Ivan Vakarchuk

Ivan Franko National University of Lviv
The properties of liquid helium are known to have brought about ample literature. Yet due to its unique characteristics, this quantum fluid keeps attracting the attention of specialists in the field of theoretical and experimental physics alike. The first principle microscopic description of thermodynamic and structure functions as well as the phenomenon of Bose-Einstein condensation at a considerable distance from the absolute zero, in particular in the vicinity of the λ-transition, is still an open problem which has not been completely solved. The latter statement can be supported by a fairly instructive study of heat capacity in the vicinity of the λ-transition. The λ-like form of the heat capacity in the vicinity of the phase transition of liquid helium into the superfluid state has been taken as a logarithmic divergence with the critical exponent α tends to 0. This view has found its way into both text-books and monographs. Precise experiments helped to determine that in fact there is no divergence in the heat capacity even though the exponent α is indeed a small but negative number, α = -0.01056[1]. The studies of the λ-transition based on the renormalization group method provide us with a possibility to carry out the correct calculus for the so-called universal characteristics solely, i. e., the critical characteristics of the thermodynamic functions and the relations of the amplitudes of their leading asymptotics at the temperature on either side tending to the phase transition point. Even though the thermodynamic potential functional from the two-component order parameter for liquid He4 was calculated precisely owing to the coherent states depiction, yet the subsequent simplifications necessary for the implementation of the renormalized group approach make it impossible to describe the system's characteristics outside the closest vicinity of the phase transition point using the same method. Notwithstanding the tangible efforts researchers, the renormalization group method has not yielded the logarithmic divergence of the heat capacity (the α exponent was received as a small but still finite positive number, the power divergence having been obtained). Only in subsequent studies, which have made use of the summation procedure of the Borel perturbation theory has a divergent series established the negative value of the exponent: α = -0.01294[2], α = -0.0150[3], α = -0.01126[4].
[1] J.A. Lipa et al. Specific heat of Helium confined to a 57-μ ν planar geometry near the lambda point. Phys. Rev. Lett. 84 (2000) 4894.
[2] H. Kleinert. Critical exponents from seven-loop strong-coupling ϕ 4 theory in three dimensions. Phys. Rev. D, 60 (1999) 085001.
[3] M. Campostrini, A. Pelissetto, P. Rossi, E. Vicari. Determination of the critical exponents for the λ transition of 4He by high-temperature expansion. Phys. Rev. B, 61 (2000) 5905.
[4] H. Kleinert. Theory and satellite experiment for critical exponent α of λ-transition in superfluid helium. Phys. Lett. A, 277 (2000) 205.

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