We develop a nonequilibrium multi-orbital extension of the two-particle self-consistent theory and apply it to the bilayer Hubbard model as a minimal platform to investigate correlation effects in the presence of interlayer interactions and tunneling. The method determines vertex corrections in the spin and charge channels self-consistently at the two-particle level, thereby avoiding the spurious finite-temperature phase transitions that limit dynamical mean-field theory in two dimensions. We derive the spectral self-energy and implement the framework directly on the real-frequency axis within the Keldysh nonequilibrium Green's function formalism, enabling the treatment of both equilibrium and non-equilibrium steady states without relying on numerical analytic continuation. As an application, we demonstrate that a pseudogap can emerge in the bilayer Hubbard model when spin, charge, or excitonic fluctuations become sufficiently strong. Instabilities in different channels are also evaluated in an unbiased manner across the parameter space. Remarkably, we find that the excitonic susceptibility grows with increasing interlayer bias, before it gets suppressed at large biases by the charge imbalance between the layers. This work establishes a versatile and computationally efficient framework for investigating correlated multi-orbital systems under nonequilibrium conditions.
We develop a nonequilibrium multi-orbital extension of the two-particle self-consistent theory and apply it to the bilayer Hubbard model as a minimal platform to investigate correlation effects in the presence of interlayer interactions and tunneling. The method determines vertex corrections in the spin and charge channels self-consistently at the two-particle level, thereby avoiding the spurious finite-temperature phase transitions that limit dynamical mean-field theory in two dimensions. We derive the spectral self-energy and implement the framework directly on the real-frequency axis within the Keldysh nonequilibrium Green's function formalism, enabling the treatment of both equilibrium and non-equilibrium steady states without relying on numerical analytic continuation. As an application, we demonstrate that a pseudogap can emerge in the bilayer Hubbard model when spin, charge, or excitonic fluctuations become sufficiently strong. Instabilities in different channels are also evaluated in an unbiased manner across the parameter space. Remarkably, we find that the excitonic susceptibility grows with increasing interlayer bias, before it gets suppressed at large biases by the charge imbalance between the layers. This work establishes a versatile and computationally efficient framework for investigating correlated multi-orbital systems under nonequilibrium conditions.
We present a nonequilibrium steady-state implementation of the two-particle self-consistent method. This approach respects the Mermin-Wagner theorem and incorporates nonlocal spatial fluctuations through self-consistent static vertices. The real-frequency implementation allows one to compute spectral properties without analytical continuation in both equilibrium and nonequilibrium. As an interesting application, we investigate spin correlations in the bilayer square lattice Hubbard model under a perpendicular static electric field. In equilibrium, the result yields spin correlations which are in good agreement with recent optical lattice experiments. Under a large enough static electric field, the interlayer spin correlations switch from antiferromagnetic to ferromagnetic. We clarify how this phenomenon is linked to the nonequilibrium modifications of the spin excitation spectrum.
The dynamical mean field theory is extended using the Keldysh Green's function formalism to study strongly correlated systems away from equilibrium. This technique has been applied to two different systems, whose insulating behavior is due to electronic correlation effects: excitonic insulators and Mott insulators. The real-time development of the electronic properties of excitonic insulators, such as the distribution function and the gap, after an ultrashort intense is studied. The appropriate self-consistency equations turned out to be matrix equations in frequency and Keldysh space. The self-consistency equations derived for the two-plane Hubbard model with a periodic time-dependent coupling between the two planes are matrix equations in energy sidebands due to quasi-energy conservation modulo the external frequency. Numerical results are shown for the local density of states and the nonequilibrium optical conductivity, both exhibiting a sideband structure, with the number and weight of the sidebands depending on the strength of the perpendicular coupling.
We present improvements of a recently introduced numerical method [E. Arrigoni et al., Phys. Rev. Lett. 110, 086403 (2013)] to compute steady-state properties of strongly correlated electronic systems out of equilibrium. The method can be considered as a nonequilibrium generalization of exact diagonalization based dynamical mean-field theory (DMFT). The key modification for the nonequilibrium situation consists in addressing the DMFT impurity problem within an auxiliary system consisting of the correlated impurity, Nb uncorrelated bath sites, and two Markovian environments (sink and reservoir). Algorithmic improvements in the impurity solver allow to treat efficiently larger values of Nb than previously in DMFT. This increases the accuracy of the results and is crucial for a correct description of the physical behavior of the system in the relevant parameter range including a semiquantitative description of the Kondo regime. To illustrate the approach, we consider a monoatomic layer of correlated orbitals, described by the single-band Hubbard model, attached to two metallic leads. The nonequilibrium situation is driven by a bias voltage applied to the leads. For this system, we investigate the spectral function and the steady-state current-voltage characteristics in the weakly as well as in the strongly interacting limit. In particular, we investigate the nonequilibrium behavior of quasiparticle excitations within the Mott gap of the correlated layer. We find for low-bias voltage Kondo-type behavior in the vicinity of the insulating phase. In particular, we observe a splitting of the Kondo resonance as a function of the bias voltage.
Nonlocal correlations play an essential role in correlated electron systems, especially in the vicinity of phase transitions and crossovers, where two-particle correlation functions display a distinct momentum dependence. In nonequilibrium settings, the effect of nonlocal correlations on dynamical phase transitions, prethermalization phenomena, and trapping in metastable states is not well understood. In this paper, we introduce a dynamical mean field theory (DMFT) extension to the nonequilibrium two-particle self-consistent (TPSC) approach, which allows to perform nonequilibrium simulations capturing short- and long-ranged nonlocal correlations in the weak-, intermediate-, and strong-correlation regimes. The method self-consistently computes local spin and charge vertices, from which a momentum-dependent self-energy is constructed. Replacing the local part of the self-energy by the DMFT result within this self-consistent scheme provides an improved description of local correlation effects. We explain the details of the formalism and the implementation, and demonstrate the versatility of DMFT+TPSC with interaction quenches and dimensional crossovers in the Hubbard model.
In systems of indistinguishable particles, correlations between the quantum degrees of freedom can give rise to collective modes where particles behave coherently together over some correlation length and time, thereby forming quasiparticles. One example of such quasiparticle would be the π-ton [66], describing the binding of two particle-hole pairs via the exchange of momentum corresponding to the antiferromagnetic (AFM) wave vector. These quasiparticles would emerge in the vicinity of an AFM ordering instability [66, 119, 120] and nonlocal correlations are crucial to describe them. Especially, in systems out of equilibrium, nonlocality can give rise to prethermal states wherein the distribution function can be momentum-dependent [92, 119], and nonthermal exotic physics can be observed and manipulated [40, 68, 79, 84, 93, 132, 147, 157–159]. To be able to model and simulate systems out of equilibrium is a challenge. A few numerical methods such as nonequilibrium Dynamical Mean Field Theory (DMFT) [44] have been used extensively [5]. Nonequilibrium DMFT is designed to treat nonperturbatively the local time-dependent single-particle correlations. Since phenomena such as quantum criticality, high-Tc superconductivity or the creation of π-tons via light-matter coupling stem from nonlocal two-particle correlations, one needs to resort to diagrammatic extensions to DMFT [103] and other methods to effectively describe them, such as the post-processing DMFT treatment of vertex corrections, which will be introduced. The Two-Particle Self-Consistent approach (TPSC) and its variants figure among those methods that include nonlocal two-particle and single-particle correlations in a self-consistent way [121, 122]. TPSC and its variants have the ability to capture the effect of the growing spin fluctuations (paramagnons) in the self-energy when close to amagnetic phase transition leading to the breakdown of the Fermi-liquid quasiparticles at the Fermi level: this would mark the onset of a pseudo-gap [145, 152]. It fulfils the Mermin-Wagner theorem in two dimensions, the Pauli principle, and various two-particle local sum-rules [152]. In the works embodied in this thesis, numerical methods have been designed to access nonequilibrium regimes, by changing model parameters across time, allowing for nonlocal correlations to be properly included. With these new algorithms at hand, the π-tons were studied in the optical conductivity and magnetic response [119, 120] as well as spin and charge fluctuations in the 2D and 3D single-band Hubbard model [121, 122]. It is demonstrated by means of those methods that nonlocal two-particle vertex corrections can underpin prethermal states and could generate transient nonthermal states such as low-energy charge excitations associated with nonthermal negative temperatures. It is also shown that states lying close to the Fermi level control the thermalization timescale and that lattice hopping quenches changing the unit-cell dimensionality could allow one to play around with the charge and spin scattering channels.
The Emery model, or three-band Hubbard model, is a Hamiltonian that is thought to contain much of the physics of cuprate superconductors. This model includes two noninteracting p orbitals and one interacting d orbital per unit cell. Few methods that can solve multiorbital interacting Hamiltonians reliably and efficiently exist. Here, we introduce an application of the two-particle self-consistent (TPSC) approach to the Emery model. We construct this method within the framework of the TPSC + DMFT method, which can be seen as a way to introduce nonlocal corrections to dynamical mean-field theory (DMFT). We show that interacting orbital densities, rather than the noninteracting ones, must be used in the calculations. For the Emery model, we find that at constant bare interaction U, the vertex for spin fluctuations, Usp, decreases rapidly with filling. This may be one of the factors that contributes to electron-doped cuprates appearing less correlated than hole-doped ones. More generally, our work opens the road to the application of the TPSC approach to spin fluctuations in multiorbital models.