Theory of absorption in
the Shastry-Sutherland material SrCu2(BO3)2
Shin Miyahara (Fukuoka University, Japan)
Acknowledgement
Nobuo Furukawa (Aoyama Gakuin University, Japan)
Isao Maruyama (Fukuoka Institute of Technology, Japan)
1.
2.
3.
4.

OutlineIntroduction
Frustrated magnetism SrCu2(BO3)2
Dynamical magnetoelectric effects in SrCu2(BO3)2
Conclusion

Magnetoelectric(ME) effects
ferroelectricity P, ferromagnetism M

conventional materials
P can be tuned by electric field E
M can be tuned by magnetic field HEPHP

multiferroics
coexistence of multi (anti)ferroic orders
magnetoelectric multiferroics
coexistence of P and M
P can be tuned by magnetic field H
M can be tuned by electric field EEPM
Hcross correlation
coupling between P and M
e.g. spin current mechanism

HMEM

𝑃𝑃𝑖𝑖𝑖𝑖sc = 𝑑𝑑𝒆𝒆𝑖𝑖𝑖𝑖 × (𝑺𝑺𝑖𝑖 × 𝑺𝑺𝑗𝑗 )Electric polarization due to spin current mechanism
Electric polarization due to spin-current (inverse DM interaction) mechanism
𝑷𝑷sc𝑖𝑖𝑖𝑖 = 𝑑𝑑𝒆𝒆𝑖𝑖𝑖𝑖 × (𝑺𝑺𝑖𝑖 × 𝑺𝑺𝑗𝑗 )coupling between P and spin-pair𝑖𝑖

𝑷𝑷sc𝑖𝑖𝑖𝑖

𝒆𝒆𝑖𝑖𝑖𝑖𝑗𝑗

H. Katsura et al., Phys. Rev. Lett. 95 057205 (2005),
M. Mostovoy, Phys. Rev. Lett. 96 067601 (2006)
I.A. Sergienko and E. Dagotto, Phys. Rev. Lett. 96 067601 (2006)

cycloidal plane flops by H // c(ab-cycloid bc-cycloid)

electric polarization flop
(P // a P // c)Dynamical magentoelectric effects
electromagnon in RMnO3

dynamical ME effect:cross correlation by time-dependent fields

𝑴𝑴(𝝎𝝎) is induced by electric field 𝐸𝐸(𝜔𝜔)𝑷𝑷(𝝎𝝎) is induced by magnetic field 𝐻𝐻(𝜔𝜔)𝐻𝐻(𝜔𝜔)

electromagnon
electro-active magnon(mostly in GHz～THz region)

ε2𝐸𝐸(𝜔𝜔)

3𝑷𝑷(𝝎𝝎)𝑴𝑴(𝝎𝝎)

𝐸𝐸(𝜔𝜔)

spin wave

N. Kida et al., Phys. Rev. B 78, 104414 (2008)

exploring other cases of the electro-active magnetic excitation

2100

102030-140

Wavenumber (cm )A. Pimenov et al., Nat. Phys. 2, 97 (2006)

spin gap system SrCu2(BO3)2

Two-dimensional spin gap system SrCu2(BO3)2

Cu: 𝑆𝑆 =12Cu(BO3) plane

spin singlet ground statespin gap ∆~35K, 730 GHz
magnetization plateauxorthogonal dimer model

Shastry-Sutherland model
exact dimer singlet states
B.S. Shastry and B. Sutherland, Physica B 108 1069 (1981)
H. Kageyama et al., Phys. Rev. Lett. 82 3168 (1999)
S. Miyahara and K. Ueda, Phys. Rev. Lett. 82 3701 (1999)

Magnetic features of SrCu2(BO3)2
ESR

spin gap ∆~700 GHz
splitting of the triplet excitation due to DM interactions

H. Nojiri et al, J. Phys. Soc. Jpn. 68 2906 (1999) , ibid. 72 3243 (2003)
T. Rõõm et al, Phys. Rev. B 61 14342 (2000)
O. Cepas et al, Phys. Rev. Lett. 87 167205 (2001)

magnetization plateauxmagnetization plateaux at 1/4, 1/3, 1/2
Y.H. Matsuda et al, Phys. Rev. Lett. 111 137204 (2013)

Exact ground state and spin gap of the Shastry-Sutherland model
Shastry-Sutherland model

1𝑆𝑆 =2𝐽𝐽′𝐽𝐽ground state′= 𝑱𝑱𝑱𝑱𝑱1 � 𝑺𝑺3 + 𝑱𝑱𝑱𝑱𝑱2 � 𝑺𝑺3 = 𝑱𝑱𝑱(𝑺𝑺1 + 𝑺𝑺2 ) � 𝑺𝑺3ℎ𝑎𝑎𝑎𝑎

𝑏𝑏𝑎𝑎
product of dimer singlet states|GS⟩ = �𝛼𝛼12|↑↓⟩𝛼𝛼 − |↓↑⟩𝛼𝛼′ | ⟩ | ⟩𝑠𝑠 𝑎𝑎 𝑠𝑠 𝑏𝑏 = 0ℎ𝑎𝑎𝑎𝑎|𝑠𝑠⟩𝛼𝛼 =exact ground state when12|↑↓⟩𝛼𝛼 − |↓↑⟩𝛼𝛼𝐽𝐽′< 0.675
𝐽𝐽B.S. Shastry and B. Sutherland, Physica B 108 1069 (1981)
A. Koga and N. Kawakami, PRL 84 4461 (2000), F. Corboz and F. Mila, PRB 87 115144 (2013)

𝑦𝑦spin gap
one of the dimer singlet𝑥𝑥

� = 𝐽𝐽 � 𝑺𝑺𝑖𝑖 � 𝑺𝑺𝑗𝑗 + 𝐽𝐽𝐽 � 𝑺𝑺𝑖𝑖 � 𝑺𝑺𝑗𝑗
𝐻𝐻n.n.

n.n.n.

|𝑡𝑡1 ⟩𝛼𝛼 = |↑↑⟩𝛼𝛼 |𝑡𝑡0 ⟩𝛼𝛼 =12triplet|↑↓⟩𝛼𝛼 + |↓↑⟩𝛼𝛼 |𝑡𝑡−1 ⟩𝛼𝛼 = |↓↓⟩𝛼𝛼

3rd order of perturbation

∆𝐽𝐽′
=1−𝐽𝐽
𝐽𝐽

21 𝐽𝐽′−2 𝐽𝐽

3|𝑇𝑇⟩
|𝑆𝑆⟩

spin gap
𝐵𝐵Absorption of spin excitation in SrCu2(BO3)2
spin gap excitationtransition from singlet g.s. to triplet excitations forbidden transition in Heisenberg model
c.f. Anisotropic terms induce magneto active mode. e.g. M. Oshikawa , JPSJ 72 Suppl. B 36 (2003), T. Saka and H. Shiba JPSJ 63 867 (1994)
Experimentally, spin gap and bound states of triplet excitations have been observed in absorption
FIR
𝑬𝑬(𝝎𝝎) ⊥ 𝑧𝑧
ESR
𝑬𝑬(𝝎𝝎) ∥ 𝑧𝑧

electro-active magnetic excitations
𝑬𝑬(𝝎𝝎) ⊥ 𝑧𝑧

spin gap excitation
𝑬𝑬(𝝎𝝎) ⊥ 𝑧𝑧
24.2 cm−1
25.7 cm−1𝑬𝑬(𝝎𝝎) ∥ 𝑧𝑧

bound states of two triplets
37.5, 43.0, 52.3, 53.5 cm−1𝑬𝑬(𝝎𝝎) ⊥ 𝑧𝑧

H. Nojiri et al, J. Phys. Soc. Jpn. 68 2906 (1999) , ibid. 72 3243 (2003)
T. Rõõm et al, Phys. Rev. B 61 14342 (2000), ibid 70 14417 (2004) , O. Cepas et al, Phys. Rev. Lett. 87 167205 (2001)

Purpose : clarify the origin of the spin gap excitation (selection rule of absorption)

Dynamical electric susceptibility
the continued fraction method by using Lanczos method

Dynamical electric susceptibility
β𝛽𝛽†𝜀𝜀𝛼𝛼𝛼𝛼 (𝜔𝜔) ∝ 0 𝑃𝑃�𝛼𝛼1�𝛼𝛼𝛽𝛽 0𝑃𝑃�𝜔𝜔 + 𝐸𝐸0 + 𝑖𝑖𝜀𝜀 − 𝐻𝐻spin current mechanism

𝛼𝛼 = 𝑥𝑥, 𝑦𝑦, 𝑧𝑧

� AS = � 𝑑𝑑𝒆𝒆𝑖𝑖𝑖𝑖 × (𝑺𝑺𝑖𝑖 × 𝑺𝑺𝑗𝑗 )𝑷𝑷n.n.

𝑬𝑬(𝝎𝝎) ⊥ 𝑧𝑧

E. Dagotto, Rev. Mod. Phys. 66 763 (1994)

β = AS, S𝐵𝐵ex ⁄𝐽𝐽 = 0.1

𝑬𝑬(𝝎𝝎) ∥ 𝑧𝑧

exchange striction mechanism
� S = � 𝛱𝛱𝒆𝒆𝑖𝑖𝑖𝑖 𝑺𝑺𝑖𝑖 � 𝑺𝑺𝑗𝑗
𝑷𝑷n.n.n.

𝑬𝑬(𝝎𝝎) ⊥ 𝑧𝑧

𝑬𝑬(𝝎𝝎) ∥ 𝑧𝑧

𝑬𝑬(𝝎𝝎) ⊥ 𝑧𝑧

𝑬𝑬(𝝎𝝎) ⊥ 𝑧𝑧

Comparison with the perturbation calculation
𝐸𝐸(𝜔𝜔) ∥ 𝑥𝑥

𝐸𝐸(𝜔𝜔) ∥ 𝑥𝑥

3rd order of perturbation
spin gap
23∆
𝐽𝐽′
1 𝐽𝐽′
−=1−𝐽𝐽𝐽𝐽
2 𝐽𝐽
bound sate of two triplets

𝐸𝐸(𝜔𝜔) ∥ 𝑧𝑧
𝐽𝐽𝐽⁄𝐽𝐽 = 0.2
spin gap and 𝑆𝑆 𝑇𝑇 = 0 bound states are electro active.

𝑆𝑆 T = 0

4 states (one of them 𝐸𝐸(𝜔𝜔) ⊥ 𝑧𝑧 active)
K. Totsuka et al, Phys. Rev. Lett. 86 520 (2001)

Comparison with the perturbation calculation
3rd order of perturbation
𝐽𝐽𝐽⁄𝐽𝐽 = 0.2

bound sate of two triplets
𝐸𝐸(𝜔𝜔) ∥ 𝑥𝑥 𝐸𝐸(𝜔𝜔) ∥ 𝑧𝑧

𝑆𝑆 T = 1
𝑆𝑆 𝑇𝑇 = 1 bound state are electro active.

4 statestwo of them 𝐸𝐸(𝜔𝜔) ⊥ 𝑧𝑧, 𝐸𝐸(𝜔𝜔) ∥ 𝑧𝑧 active
K. Totsuka et al, Phys. Rev. Lett. 86 520 (2001)

Absorption of spin excitation in SrCu2(BO3)2
spin gap excitationtransition from singlet g.s. to triplet excitations forbidden transition in Heisenberg model
c.f. Anisotropic terms induce magneto active mode. e.g. M. Oshikawa , JPSJ 72 Suppl. B 36 (2003), T. Saka and H. Shiba JPSJ 63 867 (1994)
Experimentally, spin gap and bound states of triplet excitations have been observed in absorption
FIR𝑬𝑬(𝝎𝝎) ⊥ 𝑧𝑧
ESR𝑬𝑬(𝝎𝝎) ∥ 𝑧𝑧

electro-active magnetic excitations
𝑬𝑬(𝝎𝝎) ⊥ 𝑧𝑧

spin gap excitation
𝑬𝑬(𝝎𝝎) ⊥ 𝑧𝑧
24.2 cm−1
25.7 cm−1𝑬𝑬(𝝎𝝎) ∥ 𝑧𝑧

bound states of two triplets
37.5, 43.0, 52.3, 53.5 cm−1
𝑬𝑬(𝝎𝝎) ⊥ 𝑧𝑧

H. Nojiri et al, J. Phys. Soc. Jpn. 68 2906 (1999) , ibid. 72 3243 (2003)
T. Rõõm et al, Phys. Rev. B 61 14342 (2000), ibid 70 14417 (2004) , O. Cepas et al, Phys. Rev. Lett. 87 167205 (2001)

Purpose : clarify the origin of the spin gap excitation (selection rule of absorption)

Effects of Dzyaloshinskii-Moriya interaction
Shastry-Sutherland model1
𝑆𝑆 =2𝐽𝐽′𝐽𝐽�DM = � 𝑫𝑫𝑖𝑖𝑗𝑗 � 𝑺𝑺𝑖𝑖 × 𝑺𝑺𝑗𝑗 + � 𝑫𝑫′𝑖𝑖𝑖𝑖 � 𝑺𝑺𝑖𝑖 × 𝑺𝑺𝑗𝑗
𝐻𝐻n.n.

𝑫𝑫𝑖𝑖𝑖𝑖 = �|𝑇𝑇⟩𝑦𝑦𝑥𝑥

� = 𝐽𝐽 � 𝑺𝑺𝑖𝑖 � 𝑺𝑺𝑗𝑗 + 𝐽𝐽𝐽 � 𝑺𝑺𝑖𝑖 � 𝑺𝑺𝑗𝑗
𝐻𝐻n.n.

n.n.n.

|𝑆𝑆⟩

0, 𝐷𝐷, 0
𝐷𝐷, 0, 0n.n.n.

A bondB bond𝐷𝐷 = 𝐷𝐷 ′ ≠ 0spin gap′

𝐷𝐷 = 𝐷𝐷 = 0

𝐵𝐵𝑫𝑫𝑫𝑖𝑖𝑖𝑖 = 0, 0, ±𝐷𝐷𝐷

|𝑇𝑇⟩
|𝑆𝑆⟩𝐵𝐵
splitting of the triplet excitation

O. Cepas et al, Phys. Rev. Lett. 87 167205 (2001), K. Kodama et al, J. Phys.: Condens. Matter 17 L61 (2005)

Electro active magnetic excitation in SrCu2(BO3)2The effects of DM interactions
�DM = � 𝑫𝑫𝑖𝑖𝑗𝑗 � 𝑺𝑺𝑖𝑖 × 𝑺𝑺𝑗𝑗 + � 𝑫𝑫′𝑖𝑖𝑖𝑖 � 𝑺𝑺𝑖𝑖 × 𝑺𝑺𝑗𝑗
𝐻𝐻n.n.

n.n.n.

𝑫𝑫𝑖𝑖𝑖𝑖 = �0, 𝐷𝐷, 0
𝐷𝐷, 0, 0

𝑫𝑫𝑫𝑖𝑖𝑖𝑖 = 0, 0, ±𝐷𝐷𝐷

A bondB bondFIR

T. Rõõm et al, PRB 61 14342 (2000), ibid 70 14417 (2004)

𝐽𝐽′𝐷𝐷
𝐷𝐷 ′−1𝐽𝐽 = 85K~59 cm , = 0.635, = 0.034, = 0.02,
𝐽𝐽
𝐽𝐽
𝐽𝐽
K. Kodama et al, J. Phys.: Condens. Matter 17 L61 (2005)

𝑁𝑁 = 20𝐵𝐵ex ⁄𝐽𝐽 = 0𝐸𝐸 ∥ 𝑥𝑥

� AS = � 𝑑𝑑𝒆𝒆𝑖𝑖𝑖𝑖 × (𝑺𝑺𝑖𝑖 × 𝑺𝑺𝑗𝑗 )𝑷𝑷n.n.

𝐸𝐸 ∥ 𝑧𝑧

𝐸𝐸 ∥ 𝑥𝑥� S = � 𝛱𝛱𝒆𝒆𝑖𝑖𝑖𝑖 𝑺𝑺𝑖𝑖 � 𝑺𝑺𝑗𝑗
𝑷𝑷n.n.n.

Π ≫ 𝑑𝑑𝑬𝑬 ⊥ 𝑧𝑧

𝑬𝑬 ⊥ 𝑧𝑧

𝑬𝑬 ∥ 𝑧𝑧

The peak positions and selection rules of the main peaks are consistent with FIR experiment.

The magneto active excitations
DM interactions allow magneto active excitation
T. Sakai et al. J. Phys. Soc. Jpn. 69 3521 (2000)

dynamical magnetic susceptibilities
�𝛼𝛼†𝜇𝜇𝛼𝛼𝛼𝛼 (𝜔𝜔) ∝ 0 𝑀𝑀
� = � 𝑺𝑺𝑖𝑖𝑴𝑴1
�𝛼𝛼 0𝑀𝑀�𝜔𝜔 + 𝐸𝐸0 + 𝑖𝑖𝜀𝜀 − 𝐻𝐻
𝛼𝛼 = 𝑥𝑥, 𝑦𝑦, 𝑧𝑧

dynamical electric susceptibility
β𝛽𝛽†𝜀𝜀𝛼𝛼𝛼𝛼 (𝜔𝜔) ∝ 0 𝑃𝑃�𝛼𝛼𝐽𝐽 = 85K~59 cm−1 ,1�𝛼𝛼𝛽𝛽 0𝑃𝑃�𝜔𝜔 + 𝐸𝐸0 + 𝑖𝑖𝜀𝜀 − 𝐻𝐻
𝛼𝛼 = 𝑥𝑥, 𝑦𝑦, 𝑧𝑧

β = AS, S𝐽𝐽′
𝐷𝐷𝐷𝐷 ′= 0.635, = 0.034, = 0.02,
𝐽𝐽
𝐽𝐽
𝐽𝐽

Magnetic field dependence 𝐵𝐵ex ∥ 𝑧𝑧(𝜔𝜔) ∥ 𝑧𝑧
𝜇𝜇𝑧𝑧𝑧𝑧𝐻𝐻(𝜈𝜈)singlet-triplet excitation allowed due DM interaction

AS𝜖𝜖𝑧𝑧𝑧𝑧(𝜔𝜔)𝐸𝐸(𝜈𝜈) ∥ 𝑧𝑧

singlet-triplet excitation induced by electric components of lightbound states of two triples induced by electric components of lightComparison with ESR
magnetic-active

peak positions

ESR
external magnetic field 𝐵𝐵 ∥ 𝑧𝑧

𝐻𝐻(𝜈𝜈) ∥ 𝑧𝑧

electro-active

electro-active
magnetic-active𝐸𝐸(𝜈𝜈) ∥ 𝑧𝑧

H. Nojiri et al, J. Phys. Soc. Jpn. 72 3243 (2003)

Comparison with FIRAS𝜖𝜖𝑧𝑧𝑧𝑧(𝜔𝜔)linearly polarized light

AS𝜖𝜖𝑧𝑧𝑧𝑧(𝜔𝜔)𝐵𝐵ex ∥ 𝑧𝑧

T. Rõõm et al, PRB 70 14417 (2004)

𝐵𝐵ex ∥ 𝑥𝑥

ConclusionConclusion The spin-electron coupling and DM interactions allow an absorption in SrCu2(BO3)2.
 The selection rule is consistent with experimental observation
 Magnetic excitations are allowed by the spin-electron coupling even in Heisenberg model.

c.f. TlCuCl3 : S. Kimura et al. J. Magn. Magn. Mater. 310 1218 (2007), KCuCl3 : S. Kimura et al. Appl. Magn Reson. 46 1035 (2015)
The clarification of magnetic excitation processes induced by 𝐸𝐸(𝜔𝜔) can play an important rolein the investigation of magnetic excitation in non-magnetic ground states.
Open issues
 The selection rule in the plaquette singlet statepressure induced phase transition in in SrCu2(BO3)2 T. Sakurai et al, J. Phys. Soc. Jpn. 87 033701 (2018)
 The proposal of the novel magnetic excitations and magnetoelectric effects in spin systems