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

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 Ronnow – ESM Cargese 2017 Slide 1

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 Ronnow – ESM Cargese 2017 Slide 2

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Aspen Center for Physics 2000: Workshop on Quantum Magnetism • David Thouless: • My laptop, just broken Transition without broken symmetry Ronnow – ESM Cargese 2017 Slide 4

ICCMP Brasilia 2009: Workshop on Heisenberg Model (80+1 year anniversary) • Duncan Haldane • 4h bus ride with Bethe chatter Ronnow – ESM Cargese 2017 Slide 5

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Topological Topological phase transitions phases of matter Ronnow – ESM Cargese 2017 Slide 7

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 4+4=6+2 6+8=12+2 8+6=12+2 • https://en.wikipedia.org/wiki/Topology Ronnow – ESM Cargese 2017 Slide 8

Proof that Euler was wrong ! But, need long distance and long time ! Ronnow – ESM Cargese 2017 Slide 9

The hairy ball theorem • "you can't comb a hairy ball flat without creating a cowlick“ • Topology concern non-local properties ! Ronnow – ESM Cargese 2017 Slide 10

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 Ronnow – ESM Cargese 2017 Slide 11

Mean field theory of magnetic order Kittel’s Solid State Physics, for pedagogic introduction GS of a many-body Hamiltonian • H=- ∑ ij J ij S i ·S j + g μ B S i ·B • Mean-field approx. ∑ J ij S i ·S j ≈ S i ·(∑ j J ij <S j >)  H=g μ B ∑ i S i ·B eff where B eff =B+ ∑ j J ij <S j >/g μ B =B+ λ M Solution • Eigen states H|S z =m>=E m |S z =m>, E m =g μ B mB eff Magnetization M=N<S z >= ∑ m m e -E m /k B T / ∑ m e -E m /k B T  B J Brillouin function Self-consistency • M=M s B J (g μ B B+ λ M / k B T) y Ronnow – ESM Cargese 2017 Slide 12

Order in Ferromagnet M=M s B J (g μ B B+ λ M / k B T), B J (y) ≈ (J+1)y/3J for y<<1 self-consistency equation T<T c : solution M>0, k B T c =2zJS(S+1)/3 T c <T: solution M=0 Susceptibility: χ =lim B →0 μ 0 M/B  χ ~ C/(T-T c ) Curie Weiss susceptibility Diverge at Tc T near 0: M(T)~M s -e -2Tc/T c -T)  T near T c : M(T)~(T Ronnow – ESM Cargese 2017 Slide 13

Order in Antiferromagnet Two sublattices with <S a >=-<S b > selfconsistency  M=M s B J (g μ B B- λ M / k B T) χ (T) Same solutions: antiferromagnetic order at k B T N =2zJS(S+1)/3 Susceptibility χ ~ 1/(T+T N ) Ө =T c Ө =0 General: χ ~ 1/(T- θ ), θ =0 Paramagnet θ >0 Ferromagnet Ө =T N θ <0 Antiferromagnet T C/N T χ q ~ 1/(T- θ ) diverges at T c Generalisation: J ij  J d (q) and <S d (q)> Fourier So always order at finite T? Allow meanfield of incommensurate order and No, mean-field neglects multiple magnetic sites, d, in unit cell fluctuations ! Ronnow – ESM Cargese 2017 Slide 14

Spin waves in ferromagnet H=- ∑ rr’ J rr’ S r ·S r ’ = -J ∑ <r,r ’= r+d> S z r S z r ’ + ½(S + r S - r ’ +S - r S + r ’ )  nearest neighbour  Ordered ground state, all spin up: H|g> = E g |g>, E g =-zNS 2 J Single spin flip not eigenstate: |r> = (2S) -½ S - r ’ |r> = 2S|r’> r |g>, S - r S + H|r>=(-zNS 2 J+2zSJ)|r> - 2SJ∑ d |r+d> flipped spin moves to neighbours Periodic linear combination: |k> = N -½ Σ r e ikr |r> plane wave Is eigenstate: H|k> = E g +E k |k>, E k =SJ Σ d 1-e ikd dispersion = 2SJ (1-cos(kd)) in 1D Time evolution: |k(t)> = e iHt |k> = e iE k t |k> sliding wave Dispersion: relation between time- and space- modulation period Same result in classical calculation  precession: Ronnow – ESM Cargese 2017 Slide 15

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 Ronnow – ESM Cargese 2017 Slide 16

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." Allowed because p is integrated to infinity Z does not depend on A (and hence not B) https://en.wikipedia.org/wiki/Bohr%E2%80%93van_Leeuwen_theorem Ronnow – ESM Cargese 2017 Slide 17

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 “ Ronnow – ESM Cargese 2017 Slide 18

General Mermin Wagner https://itp.uni-frankfurt.de/~valenti/TALKS_BACHELOR/mermin-wagner.pdf Ronnow – ESM Cargese 2017 Slide 19

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 Ronnow – ESM Cargese 2017 Slide 20

So how does the system behave at finite temperature? Example: 2D Heisenberg anti-ferromagnet H = J  S i  S j Correlations decay exponentially with r Correlation length diverge as T  0  (T)  exp(J/T) Ronnow – ESM Cargese 2017 Slide 21

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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) <S 0 S r >  exp(-r/  ) <S 0 S r >  r-  • What happens in between? Ronnow – ESM Cargese 2017 Slide 24

Different types of defects Ronnow – ESM Cargese 2017 Slide 25

2D XY – spins live in the plane • How does a defect in almost ordered system look? “Repairable” smooth “non - repairable” singular A vortex changes the phase also far from the defect. https://abeekman.nl Ronnow – ESM Cargese 2017 Slide 26

Topological defects Ronnow – ESM Cargese 2017 Slide 27

Energy of a vortex Ronnow – ESM Cargese 2017 Slide 28

Energy of a vortex Ronnow – ESM Cargese 2017 Slide 29

Free energy of a vortex Ronnow – ESM Cargese 2017 Slide 30

Free energy of a vortex Ronnow – ESM Cargese 2017 Slide 31

Energy of a vortex Ronnow – ESM Cargese 2017 Slide 32

Vortex anti-vortex pairs • Does not destroy algebraic correlations <S0Sr>  Ronnow – ESM Cargese 2017 Slide 33

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Vortex anti-vortex pairs Ronnow – ESM Cargese 2017 Slide 35

Vortex anti-vortex pairs • Are created already T<T KT , • But does not destroy algebraic correlations <S 0 S r >  r -  Ronnow – ESM Cargese 2017 Slide 36

Unbound vortices create global disorder: <S R S R+r >  exp(-r/  ) • The Kosterlitz-Thouless transition occur when vortices bind/unbind Ronnow – ESM Cargese 2017 Slide 37

Kosterliz-Thouless T KT unbound vortex- gas of pairs T=0 LRO vortices antivortex pair Ronnow – ESM Cargese 2017 Slide 38

Correlation lengths Heisenberg  (T)  e J/T Kosterlitz-Thouless:  (T)  e b/  t t=(T-T KT )/T KT Anisotropic Heisenberg cross-over :  (T)  e b/  t for  > 100,  (T)  e b/t for  < 100, Ronnow – ESM Cargese 2017 Slide 39

Measuring correlations with neutrons Dynamic structure factor Instantaneous equal-time structure factor:  Width  Correlation length ξ J. Mag. Mag. Mat. 236, 4 (2001) PRL 82 , 3152 (1999); 87 , 037202 (2001) Ronnow – ESM Cargese 2017 Slide 40

Heisenberg system • Scales as predicted • No cross-over to Quantum Critical yet PRL 82 , 3152 (1999); Ronnow – ESM Cargese 2017 Slide 41

XY system Conclusion: We can see KT scaling of  But in quasi-2D TKT always forestalled by 3D order Ronnow – ESM Cargese 2017 Slide 42