Pyqcm: An open-source Python library for quantum cluster methods

Pyqcm is a Python/C++ library that implements a few quantum cluster methods with an exact diagonalization impurity solver. Quantum cluster methods are used in the study of strongly correlated electrons to provide an approximate solution to Hubbard-like models. The methods covered by this library are Cluster Perturbation Theory (CPT), the Variational Cluster Approach (VCA) and Cellular (or Cluster) Dynamical Mean Field Theory (CDMFT). The impurity solver (the technique used to compute the cluster’s interacting Green function) is exact diagonalization from sparse matrices, using the Lanczos algorithm and variants thereof. The core library is written in C++ for performance, but the interface is in Python, for ease of use and inter-operability with the numerical Python ecosystem. The library is distributed under the GPL license.

We investigate the electronic structure of the strongly anisotropic, quasi-low-dimensional purple bronze Li${}_{0.9}$Mo${}_{6}$O${}_{17}$. Building on all-electron ab initio band-structure calculations, we obtain an effective model in terms of four maximally localized Wannier orbitals, which turn out to be far from atomiclike. We find two half-filled orbitals arranged in chains running along one crystallographic direction and two full orbitals in perpendicular directions, respectively. The possibility to reduce this model to only two orbitals forming two chains per unit cell with interchain coupling is discussed. Transport properties of these models show high anisotropy, reproducing trends of the experimentally determined values for the dc conductivity. We also consider basic effects of electron-electron interactions using the (extended) variational cluster approach and dynamical mean field theory. We find good agreement with experimental photoemission data upon adding moderate onsite interaction of the order of the bandwidth to the ab initio derived tight-binding Hamiltonian. The obtained models provide a profound basis for further investigations on low-energy Luttinger-liquid properties or to study electronic correlations within computational many-body theory.

The cluster perturbation theory is presented in the 2D Hubbard model constructed using X operators in the Hubbard-I approximation. The short-range magnetic order is taken into account by dividing the entire lattice into individual 2 × 2 clusters and solving the eigenvalue problem in an individual cluster using exact diagonalization taking into account all excited levels. The case of half-filling taking into account jumps between nearest neighbors is considered. As a result of numerical solution, a shadow zone is discovered in the quasiparticle spectrum. It is also found that a gap in the density of states in the quasiparticle spectrum at zero temperature exists for indefinitely small values of Coulomb repulsion parameter U and increases with this parameter. It is found that the presence of this gap in the spectrum is due to the formation of a short-range antiferromagnetic order. An analysis of the temperature evolution of the density of states shows that the metal-insulator transition occurs continuously. The existence of two characteristic energy scales at finite temperatures is demonstrated, the larger scale is associated with the formation of a pseudogap in the vicinity of the Fermi level, and the smaller scale is associated with the metal-insulator transition temperature. A peak in the density of states at the Fermi level, which is predicted in the dynamic mean field theory in the vicinity of the metal-insulator transition, is not observed.

We describe the use of coupled-cluster theory as an impurity solver in dynamical mean-field theory (DMFT) and its cluster extensions. We present numerical results at the level of coupled-cluster theory with single and double excitations (CCSD) for the density of states and self-energies of cluster impurity problems in the one- and two-dimensional Hubbard models. Comparison to exact diagonalization shows that CCSD produces accurate density of states and self-energies at a variety of values of $U/t$ and filling fractions. However, the low cost allows for the use of many bath sites, which we define by a discretization of the hybridization directly on the real frequency axis. We observe convergence of dynamical quantities using approximately 30 bath sites per impurity site, with our largest 4-site cluster DMFT calculation using 120 bath sites. We suggest coupled cluster impurity solvers will be attractive in ab initio formulations of dynamical mean-field theory.

The dynamical mean field theory (DMFT), which is successful in the study of strongly correlated fermions, was recently extended to boson systems [Phys. Rev. B {\textbf 77}, 235106 (2008)]. In this paper, we employ the bosonic DMFT to study the Bose-Hubbard model which describes on-site interacting bosons in a lattice. Using exact diagonalization as the impurity solver, we get the DMFT solutions for the Green's function, the occupation density, as well as the condensate fraction on a Bethe lattice. Various phases are identified: the Mott insulator, the Bose-Einstein condensed (BEC) phase, and the normal phase. At finite temperatures, we obtain the crossover between the Mott-like regime and the normal phase, as well as the BEC-to-normal phase transition. Phase diagrams on the $\mu/U-\tilde{t}/U$ plane and on the $T/U-\tilde{t}/U$ plane are produced ($\tilde{t}$ is the scaled hopping amplitude). We compare our results with the previous ones, and discuss the implication of these results to experiments.

A cellular dynamical mean field theory approach to Mottness

We investigate the proximity effect in a heterostructure of the topological insulator (TI) \BiSe\ deposited on the HTSC material BSCCO. The latter is described by the one-band Hubbard model and is treated with cluster dynamical mean field theory (CDMFT), the TI layers being included via the CDMFT self-consistency loop. The penetration of superconductivity into the TI depends on the position of the Fermi level with respect to the TI gap. We illustrate the back action of the TI layer on the HTSC layer, in particular the gradual disappearance of Mott physics with increasing tunneling amplitude.

We present a new methodology to solve the Anderson impurity model, in the\ncontext of dynamical mean-field theory, based on the exact diagonalization\nmethod. We propose a strategy to effectively refine the exact diagonalization\nsolver by combining a finite-temperature Lanczos algorithm with an adapted\nversion of the cluster perturbation theory. We show that the augmented\ndiagonalization yields an improved accuracy in the description of the spectral\nfunction of the single-band Hubbard model and is a reliable approach for a full\nd-orbital manifold calculation.\n

To analyze the physical properties arising from indirect magnetic exchange between several magnetic adatoms and between complex magnetic nanostructures on metallic surfaces, the real-space extension of dynamical mean-field theory (R-DMFT) appears attractive as it can be applied to systems of almost arbitrary geometry and complexity. While R-DMFT describes the Kondo effect of a single adatom exactly, indirect magnetic (RKKY) exchange is taken into account on an approximate level only. Here, we consider a simplified model system consisting of two magnetic Hubbard sites ("adatoms") hybridizing with a non-interacting tight-binding chain ("substrate surface"). This two-impurity Anderson model incorporates the competition between the Kondo effect and indirect exchange but is amenable to an exact numerical solution via the density-matrix renormalization group (DMRG). The particle-hole symmetric model at half-filling and zero temperature is used to benchmark R-DMFT results for the magnetic coupling between the two adatoms and for the magnetic properties induced in the substrate. In particular, the dependence of the local adatom and the nonlocal adatom-adatom static susceptibilities as well as the magnetic response of the substrate on the distance between the adatoms and on the strength of their coupling with the substrate is studied. We find both, excellent agreement with the DMRG data even on subtle details of the competition between RKKY exchange and the Kondo effect but also complete failure of the R-DMFT, depending on the parameter regime considered. R-DMFT calculations are performed using the Lanczos method as impurity solver. With the real-space extension of the two-site DMFT, we also benchmark a simplified R-DMFT variant.

We study the entanglement spectrum of the Hubbard model at half filling on a kagome lattice. The entanglement spectrum is defined by the set of eigenvalues of a reduced thermal density matrix, which is naturally obtained in the framework of the dynamical mean-field theory. Adopting the cluster dynamical mean-field theory combined with continuous-time auxiliary-field Monte Carlo method, we calculate the entanglement spectrum for a three-site triangular cluster in the kagome Hubbard model. We find that the results at the three-particle sector well capture the qualitative nature of the system. In particular, the eigenvalue of the reduced density matrix, corresponding to the chiral degrees of freedom, exhibits a characteristic temperature scale Tchiral, below which a metallic state with large quasiparticle mass is stabilized. The entanglement spectra at different particle number sectors also exhibit characteristic changes around Tchiral, implying the development of inter-triangular ferromagnetic correlations in the correlated metallic regime.

formula omitted]QIST v0.7: An open source continuous-time quantum Monte Carlo impurity solver toolkit

We analyze cellular dynamical mean-field theory (CDMFT) and the dynamical cluster approximation (DCA). We derive exact sum-rules for the hybridization functions and give examples for DMFT, CDMFT, and DCA. For impurity solvers based on a Hamiltonian, these sum-rules can be used to monitor convergence of the bath-parametrization. We further discuss how the symmetry of the cluster naturally leads to a decomposition of the bath Green matrix into irreducible components, which can be parametrized independently, and give an explicit recipe for finding the optimal bath-parametrization. As a benchmark we revisit the one-dimensional Hubbard model. We carefully analyze the evolution of the density as a function of chemical potential and find that, close to the Mott transition, convergence with cluster size is unexpectedly slow. In two dimensions we find, that we need so many bath-sites to obtain a reliable parametrization that Lanczos calculations are hardly feasible with current computers. For such large baths our symmetry-adapted approach should prove crucial for finding a reliable bath-parametrization.

We numerically study optical conductivity $\\sigma (\\omega )$ near the\n"antiferromagnetic" phase transition in the square-lattice Hubbard model at\nhalf filling. We use a cluster dynamical mean field theory and calculate\nconductivity including vertex corrections, and to this end, we have\nreformulated the vertex corrections in the antiferromagnetic phase. We find\nthat the vertex corrections change various important details in temperature-\nand $\\omega$-dependencies of conductivity in the square lattice, and this\ncontrasts sharply the case of the Mott transition in the frustrated triangular\nlattice. Generally, the vertex corrections enhance variations in the\n$\\omega$-dependence, and sharpen the Drude peak and a high-$\\omega$ incoherent\npeak in the paramagnetic phase. They also enhance the dip in $\\sigma (\\omega )$\nat $\\omega$=0 in the antiferromagnetic phase. Therefore, the dc conductivity is\nenhanced in the paramagnetic phase and suppressed in the antiferromagnetic\nphase, but this change occurs slightly below the transition temperature. We\nalso find a temperature region above the transition temperature in which the dc\nconductivity shows an insulating behavior but $\\sigma (\\omega )$ retains the\nDrude peak, and this region is stabilized by the vertex corrections. We also\ninvestigate which fluctuations are important in the vertex corrections and\nanalyze momentum dependence of the vertex function in detail.\n

High-temperature superconductivity emerges in the CuO$_2$ plane upon doping a Mott insulator. To ascertain the influence of Mott physics plus short-range correlations, we solve a three-band copper-oxide model in the charge-transfer regime using cellular dynamical mean-field theory with continuous-time quantum Monte Carlo as an impurity solver. We report the normal and superconducting phase diagram of this model as a function of doping, interaction strength and temperature. Upon hole doping of the charge-transfer insulator, the phase boundary between pseudogap and correlated metal consists of a first-order transition line at finite doping ending at a critical point, as in the one-band model. Beyond the endpoint, the phase boundary continues as a Widom crossover line, across which thermodynamic quantities peak. This phase boundary determines changes in the pairing mechanism and is an emergent phenomenon characteristic of doped Mott insulators, independent of many microscopic details. Broader implications are discussed.

An intricate interplay between superconductivity, pseudogap, and Mott transition, either bandwidth driven or doping driven, occurs in materials. Layered organic conductors and cuprates offer two prime examples. We provide a unified perspective of this interplay in the two-dimensional Hubbard model within cellular dynamical mean-field theory on a 2×2 plaquette and using the continuous-time quantum Monte Carlo method as impurity solver. Both at half filling and at finite doping, the metallic normal state close to the Mott insulator is unstable to d-wave superconductivity. Superconductivity can destroy the first-order transition that separates the pseudogap phase from the overdoped metal, yet that normal state transition leaves its marks on the dynamic properties of the superconducting phase. For example, as a function of doping one finds a rapid change in the particle-hole asymmetry of the superconducting density of states. In the doped Mott insulator, the dynamical mean-field superconducting transition temperature T(c)(d) does not scale with the order parameter when there is a normal-state pseudogap. T(c)(d) corresponds to the local pair formation temperature observed in tunneling experiments and is distinct from the pseudogap temperature.