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Physical review. B, Condensed matter

Energy-density correlation function of the Ising model in small magnetic fields for T< Tc

Mu-Lin Yan, Wenxiang Cao

Published Nov 1, 1988• 3 citations
Publisher

Abstract

We present a calculation of the energy-density correlation ${d}_{\mathrm{\ensuremath{-}}}$(r,h) of the two-dimensional Ising model in a magnetic field under the limit T\ensuremath{\rightarrow}${T}_{c}$-, H\ensuremath{\rightarrow}0, and R\ensuremath{\rightarrow}\ensuremath{\infty} where h=const\ifmmode\times\else\texttimes\fi{}H\ensuremath{\Vert}T-${T}_{c}$${\ensuremath{\Vert}}^{\mathrm{\ensuremath{-}}15/8}$ is fixed and small, while r=R\ensuremath{\Vert}T-${T}_{c}$\ensuremath{\Vert} is fixed and large. We find ${\mathit{d}}_{\mathrm{\ensuremath{-}}}$(r,h)\ensuremath{\sim}2${\mathrm{\ensuremath{\pi}}}^{\mathrm{\ensuremath{-}}1/2}$${\mathrm{hr}}^{\mathrm{\ensuremath{-}}1/2}$${\mathrm{e}}^{\mathrm{\ensuremath{-}}2}$, where ${\ensuremath{\lambda}}_{l}$ are the positive solutions of ${\mathit{J}}_{1/3}$(${\ensuremath{\lambda}}_{\mathrm{l}}$/3)+${\mathrm{J}}_{\mathrm{\ensuremath{-}}1/3}$ (${\ensuremath{\lambda}}_{1/3}$)=0.

DOI
10.1103/physrevb.38.9152

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arXiv:hep-th/9605065•Physics Letters B•Sep 1, 1996•67 citations
We present a calculation of the energy-density correlation d\ensuremath−{d}_{\mathrm{\ensuremath{-}}}d\ensuremath−​(r,h) of the two-dimensional Ising model in a magnetic field under the limit T\ensuremath{\rightarrow}Tc{T}_{c}Tc​-, H\ensuremath{\rightarrow}0, and R\ensuremath{\rightarrow}\ensuremath{\infty} where h=const\ifmmode\times\else\texttimes\fi{}H\ensuremath{\Vert}T-{T}_{c}$${\ensuremath{\Vert}}^{\mathrm{\ensuremath{-}}15/8} is fixed and small, while r=R\ensuremath{\Vert}T-Tc{T}_{c}Tc​\ensuremath{\Vert} is fixed and large. We find d\ensuremath−{\mathit{d}}_{\mathrm{\ensuremath{-}}}d\ensuremath−​(r,h)\ensuremath{\sim}2{\mathrm{\ensuremath{\pi}}}^{\mathrm{\ensuremath{-}}1/2}$${\mathrm{hr}}^{\mathrm{\ensuremath{-}}1/2}$${\mathrm{e}}^{\mathrm{\ensuremath{-}}2}, where \ensuremathλl{\ensuremath{\lambda}}_{l}\ensuremathλl​ are the positive solutions of J1/3{\mathit{J}}_{1/3}J1/3​(\ensuremathλl{\ensuremath{\lambda}}_{\mathrm{l}}\ensuremathλl​/3)+J\ensuremath−1/3{\mathrm{J}}_{\mathrm{\ensuremath{-}}1/3}J\ensuremath−1/3​ (\ensuremathλ1/3{\ensuremath{\lambda}}_{1/3}\ensuremathλ1/3​)=0.
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The energy-density---energy-density and energy-density---spin correlation functions are evaluated for the two-dimensional square Ising model. Results are obtained at zero magnetic field and near the critical point, i.e., for $\ensuremath{\epsilon}=(\frac{4J}{\mathrm{kT}})|(T\ensuremath{-}\frac{{T}_{c}}{{T}_{c}})|\ensuremath{\ll}1.$ The correlation functions are ${〈{E}_{R}{E}_{0}〉\ensuremath{-}{〈E〉}^{2},={(\frac{2J}{\ensuremath{\pi}})}^{2}{\ensuremath{\epsilon}}^{2}[{{K}_{1}}^{2}(\ensuremath{\epsilon}R)\ensuremath{-}{{K}_{0}}^{2}(\ensuremath{\epsilon}R)],}{〈{E}_{R}{\ensuremath{\sigma}}_{0}〉\ensuremath{-}〈E〉〈\ensuremath{\sigma}〉,=(\frac{2J}{\ensuremath{\pi}})\ensuremath{\epsilon}〈\ensuremath{\sigma}〉\ensuremath{\int}{2\ensuremath{\epsilon}R}^{\ensuremath{\infty}}\mathrm{ds} {s}^{\ensuremath{-}2}{e}^{\ensuremath{-}8},}$ where RRR is the distance between the points in question and K0{K}_{0}K0​ and K1{K}_{1}K1​ are modified Bessel functions of the second kind. Very similar results hold if the coupling constants are different in the two directions. The above results agree in form with the structure predicted by the "scaling-law" approach.
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