N=2,4,6) Soliton states change (does not change) the signunder the one-site translation.
11Off-diagonal Matrix Elements in conventional basis

diagonalー

local Dzyaloshinskii-Moriya interaction= a three-site termEx.

12Off-diagonal Matrix Elements in conventional basis

Left edge is shifted under the action of hi

Off-diagonal Matrix Elements in the signed basis
non-positive off-diagonal matrix elements(-1)d(n): absorbs the minus sign in the matrix elements13

(Off-diagonal matrix element with definite sign)+Irreducibility. i.e.
Theorem (Perron-Frobenius)⇒Lowest energy eigenstate in eigenspace of N soliton states

VN: set of basisfor N-soliton states
Ex. N=114

(Perron-Frobenius)

From Translation Symmetry➢ One-Soliton states (N=1) have minimum energy at k=p➢ Two-Soliton states (N=2) have minimum energy at k=0,
The above constitutes proof for the HDH model for15S=1/2.

Clues for understanding higher S cases: J/D ⇒0 limit
Soliton Number is not conserved in the DH model (HDH) for higher S .
However:➢ there exists a model (Hp:projected DH model) , which approximates HDHwell .

➢ Soliton numbers Nf (f=1,2….2S) with various heights f are conserved inthe Hp model.
➢ There exists a signed basis in which Off-diagonal Matrix Elements of the16
Hamiltonian (Hp) are non-positive.

How well Hp model approximates HDH model?

➢Approximate model Hp accounts for overall features of magnetization process inHDH.
➢Only solitons with maximum height (amplitude) contribute to the ground state17

Conserved Quantities
Nf number of soliton with height f = 1,2, ・・・2S

Ex. S=3/2,(N1, N2, N3)=(1,1,0)(N1, N2, N3)=(0,1,1) 18

Signed Basisd(n): depends on height-difference in the left side of the maximum
height of solitons.
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The relation among sign for each basis, translation, and Soliton
height and numbers

A soliton state with odd f (even f) changes (does not change) the signunder one-site translation.

(-1)2*2(-1)2*3
f=2, N2=1(-1)3*2

f=3, N3=1(-1)3*320

Off-diagonal Matrix Elements in the signed basis
non-positive off-diagonal matrix elements⇒Lowest energy eigenspace for Nf solitons with height f is given by

From Translation Symmetry21

Crystal momentum depends onheight f (amplitude) of soliton as
well as number Nf“height parity effect”

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Height Parity Effect ⇒Spin Parity effect
In the ground states,
only solitons withmaximum height f=2Scontribute.

“Height parity effect” +Energetics leads to “Spin parity effect”.
23Effect of J (Exchange Interaction)
Size dependence of relative weight of One-soliton state in the ground stateslightly below critical field Hc

One soliton states account for 80% (58%) weight in the ground state for S=1/2 (S=1).24

Summary:• Models in the limit D/J ⇒ ∞ are canonical models to understandSpin parity effect in monoaxial chiral ferromagnetic chain.
• Essential is “height parity effect”, a soliton with odd (even) height f
has the k=p (k=0) in the lowest energy state.
• In the low energy sector, only solitons with maximum height f=2S
contribute. It results in the spin parity effect in the magnetization
process.
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Acknowledgement• JSPS KAKENHI Grants Number 20K03855, 21H01032(YK) and 19K03662 (AT).

• H. Katsura, Y. Suzuki , K. Omiya, J. Kishine, Y. Togawa,
S. C. Furuya, M. Kunimi, T. Tomita, S. Sumita for theirfruitful discussions
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