Dynamical Mean Field Theory with the Density Matrix Renormalization Group

The metal–insulator transition in the paramagnetic Hubbard Model

Metal-insulator transition in correlated systems: A new numerical approach

With the development of computing technology, numerical exact diagonalization method plays a vital role in modern computational condensed matter physics, especially in the research area of strongly correlated electron systems: it becomes a benchmark for other numerical computational techniques, such as quantum Monte Carlo, numerical renormalization group, density matrix renormalization group, and dynamic mean field theory. In this paper, we first numerically exactly diagonalize the three-dimensional hydrogen atom with the combination of finite-difference method, and find that the numerical wave function of ground state is in good agreement with the analytical calculations. We then turn to discuss the space dimension confinement hydrogen system, two-dimensional hydrogen atom, and notice that the numerical wave function is no longer in agreement with the analytical calculation, where the ground state wave function has a numerical singularity as radius approaches to zero. Compared with the case of the three-dimensional hydrogen atom, this issue mainly comes from the nature of space dimension confinement. To resolve such an issue of numerical singularity in two-dimensional hydrogen atom, we need to construct a new discrete and normalized Bessel function as a basis to study the ground state behavior of dimension confinement system based on the framework of Lanczos-type numerical exact diagonalization. The constructed normalized Bessel basis is orthogonal and discrete, and thus becomes suitable for practical calculation. Besides, these prominent properties of such a Bessel basis greatly reduce the complexity and difficulty in practical calculation, and thus makes computing work efficient. In addition, Lanczos-type numerical exact diagonalization method can extremely speed up the process of solving the eigenvalue equation. As a result, such a high efficient calculation of our method demonstrates the consistence between numerical and analytical ground state energy value, and the corresponding wave function with enough truncated basis number. Since this kind of numerical singularity occurs in many space dimension confinement systems, our finding for constructing a new discrete Bessel basis function may be helpful in studying the quantum systems with numerical singularity behaviors in wavefunctions in future. On the other hand, it should be pointed out that the Bessel basis is incorporated into the linear augment plane wave method in the density functional theory to study the electronic band structure of the condensed material and obtain high accurate results, especially in the theoretical prediction of topological insulators and in experimental realization as well.

We study the ground state of the one-dimensional extended Hubbard model at half-filling using the entanglement entropy calculated by Density Matrix Renormalization Group (DMRG) techniques. We apply a novel curve fitting and scaling method to accurately identify a $2^{nd}$ order critical point as well as a Berezinskii-Kosterlitz-Thouless (BKT) critical point. Using open boundary conditions and medium-sized lattices with very small truncation errors, we are able to achieve similar accuracy to previous authors. We also report observations of finite-size and boundary effects that can be remedied with careful pinning.

The interplay of disorder and interactions is a subject of perennial interest. In this work, we have investigated the effect of disorder due to chemical substitution on the dynamics and transport properties of correlated Fermi liquids. A low-frequency analysis in the concentrated and dilute limits shows that the dynamical local potentials arising through disorder averaging generate a linear (in frequency) term in the scattering rate. Such non-Fermi-liquid behavior (nFL) is investigated in detail for Kondo hole substitution in heavy fermions within dynamical mean field theory. We find closed-form expressions for the dependence of the static and linear terms in the scattering rate on substitutional disorder and model parameters. We argue that the low-temperature resistivity will acquire a linear in temperature term, and show that the Drude peak structure in the optical conductivity will disappear beyond a certain disorder ${p}_{c}$, that marks the crossover from lattice coherent to single-impurity behavior. A full numerical solution of the dynamical mean field theory equations reveals that the nFL term will show up significantly only in certain regimes, although it is present for any nonzero disorder concentration in principle. We highlight the dramatic changes that occur in the quasiparticle scattering rate in the proximity of ${p}_{c}$. Remarkably, we find that the nFL behavior due to dynamical effects of impurity scattering has features that are distinct from those arising through Griffiths singularities or distribution of Kondo scales. Relevance of our findings to experiments on alloyed correlated systems is pointed out.

The Density Matrix Renormalization Group (DMRG) method with periodic boundary conditions is introduced for two dimensional classical spin models. It is shown that this method is more suitable for derivation of the properties of infinite 2D systems than the DMRG with open boundary conditions despite the latter describes much better strips of finite width. For calculation at criticality, phenomenological renormalization at finite strips is used together with a criterion for optimum strip width for a given order of approximation. For this width the critical temperature of 2D Ising model is estimated with seven-digit accuracy for not too large order of approximation. Similar precision is reached for critical indices. These results exceed the accuracy of similar calculations for DMRG with open boundary conditions by several orders of magnitude.

Hybrid theory of the dynamical mean field and the spin fluctuations in strongly correlated electron systems

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

In this work, we study the magnetic phases of a spatially modulated chain of spin-1 Rydberg excitons. Using the Density Matrix Renormalization Group (DMRG) technique, we study various magnetic and topologically nontrivial phases using both single-particle properties, such as local magnetization and quantum entropy, and many-body ones, such as pair-wise Néel and long-range string correlations. In particular, we investigate the emergence and robustness of the Haldane phase, a topological phase of anti-ferromagnetic spin-1 chains. Furthermore, we devise a hybrid quantum algorithm employing restricted Boltzmann machine to simulate the ground state of such a system that shows very good agreement with the results of exact diagonalization and DMRG.

We compute the spectral functions for the two-site dynamical cluster theory and for the two-orbital dynamical mean-field theory in the density-matrix renormalization group (DMRG) framework using Chebyshev expansions represented with matrix product states (MPS). We obtain quantitatively precise results at modest computational effort through technical improvements regarding the truncation scheme and the Chebyshev rescaling procedure. We furthermore establish the relation of the Chebyshev iteration to real-time evolution, and discuss technical aspects as computation time and implementation in detail.

Since its inception, the DMRG method has been a very powerful tool for the calculation of physical properties of low-dimensional strongly correlated systems. It has been adapted to obtain dynamical properties and to consider finite temperature, time-dependent problems, bosonic degrees of freedom, the treatment of classical problems and non-equilibrium systems, among others. We will briefly review the method and then concentrate on its latest developments, describing some recent successful applications. In particular we will show how the dynamical DMRG can be used together with the Dynamical Mean Field Theory (DMFT) to solve the associated impurity problem in the infinite-dimensional Hubbard model. This method is used to obtain spectral properties of strongly correlated systems. With this algorithm, more complex problems having a larger number of degrees of freedom can be considered and finite-size effects can be minimized.

The metal-insulator transition (MIT) remains among the most thoroughly studied phenomena in solid state physics, but the complexity of the phenomena, which usually involves cooperation of many degrees of freedom including orbitals, fluctuating local moments, magnetism, and the crystal structure, have resisted predictive ab-initio treatment. Here we develop ab-initio theoretical method for correlated electron materials, based on Dynamical Mean Field Theory, which can predict the change of the crystal structure across the MIT at finite temperature. This allows us to study the coupling between electronic, magnetic and orbital degrees of freedom with the crystal structure across the MIT in rare-earth nickelates. We predict the electronic free energy profile of the competing states, and the theoretical magnetic ground state configuration, which is in agreement with neutron scattering data, but is different from the magnetic models proposed before. The resonant elastic X-ray response at the K-edge, which was argued to be a probe of the charge order, is theoretically modelled within the Dynamical Mean Field Theory, including the core-hole interaction. We show that the line-shape of the measured resonant elastic X-ray response can be explained with the “site-selective” Mott scenario without real charge order on Ni sites.

The heavy fermion compound CeRu 4 Sn 6 was recently shown to exhibit a spontaneous nonlinear Hall effect, indicating its topological nature. This is consistent with the lack of inversion symmetry that allows for the existence of Weyl nodes. Here, we employ density functional theory combined with dynamical mean field theory, which is state of the art for strongly correlated materials, and study the topology of CeRu 4 Sn 6 . We find five inequivalent Weyl nodes of either type I or II, each having either 8 or 16 symmetry-related replicas. These Weyl nodes bridge the Kondo insulating gap, which is a direct but not an indirect gap. The Weyl points closest to the Fermi level are situated only 0.5 meV below it and have a very flat dispersion. Our results establish CeRu 4 Sn 6 as a model system for investigating the interplay between strong electronic correlations and nontrivial topology. These findings provide a theoretical foundation for future studies of quantum transport and interaction-driven topological phases in heavy fermion systems.

High-temperature bad-metal transport has been recently studied both theoretically and in experiments as one of the key signatures of strong electronic correlations. Here we use the dynamical mean field theory and its cluster extensions, as well as the finite-temperature Lanczos method to explore the influence of lattice frustration on the thermodynamic and transport properties of the Hubbard model at high temperatures. We consider the triangular and the square lattices at half-filling and at 15% hole doping. We find that for $T\ensuremath{\gtrsim}1.5t$ the self-energy becomes practically local, while the finite-size effects become small at lattice size $4\ifmmode\times\else\texttimes\fi{}4$ for both lattice types and doping levels. The vertex corrections to optical conductivity, which are significant on the square lattice even at high temperatures, contribute less on the triangular lattice. We find approximately linear temperature dependence of dc resistivity in doped Mott insulator for both types of lattices.

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.