Sharp phase boundaries for a lattice flux line model
The authors analyse the phase diagram of a model of mutually avoiding and self-avoiding random walks on rectangles, in the limit of the rectangles increasing to \(\mathbb Z^d\). Special cases arise in \(d=2\) in connection with dimer models and other exactly solvable models, in \(d=3\) in connection with flux lines in superconductors and directed polymers. The model is as follows. A nearest neighbor self-avoiding walk \(\omega\) in \(\mathbb Z^d\) receives the weight \[ z^\omega=\prod_{i=1}^d z_{i+}^{N_{i+}(\omega)}z_{i-}^{N_{i-}(\omega)}, \] where \(z=(z_{1+},z_{1-},\dots,z_{d+}, z_{d-})\) is a vector of \(2d\) nonnegative weights, and \(N_{i\pm}(\omega)\) denotes the number of positive resp.\ negative steps in the \(i\)-direction, respectively. The main attention will be paid to the case where the weights represent flux lines in a magnetic field, i.e. \(z_{i\pm}=e^{-\beta(\varepsilon\mp h_i)}\). The grand canonical partition function in a rectangle \(\Lambda\subset\mathbb Z^d\) is defined as \(Z(z,\Lambda)=\sum_{\Omega}\prod_{\omega\in\Omega} z^\omega\), where the sum is over all sets \(\Omega\) of (an arbitrary number of) self-avoiding, mutually avoiding nearest-neighbor walks, starting and ending at the boundary of \(\Lambda\). The \(k\)-point correlation function at \(k\) points \(x_1,\dots,x_k\in\mathbb Z^d\), \(S_\Lambda(x_1,\dots,x_k)\), is defined as \(1/Z(z,\Lambda)\) times the restriction of the above sum to those configurations in which each \( x_i\) is hit by at least one of the walks. The connectivity of \(x_1,\dots,x_k\), \(\tau_\Lambda (x_1,\dots,x_k)\), is the same with the restriction to those configurations in which at least one of the walks hits all the points \(x_1,\dots,x_k\). Define the \textit{free energy} \(f(z)= \lim_{\Lambda\uparrow\mathbb Z^d}|\Lambda|^{-1}\log Z(z,\Lambda)\), and define the infinite volume limits \(S\) of \(S_\Lambda\) and \(\tau\) of \(\tau_\Lambda\). The main results are as follows. In the case of flux lines in a magnetic field (see above), the values of \(f(z)\), \(S(z)\) and \(\tau(z)\) are characterized in terms of the values of the field parameters (\(\varepsilon\), \(\beta\), and \(h_1,\dots,h_d\)) and the exponential parameter \(\lambda(z)=\lim_{N\to\infty}\chi _N(z)^{1/N}\) of the generating function, \(\chi_N(z)=\sum_{\omega\colon |\omega|=N}z^\omega\) of the \(N\)-step self-avoiding paths starting at the origin. More precisely, it is shown among other things, that \(f(z)=0\) if and only if \(\lambda(z)\leq 1\), and that \(S\equiv\tau\equiv 0\) if \(\lambda(z)<1\). (The results for the flux line case follow from the analogous results for the general case, which are more abstract and are not recorded here.) The results imply the existence of three different phase transitions: a Meissner phase where there are no bulk lines in the flux; a frozen phase where the flux line density is maximal; and an intermediate flux liquid phase where the density of flux lines is neither zero nor maximal. They admit a precise location of the phase boundaries.
scientific article; zbMATH DE number 1486436