Condensed Matter Physics Sharp

Text: G. D. Mahan, Many Particle Physics Topics: –Magnetism: Simple basics, advanced topics include micromagnetics, spin polarized transport and itinerant magnetism (Hubbard model) –Superconductivity: BCS theory, advanced topics include RVB (resonanting valence bond) –Linear Response theory: advanced topics include the quantized Hall effect and the Berry phase. –Bose-Einstein condensation, superfluidity and atomic traps

Magnetism How to describe the physics: (1)Spin model (2)In terms of electrons

Spin model: Each site has a spin S i There is one spin at each site. The magnetization is proportional to the sum of all the spins. The total energy is the sum of the exchange energy E exch, the anisotropy energy E aniso, the dipolar energy E dipo and the interaction with the external field E ext.

Exchange energy E exch =-J  i,d S i  S i+  The exchange constant J aligns the spins on neighboring sites . If J>0 (<0), the energy of neighboring spins will be lowered if they are parallel (antiparallel). One has a ferromagnet (antiferromagnet)

Alternative form of exchange energy E exch =-J  (S i -S i+  ) 2 +2JS i 2. S i 2 is a constant, so the last term is just a constant. When S i is slowly changing S i -S i+     r S i. Hence E exch =-J  2 /V  dr  r S| 2.

Magnitude of J k B T c /zJ¼ 0.3 Sometimes the exchange term is written as A s d 3 r |r M(r)| 2. A is in units of erg/cm. For example, for permalloy, A= 1.3 £ erg/cm

Interaction with the external field E ext =-g  B H S=-HM We have set M=  B S. H is the external field,  B =e~/2mc is the Bohr magneton (9.27£ erg/Gauss). g is the g factor, it depends on the material. 1 A/m=4  times Oe (B is in units of G); units of H 1 Wb/m=(1/4  ) G cm 3 ; units of M (emu)

Dipolar interaction The dipolar interaction is the long range magnetostatic interaction between the magnetic moments (spins). E dipo =(1/4  0 )  i,j M ia M jb  ia  jb (1/|R i -R j |). E dipo =(1/4  0 )  i,j M ia M jb [  a,b /R 3 - 3R ij,a R ij,b /R ij 5 ]  0 =4   henrys/m

Anisotropy energy The anisotropy energy favors the spins pointing in some particular crystallographic direction. The magnitude is usually determined by some anisotropy constant K. Simplest example: uniaxial anisotropy E aniso =-K  i S iz 2

Relationship between electrons and the spin description

Local moments: what is the connection between the description in terms of the spins and that of the wave function of electrons ？ Itinerant magnetism:

Illustration in terms of two atomic sites: There is a hopping Hamiltonian between the sites on the left |L> and that on the right |R>: H t =t(|L> <L|). For non-interacting electrons, only H t is present, the eigenstates are |+> (|->) =[|L>+ (-) |R>]/2 0.5 with energies +(-)t.

Non magnetic electrons For two electrons labelled by 1 and 2, the eigenstate of the total system is |G 0 >=|1,- up 〉 |2,- down 〉 -|1,-down 〉 |2,-up 〉 by Pauli’s exclusion principle. Note that =0. There are no local moments, the system is non-magnetic.

Additional interaction: Hund’s rule energy In an atom, because of the Coulomb interaction, the electrons repel each other. A simple rule that captures this says that the energy of the atom is lowered if the total angular momentum is largest.

Some examples: First: single local moment

Single local moment H=  k n k  +E d (n d+ +n d- )+Un d+ n d- - +  k,  (c k  + d  +c.c.). Mean field approximation: H d  =  k n k  +E d (n d+ + n d- )+Un d+  +  k,  (c k  + d  +c.c.).

Nonmagnetic vs Magnetic case

Illustration of Hund’s rule Consider two spin half electrons on two sites. If the two electrons occupy the same site, the states must be |1, up>|2,down>- |1,down>|2,up>. This corresponds to a total angular momentum 0 and thus is higher in energy. This effect is summarized by the additional Hamiltonian H U =U  i n i,up n i,down.

Formation of local moments The ground state is determined by the sum H U +H t. This sum is called the Hubbard model. For the non-interacting state =U- 2t. Consider alternative ferromagnetic states |F,up>=|L,up>|R,up> etc and antiferromagnetic states, |AF>=(|L,up>|R,down>- |L,down>|R,up>)/2 0.5, etc. Their average energy is zero. If U>2t, they are lower in energy. These states have local moments.

Moments are partly localized Neutron scattering results for Ni: –3d spin= –3d orbital=0.055 –4s=-0.105

An example of the exchange interaction For our particular example, the interaction is antiferromagnetic. There is a second order correction in energy to the antiferromagnetic state given by J=| | 2 /  E. This energy correction is not present for the state |F>. In the limit of U>>t, J=- t 2 /U. In general, the exchange depends on the concentarion of the electrons and the magnitude of U and t.

Local Moment Details: PWA, Phys. Rev. 124, 41 (61)