Editing Superfluid helium-4 (section)

== Microscopic theory ==

=== Landau two-fluid approach ===
[[Lev Landau|L. D. Landau's]] phenomenological and semi-microscopic theory of superfluidity of helium-4 earned him the Nobel Prize in physics, in 1962. Assuming that sound waves are the most important excitations in helium-4 at low temperatures, he showed that helium-4 flowing past a wall would not spontaneously create excitations if the flow velocity was less than the sound velocity. In this model, the sound velocity is the "critical velocity" above which superfluidity is destroyed. (Helium-4 actually has a lower flow velocity than the sound velocity, but this model is useful to illustrate the concept.) Landau also showed that the sound wave and other excitations could equilibrate with one another and flow separately from the rest of the helium-4, which is known as the "condensate".

From the momentum and flow velocity of the excitations he could then define a "normal fluid" density, which is zero at zero temperature and increases with temperature. At the so-called Lambda temperature, where the normal fluid density equals the total density, the helium-4 is no longer superfluid.

To explain the early specific heat data on superfluid helium-4, Landau posited the existence of a type of excitation he called a "[[roton]]", but as better data became available he considered that the "roton" was the same as a high momentum version of sound.

The Landau theory does not elaborate on the microscopic structure of the superfluid component of liquid helium.<ref>{{Cite journal|last1=Alonso|first1=J. L.|last2=Ares|first2=F.|last3=Brun|first3=J. L.|date=2018-10-05|title=Unraveling the Landau's consistence criterion and the meaning of interpenetration in the "Two-Fluid" Model|journal=The European Physical Journal B|language=en|volume=91|issue=10|page=226|doi=10.1140/epjb/e2018-90105-x|issn=1434-6028|arxiv=1806.11034|bibcode=2018EPJB...91..226A|s2cid=53464405}}</ref> The first attempts to create a microscopic theory of the superfluid component itself were done by London<ref>{{cite journal|author=F. London|journal= Nature |volume=141|pages= 643–644 |year=1938|doi=10.1038/141643a0|title=The λ-Phenomenon of Liquid Helium and the Bose-Einstein Degeneracy|issue=3571|bibcode = 1938Natur.141..643L |s2cid= 4143290 }}</ref> and subsequently, Tisza.<ref>{{cite journal|author=L. Tisza|journal= Nature |volume= 141|doi=10.1038/141913a0|title=Transport Phenomena in Helium II|year=1938|issue=3577|page=913|bibcode = 1938Natur.141..913T |s2cid= 4116542 |doi-access=free}}</ref><ref>{{cite journal|author=L. Tisza|journal=Phys. Rev. |volume=72|pages= 838–854 |year=1947|doi=10.1103/PhysRev.72.838|title=The Theory of Liquid Helium|issue=9|bibcode = 1947PhRv...72..838T }}</ref>
Other microscopical models have been proposed by different authors. Their main objective is to derive the form of the inter-particle potential between helium atoms in superfluid state from first principles of [[quantum mechanics]].
To date, a number of models of this kind have been proposed, including: models with vortex rings, hard-sphere models, and Gaussian cluster theories.

=== Vortex ring model ===
Landau thought that vorticity entered superfluid helium-4 by vortex sheets, but such sheets have since been shown to be unstable.
[[Lars Onsager]] and, later independently, Feynman showed that vorticity enters by quantized vortex lines. They also developed the idea of [[quantum vortex]] rings.
[[Arie Bijl]] in the 1940s,<ref>{{Cite journal
| last = Bijl
| first = A
|author2=de Boer, J
|author3=Michels, A
 | title = Properties of liquid helium II
| journal = Physica
| volume = 8
| issue = 7
| pages = 655–675
|year = 1941
| doi = 10.1016/S0031-8914(41)90422-6
| bibcode=1941Phy.....8..655B}}</ref>
and [[Richard Feynman]] around 1955,<ref>{{Cite book
| editor = Braun, L. M.
| title = Selected papers of Richard Feynman with commentary
| publisher = World Scientific
| series = World Scientific Series in 20th century Physics
| volume = 27
| date = 2000
| isbn = 978-9810241315 }}
[https://books.google.com/books?id=qnwkqcVixucC&q=feynman Section IV (pages 313 to 414)] deals with liquid helium.</ref> developed microscopic theories for the roton, which was shortly observed with inelastic neutron experiments by Palevsky. Later on, Feynman admitted that his model gives only qualitative agreement with experiment.<ref>{{cite journal|author=R. P. Feynman|journal= Phys. Rev.|volume= 94|page= 262 |year=1954|doi=10.1103/PhysRev.94.262|title=Atomic Theory of the Two-Fluid Model of Liquid Helium|issue=2|bibcode = 1954PhRv...94..262F |url=http://authors.library.caltech.edu/3539/1/FEYpr54a.pdf}}</ref><ref>{{cite journal|author=R. P. Feynman|author2=M. Cohen|name-list-style=amp|journal= Phys. Rev. |volume=102|pages= 1189–1204 |year=1956|doi=10.1103/PhysRev.102.1189|title=Energy Spectrum of the Excitations in Liquid Helium|issue=5|bibcode = 1956PhRv..102.1189F |url=https://authors.library.caltech.edu/3549/1/FEYpr56.pdf}}</ref>

=== Hard-sphere models ===
The models are based on the simplified form of the inter-particle potential between helium-4 atoms in the superfluid phase. Namely, the potential is assumed to be of the hard-sphere type.<ref>{{cite journal|author=T. D. Lee|author2= K. Huang|author3=C. N. Yang|name-list-style=amp|journal= Phys. Rev. |volume=106|pages= 1135–1145 |year=1957|doi=10.1103/PhysRev.106.1135|title=Eigenvalues and Eigenfunctions of a Bose System of Hard Spheres and Its Low-Temperature Properties|issue=6|bibcode = 1957PhRv..106.1135L }}</ref><ref>{{cite journal|author=L. Liu|author2=L. S. Liu|author3=K. W. Wong|name-list-style=amp|journal= Phys. Rev. |volume=135|pages= A1166–A1172 |year=1964|doi=10.1103/PhysRev.135.A1166|title=Hard-Sphere Approach to the Excitation Spectrum in Liquid Helium II|issue=5A|bibcode = 1964PhRv..135.1166L }}</ref><ref>{{cite journal|author=A. P. Ivashin|author2=Y. M. Poluektov|name-list-style=amp|journal= Cent. Eur. J. Phys. |volume= 9|pages= 857–864 |year=2011|doi=10.2478/s11534-010-0124-7|title=Short-wave excitations in non-local Gross-Pitaevskii model|issue=3|bibcode = 2011CEJPh...9..857I |arxiv=1004.0442|s2cid=118633189}}</ref>
In these models the famous Landau (roton) spectrum of excitations is qualitatively reproduced.

=== Gaussian cluster approach ===
This is a two-scale approach which describes the superfluid component of liquid helium-4. It
consists of two [[Critical phenomena|nested models linked via parametric space]]. The short-wavelength part describes the interior structure of the [[fluid element]] using a non-perturbative approach based on the [[logarithmic Schrödinger equation]]; it suggests the [[Gaussian]]-like behaviour of the element's interior density and interparticle interaction potential. The long-wavelength part is the quantum many-body theory of such elements which deals with their dynamics and interactions.<ref>{{cite journal | url=https://link.aps.org/doi/10.1103/PhysRevLett.90.250403 | doi=10.1103/PhysRevLett.90.250403 | title=Roton-Maxon Spectrum and Stability of Trapped Dipolar Bose-Einstein Condensates | year=2003 | last1=Santos | first1=L. | last2=Shlyapnikov | first2=G. V. | last3=Lewenstein | first3=M. | journal=Physical Review Letters | volume=90 | issue=25 | page=250403 | pmid=12857119 | arxiv=cond-mat/0301474 | bibcode=2003PhRvL..90y0403S | s2cid=25309672 }}</ref> The approach provides a unified description of the [[phonon]], [[maxon excitation|maxon]] and [[roton]] excitations, and has noteworthy agreement with experiment: with one essential parameter to fit one reproduces at high accuracy the Landau roton spectrum, [[sound velocity]] and [[structure factor]] of superfluid helium-4.<ref>{{cite journal|author=K. G. Zloshchastiev|journal= Eur. Phys. J. B |volume= 85|page= 273 |year=2012|doi=10.1140/epjb/e2012-30344-3|title=Volume element structure and roton-maxon-phonon excitations in superfluid helium beyond the Gross-Pitaevskii approximation|issue=8|bibcode = 2012EPJB...85..273Z|arxiv = 1204.4652 |s2cid= 118545094 }}</ref> This model utilizes the general theory of quantum Bose liquids with logarithmic nonlinearities<ref>{{cite journal|author=A. V. Avdeenkov|author2=K. G. Zloshchastiev|name-list-style=amp|journal= J. Phys. B: At. Mol. Opt. Phys. |volume= 44|page= 195303|year=2011|doi=10.1088/0953-4075/44/19/195303|title=Quantum Bose liquids with logarithmic nonlinearity: Self-sustainability and emergence of spatial extent|issue=19|bibcode = 2011JPhB...44s5303A|arxiv = 1108.0847 |s2cid=119248001}}</ref> which is based on introducing a [[Open quantum system|dissipative]]-type contribution to energy related to the quantum [[Entropic uncertainty|Everett–Hirschman entropy function]].<ref>[[Hugh Everett]], III. The Many-Worlds Interpretation of Quantum Mechanics: the theory of the universal wave function. [https://www.pbs.org/wgbh/nova/manyworlds/pdf/dissertation.pdf Everett's Dissertation]</ref><ref name="Hirschman">[[Isidore Isaac Hirschman, Jr.|I.I. Hirschman, Jr.]], ''A note on entropy''. American Journal of Mathematics (1957) pp. 152–156</ref>