1 Localization and Critical Diffusion of Quantum Dipoles in two Dimensions U(r)-random ( =0) I.L. Aleiener, B.L. Altshuler and K.B. Efetov Quantum Particle in a Disorder Potential

2 Density of states: Density-density correlation function: Non-linear -model Replica (Wegner 1979), Supermatrix (Efetov 1982)

3 Non-linear supermatrix -model for describing localization effects General scheme: following “Supersymmetry in disorder and chaos”, K.B. Efetov, Cambridge University Press, (1997) For any correlation function O (expressed in terms of the Green functions) Due to supersymmetry

4 Physical quantities as integrals over the supermatrices! Adding magnetic or spin-orbit interactions one changes the symmetry of the supermatrices Q (orthogonal, unitary and symplectic). Depending on the dimensionality (geometry of the sample) one can study different problems (localization in wires and films, Anderson metal-insulator transition, etc.) Analogy to spin models.

5 Renormalization group in two dimensions. Integration over fast modes: fast, slow Expansion in and integration over New non-linear -model with renormalized and Gell-Mann-Low equations: A consequence of the supersymmetry Physical meaning: the density of states is constant.

6 For the orthogonal, unitary and symplectic ensembles Orthogonal: localization Unitary: localization but with a much larger localization length Symplectic: “antilocalization” Unfortunately, no exact solution for 2D has been obtained. Reason: non-compactness of the symmetry group of Q. Renormalization group (RG) equations.

7 The explicit structure of Q u,v contain all Grassmann variables All essential structure is in (unitary ensemble) Mixture of both compact and non-compact symmetries rotations: rotations on a sphere and hyperboloid glued by the anticommuting variables.

8 Explicit form Orthogonal Unitary Symplectic

9 Quantum dipoles motion Dipoles on square lattice Interaction of 2 dipoles

10 Two processes in addition to the conventional particle motion (emission and absorbtion of photons): 1) the dipole annihilates in state |r,d> and then appears in |r’,d’> 2) the second dipole in state |r’,d’> is created and then the first one in state |r,d> collapses Transition amplitude T of the processes -dipole energy -tunneling to the nearest neighbor

11 Hamiltonian of the model Random potential : For time reversal invariant systems -Pauli matrices in “dipole space” Rotational symmetry after averaging:Also:

12 Regular part of the Hamiltonian in Fourier space h-magnetic field -regular functions at k=0 -density of states

13 Conventional transformation: 1) Writing the partition function in terms of an integral over 16-component supervectors X. 2) Decoupling by an integral over supermatrix Q. 3) Saddle point an expansion in gradients. As a result, one obtains for physical quantities: However, now due to the singular form of an important contribution comes also from slowly varying X(r)! Usually supervectors X(r) oscillate fast (the reason for introducing slow Q).

14 One more transformation removing the singularity in -32x32 supermatrix

15 Final form of the -model The -term is new. It adds an additional “charge” in the renormalization group! Method of calculation: writing Q as and integrating over.

16 Properties of RG: 1) The form of the model is reproduced after integration over fast variables. 2) The -term does not change but contributes to renormalization of the conductance. First order correction to G. is positive for the unitary ensemble but still negative for the orthogonal one.

17 Two-loop approximation. The limit corresponds to the case when the long-range hopping of the dipoles essentially exceeds the short-range one. The limit is opposite in this sense. -orthogonal -unitary Long-range hopping of dipoles drastically changes the localization picture: 1) Existence of a stable fixed point for the orthogonal ensemble. 2) Existence of an unstable fixed point for the unitary ensemble.

18 Orthogonal ensemble: universal conductance (independent of disorder) Unitary ensemble: metal-insulator transition

19 Conclusions. 1. Dipoles move easier than particles due to long-range hops. 2. Non-linear sigma-model acquires a new term contributing to RG. 3. RG analysis demonstrates criticality for any disorder for the orthogonal ensemble and existence of a metal-insulator transition for the unitary one.