A Marginal Dimension of a Weakly Diluted Quenched m-Vector Model

TL;DR

This paper calculates a critical marginal order parameter dimension in a weakly diluted quenched m-vector model, revealing that weak disorder does not alter the critical exponents of the XY model, supported by advanced series resummation techniques.

Contribution

It provides a precise five- and six-loop field-theoretical calculation of the marginal dimension $m_c$, demonstrating that weak quenched disorder does not affect XY-model critical exponents.

Findings

Estimated $m_c=1.912 extpm0.004$ from pseudo-$oldsymbol{ extepsilon}$ expansion.

Weak quenched disorder does not change XY-model critical exponents.

Results align with experimental data on ${ m He}^4$ in porous media.

Abstract

We calculate a marginal order parameter dimension $m_c$ which in a weakly diluted quenched $m$-vector model controls the crossover from a universality class of a ``pure'' model ($m>m_c$) to a new universality class ($m<m_c$). Exploiting the Harris criterion and the field-theoretical renormalization group approach allows us to obtain $m_c$ as a five-loop $\epsilon$-expansion as well as a six-loop pseudo-$\epsilon$ expansion. In order to estimate the numerical value of $m_c$ we process the series by precisely adjusted Pad\'e--Borel--Leroy resummation procedures. Our final result $m_c=1.912\pm0.004<2$ stems from the longer and more reliable pseudo-$\epsilon$ expansion, suggesting that a weak quenched disorder does not change the values of $xy$-model critical exponents as it follows from the experiments on critical properties of ${\rm He}^4$ in porous media.