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Physical Review Letters

Transfer-Matrix Spectrum for Systems with Interfaces

Vladimir Privman A, Nenad M. Svrakic

Published Feb 6, 1989• 12 citations
Publisher

Abstract

We consider Ising-type systems in the finite cross-section cylindrical geometry, with an interface imposed by the antiperiodic or + - boundary conditions. Closed forms are derived for the largest, asymptotically degenerate transfer-matrix eigenvalues. For rough interfaces ($0<T<{T}_{c}$ in 2D, ${T}_{R}<T<{T}_{c}$ in 3D), the eigenvalues are split algebraically, and the spectral gaps are governed by the surface stiffness coefficient. For "rigid" interfaces ($0<T<{T}_{R}$ in 3D), the splitting is exponential, and determined by the step-free energy.

DOI
10.1103/physrevlett.62.633

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We consider Ising-type systems in the finite cross-section cylindrical geometry, with an interface imposed by the antiperiodic or + - boundary conditions. Closed forms are derived for the largest, asymptotically degenerate transfer-matrix eigenvalues. For rough interfaces (0<T<Tc0<T<{T}_{c}0<T<Tc​ in 2D, TR<T<Tc{T}_{R}<T<{T}_{c}TR​<T<Tc​ in 3D), the eigenvalues are split algebraically, and the spectral gaps are governed by the surface stiffness coefficient. For "rigid" interfaces (0<T<TR0<T<{T}_{R}0<T<TR​ in 3D), the splitting is exponential, and determined by the step-free energy.
We compute by numerical transfer-matrix methods the surface free energy \ensuremath{\tau}(T), the surface stiffness coefficient \ensuremath{\kappa}(T), and the step free energy s(T) for Ising ferromagnets with (\ensuremath{\infty}\ifmmode\times\else\texttimes\fi{}L) square-lattice and (\ensuremath{\infty}\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}M) cubic-lattice geometries, into which an interface is introduced by imposing antiperiodic or plus/minus boundary conditions in one transverse direction. These quantities occur in expansions of the angle-dependent surface tension for either rough or smooth interfaces. The finite-size scaling behavior of the interfacial correlation length provides the means of investigating \ensuremath{\tau}(T), \ensuremath{\kappa}(T), and s(T). The resulting transfer-matrix estimates are fully consistent with previous series and Monte Carlo studies, although current computational technology does not permit transfer-matrix studies of sufficiently large systems to show quantitative improvement over the previous estimates.
For arbitrary Ising-like models of any dimension and Hamiltonians with a finite support with all possible multispin interactions and boundary conditions with a shift, the exact value of the free energy in the thermodynamic limit is obtained at some parametrically specified set of multispin interaction coefficients. In this case, half of the multispin interaction coefficients and the coordinates of the special eigenvector corresponding to the largest eigenvalue of the elementary transfer matrix are parameters, and the second half of the multispin coefficients is calculated using simple explicit formulas. For models with Hamiltonians invariant under the reversal of signs of all spins, the formulas are simplified. As examples of independent interest, solutions are written for the cases when the support of the Hamiltonian is a simplex, a cube, the support of the ANNNI model in spaces of 2, 3 and arbitrary dimensions.
The eigenvalue problem for the transfer matrix of the two‐dimensional Ising lattice, with free boundaries perpendicular to the transfer direction, is solved. The mathematical mechanism which produces the asymptotic degeneracy of the eigenvalue spectrum below the bulk transition temperature is different from the case with cyclic boundary conditions. The results are used to calculate the two‐spin correlation function between two spins in the same edge of the lattice, and between two spins in opposite edges.
For the title system the column-column transfer matrix is set up and diagonalized by exact calculation. The free energy and correlation length follow exactly from the eigenvalues of the auxiliary problem, whereas other quantities such as magnetization profiles, heat capacities, etc., are calculable numerically with arbitrary accuracy. The case of surface fields with the opposite signs is studied in some detail. The transition from exponentially decaying to oscillatory eigenvectors is discussed and related to wetting transitions. For the important case of the perfect asymmetry system the finite-size scaling near the wetting temperature Tw{\mathit{T}}_{\mathit{w}}Tw​ of surface tension and correlation length parallel to the walls, is derived. Also the scaling of magnetization profiles is found. \textcopyright{} 1996 The American Physical Society.