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arXiv:2007.07150 · Physical review. D/Physical review. D.

Energy trapped Ising model

Andrea Amoretti, Gianluca Costagliola, Nicodemo Magnoli, Marcello Scanavino

Published Aug 21, 2020• 1 citation
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Abstract

The authors explore a method for computing correlation functions that is complementary to the well-known bootstrap program and particularly well suited for determining the behavior of off-critical systems, i.e. systems perturbed around their conformal point. They do so for a class of perturbations of the 3$D$ Ising model which are relevant for real experiments on systems in a so-called $t\phantom{\rule{0}{0ex}}h\phantom{\rule{0}{0ex}}e\phantom{\rule{0}{0ex}}r\phantom{\rule{0}{0ex}}m\phantom{\rule{0}{0ex}}a\phantom{\rule{0}{0ex}}l$ $t\phantom{\rule{0}{0ex}}r\phantom{\rule{0}{0ex}}a\phantom{\rule{0}{0ex}}p$.

DOI
10.1103/physrevd.102.036018

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We explain how the axioms of Conformal Field Theory are used to make predictions about critical exponents of continuous phase transitions in three dimensions, via a procedure called the conformal bootstrap. The method assumes conformal invariance of correlation functions, and imposes some relations between correlation functions of different orders. Numerical analysis shows that these conditions are incompatible unless the critical exponents take particular values, or more precisely that they must belong to a small island in the parameter space.
Recently the operator product expansion coefficients of the 3D Ising model universality class have been calculated by studying via Monte Carlo simulation the two-point functions perturbed from the critical point with a relevant field. We show that this method can be applied also when the perturbation is performed with a relevant field coupled to a nonuniform potential acting as a trap. This setting is described by the trap size scaling ansatz, which can be combined with the general framework of the conformal perturbation in order to write down the correlators $⟨\ensuremath{\sigma}(\mathbf{r})\ensuremath{\sigma}(0)⟩$, $⟨\ensuremath{\sigma}(\mathbf{r})\ensuremath{\epsilon}(0)⟩$ and $⟨\ensuremath{\epsilon}(\mathbf{r})\ensuremath{\epsilon}(0)⟩$, from which the operator product expansion coefficients can be estimated. We find ${C}_{\ensuremath{\sigma}\ensuremath{\epsilon}}^{\ensuremath{\sigma}}=1.051(3)$, in agreement with the results already known in the literature, and ${C}_{\ensuremath{\epsilon}\ensuremath{\epsilon}}^{\ensuremath{\epsilon}}=1.32(15)$, confirming and improving the previous estimate obtained in the uniform perturbation case.