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arXiv:1001.1059 · Journal of Statistical Mechanics Theory and Experiment

Bethe ansatz solution for one-dimensional directed polymers in random media

Victor S Dotsenko, B. A. Klumov

Published Mar 24, 2010• 45 citations
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Abstract

We study the statistical properties of one-dimensional directed polymers in a short-range random potential by mapping the replicated problem to a many-body quantum boson system with attractive interactions. We find the full set of eigenvalues and eigenfunctions of the many-body system and perform the summation over the entire spectrum of excited states. The analytic continuation of the obtained exact expression for the replica partition function from integer to non-integer replica parameter N turns out to be ambiguous. Performing the analytic continuation simply by assuming that the parameter N can take arbitrary complex values, and going to the thermodynamic limit of the original directed polymer problem, we obtain the explicit universal expression for the probability distribution function of free energy fluctuations.

DOI
10.1088/1742-5468/2010/03/p03022

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We study the statistical properties of one-dimensional directed polymers in a short-range random potential by mapping the replicated problem to a many-body quantum boson system with attractive interactions. We find the full set of eigenvalues and eigenfunctions of the many-body system and perform the summation over the entire spectrum of excited states. The analytic continuation of the obtained exact expression for the replica partition function from integer to non-integer replica parameter N turns out to be ambiguous. Performing the analytic continuation simply by assuming that the parameter N can take arbitrary complex values, and going to the thermodynamic limit of the original directed polymer problem, we obtain the explicit universal expression for the probability distribution function of free energy fluctuations.
The energy gap between the ground state and the first excited state of the one-dimensional attractive Bose-Hubbard Hamiltonian is investigated in connection with directed polymers in random media. The excitation gap Δis obtained by exact diagonalization of the Hamiltonian in the two- and three-particle sectors and also by an exact Bethe Ansatz solution in the two-particle sector. The dynamic exponent z is found to be 2. However, in the intermediate range of the size L where UL~O(1), U being the attractive interaction, the effective dynamic exponent shows an anomalous peak reaching high values of 2.4 and 2.7 for the two- and the three-particle sectors, respectively. The anomalous behavior is related to a change in the sign of the first excited-state energy. In the two-particle sector, we use the Bethe Ansatz solution to obtain the effective dynamic exponent as a function of the scaling variable UL/π. The continuum version, the attractive delta-function Bose-gas Hamiltonian, is integrable by the Bethe Ansatz with suitable quantum numbers, the distributions of which are not known in general. Quantum numbers are proposed for the first excited state and are confirmed numerically for an arbitrary number of particles.
We investigate the universal fluctuations of localized wave functions in the Fock space of two interacting particles in one-dimensional disordered systems, focusing on the interplay between random potentials and random long-range interactions. By mapping the system onto a directed polymer problem, we show that random potentials alone produce correlated energies for the sites in the Fock space, giving rise to the fluctuation growth exponent 1 / 2 . Introducing random long-range interactions alters these correlations and drives the system's fluctuations into the Kardar-Parisi-Zhang universality class in (1+1)D with the exponent 1 / 3 . To validate the universality of the observed fluctuation scaling, we study a complex directed polymer model with competing point and columnar disorder. Our results confirm that columnar disorder corresponds to on-site energies in the Fock space from the random potentials, while point disorder models the effects of random long-range interactions between the two particles. These findings provide new insights into the Fock-space perspective for examining disordered quantum many-body systems, and they emphasize the critical role of disorder structure in determining the universality class of fluctuations in localized quantum systems.
A finite density of mutually avoiding directed polymers in a quenched random medium is a minimal model of glassy line matter. The dilute theory, solved by replica Bethe ansatz, predicts an interaction free energy proportional to ρ2ρ^2ρ2 and disorder cumulants with distinct power-law dependences on the density ρρρ, but direct numerical tests have been hindered by the combinatorially large many-polymer transfer matrix. We recast the problem as filling logarithmic eigenvalues of a single-polymer transfer-matrix product, obtaining the quenched free energy, its cumulants, and a disorder-induced linear spectral edge consistent with the replica prediction.
Anderson localization - the suppression of wave propagation by interference in a sufficiently disordered medium - is one of the most remarkable phenomena in condensed matter physics. These lecture notes provide a pedagogical introduction to Anderson localization on the Bethe lattice, an infinite tree with fixed connectivity and no loops, and one of the few settings in which the transition can be studied exactly. The problem is important for two main reasons. First, the Anderson model on the Bethe lattice provides a mean-field, infinite-dimensional limit in which the transition point and critical behavior can be determined exactly. Second, it provides a simplified description of the high-dimensional configuration spaces of interacting quantum systems, and thus a useful toy model for many-body localization. We focus on the Green's function formalism and the cavity method. After recalling standard diagnostics of localization based on transport, spectral statistics, and eigenfunction moments, we show that they are encoded in the statistics of the Green's functions, and that the full distribution of the local density of states is the natural order parameter of the transition. We derive the exact self-consistent cavity equations on the Bethe lattice, use them to characterize the two phases and critical behavior - including an analytic solution at large connectivity - and describe population-dynamics algorithms that solve them with high precision. We explain how an exponentially large correlation volume and the localized nature of the critical point account for the strong finite-size effects on random regular graphs, which led to a long debate over a possible intermediate non-ergodic phase. Finally, we discuss the mapping onto directed polymers in random media, interpreting localization as a freezing transition dominated by a few rare resonant paths.
The impact of polymer-polymer interactions of various types on the thermodynamics, structure, and accommodation of topological constraints is addressed for systems comprising many directed polymers in two spatial dimensions. The approach is predicated on the well-known equivalence between the classical equilibrium statistical mechanics of directed polymers in two spatial dimensions and the imaginary-time quantum dynamics of particles in one spatial dimension, originally exploited by P.-G. de Gennes [P.-G. de Gennes, J. Chem. Phys. 48, 2257 (1968)]. Known results concerning two exactly solvable microscopic models of quantum particles moving in one spatial dimension---the Lieb-Liniger model of contact interactions and the Calogero-Sutherland model of long-range interactions---are used to shed light on the behavior of the corresponding polymeric systems. In addition, the technique of bosonization is used to reveal how generic polymer interactions give rise to an emergent polymer fluid that has universal collective excitations. Additionally, the response of the system to topological constraints such as pins though which polymers cannot pass is explored. Immediately on the compressed side of a pin there is a divergent pile-up in polymer density, while on the other side there is a gap of finite area in which polymer density is negligible. Comparison of this response to that of a system of simply noncrossing (i.e., noncrossing but otherwise noninteracting) directed polymers, explored in a companion paper, reveals that generic interactions leave the structure quantitatively unchanged on the line transverse to the pin, and leave it qualitatively unchanged throughout the two dimensions of the system's extent. Furthermore, the free-energy cost associated with a pin that partitions a system having generic interactions is found to be proportional to the pin-partitioning cost for a system of simply noncrossing polymers.