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# Strong interactions and Fermi-Hubbard model

Reza Nourafkan

Photo : IQA recent publication in Science illustrates perfectly how team work is essential to scientific research. Members of IQ Reza Nourafkan, Alexis Reymbaut, Charles-David Hébert and Simon Bergeron, all part of Prof. André-Marie Tremblay’s research group, provided the theoretical part to *Bad metallic transport in a cold atom Fermi-Hubbard system*, while the experiments were done in Princeton by Prof. W. S. Bakr team. Prof J. Kokalj from Ljubljana also provided complementary theoretical work.

### Materials simulation with predictive power

Reza Nourafkan works as a research professional at IQ. He was born and raised in Isfahan, Iran. His vision is obtaining predictive power in materials simulation.

“I employ a quantum mechanical description of the interactions between electrons and atomic nuclei to understand and explain experimental results, to identify crucial electronic structure characteristics for a given property and to design materials for technology. I have also developed the necessary formalism and computer codes.”

### A discussion with Prof. André Marie Tremblay convinced him

Reza moved to Sherbrooke in 2013 after a one-year postdoc at the University of Alberta in Edmonton followed by another postdoc at Rutgers University in New Jersey, collaborating with Prof. Gabriel Kotliar.

When asked what convinced him to come to Sherbrooke, Reza spontaneously answers : “When I was in Rutgers, André-Marie came for a visit. He is renowned in our field, so I discussed physics with him. I was familiar with his research and the quality of the work that he is doing and I liked his personality as well. That’s why I came.” Reza works mainly with Prof. Tremblay, but he also helps with student supervision and assists other professors with their projects.

### The publication in Science

Electrical resistivity determines if a material is a conductor or an insulator, depending on whether it transports electrical current or not. Electrons carry current, but they usually scatter from one state to another, that is what causes electrical resistivity, ρ. The average distance covered by electrons between scatterings is called mean free path, l. The larger the scattering rate, the smaller is the mean free path.

According to the Mott-Ioffe-Regel limit (MIR limit), this mean free path cannot become shorter than the distance between two atoms. When this happens, resistivity should saturate, reach its limit. Furthermore, the conventional quasi-particle picture for electrons predicts that the temperature-dependent resistivity should scale like the square of temperature, T^{2}, at low temperature.

However, a wide variety of materials, including cuprate superconductors, do not meet the above expectations: resistivity is linear in T and keeps rising with increasing T, violating of the resistivity bound imposed by the MIR limit. Materials that exhibit this property are called strange metals.

Recent work of Prof. Louis Taillefer’s team published in Nature Physics, *Universal T-linear resistivity and Planckian dissipation in overdoped cuprates* explains the T-linear resistivity using the assumption that the scattering rate 1/τ of charge carriers reaches the Planckian limit, whereby ħ/τ = k_{B}T. The scattering rate for electrons, 1/τ, is related to the mean free path by l=v_{F} τ where v_{F} is the velocity of carriers. Such a scenario does not impose a saturation of the mean free path.

### The Princeton team

In real materials, electrons interact not only with each other but also with vibrations of nuclei and with impurities. This complicates their response to an electric field. In order to simplify the problem, Prof. W. S. Bakr team created a quantum simulator from cold atom systems, which can be described by a simple model, the so-called Hubbard model. The model is widely used to describe cuprate superconductors as well. The Princeton group also found an ingenious way to measure electrical conductivity despite the fact that the carriers are neutral in this case. They employ the Nernst-Einstein equation to relate the conductivity (inverse of resistivity) to the compressibility. The diffusion constant gives the proportionality constant.

The results from the quantum simulator show that resistivity exhibits a linear temperature dependence and shows no evidence of saturation, similar to what seen in the strange metal phase of cuprate superconductors.

However, the T-linearity of ρ does not come from the Planckian limit of the scattering rate. Indeed, the scattering rate saturates in agreement with MIR limit. Instead, the compressibility falls at 1/T at high temperature, leading to linear temperature dependence of the resistivity.

### The expertise of Prof. Tremblay’s group

Intuitively, the T-linearity of ρ is attributed to the strong interaction between carriers, which may destroy the conventional quasiparticle picture and invalidate its predictions. However, a quantitative understanding of this problem requires theoretical simulation of the electrical conductivity of the Hubbard model, a subject where Prof. Tremblay’s group has extensive expertise.

Reza Nourafkan used a state of the art method, dubbed dynamical mean field theory (DMFT) and its extension, to solve the Hubbard model and calculate the resistivity. This was complemented by the work of postdoc Alexis Reymbaut, undergraduate student Simon Bergeron and PhD student Charles-David Hébert. The calculated scattering rate does saturate and the compressibility depends on temperature like 1/T, in excellent agreement with experimental data. This confirms that the T-linear resistivity comes from the 1/T dependence of the compressibility.

Surprisingly, the linear slope of the resistivity is however quite close to that predicted by the Planckian limit. Solving a problem often raises new questions…