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arXiv:1209.5827 · Modern Physics Letters B

PHASE TRANSITION OF THE DISSIPATIVE DOUBLE-WELL QUANTUM MECHANICS

KEN-ICHI AOKI, TAMAO KOBAYASHI

Published Oct 22, 2012• 2 citations
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Abstract

We investigate the critical dissipation of the double-well quantum mechanics. We adopt two-state approximation to define effective Ising models and apply the block decimation renormalization group and the finite range scaling method recently proposed for the long range Ising model. We briefly report the numerical results of the critical dissipation for various model parameters.

DOI
10.1142/s0217984912502028

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Dimensional correspondences have a long history in critical phenomena. Here, we review the effective dimension approach, which relates the scaling exponents of a critical system in d spatial dimensions with power-law decaying interactions r^{d+σ} to a local system, i.e., with finite-range interactions, in an effective fractal dimension d_{eff}. This method simplifies the study of long-range models by leveraging known results from their local counterparts. While the validity of this approximation beyond the mean-field level has been long debated, we demonstrate that the effective dimension approach, while approximate for non-Gaussian fixed points, accurately estimates the critical exponents of long-range models with an accuracy typically larger than 97%. To do so, we review perturbative renormalization-group (RG) results, extend the approximation's validity using functional RG techniques, and compare our findings with precise numerical data from conformal bootstrap for the two-dimensional Ising model with long-range interactions.
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