arXiv:cond-mat/9608067

Reflection equations and surface critical phenomena

Murray T. Batchelor

Published Aug 16, 1996• 5 citations

Abstract

A brief review is given of recent developments in the study of surface critical phenomena from the viewpoint of exactly solved lattice models. These developments include exact results for the polymer adsorption transition and the surface critical exponents of the eight-vertex model.

DOI: 10.48550/arxiv.cond-mat/9608067

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arXiv:cond-mat/9608067

Reflection equations and surface critical phenomena

Murray T. Batchelor

Published Aug 16, 1996• 5 citations

arXiv HTML PDF

Abstract

DOI: 10.48550/arxiv.cond-mat/9608067

Related papers

Surface Critical Phenomena and Scaling in the Eight-Vertex Model [Phys. Rev. Lett. 76, 14 (1996)]

Murray T. Batchelor, Yu-Sen Zhou

Physical Review LettersApr 8, 199611 citations

Surface Critical Exponents in Terms of Bulk Exponents

Alan J. Bray, M A Moore

The surface exponents associated with critical phenomena in semi-infinite systems are derived exactly in terms of bulk exponents. Results are ${\ensuremath{\gamma}}_{1,1}=\ensuremath{\nu}\ensuremath{-}1$, ${\ensuremath{\gamma}}_{1}=\ensuremath{\nu}+\frac{(\ensuremath{\gamma}\ensuremath{-}1)}{2}$, ${\ensuremath{\beta}}_{1}=\frac{(3\ensuremath{-}\ensuremath{\alpha})}{2}\ensuremath{-}\ensuremath{\nu}$, ${\ensuremath{\Delta}}_{1}=\frac{(1\ensuremath{-}\ensuremath{\alpha})}{2}$, ${\ensuremath{\eta}}_{\ensuremath{\parallel}}=\frac{1}{\ensuremath{\nu}}$, and ${\ensuremath{\eta}}_{\ensuremath{\perp}}=1\ensuremath{-}\frac{(\ensuremath{\gamma}\ensuremath{-}1)}{2\ensuremath{\nu}}$, where $\ensuremath{\nu}$, $\ensuremath{\gamma}$, $\ensuremath{\alpha}$, etc., are the standard bulk exponents. The bulk-surface crossover exponent is ${\ensuremath{\phi}}_{s}=1\ensuremath{-}\ensuremath{\nu}$. For $\ensuremath{\nu}>1$ no surface phase can exist.The surface exponents associated with critical phenomena in semi-infinite systems are derived exactly in terms of bulk exponents. Results are ${\ensuremath{\gamma}}_{1,1}=\ensuremath{\nu}\ensuremath{-}1$, ${\ensuremath{\gamma}}_{1}=\ensuremath{\nu}+\frac{(\ensuremath{\gamma}\ensuremath{-}1)}{2}$, ${\ensuremath{\beta}}_{1}=\frac{(3\ensuremath{-}\ensuremath{\alpha})}{2}\ensuremath{-}\ensuremath{\nu}$, ${\ensuremath{\Delta}}_{1}=\frac{(1\ensuremath{-}\ensuremath{\alpha})}{2}$, ${\ensuremath{\eta}}_{\ensuremath{\parallel}}=\frac{1}{\ensuremath{\nu}}$, and ${\ensuremath{\eta}}_{\ensuremath{\perp}}=1\ensuremath{-}\frac{(\ensuremath{\gamma}\ensuremath{-}1)}{2\ensuremath{\nu}}$, where $\ensuremath{\nu}$, $\ensuremath{\gamma}$, $\ensuremath{\alpha}$, etc., are the standard bulk exponents. The bulk-surface crossover exponent is ${\ensuremath{\phi}}_{s}=1\ensuremath{-}\ensuremath{\nu}$. For $\ensuremath{\nu}>1$ no surface phase can exist.

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We derive the exact critical couplings $({x}^{*}{,y}_{a}^{*})$ , where ${y}_{a}^{*}{/x}^{*}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\sqrt{1+\sqrt{2}}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.533\dots{}$ , for the polymer adsorption transition on the honeycomb lattice, along with the universal critical exponents, from the Bethe ansatz solution of the $O\left(n\right)$ loop model at the special transition. Our result for the thermal scaling dimension, and thus the crossover exponent $\ensuremath{\varphi}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\frac{1}{2}$, is in agreement with an earlier result based on conformal invariance arguments. Our result for the geometric scaling dimensions confirms recent conjectures that they are given by ${h}_{\ensuremath{\ell}+1,3}$ in the Kac formula.We derive the exact critical couplings $({x}^{*}{,y}_{a}^{*})$, where ${y}_{a}^{*}{/x}^{*}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\sqrt{1+\sqrt{2}}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.533\dots{}$ , for the polymer adsorption transition on the honeycomb lattice, along with the universal critical exponents, from the Bethe ansatz solution of the $O\left(n\right)$ loop model at the special transition. Our result for the thermal scaling dimension, and thus the crossover exponent $\ensuremath{\varphi}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\frac{1}{2}$, is in agreement with an earlier result based on conformal invariance arguments. Our result for the geometric scaling dimensions confirms recent conjectures that they are given by ${h}_{\ensuremath{\ell}+1,3}$ in the Kac formula.

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We derive the exact critical couplings

({x}^{*}{,y}_{a}^{*})

, where

{y}_{a}^{*}{/x}^{*}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\sqrt{1+\sqrt{2}}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.533\dots{}

, for the polymer adsorption transition on the honeycomb lattice, along with the universal critical exponents, from the Bethe ansatz solution of the

O\left(n\right)

loop model at the special transition. Our result for the thermal scaling dimension, and thus the crossover exponent $\ensuremath{\varphi}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\frac{1}{2}$, is in agreement with an earlier result based on conformal invariance arguments. Our result for the geometric scaling dimensions confirms recent conjectures that they are given by ${h}_{\ensuremath{\ell}+1,3}$ in the Kac formula.