arXiv2018
Despite much theoretical effort, there is no complete theory of the 'strange' metal state of the high temperature superconductors, and its linear-in-temperature, $T$, resistivity. Recent experiments showing an unexpected linear-in-field, $B$, magnetoresistivity have deepened the puzzle. We propose a simple model of itinerant electrons, interacting via random couplings with electrons localized on a lattice of quantum 'dots' or 'islands'. This model is solvable in a large-$N$ limit, and can reproduce observed behavior. The key feature of our model is that the electrons in each quantum dot are described by a Sachdev-Ye-Kitaev model describing electrons without quasiparticle excitations. For a particular choice of the interaction between the itinerant and localized electrons, this model realizes a controlled description of a diffusive marginal-Fermi liquid (MFL) without momentum conservation, which has a linear-in-$T$ resistivity and a $T \ln T$ specific heat as $T\rightarrow 0$. By tuning the strength of this interaction relative to the bandwidth of the itinerant electrons, we can additionally obtain a finite-$T$ crossover to a fully incoherent regime that also has a linear-in-$T$ resistivity. We show that the MFL regime has conductivities which scale as a function of $B/T$; however, its magnetoresistance saturates at large $B$. We then consider a macroscopically disordered sample with domains of MFLs with varying densities of electrons. Using an effective-medium approximation, we obtain a macroscopic electrical resistance that scales linearly in the magnetic field $B$ applied perpendicular to the plane of the sample, at large $B$. The resistance also scales linearly in $T$ at small $B$, and as $T f(B/T)$ at intermediate $B$. We consider implications for recent experiments reporting linear transverse magnetoresistance in the strange metal phases of the pnictides and cuprates.