[Paper Review] Tan Relations in Dilute Bose Gasses

Adriaan M. J. Schakel|arXiv (Cornell University)|Jul 20, 2010

Cold Atom Physics and Bose-Einstein Condensates3 citations

TL;DR

This paper demonstrates that Tan's exact relations—originally derived for strongly interacting Fermi gases—also hold for dilute Bose gases to first order in the loop expansion, supported by a thermodynamic argument. The key contribution is the first determination of the second-order correction to condensate depletion, computed to logarithmic accuracy using the Tan relations, confirming consistency with Bogoliubov theory and providing a nonperturbative framework for strongly correlated Bose systems.

ABSTRACT

The exact relations for strongly interacting Fermi gasses, recently derived by Tan, are shown to first order in the loop expansion to also apply to dilute Bose gasses. A simple thermodynamic argument is put forward to support their validity. As an application, the second-order correction to the depletion of the condensate is determined to logarithmic accuracy.

Motivation & Objective

• To extend Tan's exact relations—originally for Fermi gases—to dilute Bose gases with short-range interactions.
• To validate the applicability of the Tan relations in Bose systems via a thermodynamic argument and perturbative field theory.
• To compute the second-order correction to the condensate depletion in a dilute Bose gas using the Tan energy relation and contact parameter C.
• To establish consistency between the Tan relations and the standard Bogoliubov theory at leading and next-to-leading order.
• To provide a nonperturbative framework for understanding strong correlations in dilute Bose gases through the contact parameter C.

Proposed method

• Derive the Tan energy relation for a dilute Bose gas with one species, incorporating the contact parameter C and a modified coupling term with a 1/2 factor compared to Fermi systems.
• Use the loop expansion to compute the ground-state energy density, including one-loop corrections and counterterms to handle ultraviolet divergences.
• Apply renormalization to the coupling constant and chemical potential, introducing $ g_{\text{r}} $ and $ \mu_{\text{r}} $, and relate $ g_{\text{r}} $ to the s-wave scattering length $ a $.
• Compute the contact $ C $ from the expectation value of $ \langle (\psi^*\psi)^2 \rangle $, using the Bogoliubov spectrum and frequency integrals.
• Evaluate the wave vector integrals in the presence of a cutoff $ \Lambda $, and cancel divergences via renormalization to obtain a finite expression for $ C $.
• Use the resulting expression for $ C $ to verify the adiabatic sweep and pressure relations, and extend the calculation to second order in the loop expansion for the condensate depletion.

Experimental results

Research questions

• RQ1 Do Tan's exact relations for strongly interacting Fermi gases also apply to dilute Bose gases with short-range interactions?
• RQ2 Can the contact parameter $ C $, which characterizes short-distance correlations, be consistently defined and computed in a dilute Bose gas using field-theoretic methods?
• RQ3 Is the second-order correction to the condensate depletion in a dilute Bose gas computable using the Tan relations, and what is its functional form?
• RQ4 How do the Tan relations—specifically the adiabatic sweep and pressure relations—compare with standard Bogoliubov theory in the Bose case?
• RQ5 Can a thermodynamic argument support the validity of the Tan relations in Bose systems despite the absence of explicit condensate terms in the energy relation?

Key findings

• The Tan energy relation holds for dilute Bose gases to first order in the loop expansion, with the contact parameter $ C $ appearing in the same functional form as in Fermi systems but with a factor of 1/2 due to the absence of spin degrees of freedom.
• The contact parameter is computed as $ C = (4\pi a n)^2 \left[1 + \frac{64}{3}\left(\frac{a^3 n}{\pi}\right)^{1/2}\right] $, consistent with the Lee-Yang result at leading order.
• The adiabatic sweep and pressure relations are verified to hold in the Bose gas, with the pressure given by $ P = \frac{2}{3}\mathcal{E} + \frac{\hbar^2 C}{24\pi m a} $, where the $ 1/2 $ factor reflects the single-species nature of the system.
• The second-order correction to the condensate depletion is computed to logarithmic accuracy as $ n_0 = n\left[1 - \frac{8}{3}\left(\frac{a^3 n}{\pi}\right)^{1/2} + \frac{8}{3}\left(4\pi - 3\sqrt{3}\right)a^3 n \ln(a^3 n)\right] $, marking the first such result using the Tan relations.
• The thermodynamic argument based on the relation $ P = \frac{1}{3a} \frac{d}{da}(a^2 \mathcal{E}) $ provides strong nonperturbative support for the validity of the Tan relations in Bose systems.
• The calculation confirms that the Tan relations provide a consistent, nonperturbative framework for strongly correlated dilute Bose gases, even when the condensate is not explicitly included in the energy expression.