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Solitons and Instantons in Condensed Matter
Alexei Kolezhuk
Institut für Theoretische Physik C, RWTH Aachen
Heisenberg Program
Alexei Kolezhuk (RWTH Aachen) Solitons and Instantons in Condensed Matter 11 Mar 2009, NBI 1 / 40
Outline
1 from a “great wave” to a “soliton”
2 topological solitons
3 instantons
4 a few examples
5 summary
Alexei Kolezhuk (RWTH Aachen) Solitons and Instantons in Condensed Matter 11 Mar 2009, NBI 2 / 40
from a “great wave” to a “soliton”
Great Wave of Translation: The Discovery
John Scott Russell 1834: wave in a narrow channel
1-2 miles without changing shapelinear wave theory: packets disperse !
10 years of experiments30-feet tank in the back yard
⇓
“Report on Waves” (1844)“The Wave of Translation
in the Oceans of Water, Air and Ether ” (1885, posthumous)
Alexei Kolezhuk (RWTH Aachen) Solitons and Instantons in Condensed Matter 11 Mar 2009, NBI 3 / 40
from a “great wave” to a “soliton”
Waves in Shallow WaterD. Korteweg, G. de Vries (1895)Boussinesq (1871), Rayleigh (1876)
<<A/h 1 <<h/ 1B
A
y
xB
hnonlinearityuτ + uξ + 6uuξ + uξξξ = 0
dispersion
u ∝ (y − h)/h, c =√
gh τ = (ct/h), ξ = (x/h)
Solitary Wave: propagation with a constant shape
u =α
2 cosh2 {(x − Vt)/B
} , B =2h√α
, α =Vc
− 1 ≪ 1
N.J.Zabusky, M.D.Kruskal 1965:no momentum transfer in a collision
1965: the word “SOLITON” is coinedAlexei Kolezhuk (RWTH Aachen) Solitons and Instantons in Condensed Matter 11 Mar 2009, NBI 4 / 40
from a “great wave” to a “soliton”
“Tsu-nami” = Harbor Wave
Katsushika Hokusai,“The Great Wave off the Coast of Kanagawa”
... on a radar screen:
in the ocean: A ∼ 1 m, B ∼ 100 km ⇒ “invisible”; V ∼ 800 km/h
at the coast: A reaches ≈ 30 m; record: A = 525 m (Alaska 1958)
Alexei Kolezhuk (RWTH Aachen) Solitons and Instantons in Condensed Matter 11 Mar 2009, NBI 5 / 40
topological solitons
Kink: a very different Solitontoy model: “string” in a two-well potential
L =12
∫dx
{(∂ϕ
∂t
)2−
(∂ϕ
∂x
)2− U(ϕ)
}
U
ϕ
x
“Kink” solution: e.g., ϕ4-model, U(ϕ) = 12U0(ϕ
2 − ϕ20)
2
ϕ = ±ϕ0 tanh(x − x0
δ
), 1/δ = ϕ0
√U0
“topological charge”Q =1
2ϕ0
∫dx
(∂ϕ
∂x
)= integer
stable: nontrivial b.c. at ±∞, can only annihilate with antikink
Kink = Domain Walltopological defect
M z
x
δ
Alexei Kolezhuk (RWTH Aachen) Solitons and Instantons in Condensed Matter 11 Mar 2009, NBI 6 / 40
topological solitons
Higher-Dimensional Topological Solitons (in Magnets)“delocalized” (nontrivial b.c. at ∞): stable even on a lattice
2D (easy-plane): Bloch Linemagnetic Vortex
3D (isotropic):
Hedgehog / Monopole
Bloch Point
Alexei Kolezhuk (RWTH Aachen) Solitons and Instantons in Condensed Matter 11 Mar 2009, NBI 7 / 40
topological solitons
“Weak” Topological Solitons“localized”: metastable on a lattice
1D (easy-plane):KinkS1 7→ S1: winding number
2D (easy-axis):“Skyrmion”S2 7→ S2: Pontryagin index
3D (easy-axis):Hopf SolitonS3 7→ S2: Hopf index
Alexei Kolezhuk (RWTH Aachen) Solitons and Instantons in Condensed Matter 11 Mar 2009, NBI 8 / 40
topological solitons
Topological Solitons and Phase Transitions
Landau:no order in 1D at T 6= 0
kink energy E0 = finitedensity n ∝ e−E0/T
correlations 〈M(x)M(0)〉 ∝ e−2nx
+m
x−m
2D, easy-plane symmetry: vortices/antivorticesKosterlitzargument: special phase transition
vortex energy diverges as E = A ln(L/a)entropy S ∼ ln(L2/a2)free energy F = E − TS ∼ (A − 2T ) ln(L/a)T > Tc ∼ A/2 ⇒ unbinding of vortex pairs
L
Alexei Kolezhuk (RWTH Aachen) Solitons and Instantons in Condensed Matter 11 Mar 2009, NBI 9 / 40
instantons
Quantum Tunneling and Instantonsnanoscale two-state system
Mn12 cluster: S = 10
=
classically: or , degenerate in energy
quantum-mechanically: 1/√
2{
±}
, tunnel splitting
effective description 7→ particle on a circlem
R ϕ
0 π ϕ
U
L =12
mR2(
dϕ
dt
)2
−U(ϕ)+αdϕ
dt
U(ϕ) = U0(1 − cos(2ϕ))
Alexei Kolezhuk (RWTH Aachen) Solitons and Instantons in Condensed Matter 11 Mar 2009, NBI 10 / 40
instantons
Instantons: solitons in “imaginary time” I
QM: a trajectory contributes ∝ eiA/~ to the transition amplitude
“imaginary time”: t 7→ τ = it ⇒ iA 7→ −AE , U 7→ −U
0 π ϕ
UA =
∫dt
{12
mR2(
dϕ
dt
)2
−U(ϕ)+αdϕ
dt
}
U(ϕ) = U0(1 − cos(2ϕ))
0 π ϕ
-UAE =
∫dτ
{12
mR2(
dϕ
dτ
)2
+U(ϕ)+iαdϕ
dτ
}
Instanton solution: minAE
ϕττ +12ω2
0 sin 2ϕ = 0, ω20 =
4U0
mR2
“sine-Gordon model”
Alexei Kolezhuk (RWTH Aachen) Solitons and Instantons in Condensed Matter 11 Mar 2009, NBI 11 / 40
instantons
Instantons: solitons in “imaginary time” IIInstanton: cos ϕ = ± tanh[ω0(τ − τ0)], action A0
approximation: “dilute gas” of instantons + small fluctuations⇓
tunnel splitting:
E± =~ω0 ± ∆
2, ∆ = C cos
πα
~~ω0
(A0
~
)1/2
e−A0/~
instantons/anti-instantons: e±iπα/~ ⇒ cos(πα/~) π 0oscillations vs “flux” α ↔ Aharonov-Bohm effect
Wernsdorfer et al. (2000)oscillations in Fe8 cluster
Alexei Kolezhuk (RWTH Aachen) Solitons and Instantons in Condensed Matter 11 Mar 2009, NBI 12 / 40
instantons
Instantons in d dimensions ↔ Solitons in d + 1
tunneling in 2D vortex or 3D monopole
τimaginary time
vortices
monopole
important for quantum phase transitions in magnets
Alexei Kolezhuk (RWTH Aachen) Solitons and Instantons in Condensed Matter 11 Mar 2009, NBI 13 / 40
a few examples
Signatures of Solitons in a Quantum Spin Chain
Pyrimidine− Cu(NO3)2(H2O)2: quantum S=1/2 chain materialstaggered g-tensor ⇒ staggered local field h = cH
H = J∑
n
~Sn · ~Sn+1 − gµB
{H
∑
n
Szn + h
∑
n
(−1)nSxn
}
theory: quantum sine-Gordon modelexcitations: solitons and their bound states (breathers)
Alexei Kolezhuk (RWTH Aachen) Solitons and Instantons in Condensed Matter 11 Mar 2009, NBI 14 / 40
a few examples
Signatures of Solitons in a Quantum Spin Chain
Pyrimidine− Cu(NO3)2(H2O)2: quantum S=1/2 chain materialstaggered g-tensor ⇒ staggered local field h = cH
H = J∑
n
~Sn · ~Sn+1 − gµB
{H
∑
n
Szn + h
∑
n
(−1)nSxn
}
theory: quantum sine-Gordon modelexcitations: solitons and their bound states (breathers)
Alexei Kolezhuk (RWTH Aachen) Solitons and Instantons in Condensed Matter 11 Mar 2009, NBI 14 / 40
a few examples
Signatures of Solitons in a Quantum Spin Chain
Pyrimidine− Cu(NO3)2(H2O)2: quantum S=1/2 chain materialstaggered g-tensor ⇒ staggered local field h = cH
H = J∑
n
~Sn · ~Sn+1 − gµB
{H
∑
n
Szn + h
∑
n
(−1)nSxn
}
theory: quantum sine-Gordon modelexcitations: solitons and their bound states (breathers)
Alexei Kolezhuk (RWTH Aachen) Solitons and Instantons in Condensed Matter 11 Mar 2009, NBI 14 / 40
a few examples
Signatures of Solitons in a Quantum Spin ChainPyrimidine− Cu(NO3)2(H2O)2: quantum S=1/2 chain materialstaggered g-tensor ⇒ staggered local field h = cH
H = J∑
n
~Sn · ~Sn+1 − gµB
{H
∑
n
Szn + h
∑
n
(−1)nSxn
}
theory: quantum sine-Gordon modelexcitations: solitons and their bound states (breathers)
0 2 4 6 8 10 12 14 16 18 20 22 24Magnetic field H [T]
0
100
200
300
400
500
600
700
800
ESR
fre
quen
cy [
GH
z]
B1
B2
B3
S
PM-Cu(NO3)2(H
2O)
2
Electron spin resonance in
S.Zvyagin, AK et al., PRL (2004)
Alexei Kolezhuk (RWTH Aachen) Solitons and Instantons in Condensed Matter 11 Mar 2009, NBI 14 / 40
a few examples
Monopoles in Spin Ice
Dy2Ti2O7: pyrochlore lattice, µDy = 10µB
strong dipole-dipole interaction ⇒ ice rules “2-in, 2-out”
topological defects: “3-in, 1-out”Coulomb-like interaction (Castelnovo, Moessner, Sondhi 2007)
Alexei Kolezhuk (RWTH Aachen) Solitons and Instantons in Condensed Matter 11 Mar 2009, NBI 15 / 40
a few examples
Monopoles in Spin Ice
Dy2Ti2O7: pyrochlore lattice, µDy = 10µB
strong dipole-dipole interaction ⇒ ice rules “2-in, 2-out”
topological defects: “3-in, 1-out”Coulomb-like interaction (Castelnovo, Moessner, Sondhi 2007)
magnetic field = chemical potential ⇒ phase transition
Alexei Kolezhuk (RWTH Aachen) Solitons and Instantons in Condensed Matter 11 Mar 2009, NBI 15 / 40
a few examples
Solitons in Metallic Nanowires
nanowires: “necking down” under stressclassically: unstable under surface tension
QM: electron energydepends on the “trap” radiusBürki, Goldstein, Stafford (2003)
kinks transport atoms away
Alexei Kolezhuk (RWTH Aachen) Solitons and Instantons in Condensed Matter 11 Mar 2009, NBI 16 / 40
a few examples
Solitons in conductive polymers
Nobel prize in Chemistry 2000H.Shirakawa, A.G.MacDiarmid, A.J.Heeger, (1977)
trans-Polyacetylene (CH)x
doped with I2
−>3 I 2 2 I3−
CH
neutral chain
2e−
Polaron
+e
+e
charged solitons
+2e
C
C
C
C
C
C
C
C
C
C
neutral solitonsH
H
H H H H
H H H H
S=1/2
S=−1/2
undoped polyacetylene
Alexei Kolezhuk (RWTH Aachen) Solitons and Instantons in Condensed Matter 11 Mar 2009, NBI 17 / 40
a few examples
Conductivity of Polyacetylene
1977: σ = 38 S/cm σmax ≃ 105 S/cm σCu,Ag ≃ 106 S/cm
Alexei Kolezhuk (RWTH Aachen) Solitons and Instantons in Condensed Matter 11 Mar 2009, NBI 18 / 40
a few examples
Davydov’s Solitons: Energy Transfer in Proteins
Protein = chain of aminoacids:
R1
R2
R3
C
C
N
C
C
N
C
C
N
O
O
O
H
H
H
H
H H
model (A.S.Davydov ’77):
H =∑
n
εB†nBn + t(B†
nBn+1 + h.c.) +∑
n
P2n
2M+ κ(Qn+1 − Qn)
2
Hint = λ∑
n
B†nBn(Qn+1 − Qn−1)
O=C bond inside a peptide (H-N-C=0) group: two-state system 7→ Bn
Qn, Pn: displacements of peptide groups along the chain directionintegrate out Q, ansatz |Ψ〉 =
∑n Φn(t)B
†n|0〉
⇒ nonlinear equation for the envelope w.f. Φ(x , t)
Alexei Kolezhuk (RWTH Aachen) Solitons and Instantons in Condensed Matter 11 Mar 2009, NBI 19 / 40
a few examples
Soliton-Mediated Protein FoldingS.Caspi und E.Ben-Jacob (2000)global 3D structure: torsion angles ψ, φ:
asym. double-well potential V (φ, ψ) + coupling Uint ∝ (αφ2 + βψ2) B†B
simulation:
Alexei Kolezhuk (RWTH Aachen) Solitons and Instantons in Condensed Matter 11 Mar 2009, NBI 20 / 40
summary
Summary
Solitons: ubiquitous in Nature:
◮ surfaces (of liquids, crystals, . . . )◮ magnetic materials◮ optics◮ molecular biology◮ superfluids, superconductors, Bose-Einstein condensates◮ as instantons: many strongly correlated electron systems,
quantum field theory, particle physics, cosmology,. . .
play important role in phase transitions
Alexei Kolezhuk (RWTH Aachen) Solitons and Instantons in Condensed Matter 11 Mar 2009, NBI 21 / 40
summary
Research Summary
Alexei Kolezhuk
Institut für Theoretische Physik C, RWTH Aachen
Heisenberg Program
unconventional orders in spin, electron and cold atom systems
multicomponent Bose gases in low dimensions
Alexei Kolezhuk (RWTH Aachen) Research Summary 11 Mar 2009, NBI 22 / 40
Unconventional Orders in Strongly Correlated Systems
Unconventional Orders: Spin Current States
FM: magnetization ~M 7→ ~Sn, AFM: Néel order ~L 7→ (−1)n~Sn
vector chirality ~κij = (~Si × ~Sj) = spin current
“remnant” of the conventional helical order
e.g.: frustrated spin chainJ2
J1
J2 J1/α=
QM: helical LRO 7→ U(1): destroyed by fluctuationsleft ↔ right 7→ Z2: can survive
2D, 3D: route to “vector chiral spin liquid” (J.Villain, H.Kawamura)
Alexei Kolezhuk (RWTH Aachen) Research Summary 11 Mar 2009, NBI 23 / 40
Unconventional Orders in Strongly Correlated Systems
Unconventional Orders: Spin Current Statesinduced by magnetic field in isotropic frustrated spin chains
AK, T.Vekua (2005):theory (bosonization)
I.McCulloch,. . . , AK (2008):numerics (DMRG)
0 0.2 0.4 0.6 0.8 1M=S
tot/(LS)
0.01
0.1
1 κ2
L=128, S=1L=160, S=1L=192, S=1L=256, S=3/2L=168, S=1/2L=256, S=1/2
S=1S=3/2
S=1/2
S=1/2
J2
J1
J2 J1/α=
Alexei Kolezhuk (RWTH Aachen) Research Summary 11 Mar 2009, NBI 24 / 40
Unconventional Orders in Strongly Correlated Systems
Unconventional Orders: Spin Nematic
Nαβ =12〈SαSβ + SβSα〉 − S(S + 1)
3δαβ
Example:
|0>|−>|+>S=1
Nxx = Nyy 6= Nzz :uniaxial spin nematic
favored by: frustration + strong field, or non-Heisenberg exchange:spin-1 bosons in optical lattices
Alexei Kolezhuk (RWTH Aachen) Research Summary 11 Mar 2009, NBI 25 / 40
Unconventional Orders in Strongly Correlated Systems
Garlea et al (2008,2009): SuICu2Cl4 - highly 1D materialfield-induced helimagnetismexponents neither BEC nor 3d XY ⇒ new universality class?
LiCuVO4 : also good 1D, frustrated chainhelical phase killed by magnetic field 7→ phase with no magneticorder
Alexei Kolezhuk (RWTH Aachen) Research Summary 11 Mar 2009, NBI 26 / 40
Unconventional Orders in Strongly Correlated Systems
Current-carrying states in bilayer systemsfermionic polar molecules (e.g., HCN) or atoms (e.g., 53Cr)
on a bilayer
11t , V
2
1t, V
t , V22
V12
electric/magnetic dipole moments polarized by external field
V12
/V/
11 , t /
V, t
θ,ϕp( )
l⊥
l||
strong on-dimer repulsion V ≫ t, t ′, V ′
ij
half filling (one particle per dimer)
Alexei Kolezhuk (RWTH Aachen) Research Summary 11 Mar 2009, NBI 27 / 40
Unconventional Orders in Strongly Correlated Systems
Circulating current state: polar fermions
map to 2d anisotropic AFM in mag. field
if V ′ = V ′11 − V ′
12 < 0, t ′ > t ′c ≃√
tV/8
⇒ staggered vertical current
t, Vt , V2
1
if V ′ > 0 ⇒ usual density wave
A.K., PRL 99, 020405 (2007)
Alexei Kolezhuk (RWTH Aachen) Research Summary 11 Mar 2009, NBI 28 / 40
Unconventional Orders in Strongly Correlated Systems
Spin current state : Hubbard model on a bilayer
half filling, U ≫ t ≫ t ′ ⇒ effectively only spin degrees of freedom
Heisenberg exchange J⊥ (inter-layer), J‖ (intra-layer)
+ ring exchange on vertical plaquettes J4
mapping to bond bosons: instability of AF ground state at J4 > J4c
⇒ staggered vertical spin current ~κ =∑
r(−)~r~S1~r × ~S2~r
S2
S1
t
t’
A.K., PRL 99, 020405 (2007)
Alexei Kolezhuk (RWTH Aachen) Research Summary 11 Mar 2009, NBI 29 / 40
Multicomponent Bose gases
Multicomponent Bose gases in low dimensions
multi-species Bose condensates: spinor bosonse.g. 3 lowest hf states ⇒ “hf spin” F = 1 (23Na, 87Rb, 7Li , 39K)heteronuclear mixtures: 41K − 87Rb
recently: tunable inter/intra-species ineractions87Rb− 85Rb (Papp et al. ’08), 87Rb− 41K (Thalhammer et al. ’08)
two species model, contact interaction
U(ψ1, ψ2) =12
(u11|ψ1|4+u22|ψ2|4+2u12|ψ1|2|ψ2|2
), u11, u22 > 0
mean-field stability condition: u212 < u11u22
u12 >√
u11u22 ⇒ phase separationu12 < −√
u11u22 ⇒ collapse
low dimensions - ?
Alexei Kolezhuk (RWTH Aachen) Research Summary 11 Mar 2009, NBI 30 / 40
Multicomponent Bose gases
Multicomponent Bose gas: dilute limit
Model: N species, d dimensions, T = 0, contact quartic interaction:
AN =
∫dτ
∫ddx
{ψ∗
α(∂τ − µα)ψα +|∇ψα|22mα
+gαβ,α′β′
2ψ∗
αψ∗βψα′ψβ′
}
mixture in continuum or optical latticealso: “magnon condensation” in a frustrated magnet
ε ε
kkh>hs
h<hs
critical point: µα → 0 (low density limit)
Alexei Kolezhuk (RWTH Aachen) Research Summary 11 Mar 2009, NBI 31 / 40
Multicomponent Bose gases
RG equations in low-density limitd Γ
dl= (2 − d)Γ − Γ2
Γαβ,γδ(l) = Γαβ,γδ(l) ×
[mγδ(mα+mβ)]1/2
Λ0, d=1
(mαβmγδ)1/2
π , d=2, mαβ ≡ mαmβ
(mα + mβ)
RG flow is interrupted at the scale l = l∗:
ρtotedl∗ ∼ Λ0
Λ0 7→ inverse potential range or lattice cutoff.direct generalization of the RG for one-component case (D.S. Fisher& P.C. Hohenberg ’88, D.R. Nelson & H.S. Seung ’89,E.B. Kolomeisky & J.P. Straley ’92 . . . )
A.K., arXiv:0903.1647
Alexei Kolezhuk (RWTH Aachen) Research Summary 11 Mar 2009, NBI 32 / 40
Multicomponent Bose gases
Stability of Double-Species Mixture
stability: u11(l)u22(l) − u212(l) > 0 for all scales up to l = l∗
d = 1: stability condition gets broken at the scale ℓ = ℓc :
u(1d)12 =
2u11u22√
m1m2
u11m1 + u22m2, u(c)
12 =√
u11u22.
lcl *
u12(1d) u
12(c)u
12(c)− 0
intermediate phase!two coexisting components:demixed /collapsed , Q < Qc
mixed , Q > Qc = Λ0e−ℓc
A.K., arXiv:0903.1647
Alexei Kolezhuk (RWTH Aachen) Research Summary 11 Mar 2009, NBI 33 / 40
Multicomponent Bose gases
Spin-1 bosons in optical lattice: odd-integer filling
Bose-Hubbard model with spin-dependent interaction
H = −t∑
〈ij〉
(b†i,σbj,σ + h.c) +
∑
i
{U0
2ni(ni − 1) +
U2
2(~Si)
2 − µni
}
U0 ≫ t , U2: only spin d.o.f. ⇒ effective spin-1 lattice model
H = −J1
∑
〈ij〉
(~Si · ~Sj) − J2
∑
〈ij〉
(~Si · ~Sj)2
J1, J2
J1, 2Jλ λ 1D ↔ 2D
hidden SU(3) symmetryat J1 = 0 or J1 = J2
Law, Pu, Bigelow (1998); Imambekov, Lukin, Demler (2003)
Alexei Kolezhuk (RWTH Aachen) Research Summary 11 Mar 2009, NBI 34 / 40
Multicomponent Bose gases
Effective field theory
near J1 = 0 /AF-SU(3) point/:
AE =1
2g
∫dd+1x
{|∂µ~z|2 − |~z∗∂µ~z|2−
J1
J2|~z2|2
}+ Atop
Atop =
∫dτ
∑
n
ηn(~z∗n∂τ~zn), ηn = ±1 for n ∈ sublattice A(B)
large-N analysis (N-component field ~z)
A.K., PRB78, 144428 (2008)
Alexei Kolezhuk (RWTH Aachen) Research Summary 11 Mar 2009, NBI 35 / 40
Multicomponent Bose gases
Role of topological defectsin d = 1: skyrmions ⇒ dimerization for odd nc
Atop = iπncq, q = − i2π
∫d2xǫµν(∂µ~z∗ · ∂ν~z) = integer
q 7→ “skyrmion number”
nc = 1 for our case
in d = 2: monopoles (skyrmion creation/destruction)
lead to dimerization for nc 6= 0 mod 4
dimerization ∝ skyrmion/monopole densitydegeneracy depends on nc
N.Read, S.Sachdev ’91
Alexei Kolezhuk (RWTH Aachen) Research Summary 11 Mar 2009, NBI 36 / 40
Multicomponent Bose gases
Role of SU(N)-breaking perturbation
at the mean field level: favors AF/nematic order
gcrit ∝1N
(1 + c√|J1/J2|) =
√1 + 1/λcrit
nc
J1 > 0: c = O(1)J1 < 0: c = O( 1
N )
λ
0
Dimerized
NematicAF
−J1affects the topological term!
A.K., PRB78, 144428 (2008)
Alexei Kolezhuk (RWTH Aachen) Research Summary 11 Mar 2009, NBI 37 / 40
Multicomponent Bose gases
How SU(3)-breaking affects the topological phases?
AF side: ~z = ~zAF =~e1 + i~e2√
2, ~m = ~e1 × ~e2 - Néel vector
D = (1 + 1) or 2d slice in D = (2 + 1): O(3) topological charge
Qm =1
8π
∫d2x εµν ~m · (∂µ ~m × ∂ν ~m) = integer
SU(3) top.charge for ~z = ~zAF : ~e1,2 = R(θ, ϕ, ψ)~ex ,y
q = − i2π
∫d2xǫµν(∂µ~z∗ · ∂ν~z)
7→ 12π
∫d2x sin θǫµν(∂µθ)(∂νϕ) = 2Qm
Ivanov, Khymyn, AK, PRL100, 047203 (2008)A.K., PRB78, 144428 (2008)
Alexei Kolezhuk (RWTH Aachen) Research Summary 11 Mar 2009, NBI 38 / 40
Multicomponent Bose gases
Topological pairing of Skyrmions and Monopoles
skyrmion/monopole with topological charge q at J1 = 0perturb. −J1|~z2|2, J1 < 0 (AF) ⇒ contrib. to the action ∆A = ?
q = 1 skyrmion ⇒ ∆A ∝ −J1R2 if N = 3 – metastable
q = 1 monopole ⇒ ∆A ∝ −J1 × (system volume) – collapse. . . but ∆A = 0 for N ≥ 4: q = 1 stays exact
q = 2 skyrmion/monopole “adjust” to remain exact/stable
pairing of skyrmions and monopoles for N = 3
change of the degeneracy in the dimerized phase
AK, PRB 78, 144428 (2008)
Alexei Kolezhuk (RWTH Aachen) Research Summary 11 Mar 2009, NBI 39 / 40
Multicomponent Bose gases
Summary
unconventional orders in spin, electron, and cold atom systems◮ states with broken time reversal symmetry:
current / spin current (vector chirality)◮ compete with quadrupolar /multipolar orders
multicomponent Bose gases in low dimension
◮ dilute regime, continuum/lattice:novel phases with partial phase separation
application to magnets in high external fields◮ optical lattice, integer filling: topological effects near
enhanced symmetry points
Alexei Kolezhuk (RWTH Aachen) Research Summary 11 Mar 2009, NBI 40 / 40