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Contributed by: Elliott H. Lieb (Princeton Univ.), Sept. 15, 1998.

Abstract   The well known diamagnetic inequality states that the application of a magnetic field to the orbital motion of an electron raises the electron's energy. When many electrons are present, however, the Pauli exclusion principle can cause a lowering of the energy. Indeed, it has been conjectured by several authors (cf. [1] for references) that this is so for a lattice model of independent electrons (the Hueckel model) and that the magnetic flux that gives the lowest energy is 2\pi times the electron density. This was proved in the above paper for the half-filled band (density = 1/2), even with an on-site interaction among the electrons (Hubbard model).

It is not clear just how correct the conjectured value of the flux is when we move away from the half-filled band, but there is reason to believe it is correct for the quarter filled band in 2D (density = 1/4 and flux = \pi /2).
Is this so?

  1. E. H. Lieb, Phys. Rev. Lett. 73, 2158-2161 (1994)
    (and references therein).

Some additional background can be found in

  • E. H. Lieb: `The Hubbard model -- Some Rigorous Results and Open Problems', Proceedings of the XIth International Congress of Mathematical Physics, Paris, 1994, D. Iagolnitzer ed., pp. 392-412 (International Press, 1995).

  • Further technical details are in the following TeX document.


    {\bf THE OPTIMAL FLUX FOR THE `QUARTER-FILLED BAND'.} Consider a large square lattice $\Lambda$ with $|\Lambda|$ sites and with periodic boundary conditions. We assume that $|\Lambda|$ is divisible by 4. Now consider the hopping matrix $T$ on $\Lambda$, but with a magnetic field, i.e., $T_{x,y} =\exp[i\theta (x,y)]$ if $x,y$ are neighboring sites of $\Lambda$ and $T_{x,y} =0$ otherwise. The real numbers $\theta (x,y)$ are arbitrary, except for the hermiticity condition that $\theta (x,y) =- \theta (y,x)$. Let $\lambda_1\leq \lambda_2\leq \lambda_3\leq ... \leq \lambda_{|\Lambda|}$ denote the eigenvalues of $T$. It is easy to prove that these eigenvalues depend only on the fluxes, namely the sum of the $\theta (x,y)$ around the elementary small squares (or plaquettes) of $\Lambda$. With $E = \sum_{j=1}^{ |\Lambda|/4} \lambda_j$ denoting the sum of the lowest one quarter of the eigenvalues, our goal is to find the choice of fluxes that minimize $E$. It has been conjectured that the optimum choice is when the flux through each elementary square is $\pi / 2$. (Note: The corresponding problem for the half-filled band is known to have the optimum flux $= \pi$, even when a Hubbard type on-site interaction is included; E. H. Lieb, Phys. Rev. Lett. {\it 73}, 2158-2161 (1994).)

           postscript (PS) version        PDF file

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    lieb@math.princeton.edu (contributor), aizenman@princeton.edu (editor)