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arXiv:1011.5463 · Journal of Physics A Mathematical and Theoretical

The energy density of an Ising half-plane lattice

M Assis, Barry M. McCoy

Published Feb 10, 2011• 2 citations
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Abstract

We compute the energy density at arbitrary temperature of the half-plane Ising lattice with a boundary magnetic field H b at a distance M rows from the boundary and compare limiting cases of the exact expression with recent calculations at T = T c done by means of discrete complex analysis methods.
DOI
10.1088/1751-8113/44/9/095003

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Although the physical properties of the 2D and 1D Ising models are quite different, we point out an interesting connection between their complex-temperature phase diagrams. We carry out an exact determination of the complex-temperature phase diagram for the 1D Ising model for arbitrary spin s and show that in the u_s=e^{-K/s^2} plane (i) it consists of N_{c,1D}=4s^2 infinite regions separated by an equal number of boundary curves where the free energy is non-analytic; (ii) these curves extend from the origin to complex infinity, and in both limits are oriented along the angles \\theta_n = (1+2n)\\pi/(4s^2), for n=0,..., 4s^2-1; (iii) of these curves, there are N_{c,NE,1D}=N_{c,NW,1D}=[s^2] in the first and second (NE and NW) quadrants; and (iv) there is a boundary curve (line) along the negative real u_s axis if and only if s is half-integral. We note a close relation between these results and the number of arcs of zeros protruding into the FM phase in our recent calculation of partition function zeros for the 2D spin s Ising model.