Theoretical Investigation of the Critical Temperature Shift in a Homogeneous Repulsive Weakly Interacting Bose Gas
Core Concepts
The critical temperature of a homogeneous repulsive weakly interacting Bose gas is shifted compared to the ideal Bose gas, and this shift can be expressed in a universal form that is linear with respect to the scattering length.
Abstract
The article explores the relative shift in the transition temperature of a homogeneous dilute Bose gas using the CornwallJackiwTomboulis (CJT) effective action formalism within the improved HartreeFock approximation. The key findings are:

The first correction to the enhancement of the transition temperature, as compared to that of the ideal Bose gas, is expressed in a universal form that is linear with respect to the scattering length. The slope of this linear relationship shows excellent agreement with previous exact numerical calculations.

The article also identifies nonuniversal terms contributing to the shift in the critical temperature.

The CJT effective potential is modified to incorporate an additional term to restore the Goldstone boson, which is required for the spontaneous breaking of the continuous symmetry U(1) associated with BoseEinstein condensation.

The solution of the gap and SchwingerDyson equations within the improved HartreeFock approximation leads to an expression for the critical temperature shift that is linear in the scattering length, with a coefficient of 1.414, in close agreement with previous results.

Higherorder terms in the gas parameter are also considered, showing that the contribution from these terms has a minimal effect on the shift of the critical temperature due to the condition of diluteness.

The results are compared with other theoretical approaches, highlighting the importance of the twoloop diagrams in determining the properties of the homogeneous weakly interacting Bose gas.
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Transition temperature of the homogeneous imperfect Bose gas
Stats
The critical temperature of the ideal Bose gas is given by T0 = (2πℏ^2/2mkB)[ρ/ζ(3/2)]^(2/3).
The relative shift in the critical temperature is given by ΔTC/TC = c·ρ^(1/3)·as, where c is a constant.
Quotes
"Beyond the available methods in literature, in this Letter we use the CornwallJackiwTomboulis (CJT) effective action approach [21] developed for the dilute Bose gas [22] to investigate the effect of the interatomic interaction on the shift of the critical temperature."
"It is obvious that the Goldstone is restored with the gapless dispersion relation E(k) = sqrt((ℏ^2k^2/2m)(ℏ^2k^2/2m + M))."
"The solution for Eq. (13) is approximated ρ0/ρ ≈ 1  (8/3√π)(ρa^3_s)^(1/2)  2ζ(1/2)as/λ_B  ζ(3/2)/(ρλ^3_B)."
Deeper Inquiries
How do the results of this approach compare to experimental observations of the critical temperature shift in homogeneous Bose gases?
The results derived from the Cornwall–Jackiw–Tomboulis (CJT) effective action formalism, particularly within the improved HartreeFock approximation (IHFA), show a significant alignment with experimental observations regarding the critical temperature shift in homogeneous Bose gases. The theoretical prediction of the shift in critical temperature, expressed as (\Delta T_C / T_0 = 4\zeta(1/2)/(3\zeta(3/2)^{1/3}) \rho^{1/3} a_s), yields a value of (c \approx 1.414). This result is consistent with experimental findings, such as the value (c = 1.32 \pm 0.02) reported in lattice simulations and (c = 1.27 \pm 0.11) from variational perturbation theory. The agreement between theoretical predictions and experimental data underscores the robustness of the CJT formalism in capturing the essential physics of the critical temperature shift due to interatomic interactions in dilute Bose gases. Furthermore, the framework's ability to incorporate nonuniversal terms and higherorder corrections enhances its predictive power, making it a valuable tool for understanding the behavior of Bose gases in experimental settings.
What are the limitations of the CJT effective action formalism in capturing the critical behavior of interacting Bose gases, and how could it be further improved?
While the CJT effective action formalism provides a comprehensive framework for analyzing the critical behavior of interacting Bose gases, it does have limitations. One significant limitation is its reliance on the improved HartreeFock approximation, which may not fully account for strong correlations and fluctuations present in more dense or strongly interacting systems. The formalism also struggles to accurately describe the Goldstone boson behavior at finite temperatures without the inclusion of additional phenomenological terms, which can complicate the theoretical landscape.
To improve the CJT effective action formalism, one could explore the incorporation of higherloop corrections beyond the twoloop approximation, which may provide a more accurate representation of the critical behavior and phase transitions. Additionally, employing nonperturbative techniques or advanced numerical methods, such as functional renormalization group approaches, could enhance the formalism's ability to capture the intricate dynamics of interacting Bose gases. These improvements would allow for a more nuanced understanding of the critical temperature shifts and the associated thermodynamic properties of Bose gases under various interaction regimes.
What insights can this theoretical framework provide for understanding the phase transitions and collective excitations in other manybody quantum systems beyond Bose gases?
The CJT effective action formalism offers valuable insights into the nature of phase transitions and collective excitations in various manybody quantum systems beyond just Bose gases. By emphasizing the role of spontaneous symmetry breaking and the emergence of Goldstone modes, the framework can be applied to systems exhibiting similar phenomena, such as Fermi gases, superconductors, and quantum spin systems.
The ability to derive equations of state and critical behavior from the effective potential allows researchers to explore the interplay between interactions and collective excitations in these systems. For instance, the formalism can be adapted to study the effects of interparticle interactions on the critical temperature and excitation spectra in Fermi gases, where pairing interactions lead to superfluidity.
Moreover, the insights gained from the CJT approach regarding the restoration of Goldstone bosons and the nature of phase transitions can inform the understanding of topological phases and quantum critical points in other manybody systems. This versatility makes the CJT effective action formalism a powerful tool for investigating a wide range of phenomena in condensed matter physics and quantum manybody theory, ultimately contributing to a deeper understanding of collective behavior in complex quantum systems.