What is the relationship between bad metals, non-Fermi liquids and the strange metal phase of the cuprate superconductors?
They are not the same thing, although some people interchange the terms. Here are my definitions and clarifications, with particular emphasis on temperature regimes.
A Fermi liquid is a low temperature state of a metal characterised by well-defined quasi-particles, a Drude peak in the frequency-dependent conductivity, a thermopower much less than k_B/e, and a resistivity much less than the Mott-Ioffe-Regel limit h a/e^2, and certain characteristic temperature dependences (e.g. specific heat is linear in T, resistivity ~T^2. The Fermi liquid only occurs below some coherence temperature T_coh. In some formal theoretical sense (a la the renormalisation group) it is a statement about a zero-temperature fixed point (all the interactions become irrelevant).
A non-Fermi liquid does not have have well-defined quasi-particles or has quasi-particles with different quantum numbers to a Fermi liquid (spin-1/2 and charge -e). It is also a statement about a zero-temperature fixed point.
The strange metal of the cuprates is the part of the phase diagram at optimal doping. One characteristic of it is a resistivity that is linear in temperature, and of order the Mott-Ioffe-Regel limit. Since, the low temperature properties are hidden by the superconducting dome, we can't say definitely whether or not it is a non-Fermi liquid.
A bad metal is an intermediate temperature state of a metal. It is characterised by ill-defined quasi-particles, no Drude peak in the frequency-dependent conductivity, a thermopower of order k_B/e, and a resistivity of order the Mott-Ioffe-Regel limit. This definition makes no commitment about what happens at low temperatures. A bad metal can evolve into a Fermi liquid or a non-Fermi liquid. A key issue is how low in temperature does one have to go to be in the "scaling" region of the zero-temperature fixed point.
The above view of a bad metal is distinctly different from that originally proposed by Emery and Kivelson. In their 1995 PRL, "Superconductivity in bad metals," they stated that
The failure of bad metals to exhibit resistivity saturation [at the Mott-Ioffe-Regel limit] strongly suggests that any theory based on conventional quasiparticles with more or less well-defined crystal momenta suffering occasional scattering events does not apply. Since there is no crossover in the temperature dependence of the resistivity as the temperature is lowered, this conclusion applies by continuity even at lower temperatures where the putative mean free path deduced from the measured values of the resistivity would not, of itself, rule out the possibility of quasiparticle transport. In other words, a bad metal behaves as if it is a quasiparticle insulator which is rendered metallic by collective excitations.[Italics is theirs].
i.e. They claimed that a bad metal must be a non-Fermi liquid at low temperatures.
However, Dynamical Mean-Field Theory provides a counter example to this argument. As the temperature increases one smoothly evolves from a Fermi liquid to a bad metal. This was pointed out by Jaime Merino and I in 2000. Organic charge transfer salts also provide a counter example. At low temperatures one observes beautiful Shubnikov de Haas oscillations [described by a Fermi surface and Lifshitz-Kosevich theory] and a bad metal about 50 Kelvin.
Aside: I don't think that a resistivity that is linear in T over some limited temperature range is really much of a signature of anything.
I welcome discussion of these issues.