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arXiv:2005.01369 · Physical review. B./Physical review. B

Anisotropy of electronic correlations: On the applicability of local theories to layered materials

Benjamin Klebel-Knobloch, Thomas Schäfer, A. Toschi, Jan M. Tomczak

Published Jan 19, 2021• 13 citations
arXivHTMLPDF

Abstract

Besides the chemical constituents, it is the lattice geometry that controls the most important material properties. In many interesting compounds, the arrangement of elements leads to pronounced anisotropies, which reflect into a varying degree of quasi-two-dimensionality of their low-energy excitations. Here we start by classifying important families of correlated materials according to a simple measure for the tetragonal anisotropy of their ab initio electronic (band) structure. Second, we investigate the impact of a progressively larger anisotropy in driving the nonlocality of many-body effects. To this end, we tune the Hubbard model from isotropic cubic in three dimensions to the two-dimensional limit and analyze it using the dynamical vertex approximation. For sufficiently isotropic hoppings, we find the low-energy self-energy to be well separable into a static nonlocal and a dynamical local contribution. While the latter could potentially be obtained from dynamical mean-field approaches, we find the former to be nonnegligible in all cases. Further, by increasing the model's anisotropy, we quantify the degree of quasi-two-dimensionality which causes this ``space-time separation'' to break down. Our systematic analysis improves the general understanding of electronic correlations in anisotropic or layered materials and heterostructures and provides useful guidance for future realistic studies.

DOI
10.1103/physrevb.103.045121

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Besides the chemical constituents, it is the lattice geometry that controls the most important material properties. In many interesting compounds, the arrangement of elements leads to pronounced anisotropies, which reflect into a varying degree of quasi-two-dimensionality of their low-energy excitations. Here we start by classifying important families of correlated materials according to a simple measure for the tetragonal anisotropy of their ab initio electronic (band) structure. Second, we investigate the impact of a progressively larger anisotropy in driving the nonlocality of many-body effects. To this end, we tune the Hubbard model from isotropic cubic in three dimensions to the two-dimensional limit and analyze it using the dynamical vertex approximation. For sufficiently isotropic hoppings, we find the low-energy self-energy to be well separable into a static nonlocal and a dynamical local contribution. While the latter could potentially be obtained from dynamical mean-field approaches, we find the former to be nonnegligible in all cases. Further, by increasing the model's anisotropy, we quantify the degree of quasi-two-dimensionality which causes this ``space-time separation'' to break down. Our systematic analysis improves the general understanding of electronic correlations in anisotropic or layered materials and heterostructures and provides useful guidance for future realistic studies.
While second-order phase transitions always cause strong nonlocal fluctuations, their effect on spectral properties crucially depends on the dimensionality. For the important case of three dimensions, we show that the electron self-energy is well separable into a local dynamical part and static nonlocal contributions. In particular, our nonperturbative many-body calculations for the three-dimensional Hubbard model at different fillings demonstrate that the quasiparticle weight remains essentially momentum independent, including in the presence of overall large nonlocal corrections to the self-energy. Relying on this insight, we propose a ``space-time-separated'' scheme for many-body perturbation theory that is up to ten times more efficient than current implementations. Besides these far-reaching implications for state-of-the-art electronic structure schemes, our analysis will also provide guidance to the quest of going beyond them.
In this paper, we investigate how nonlocal correlations affect, selectively, the physics of correlated electrons over different energy scales, from the Fermi level to the band edges. This goal is achieved by applying a diagrammatic extension of dynamical mean field theory (DMFT), the dynamical vertex approximation $(\mathrm{D}\mathrm{\ensuremath{\Gamma}}\mathrm{A})$, to study several spectral and thermodynamic properties of the unfrustrated Hubbard model in two and three dimensions. Specifically, we focus first on the low-energy regime by computing the electronic scattering rate and the quasiparticle mass renormalization for decreasing temperatures at a fixed interaction strength. This way, we obtain a precise characterization of the several steps through which the Fermi-liquid physics is progressively destroyed by nonlocal correlations. Our study is then extended to a broader energy range, by analyzing the temperature behavior of the kinetic and potential energy, as well as of the corresponding energy distribution functions. Our findings allow us to identify a smooth but definite evolution of the nature of nonlocal correlations by increasing interaction: They either increase or decrease the kinetic energy w.r.t. DMFT depending on the interaction strength being weak or strong, respectively. This reflects the corresponding evolution of the ground state from a nesting-driven (Slater) to a superexchange-driven (Heisenberg) antiferromagnet (AF), whose fingerprints are, thus, recognizable in the spatial correlations of the paramagnetic phase. Finally, a critical analysis of our numerical results of the potential energy at the largest interaction allows us to identify possible procedures to improve the ladder-based algorithms adopted in the dynamical vertex approximation.
An interacting lattice model describing the subspace spanned by a set of strongly correlated bands is rigorously coupled to density functional theory to enable ab initio calculations of geometric and topological material properties. The strongly correlated subspace is identified from the occupation number band structure as opposed to a mean-field energy band structure. The self-consistent solution of the many-body model Hamiltonian and a generalized Kohn-Sham equation exactly incorporates momentum-dependent and crystal-symmetric correlations into electronic structure calculations in a way that does not rely on a separation of energy scales. Calculations for a multiorbital Hubbard model demonstrate that the theory accurately reproduces the many-body macroscopic polarization.
We consider tight-binding electrons on a square lattice, with an anisotropic hopping intensity, and in a periodic external potential. Breaking of the translational symmetry of the system results in two bands, separated by a gap of width proportional to the unique energy parameter of the model. The off-diagonal matrix elements of the one-body reduced density matrix are exactly expressed by product of Euler's Γ-functions and Appell function. Their large-distance (σ) decay rate is derived to be of the form σ −1 exp (−σ/ξ). A control of the crossover from 2D to 1D system is achieved. In comparison with the isotropic case, the correlation length ξ varies with the gap in a strikingly different manner. In particular, it remains nonzero as the gap vanishes, in all the directions except the direction of stronger hopping intensity.
Materials with strongly correlated electrons show a multitude of unconventional phenomena, such as superconductivity or magnetism, that cannot be explained in a single-particle picture and necessitate to take many-body effects into account. A route to explain the origin of these phenomena consists in constructing effective models for these materials that are then solved, often using numerical techniques. Paradigmatic models for two different classes of strongly correlated electron systems are investigated in this thesis: The Hubbard model is a paradigmatic model to describe the Mott metal-insulator transition (MIT) and the Kondo lattice model describes the low-energy physics in heavy fermion compounds. Both models are analyzed numerically using different variations of a quantum cluster technique, the variational cluster approximation (VCA). The first part focuses on the MIT, as it is investigated experimentally in quasi-two-dimensional charge-transfer salts. Its nature and associated universality class are still heavily discussed and both experiments and theoretical approaches come to different conclusions. In this thesis, the Mott transition is investigated on an anisotropic two-dimensional Hubbard model at half-filling. By using a control parameter, which induces this anisotropy, strong evidence for Mott quantum criticality is found in weakly coupled Hubbard chains. The results at zero temperature show that the second-order critical end point Tc of the interaction-driven metal-insulator transition can be tuned down to zero at strong anisotropy. Further results for the antiferromagnetic phase suggest a similar picture and motivate adding the anisotropy as a new axis in the phase diagram to account for a low-temperature critical end point of the Mott transition. The second part of this thesis focuses on unconventional phases that emerge in a paradigmatic model for heavy fermion systems, the Kondo lattice model. Using the VCA, the two-dimensional Kondo lattice model is investigated in the paramagnetic phase as well as in phases with broken symmetry. An antiferromagnetic phase at weak coupling and a phase with Kondo-singlet formation at strong coupling are analyzed. Within the antiferromagnetically ordered region, two phases with different Fermi-surface topology, which are separated by a discontinuous transition, are identified. The model is also tested for s-wave superconductivity as found recently using dynamical mean-field theory, but no indications for robust local pairing are seen within VCA. Instead, nodal d-wave superconductivity is found and analyzed in a large range of couplings and electron fillings, and its interplay with antiferromagnetic order at weak coupling is discussed. This motivates further studies on extended models that might allow for a close comparison with experiments on heavy fermion systems. Understanding the formation of unconventional phases in strongly correlated electron systems is not only of fundamental interest, but it is also essential for designing new functional materials. Although the results of approximate cluster techniques have to be handled with care, the new aspects presented in this thesis pave the way for further investigations in this direction.
Recent trends of ab initio studies and progress in methodologies for electronic structure calculations of strongly correlated electron systems are discussed. The interest for developing efficient methods is motivated by recent discoveries and characterizations of strongly correlated electron materials and by requirements for understanding mechanisms of intriguing phenomena beyond a single-particle picture. A three-stage scheme is developed as renormalized multi-scale solvers (RMS) utilizing the hierarchical electronic structure in the energy space. It provides us with an ab initio downfolding of the global band structure into low-energy effective models followed by low-energy solvers for the models. The RMS method is illustrated with examples of several materials. In particular, we overview cases such as dynamics of semiconductors, transition metals and its compounds including iron-based superconductors and perovskite oxides, as well as organic conductors of kappa-ET type.