Bose Quantum Fluids and Lattice Bosons

Bose quantum fluids connect macroscopic occupation to collective excitations, phase rigidity, and lattice insulating phases. These properties depend differently on dimension, interactions, temperature, and the thermodynamic limit.

Enter this chapter

Begin with The Ideal Bose Gas and Bose–Einstein Condensation to derive when excited states can no longer accommodate all particles.

Helpful background. The Ideal Bose Gas and Bose–Einstein Condensation and Short-Range Scattering Data as Many-Body Inputs develop the definitions used by the chapter’s opening pages.

Topics

1. The Ideal Bose Gas and Bose–Einstein Condensation — Derives occupations, critical density, and condensate fraction, with finite-volume and dimensional qualifications.
2. Condensation, Off-Diagonal Order, and Superfluidity — Defines four nearby but inequivalent notions and supplies counterexamples.
3. The Weakly Interacting Bose Gas — Builds the Gross–Pitaevskii saddle from a scattering-length-matched interaction.
4. Bogoliubov Theory and Bose Quasiparticles — Diagonalizes quadratic fluctuations and derives the phonon-to-particle crossover and depletion.
5. Beyond Bogoliubov Theory in the Dilute Expansion — Obtains the Lee–Huang–Yang term and makes ultraviolet cancellation explicit.
6. Phase–Density EFT for Bose Superfluids — Integrates density fluctuations to obtain the compact phase theory, sound speed, and stiffness.
7. Low-Dimensional Bose Gases and BKT Physics — Explains algebraic order, vortex energetics, and the universal stiffness jump.
8. Strongly Correlated One-Dimensional Bose Fluids — Connects the Lieb–Liniger and Tonks–Girardeau limits to Luttinger-liquid observables.
9. The Bose–Hubbard Model and Controlled Limits — Derives atomic Mott gaps and their leading hopping corrections.
10. The Superfluid–Mott Quantum Phase Transition — Distinguishes the relativistic lobe tip from the density-driven edge.
11. Self-Bound Quantum Droplets — Balances mean-field attraction against fluctuation pressure while retaining range, surface, and loss limits.

How the ideas fit together

Bogoliubov theory explains how weak repulsion converts the ideal gas’s quadratic low-momentum spectrum into sound. The phase–density description isolates that collective motion, while its compact phase also admits vortices. In two dimensions, vortex unbinding can destroy superfluid response even though a finite-temperature thermodynamic condensate is absent on both sides Pitaevskii and Stringari 2016, chs. 2–9.

On a lattice, suppressing number fluctuations instead produces a Mott state. Neither macroscopic occupation nor a gap alone identifies all these regimes; correlations, stiffness, compressibility, and excitation spectra supply distinct tests.

The comparison below separates the assumptions and tests needed for these different conclusions.

Bose-phase validity table

Regime	Dimension and order	Stiffness and compressibility	Excitation test	Control parameter	Finite-size signature	Negative test or model boundary
Ideal condensation	Homogeneous $d>2$; macroscopic one-body eigenvalue below $T_c$	Condensation alone does not determine interacting-fluid stiffness	Quadratic particles; saturation of excited states	Thermodynamic limit at fixed density	Rounded occupation crossover with $N_0=O(N)$ only asymptotically	No superfluid claim without an independent response test
Weak three-dimensional superfluid	3D ODLRO with perturbative depletion	$\rho_s>0$ and positive compressibility	$E_k=\sqrt{\epsilon_k(\epsilon_k+2gn)}$ from phonon to particle	$na^3\ll1$ and $kr_0\ll1$	Depletion and low-$k$ phonon scaling stabilize with volume	Large depletion, range sensitivity, or a violated sum rule
BKT fluid	2D algebraic $g_1(r)$ below $T_{\mathrm{BKT}}$; no finite-$T$ ODLRO	Finite renormalized helicity modulus with the universal jump	Phonons plus bound vortex–antivortex pairs	Long wavelengths and controlled initial $K,y$	Size drift of stiffness crossings and the BKT correlation length	Exponential correlations or failure of BKT finite-size scaling
One-dimensional Bose fluid	1D algebraic order at $T=0$; no extensive condensate eigenvalue	Finite phase rigidity and compressibility in the Luttinger theory	Sound mode with parameter $K$	Low energy below the microscopic cutoff; algebraic correlations at zero temperature or within the thermal length; perturbations must leave the gapless phase stable	Algebraic occupations and correlators with one consistent $K$	Finite-$T$ exponential decay or incompatible exponents
Commensurate Bose–Hubbard Mott state	Integer-filled lattice; no thermodynamic condensate	$\kappa=0$ and $\rho_s=0$	Nonzero particle and hole gaps	$t/U$ near the atomic limit, with tip or edge scaling declared	Gap and stiffness crossings follow the chosen $z$ and aspect ratio	Nonzero thermodynamic compressibility or closing charge gap
Self-bound droplet	3D finite-density stationary state without external confinement	Positive compressibility; superfluid response is a separate test	Real collective modes below the emission threshold	Dilute matched functional plus range, surface, and loss scales	Minimum particle number and surface-to-volume drift	Collapse, evaporation, strong range drift, or lifetime below equilibration

Where to go next

Continue to Fermi Surfaces and Fermi Liquids to relate particle density to Fermi-surface geometry and phase space.

References

• Lev P. Pitaevskii and Sandro Stringari, Bose–Einstein Condensation and Superfluidity, Oxford University Press (2016), chs. 2–9, doi:10.1093/acprof:oso/9780198758884.001.0001.