D. J. Thouless

Ordering, metastability and phase transitions in two-dimensional systems

A new definition of order called topological order is proposed for two-dimensional systems in which no long-range order of the conventional type exists. The possibility of a phase transition characterized by a change in the response of the system to an external perturbation is discussed in the context of a mean field type of approximation. The critical behaviour found in this model displays very weak singularities. The application of these ideas to the xy model of magnetism, the solid-liquid transition, and the neutral superfluid are discussed. This type of phase transition cannot occur in a superconductor nor in a Heisenberg ferromagnet.

Stability of the Sherrington-Kirkpatrick solution of a spin glass model

The stationary point used by Sherrington and Kirkpatrick (1975) in their evaluation of the free energy of a spin glass by the method of steepest descent is examined carefully. It is found that, although this point is a maximum of the integrand at high temperatures, it is not a maximum in the spin glass phase nor in the ferromagnetic phase at low temperatures. The instability persists in the presence of a magnetic field. Results are given for the limit of stability both for a partly ferromagnetic interaction in the absence of an external field and for a purely random interaction in the presence of a field.

Electrons in disordered systems and the theory of localization

Abstract This paper gives a review of the theory of noninteracting electrons in a static disordered lattice. The introductory section gives a brief survey of the main aspects of the problem and of its relevance to the physics of amorphous and disordered crystalline solids. The second section is concerned with the methods which can be used to find the density of states, both in the main part of the band, where the coherent potential approximation can be used, and in the tail of the band, where other methods must be used. The third section gives a survey of the theory of localization. There is a detailed discussion of the qualitative differences between localized and extended states which enable a sharp distinction to be made between them. There is a brief survey of the theory of one-dimensional systems and of the percolation problem, and then the Anderson model and its self-consistent modification are discussed. There is also a discussion of numerical work on the Anderson model and the use of path-integral methods. In the final section a tentative theory is proposed to combine various features of the problem which have been revealed by some of the different approaches.

Solution of 'Solvable model of a spin glass'

Abstract The Sherrmgton-Kirkpatrick model of a spin glass is solved by a mean field technique which is probably exact in the limit of infinite range interactions. At and above T c the solution is identical to that obtained by Sherrington and Kirkpatrick (1975) using the n → O replica method, but below T c the new result exhibits several differences and remains physical down to T = 0.

Long range order and metastability in two dimensional solids and superfluids. (Application of dislocation theory)

Dislocation theory is used to define long range order for two dimensional solids. An ordered state exists at low temperatures, and the rigidity modulus is nonzero at the transition temperature. Similar arguments show that the superfluid density is nonzero at the transition temperature of a two dimensional superfluid.

Numerical studies of localization in disordered systems

The results of numerical work on the Anderson model of disordered systems are presented. The sensitivity of the eigenvalues to the choice of periodic or antiperiodic boundary conditions is used as a criterion for localization, and the theory of this criterion is discussed. For the two dimensional square lattice this criterion gives a reasonably sharp result for the onset of localization which is not in conflict with other criteria of localization, and it is found that localization occurs far more easily than Andersons theory suggests. For the diamond lattice the onset of localization is less sharply defined, and localization occurs less easily than for the square lattice, but more easily than in Andersons theory.

A relation between the density of states and range of localization for one dimensional random systems

The formula of Herbert and Jones (1971) relating the distribution of eigenvalues to the range of localization of an eigenstate for the Anderson model in one dimension is discussed. An explicit formula for the localization distance is given for Lloyds model in one dimension. The formula, which is essentially a dispersion relation is generalized to the case of the Schrodinger equation in one dimension.

Conductivity of the disordered linear chain

The authors develop a fast algorithm for evaluating the Kubo formula for the conductivity of a linear chain, and use it to study the dependence of the conductivity as a function of imaginary frequency. The results for the Anderson model with different degrees of disorder and different energies can all be scaled onto the same curve, which is of the form expected from the theory of localised states. The universal curve obtained provides a simple connection between tight-binding model results and the conductivity which can be calculated for an electron in a white noise potential. Similar, but not identical, results are obtained for tight-binding chains with a Cauchy distribution of site energies.

Structure of stochastic dynamics near fixed points

We analyze the structure of stochastic dynamics near either a stable or unstable fixed point, where the force can be approximated by linearization. We find that a cost function that determines a Boltzmann-like stationary distribution can always be defined near it. Such a stationary distribution does not need to satisfy the usual detailed balance condition but might have instead a divergence-free probability current. In the linear case, the force can be split into two parts, one of which gives detailed balance with the diffusive motion, whereas the other induces cyclic motion on surfaces of constant cost function. By using the Jordan transformation for the force matrix, we find an explicit construction of the cost function. We discuss singularities of the transformation and their consequences for the stationary distribution. This Boltzmann-like distribution may be not unique, and nonlinear effects and boundary conditions may change the distribution and induce additional currents even in the neighborhood of a fixed point.

Conductivity and mobility edges for two-dimensional disordered systems

The energy levels of an electron in a disordered two-dimensional lattice are evaluated numerically for the honeycomb, square and triangular lattices with between 36 and 196 sites. The density of states and their localization properties are deduced from the results and the dependence of the positions of the mobility edges on disorder is studied. An analysis of the conductivity confirms the claim that the minimum metallic conductivity should have a universal value in two dimensions and that value was found to be (0.12+or-0.03)e2/h. A comparison of the predictions for the conductivity with the weak scattering limit is given.