Bose Quantum Fluids and Lattice Bosons

A Bose-matter statement is justified only after four questions have been answered separately: what long-distance order exists, what response is nonzero, in what dimension and order of limits the statement is made, and which small parameter controls the calculation. This chapter develops that discipline from the ideal gas through dilute superfluids, low-dimensional fluids, the Bose–Hubbard model, and self-bound droplets Pitaevskii and Stringari 2016, chs. 2–9.

The chapter covers Bose-fluid and lattice-boson realizations. General equilibrium and hydrodynamic formalisms are developed in Volume XI, while generic renormalization-group constructions are developed in Volume V. Numerical benchmarks should publish their code, frozen inputs, and checks; changing platform records belong in Research.

Helpful background. Ideal-gas condensation supplies Bose occupations and thermodynamic-limit counting; contact interactions and the scattering length supplies the low-energy matching used for dilute gases.

Enter this chapter

You are ready for the main route if you can (i) obtain Bose occupation numbers from a grand-canonical ensemble, (ii) distinguish a one-body density matrix from a response coefficient, and (iii) replace a bare three-dimensional contact coupling by the scattering length. If the third step is unfamiliar, begin with short-range scattering data. For the quickest conceptual entry, start with the ideal gas and then separate condensation from superfluidity.

Reader goal	Suggested route	Capability at the end
Graduate core	Ideal gas → distinctions → weak gas → Bogoliubov theory → phase–density EFT	Derive the equation of state, sound mode, depletion, and stiffness without conflating them
Low dimensions	Phase–density EFT → BKT → one-dimensional fluids	Decide whether order is true long-range, algebraic, or absent and identify the controlling infrared theory
Lattice bosons	Bose–Hubbard limits → superfluid–Mott criticality	Separate an atomic charge gap from critical scaling at a lobe tip or edge
Beyond mean field	Bogoliubov theory → dilute expansion → droplets	Track the gas parameter, ultraviolet matching, and metastability ceiling

Order, response, and control

The diagram should be read from left to right: a microscopic Bose Hamiltonian supplies a state and dimension; correlation functions establish condensation or algebraic order; free-energy response establishes stiffness; spectra and compressibility diagnose gapless or insulating behavior. None of these arrows is reversible without extra hypotheses.

Independent tests supporting Bose-matter claims. The figure is schematic and not to scale: a momentum-space peak alone does not establish stiffness, while a nonzero stiffness does not by itself establish three-dimensional off-diagonal long-range order.

The second map organizes the approximation boundaries. Inspect where the expansion parameter changes: $\sqrt{na^3}$ for a dilute three-dimensional gas, vortex fugacity and stiffness for BKT flow, $t/U$ near an atomic Mott state, and competing energy-density terms plus loss scales for a droplet. The continuum and lattice limits are developed respectively in Altland and Simons 2023, §§ 5.2 and 6.5 and Fisher et al. 1989, §§ IV–V, pp. 555–562.

Control parameters and decisive failure tests across the chapter. The map is schematic: finite size, traps, disorder, finite range, and loss can cut off the displayed asymptotic regimes.

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 $K$	Energy below microscopic, thermal, and integrability-breaking scales	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

Together with the preceding relationship-centered explanations and alt text, this table supplies the nonvisual account of both diagrams. It deliberately keeps dimension, order, response, spectrum, finite-size behavior, control, and falsification in separate columns.

Guide to the pages

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.

Conventions that recur

We use $\hbar=k_{\mathrm B}=1$. In the continuum, $[\psi(\mathbf x),\psi^\dagger(\mathbf y)]=\delta^{(d)}(\mathbf x-\mathbf y)$ and $\int_{\mathbf k}\equiv\int \mathrm d^d k/(2\pi)^d$. In three dimensions the low-energy coupling is $g=4\pi a/m$ only after matching to the $s$-wave scattering length $a$; it is not a bare ultraviolet parameter. On a lattice, $t>0$ appears as $-t\sum_{\langle ij\rangle}b_i^\dagger b_j+\mathrm{h.c.}$, and filling means particles per site. Page-local dimensions, ensembles, and orders of limits are stated before they matter.

Review the chapter

Classification. Can a uniform ideal Bose gas be condensed but fail an interacting-fluid stiffness test? Can a two-dimensional BKT phase be superfluid without true ODLRO? A successful answer names the thermodynamic limit and uses independent definitions rather than vocabulary alone.

Answer criterion

Yes to both. Ideal-gas macroscopic occupation is an eigenvalue statement about $\rho_1$ and does not by itself supply the metastable current response of an interacting superfluid. A BKT phase has algebraic one-body correlations and finite renormalized helicity modulus, but no nonzero infinite-distance limit of $\rho_1$ at $T>0$.

Control assessment. For a proposed droplet calculation, list the minimum checks that connect it back to the dilute-gas expansion. The answer must include scattering-data matching, a gas/range parameter, a positive-compressibility or mode-stability test, a finite-size surface term, and a lifetime comparison.

Translation. Starting from the phase action $S=\frac12\int(\kappa\dot\theta^2-\rho_s(\nabla\theta)^2)$, recover $c^2=\rho_s/\kappa$ and explain why neither coefficient alone equals a condensate fraction. This tests dimensions and the distinction between order and response.

References

• Alexander Altland and Ben Simons, Condensed Matter Field Theory, 3rd ed., Cambridge University Press (2023), §§ 5.2 and 6.5, doi:10.1017/9781108781244.
• Michael P. A. Fisher, Peter B. Weichman, Geoffrey Grinstein, and Daniel S. Fisher, “Boson Localization and the Superfluid–Insulator Transition,” Physical Review B 40 (1989) 546–570, doi:10.1103/PhysRevB.40.546.
• 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.