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arXiv:2504.17668 · Progress of Theoretical and Experimental Physics

FRG analysis for a relativistic BEC in arbitrary spatial dimensions

Fumio Terazaki, Kazuya Mameda

Published Jun 17, 2025• 0 citations
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Abstract

Abstract A relativistic Bose–Einstein condensate (BEC) is studied within the complex scalar field theory using the functional renormalization group (FRG) under the local potential approximation. We investigate fluctuation effects on the relativistic BEC through numerical analyses for various spatial dimensions and chemical potentials. Our numerical results are consistent with the Mermin–Wagner theorem, and this consistency is also analytically confirmed from the flow equation. We also discuss a numerical instability of the FRG in lower spatial dimensions, which is avoidable for certain parameter choices.

DOI
10.1093/ptep/ptaf088

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Abstract A relativistic Bose–Einstein condensate (BEC) is studied within the complex scalar field theory using the functional renormalization group (FRG) under the local potential approximation. We investigate fluctuation effects on the relativistic BEC through numerical analyses for various spatial dimensions and chemical potentials. Our numerical results are consistent with the Mermin–Wagner theorem, and this consistency is also analytically confirmed from the flow equation. We also discuss a numerical instability of the FRG in lower spatial dimensions, which is avoidable for certain parameter choices.
This dissertation focuses on theoretical studies of relativistic Bose-Einstein condensation (BEC).Based on the complex scalar field theory, which serves as a fundamental model for relativistic BEC, we provide a detailed explanation of the analyses using the functional renormalization group (FRG), including both its formalism and the underlying physics.We also present the numerical implementation of the FRG flow equation and the results obtained from it, as well as the derivations of various related equations.Although previous studies rely on approaches that are not well justified, we develop a novel and physically consistent FRG framework.In this framework, we impose a condition on the FRG flow equation, analogous to that in the non-interacting case.By solving the equation under this condition, we find that quantum fluctuations significantly affect the phase transition of relativistic BEC in the strong coupling regime.We further investigate the dependence of the condensate on spatial dimensions, and in particular, confirm that the FRG analysis is consistent with the Mermin-Wagner theorem from both numerical and analytical perspectives.These results contribute to the development of the theoretical foundation of the FRG, and uncover an aspect of relativistic BEC under non-perturbative interactions and for various spatial dimensions.
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We calculate the temperature-dependent condensate density rho0(T) of interacting bosons in three dimensions using the functional renormalization group (FRG). From the numerical solution of suitably truncated FRG flow equations for the irreducible vertices we obtain rho0(T) for arbitrary temperatures. We carefully extrapolate our numerical results to the critical point and determine the order parameter exponent beta approximately 0.32 in reasonable agreement with the expected value 0.345 associated with the XY -universality class. We also calculate the condensate density in two dimensions at zero temperature using a truncation of the FRG flow equations based on the derivative expansion including cubic and quartic terms in the expansion of the effective potential in powers of the density. As compared with the widely used quadratic approximation for the effective potential, the coupling constants associated with the cubic and quartic terms lead to small corrections of the condensate density.