Two-Particle-Self-Consistent Approach for the Hubbard Model

Even at weak to intermediate coupling, the Hubbard model poses a formidable challenge. In two dimensions in particular, standard methods such as the Random Phase Approximation are no longer valid since they predict a finite temperature antiferromagnetic phase transition prohibited by the Mermin-Wagner theorem. The Two-Particle-Self-Consistent (TPSC) approach satisfies that theorem as well as particle conservation, the Pauli principle, the local moment and local charge sum rules. The self-energy formula does not assume a Migdal theorem. There is consistency between one- and two-particle quantities. Internal accuracy checks allow one to test the limits of validity of TPSC. Here I present a pedagogical review of TPSC along with a short summary of existing results and two case studies: a) the opening of a pseudogap in two dimensions when the correlation length is larger than the thermal de Broglie wavelength, and b) the conditions for the appearance of d-wave superconductivity in the two-dimensional Hubbard model.Comment: Chapter in "Theoretical methods for Strongly Correlated Systems", Edited by A. Avella and F. Mancini, Springer Verlag, (2011) 55 pages. Misprint in Eq.(23) corrected (thanks D. Bergeron

Presynaptic NMDA Receptors Mediate IPSC Potentiation at GABAergic Synapses in Developing Rat Neocortex

NMDA receptors are traditionally viewed as being located postsynaptically, at both synaptic and extrasynaptic locations. However, both anatomical and physiological studies have indicated the presence of NMDA receptors located presynaptically. Physiological studies of presynaptic NMDA receptors on neocortical GABAergic terminals and their possible role in synaptic plasticity are lacking.We report here that presynaptic NMDA receptors are present on GABAergic terminals in developing (postnatal day (PND) 12-15) but not older (PND21-25) rat frontal cortex. Using MK-801 in the recording pipette to block postsynaptic NMDA receptors, evoked and miniature IPSCs were recorded in layer II/III pyramidal cells in the presence of AMPA/KA receptor antagonists. Bath application of NMDA or NMDA receptor antagonists produced increases and decreases in mIPSC frequency, respectively. Physiologically patterned stimulation (10 bursts of 10 stimuli at 25 Hz delivered at 1.25 Hz) induced potentiation at inhibitory synapses in PND12-15 animals. This consisted of an initial rapid, large increase in IPSC amplitude followed by a significant but smaller persistent increase. Similar changes were not observed in PND21-25 animals. When 20 mM BAPTA was included in the recording pipette, potentiation was still observed in the PND12-15 group indicating that postsynaptic increases in calcium were not required. Potentiation was not observed when patterned stimulation was given in the presence of D-APV or the NR2B subunit antagonist Ro25-6981.The present results indicate that presynaptic NMDA receptors modulate GABA release onto neocortical pyramidal cells. Presynaptic NR2B subunit containing NMDA receptors are also involved in potentiation at developing GABAergic synapses in rat frontal cortex. Modulation of inhibitory GABAergic synapses by presynaptic NMDA receptors may be important for proper functioning of local cortical networks during development