Nonuniversal beyond-LHY corrections to thermodynamic properties of a weakly interacting Bose gas

1. Problem Statement

The study addresses the thermodynamic properties of weakly interacting Bose gases, specifically focusing on corrections beyond the well-established Lee-Huang-Yang (LHY) theory.

• Context: Standard LHY theory describes the ground-state energy of a dilute Bose gas with corrections scaling as $(\rho a_s^3)^{1/2}$, where $\rho$ is density and $a_s$ is the s-wave scattering length. This assumes a zero-range (contact) interaction model.
• Gap: Recent experimental advances in tunable quantum gas mixtures have highlighted the need to account for finite-range interatomic interactions. Previous studies using the Cornwall–Jackiw–Tomboulis (CJT) effective action formalism had reported corrections up to order $(\rho a_s^3)^{3/2}$, but the specific impact of finite-range effects on nonuniversal thermodynamic quantities (beyond the standard LHY terms) required further systematic investigation.
• Objective: To derive analytical expressions for thermodynamic quantities (quantum depletion, chemical potential, pressure, and ground-state energy density) that include nonuniversal corrections arising from finite-range interactions within a self-consistent framework.

2. Methodology

The authors employ the Cornwall–Jackiw–Tomboulis (CJT) effective action formalism at the two-loop level, utilizing an Improved Hartree–Fock (HF) approximation.

• Lagrangian Formulation: The system is described by a Lagrangian density including a contact interaction term ($g$) and a finite-range interaction term ($\lambda$). The parameter $\lambda$ is defined as $\lambda = 2\pi\hbar^2 a_s^2 r_s / m$, where $r_s$ is the effective range of the interaction.
• Effective Potential Density: The CJT effective potential $V_{eff}[\psi_0, G]$ is constructed as a functional of the field expectation value $\psi_0$ and the full propagator $G$.
• Self-Consistency and Symmetry Restoration:
  • The authors derive self-consistent Schwinger-Dyson (SD) equations by minimizing the effective potential with respect to $\psi_0$ and $G$.
  • A critical step involves addressing the violation of Goldstone's theorem in the standard HF approximation (which predicts an unphysical energy gap). To restore the Nambu-Goldstone boson (gapless excitation), the effective potential is modified by adding a specific term, and the self-energy definitions ($\Sigma_1, \Sigma_2$) are interchanged.
• Effective Mass and Dispersion: The finite-range interaction leads to a modified effective mass $m^*$ and a gapless excitation spectrum $E(k) = \sqrt{\epsilon_k (\epsilon_k^* + M^2)}$, where $M$ is a mass parameter determined self-consistently.
• Perturbative Expansion: The mass parameter $M$ is solved using perturbation theory in terms of a small parameter related to the gas parameter $\gamma = \rho a_s^3$. The solution is expanded up to second order to capture higher-order corrections.
• Thermodynamic Derivation: Once $M$ and the self-energies are determined, the authors calculate:
  • Quantum depletion ($\rho_{ex}$)
  • Chemical potential ($\mu$)
  • Pressure ($P$)
  • Ground-state energy density ($E$) via the Legendre transform.

3. Key Contributions

• Systematic Derivation of Nonuniversal Terms: The paper provides explicit analytical expressions for thermodynamic quantities that include terms dependent on the ratio of the effective range to the scattering length ($r_s/a_s$). These terms represent nonuniversal corrections that vanish in the zero-range limit.
• Resolution of Discrepancies: The authors identify and correct a discrepancy in previous literature (specifically Ref. [11]) regarding the order $\gamma^{3/2}$ terms. They attribute this to the omission of the second-order correction term ($M_2$) in the perturbative expansion of the mass parameter $M$ in earlier works.
• Improved Accuracy: By retaining higher-order terms in the gas parameter expansion and rigorously treating the finite-range parameter $\lambda$, the derived equations of state (EoS) offer higher accuracy compared to standard contact-interaction models.

4. Key Results

The study yields the following analytical results (expressed in terms of the gas parameter $\gamma = \rho a_s^3$ and the ratio $r_s/a_s$):

• Quantum Depletion ($\rho_{ex}$): Includes corrections of order $\gamma^{1/2}$, $\gamma$, $\gamma^{3/2}$, and terms involving $r_s/a_s$.
• Chemical Potential ($\mu$): The expansion reveals nonuniversal corrections at order $\gamma^{3/2}$ and $\gamma^2$ dependent on $r_s$.
• Ground-State Energy Density ($E$):
  $$E = \frac{g\rho^2}{2} \left[ 1 + \frac{128\sqrt{\gamma}}{15\sqrt{\pi}} - \frac{1024\gamma}{3\pi} + \frac{81920\gamma^{3/2}}{9\pi^{3/2}} - \frac{2048\sqrt{\pi}r_s}{15a_s}\gamma^{3/2} + \dots \right]$$
  The term proportional to $r_s/a_s$ represents the specific nonuniversal beyond-LHY correction.
• Numerical Analysis (Li-7 System):
  • Using parameters for Lithium-7 ($^7$Li), the authors modeled the ground-state energy density as a function of $\gamma$ for varying $r_s/a_s$.
  • Finding: For a ratio $r_s/a_s = 1.0$, the ground-state energy deviates by over 8% compared to the zero-range case ($r_s/a_s = 0$).
  • This deviation is well within the reach of current experimental precision.

5. Significance

• Experimental Relevance: The results establish the ground-state energy density as a promising observable for experimentally probing nonuniversal effects in ultracold Bose gases. The predicted 8% deviation suggests that finite-range effects are not merely theoretical curiosities but measurable physical phenomena.
• Theoretical Framework: The work demonstrates the robustness of the CJT effective action approach in handling spontaneous symmetry breaking and finite-range interactions simultaneously. It provides a complementary perspective to functional path-integral methods.
• Future Directions: The formalism developed here can be extended to investigate strongly correlated regimes, quantum droplets in Bose mixtures, and ultracold Fermi systems, offering a pathway to understand novel quantum phases driven by finite-range interactions.

In summary, this paper successfully bridges the gap between theoretical predictions of finite-range effects and experimental observability in ultracold Bose gases, providing a refined equation of state that captures nonuniversal quantum corrections beyond the standard LHY approximation.