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Two-dimensional layered Ising models: Exact variational formulation and analysis

1994; American Physical Society; Volume: 49; Issue: 1 Linguagem: Inglês

10.1103/physrevb.49.378

1095-3795

Lev V. Mikheev, Michael E. Fisher,

Quantum many-body systems

Ising models on the plane square lattice with an arbitrary variation of the bond strengths, ${\mathit{J}}^{\mathrm{\ensuremath{\parallel}}}$(z) and ${\mathit{J}}^{\mathrm{\ensuremath{\perp}}}$(z), with one of the two axial coordinates, z, are considered. The total entropy is exactly represented as a functional of contributions ${\mathrm{\ensuremath{\varepsilon}}}_{\mathit{q}}$(z) to the local energy density arising from the Onsager fermions with wave vector q parallel to the layer axis, y. The resulting explicit local expression provides an effective variational principle for the free energy and energy-density profiles. In the scaling limit the problem reduces to a set of independent second-order differential equations for each ${\mathrm{\ensuremath{\varepsilon}}}_{\mathit{q}}$(z). The power of the method is demonstrated by application to an interface between two uniform but distinct regions; this includes the problem of a wall with a surface field, ${\mathit{h}}_{1}$, as a special case. Previous results for the bulk and surface exponents and for the energy-energy correlation function are easily recovered. Near criticality the method yields, in addition, universal, scaled energy-density profiles, \ensuremath{\varepsilon}(z;T), which exhibit rich crossovers and nonmonotonic variation with z.