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Continued Functions and Perturbation Series: Simple Tools for Convergence of Diverging Series in O(n)-Symmetric \(\phi ^4\) Field Theory at Weak Coupling Limit

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Abstract

We determine universal critical exponents that describe the continuous phase transitions in different dimensions of space. We use continued functions without any external unknown parameters to obtain analytic continuation for the recently derived 7-loop weak coupling \(\epsilon \)-expansions from O(n)-symmetric \(\phi ^4\) field theory. Employing a new blended continued function, we obtain critical exponent \(\alpha =-0.0121(22)\) for the phase transition of superfluid helium which matches closely with the most accurate experimental value. This result addresses the long-standing discrepancy between the theoretical predictions and precise experimental result of O(2) \(\phi ^4\) model known as ”\(\lambda \)-point specific heat experimental anomaly”. Further we have also examined the applicability of such continued functions in other examples of field theories.

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  1. Department of Physics, National Institute of Technology, Tiruchirapalli, 620015, India

    Venkat Abhignan & R. Sankaranarayanan

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  1. Venkat Abhignan
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6. Appendix

6. Appendix

Table 2 Critical exponents \(\nu \), \(\alpha \), \(\omega \), \(\gamma \) and \(\beta \) compared with literature for \(d=3\), \(0\le n \le 3\)
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Table 3 Critical exponents \(\nu \), \(\alpha \), \(\omega \), \(\gamma \) and \(\beta \) compared with other theoretical results for \(d=3\), \(n=-2\) and \(n>3\). CE values denoted by superscript\(^\#\) and CEF values denoted by superscript\(^*\)
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Table 4 Critical exponents \(\nu \) and \(\gamma \) for Ising model on fractal lattice with \(n=1\) for \(d<2\) compared with literature
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Table 5 Critical exponents \(\nu \) and \(\gamma \) for non-integer dimensions with \(n=1\) compared with literature
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Table 6 Critical exponents \(\nu \) and \(\gamma \) for non-integer dimensions compared with literature. CE values denoted by superscript\(^\#\) and CEF values denoted by superscript\(^*\)
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Abhignan, V., Sankaranarayanan, R. Continued Functions and Perturbation Series: Simple Tools for Convergence of Diverging Series in O(n)-Symmetric \(\phi ^4\) Field Theory at Weak Coupling Limit. J Stat Phys 183, 4 (2021). https://doi.org/10.1007/s10955-021-02719-z

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