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 $\epsilon$ expansion 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.
Highlights
Perturbation methods [1, 2] are the most commonly used techniques in condensed matter physics to obtain theoretical results comparable with experimentally obtained values
We propose simple tools using continued functions to obtain meaningful answers from divergent power series where limited number of successive approximations are known for the physical quantity
The errors in calculation of these parameters can reflect upon the calculation of critical exponents
Summary
Perturbation methods [1, 2] are the most commonly used techniques in condensed matter physics to obtain theoretical results comparable with experimentally obtained values. The desired quantity is calculated as a power series of a small perturbation parameter associated with the system, and generally the level of computational complexity to calculate the quantity increases at higher powers. The continued exponential was chosen because its convergence properties were studied previously by Bender and Vinson [10] It was used in certain applications related to statistical physics by Poland to obtain convergence [11]. We implement combination of continued functions and Shanks transformation to show empirically that convergence can be obtained in divergent perturbation series encountered in field theories.
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