Phys. Rev. B 101, 075109 (2020) We present a new method to treat the two-dimensional (2D) Hubbard model for
parameter regimes which are relevant for the physics of the high-$T_c$
superconducting ...cuprates. Unlike previous attempts to attack this problem, our
new approach takes into account all fluctuations in different channels on equal
footing and is able to treat reasonable large lattice sizes up to 32x32. This
is achieved by the following three-step procedure: (i) We transform the
original problem to a new representation (dual fermions) in which all purely
local correlation effects from the dynamical mean field theory are already
considered in the bare propagator and bare interaction of the new problem. (ii)
The strong $1/(i\nu)^2$ decay of the bare propagator allows us to integrate out
all higher Matsubara frequencies besides the lowest using low order diagrams.
The new effective action depends only on the two lowest Matsubara frequencies
which allows us to, (iii) apply the two-particle self-consistent parquet
formalism, which takes into account the competition between different
low-energy bosonic modes in an unbiased way, on much finer momentum grids than
usual. In this way, we were able to map out the phase diagram of the 2D Hubbard
model as a function of temperature and doping. Consistently with the
experimental evidence for hole-doped cuprates and previous dynamical cluster
approximation calculations, we find an antiferromagnetic region at low-doping
and a superconducting dome at higher doping. Our results also support the role
of the van Hove singularity as an important ingredient for the high value of
$T_c$ at optimal doping. At small doping, the destruction of antiferromagnetism
is accompanied by an increase of charge fluctuations supporting the scenario of
a phase separated state driven by quantum critical fluctuations.
We derive an analytical expression for the local two-particle vertex of the Falicov-Kimball model, including its dependence on all three frequencies, the full vertex and all reducible vertices. This ...allows us to calculate the self energy in diagrammatic extensions of dynamical mean field theory, specifically in the dual fermion and the one-particle irreducible approach. Non-local correlations are thence included and originate here from charge density wave fluctuations. At low temperatures and in two dimensions, they lead to a larger self energy contribution at low frequencies and a more insulating spectrum.
Vertex functions are a crucial ingredient of several forefront many-body algorithms in condensed matter physics. However, the full treatment of their frequency and momentum dependence severely ...restricts numerical calculations. A significant advancement requires an efficient treatment of the high-frequency asymptotic behavior of the vertex functions. In this work, we first provide a detailed diagrammatic analysis of the high-frequency structures and their physical interpretation. Based on these insights, we propose a parametrization scheme, which captures the whole high-frequency domain for arbitrary values of the Coulomb interaction and electronic density, and we discuss the details of its algorithmic implementation in many-body solvers based on parquet-equations as well as functional renormalization group schemes. Finally, we assess its validity by comparing our results for a single impurity Anderson model with exact diagonalization calculations. The proposed parametrization is pivotal for the algorithmic development of all quantum many-body methods based on vertex functions arising from both local and non-local static microscopic interactions as well as effective dynamic interactions which uniformly approach a static value for large frequencies. In this way, our present technique can substantially improve vertex-based diagrammatic approaches including spatial correlations beyond dynamical mean-field theory.
We present an approach which is based on the one-particle irreducible (1 PI) generating functional formalism and includes electronic correlations on all length scales beyond the local correlations of ...dynamical mean-field theory (DMFT). This formalism allows us to unify aspects of the dynamical vertex approximation (DGammaA) and the dual fermion (DF) scheme, yielding a consistent formulation of nonlocal correlations at the one- and two-particle level beyond DMFT within the functional integral formalism. In particular, the considered approach includes one-particle reducible contributions from the three- and more-particle vertices in the dual fermion approach, as well as some diagrams not included in the ladder version of DGammaA. To demonstrate the applicability and physical content of the 1 PI approach, we compare the diagrammatics of 1 PI, DF, and DGammaA, as well as the numerical results of these approaches for the half-filled Hubbard model in two dimensions.
Electrons on a two-dimensional (2\(d\)) lattice which is exposed to a strong uniform magnetic field show intriguing physical phenomena. The spectrum of such systems exhibits a complex (multi-)band ...structure known as Hofstadter's butterfly. For fillings at which the system is a band insulator one observes a quantized integer-valued Hall conductivity \(\sigma_{xy}\) corresponding to a topological invariant, the first Chern number \(\mathcal{C}_1\). This is robust against many-body interactions as long as no changes in the gap structure occur. Strictly speaking, this stability holds only at zero temperatures \(T\) while for \(T>0\) correlation effects have to be taken into account. In this work, we address this question by presenting a dynamical mean field theory (DMFT) study of the Hubbard model in a uniform magnetic field. The inclusion of local correlations at finite temperature leads to (i) a shrinking of the integer plateaus of \(\sigma_{xy}\) as a function of the chemical potential and (ii) eventually to a deviation from these integer values. We demonstrate that these effects can be related to a correlation-driven narrowing and filling of the band gap, respectively.
Starting from the (Hubbard) model of an atom, we demonstrate that the uniqueness of the mapping from the interacting to the noninteracting Green's function, \(G\to G_0\), is strongly violated, by ...providing numerous explicit examples of different \(G_0\) leading to the same physical \(G\). We argue that there are indeed infinitely many such \(G_0\), with numerous crossings with the physical solution. We show that this rich functional structure is directly related to the divergence of certain classes of (irreducible vertex) diagrams, with important consequences for traditional many-body physics based on diagrammatic expansions. Physically, we ascribe the onset of these highly non-perturbative manifestations to the progressive suppression of the charge susceptibility induced by the formation of local magnetic moments and/or RVB states in strongly correlated electron systems.
We study the relation between the microscopic properties of a many-body system and the electron spectra, experimentally accessible by photoemission. In a recent paper Phys. Rev. Lett. 114, 236402 ...(2015), we introduced the "fluctuation diagnostics" approach, to extract the dominant wave vector dependent bosonic fluctuations from the electronic self-energy. Here, we first reformulate the theory in terms of fermionic modes, to render its connection with resonance valence bond (RVB) fluctuations more transparent. Secondly, by using a large-U expansion, where U is the Coulomb interaction, we relate the fluctuations to real space correlations. Therefore, it becomes possible to study how electron spectra are related to charge, spin, superconductivity and RVB-like real space correlations, broadening the analysis of an earlier work Phys. Rev. B 89, 245130 (2014). This formalism is applied to the pseudogap physics of the two-dimensional Hubbard model, studied in the dynamical cluster approximation. We perform calculations for embedded clusters with up to 32 sites, having three inequivalent K-points at the Fermi surface. We find that as U is increased, correlation functions gradually attain values consistent with an RVB state. This first happens for correlation functions involving the antinodal point and gradually spreads to the nodal point along the Fermi surface. Simultaneously a pseudogap opens up along the Fermi surface. We relate this to a crossover from a Kondo-like state to an RVB-like localized cluster state and to the presence of RVB and spin fluctuations. These changes are caused by a strong momentum dependence in the cluster bath-couplings along the Fermi surface. We also show, from a more algorithmic perspective, how the time-consuming calculations in fluctuation diagnostics can be drastically simplified.
We present a new method to treat the two-dimensional (2D) Hubbard model for parameter regimes which are relevant for the physics of the high-\(T_c\) superconducting cuprates. Unlike previous attempts ...to attack this problem, our new approach takes into account all fluctuations in different channels on equal footing and is able to treat reasonable large lattice sizes up to 32x32. This is achieved by the following three-step procedure: (i) We transform the original problem to a new representation (dual fermions) in which all purely local correlation effects from the dynamical mean field theory are already considered in the bare propagator and bare interaction of the new problem. (ii) The strong \(1/(i\nu)^2\) decay of the bare propagator allows us to integrate out all higher Matsubara frequencies besides the lowest using low order diagrams. The new effective action depends only on the two lowest Matsubara frequencies which allows us to, (iii) apply the two-particle self-consistent parquet formalism, which takes into account the competition between different low-energy bosonic modes in an unbiased way, on much finer momentum grids than usual. In this way, we were able to map out the phase diagram of the 2D Hubbard model as a function of temperature and doping. Consistently with the experimental evidence for hole-doped cuprates and previous dynamical cluster approximation calculations, we find an antiferromagnetic region at low-doping and a superconducting dome at higher doping. Our results also support the role of the van Hove singularity as an important ingredient for the high value of \(T_c\) at optimal doping. At small doping, the destruction of antiferromagnetism is accompanied by an increase of charge fluctuations supporting the scenario of a phase separated state driven by quantum critical fluctuations.
We analyze the highly non-perturbative regime surrounding the Mott-Hubbard metal-to-insulator transition (MIT) by means of dynamical mean field theory calculations at the two-particle level. By ...extending the results of Sch\"afer, et al. Phys. Rev. Lett. 110, 246405 (2013) we show the existence of infinitely many lines in the phase diagram of the Hubbard model where the local Bethe-Salpeter equations, and the related irreducible vertex functions, become singular in the charge as well as the particle-particle channel. These divergence lines accumulate around the critical Mott endpoint in accordance with the interpretation as precursors of the MIT. By comparing our numerical data with analytical calculations of increasing complexity, such as for the disordered Binary Mixture and Falicov-Kimball (FK) models, as well as for the atomic limit (AL) case, (i) we identify two different kinds of divergences lines; (ii) we classify them in terms of the frequency-structure of the associated singular eigenvectors; (iii) we investigate their relation to the multiple branches in the Luttinger-Ward formalism. Moreover, we could distinguish the situations where the multiple divergences simply reflect the emergence of an underlying, unique energy scale \(\nu^*\) below which perturbation theory does no longer apply, from those where the breakdown of perturbation theory affects, not trivially, different energy regimes. Finally, we discuss the implications of our results on the theoretical understanding of the non-perturbative physics around the MIT and for future developments of many-body algorithms applicable in this regime.
We investigate the influence of self-energy diagrams beyond the two-particle vertex level within dual fermion theory. Specifically, we calculate the local three-particle vertex and construct from it ...selected dual fermion self-energy corrections to dynamical mean field theory. For the two-dimensional Hubbard model, the thus obtained self-energy corrections are small in the parameter space where dual fermion corrections based on the two-particle vertex only are small. However, in other parts of the parameter space, they are of a similar magnitude and qualitatively different from standard dual fermion theory. The high-frequency behaviour of the self-energy correction is - surprisingly - even dominated by corrections stemming from the three-particle vertex.
