# Research

The page for the layman gives a general idea of the type of problems and of ongoing research.

## Abstract

Standard methods of solid-state physics that were highly successful historically are helpless to understand materials where strong electron-electron interactions are important1-4. One must take a broader perspective and understand how the laws of quantum mechanics allow electrons to create a bad metal, an antiferromagnet and a superconductor. Optical lattices of cold atoms are an ideal laboratory to explore these kinds of problems.

**Introduction**

According to a Science magazine5 editorial “Two decades after their discovery, high-temperature superconductors are viewed less as a singular mystery and more as a threshold to new realms of physics.” Several recent breakthroughs, both on the experimental6,7 and theoretical sides, suggest this. On the theoretical side, the availability of powerful computers and of new algorithms has led to spectacular advances where theory is finally beginning to explain experimental results.8-11

The phase diagram of high temperature superconductors is illustrated12 in Fig. 1. The horizontal axis is the average number of electrons per unit cell. Electron-doped compounds appear on the right-hand side, hole-doped on the left. We are interested in the metallic state above the superconducting dome, and in the state below the lines noted *T**. Below those lines, one enters a state with unusual properties, the so-called pseudogap state. In that state, the scattering rate of electrons becomes highly anisotropic and states at the Fermi energy seem to disappear. Above optimal doping, the metallic state is referred to as a “strange metal” because of the unusual temperature dependence of the resistivity and other physical properties.

Figure 1: Temperature-filling phase diagram of high-temperature superconductors. SC refers to d-wave superconducting phases, AF to antiferromagnetic phases. From Ref.12.

**Methodology**

The simplest model that embodies the Physics of strongly correlated electrons, namely kinetic band effects competing with a screened interaction, is the Hubbard model13,14. It is characterized by a hopping integral matrix *tij* and by an interaction term *U*. This model is not only the most studied model in the context of high-temperature superconductivity, it is also the prototype of interacting electrons. This is the main model that I am studying. I have developed a combination of numerical and analytical approaches to tackle this problem.

For numerical simulations, my group is now mostly using new powerful numerical approaches, such as Cellular Dynamical Mean-Field Theory15 (CDMFT) in collaboration with my colleague Sénéchal and others. The name of these methods refers to the self-consistency condition. Generally, in these “Quantum Cluster Methods”,16,17 a cluster is immersed in a self-consistent bath of non-interacting electrons. Through collaborations with K. Haule,17 we have gained access to Continuous Time Quantum Monte Carlo hybridization expansion solvers. These are state of the art codes. A more efficient version of this code has by now been developed in my group.

While Quantum Cluster approaches such as CDMFT describe well the physics at strong coupling, they suffer from the lack of spatial correlations, especially at weak to intermediate coupling where these spatial correlations are important to yield non-trivial physics. We have developed the Two-Particle-Self-Consistent Approach18-20 (TPSC) that is the best known non-perturbative approach to the single-band Hubbard model at weak to intermediate coupling.

## Bibliography

[1] R. Joynt and L. Taillefer, Rev. Mod. Phys. **74**, 235 (2002).

[2] Myron B. Salamon and Marcelo Jaime, Rev. Mod. Phys. **73**, 583 (2001).

[3] E. Dagotto et al. "*Nanoscale Phase Separation and Colossal Magnetoresistance"* (Springer, N.Y. 2003).

[4] B.J. Powell, and R.H. McKenzie, J. Phys. Cond. Matt. **18**, R827-R866 (2006).

[5] Adrian Cho, Science **314**, 1072 (2006).

[6] N. Doiron-Leyraud, *et al. *Nature **447**, 565 (2007)*.*

[7] D. LeBoeuf, *et al. *Nature **450, **533 (2007).

[8] S. S. Kancharla, M. Civelli, M. Capone, B. Kyung, D. Sénéchal, G. Kotliar,

A.-M.S. Tremblay, Phys. Rev. B **77**,184516/1-12 (2008)

[9] David Sénéchal, P.-L. Lavertu, M.-A. Marois, and A.-M.S. Tremblay,

Phys. Rev. Lett. **94**, 156404/1-4 (2005) (4 pages).

[10] T.A. Maier, Phys. Rev. Lett. **95**, 237001 (2005);

[11] K. Haule *et al.*, Phys. Rev. B **76**, 104509 (2007).

[12] N.P. Armitage, P. Fournier, R.L. Greene, Rev. Mod. Phys. **82**, 2421 (2010).

[13] N. F. Mott, Metal-Insulator Transitions (Taylor & Francis, London, 1974).

[14] M. Imada, A. Fujimori, and Y. Tokura, Rev. Mod. Phys. **70**, 1039 (1998).

[15] G. Kotliar *et al.* Rev. Mod. Phys., **78**, 865 (2006) .

[16] Th. Maier, M. Jarrell, Th. Pruschke, M. H. Hettler, Rev. Mod. Phys. **77**, 1027 (2005).

[17] K. Haule, Phys. Rev. B **75**, 155113 (2007).

[18] Y.M. Vilk, Liang Chen et A.-M.S. Tremblay Phys. Rev. B **49**, 13267(R) (1994).

[19] Y.M. Vilk et A.-M.S. Tremblay, J. Phys. I France **7**, 1309 (1997).

[20] S. Allen, A.-M.S. Tremblay, Y.M. Vilk in *"Theoretical Methods for *

* Strongly Correlated Electrons" *David Sénéchal, André-Marie Tremblay and Claude

Bourbonnais (eds.) CRM Series in Mathematical Physics, (Springer, New York, 2003),

p.341-355.