Colloquium: Linear in temperature resistivity and associated mysteries including high temperature superconductivity

Immediately after the discovery of high temperature superconductivity in the cuprates in 1987, properties in the metallic state above $T_c$ were discovered that violated the reigning paradigm in condensed matter physics: the quasiparticle concept due to Landau. The most discussed of such properties is the linear in temperature resistivity down to asymptotically low temperatures, sometimes called Planckian resistivity, above the region of the highest $T_c$. Similar anomalies have since also been discovered in the heavy-fermion compounds and in the Fe-based superconducting metals, and most recently in twisted bilayer graphene. Innumerable papers in the past three decades have pointed out that the linear in $T$ resistivity and associated properties are a mystery and the most important unsolved problem in condensed matter physics; superconductivity itself is a corollary to the normal state properties. Even in this prolifically investigated field, quantitative experimental results on crucial normal state and superconducting state properties have only recently become available. It is now possible to compare some of the detailed predictions of a theory for the normal and superconductive state in cuprates and in heavy fermions with the experiments. The theory gives the frequency and temperature dependence of various normal state properties and also their measured magnitudes in terms of the same values of two parameters. It also resolves the paradox of $d$-wave symmetry of superconductivity in the cuprates given that the scattering rate of fermions in the normal state is nearly momentum independent. The same parameters that govern the normal state anomalies are also deduced from the quantitative analysis of data in the superconducting state in cuprates. The simplicity of the results depends on the discovery of a new class of quantum-critical fluctuation in which orthogonal topological excitations in space and time determine the spectra, such that the correlations of the critical spectra are a product of a function of space and a function of time with the spatial correlation length proportional to the logarithm of the temporal correlation length. The fermions scattering with such fluctuations form a marginal Fermi liquid.

Figure 1

Schematic universal phase diagrams (left panel) of hole-doped cuprates and (right panel) of the Fe-based compound, based on the data on $\mathrm{Ba}{({\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x)}_2{\mathrm{As}}_2$ from [18]. The quantum-critical point discussed in the text for both is indicated. The wide crossover to Fermi liquid is indicated only roughly for each. In the cuprates, depending on the compound a variety of other thermodynamically small transitions are seen in the pseudogap state below the phase boundary marked $T^{*}$ as well as a spin-glass state at low temperatures from the antiferromagnetic region to near the quantum-critical point. These are not shown. It has not been clear from experiments where the line marked $T^x$ ends at low doping. In theory, it persists to zero doping.

Figure 2

The electronic specific heat in ${\mathrm{La}}_{2-p}{A}_p{\mathrm{CuO}}_4$. (Left panel) Data at 0.5 K, the lowest temperature measured (in a magnetic field to remove superconductivity). (Right panel) Temperature dependence of the specific heat nearest the critical composition $p\approx p_c$. From [39].

Figure 3

Single-particle scattering rate measured by ARPES. (Top panels) Momentum distribution curves at various energies with a fit from which the parameters of the self-energy are extracted. The data are fit by red curves that are Lorentzians, proving that the scattering rate is independent of momentum perpendicular to the Fermi surface. (Bottom left panel) The points on the Fermi surface and the directions in which the data were taken. (Bottom right panel) The extracted parameters for the self-energy. From [29]

Figure 4

The full width of the Lorentizian momentum distribution curves as a function of energy up to high energy for the three compounds that were measured by ARPES. The open black circles are data for optimally doped Bi2201 [nodal cut, from 38]. The red crosses are for “optimally doped” Bi2212 [nodal cut, from 22]. The solid green circles are for ${\mathrm{La}}_{2-p}{\mathrm{Sr}}_p{\mathrm{CuO}}_4$ for $p=0.17$ at about 20° from the nodal direction, and the blue squares are for the same compound at $p=0.145$ from the nodal direction. The last two set of points are from [16].

Figure 5

The resistivity “phase diagram” for ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$. The temperature dependence of the resistivity begins to show departure from linearity below the lines marked $T^{*}(x)$ and $T_{\mathrm{coh}}$. In this Colloquium, we are concerned with the latter. This line may be fitted with $(x-x_c{)}^{0.5}$, just as the crossover line in the previously mentioned specific heat data. From [26].

Figure 6

The imaginary part of the “neutral” density-density correlation function for the region $v_{F}q/\omega \lesssim 1$ showing the fit to the square of this quantity whose coefficient is related to the compressibility and the scattering rate using the Einstein relation. From [40].

Figure 7

Resistivity of $\mathrm{Ce}{\mathrm{Cu}}_{5.9}{\mathrm{Au}}_{0.1}$ and specific heat at various pressures and dopings of $\mathrm{Ce}{\mathrm{Cu}}_6$ across the antiferromagnetic quantum-critical point. From [70].

Figure 8

(Left panel) Phase diagram of $\mathrm{Ba}{\mathrm{Fe}}_2({\mathrm{As}}_{1-x}{P}_x{)}_2$ and resistivity at criticality ($x\approx 0.3$). From [18]. (Right panel) Thermopower $S$ divided by $T$ across AFM quantum criticality in ${\text{K}}_x{\mathrm{Sr}}_{1-x}{\mathrm{Fe}}_2{\mathrm{As}}_2$, effectively showing the $T\ln T$ dependence of the specific heat at quantum criticality. Lv et al. gave resistivity showing $\propto T$ behavior in a similar range in temperature and composition. From [36].

Figure 9

(Left panel) The order parameter depicted by the vector $\mathbf{\Omega }$ representing the magnetoelectric order parameter of Eq. (12). $\mathbf{\Omega }$ is odd in both time-reversal and inversion and preserves their product. These symmetries come from a pair of spontaneously generated current loops in a $\mathrm{Cu}\text{−}{\mathrm{O}}_2$ unit cell. (Right panel) Various experiments showing the onset temperature of symmetries consistent with that mentioned in the compound ${\mathrm{YBaCu}}_3{\mathrm{O}}_{6+x}$. The neutron scattering experiments are from [21], the polarimetry experiments from [35], the second harmonic generation from [75], the $\mu \mathrm{SR}$ from [74], and the ultrasound measurements from [55].

Figure 10

The Kubo formula for the current-current correlation in terms of the bare current operator, the renormalized current operator and the exact single-particle Green’s functions. The dc conductivity is $(1/\epsilon )$ times the imaginary part of the current-current correlation at $\mathbf{q}\to 0$.

Figure 11

The skeletal diagrams for the normal self-energy and the pairing self-energy for scattering fermions from fluctuations whose propagator is depicted as a wavy line. The former has the ordinary part of the Gorkov Green’s function in the intermediate state and the latter the pairing part. The vertices in the two diagrams are closely related.