Beyond Bogoliubov Theory in the Dilute Expansion

Beyond quadratic Bogoliubov theory, the three-dimensional zero-temperature Bose gas has an asymptotic expansion in $x=\sqrt{na^3}$. After matching the contact interaction to the scattering length, the leading fluctuation contribution is the Lee–Huang–Yang (LHY) term, $E/E_{\mathrm{MF}}=1+128x/(15\sqrt\pi)+\cdots$.

Required background. Use Bogoliubov quasiparticles and the scattering-length matching relation. Helpful background. Finite-density diagrammatics organizes the loop expansion.

Renormalized zero-point energy

For a uniform gas, the quadratic zero-point contribution is ultraviolet divergent if the coupling is treated as a physical constant at all momenta. Expressing the bare coupling through the two-body $T$ matrix produces the finite combination

$$\frac{E}{V}=\frac{gn^2}{2} +\frac12\int_{\mathbf k}\left[ E_k-\epsilon_k-gn+\frac{(gn)^2}{2\epsilon_k} \right]+\cdots,$$

where $E_k=\sqrt{\epsilon_k(\epsilon_k+2gn)}$ and $g=4\pi a/m$. The final counterterm is precisely the second Born subtraction. At large $k$, the first three terms leave $-(gn)^2/(2\epsilon_k)+O(k^{-4})$, which the counterterm cancels. Rescaling $k=\sqrt{2mgn}\,q$ makes the remaining integral proportional to $(mgn)^{5/2}/m$, and evaluation gives

$$\frac{E}{N}=\frac{gn}{2} \left[1+\frac{128}{15\sqrt\pi}\sqrt{na^3} +O\!\left(na^3\ln na^3\right)\right].$$

Differentiation at fixed volume yields

$$\mu=gn\left[1+\frac{32}{3\sqrt\pi}\sqrt{na^3}+\cdots\right].$$

The original many-body calculation and coefficient are in Lee, Huang, and Yang 1957, pp. 1135–1145.

Expansion order and infrared consistency

The condensate depletion is relative order $x$, while replacing $n_0$ by $n$ inside the mean-field term also changes the energy at that order. A consistent calculation therefore holds the ensemble and density definition fixed and includes all diagrams and counterterms at one order. Individual anomalous diagrams can be infrared singular although the physical equation of state is finite; the cancellation follows from the gapless Ward identity, not from discarding the singular pieces separately.

The next correction is not simply $na^3$: logarithms and three-body information enter. Effective range contributes when $nr_e^3$ or $k r_e$ is no longer negligible. Thus the LHY formula is a controlled asymptotic result, not an interpolation to unitary Bose matter Pitaevskii and Stringari 2016, ch. 7.

Independent checks

Dimensional analysis fixes the fluctuation energy density to scale as $m^{3/2}(gn)^{5/2}$. The pressure $P=n\mu-E/V$ is positive for $g>0$. Differentiating once more gives a positive compressibility in the dilute repulsive regime. These checks catch sign, ensemble, and normalization errors independently of the integral.

Exercises

Use $E/V=A n^2(1+B\sqrt{na^3})$ with $A=g/2$ and $B=128/(15\sqrt\pi)$ to derive the LHY coefficient in $\mu$.

Solution

$\mu=\partial(E/V)/\partial n=2An+(5/2)ABn\sqrt{na^3}$. Factoring out $gn=2An$ gives $1+(5/4)B\sqrt{na^3}=1+32\sqrt{na^3}/(3\sqrt\pi)$.

References

• T. D. Lee, Kerson Huang, and C. N. Yang, “Eigenvalues and Eigenfunctions of a Bose System of Hard Spheres and Its Low-Temperature Properties,” Physical Review 106 (1957) 1135–1145, doi:10.1103/PhysRev.106.1135.
• Lev P. Pitaevskii and Sandro Stringari, Bose–Einstein Condensation and Superfluidity, Oxford University Press (2016), ch. 7, doi:10.1093/acprof:oso/9780198758884.001.0001.