SCALING PHENOMENA IN A UNITARY MODEL OF DIRECTED PROPAGATING WAVES WITH APPLICATIONS TO ONE-DIMENSIONAL ELECTRONS IN A TIME-VARYING POTENTIAL

We study a two-dimensional (2D) lattice model of forward-directed wave s in which the integrated intensity for classical waves (or probabilit y for quantum mechanical particles) is conserved. The model describes the time evolution of a 1D quantum particle in a time-varying potentia l and also applies to propagation of electromagnetic waves in two dime nsions within the parabolic approximation. We present a closed-form so lution for propagation in a uniform system. Motivated by recent studie s of nonunitary directed models for localized 2D electrons tunneling i n a magnetic field, we then address related theoretical questions of h ow the interference pattern between constrained forward paths in this unitary model is affected by the addition of phases corresponding to s uch a magnetic field. The behavior is found to depend sensitively on t he value of PHI/PHI0, where PHI is flux per plaquette and PHI0 is the unit of flux quantum. For PHI/PHI0 = p/q we find the amplitude to be m ore collimated the larger the value of q is. We next consider propagat ion in random forward-scattering media. In particular, the scaling pro perties associated with the transverse width x of the wave, as a funct ion of its distance t from the point source, are addressed. We find th e moments of x to scale with t in a very different way from what is kn own for either off-lattice unitary or on-lattice nonunitary systems. T he scaling of the moments of the probability [P(n)(x,t)] (or intensity ) at a point (x = 0,t) is found to be consistent with a simple behavio r [P(n)(0,t)] approximately t(-n/2). Implications for the behavior of one-dimensional lattice quantum particles in a dynamically fluctuating random potential are discussed.