スピン軌道相互作⽤を有する冷却原⼦系における
𝟐𝑪𝑷 スキルミオン結晶とその派⽣相Yuki Amari

Keio University
amari.yuki.ph@gmail.com
Based on PRB 106, L100406 (2022) and arXiv:23XX.XXXX
Collaborators：Yutaka Akagi, Sven Gudnason, Muneto Nitta, Yakov Shnir(U. Tokyo)(Henan)(Keio)

熱場の量⼦論とその応⽤ @ KEK, 2023年8⽉28⽇(JINR)

Summary of my talk2/12

Hamiltonian = 𝑆𝑈 3 Heisenberg + 𝑆𝑈 2 Heisenberg + Zeeman + Generalized DM
Double-Skyrmion crystal (D-SkX)
Helimagnetic (HM)
Coupling of the 𝑆𝑈 2 Heisenberg

Skyrmionium crystal

7LiSkyrmion crystal (SkX)FMD-SkXMeron crystal (MX)HM
MeX-SkX
crossover
Zeeman coupling

41K

87Rb23Na

Magnetic SkyrmionsExperiments on a thin film of Fe!.# Co!.# Si[X. Z. Yu et.al., Nature 465, 901(2010)]

3/12

Magnetic Skyrmions
Recent trends

Experiments on a thin film of Fe!.# Co!.# Si[X. Z. Yu et.al., Nature 465, 901(2010)]

n 3D topological soliton• Skyrmion string• Hopfion⋮n Composite & Constituent• multi-Skyrmion• Skyrmionium• fractional Skyrmion (meron)⋮n Different surroundings• anti-ferromagnets• ferrimagnets• 𝑺𝑼 𝑵 magnets
These figures are taken from
[B. Göbel, I. Mertig, & O. Tretiakov, Phys. Rep. 895, 1 (2021)]

3/12

𝑺𝑼 𝑵 magnets
n 𝑺𝑼 𝟐𝑰 + 𝟏 magnets have been realized using cold atoms with nuclear spin 𝑰.
•••

173Yb → 𝑆𝑈 687Sr → 𝑆𝑈 1087Rb → 𝑆𝑈 3

[T. Fukuhara et.al, PRL 98, 030401 (2007)][B. J. DeSalvo et.al, PRL 105, 030402 (2010)][S. Will el.al., Nature 465, 197–201 (2010)]

n Spin-1 systems can be viewed as 𝑺𝑼 𝟑 magnets
Bilinear Biquadratic (BBQ) model'%%%%𝐻!!" = # 𝐽& 𝑺# ⋅ 𝑺% + 𝐽' 𝑺# ⋅ 𝑺%#,%𝐽' ( (ℋ!!" = #𝑇+# 𝑇+% + 𝐽& − 𝐽' 𝑆.#) 𝑆.%)
2#,%𝑆.#) : Spin-1 operator, 𝑇+#( : 𝑆𝑈 3 spin operator
𝑇+#& + 𝑇+#* 𝑇+#' + 𝑇+#+ 𝑇+#, + 3𝑇+#%# =𝑺,,2
224/12

𝑺𝑼 𝑵 magnets

4/12

n 𝑺𝑼 𝟐𝑰 + 𝟏 magnets have been realized using cold atoms with nuclear spin 𝑰.
•••173Yb → 𝑆𝑈 687Sr → 𝑆𝑈 1087Rb → 𝑆𝑈 3

[T. Fukuhara et.al, PRL 98, 030401 (2007)][B. J. DeSalvo et.al, PRL 105, 030402 (2010)][S. Will el.al., Nature 465, 197–201 (2010)]

n Spin-1 systems can be viewed as 𝑺𝑼 𝟑 magnets

Mean-field phase diagram (square lattice)[N. Papanicolaou, Nucl. Phys. B 305, 367]

𝐽% /𝐽

Bilinear Biquadratic (BBQ) model

𝑆𝑈 3'%%%%
𝐻!!" = # 𝐽& 𝑺# ⋅ 𝑺% + 𝐽' 𝑺# ⋅ 𝑺%

FM

#,%

𝐽' ( (ℋ!!" = #𝑇+# 𝑇+% + 𝐽& − 𝐽' 𝑆.#) 𝑆.%)
2

𝐽$ /𝐽#,%𝑆.#) : Spin-1 operator, 𝑇+#( : 𝑆𝑈 3 spin operator
𝑇+#& + 𝑇+#* 𝑇+#' + 𝑇+#+ 𝑇+#, + 3𝑇+#%# =𝑺,,222SO

𝑆𝑈 3

𝑆𝑈 3FQ

AFM
𝑆𝑈 3%% ⊗ 𝑺%%. − ' 𝟏Quadrupolar tensor : 𝑄+% = 𝑺,How to introduce a stabilizing term?
Dzyaloshinskii-Moriya (DM) interaction
𝑫#,%./ ×𝑺.1
𝐻-. = + 𝑫/,1 ⋅ 𝑺/,1n It favors to twist the spins.
n An effect of the spin-orbit coupling.

𝑺#𝑺%

5/12

How to introduce a stabilizing term?

5/12

Dzyaloshinskii-Moriya (DM) interaction
𝑫#,%./ ×𝑺.1
𝐻-. = + 𝑫/,1 ⋅ 𝑺/,1n It favors to twist the spins.
n An effect of the spin-orbit coupling.

𝑺#𝑺%Spin-1 Bose-Hubbard model with spin-orbit coupling
15/6!,#22.9/ + 2𝑛; /𝐻23 = −𝑡 + 𝑏/,4 𝑒𝑏+H.
c.
++𝑈𝑛;
𝑛;
−1+𝑈
𝑺
1,78//
947
2/1− ℎ + 𝑆>/:/

𝐴/,/±𝒆$ = ±𝜅2

𝜏 =,𝐴/,/±𝒆% = ±

𝜅2

𝜏>

/

𝜏 ? : Spin-1 matrix[Juzeliūnas, Ruseckas, Dalibard, PRA 81, 053403 (2010)]

Our model

6/12

1G H
𝐻 = + 𝐽 𝑇2/F 𝑇21F + 𝐾 𝑆>/? 𝑆>1? + 2𝑓FGH 𝐴F/,1 𝑇2/ 𝑇21 − ℎ + 𝑆>/:
2/,1𝑆𝑈 3𝑆𝑈 2

Generalized DM( 𝐽 <0)/+ E, E𝐻&' = B 𝜀+,- 𝐷(,*𝑆( 𝑆*(,*Our model

6/12

1G H
𝐻 = + 𝐽 𝑇2/F 𝑇21F + 𝐾 𝑆>/? 𝑆>1? + 2𝑓FGH 𝐴F/,1 𝑇2/ 𝑇21 − ℎ + 𝑆>/:
2/,1Generalized DM

𝑆𝑈 2𝑆𝑈 3( 𝐽 <0)/+ E, E𝐻&' = B 𝜀+,- 𝐷(,*𝑆( 𝑆*𝑺𝑼 𝟑 operators

𝑇21F = 𝜆F 47 𝜎 1 𝜌 1 where 𝑆>1: 𝜎 1 ≡ 𝜎 𝜎 1

SU 𝟑 spin coherent state𝑍 =⊗1 𝑍1 with

𝑍1 = 𝑍14 𝜎 1

n At the single site level, 𝑍1 can describe any spin-1 state at site 𝑗.
n 𝒁1 =IJ 8 KJ L
𝑍1 , 𝑍1 , 𝑍1
takes its value on 𝑆 M/𝑆 J = 𝑆𝑈(3)/𝑈(2) = 𝐶𝑃9.

Topological charge

1H
𝑁=−+ 𝑓FGH 𝑇2/F 𝑇21G 𝑇2N ∈ 𝜋9 𝐶𝑃9 = ℤ
64𝜋/1N(,*Ground state phase diagram

7/12

𝐶𝑃% Double-Skyrmion crystal (D-SkX)
𝐶𝑃% Skyrmion crystal (SkX)

𝐽 = −1, 𝜅 = −0.4
7Li

𝐶𝑃% Helimagnetic (HM)
𝐶𝑃% Meron crystal (MeX)

FM−𝐾
D-SkX
𝐶𝑃% Skyrmionium crystal[YA, Y. Akagi, et al.,
PRB 106, L100406 (2022)]

HM
41K

MeX-SkX
crossover

87Rbℎ23Na

𝑪𝑷 Helical structure
𝟐8/12

n Helical structure with small modulation along the stripesn Single 𝑞-state both in 𝒮 𝒒 and 𝒬 𝒒
𝐽 = −1, 𝜅 = −0.4
7Li%#
Magnetization vectors 𝑺𝑆!!"

FM−𝐾
D-SkX
𝑦𝑥

HM
41K

MeX-SkX
crossover

87Rbℎ23Na

𝑪𝑷 Helical structure𝟐

8/12

n Helical structure with small modulation along the stripesn Single 𝑞-state both in 𝒮 𝒒 and 𝒬 𝒒
𝐽 = −1, 𝜅 = −0.4
Spin Structure FactorQuadrupole Structure Factor

𝒮 𝒒

7Li𝒬 𝒒𝑞/FM−𝐾D-SkXHM𝑞.
Z* ⋅ 𝑺Z1 𝑒 (𝒒⋅ 𝒓!0𝒓"
𝒮 𝒒 = 𝒩 0% B 𝑺*,1𝒬 𝒒= 𝒩 0% B*,11
Tr 𝑄a* 𝑄a12𝑒(𝒒⋅ 𝒓! 0𝒓"

41K

MeX-SkX
crossover

87Rbℎ23Na

𝑪𝑷 Double-Skyrmion crystal
𝟐n Triangular lattice of 𝑁 = −2 𝐶𝑃9 Skyrmionsn Magnetic Skyrmion-like magnetic structuren But, non-trivial quandrupole structure

9/12

𝐽 = −1, 𝜅 = −0.4
7Li

Energy density

Top. charge density
𝐸(𝑁(FM−𝐾𝑦D-SkX
HM𝑥41K

MeX-SkXcrossover

87Rbℎ23Na

𝑪𝑷 Double-Skyrmion crystal
𝟐n Triangular lattice of 𝑁 = −2 𝐶𝑃9 Skyrmionsn Magnetic Skyrmion-like magnetic structuren But, non-trivial quandrupole structure

9/12

𝐽 = −1, 𝜅 = −0.4
7Li

Norm of spins
Z(𝑺%Magnetization vectors
𝑆!!"

FM−𝐾
D-SkX
𝑦𝑥

HM
41K

MeX-SkX
crossover

87Rbℎ23Na

𝑪𝑷 Double-Skyrmion crystal
𝟐n Triangular lattice of 𝑁 = −2 𝐶𝑃9 Skyrmionsn Magnetic Skyrmion-like magnetic structuren But, non-trivial quandrupole structure

9/12

𝐽 = −1, 𝜅 = −0.4
7Li

Spin structure factor
𝒮 𝒒

Quadrupole structure factor
𝒬 𝒒

FM−𝐾
D-SkX𝑞/HM𝑞.

41K

MeX-SkX
crossover

87Rbℎ23Na𝑪𝑷 Skyrmionium crystal
𝟐10/12=Skyrmion surrounded by an anti-Skyrmionn Square lattice of 𝐶𝑃9 Skyrmioniums
n Spin nematic realize outside of Skyrmioniums
n Double 𝑞-structure in 𝒮 𝒒 and 𝒬 𝒒

𝐽 = −1, 𝜅 = −0.4
7Li

Top. charge density

FM−𝐾𝑦D-SkXHM𝑥𝐶𝑃9 Skyrmionium

41K

MeX-SkX
crossover

87Rbℎ23Na𝑪𝑷 Skyrmionium crystal
𝟐10/12=Skyrmion surrounded by an anti-Skyrmionn Square lattice of 𝐶𝑃9 Skyrmioniums
n Spin nematic realize outside of Skyrmioniums
n Double 𝑞-structure in 𝒮 𝒒 and 𝒬 𝒒

𝐽 = −1, 𝜅 = −0.4
7Li

Norm of spins
Z(𝑺%FM−𝐾𝑦D-SkXHM𝑥41K

MeX-SkX
crossover

87Rb

ℎ23Na

𝑪𝑷 Skyrmionium crystal
𝟐

10/12

=Skyrmion surrounded by an anti-Skyrmionn Square lattice of 𝐶𝑃9 Skyrmioniums
n Spin nematic realize outside of Skyrmioniums
n Double 𝑞-structure in 𝒮 𝒒 and 𝒬 𝒒

𝐽 = −1, 𝜅 = −0.4
7Li

Spin Structure Factor
𝒮 𝒒

Quadrupole Structure Factor
𝒬 𝒒FM−𝐾
D-SkX𝑞/HM𝑞.

41K

MeX-SkX
crossover

87Rbℎ23Na

Meron crystal – Skyrmion crystal crossover 11/12
n Meron crystal = honeycomb lattice of meronsn Skyrmion crystal = triangular lattice of Skyrmionsn Spin nematic state is realized at the core ofSkyrmions and outside of merons.

𝐽 = −1, 𝜅 = −0.4
7Li

n Triple 𝑞-structure in 𝒮 𝒒 and 𝒬 𝒒
n These state are smoothly connected.
FM−𝐾

Meron crystal

D-SkX

Skyrmion crystal
𝑁(𝑁(HM41K

MeX-SkX
crossoverℎ = 0.05ℎ = 0.45

87Rbℎ23Na

Meron crystal – Skyrmion crystal crossover 11/12
n Meron crystal = honeycomb lattice of meronsn Skyrmion crystal = triangular lattice of Skyrmionsn Spin nematic state is realized at the core ofSkyrmions and outside of merons.

𝐽 = −1, 𝜅 = −0.4
7Li

n Triple 𝑞-structure in 𝒮 𝒒 and 𝒬 𝒒
n These state are smoothly connected.
FM−𝐾Meron crystal

Z(𝑺%Skyrmion crystal

D-SkXZ(𝑺%HM
41K

MeX-SkXcrossoverℎ = 0.05ℎ = 0.45

87Rbℎ23Na

Meron crystal – Skyrmion crystal crossover 11/12
n Meron crystal = honeycomb lattice of meronsn Skyrmion crystal = triangular lattice of Skyrmionsn Spin nematic state is realized at the core ofSkyrmions and outside of merons.

𝐽 = −1, 𝜅 = −0.4
7Li

n Triple 𝑞-structure in 𝒮 𝒒 and 𝒬 𝒒
n These state are smoothly connected.
Meron crystal
𝒮 𝒒Skyrmion crystal

FM−𝐾D-SkX

𝒮 𝒒HM

𝒬 𝒒

𝒬 𝒒

41K

MeX-SkX
crossover

87Rbℎ23Na

Meron crystal – Skyrmion crystal crossover 11/12
n Meron crystal = honeycomb lattice of meronsn Skyrmion crystal = triangular lattice of Skyrmionsn Spin nematic state is realized at the core ofSkyrmions and outside of merons.

𝐽 = −1, 𝜅 = −0.4
7Li

n Triple 𝑞-structure in 𝒮 𝒒 and 𝒬 𝒒
n These state are smoothly connected.
FM𝐾Top. chargedensity

D-SkXHM41K

MeX-SkX
crossover

Energydensity

87Rb23Naℎℎ = 0.05ℎ = 0.15ℎ = 0.18ℎ = 0.25ℎ = 0.31ℎ = 0.33ℎ = 0.45

Summary12/12

ü We have studied the ground states in an 𝑆𝑈 3 spin system obtained as an effective
theory of the spin-orbit coupled spin-1 Bose-Hubbard model.
ü The 𝑺𝑼 𝟑 spin systems host various exotic phases:
𝐶𝑃% Meron crystal

𝐶𝑃% Skyrmion crystal 𝐶𝑃% Skyrmionium crystal 𝐶𝑃% Double-Skyrmion crystal

𝐶𝑃% Helixü They possess non-trivial dipole and quadrupole moment structures, unlike the standard
magnetic Skyrmions.

Thank you for your attention!