Abstract
In the previous chapters, we presented various methods that allow for taking the electron–electron interaction into account and studied some of its effects. One important finding was that a low-order perturbative treatment is not sufficient for a quantitatively correct description of the properties of even the simplest metals since the Coulomb repulsion between electrons is strong and long ranged. The whole panoply of many-body physics is needed. In spite of this, experiments show that – rather surprisingly – simple metals behave in many respects as the noninteracting chargeless fermion gas: the heat capacity varies linearly with T and the Pauli susceptibility is independent of temperature. Similar behavior is observed in the normal liquid phase of \(^3{\textrm {He}}\), which is a fermionic system, while \(^4{\textrm {He}}\) is a Bose liquid. There are of course exceptions. The most notable are superconductors, whose thermal and magnetic properties differ drastically from those of normal metals.
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Notes
- 1.
Ya. I. Pomeranchuk , 1958.
- 2.
V. P. Silin , 1957.
- 3.
Sin-Itiro Tomonaga (1906–1979) shared the Nobel Prize with J. Schwinger and R. P. Feynman in 1965 “for their fundamental work in quantum electrodynamics, with deep-ploughing consequences for the physics of elementary particles”.
- 4.
That the electron–hole excitations in the one-dimensional noninteracting electron gas can be described as sound waves was discovered by F. Bloch already in 1934.
- 5.
The continuum of particle–hole excitations is similar to the spectrum shown in Fig. 15.13 except that the soft modes are shifted from \(\pm \pi/a\) to \(\pm 2k_{\textrm{F}}\).
- 6.
Note that another convention is also quite common in the literature. The bosons with negative q formed from left-moving particle–hole pairs are replaced by another boson branch with positive q defined via \(\tilde{b}_{q\sigma} = b_{-q \sigma}\) and \(\tilde{b}_{q \sigma}^{\dagger} = b_{-q \sigma}^{\dagger}\).
- 7.
The chirality of the particle tells whether it is right or left moving, that is whether its wave number is close to \(+k_{\textrm{F}}\) or \(-k_{\textrm{F}}\).
- 8.
The notations ρ and σ are also common.
- 9.
J. C. Ward , 1950.
- 10.
Unlike in (28.1.42), the notation U is used in this chapter instead of \(U_{\textrm{H}}\) for the on-site Coulomb repulsion.
- 11.
In the spin-1/2 Heisenberg chain, where spins on neighboring sites interact, the Bethe ansatz leads to an extra condition on the wavefunction when spins are reversed on neighboring sites. In the Hubbard model with on-site interaction, the Bethe ansatz provides a constraint for the case when two electrons sit on the same site.
- 12.
E. H. Lieb and F. Y. Wu , 1968.
- 13.
The \(S_z=-1\) state, where more than half of the particles have their spins oriented down, can be described by a Bethe-ansatz wavefunction by reversing the role of up- and down-spin electrons. The motion of up-spin particles is considered through the background of down-spin particles.
- 14.
The states close to the zone boundary would be filled for \(J < 0\). They can be shifted to the center of the Brillouin zone by the transformation \(k \rightarrow k+ \pi/a\).
- 15.
The difference compared to (15.5.119) is due to a redefinition of the model. The antiferromagnetic side corresponds to \({\mathnormal{\Delta}} < 0\) in the present treatment, while we used the convention \({\mathnormal{\Delta}} > 0\) there.
- 16.
A. Luther and V. J. Emery , 1974.
- 17.
See the footnote on page 6 of Volume 1.
- 18.
See footnote on page 6 of Volume 1.
- 19.
R. Jastrow, 1955.
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Sólyom, J. (2010). Fermion Liquids. In: Fundamentals of the Physics of Solids. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04518-9_5
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DOI: https://doi.org/10.1007/978-3-642-04518-9_5
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