Fermions in a Random Medium

We study the continuum field theory for an ensemble of directed polymers ri(t) in 1+d ′ dimensions that live in a medium with quenched point disorder and interact via short-ranged pair forces gΨ(ri − rj). In the strong-disorder (or low-temperature) regime, such forces are found to be relevant in any dimension d′ below the upper critical dimension for a single line. Attractive forces generate a bound state with localization length ξ⊥ ∼ |g| −ν⊥ ; repulsive forces lead to mutual avoidance with a pair distribution function P(ri−rj) ∼ |ri−rj| θ reminiscent of interacting fermions. In the experimentally important dimension d′ = 2, we obtain ν⊥ ≈ 0.8 and θ ≈ 2.4. PACS numbers: 74.40, 64.60A, 5.40 ∗ Electronic mail: [email protected] ∗∗ Electronic mail: [email protected] Flux lines in dirty type-II superconductors [1] are a well-known example of lowdimensional manifolds embedded in media with quenched disorder (reviewed e.g. in [2]). Such systems are of interest also because of their links to more complicated random systems such as spin glasses [3], to surface growth [4, 5], and to randomly driven hydrodynamics [6]. A well-studied case is the low-density limit of a single line in a medium with quenched point impurities. The statistical properties of such a line differ from those of a free, thermally fluctuating line. Moreover, the disorder modifies the interactions of the line with other objects. For example, the effect of a rigid line defect (“columnar” defect) parallel to the preferred axis of the lines turns out to be weaker than in a pure system. A weakly attractive defect localizes the line only up to the borderline dimension d⋆ = 1; in higher dimensions, the transition to a localized state takes place at finite coupling strength [7, 8]. In a pure system, the borderline dimension is d⋆ = 2 [9]. Mutual interactions between several identical lines in a disordered medium are the subject of this letter. We study their effects by the methods of continuum field theory. Related aspects of this system have been treated by Bethe ansatz methods in d = 1 [10], on a lattice [11, 12], and in a Wilson renormalization group [13, 14], see the discussion below. In a pure system, weakly attractive short-ranged forces generate a bound state up to the same dimension d⋆ = 2 as do columnar defects. Unlike in the case of columnar defects, however, the effect of pair interactions between identical lines is enhanced by the disorder, which (at sufficiently low temperature) leads to a bound state in any dimension where the low-temperature behavior of a single line is governed by non-thermal scaling exponents. Of equal importance are repulsive forces; for example, the magnetic interaction between flux lines can be treated as a contact interaction at sufficiently low densities [1, 13]. In a pure system, such forces are important only in d = 1, where they act as an effective constraint on the fluctuations that is equivalent to the Pauli principle: the lines behave like the