Duality and form factors in the thermally deformed two-dimensional tricritical Ising model

The thermal deformation of the critical point action of the 2D tricritical Ising model gives rise to an exact scattering theory with seven massive excitations based on the exceptional E_7 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>E</mml:mi> <mml:mn>7</mml:mn> </mml:msub> </mml:math> Lie algebra. The high and low temperature phases of this model are related by duality. This duality guarantees that the leading and sub-leading magnetisation operators, \sigma(x) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>σ</mml:mi> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> and \sigma'(x) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>σ</mml:mi> <mml:mi>′</mml:mi> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> , in either phase are accompanied by associated disorder operators, \mu(x) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>μ</mml:mi> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> and \mu'(x) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>μ</mml:mi> <mml:mi>′</mml:mi> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> . Working specifically in the high temperature phase, we write down the sets of bootstrap equations for these four operators. For \sigma(x) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>σ</mml:mi> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> and \sigma'(x) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>σ</mml:mi> <mml:mi>′</mml:mi> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> , the equations are identical in form and are parameterised by the values of the one-particle form factors of the two lightest \mathbb{Z}_2 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mstyle mathvariant="double-struck"> <mml:mi>ℤ</mml:mi> </mml:mstyle> <mml:mn>2</mml:mn> </mml:msub> </mml:math> odd particles. Similarly, the equations for \mu(x) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>μ</mml:mi> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> and \mu'(x) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>μ</mml:mi> <mml:mi>′</mml:mi> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> have identical form and are parameterised by two elementary form factors. Using the clustering property, we show that these four sets of solutions are eventually not independent; instead, the parameters of the solutions for \sigma(x)/\sigma'(x) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>σ</mml:mi> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false" form="postfix">)</mml:mo> <mml:mi>/</mml:mi> <mml:mi>σ</mml:mi> <mml:mi>′</mml:mi> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> are fixed in terms of those for \mu(x)/\mu'(x) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>μ</mml:mi> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false" form="postfix">)</mml:mo> <mml:mi>/</mml:mi> <mml:mi>μ</mml:mi> <mml:mi>′</mml:mi> <mml:mo stretchy="false" form="prefix">(</mml:mo>

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