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arXiv:2211.16170 · Physical review. B./Physical review. B

Cubic fixed point in three dimensions: Monte Carlo simulations of the ϕ4 model on the simple cubic lattice

Martin Hasenbusch B

Published Jan 10, 2023• 23 citations
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Abstract

We study the cubic fixed point for $N=3$ and 4 by using finite-size scaling applied to data obtained from Monte Carlo simulations of the $N$-component ${\ensuremath{\phi}}^{4}$ model on the simple cubic lattice. We generalize the idea of improved models to a two-parameter family of models. The two-parameter space is scanned for the point, where the amplitudes of the two leading corrections to scaling vanish. To this end, a dimensionless quantity is introduced that monitors the breaking of the $O(N)$ invariance. For $N=4$, we determine the correction exponents ${\ensuremath{\omega}}_{1}=0.763(24)$ and ${\ensuremath{\omega}}_{2}=0.082(5)$. In the case of $N=3$, we obtain ${Y}_{4}=0.0142(6)$ for the renormalization group exponent of the cubic perturbation at the $O(3)$-invariant fixed point, while the correction exponent ${\ensuremath{\omega}}_{2}=0.0133(8)$ at the cubic fixed point. Simulations close to the improved point result in the estimates $\ensuremath{\nu}=0.7202(7)$ and $\ensuremath{\eta}=0.0371(2)$ of the critical exponents of the cubic fixed point for $N=4$. For $N=3$, at the cubic fixed point, the $O(3)$ symmetry is only mildly broken and the critical exponents differ only by little from those of the $O(3)$-invariant fixed point. We find $\ensuremath{-}0.000\phantom{\rule{0.16em}{0ex}}01⪅{\ensuremath{\eta}}_{\text{cubic}}\ensuremath{-}{\ensuremath{\eta}}_{O(3)}⪅0.000\phantom{\rule{0.16em}{0ex}}07$ and ${\ensuremath{\nu}}_{\text{cubic}}\ensuremath{-}{\ensuremath{\nu}}_{O(3)}=\ensuremath{-}0.000\phantom{\rule{0.16em}{0ex}}61(10)$.

DOI
10.1103/physrevb.107.024409

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