New field theory formulation of localised states in disordered systems

We present a new field theoretic formulation of the quantun mechanics of disordered systems. The problem is converted to an explicit field theory by changing variables in the functional integral over all random potentials to an integral over all possible wavefunctions. Unlike previous formulations this field theory has the novel feature of having the 'right' sign for the coefficient of the 'p" term. Thus the important excitations are kinks rather than instantons. In addition, it exhibits singular solutions with finite action. The method is illustrated with an exact calculation of the asymptotic density of states in a one-dimensional gaussian white noise potential. There has been a great interest in recent years in the quantum mechanics of random potentials. Typically one would like to calculate some property of the eigenvalues and eigenfunctions averaged over the ensemble of random potentials. For example, the average Green's function has spectral density where P is the probability distribution for the potential V , Yi is the ith eigenfunction with eigenvalue Ei and Z is the partition function. Considerable effort has recently gone into the question of how to apply the techniques of quantum field theory and statistical mechanics to this type of problem. In a typical phase transition problem one is interested in correlation functions for some field S of the form G, = 2-' DS P ( S ) S ( x ) Sol). I Unfortunately the random potential problem does not have this form since the functional integral is over the potential V , whereas the correlations of interest are between wavefunctions. The main thrust of efforts to circumvent this difficulty have taken advantage of the formal equivalence (Ma 1972, Thouless 1975) (term by term in perturbation theory) between the random (gaussian distributed) potential problem and the n + 0 limit (replica trick) of an n-component Landau-Ginzburg field theory. Cardy (1978) and BrCzin and Parisi (1980) have used this equivalence to calculate the ensemble averaged density of states in the weak coupling regime (deeply localised states). Wegner (1979a, 5 NRCNBS Postdoctoral Fellow 0022-3719/81/290881 + 06 $01.50 @ 1981 The Institute of Physics L88 1 L882 Letter to the Editor b, 1980) and others (Harris and Lubensky 1980,1981, McKane and Stone 1981) have extensively developed the replica field theory formulation of calculations for the fourpoint correlation function which yields information about localisation and transport. There has been much interest in the fact that the coefficient of the q4 term in the Lagrangian density has the 'wrong' sign, so that the important excitations in this problem are non-perturbative instanton solutions of the classical Landau-Ginzburg equation. The use of non-perturbative instanton solutions was justified (at least in the density of states calculation) by Houghton and Schafer (HS) (1979) (see also Houghton et a1 1980). They avoided the n + 0 trick by formulating a direct variational principle for the problem. Variational formulations have been presented previously (Zittartz and Langer 1966, Langer 1967, Halperin and Lax 1966,1967, Edwards 1970, Abram and Edwards 1972, Thouless 1975, Thouless and Elzain 1978); however, the HS method takes advantage of field theoretic techniques for dealing with nonlinear quantum systems. Because they were forced to invoke second-order perturbation theory in evaluating fluctuations about the extrema1 potential, HS did not obtain an explicit field theory. It is thus difficult to develop a systematic expansion in the coupling constant with their scheme. Schafer and Wegner (1980) pointed out the importance of this difficulty and show that the perturbation theory is at least somewhat simpler in lattice models. We present here preliminary results of a new approach to the problem which avoids the replica trick and the necessity of perturbation theory and reduces the problem to an explicit field theory which is amenable to analysis using standard techniques. One of the novel features of this field theory is that the coefficient of the q4 term has the 'right' sign. There are also several other extremely interesting features which we shall discuss. As a simple illustration of the procedure we calculate the asymptotically exact density of states in a one-dimensional gaussian white noise potential. More extensive results will be reported elsewhere (Rendell et a1 1981). Consider the density of states in one dimension for asymptotically deep energies, o-, CQ . The ensemble average is where L is the sample length, Ei is the ith eigenvalue of the random potential V , 3 is the action 1 Li2 3 1 dXVZ(x) 202 -L/Z and 2 is the partition function. The mean square potential c? is a measure of the disorder. The main difficulty in evaluating equation (1) is the delta function which requires solving the Schrodinger equation with a random potential in order to determine the eigenvalues. The key to our approach is the observation that given a wavefunction Y and an eigenvalue w it is trivial to invert the Schrodinger equation to find the corresponding potential (h2/2m = 1):