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Duality and Form Factors in the Thermally Deformed Two-Dimensional Tricritical Ising Model

by A. Cortés Cubero, R. M. Konik, M. Lencsés, G. Mussardo, G. Takács

This Submission thread is now published as SciPost Phys. 12, 162 (2022)

Submission summary

As Contributors: Máté Lencsés · Gabor Takacs
Arxiv Link: (pdf)
Date accepted: 2022-04-25
Date submitted: 2022-04-11 09:00
Submitted by: Takacs, Gabor
Submitted to: SciPost Physics
Academic field: Physics
  • Condensed Matter Physics - Theory
  • High-Energy Physics - Theory
  • Mathematical Physics
Approaches: Theoretical, Computational


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 $E_7$ 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, $\sigma(x)$ and $\sigma'(x)$, in either phase are accompanied by associated disorder operators, $\mu(x)$ and $\mu'(x)$. Working specifically in the high temperature phase, we write down the sets of bootstrap equations for these four operators. For $\sigma(x)$ and $\sigma'(x)$, the equations are identical in form and are parameterised by the values of the one-particle form factors of the two lightest $\mathbb{Z}_2$ odd particles. Similarly, the equations for $\mu(x)$ and $\mu'(x)$ 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 $\sigma(x)/\sigma'(x)$ are fixed in terms of those for $\mu(x)/\mu'(x)$. We use the truncated conformal space approach to confirm numerically the derived expressions of the matrix elements as well as the validity of the $\Delta$-sum rule as applied to the off-critical correlators. We employ the derived form factors of the order and disorder operators to compute the exact dynamical structure factors of the theory, a set of quantities with a rich spectroscopy which may be directly tested in future inelastic neutron or Raman scattering experiments.

Published as SciPost Phys. 12, 162 (2022)

Author comments upon resubmission

We thank the referee for his thorough review of our work. We have clarified the points raised in his review (c.f. the list of changes).

List of changes

- We have added a discusson of $\mathbb{Z}_2$ symmetries on page 10.

- Indeed, it is the spin of the charges, not of the currents that is referred to on page 20. We corrected the statement accordingly.

- We added the required clarification in Appendix A.

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