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Within [[Hartree–Fock]] theory, the ferromagnetic fluid abruptly becomes more stable than the paramagnetic fluid at a density parameter of <math>r_s=5.45</math> in three dimensions (3D) and <math>2.01</math> in two dimensions (2D).<ref>{{cite book|last=Giuliani|first=Gabriele|author2=Vignale |author3=Giovanni |title=Quantum Theory of the Electron Liquid|url=https://archive.org/details/quantumtheoryofe0000giul|url-access=registration|publisher=Cambridge University Press|year=2005|isbn=978-0-521-82112-4}}</ref> However, according to Hartree–Fock theory, Wigner crystallization occurs at <math>r_s=4.5</math> in 3D and <math>1.44</math> in 2D, so that jellium would crystallise before itinerant ferromagnetism occurs.<ref>{{cite journal|author1=J. R. Trail |author2=M. D. Towler |author3=R. J. Needs |title= Unrestricted Hartree-Fock theory of Wigner crystals|journal= Phys. Rev. B |volume= 68 |issue=4 |pages= 045107 |year=2003|doi= 10.1103/PhysRevB.68.045107|arxiv = 0909.5498 |bibcode = 2003PhRvB..68d5107T }}</ref> Furthermore, Hartree–Fock theory predicts exotic magnetic behavior, with the paramagnetic fluid being unstable to the formation of a spiral spin-density wave.<ref>{{cite journal|author= A. W. Overhauser|title= Giant Spin Density Waves|journal= Phys. Rev. Lett. |volume= 4 |issue= 9|pages= 462–465 |year=1960|doi= 10.1103/PhysRevLett.4.462|bibcode = 1960PhRvL...4..462O }}</ref><ref>{{cite journal|author= A. W. Overhauser|title=Spin Density Waves in an Electron Gas |journal= Phys. Rev. |volume= 128 |issue=3 |pages= 1437–1452 |year=1962|doi= 10.1103/PhysRev.128.1437|bibcode = 1962PhRv..128.1437O }}</ref> Unfortunately, Hartree–Fock theory does not include any description of correlation effects, which are energetically important at all but the very highest densities, and so a more accurate level of theory is required to make quantitative statements about the phase diagram of jellium.
[[Quantum Monte Carlo]] (QMC) methods, which provide an explicit treatment of electron correlation effects, are generally agreed to provide the most accurate quantitative approach for determining the zero-temperature phase diagram of jellium. The first application of the [[diffusion Monte Carlo]] method was Ceperley and Alder's famous 1980 calculation of the zero-temperature phase diagram of 3D jellium.<ref name=Ceperley>{{cite journal|author1=D. M. Ceperley |author2=B. J. Alder |title= Ground State of the Electron Gas by a Stochastic Method|journal= Phys. Rev. Lett. |volume= 45 |issue=7 |pages=
In 2D, QMC calculations indicate that the paramagnetic fluid to ferromagnetic fluid transition and Wigner crystallization occur at similar density parameters, in the range <math>30<r_s<40</math>.<ref>{{cite journal|author1=B. Tanatar |author2=D. M. Ceperley |title= Ground state of the two-dimensional electron gas |journal= Phys. Rev. B |volume= 39 |issue=8 |pages= 5005 |year=1989|doi= 10.1103/PhysRevB.39.5005|pmid=9948889 |bibcode = 1989PhRvB..39.5005T }}</ref><ref>{{cite journal|author1=F. Rapisarda |author2=G. Senatore |title= Diffusion Monte Carlo Study of Electrons in Two-dimensional Layers|journal= Aust. J. Phys. |volume= 49 |pages= 161 |year=1996|doi= 10.1071/PH960161 |bibcode = 1996AuJPh..49..161R }}</ref> The most recent QMC calculations indicate that there is no region of stability for a ferromagnetic fluid.<ref>{{cite journal|author1=N. D. Drummond |author2=R. J. Needs |title= Phase Diagram of the Low-Density Two-Dimensional Homogeneous Electron Gas|journal= Phys. Rev. Lett. |volume= 102 |issue=12 |pages= 126402 |year=2009|doi= 10.1103/PhysRevLett.102.126402|arxiv = 1002.2101 |bibcode = 2009PhRvL.102l6402D |pmid=19392300}}</ref> Instead there is a transition from a paramagnetic fluid to a hexagonal Wigner crystal at <math>r_s=31(1)</math>. There is possibly a small region of stability for a (frustrated) antiferromagnetic Wigner crystal, before a further transition to a ferromagnetic crystal. The crystallization transition in 2D is not first order, so there must be a continuous series of transitions from fluid to crystal, perhaps involving striped crystal/fluid phases.<ref>{{cite journal|author1=B. Spivak |author2=S. A. Kivelson |title= Phases intermediate between a two-dimensional electron liquid and Wigner crystal|journal=Phys. Rev. B |volume= 70 |issue=15 |pages= 155114 |year=2004|doi=10.1103/PhysRevB.70.155114|bibcode = 2004PhRvB..70o5114S }}</ref> Experimental results for a 2D hole gas in a GaAs/AlGaAs heterostructure (which, despite being clean, may not correspond exactly to the idealized jellium model) indicate a Wigner crystallization density of <math>r_s=35.1(9)</math>.<ref>{{cite journal|author1=J. Yoon |author2=C. C. Li |author3=D. Shahar |author4=D. C. Tsui |author5=M. Shayegan |title= Wigner Crystallization and Metal-Insulator Transition of Two-Dimensional Holes in GaAs at <math>B=0</math>|journal= Phys. Rev. Lett. |volume= 82 |issue=8 |pages= 1744 |year=1999|doi= 10.1103/PhysRevLett.82.1744|arxiv = cond-mat/9807235 |bibcode = 1999PhRvL..82.1744Y }}</ref>
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