Past exam of the quantum condensed matter course of the University of Cambridge / 2015 / Paper 1 / 2

Heavy fermion materials are correlated metals whose low-energy quasiparticles have exceptionally large effective masses. Partly localized $f$-electron moments hybridize with broader conduction states. The Kondo effect and coherence below the Kondo-lattice coherence temperature produce a narrow low-energy band, so a small energy change corresponds to a large momentum-space state density.

For the electronic heat capacity of a heavy Fermi liquid, the resulting density of states gives a large low-temperature electronic specific-heat coefficient,and often an enhanced magnetic susceptibility. Here $g$ includes both spins and must be expressed per sample, mole or volume consistently with $\gamma$. A coherent Fermi-liquid regime often has $\rho ={\rho }_0+A{T}^2$ with an enhanced coefficient $A$. Above coherence, scattering by local moments can instead give a Kondo-like resistivity upturn.

The competition between moment screening and intersite magnetic interactions can yield a heavy paramagnetic metal, magnetic order, quantum-critical behaviour or unconventional superconductivity. Ce- and U-based compounds supply familiar examples. “Heavy” describes the emergent electronic quasiparticles, not the ionic mass; the simple heavy-band picture need not survive at a quantum critical point.^[ref]

The Hubbard model combines hopping with a local repulsion,Hopping favours a broad itinerant band; the interaction penalizes opposite-spin double occupancy. For uniform Stoner decoupling of the Hubbard interaction, let $s=n_{\uparrow }-n_{\downarrow }$ be the spin imbalance per site. The interaction is $U(n^2-s^2)/4$, so polarization lowers it by $U{s}^2/4$, at the expense of kinetic energy.

If $D(E_F)$ is the noninteracting density of states per spin per site, small polarization costs $s^2/[4D(E_F)]$. Thereforeis the mean-field ferromagnetic instability criterion. The exchange splitting is $\Delta =Us$, and the paramagnetic response is enhanced as $\chi ={\chi }_0/[1-UD(E_F)]$. A density of states including both spins changes the written factor in the criterion.

The model explains how itinerant bands can polarize without imposing pre-existing localized atomic spins. It is not a claim that every large-$U$ metal becomes ferromagnetic: near half filling the Hubbard model can instead form a Mott insulator with antiferromagnetic superexchange of scale $4t^2/U$. Band filling, bandwidth and fluctuations matter.^[ref] The interaction-energy and kinetic-cost expansion above is the stated mean-field derivation of the enhancement.^[ref]

For an electron wavepacket in one isolated band under a uniform electric field,Because $ϵ(k+2\pi /a)=ϵ(k)$, acceleration through successive Brillouin zones makes the velocity periodic rather than increasing indefinitely. For $ϵ={ϵ}_0-2J\cos (ka)$,for a positive field along the chosen axis. Integration givesThe excursion amplitude of a Bloch-oscillating wavepacket is $2|J|/(e|E|)$ about its mean and peak-to-peak extent $4|J|/(e|E|)$. The turning of velocity corresponds semiclassically to Bragg reflection near a zone boundary.

Coherent Bloch oscillations require scattering/dephasing times long compared with $T_B$ and a field weak enough to avoid substantial interband tunnelling. Ordinary metals usually scatter too rapidly; semiconductor superlattices and cold atoms in periodic optical potentials provide cleaner realizations. A completely filled band has canceling velocities, so its net current does not oscillate simply because individual wavepackets can.^[ref]