André-Marie Tremblay

Recherche

• Électrons fortement corrélés, supraconductivité et matériaux quantiques.
  1. Problématique
  2. Objectifs
  3. Méthodologie
  4. Réalisations
  5. Collaborateurs
  6. English Summary
• Qubits supraconducteurs
• Projets étudiants

6. English Summary

We are studying the properties of models for strongly correlated electronic systems that are regarded as possible candidates for explaining high-temperature superconductivity as well as organic conductors.

i) Earlier work (context)

With Yury Vilk, we have developed a new approach to the Hubbard model. A detailed account was published in 1997. It is based on enforcing conservation laws, the Pauli principle and a number of crucial sum rules. More specifically, spin and charge susceptibilities are expressed, in a conserving approximation, as a function of two irreducible vertices whose values are found by imposing the local Pauli principle as well as the local-moment sum rule and consistency with the equations of motion in a local-field approximation. The Mermin-Wagner theorem in two dimensions is automatically satisfied. The effect of collective modes on single-particle properties is then obtained by a paramagnon-like formula that is consistent with the two-particle properties. Since there is no Migdal theorem controlling the effect of spin and charge fluctuations on the self-energy, the required vertex corrections are included. It was shown that the theory is in quantitative agreement with Monte Carlo simulations for both single-particle and two-particle properties. The theory predicts a magnetic phase diagram where magnetic order persists away from half-filling but where ferromagnetism is completely suppressed. Both quantum-critical and renormalized-classical behaviour can occur in certain parameter ranges. We have shown that in the renormalised classical regime, spin fluctuations lead to precursors of antiferromagnetic bands (shadow bands) and to the destruction of the Fermi-liquid quasiparticles in a wide temperature range above the zero-temperature phase transition. This is illustrated in our 1996 report. The upper critical dimension for this phenomenon is three. Pair fluctuations in attractive models can lead to analogous physical effects. These precursor effects are directly related to the existence of the famous "pseudogap regime", a subject of intense debate in the high T_c literature.

We also continue to develop and use our Monte Carlo simulations programs.The program has been rewritten in Fortran 90 in 1996 and it runs on the SP parallel computer of the RQCHP and on Beowulf clusters. We have also developed code that allows us to extract real-frequency properties from imaginary time data. These Maximum Entropy programs are allowing us to investigate dynamical properties and to establish the difference between thermal and strong-coupling induced pseudogaps. These programs are also used in a number of projects described below. A grant from the Canadian Foundation for Innovation and the Canada Research Chair Program will allow us to build a 200 CPU cluster.

ii) Completed in 2001

Conserving approximations vs Two-Particle Self-Consistent Approach

by S. Allen, A.-M.S. Tremblay and Y.M. Vilk.

In this rather formal work, we present a derivation of our non-perturbative approach that highlights similarities and differences with so-called conserving approximations. We use the source field approach (or functional-derivative approach) developed by Martin, Schwinger, Kadanoff and Baym.

Comment on "Absence of a Slater Transition in Two-dimensional Hubbard Model"

by B. Kyung, J.S. Landry, D. Poulin and A.-M.S. Tremblay.

We have shown how to correctly interpret the results of numerical simulations that were claiming to show the existence of a Mott transition even at weak coupling in the two-dimensional Hubbard model. The validity of our viewpoint is experimentally verifiable by measuring the relative size of the antiferromagnetic correlation length and the thermal de Broglie wavelength.

d-wave Pair Susceptibility for the two-dimensional repulsive Hubbard model

by J.S. Landry and A.-M.S. Tremblay.

With coop student Jean-Sébastien Landry , we began a systematic study of the superconducting d-wave susceptibility for the attractive Hubbard model. Thanks to exceptionally large computing power, our results cover a broad range of parameters. We observe a maximum at finite doping. A generalization of the approach developed with Vilk allows us to explain these numerical results from weak to intermediate coupling (see below).

iii) Ongoing projects, begun before January 2002

Spin and charge susceptibility in the attractive Hubbard model.

Despite the fact that the method developed with Yury Vilk and Steve Allen allows a satisfactory description of dominant fluctuations, such as pair fluctuations in the attractive Hubbard model, non-singular channels are more difficult to compute in a reliable analytical way. With Bumsoo Kyung and Steve Allen, we have found a way to use crossing symmetry to compute the spin and charge susceptibilities of the attractive model. These results can now be generalized to the case of the repulsive Hubbard model, as described in the following paragraph.

How antiferromagnetic fluctuations can both help and hinder d-wave superconductivity.

Sub-dominant channels in the repulsive Hubbard model are superconducting channels. We are touching the heart of the high-Tc problem. Analytical results of Bumsoo Kyung are in quantitative agreement with simulations of Jean-Sébastien Landry, mentioned above. Our approach allows us to obtain an estimate of the superconducting transition temperature. Far from half-filling Tc increases as one decreases doping because of the enhanced antiferromagnetic fluctuations. Closer to half-filling, antiferromagnetic fluctuations create a pseudogap that leads to a decrease in Tc. Adding three-dimensional effects will allow us to obtain more quantitative results, in particular for the position of the antiferromagnetic phase boundary.

Phenomenological description of the competition between antiferromagnetism and d-wave superconductivity.

Results of the preceding paragraph were obtained in the weak to intermediate coupling regime. But high-temperature superconductors are rather in the intermediate to strong coupling regime. It is possible to obtain a phenomenological description of this regime by starting directly with thet-J model to calculate effective interactions through sum rules. Correlation functions entering these sum rules are at equal time and short range in space. They can be estimated from mean-field theory, as shown by earlier studies. This approach, suggested by Bumsoo Kyung, allows one to find good estimates for the pseudogap and many other physical quantities but is not quantitative for all observable quantities.

Thermodynamic consistency and temperature-induced localization in interacting systems.

Sébastien Roy, in his Master's thesis, is generalizing the approach of Yury Vilk to compute thermodynamic properties of the Hubbard model. There are two ways to obtain the thermodynamics in this approach. In general the results agree with each other within a few percent. When there is large disagreement, analytical methods are no longer reliable, as can be checked by comparing with Quantum Monte Carlo simulations. The calculation of the entropy will allow to interpret various measurements as well as the temperature and interaction induced localization phenomenon. The latter phenomenon had been discovered for all fillings in two dimensions by François Lemay in this PhD thesis and had been confirmed by Quantum Monte Carlo simulations begun by Jean-Sébastien Landry. Sébastien Roy is also checking, with David Sénéchal, the accuracy of the thermodynamics obtained from the Cluster Perturbation Theory method developed by Sénéchal.

Impurities in interacting systems.

Coop student Alexis Gagné-Lebrun has begun computing the local density of states around an impurity using Quantum Monte Carlo simulations of the Hubbard model. The local density of states is measurable through Scanning Tunnelling Spectroscopy. The way in which the impurity modifies the local density of states as a function of distance from the impurity gives detailed information on the nature of impurities and interactions in the surrounding medium.

Fermi surface "melting".

David Sénéchal has developed a numerical method, Cluster Perturbation Theory , that is based on a strong-coupling perturbation theory that we have developed a few years back. We are presently using this approach to understand how are modified the concepts related to the Fermi surface when the proximity to half-filling and the strength of interactions invalidates the usual concepts, in particular those attached to the Fermi liquid.

Effect of superconducting fluctuations on ultrasonic attenuation in organic quasi two-dimensional superconductors.

With Maxim Mar'enko and Claude Bourbonnais, we are are trying to interpret experiments made by the group of Poirier on ultrasonic absorption in organic conductors of the (BEDT-TTF)₂ X family (kappa phase). Despite the vast literature on the effect of superconducting fluctuations on conductivity, very few works have been devoted to ultrasonic absorption.

5. Collaborateurs	Recherche	Qubits supraconducteurs