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a phenomenological account Wednesday: vortices Friday: skyrmions - - PowerPoint PPT Presentation

Jun 26, 2023 597 likes •1.17k views

Topology in Magnetism a phenomenological account Wednesday: vortices Friday: skyrmions Henrik Moodysson Rnnow Laboratory for Quantum Magnetism (LQM), Institute of Physics, EPFL Switzerland Many figures copied from internet Ronnow ESM

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Topology in Magnetism – a phenomenological account Wednesday: vortices Friday: skyrmions

Henrik Moodysson Rønnow

Laboratory for Quantum Magnetism (LQM), Institute of Physics, EPFL Switzerland Many figures copied from internet

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Topology in Magnetism

  • 2016 Nobel Prize: Kosterlitz, Thouless and Haldane
  • The Kosterlitz-Thouless transition

– Phase transitions: Broken symmetry, Goldstone mode – Mermin-Wagner theorem – Kosterlitz-Thouless transition – Correlation lengths and neutron scattering

  • The Haldane chain

– Quantum fluctuations suppress order – S=1/2 chain: Bethe solution, spinons – S=1 chain: Haldane gap, hidden order – Inelastic neutron scattering

  • Hertz-Millis
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Aspen Center for Physics 2000: Workshop on Quantum Magnetism

  • My laptop, just broken
  • David Thouless:

Transition without broken symmetry

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ICCMP Brasilia 2009: Workshop on Heisenberg Model (80+1 year anniversary)

  • Duncan Haldane
  • 4h bus ride with Bethe chatter
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Topological phase transitions Topological phases of matter

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Topology

  • In mathematics, topology (from the Greek τόπος, place, and λόγος,

study) is concerned with the properties of space that are preserved under continuous deformations.

  • Euler

– 1736: 7 bridges of Konigsberg – 1750: Polyhedara: vertices+faces=edges+2

  • https://en.wikipedia.org/wiki/Topology

4+4=6+2 6+8=12+2 8+6=12+2

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Proof that Euler was wrong !

But, need long distance and long time !

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The hairy ball theorem

  • "you can't comb a hairy ball flat

without creating a cowlick“

  • Topology concern non-local properties !
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Topological phase transitions

  • Driven by topological defects
  • Vortices (for spins rotating on 2D circle)

– The Kosterlitz Thouless transition in 2D XY model – Superfluid films – Josephson junction arrays

  • Skyrmions (for spins rotating on 3D sphere)

– Lecture on Friday

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Mean field theory of magnetic order

  • GS of a many-body Hamiltonian
  • Mean-field approx.
  • Solution

H=-∑ijJij Si·Sj + gμBSi·B

∑ JijSi·Sj ≈ Si·(∑jJij<Sj>)  H=gμB∑iSi·Beff where Beff=B+∑jJij<Sj>/gμB=B+λM Eigen states H|Sz=m>=Em|Sz=m>, Em=gμBmBeff Magnetization M=N<Sz>= ∑mm e-Em/kBT / ∑m e-Em/kBT  BJ Brillouin function M=MsBJ(gμBB+λM / kBT)

  • Self-consistency

Kittel’s Solid State Physics, for pedagogic introduction y

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Order in Ferromagnet

M=MsBJ(gμBB+λM / kBT), BJ(y) ≈ (J+1)y/3J for y<<1 self-consistency equation T<Tc: solution M>0, kBTc=2zJS(S+1)/3 Tc<T: solution M=0 T near 0: M(T)~Ms-e-2Tc/T T near T

c: M(T)~(T c-T)

Susceptibility: χ=limB→0 μ0M/B  χ~ C/(T-T

c)

Curie Weiss susceptibility Diverge at Tc

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Order in Antiferromagnet

Two sublattices with <Sa>=-<Sb>

selfconsistency  M=MsBJ(gμBB-λM / kBT) Same solutions: antiferromagnetic order at kBTN=2zJS(S+1)/3 Susceptibility χ~ 1/(T+TN) General: χ~ 1/(T-θ), θ=0 Paramagnet θ>0 Ferromagnet θ<0 Antiferromagnet Generalisation: Jij  Jd(q) and <Sd(q)> Fourier Allow meanfield of incommensurate order and multiple magnetic sites, d, in unit cell χ(T) T TC/N

Ө=T

c

Ө=TN Ө=0

χq~ 1/(T-θ) diverges at Tc So always order at finite T? No, mean-field neglects fluctuations !

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Spin waves in ferromagnet

H=-∑rr’Jrr’Sr·Sr’ = -J ∑<r,r’=r+d> Sz

rSz r’+ ½(S+ rS- r’+S- rS+ r’)

Ordered ground state, all spin up: H|g> = Eg|g>, Eg=-zNS2J Single spin flip not eigenstate: |r> = (2S)-½ S-

r|g>, S- rS+ r’|r> = 2S|r’>

H|r>=(-zNS2J+2zSJ)|r> - 2SJ∑d |r+d> flipped spin moves to neighbours Periodic linear combination: |k> = N-½Σreikr|r> plane wave Is eigenstate: H|k> = Eg+Ek|k>, Ek=SJΣd1-eikd dispersion = 2SJ (1-cos(kd)) in 1D Time evolution: |k(t)> = eiHt|k> = eiEkt|k> sliding wave Dispersion: relation between time- and space- modulation period Same result in classical calculation  precession:

 nearest neighbour 

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Magnetic order - Against all odds

  • Bohr – van Leeuwen theorem:

(cf Kenzelmann yesterday)

– No FM from classical electrons

  • <M>=0 in equilibrium

(cf Canals yesterday)

  • Mermin – Wagner theorem:

– No order at T>0 from continuous symmetry in D2

  • No order even at T=0 in 1D
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Bohr – van Leeuwen theorem

  • "At any finite temperature, and in all finite applied electrical or

magnetical fields, the net magnetization of a collection of electrons in thermal equilibrium vanishes identically." https://en.wikipedia.org/wiki/Bohr%E2%80%93van_Leeuwen_theorem

Allowed because p is integrated to infinity Z does not depend on A (and hence not B)

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Mermin, Wagner, Berezinskii (Stat Phys); Coleman (QPT)

  • Generalized to:

“Continuous symmetries cannot be spontaneously broken at finite temperature in systems of dimension d ≤ 2 with sufficiently short-range interactions “

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General Mermin Wagner

https://itp.uni-frankfurt.de/~valenti/TALKS_BACHELOR/mermin-wagner.pdf

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Specific case of ferromagnet in 2D:

  • Magnetization reduced by thermally excited spin waves
  • Dispersion:

  • Volume element in d-dimensional k space:

=

  • Density of states:

For n=2 and d=2

  • Diverges logarithmically  M(T)=M(T=0)-M(T)  0 for any T>0
  • Also works for anti-ferromagnet ; Does not diverge for d>n
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So how does the system behave at finite temperature?

Example: 2D Heisenberg anti-ferromagnet Correlations decay exponentially with r Correlation length diverge as T  0 (T)  exp(J/T)

H = J  Si  Sj

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Lets look at 2D XY model: spins rotate only in the plane

  • Mermin-Wagner: No ordered symmetry broken state for T>0
  • Calculations of correlation function

For high T: For low T: (assuming smooth rotations) <S0Sr>exp(-r/) <S0Sr>  r-

  • What happens in between?
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Different types of defects

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2D XY – spins live in the plane

  • How does a defect in almost ordered system look?

“Repairable” smooth “non-repairable” singular

https://abeekman.nl

A vortex changes the phase also far from the defect.

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Topological defects

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Energy of a vortex

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Energy of a vortex

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Free energy of a vortex

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Free energy of a vortex

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Energy of a vortex

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Vortex anti-vortex pairs

  • Does not destroy algebraic correlations <S0Sr>
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Vortex anti-vortex pairs

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Vortex anti-vortex pairs

  • Are created already T<TKT,
  • But does not destroy algebraic correlations <S0Sr>  r-
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Unbound vortices create global disorder: <SRSR+r>  exp(-r/)

  • The Kosterlitz-Thouless transition
  • ccur when vortices bind/unbind
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Kosterliz-Thouless

T=0 LRO vortex- antivortex pair gas of pairs TKT unbound vortices

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Correlation lengths

Heisenberg (T)eJ/T Kosterlitz-Thouless: (T)eb/t

t=(T-TKT)/TKT

Anisotropic Heisenberg cross-over : (T)eb/t

for  > 100,

(T)eb/t for  < 100,

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Measuring correlations with neutrons

Width  Correlation length ξ

  • J. Mag. Mag. Mat. 236, 4 (2001) PRL 82, 3152 (1999); 87, 037202 (2001)

 Dynamic structure factor Instantaneous equal-time structure factor:

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Heisenberg system

  • Scales as predicted
  • No cross-over to Quantum

Critical yet

PRL 82, 3152 (1999);

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XY system

Conclusion: We can see KT scaling of  But in quasi-2D TKT always forestalled by 3D order

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Correlation lengths Real materials are quasi-2D: Interlayer coupling J’<<J

3D order: TN ~ J’ (TN)2  (TN) ~100 if J’=10-4J So Kosterlitz-Thouless transition never really reached in magnetic materials !

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Topological phase transitions

  • Driven by topological defects
  • Vortices (for spins rotating on 2D circle)

– The Kosterlitz Thouless transition in 2D XY model – Superfluid films – Josephson junction arrays

  • Skyrmions (for spins rotating on 3D sphere)

– Lecture on Friday

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What about Duncan ? – T=0 and quantum states

Topological phases of matter

  • The Haldane S=1 chain
  • Quantum Hall states
  • Topological Quantum Spin Liquids
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AFM spin waves

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Spin waves in antiferromagnet

  • Up sites (A) and down sites (B) – bipartite lattice
  • Holstein-Primakoff bosonisation
  • Linearization
  • Fourier transformation: decouple from r,r’ to q

Hamiltonian still mix A and B, r and r’

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  • Bogoluibov trans.

to decouple a,b

  • Diagonalise:

Ground state excitations = spin waves dispersion

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AFM spin wave dispersion

Average spin-wave population = zero point fluctuations reduced moment: 60% left in 2D ≈0.078 <<1 in D=3 ≈0.197 in D=2 Diverges in D=1 ! Quantum fluctuations destroy order in 1D

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antiferromagnetic spin chain

Ferro

Ground state (Bethe 1931) – a soup of domain walls

Classical AF Quantum AF = 0 <<S2 ~ S2

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Spinon excitations

Energy: E(q) = E(k1) + E(k2) Momentum: q = k1 + k2 Spin: S = ½  ½ Continuum of scattering  Elementary excitations: – “Spinons”: spin S = ½ domain walls with respect to local AF ‘order’ – Need 2 spinons to form S=1 excitation we can see with neutrons

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The antiferromagnetic spin chain

Mourigal, Enderle, HMR, Caux

H=5T H=0 FM: ordered ground state (in 5T mag. field)

  • semiclassical spin-wave excitations

AFM: quantum disordered ground state

  • Staggered and singlet correlations
  • Spinon excitations

– Agebraic Bethe ansatz for inelastic lineshape – Beyond Müller-conjecture  H=5T H=0

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Surprise: 1D S=1 chain has a gap !

  • Haldane’s conjecture 1983:

“Integer spin chains have a gap”

  • No classical order
  • Hidden topological order
  • See lecture by Kenzelmann

+ +

  • +
  • coupled S=1 model with string order
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Hertz-Millis

  • A quantum system in D dimensions



  • A classical system in D+1 dimensions
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Topological phases transition

  • Topological defects
  • 2D XY model, BKT transition

Topological phases

  • The Haldane S=1 chain – confirmed by neutron spectroscopy
  • Quantum Hall states – theory and experiments
  • 2D and 3D topological spin liquids?

– Found in constructed models – Can we find them in real materials?

Friday: Skyrmions

– Local topological defects

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