Presentation on theme: "Presentation in course Advanced Solid State Physics By Michael Heß"— Presentation transcript:
1 Presentation in course Advanced Solid State Physics By Michael Heß Ising ModelPresentation in courseAdvanced Solid State PhysicsBy Michael Heß
2 History of critical point behavior history: three diff periods in the development of the theory of critical point behavior1. v.d.Waals mean field theory to liquid-gas phase transition but after 1965 numerical calculations and experiments proved that this mean field theory was quantitatively incorrect around the critical point2. after 1965 more phenomenological theories based on scaling invariance3. same period: another key in critical phenomena theory based on universality (universality: dissimilar systems show similarities near their critical points)/72: K.Wilson took semi-phenomenological concepts of scaling and universality and converted these ideas into real calculations of critical point behavior
3 Ernst Ising Student of Wilhelm Lenz in Hamburg. PhD 1924, in Köln – in Peoria (IL)Student of Wilhelm Lenz in Hamburg. PhD 1924,Thesis work on linear chains of coupled magnetic moments. This is known as the Ising model.The name ‘Ising model’ was coined by Rudolf Peierls in his 1936 publication ‘On Ising’s model of ferromagnetism’.He survived World War II but it removed him from research. He learned in 1949 (25 years after the publication of his model) that his model had become famous.S. G. Brush, History of the Lenz-Ising Model, Rev. Mod. Phys 39, (1962)
4 Solving of Ising modelinvented by W. Lenz and his student E. Ising (1920)1D: solved analytically by Ising (1925): no phase transition in 1D and he concluded incorrectly that in higher D also no phase transition2D square lattice: solved by L. Onsager (1944), exhibits phase transitionAlso in higher dimensions phase transition can be modeledIstrail showed that computation of the free energy of an arbitrary subgraph based on Ising model will not be approximated computationally intractable (not solvable) by any method for the case 3D and higher- impossible to efficiently compute all possible thermodynamic quantities with arbitrary external fields- it does not mean that the critical exponents or spin-spin correlations cannot be computed near criticality.
5 1. Critical point and phase transition Critical point of fluid (p=const)1. temp dependencePhases for SF6 the curve isMagnetization for DyAlO3rho = densityTc = critical tempFig 1: phase diagram of fluid at p=constFig 2: Onset of magnetization in ferromagnet
6 2. Ising model Ising model: for ferromagnetics invented Simple Hamiltonian of spins s(r) at lattice point rAssumptions: - only two states- only nearest neighbor interactions- every pair counts only one time- Hamiltonian:Ising model in 2DCoupling to field pair interaction 3-body interactions = spinsJ = energy of interaction (<0 if σi = σj , >0 if σi = -σj )H = external magnetic field (decreasing H if spinslined up, increasing H if not)
7 2. Ising model Magnetic interactions: - seek to align spins relative to one another.- spins become effectively "randomized" when thermal energy is greater than the strength of the interaction.- w/o magnetic field the Ising model is symmetric for interchange of ± but magnetic field breaks this symmetryEnergy of interaction J:- Jij > 0 the interaction is called ferromagnetic (aligned spins)- Jij < 0 the interaction is called antiferromagnetic (antialigned spins)- Jij = 0 the spins are noninteracting
8 2. Ising model Thermodynamics of Ising model can be obtained: for this system, the operation Tr means:So the free energy is given byThermodynamic properties derived by differentiation e.g. average magnetization at site i is derived by δF/ δHi(method of sources)
9 Ising Model Free energy functional: Goal: - model phase changes of real lattices- the 2D square lattice Ising model is simplest model to showphase changesExtensive operators (entire lattice)Local operatorsInteraction scalars
10 Ising model in 1DWe defined h=H/kT and K=J/kT. The partition function is given bycan be calculated exacty.Ising model in 1D:In the following, we will take a look at boundary conditions, thermodynamics and correlations.
11 Ising model in 1D: Periodic boundaries Periodic boundary conditions are defined byIsing model in 1D with PBC:We assume that there is no external field (h=0). Then, we haveWe can solve this. Define where i=1…N-1. Then we haveSubstitution to the partition function gives
12 Ising model in 1D: Free boundaries Ising model in 1D with free b.c.:Again, we assume that there is no external field (h=0). Then, we haveUsing the same transformation as before, i.e. where i=1…N-1.We getWe have the partition function now. Next, we take a look at free energy and thermodynamics.
13 Ising model in 1D: Free energy Since we have the partition function, we also have the free energya) For PBCb) For free b.c.The difference between boundary conditions becomes negligible at the thermodynamic limit.The more general way is do this with transfer matrix. Works also for nonzero field.Thermodyn limit
14 Ising Model Visualization of Ising model during phase transition Ising model with external field h
15 Universality of Ising Model - the fluctuations close to the phase transition are described by a continuum field with a free energy or Lagrangian as a function of the field values.- Ising model decribes exactly the fluctuations around the critical pointIn contrast: the mean field doesn’t describe fluctuation at Tc
16 Applications Magnetism: In 19th century: two theories: Ampere postulated that permanent magnets are due to permanent internal atomic currents vs. theroy of permanent magnetic momentElectron spin discovered to describe magnetismIsing model: investigate if electrons could made tospin in same direction by simple local forcesLattice gas:Interpret Ising model as a statistical modelB = [0,1],[unoccupied,occupied] B = (σ + 1)/2 σ = [-1,1]The density of atoms can be controlled by chem pot
17 Applications Biologie – neurons in brain: states: firing, not firing To reproduce average firing rate for each neuron includes activity of each neuron (statistically independent)To allow for pair interactions when a neuron tends to fire along with anotherThis energy function only introduces probability biases for a spin having a value and for a pair of spins having the same value.J – NN interaction of firing rateh – self-firing rate