Session D16: Strongly Correlated Numerics and Theory

Local and non-local correlations in nanoscopic systems

Tools for reliably treating nanoscopic systems, like coupled quantum-dots, ad-atoms on surfaces, macromolecules, etc., in the presence of electronic correlations are either missing or prohibitively expensive. We have implemented a new computational scheme based on a self-consistently defined set of local problems [1]. Our method scales linearly with the number of sites and allows us to perform large-scale sign-problem free Quantum Monte-Carlo simulations. We have studied the behavior of a single-atom junction formed upon stretching a metallic wire and found that a metal-insulator crossover is induced when the wire is about to break up. The combination with ab-initio techniques allowed us to study size-dependent properties of Manganite nano-clusters [2]. The simplest implementation of our method includes only local self-energy effects. We recently went beyond this and applied the resulting more sophisticated version of our method to an exactly solvable model finding results in remarkable agreement with the exact solution. \\ \\ \noindent [1] A. Valli, G. Sangiovanni, O. Gunnarsson, A. Toschi and K. Held, PRL {\bf 104}, 246402 (2010) \\ \noindent [2] H. Das, G. Sangiovanni, A. Valli, K. Held and T. Saha-Dasgupta, PRL {\bf 107}, 197202 (2011)   

Quantum impurity solver based on truncated ED (RASCI) wave function expansion

Quantum impurity models appear in many applications, including nanoscience and the dynamical mean field approximation (DMFT). Many physically relevant impurity models are too large to be solved by exact diagonalization (ED), lack the interaction and hybridization structure required for quantum Monte Carlo (QMC) simulations, or suffer from a severe sign problem. We present an alternative impurity solver inspired by configuration interaction (RASCI) techniques of quantum chemistry and based on a controlled truncation of a wave function expansion. The method can access larger impurity models (impurities with 5 $d$-orbitals and 20 bath orbitals can be easily calculated on a single processor) than can ED and avoids the sign problems of QMC methods. The performance is demonstrated for a cluster DMFT approximation to the two dimensional Hubbard model and for the problem of a Co adatom on a Cu(111) surface.   

Modified Iterated perturbation theory in the strong coupling regime and its application to the 3d FCC lattice

The Dynamical Mean-Field theory(DMFT) approach to the Hubbard model requires a method to solve the problem of a quantum impurity in a bath of non-interacting electrons. Iterated Perturbation Theory(IPT)[1] has proven its effectiveness as a solver in many cases of interest. Based on general principles and on comparisons with an essentially exact Continuous-Time Quantum Monte Carlo (CTQMC)[2], here we show that the standard implementation of IPT fails when the interaction is much larger than the bandwidth. We propose a slight modification to the IPT algorithm by requiring that double occupancy calculated with IPT gives the correct value. We call this method IPT-$D$. We show how this approximate impurity solver compares with respect to CTQMC. We consider a face centered cubic lattice(FCC) in 3d for different physical properties. We also use IPT-$D$ to study the thermopower using two recently proposed approximations[3]$S^*$ and $S_{Kelvin}$ that do not require analytical continuation and show how thermopower is essentially the entropy per particle in the incoherent regime but not in the coherent one.[1]H.Kajueter et al. Phys. Rev. Lett. 77, 131(1996)[2]P. Werner, et al. Phys. Rev. Lett. 97, 076405(2006)[3]B.S. Sriram Shastry Rep. Prog. Phys. 72 016501(2009)   

Magnetic properties of the Hubbard model on the fcc lattice

As a possible model for ferromagnetism, we study the magnetic properties of the Hubbard model on an fcc lattice. Near-neighbor and next-near-neighbor hopping parameters are included to examine the effect of band structure. We use exact diagonalization and the Constraint Path Monte Carlo (CPMC) \footnote{S.~Zhang, J.~Carlson, and J.~Gubernatis, Phys.~Rev.~B {\bf 55}, 7464 (1997); C.-C.~Chang and S.~Zhang, Phys.~Rev.~B {\bf 78}, 165101 (2008).} methods. Several methodological improvements in CPMC, for example the release of the constraint, will be discussed. We present benchmark quality results on the paramagnetic ground state and partially polarized states, as a function of interaction strength. A magnetic phase diagram is obtained from our many-body calculations, and comparison will be made with results from Dynamical Mean Field theory \footnote{ M.~Ulmke, The Eur. Phys. J.~B.~{\bf 1}, 301 (1998) }.   

Dual fermion dynamical cluster approach for strongly correlated systems

A multi-scale many-body approach is developed for strongly-correlated electron systems by combining the dynamical cluster approximation (DCA) and the recently introduced dual fermion formalism. This approach systematically incorporates non-local corrections to the DCA by employing an exact mapping from a real lattice to a DCA cluster of linear size Lc embedded in a dual fermion lattice. The Green function in the dual space serves as a small parameter enabling the use of a diagrammatic perturbation calculation on the dual fermion lattice. For example, the dual fermion self-energy calculated with simple second-order perturbation theory scales as ${\cal{O}}(1/L_c^3)$. We demonstrate the effectiveness of the approach by applying it to the 2D Hubbard model.   

Extended Correlation in Strongly Correlated Systems, Beyond Dynamical Cluster Approximation

We present a new multi-scale approach for strongly correlated systems that combines the Dynamical Cluster Approximation and the recently introduced dual-fermion formalism. This approach employs an exact mapping from a real lattice to a DCA cluster of linear size $L_c$ embedded in a dual fermion lattice. The short-length-scale physics is addressed by DCA cluster calculations, while the longer-length-scale physics is addressed diagrammatically using dual fermions. The bare and dressed dual fermionic Green functions scale as ${\cal{O}}(1/L_c)$, so perturbation theory on the dual lattice converges very quickly. E.g., the dual Fermion self-energy calculated with simple second order perturbation theory is of order ${\cal{O}}(1/L_c^3)$, with third order and three-body corrections down by an additional factor of ${\cal{O}}(1/L_c)$.   

Extension of dual-fermion formalism towards disordered systems

To study the correlation effects in disordered materials, we extend the recently developed dual fermion approach [1] to include systems with disorder. In particular, we consider the effect of nonlocal disorder-induced correlations in the non-interacting Anderson model. Within this method, such nonlocal effects are included in a systematic way as an expansion to the coherent potential approximation (CPA). The ability to properly treat the nonlocal correlations and provide non-local corrections to the CPA, is crucial for the description of the electron localization. In our analysis, we consider the density of sates and localization effects and compare them with the existing results. [1] A.N. Rubtsov, et. al., Phys. Rev. B 79, 045133 (2009).   

Application of the Dual Fermion-Dynamical Cluster Approach to the 1D Falicov Kimball Model

The Falicov Kimball model is the simplest model for correlated electrons. It was introduced to study metal-insulator transitions. In one dimension, it is known to possess a charge density wave (CDW) instability at zero transition temperature ($T_c$). However, finite cluster methods like Dynamical Mean Field Theory (DMFT), Dynamical Cluster Approximation (DCA), Cellular Dynamical Mean Field Theory (CDMFT) , etc. show finite temperature CDW transition. In this paper, we study the model using the recently developed Dual Fermion-Dynamical Cluster approach that takes into account large length scale correlations through the auxiliary particles known as dual Fermions. We find that $T_c$ obtained from this method is lower than that obtained from the cluster methods. In particular, we study the scaling behavior of $T_c$ with the linear cluster size and also the scaling of other one-particle and two-particle quantities near the criticality.   

Phase Separation and Charge-Ordered Phases of the d = 3 Falicov-Kimball Model at T$>$0: Temperature-Density-Chemical Potential Global Phase Diagram from RG Theory

The global phase diagram of the spinless FK model in d=3 is obtained by renormalization-group theory, exhibiting 5 distinct phases. Four of these phases are charge-ordered (CO) phases, in which the system forms two sublattices with different electron densities. The CO phases occur near half filling of the conduction electrons, for the entire range of localized electron densities. Phase boundaries are second order, except for the intermediate and large interaction regimes, where a first-order phase boundary occurs in the central region of the phase diagram, resulting in phase coexistence near half filling of both localized and conduction electrons. These two-phase or three-phase coexistence regions are between different charge ordered phases, between charge-ordered and disordered phases, and between dense and dilute disordered phases. The second-order phase boundaries terminate on the first-order transitions via critical endpoints and double critical endpoints. The first-order phase boundary is delimited by critical points. The phase diagram cross-sections with respect to the chemical potentials and densities of the localized and conduction electrons, at all representative interaction strengths, hopping strengths, temperatures, are calculated and exhibit 10 distinct topologies.   

Collective Excitations and Stability of the Excitonic Phase in the Extended Falicov--Kimball Model

We consider the excitonic insulator state (often associated with electronic ferroelectricity), which arises on the phase diagram of an extended spinless Falicov--Kimball model (FKM) at half-filling. Within the Hartree--Fock approach, we calculate the spectrum of low-energy collective excitations in this state up to second order in the narrow-band hopping and/or hybridisation. This allows to probe the mean-field stability of the excitonic insulator. The latter is found to be unstable when the case of the pure FKM (no hybridisation with a fully localised band) is approached. The instability is due to the presence of another, lower-lying ground state and {\it not} to the degeneracy of the excitonic phase in the pure FKM. The excitonic phase, however, may be stabilised further away from the pure FKM limit. In this case, the low-energy excitation spectrum contains new information about the properties of the excitonic condensate (likely including the critical temperature).   

Propagation of entanglement in one-dimensional models of many-body localization

An important and still unanswered fundamental question is whether a closed quantum system of many interacting particles can be localized by disorder. The time evolution of an initially unentangled state is studied for a 1D random-field XXZ Hamiltonian. Even for weak interactions, when the system is thought to be in a many-body localized phase, entanglement shows neither localized nor diffusive behavior: interactions act as a singular perturbation on the truly localized state with no interactions. The logarithmic time dependence of entanglement observed rather suggests a broad range of time scales typical of glassy behavior.   

Fidelity Spectrum in Quantum Phase Transitions

A quantum phase transition (QPT) is incarnated by an abrupt change in the qualitative structure in the ground state wavefunction of a many-body system as the external driving parameter varies. The ground state fidelity, which is a measure of similarity between two states, is expected to show a sudden drop across the transition point and its possibility as a witness to QPTs has raised much interest in recent years. However, the ground state fidelity does not capture much information about the contribution of the low-lying excitations. In this presentation, we introduce the concept of fidelity spectrum, i.e. the matrix elements of $M=|\Psi(\lambda)\rangle\langle\Psi(\lambda+\delta\lambda)|$, where $\lambda$ is the external driving parameter and $\Psi(\lambda)$ is the wavefunction of the system at $\lambda$. By studying the fidelity spectrum, we hope to shed light on the role of excited states played in QPTs. We investigate the fidelity spectrum in two many-body systems, namely the one-dimensional transverse-field Ising model and the two-dimensional Kitaev model defined on a honeycomb lattice. We found that in different phases, as well as at the critical points, the fidelity spectrum shows significant different behaviors.   

Quantum Monte Carlo calculation of reduced density matrices

Quantum Monte Carlo(QMC) methods offer an efficient way to approximate the interacting ground state and some excited states of realistic model Hamiltonians based on the fundamental Coulomb interaction between electrons and nuclei. Many highly accurate results have been obtained using this method; however, it is often a challenge to extract the important correlations that the QMC wave function contains. I will describe some new results using the reduced density matrices(RDM's) to understand the electron correlation in the many-body wave function. The RDM's have both informative usage for describing correlation and pragmatic uses in further improving the variational wave function.