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arXiv:cond-mat/9411126

Fermions in a random medium

Harald Kinzelbach, Michael Lässig

Published Nov 29, 1994• 0 citations
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Abstract

We study the continuum field theory for an ensemble of directed polymers r_i (t) in 1+d' dimensions that live in a medium with quenched point disorder and interact via short-ranged pair forces g Ψ(r_i - r_j). 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_\perp \sim |g|^{-ν_\perp}; repulsive forces lead to mutual avoidance with a pair distribution function P(r_i - r_j) \sim |r_i - r_j|^θreminiscent of interacting fermions. In the experimentally important dimension d' = 2, we obtain ν_\perp \approx 0.8 and θ\approx 2.4 .

DOI
10.48550/arxiv.cond-mat/9411126

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We study the continuum field theory for an ensemble of directed polymers r_i (t) in 1+d' dimensions that live in a medium with quenched point disorder and interact via short-ranged pair forces g Ψ(r_i - r_j). 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_\perp \sim |g|^{-ν_\perp}; repulsive forces lead to mutual avoidance with a pair distribution function P(r_i - r_j) \sim |r_i - r_j|^θreminiscent of interacting fermions. In the experimentally important dimension d' = 2, we obtain ν_\perp \approx 0.8 and θ\approx 2.4 .
For the quantum Ising model with ferromagnetic random couplings J i , j > 0 and random transverse fields h i > 0 at zero temperature in finite dimensions d > 1, we consider the lowest order contributions in perturbation theory in ( J i , j / h i ) to obtain some information on the statistics of various observables in the disordered phase. We find that the two-point correlation scales as , where ξ typ is the typical correlation length, u is a random variable and ω coincides with the droplet exponent ω DP ( D = d − 1) of the directed polymer with D = ( d − 1) transverse directions. Our main conclusions are as follows. (i) Whenever ω > 0, the quantum model is governed by an infinite-disorder fixed point: there are two distinct correlation length exponents related by ν typ = (1 − ω)ν av ; the distribution of the local susceptibility χ loc presents the power-law tail P (χ loc ) ∼ 1/χ 1 + μ loc , where μ vanishes as ξ −ω av so that the averaged local susceptibility diverges in a finite neighborhood 0 < μ < 1 before criticality (Griffiths phase); the dynamical exponent z diverges near criticality as z = d /μ ∼ ξ ω av . (ii) In dimensions d ⩽ 3, any infinitesimal disorder flows toward this infinite-disorder fixed point with ω( d ) > 0 (for instance ω( d = 2) = 1/3 and ω( d = 3) ∼ 0.24). (iii) In finite dimensions d > 3, a finite disorder strength is necessary to flow toward the infinite-disorder fixed point with ω( d ) > 0 (for instance ω( d = 4) ≃ 0.19), whereas a finite-disorder fixed point remains possible for a small enough disorder strength. For the Cayley tree of effective dimension d = ∞, where ω = 0, we discuss the similarities and differences with the case of finite dimensions.
We analyze, via Imry-Ma scaling arguments, the strong disorder phases that exist in low dimensions at all temperatures for directed polymers and interfaces in random media. For the uncorrelated Gaussian disorder, we obtain that the optimal strategy for the polymer in dimension 1+d with 0<d<2 involves at the same time (i) a confinement in a favorable tube of radius R(S) approximately L (nu(S) ) with nu(S) =1/(4-d)<1/2 (ii) a superdiffusive behavior R approximately Lnu with nu=(3-d)/(4-d)>1/2 for the wandering of the best favorable tube available. The corresponding free energy then scales as F approximately Lomega with omega=2nu-1 and the left tail of the probability distribution involves a stretched exponential of exponent eta=(4-d)/2. These results generalize the well known exact exponents nu=2/3, omega=1/3, and eta=3/2 in d=1, where the subleading transverse length R(S) approximately L(1/3) is known as the typical distance between two replicas in the Bethe ansatz wave function. We then extend our approach to correlated disorder in transverse directions with exponent alpha and/or to manifolds in dimension D+d= d(t) with 0<D<2. The strategy of being both confined and superdiffusive is still optimal for decaying correlations ( alpha<0 ), whereas it is not for growing correlations ( alpha>0 ). In particular, for an interface of dimension ( d(t) -1) in a space of total dimension 5/3< d(t) <3 with random-bond disorder, our approach yields the confinement exponent nu(S) =( d(t) -1)(3- d(t) )/(5 d(t) -7). Finally, we study the exponents in the presence of an algebraic tail 1/ V1+micro in the disorder distribution, and obtain various regimes in the (micro,d) plane.
One-dimensional massive quantum particles [or (1 + 1)-dimensional random walks] with short-ranged multiparticle interactions are studied by exact renormalization group methods. With repulsive pair forces, such particles are known to scale as free fermions. With finite mmm-body forces (m=3,4,…m=3,4,\dots{}m=3,4,…), a critical instability is found, indicating the transition to a fermionic bound state. These unbinding transitions represent new universality classes of interacting fermions relevant to polymer and membrane systems. Implications for massless fermions, e.g., in the Hubbard model, are also noted.
We investigate the localization of stiff directed lines with bending energy by a short-range random potential. We apply perturbative arguments, Flory scaling arguments, a variational replica calculation, and functional renormalization to show that a stiff directed line in 1+d dimensions undergoes a localization transition with increasing disorder for d>2/3. We demonstrate that this transition is accessible by numerical transfer matrix calculations in 1+1 dimensions and analyze the properties of the disorder-dominated phase in detail. On the basis of the two-replica problem, we propose a relation between the localization of stiff directed lines in 1+d dimensions and of directed lines under tension in 1+3d dimensions, which is strongly supported by identical free-energy distributions. This shows that pair interactions in the replicated Hamiltonian determine the nature of directed line localization transitions with consequences for the critical behavior of the Kardar-Parisi-Zhang equation. We support the proposed relation to directed lines via multifractal analysis, revealing an analogous Anderson transition-like scenario and a matching correlation length exponent. Furthermore, we quantify how the persistence length of the stiff directed line is reduced by disorder.
We consider a pair of identical fermions with a short-range attractive interaction on a finite lattice cluster in the presence of strong site disorder. This toy model imitates a low-density regime of the strongly disordered Hubbard model. In contrast to spinful fermions, which can simultaneously occupy a site with minimal energy and thus always form a bound state resistant to disorder, for the identical fermions the probability of pairing on neighboring sites depends on the relation between the interaction and the disorder. The complexity of ``brute-force'' calculations (both analytical and numerical) of this probability grows rapidly with the number of sites even for the simplest cluster geometry in the form of a closed chain. Remarkably, this problem is related to the old mathematical task of computing the volume of a polyhedron, known as NP-hard. However, we have found that the problem in the chain geometry can be exactly solved by the transfer matrix method. Using this approach, we have calculated the pairing probability in the long chain for an arbitrary relation between the interaction and the disorder strengths and completely describe the crossover between the regimes of coupled and separated fermions.
The localization of a directed polymer onto an extended defect (such as a line or a plane) in the presence of competing bulk disorder is examined. Based on scaling ideas and exact analysis on a hierarchical lattice, we develop a new renormalization scheme to study the directed polymer localization problem. We establish absence of delocalization transition for attractive columnar defect in the marginal dimension dc{\mathit{d}}_{\mathit{c}}dc​=2, and for attractive planar defect in d=3. For columnar defect in three dimensions, our simulations yield a localization length exponent ${\ensuremath{\nu}}_{\mathrm{\ensuremath{\perp}}}$=1.8\ifmmode\pm\else\textpm\fi{}0.6.