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Physical review. B, Condensed matter

Replica-scaling analysis of a directed manifold in a random medium, and ground-state properties of attractive and repulsive Bose systems

Eugene B. Kolomeisky

Published Apr 1, 1992• 6 citations
Publisher

Abstract

A replica theory for the scaling properties of directed manifolds of internal dimension D immersed in a random (D+d)-dimensional medium and the related theory for Bose systems with pairwise interaction are presented. For zero temperature and disorder decreasing with distance, the wandering exponent is given by the Nattermann formula [J. Phys. C 18, 6661 (1985)] \ensuremath{\zeta}=(4-D)/4 (which implies that the correlation-length exponent of the percolation problem in the 1+\ensuremath{\epsilon} approximation is given by ${\ensuremath{\nu}}^{\mathrm{\ensuremath{-}}1}$=\ensuremath{\epsilon}/2). For nonzero temperature \ensuremath{\zeta}=(2-D)/2 for 0D2d/(2+d), and \ensuremath{\zeta}=2/(2+d) for D>2d/(2+d). The ground-state energy of a d-dimensional attractive Bose system with hard-core interaction and particle number n\ensuremath{\gg}1 scales for d2 as ${\mathit{E}}_{\mathit{n}}$\ensuremath{\simeq}-${\mathit{n}}^{(4\mathrm{\ensuremath{-}}\mathit{d})/(2\mathrm{\ensuremath{-}}\mathit{d})}$${\mathit{c}}^{2/(2\mathrm{\ensuremath{-}}\mathit{d})}$ when (nc${)}^{1/(\mathit{d}\mathrm{\ensuremath{-}}2)}$\ensuremath{\ge}a, and ${\mathit{E}}_{\mathit{n}}$\ensuremath{\simeq}-${\mathit{cn}}^{2}$/${\mathit{a}}^{\mathit{d}}$ when (nc${)}^{1/(\mathit{d}\mathrm{\ensuremath{-}}2)}$\ensuremath{\le}a, where c is the attraction strength and a is the core radius. For d>2 a bound state arises when (nc${)}^{1/(\mathit{d}\mathrm{\ensuremath{-}}2)}$\ensuremath{\ge}a and the corresponding ground-state energy scales as ${\mathit{E}}_{\mathit{n}}$\ensuremath{\simeq}-${\mathit{cn}}^{2}$/${\mathit{a}}^{\mathit{d}}$. For the case of repulsive interactions, a connection is made to other treatments of the problem.

DOI
10.1103/physrevb.45.7094

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