| Landau criterion | |
|---|---|
| Name | Landau criterion |
| Field | Quantum mechanics; Condensed matter physics |
| Introduced | 1941 |
| Inventor | Lev Landau |
| Related | superfluidity, Bose–Einstein condensate, quasiparticle |
Landau criterion
The Landau criterion is a theoretical condition that predicts the critical velocity above which a moving object or flow in a quantum fluid can create excitations and thus dissipate energy. It provides a link between the spectrum of elementary excitations and the onset of frictionless flow in systems such as superfluids and Bose–Einstein condensates and is central to understanding macroscopic quantum phenomena in quantum many-body systems.
The Landau criterion states that dissipationless flow is possible only if the velocity v of an object relative to a quantum fluid is below the minimum of ε(p)/p, where ε(p) is the energy of an excitation with momentum p. Formally, v_c = min_p [ε(p)/p]. If v>v_c, creating excitations (phonon, roton, or other quasiparticle modes) becomes energetically allowed, enabling drag and energy loss. This criterion connects microscopic properties — the excitation spectrum determined by interactions, symmetry breaking, and confinement — to emergent macroscopic behavior like persistent currents in liquid helium or superflow in ultracold atomic gases. The criterion is widely used in condensed matter physics and informs design and interpretation of experiments at institutions such as Cavendish, Max Planck Institutes, and national laboratories.
Derivations begin with energy–momentum conservation in the frame of the moving impurity or container. For a weakly interacting Bose gas described by the Gross–Pitaevskii equation, low-energy excitations are collective Bogoliubov modes with dispersion ε(p)=sqrt[(cp)^2+(p^2/2m)^2], giving a sound velocity c and a Landau critical velocity approximately equal to c for long-wavelength phonons. In strongly interacting Helium-4 the dispersion features a local minimum — the roton — which reduces v_c to ε_roton/p_roton. The Bogoliubov and Landau approaches are linked by the concept of spontaneous symmetry breaking of global U(1) symmetry and the emergence of a Nambu–Goldstone boson (phonon). Key theoretical tools include second quantization, the Bogoliubov transformation, and linear response theory as developed in works by Lev Landau, Nikolay Bogoliubov, and others.
The Landau criterion generalizes to charged superfluids (superconductors) and fermionic paired systems by replacing neutral excitation spectra with those appropriate to Cooper pair formation and quasiparticle excitations in the BCS framework. In conventional superconductors the relevant excitations are Bogoliubov quasiparticles with an energy gap Δ, giving a critical velocity v_c~Δ/p_F set by the Fermi energy and Fermi momentum p_F. In superfluid 3He and unconventional superconductors (e.g., cuprates, Sr2RuO4), gap anisotropy and nodes modify the spectrum and lower v_c along certain directions. Extensions incorporate lattice effects, broken time-reversal symmetry, and spin–orbit coupling; theoretical developments have involved researchers at Cambridge University and Stanford University studying topological superfluids and Majorana modes where the notion of critical velocity acquires new nuance.
Experiments probe v_c by moving impurities, laser beams, or obstacles through condensates, measuring onset of heating, drag, or vortex nucleation. Landmark studies in MIT and JILA observed critical velocities in dilute atomic Bose gases using focused laser beams and time-of-flight imaging. In liquid helium, classic experiments at low-temperature facilities revealed roton-limited critical velocities and flux quantization in toroidal vessels. Modern cold-atom platforms at Institut d'Optique, École Normale Supérieure, and national labs allow tunable interactions via Feshbach resonance and optical lattices, enabling systematic tests of Landau predictions and observation of deviations due to finite-size effects, dimensional crossover (1D, 2D), and integrability.
Landau's picture underpins quantum hydrodynamics: the two-fluid model of Helium II separates superfluid and normal components and uses v_c to characterize when the normal fraction grows. Above v_c, coherent motion breaks down via excitations or vortex nucleation, seeding quantum turbulence composed of quantized vortices. Understanding the transition from ordered superflow to turbulent cascades engages research in non-equilibrium dynamics, statistical mechanics, and connections to turbulent phenomena in classical fluids. Experiments on vortex rings and decay at facilities studying quantum turbulence (e.g., University of Maryland groups) test how Landau-based thresholds relate to vortex-mediated dissipation.
The Landau criterion is necessary but not always sufficient: mechanisms like thermally assisted processes, finite system size, and dynamical instabilities can cause dissipation below the formal v_c. Impurities, disorder, and pinning alter local excitation spectra and can either suppress or enhance dissipation; studies in disordered superconductors and dirty Bose gases reveal subcritical onset of resistance. Mesoscopic and nanoscale systems exhibit additional channels (e.g., surface excitations, Andreev scattering) that violate simple Landau assumptions. Critics note the criterion's reliance on equilibrium excitation spectra and call for inclusion of non-equilibrium and many-body localization effects addressed in contemporary theoretical work.
The criterion originated in Lev Landau's 1941 analysis of superfluidity in Landau's seminal papers, which introduced ideas of elementary excitations and the two-fluid model. Landau's framework transformed the field of low-temperature physics and influenced successors such as Ilya Lifshitz, Nikolay Bogoliubov, and experimentalists including Pyotr Kapitsa and John F. Allen. The concept remains foundational in condensed matter physics curricula and continues to inform equitable access to scientific infrastructure: modern cold-atom and low-temperature research communities emphasize open collaboration, sharing of facilities, and diversity in participation to ensure broad benefit from advances in quantum technologies.
Category:Quantum mechanics Category:Superfluidity Category:Condensed matter physics