Universal resistivity: From strange metals to high temperature superconductors

People call things “strange” when they don’t understand them. This is also the case for the so called “strange metals”. Not only they exhibit some unusual normal state properties which cannot be explained by the existing theories, but also the transition to superconducting state occurs at temperatures higher than the boiling point of liquid nitrogen, which was thought to be impossible before [1]. Cuprates and ferropnictides are some prominent examples of strange metals exhibiting the high temperature superconductivity along with all other peculiarities. If we can understand the physics of normal and superconducting states of strange metals, this will be a major breakthrough in the condensed matter physics, as it can pave the way to achieve even room temperature superconductivity someday.


So, what specifically is strange about these strange metals? First of all, they are unusually resistive than conventional metals, yet exhibit superconductivity at much higher temperatures than the latter. Unexpectedly, their resistivity does not saturate at high temperatures and violate the Mott-Ioffe-Regel limit. Resistivity of strange metals increases linearly with temperature spanning the whole possible ranges of temperatures in their normal state. As of today, no acceptable theory has been found to explain these uncommon and anomalous behaviors. Many scientists believe these anomalous normal state behaviors are key to understand the high temperature superconductivity, of which the mechanism is unknown [2]. 



Figure 1: Resistivity versus temperature for strange metals (red curve) and conventional metals (dashed black curve) Strange metals have high superconducting transition temperature. They have orders of magnitude larger resistivity than normal metals, there is no saturation at high temperatures observed, it rises perfectly linear. 


Based on the theoretical and experimental results, researchers have some idea about the possible origins of these effects. The resistivity of a metal is related with the behavior of electrons which elicit conduction. Electrons in strange metals are thought to be strongly correlated, flowing like a liquid molecule rather than moving ballistically until they collide [3]. This picture is motivated by some results from the experiments and guided by the failure of existing theories. Temperature-doping phase diagrams are proposed to interpret the behaviors of different regimes and critical points. Apparently, strange metal (non-Fermi liquid) phase sits right above the superconducting dome and has a V shape [2]. 



Figure 2: Temperature versus doping (hole) phase diagram. There is an optimal doping condition at which the superconducting transition temperature is at its highest. Strange metal phase is sandwiched between the antiferromagnetic insulator and pseudogap regions on the left and Fermi liquid phase on the right. 


Explanations for the “strange” properties of strange metals have been searched not only within condensed matter physics (which deals with the properties of solid and liquid phases of matter) but also various seemingly unrelated fields such as string theory (through the holographic principle) and quantum information theory [1]. Apparently, strange metals behave at the edge of certain physical laws so that similarities can be found in various other areas of physics.



According to Nobel laureate Robert B. Laughlin, peculiar behavior of linear resistivity in strange metals can only be explained by a simple theory [1]. His intuition has been backed by recent observations. Particularly, in 2013, universality in the scattering rates of strange metals has been shown [4]. The scattering rate of electrons in strange metals obeys the so called Planckian dissipation rate which is τ~ħ/kBT . Here ħ is the Planck constant, a physical parameter indicates the quantumness, kB is Boltzmann constant indicating the role of thermal processes and T is temperature. Universality of the scattering rate makes it possible to use the good old Drude model to interpret the resistivity results because the underlying scattering mechanism seems to be unimportant [5]. There are abundant attempts to crack the problem of the universal resistivity behavior. Careers have been spent on this, since 1987, when the high temperature superconductivity has been discovered. Yet, we have no consensus on or even a bird’s-eye view of the complete picture so far.



We think we are close to cracking the case. Recently, Heller group showed that electron-phonon interaction can be modelled by a coherent state representation of acoustic phonons [6]. Conventionally, phonons are interpreted as number states within the framework of second quantization and perturbation theory. This approach deals with phonons as particles and it works extremely well for most cases. On the other hand, phonons are also waves and nonperturbative treatment of deformation potential by a coherent state picture recovers the literature results in electrical resistivity of metals. This new insight goes beyond the perturbation theory and computationally not expensive. Moreover, it can reveal the important missing pieces of universal resistivity. 


Figure 3: Density plot of a frozen deformation potential at some temperature. From electrons’ point of view, they travel on a potential like this which is generated by phonons. Below the Debye temperature both disorder (density of local wrinkles) and strength (magnitude of the potential) increase with increasing temperature. After the Debye temperature the potential only scales with strength.

Click here to see more about deformation potentials.


An answer to the universal resistivity problem should include both quantum and thermal effects, as it is clearly implied in the Planckian scattering rate. Because quantum and thermal effects do not usually like coexist and since it is not particularly easy to properly deal simultaneously with both, the problem stalled physicists off quite some time. The answer seems to loom on the horizon but it will only reveal itself by approaching to the problem in a genuine way rather than looking from the same angles over and over again.



1- J. Zaanen, "Planckian dissipation, minimal viscosity and the transport in cuprate strange metals", SciPost Phys. 6, 061 (2019).

2- C. M. Varma, "Colloquium: Linear in temperature resistivity and associated mysteries including high temperature superconductivity", Rev. Mod. Phys. 92, 031001 (2020).

3- S. A. Hartnoll, "Theory of universal incoherent metallic transport", Nat. Phys. 11, 54–61 (2015).

4- J. A. N. Bruin et al. "Similarity of Scattering Rates in Metals Showing T-Linear Resistivity", Science, 339, 804-807 (2013).

5- A. Legros et al. "Universal T-linear resistivity and Planckian dissipation in overdoped cuprates", Nat. Phys. 15 142–147 (2019).

6- E. J. Heller et al. "Phonons as waves: the semiclassical theory of electrical resistivity", arXiv:2005.14239 (2020).