Self-energy Functional Theory Research Articles

Variational cluster approach to thermodynamic properties of interacting fermions at finite temperatures: A case study of the two-dimensional single-band Hubbard model at half filling

We formulate a finite-temperature scheme of the variational cluster approximation (VCA) particularly suitable for an exact-diagonalization cluster solver. Based on the analytical properties of the single-particle Green's function matrices, we explicitly show the branch-cut structure of logarithm of the complex determinant functions appearing in the self-energy-functional theory (SFT) and whereby construct an efficient scheme for the finite-temperature VCA. We first apply the method to explore the antiferromagnetic order in a half-filled Hubbard model by calculating the entropy, specific heat, and single-particle excitation spectrum for different values of on-site Coulomb repulsion $U$ and temperature $T$. We also calculate the $T$ dependence of the single-particle excitation spectrum in the strong coupling region, and discuss the overall similarities to and the fine differences from the spectrum obtained by the spin-density-wave mean-field theory at low temperatures and the Hubbard-I approximation at high temperatures. On the basis of the thermodynamic properties, we obtain a crossover diagram in the $(U,T)$-plane which separates a Slater-type insulator and a Mott-type insulator. Next, we demonstrate the finite-temperature scheme in the cluster-dynamical-impurity approximation (CDIA) and study the paramagnetic Mott metal-insulator transition in the half-filled Hubbard model. Formulating the finite-temperature CDIA, we first address a subtle issue regarding the treatment of the artificially introduced bath degrees of freedom which are absent in the originally considered Hubbard model. We then apply the finite-temperature CDIA to calculate the finite-temperature phase diagram in the $(U,T)$-plane. We find that the Mott transition at low temperatures is discontinuous, and the coexistence region of the metallic and insulating states persists down to zero temperature.

Excitonic Order and Superconductivity in the Two-Orbital Hubbard Model: Variational Cluster Approach

Using the variational cluster approach based on the self-energy functional theory, we study the possible occurrence of excitonic order and superconductivity in the two-orbital Hubbard model with intra- and inter-orbital Coulomb interactions. It is known that an antiferromagnetic Mott insulator state appears in the regime of strong intra-orbital interaction, a band insulator state appears in the regime of strong inter-orbital interaction, and an excitonic insulator state appears between them. In addition to these states, we find that the $s^{\pm}$-wave superconducting state appears in the small-correlation regime, and the $d_{x^2 - y^2}$-wave superconducting state appears on the boundary of the antiferromagnetic Mott insulator state. We calculate the single-particle spectral function of the model and compare the band gap formation due to the superconducting and excitonic orders.

Comparison of Green’s functions for transition metal atoms using self-energy functional theory and coupled-cluster singles and doubles (CCSD)

We demonstrate in the present study that self-consistent calculations based on the self-energy functional theory (SFT) are possible for the electronic structure of realistic systems in the context of quantum chemistry. We describe the procedure of a self-consistent SFT calculation in detail and perform the calculations for isolated 3d transition metal atoms from V to Cu as a preliminary study. We compare the one-particle Green's functions obtained in this way and those obtained from the coupled-cluster singles and doubles method. Although the SFT calculation starts from the spin-unpolarized Hartree-Fock state for each of the target systems, the self-consistency loop correctly leads to degenerate spin-polarized ground states. We examine the spectral functions in detail to find their commonalities and differences among the atoms by paying attention to the characteristics of the two approaches. It is demonstrated via the two approaches that calculations based on the density functional theory (DFT) can fail in predicting the orbital energy spectra for spherically symmetric systems. It is found that the two methods are quite reliable and useful beyond DFT.

Self-energy functional theory with symmetry breaking for disordered lattice bosons

We extend the self-energy functional theory to the case of interacting lattice bosons in the presence of symmetry breaking and quenched disorder. The self-energy functional we derive depends only on the self-energies of the disorder-averaged propagators, allowing for the construction of general non-perturbative approximations. Using a simple single-site reference system with only three variational parameters, we are able to reproduce numerically exact quantum Monte Carlo (QMC) results on local observables of the Bose–Hubbard model with box disorder with high accuracy. At strong interactions, the phase boundaries are reproduced qualitatively but shifted with respect to the ones observed with QMC due to the extremely low condensate fraction in the superfluid phase. Deep in the strongly-disordered weakly-interacting regime, the simple reference system employed is insufficient and no stationary solutions can be found within its restricted variational subspace. By systematically analyzing thermodynamical observables and the spectral function, we find that the strongly interacting Bose glass is characterized by different regimes, depending on which local occupations are activated as a function of the disorder strength. We find that the particles delocalize into isolated superfluid lakes over a strongly localized background around maximally-occupied sites whenever these sites are particularly rare. Our results indicate that the transition from the Bose glass to the superfluid phase around unit filling at strong interactions is driven by the percolation of superfluid lakes which form around doubly occupied sites.

Bosonic self-energy functional theory

We derive the self-energy functional theory for bosonic lattice systems with broken $U(1)$ symmetry by parametrizing the bosonic Baym-Kadanoff effective action in terms of one- and two-point self-energies. The formalism goes beyond other approximate methods such as the pseudoparticle variational cluster approximation, the cluster composite boson mapping, and the Bogoliubov+U theory. It simplifies to bosonic dynamical-mean field theory when constraining to local fields, whereas when neglecting kinetic contributions of non-condensed bosons it reduces to the static mean-field approximation. To benchmark the theory we study the Bose-Hubbard model on the two- and three-dimensional cubic lattice, comparing with exact results from path integral quantum Monte Carlo. We also study the frustrated square lattice with next-nearest neighbor hopping, which is beyond the reach of Monte Carlo simulations. A reference system comprising a single bosonic state, corresponding to three variational parameters, is sufficient to quantitatively describe phase-boundaries, and thermodynamical observables, while qualitatively capturing the spectral functions, as well as the enhancement of kinetic fluctuations in the frustrated case. On the basis of these findings we propose self-energy functional theory as the omnibus framework for treating bosonic lattice models, in particular, in cases where path integral quantum Monte Carlo methods suffer from severe sign problems (e.g. in the presence of non-trivial gauge fields or frustration). Self-energy functional theory enables the construction of diagrammatically sound approximations that are quantitatively precise and controlled in the number of optimization parameters, but nevertheless remain computable by modest means.

Time-dependent Mott transition in the periodic Anderson model with nonlocal hybridization

The time-dependent Mott transition in a periodic Anderson model with off-site, nearest-neighbor hybridization is studied within the framework of nonequilibrium self-energy functional theory. Using the two-site dynamical-impurity approximation, we compute the real-time dynamics of the optimal variational parameter and of different observables initiated by sudden quenches of the Hubbard-U and identify the critical interaction. The time-dependent transition is orbital selective, i.e., in the final state, reached in the long-time limit after the quench to the critical interaction, the Mott gap opens in the spectral function of the localized orbitals only. We discuss the dependence of the critical interaction and of the final-state effective temperature on the hybridization strength and point out the various similarities between the nonequilibrium and the equilibrium Mott transition. It is shown that these can also be smoothly connected to each other by increasing the duration of a U-ramp from a sudden quench to a quasi-static process. The physics found for the model with off-site hybridization is compared with the dynamical Mott transition in the single-orbital Hubbard model and with the dynamical crossover found for the real-time dynamics of the conventional Anderson lattice with on-site hybridization.

Nonequilibrium self-energy functional approach to the dynamical Mott transition

The real-time dynamics of the Fermi-Hubbard model, driven out of equilibrium by quenching or ramping the interaction parameter, is studied within the framework of the nonequilibrium self-energy functional theory. A dynamical impurity approximation with a single auxiliary bath site is considered as a reference system and the time-dependent hybridization is optimized as prescribed by the variational principle. The dynamical two-site approximation turns out to be useful to study the real-time dynamics on short and intermediate time scales. Depending on the strength of the interaction in the final state, two qualitatively different response regimes are observed. For both weak and strong couplings, qualitative agreement with previous results of nonequilibrium dynamical mean-field theory is found. The two regimes are sharply separated by a critical point at which the low-energy bath degree of freedom decouples in the course of time. We trace the dependence of the critical interaction of the dynamical Mott transition on the duration of the interaction ramp from sudden quenches to adiabatic dynamics, and therewith link the dynamical to the equilibrium Mott transition.

Nonequilibrium variational-cluster approach to real-time dynamics in the Fermi-Hubbard model

The nonequilibrium variational-cluster approach is applied to study the real-time dynamics of the double occupancy in the one-dimensional Fermi-Hubbard model after different fast changes of hopping parameters. A simple reference system, consisting of isolated Hubbard dimers, is used to discuss different aspects of the numerical implementation of the approach in the general framework of nonequilibrium self-energy functional theory. Opposed to a direct solution of the Euler equation, its time derivative is found to serve as numerically tractable and stable conditional equation to fix the time-dependent variational parameters.

Frustration and magnetic orderings in the square-lattice Hubbard model with anisotropic next-nearest-neighbor hopping

Magnetic orders of the square-lattice Hubbard model with the anisotropic next-nearest-neighbor hopping parameters at half filling are studied by the variational cluster approximation based on the self-energy functional theory. We show that in the strong correlation region at zero temperature the disordered phase appears between the collinear-ordered and 120°-ordered phases due to frustration in the spin degrees of freedom of the model. We also show that the phase transitions to the two ordered phases from the disordered phase are of the first order.

Variational cluster approximation to the thermodynamics of quantum spin systems

We derive a variational cluster approximation for Heisenberg spin systems at finite temperature based on the ideas of the self-energy functional theory by Potthoff for fermionic and bosonic systems with local interactions. Partitioning the real system into a set of clusters, we find an analytical expression for the auxiliary free energy, depending on a set of variational parameters defined on the cluster, whose stationary points provide approximate solutions from which the thermodynamics of spin models can be obtained. We explicitly describe the technical details of how to evaluate the free energy for finite clusters and remark on specific problems and possible limitations of the method. To test the approximation we apply it to the antiferromagnetic spin chain and compare the results for varying cluster sizes and choices of variational parameters with the exact Bethe ansatz solution.

Nonequilibrium self-energy functional theory

The self-energy functional theory (SFT) is generalized to describe the real-time dynamics of correlated lattice-fermion models far from thermal equilibrium. This is achieved by means of a reformulation of the original equilibrium theory in terms of double-time Green's functions on the Keldysh-Matsubara contour. We construct a functional which is stationary at the physical (nonequilibrium) self-energy and which yields the grand potential of the initial thermal state at the physical point. Non-perturbative approximations can be defined by specifying a reference system that serves to generate trial self-energies. These self-energies are varied by varying the reference system's one-particle parameters on the Keldysh-Matsubara contour. In case of thermal equilibrium, the new approach reduces to the conventional SFT. However, "unphysical" variations, i.e., variations that are different on the upper and the lower branch of the Keldysh contour, must be considered to fix the time-dependence of the optimal physical parameters via the variational principle. Functional derivatives in the nonequilibrium SFT Euler equation are carried out analytically to derive conditional equations for the variational parameters that are accessible to a numerical evaluation via a time-propagation scheme. Approximations constructed by means of the nonequilibrium SFT are shown to be inherently causal, internally consistent and to respect macroscopic conservation laws resulting from gauge symmetries of the Hamiltonian. This comprises the nonequilibrium dynamical mean-field theory but also dynamical-impurity and variational-cluster approximations that are specified by reference systems with a finite number of degrees of freedom. In this way, non-perturbative and consistent approximations can be set up, the numerical evaluation of which is accessible to an exact-diagonalization approach.

Theoretical studies of a three-band Hubbard model with a strong spin-orbit coupling for 5d transition metal oxide Sr2IrO4

Motivated by recent experiments on Sr2IrO4, we study the ground state properties of a two-dimensional three-band Hubbard model with a strong relativistic spin-orbit coupling. Using the exact diagonalization technique, the dynamical magnetic structure factor M(q, ω) is calculated to examine the low-energy magnetic excitations. We find that the low-energy excitations in M(q, ω) are well described by an effective Heisenberg model composed of a local Kramers doublet of an effective total angular momentum Jeff = |s⃗ − ℓ⃗| = 1/2. The antiferromagnetic exchange interaction estimated from M(q, ω) is as large as ~ 80 meV, which is in good quantitative agreement with experiments. To study a possible long-range ordered state in the thermodynamic limit, we use the variational cluster approximation based on the self-energy functional theory, which is parallelized to accelerate the calculations. We find the ground state where the local Kramers doublet is in-plane antiferromagnetic ally ordered.

Variational cluster approximation study of Mott transition with strong spin-orbit coupling

Motivated by recent experiments on Sr2IrO4, the ground state magnetic and electronic structures are studied theoretically for a two-dimensional three-band Hubbard model with strong spin-orbit coupling. To treat spin-orbit coupling, local Coulomb interactions, and band structure effects on the same footing, the variational cluster approximation based on the self-energy functional theory is employed. It is found that for a relatively large coupling region, the ground state is an anisotropic antiferromagnetic Mott insulator of an effective local angular momentum Jeff = 1/2 with xy plane as an easy plane. This anisotropy is caused by the strong spin-orbit coupling along with the inter-orbital Hund's coupling. The momentum resolved one-particle excitations are also studied for the Mott insulating phase. It is found that the low-energy one-particle excitations consist mostly of the Jeff = 1/2 state, a direct evidence of a novel Jeff = 1/2 Mott insulator.

Antiferromagnetism and Kondo screening in the periodic anderson model: Variational cluster approach

The variational cluster approach based on the self-energy functional theory is applied to consider the two-dimensional symmetric periodic Anderson model at half filling We calculate a variety of physical quantities including the single-particle spectra in the thermodynamic limit at zero temperature and show that the symmetry breaking due to antiferromagnetic order of localized spins occurs in the strong coupling region, whereas in the weak coupling region, the Kondo insulating state without symmetry breaking is observed. We thus discuss the competition between antifomagnetim and Kondo seening in heavy fmion ystems.

Antiferromagnetism versus Kondo screening in the two-dimensional periodic Anderson model at half filling: Variational cluster approach

The variational cluster approach (VCA) based on the self-energy functional theory is applied to the two-dimensional symmetric periodic Anderson model at half filling. We calculate a variety of physical quantities including the staggered moments and single-particle spectra at zero temperature to show that the symmetry breaking due to antiferromagnetic ordering occurs in the strong coupling region, whereas in the weak-coupling region, the Kondo insulating state without symmetry breaking is realized. The critical interaction strength is estimated. We thus demonstrate that the phase transition due to competition between antiferromagnetism and Kondo screening in the model can be described quantitatively by VCA.

Self-energy-functional theory for systems of interacting electrons with disorder

Based on a functional-integral formalism, a generalization of the self-energy-functional theory (SFT) is proposed which is applicable to systems of interacting electrons with disorder. Similar to the pure case without disorder, a variational principle is set up which gives the physical (disorder) self-energy as a stationary point of the (averaged) grand potential. Although the resulting self-energy functional turns out to be more complicated, the formal structure of the theory can be retained since the unknown part of the functional is universal. This allows to construct non-perturbative and thermodynamically consistent approximations via searching for a stationary point on a restricted domain of the functional. The theory and the possible approximations are worked out for models with local interactions and local disorder. This results in a derivation of different mean-field approaches and various cluster extensions, including well-known concepts as the statistical dynamical mean-field theory, the molecular coherent-potential approximation and the dynamical cluster approximation. Due to the common formal framework provided by the SFT, one achieves a general systematization of dynamical approaches, i.e. approaches based on the spectrum of one-particle excitations. New mean-field and new cluster schemes naturally appear in this framework and complement the existing ones. Their prospects for future applications are discussed.