Institut für Theoretische Physik C, RWTH Aachen Heisenberg ...iht.univ.kiev.ua/Kolezhuk/presentations/cph.pdf · Solitons and Instantons in Condensed Matter Alexei Kolezhuk Institut

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


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)



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


“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)


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



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



“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



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


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))


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”


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


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


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)


a few examples



H = J∑

n


{H

∑

n

Szn + h

∑

n

(−1)nSxn

}



a few examples



H = J∑

n


{H

∑

n

Szn + h

∑

n

(−1)nSxn

}



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


{H

∑

n

Szn + h

∑

n

(−1)nSxn

}


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)


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)


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


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


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


a few examples

Conductivity of Polyacetylene

1977: σ = 38 S/cm σmax ≃ 105 S/cm σCu,Ag ≃ 106 S/cm


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)


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:


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


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)



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=



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



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



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)



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)



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)


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 - ?



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)



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



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



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)



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)



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



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)



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)



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)



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


Documents

Institut für Theoretische Physik C, RWTH Aachen Heisenberg ...iht.univ.kiev.ua/Kolezhuk/presentations/cph.pdf · Solitons and Instantons in Condensed Matter Alexei Kolezhuk Institut