Mean-field theory. Alessandro Vindigni. ETH October 29, Laboratorium für Festkörperphysik, ETH Zürich

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1 Alessandro Vindigni Laboratorium für Festkörperphysik, ETH Zürich ETH October 29, 2012

2 Lecture plan N-body problem Lecture plan 1. Atomic magnetism (Pescia) 2. Magnetism in solids (Pescia) 3. Magnetic order at finite temperature (A. V.) Mean-filed theory and Landau approach Classical spin models 4. Magnetic domains and domain walls (A. V.) Magnetic anisotropy and domain walls Dipolar interaction and magnetic domains

3 Lecture plan N-body problem Characteristic energy scales intra-atomic exchange interaction (Hund s rules) 2 3 ev 10 4 K gµ B Sh: Zeeman splitting for B-field=1 T 0.1 mev 1 K J: inter-atomic exchange interaction (RKKY) mev K T c [K] C[K] µ[µ B ] J[eV] J[k] M(0, 0)[Gauss] Fe Co Ni Gd Dy

4 Lecture plan N-body problem N-body problem Magnetic moment of an isolated paramagnetic atom [ ] m = gµ B Ŝ z gµb S B = gµ B S B S k B T where B S is the Brillouin function Heisenberg Hamiltonian for N-coupled spins H = 1 2 J Ŝ(n) Ŝ(n ) + gµ B B n n =1 n Ŝ z (n) Hilbert space (2S + 1) N

5 Lecture plan N-body problem State-of-the-art of numerical methods Actual number of spin 1/ D 3D 2D 1D Quantum Classical Monte Carlo Monte Carlo 10 0 ED ED: exact diagonalization Linear size of the system

6 Lecture plan N-body problem Possible ways to simplify the problem 1. Reduce the many-body problem to a s.p. problem (MFA) 2. Simplify the problem replacing the quantum-spin operators by classical vectors 3. Use symmetry arguments 4. Consider only a selected family of excitations: localized: domain walls delocalized: spin waves

7 Critical exponets MFA: single-particle Hamiltonian H sp (n) = gµ B (B + B W ) Ŝ z (n) with B W = zj Ŝ z (n) Hsp /gµ B Weiss field Many-body problem Single-particle problem MFA effective field

8 Critical exponets By analogy with an isolated atom Ŝ z (n) Hsp = SB S (α) with α = gµ B S B t k B T = gµ B S B zjs Ŝ z (n) Hsp k B T. Setting m = gµ B Ŝ z (n) Hsp, one gets [ gµb S B m = gµ B S B S k B T + zj S m ] gµ B k B T

9 Critical exponets For graphic visualization { σ = m gµ B S = B S(α) σ = k BT α gµ BB zjs 2 zjs When B = 0 and α 0 B S (α) S S α +... so that a spontaneous magnetization (σ 0) only arises for T < T C with T C = S + 1 zjs 2 3 S k B

10 Critical exponets Curie-Weiss law For T > T C and α 0 { σ = S+1 by using the definition of T C σ or equivalently (using T C ) m = gµ B Sσ = (gµ B) 2 zj 3 S α = k BT α gµ BB zjs 2 zjs (1 T T C )σ = gµ B zjs B T C B = (gµ B) 2 S(S + 1) 1 B T T C 3k B T T C The pre-factor of h on the right-hand side is right the susceptibility C χ = with C = (gµ B) 2 S(S + 1) T T C 3k B Curie-Weiss law.

11 Critical exponets Critical exponents in general Being τ := T T C 1 the reduced temperature, α, β, γ and δ critical exponents are defined as follows: C(τ, B = 0) τ α M(τ, B = 0) ( τ) β, τ < 0 χ(b = 0, τ) τ γ M(τ = 0, B) B 1/δ. MFA 3d-Heisenberg α 0 (Jump) 0.11 ± β ± γ ± δ

12 Critical exponets Experiments on a Fe film on W(110)