Session T21: Numerical Methods for Correlated Electrons

Quantum quenches in 2D via arrays of coupled chains

Matrix product state (MPS) methods are extremely powerful when applied to strongly correlated systems in 1D. However they are less efficacious in 2D due to the `area law' growth of entanglement, limiting the system sizes that can be studied. We combine MPS methods with analytical results for integrable chains to build an algorithm that can study large (anisotropic) 2D many body quantum systems, because it reduces the need for a large `area'. As an example we describe the application of our method to quantum quenches of the $2+1$ dimensional quantum Ising model.   

Exact transient dynamics of the Anderson impurity

We study dynamics of a single Anderson impurity model subject to voltage and thermal quenches. We develop a hybridization expansion diagrammatic Monte Carlo algorithm to describe the exact dynamics of the problem. By including the initial correlations into the problem we describe the destruction of the Kondo state and infer the characteristic time scales of the problem. An interplay between different time scales of spin and charge excitations of the strongly correlated setup as visible in the ultra-fast response is in the focus of our study. We compare our results with frequently employed non-crossing and one-crossing approximations.   

Time Evolution of a Quantum Impurity System following a Sudden Quench and General Pulse at Finite Temperatures - a td-NRG Study

To study the time evolution of an observable of a quantum impurity system after a sudden quench at an arbitrary temperature, we apply the recently developed time-dependent numerical renormalization group approach (td-NRG) to the Anderson model and resonant level model.\footnote{H. T. M. Nghiem and T. A. Costi, Phys. Rev. B \textbf{89}, 075118 (2014)} In the application to the Anderson model, we quantify the results in the short and long time limits by comparing them to the thermodynamic values in the initial and final states. In the case of the resonant level model, we compare the time evolution of the local occupancy calculated by td-NRG to the exact analytic result. We also present the time evolution in response to general continuous pulses, acting in a finite time interval, and in response to periodic driving fields.\footnote{H. T. M. Nghiem and T. A. Costi, Phys. Rev. B \textbf{90}, 035129 (2014)} The study is accomplished by a generalization of the single-quench formalism to multiple quenches and by approximating smooth pulses (or periodic trains of pulses) by a sufficient number of smaller quenches.   

Non-equilibrium Hybridization Expansion Impurity-solver

The study of non-equilibrium phenomena in strongly correlated systems has developed into one of the most active and exciting branches of condensed matter physics. Meanwhile, quantum impurity models play a prominent role as mathematical representations of quantum dots, single-molecule devices, and effective models for the dynamical mean field theory. We show results for a generalization of the hybridization expansion diagrammatic Monte Carlo technique for the Anderson impurity model. And we perform non-equilibrium calculations on the full Keldysh contour, where a dynamical sign problem vastly increases the complexity of real-time simulation. By further combining this method with a non-crossing approximation, our ``bold-line'' Monte Carlo can reach substantially longer times out of equilibrium than previously accessible, and provides an accurate description of quench and driven dynamics of correlated systems.   

Electric-field-driven resistive switching in dissipative Hubbard model

Understanding of solids driven out of equilibrium by external fields has been one of the central goals in condensed matter physics for the past century and is relevant to nanotechnology applications such as resistive transitions. We study how strongly correlated electrons on a dissipative lattice evolve from equilibrium when driven by a constant electric field, focusing on the extent of the linear regime and hysteretic non-linear effects at higher fields. We access the non-equilibrium steady states\footnote{J. E. Han and Jiajun Li, Phys. Rev. B \textbf{88}, 075113 (2013)}, non-perturbatively in both the field and the electronic interactions, by means of a non-equilibrium dynamical mean-field theory in the Coulomb gauge. The linear response regime is limited by Joule heating effects and breaks down at fields orders of magnitude smaller than the quasi-particle energy scale. For large electronic interactions, strong but experimentally accessible electric fields can induce a resistive switching by driving the strongly correlated metal into a Mott insulator. Hysteretic $I$-$V$ curves suggest that the non-equilibrium current is carried through a spatially inhomogeneous metal-insulator mixed state.   

The Actinide Transition Revisited by Gutzwiller Approximation

We revisit the problem of the actinide transition using the Gutzwiller approximation (GA) in combination with the local density approximation (LDA). In particular, we compute the equilibrium volumes of the actinide series and reproduce the abrupt change of density found experimentally near plutonium as a function of the atomic number. We discuss how this behavior relates with the electron correlations in the $5f$ states, the lattice structure, and the spin-orbit interaction. Our results are in good agreement with the experiments.   

Magnetic formfactor and dynamic magnetic susceptibility within DMFT for $\alpha - \gamma$ transition in Cerium and $\delta$- Plutonium

Using LDA+DMFT we study the magnetic properties of the isostructural volume collapse transition between $\alpha$ and $\gamma$ Cerium. We compute the magnetic formfactor F(q), and show that it is very close to free ion behavior in both the local moment $\gamma$ phase as well as the more itinerant $\alpha$ phase, in excellent agreement with neutron scattering experiments. In sharp contrast, the dynamic local magnetic susceptibility $\chi_{loc}(\omega)$ of the two phases is strikingly different. In the $\gamma$ phase, the spectra is dominated by the sharp low energy peak due to local moment formation,whereas in the $\alpha$ phase we see two broad peaks, the first due to Kondo screening and the second due to Hund's coupling. We also calculate the magnetic spectral function $S(q,\omega)$ where we achieve excellent agreement with experiment. This shows that hybridization plays a central role in the $\alpha$ - $\gamma$ transition in cerium, and that the 4f electrons are strongly correlated in both phases. We also study the magnetic properties of $\delta$-Plutonium where our results give us important clues about the magnetic excitations of the system.   

Geometry Dependence of the Sign Problem

The sign problem is a fundamental limitation to Quantum Monte Carlo (QMC) simulations of the statistical mechanics of interacting fermions and frustrated quantum spins. We produced a comprehensive dataset on the geometry dependence of the sign problem for different densities, interaction strengths, inverse temperatures and spatial lattice sizes. We supplement this data with several observations concerning general patterns/trends in the data, including the dependence on spatial volume and how this can be probed by examining decoupled clusters, the scaling of the sign in the vicinity of a particle-hole symmetric point, and the correlation between the total sign and the signs of the individual spin up and spin down components.   

Quasi-continuous-time impurity solver for the cluster dynamical mean-field theory with linear scaling in the inverse temperature

We present an extension to the dynamical cluster approximation (DCA) of an recently developed unbiased quantum Monte Carlo (QMC) impurity solver for single-site DMFT [1]. The novel algorithm is based on a multigrid version of BSS-QMC [2,3], which yields Green functions free of significant Trotter errors, and scales linearly with the inverse temperature $\beta = 1/T$ and cubically in the cluster size $N$. We use the superior scaling to explore ultra-low temperature regimes at moderate cluster sizes, not reachable with state-of-the-art continuous time QMC impurity solvers that scale cubically in $\beta$. Benchmark results for the the two-dimensional (2d) Hubbard model, compared with complementary methods (unbiased lattice QMC, dynamical vertex approximation (D$\Gamma$A) [4]), are presented as well as a study of the 2d doped Kondo lattice model. \\[4pt] [1] D. Rost, F. Assaad and N. Bl\"{u}mer, PRE 87, 053305 (2013).\\[0pt] [2] R. Blankenbecler, D. Scalapino, R. Sugar, PRD 24, 2278 (1981). \\[0pt] [3] E. Khatami, C. Lee, Z. Bai, R. Scalettar, and M. Jarrell, PRE 81, 056703 (2010).\\[0pt] [4] T. Sch\"{a}fer, F. Geles, D. Rost, G. Rohringer, E. Arrigoni, K. Held, N. Bl\"{u}mer, M. Aichhorn, A. Toschi, arXiv:1405.7250 (2014).   

The density matrix renormalization group as a solver for cluster perturbation theory

Cluster Perturbation Theory (CPT) provides an approximation for the single particle Green's function of strongly correlated models in the thermodynamic limit by coupling clusters of small size using a variation of strong coupling perturbation theory. The method itself cannot account for the effects of symmetry breaking, such as in the presence of antiferromagnetic long range order, since it relies on the exact solution of clusters that are too small. The DMRG method provides a path toward a more reliable application of the CPT in the presence of long range order since it is able to calculate the single particle Green's function of an infinite (very large) one dimensional chain, or ladder. By coupling these chains and ladders in the perpendicular direction using CPT we recover the spectral functions of the 2D lattice in the thermodynamic limit. A remarkable advantage of this approach is that unlike small clusters, the one-dimensional systems are already \textbraceleft $\backslash $it infinite\textbraceright . We can study the effects of the onset of long range order and its spectral signatures by extending our study to multi-leg ladders..   

Typical medium dynamical cluster approximation applied to Migdal-Eliashberg theory

We use the recently developed typical medium dynamical cluster approximation (TMDCA) to study Anderson localization and the superconductor-insulator transition. In our analysis both phonons and disorder are treated on equal footing. For phonons we use the Holstein model Hamiltonian and perform analysis for different types of disorder distributions, i.e. binary or box distribution. It is of interest to see how phonons and disorder compete in fine-tuning of this phase transition by re-normalizing the gap parameter. For weak disorder we find that the size of the gap depends on the phonon frequency. Since for large phonon frequencies the Holstein model maps onto an attractive Hubbard model, we focus on the region where the phonon frequency is small and intermediate for both weak and strong disorders.   

Correlations in lacunar spinels: dynamical mean-field study with configuration interaction based impurity solver

Density functional plus dynamical mean field methods are used to study the role of correlations in in lacunar spinel compounds GaM$_4$X$_8$ (M=Nb, Mo, Ta and and X=S, Se and Te) to investigate the interplay of correlations and topology in materials with strong spin-orbit coupling. A novel configuration-interaction exact diagonalization solver enables inclusion of more bath orbitals, enabling a better treatment of spectral functions and more accurate computations of phase boundaries. Focussing on GaTa$_4$Se$_8$, we discuss how the correlation induces metal-insulator transition in presence of the spin-orbit coupling, based on spectral functions and optical conductivities.