Two-particle Self-consistent Approach Research Articles

Two-particle self-consistent approach for multiorbital models: Application to the Emery model

Two-particle self-consistent approach for multiorbital models: Application to the Emery model

Improved two-particle self-consistent approach for the single-band Hubbard model in two dimensions

Improved two-particle self-consistent approach for the single-band Hubbard model in two dimensions

Dynamical mean field theory extension to the nonequilibrium two-particle self-consistent approach

Dynamical mean field theory extension to the nonequilibrium two-particle self-consistent approach

Improved effective vertices in the multiorbital two-particle self-consistent method from dynamical mean-field theory

In this paper, we present a multiorbital form of the two-particle self-consistent approach (TPSC) where the effective local and static irreducible interaction vertices are determined by means of the dynamical mean-field theory (DMFT). This approach replaces the approximate ansatz equations for the double-occupations $\ensuremath{\langle}{n}_{\ensuremath{\alpha},\ensuremath{\sigma}}^{}{n}_{\ensuremath{\beta},{\ensuremath{\sigma}}^{\ensuremath{'}}}^{}\ensuremath{\rangle}$ by sampling them directly for the same model using DMFT. Compared to the usual Hartree-Fock-like ansatz, this leads to more accurate local vertices in the weakly correlated regime and provides access to stronger correlated systems that were previously out of reach. This approach is extended by replacing the local component of the TPSC self-energy by the DMFT impurity self-energy, which results in an improved self-energy that incorporates strong local correlations but retains a nontrivial momentum dependence. We find that this combination of TPSC and DMFT provides a significant improvement over the multiorbital formulation of TPSC as it allows to determine the components of the spin vertex without artificial symmetry assumptions and opens the possibility to include the transversal particle-hole channel. The new approach is also able to remove unphysical divergences in the charge vertices in TPSC. We find a general trend that lower temperatures can be accessed in the calculation. Benchmarking single-particle quantities, such as the local spectral function with other many-body methods, we find significant improvement in the more strongly correlated regime.

Nonequilibrium two-particle self-consistent approach

We present the nonequilibrium implementation of the two-particle self-consistent (TPSC) approach, which has been shown to provide a reliable equilibrium description of interacting lattice systems in the weak- and intermediate-correlation regime. This method captures the effects of local and nonlocal correlations in two- and higher-dimensional systems and satisfies the Mermin-Wagner theorem. We demonstrate the versatility of nonequilibrium TPSC with calculations of the time-dependent spin and charge response functions and the evolution of effective temperatures extracted from different correlation functions, after interaction ramps in the two-dimensional Hubbard model.

Disorder effects on hot spots in electron-doped cuprates

Antiferromagnetic fluctuations in two dimensions cause a decrease in spectral weight at so-called hot spots associated with the pseudogap in electron-doped cuprates. In the $2\text{D}$ Hubbard model, these hot spots occur when the Vilk criterion is satisfied, namely, when the spin correlation length exceeds the thermal de Broglie wavelength. Using the two-particle self-consistent approach, here we show that this criterion may be violated near the antiferromagnetic quantum critical point when sufficient disorder is added to the model. Static disorder decreases inelastic scattering, in contradiction with Matthiessen's rule, leading to a shift in the position of the quantum critical point and a modification of the conditions for the appearance of hot spots. This opens the road to a study of the interplay between disorder, antiferromagnetic fluctuations, and superconductivity in electron-doped cuprates.

Self-energy dispersion in the Hubbard model

We introduce the concept of self-energy dispersion as an error bound on local theories and apply it to the two-dimensional Hubbard model on the square lattice at half-filling. Since the self-energy has no single-particle analog and is not directly measurable in experiments, its general behavior as a function of momentum is an open question. In this article we benchmark the momentum dependence with the Two-Particle Self-Consistent approach together with analytical and numerical considerations and we show that through the addition of a local single-particle potential to the Hubbard model the self-energy can be flattened, such that it is essentially described by only a frequency-dependent term. We use this observation to motivate that local theories, such as the dynamical mean-field theory, should be expected to give very accurate results in the presence of a potential of this kind. Finally, we propose a simple energy argument as an estimator for the crossover from non-local to local self-energies, which can be computed even by local theories such as dynamical mean-field theory.

Microscopic theory of the superconducting gap in the quasi-one-dimensional organic conductor (TMTSF)2ClO4 : Model derivation and two-particle self-consistent analysis

We present a first-principles band calculation for the quasi-one-dimensional (Q1D) organic superconductor (TMTSF)$_2$ClO$_4$. An effective tight-binding model with the TMTSF molecule to be regarded as the site is derived from a calculation based on maximally localized Wannier orbitals. We apply a two-particle self-consistent (TPSC) analysis by using a four-site Hubbard model, which is composed of the tight-binding model and an on-site (intramolecular) repulsive interaction, which serves as a variable parameter. We assume that the pairing mechanism is mediated by the spin fluctuation, and the sign of the superconducting gap changes between the inner and outer Fermi surfaces, which correspond to a d-wave gap function in a simplified Q1D model. With the parameters we adopt, the critical temperature for superconductivity estimated by the TPSC approach is approximately 1K which is consistent with experiment.

Superconductivity in correlated BEDT-TTF molecular conductors: Critical temperatures and gap symmetries

Starting from an {\it ab initio}-derived two-site dimer Hubbard hamiltonian on a triangular lattice, we calculate the superconducting gap functions and critical temperatures for representative $\kappa$-(BEDT-TTF)$_2$X superconductors by solving the linearized Eliashberg equation using the Two-Particle Self-Consistent approach (TPSC) extended to multi-site problems. Such an extension allows for the inclusion of molecule degrees of freedom in the description of these systems. We present both, benchmarking results for the half-filled dimer model as well as detailed investigations for the 3/4-filled molecule model. Remarkably, we find in the latter model that the phase boundary between the two most competing gap symmetries discussed in the context of these materials -d$_{xy}$ and the recently proposed eight-node gap s+d$_{x^2-y^2}$ symmetry- is located within the regime of realistic model parameters and is especially sensitive to the degree of in-plane anisotropy in the materials as well as to the value of the on-site Hubbard repulsion. We show that these results provide a more complete and accurate description of the superconducting properties of $\kappa$-(BEDT-TTF)$_2$X than previous Random Phase Approximation (RPA) calculations and, in particular, we discuss predicted critical temperatures in comparison to experiments. Finally, our findings suggest that it may be even easier to experimentally switch between the two pairing symmetries as previously anticipated by invoking pressure, chemical doping or disorder effects.

Self-Doping Effect Arising from Electron Correlations in Multilayer Cuprates

A self-doping effect between outer and inner CuO$_2$ planes (OPs and IPs) in multi-layer cuprate superconductors is studied. When one considers a three-layer tight-binding model of the Hg-based three-layer cuprate derived from the first principle calculations, the electron concentration gets to be large in the OP compared to IP. This is inconsistent with the experimental fact that more hole carriers tend to be introduced into the OP than IP.We investigate a three-layer Hubbard model with the two-particle self-consistent approach for multi-layer systems to incorporate electron correlations. We observe that the double occupancy (antiferromagnetic instability) in the IP decreases (increases) more than the OP, and also reveal that more electrons tend to be introduced into the IP than OP to obtain the energy gain from the on-site Hubbard interaction. These results are consistent with the experimental facts, and this electron distribution between the OP and IP can be interpreted as a self-doping effect arising from strong electron correlations.

Two-particle self-consistent approach to unconventional superconductivity

A nonperturbative approach to anisotropic superconductivity is developed based on the idea of two-particle self-consistent (TPSC) theory by Vilk and Tremblay. A sum-rule which the momentum-dependent pairing susceptibility satisfies is derived. An effective pairing interaction between quasiparticles is determined so that the susceptibility should fulfill this exact sum-rule, in which fluctuations belonging to different symmetries couple at finite momentum. It is demonstrated that the mode coupling between d-wave and s-wave pairing fluctuations leads to suppression of the d-wave fluctuation near the Mott insulator.

Optical and dc conductivity of the two-dimensional Hubbard model in the pseudogap regime and across the antiferromagnetic quantum critical point including vertex corrections

The conductivity of the two-dimensional Hubbard model is particularly relevant for high-temperature superconductors. Vertex corrections are expected to be important because of strongly momentum dependent self-energies. We use the Two-Particle Self-Consistent approach that satisfies crucial constraints such as the Mermin-Wagner theorem, the Pauli principle and sum rules in order to reach non-perturbative regimes. This approach is reliable from weak to intermediate coupling. A functional derivative approach ensures that vertex corrections are included in a way that satisfies the f sum-rule. The two types of vertex corrections that we find are the antiferromagnetic analogs of the Maki-Thompson and Aslamasov-Larkin contributions of superconducting fluctuations to the conductivity but, contrary to the latter, they include non-perturbative effects. The resulting analytical expressions must be evaluated numerically. The calculations are impossible unless a number of advanced numerical algorithms are used. A maximum entropy approach is specially developed for analytical continuation of our results. The numerical results are for nearest neighbor hoppings. In the pseudogap regime induced by two-dimensional antiferromagnetic fluctuations, the effect of vertex corrections is dramatic. Without vertex corrections the resistivity increases as we enter the pseudogap regime. Adding vertex corrections leads to a drop in resistivity, as observed in some high temperature superconductors. At high temperature, the resistivity saturates at the Ioffe-Regel limit. At the quantum critical point and beyond, the resistivity displays both linear and quadratic temperature dependence and there is a correlation between the linear term and the superconducting transition temperature. A hump is observed in the mid-infrared range of the optical conductivity in the presence of antiferromagnetic fluctuations.

Competition between charge and spin order in thet−U−Vextended Hubbard model on the triangular lattice

Several new classes of compounds can be modeled in first approximation by electrons on the triangular lattice that interact through on-site repulsion $U$ as well as nearest-neighbor repulsion $V$. This extended Hubbard model on a triangular lattice has been studied mostly in the strong coupling limit for only a few types of instabilities. Using the extended two-particle self-consistent approach (ETPSC), that is valid at weak to intermediate coupling, we present an unbiased study of the density and interaction dependent crossover diagram for spin- and charge-density wave instabilities of the normal state at arbitrary wave vector. When $U$ dominates over $V$ and electron filling is large, instabilities are chiefly in the spin sector and are controlled mostly by Fermi surface properties. Increasing $V$ eventually leads to charge instabilities. In the latter case, it is mostly the wave vector dependence of the vertex that determines the wave vector of the instability rather than Fermi surface properties. At small filling, nontrivial instabilities appear only beyond the weak coupling limit. There again, charge-density wave instabilities are favored over a wide range of dopings by large $V$ at wave vectors corresponding to $\sqrt{(3)}\ifmmode\times\else\texttimes\fi{}\sqrt{(3)}$ superlattice in real space. Commensurate fillings do not play a special role for this instability. Increasing $U$ leads to competition with ferromagnetism. At negative values of $U$ or $V$, neglecting superconducting fluctuations, one finds that charge instabilities are favored. In general, the crossover diagram presents a rich variety of instabilities. We also show that thermal charge-density wave fluctuations in the renormalized-classical regime can open a pseudogap in the single-particle spectral weight, just as spin or superconducting fluctuations.

Conditions for magnetically induced singletd-wave superconductivity on the square lattice

It is expected that at weak to intermediate coupling, $d$-wave superconductivity can be induced by antiferromagnetic fluctuations. However, one needs to clarify the role of Fermi surface topology, density of states, pseudogap, and wave vector of the magnetic fluctuations on the nature and strength of the induced $d$-wave state. To this end, we study the generalized phase diagram of the two-dimensional half-filled Hubbard model as a function of interaction strength $U∕t$, frustration induced by second-order hopping ${t}^{\ensuremath{'}}∕t$, and temperature $T∕t$. In experiment, $U∕t$ and ${t}^{\ensuremath{'}}∕t$ can be controlled by pressure. We use the two-particle self-consistent approach, valid from weak to intermediate coupling. We first calculate as a function of ${t}^{\ensuremath{'}}∕t$ and $U∕t$ the temperature and wave vector at which the spin response function begins to grow exponentially. $d$-wave superconductivity in a half-filled band can be induced by such magnetic fluctuations at weak to intermediate coupling, but only if they are near commensurate wave vectors and not too close to perfect nesting conditions where the pseudogap becomes detrimental to superconductivity. For given $U∕t$, there is thus an optimal value of frustration ${t}^{\ensuremath{'}}∕t$ where the superconducting ${T}_{c}$ is maximum. The noninteracting density of states plays little role. The symmetry ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ vs ${d}_{xy}$ of the superconducting order parameter depends on the wave vector of the underlying magnetic fluctuations in a way that can be understood qualitatively from simple arguments.

Non-perturbative treatment of charge and spin fluctuations in the two-dimensional extended Hubbard model: Extended two-particle self-consistent approach

We study the spin and charge fluctuations of the extended Hubbard model (EHM) with on-site interaction U and first neighbor interaction V on the two-dimensional square lattice in the weak to intermediate coupling regime. We propose an extension of the two-particle self-consistent (ETPSC) approximation that includes the effect of functional derivatives of the pair correlation functions on irreducible spin and charge vertices. These functional derivatives were ignored in our previous work. We evaluate them assuming particle-hole symmetry. The resulting theory satisfies conservations laws and the Mermin-Wagner theorem. Our current results are in much better agreement with benchmark Quantum Monte-Carlo (QMC) results. As a function of U and V, we can determine the crossover temperatures towards renormalized classical regimes where either spin or charge fluctuations dominate. The dominant wave vector is self-determined by the approach.

Interaction-induced adiabatic cooling for antiferromagnetism in optical lattices

In the experimental context of cold-fermion optical lattices, we discuss the possibilities to approach the pseudogap or ordered phases by manipulating the scattering length or the strength of the laser-induced lattice potential. Using the Two-Particle Self-Consistent Approach as well as Quantum Monte Carlo simulations, we provide isentropic curves for the two- and three-dimensional Hubbard models at half-filling. These quantitative results are important for practical attempts to reach the ordered antiferromagnetic phase in experiments on optical lattices of two-component fermions. We find that adiabatically turning on the interaction in two dimensions to cool the system is not very effective. In three dimensions, adiabatic cooling to the antiferromagnetic phase can be achieved in such a manner although the cooling efficiency is not as high as initially suggested by Dynamical Mean-Field Theory. Adiabatic cooling by turning off the repulsion beginning at strong coupling is possible in certain cases.

Nearest-neighbor repulsion and competing charge and spin order in the extended Hubbard model

We generalize the two-particle self-consistent approach (TPSC) to study the extended Hubbard model, where nearest-neighbor interaction is present in addition to the usual local screened interaction. Similarities and differences between the TPSC approach and the Singwi Tosi Land Sj\"{o}lander (STLS) approximation for the electron gas are discussed. The accuracy of our extension of TPSC is assessed by comparisons with Quantum Monte Carlo calculations of Y. Zhang and J. Callaway, Phys. Rev. B \textbf{39}, 9397 (1989). We observe that a positive off-site interaction enhances staggered charge fluctuations and reduces staggered magnetic order.

Pseudogap and high-temperature superconductivity from weak to strong coupling. Towards a quantitative theory (Review Article)

This is a short review of the theoretical work on the two-dimensional Hubbard model performed in Sherbrooke in the last few years. It is written on the occasion of the twentieth anniversary of the discovery of high-temperature superconductivity. We discuss several approaches, how they were benchmarked and how they agree sufficiently with each other that we can trust that the results are accurate solutions of the Hubbard model. Then comparisons are made with experiment. We show that the Hubbard model does exhibit d-wave superconductivity and antiferromagnetism essentially where they are observed for both hole- and electron-doped cuprates. We also show that the pseudogap phenomenon comes out of these calculations. In the case of electron-doped high temperature superconductors, comparisons with angle-resolved photoemission experiments are nearly quantitative. The value of the pseudogap temperature observed for these compounds in recent photoemission experiments had been predicted by theory before it was observed experimentally. Additional experimental confirmation would be useful. The theoretical methods that are surveyed include mostly the two-particle self-consistent approach, variational cluster perturbation theory (or variational cluster approximation), and cellular dynamical mean-field theory.

Strong- and weak-coupling mechanisms for pseudogap in electron-doped cuprates

Using the two-particle self-consistent approach and cluster perturbation theory for the two-dimensional t-t'-t''-U Hubbard model, we discuss weak- and strong-coupling mechanisms for the pseudogap observed in recent angle resolved photoemission spectroscopy on electron-doped cuprates. In the case of the strong-coupling mechanism, which is more relevant near half-filling, the pseudogap can be mainly driven by short range correlations near the Mott insulator. In the vicinity of optimal doping, where weak-coupling physics is more relevant, large antiferromagnetic correlation lengths, seen in neutron measurements, are the origin of the pseudogap. The t-J model is not applicable in the latter case.

Minimally self-consistent T-matrix approximation to describe the low-temperature properties of the Hubbard model in the atomic limit

The atomic limit of the Hubbard model is a simple single-site problem which can be solved exactly, and all one- and two-particle Green's functions can be obtained analytically. These solutions can thus serve as a means of critiquing the success of various approximate theories which might be applied to the full Hubbard model. In particular, we have examined the $T$-matrix approximation for the attractive Hubbard model in the atomic limit, which should give reasonable results at low electronic densities, if one can avoid the spurious phase transition that results when a fully non-self-consistent $T$-matrix approximation is employed---previously we have shown that any level of self-consistency guarantees that this phase transition is correctly suppressed to zero temperature in two dimensions or less. Here, a minimally self-consistent $T$-matrix approximation is shown to be successful in reproducing the exact results for the atomic limit, while fully self-consistent $T$-matrix results do not agree with the known solutions. Of particular note is that the minimally self-consistent $T$-matrix approximation reproduces not only one- and two-particle (static) thermodynamic quantities, but it also exactly reproduces the one-particle spectral function at low but nonzero temperatures. We also make a comparison to the two-particle self-consistent approach of Vilk and Tremblay, and find that the minimally self-consistent $T$-matrix theory can give better results over a broader temperature range.