Numerical Renormalisation Group
 and the
Metal-Insulator Transition
in Disordered Systems

	Angus MacKinnon
	Imperial College London

Acknowledgements

	Bernhard Kramer, Michael Schreiber, Keith Slevin, Tomi Ohtsuki, Isa Zarakheshev, Rudolf Römer, Kohei Itoh
	Etienne Hofstetter, John Taylor, Alex Taylor, Jonathan Carter

The Anderson Hamiltonian

Mobility Edge

Universality Classes

	The original 3
		Orthogonal
			spinless
			time-reversal symmetric
		Unitary
			broken time-reversal symmetry
			Static or random magnetic field
		Symplectic
			spin-orbit coupling
	The others
		Bogoliubov-de Gennes
			Superconductivity
		……

Critical Behaviour

Numerical Methods (a)

	Transfer Matrix

Data Analysis

Fitted Data

Conductance Fit

Numerical Methods (b)

	Level Spacing

Scaling of level spacing distribution

Experiment

	Neutron transmutation doping (Kohei Itoh et al.)

Experiment
Neutron Transmutation Doping

Experiment 2

	Same Ge:Ga
	Boundary depends on compensation.

Summary so far

	Critical exponent can be calculated accurately: s = 1.58
	Experiments now give good temperature driven scaling: s=1.0
	What about 2D at low density?
	What is wrong?
		Theory and/or Numerics wrong?
		Experiments wrong?
	Something missing?
		Interactions?
	Good agreement between theory and experiment for QHE system only.

Interacting Electrons

	Number of states scales as 2N
	rather than N
	Possible Approaches:
		Recursive Green’s function like
			Exact representation including all states
			Not yet possible
		Density Matrix Renormalisation Group like
			Some results
		Hubbard-Stratonovich like
			d+1 dimensions

Spinless Electrons in 1D

Delocalisation in 1D

Conclusions

	Critical exponent for non-interacting systems can be evaluated accurately and reproducibly:  1.58
	Experimental results becoming more reliable, but critical exponent = 1.0
	Need calculations with disorder and interactions.
		Initial steps being made.

Slide 21