What is a "bad metal"? What is a "strange metal"?

Way back in the mists of time, I wrote about what what physicists mean when they say that some material is a metal.  In brief, a metal is a material that has an electrical resistivity that decreases with decreasing temperature, and in bulk has low energy excitations of the electron system down to arbitrarily low energies (no energy gap in the spectrum).  In a conventional or good metal, it makes sense to think about the electrons in terms of a classical picture often called the Drude model or a semiclassical (more quantum mechanical) picture called the Sommerfeld model.  In the former, you can think of the electrons as a gas, with the idea that the electrons travel some typical distance scale, \(\ell\), the mean free path, between scattering events that randomize the direction of the electron motion.  In the latter, you can think of a typical electronic state as a plane-wave-like object with some characteristic wavelength (of the highest occupied state) \(\lambda_{\mathrm{F}}\) that propagates effortlessly through the lattice, until it comes to a defect (break in the lattice symmetry) causing it to scatter.  In a good metal, \(\ell >> \lambda_{\mathrm{F}}\), or equivalently \( (2\pi/\lambda_{\mathrm{F}})\ell >> 1\).  Electrons propagate many wavelengths between scattering events.  Moreover, it also follows (given how many valence electrons come from each atom in the lattice) that \(\ell >> a\), where \(a\) is the lattice constant, the atomic-scale distance between adjacent atoms.

Another property of a conventional metal:  At low temperatures, the temperature-dependent part of the resistivity is dominated by electron-electron scattering, which in turn is limited by the number of empty electronic states that are accessible (e.g., not already filled and this forbidden as final states due to the Pauli principle).    The number of excited electrons (that in a conventional metal called a Fermi liquid act roughly like ordinary electrons, with charge \(-e\) and spin 1/2) is proportional to \(T\), and therefore the number of empty states available at low energies as "targets" for scattering is also proportional to \(T\), leading to a temperature-varying contribution to the resistivity proportional to \(T^{2}\).

A bad metal is one in which some or all of these assumptions fail, empirically.  That is, a bad metal has gapless excitations, but if you analyze its electrical properties and tried to model them conventionally, you might find that the \(\ell\) that you infer from the data might be small compared to a lattice spacing.   This is called violating the Ioffe-Mott-Regel limit, and can happen in metals like rutile VO₂ or LaSrCuO₄ at high temperatures.

A strange metal is a more specific term.  In a variety of systems, instead of having the resistivity scale like \(T^{2}\) at low temperatures, the resistivity scales like \(T\).  This happens in the copper oxide superconductors near optimal doping.  This happens in the related ruthenium oxides.  This happens in some heavy fermion metals right in the "quantum critical" regime.  This happens in some of the iron pnictide superconductors.  In some of these materials, when some technique like photoemission is applied, instead of finding ordinary electron-like quasiparticles, a big, smeared out "incoherent" signal is detected.  The idea is that in these systems there are not well-defined (in the sense of long-lived) electron-like quasiparticles, and these systems are not Fermi liquids.

There are many open questions remaining - what is the best way to think about such systems?  If an electron is injected from a boring metal into one of these, does it "fractionalize", in the sense of producing a huge number of complicated many-body excitations of the strange metal?  Are all strange metals the same deep down?  Can one really connect these systems with quantum gravity?  Fun stuff.