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Directed polymer melts and quantum critical phenomena

Randall D. Kamien, David R. Nelson

arXiv:cond-mat/9206006Journal of Statistical PhysicsApr 1, 199317 citations

On the Bethe ansatz for random directed polymers

J. P. Bouchaud, Henri Orland

Journal of Statistical PhysicsNov 1, 199054 citations

Directed polymer inside a parabola: Exact solution

Ferenc Iglói

We consider a directed polymer on a square lattice confined inside a parabola with boundaries Y=\ifmmode\pm\else\textpm\fi{}${\mathit{Cx}}^{\mathrm{\ensuremath{\alpha}}}$. Generalizing standard transfer-matrix methods, an effective transfer operator was introduced that takes into account slow variation of the size of the system in the y direction. By an exact calculation the effect of the confinement is found to be relevant and irrelevant for \ensuremath{\alpha}1/2 and >1/2, respectively; in the former case the thermodynamical functions have a stretched-exponential behavior around the critical point. For \ensuremath{\alpha}=1/2 the geometrical constraint is marginal, and the critical exponents are nonuniversal.We consider a directed polymer on a square lattice confined inside a parabola with boundaries Y=\ifmmode\pm\else\textpm\fi{}${\mathit{Cx}}^{\mathrm{\ensuremath{\alpha}}}$. Generalizing standard transfer-matrix methods, an effective transfer operator was introduced that takes into account slow variation of the size of the system in the y direction. By an exact calculation the effect of the confinement is found to be relevant and irrelevant for \ensuremath{\alpha}1/2 and >1/2, respectively; in the former case the thermodynamical functions have a stretched-exponential behavior around the critical point. For \ensuremath{\alpha}=1/2 the geometrical constraint is marginal, and the critical exponents are nonuniversal.

Physical Review AMay 1, 199215 citations

Critical behaviour in aperiodic systems

Ferenc Iglói

The author considers layered systems with aperiodically modulated couplings and study the effect of inhomogeneity on the critical behaviour. Using scaling arguments a relevance/irrelevance-type criterion is formulated and for relevant inhomogeneities the scaling form of singular quantities is determined. The author has performed exact calculations for the surface magnetization of aperiodic quantum Ising chains and for a (1+1)-dimensional directed polymer in a longitudinal aperiodic environment. The results for both systems are in accord with scaling considerations. For a marginal aperiodicity the critical exponents are found to be non-universal.The author considers layered systems with aperiodically modulated couplings and study the effect of inhomogeneity on the critical behaviour. Using scaling arguments a relevance/irrelevance-type criterion is formulated and for relevant inhomogeneities the scaling form of singular quantities is determined. The author has performed exact calculations for the surface magnetization of aperiodic quantum Ising chains and for a (1+1)-dimensional directed polymer in a longitudinal aperiodic environment. The results for both systems are in accord with scaling considerations. For a marginal aperiodicity the critical exponents are found to be non-universal.

Journal of Physics A Mathematical and GeneralAug 7, 199377 citations

Directed polymers in Hartree-Fock approximation

Yi‐Cheng Zhang

Journal of Statistical PhysicsDec 1, 198922 citations

Fermions in a random medium

Harald Kinzelbach, Michael Lässig

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 .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 .

arXiv:cond-mat/9411126Nov 29, 19940 citations

Directed Polymers with Random Interaction: Marginal Relevance and Novel Criticality

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