Session K20: Quantum Many-Body Systems and Methods I

Continuous-time auxiliary field~quantum Monte Carlo study of charge ordering in two-dimensional extended Hubbard model.

The competition between local and non-local long-range Coulomb repulsions in strongly interacting electron systems leads to emergence of complex charge ordering phases. To perform the quantum many-body simulations of such effects, we extend the~existing continuous-time auxiliary field quantum Monte Carlo method to incorporate the non-local density-density interactions.~We apply the developed method to the two-dimensional extended Hubbard model (with all range of non-local interactions) at~and away from the half-filling. We explore the properties of the model in different parameter regimes of short and long range interactions, temperature and fillings.   

Diagrammatic Monte Carlo for Dual Fermions

The dual fermion series is a diagrammatic extension of the dynamical mean field theory that includes non-local dynamic correlations. Evaluating this series analytically has proven to be challenging. In this talk we show results for a diagrammatic Monte Carlo method that stochastically samples two-particle vertex diagrams of the dual fermion perturbation series. We present an introduction to the method and show applications to correlated systems.   

The numerical renormalization group and multi-orbital impurity models

The numerical renormalization group (NRG) is a highly versatile and accurate method for the simulation of (effective) fermionic impurity models. Despite that the cost of NRG is exponential in the number of orbitals, by now, symmetric three-band calculations have become available on a routine level. [1] Here we present a recent detailed study on the spin-orbital separation in a three-band Hund metal with relevance for iron-pnictides via the dynamical mean field theory (DMFT). [2] In cases, finally, where the orbital symmetry is broken, we demonstrate that interleaved NRG [3] still offers an accurate alternative approach within the NRG with dramatically improved numerical efficiency at comparable accuracy relative to conventional NRG. \\[2ex] [1] Weichselbaum, Annals of Physics {\bf 327}, 2972 (2012) \\ [2] Stadler et al, PRL {\bf 115}, 136401 (2015) \\ [3] Mitchell et al, PRB {\bf 89}, 121105(R) (2014)   

DFT$+$DMFT calculation of band gaps for the transition metal monoxides NiO, CoO, FeO and MnO.

We report calculated spectral functions of the four late transition metal monoxides MnO, FeO, CoO and NiO in the paramagnetic phase. We used density functional theory (DFT) in combination with dynamic mean field theory (DMFT), which gives much better description of band gaps. Both projected Wannier orbitals and the on-site screened Coulomb interactions are obtained from DFT ground states to ensure consistency. Because of the p-d hybridization in these materials, we calculated Coulomb interactions for the dp model as well as the d-dp model using the cRPA method. With the standard fully localized limit double counting correction, we found that the d-dp model gives results in better agreement with experiments. This work was supported by the US Department of Energy (DOE), Office of Basic Energy Sciences (BES), under Contract No. DE-FG02-02ER45995.   

Probing many body effects using Fourier Transform Scanning Tunneling Spectroscopy: Can spin-orbit splitting in dispersion be observed in q-space?

Well studied surface systems such as Ag and Cu provide a safe platform to test novel spectroscopy methods that can have extended applications in near future. Our current focus is given to Fourier Transform Scanning Tunneling Spectroscopy (FT-STS) that allows us to study scattering effects (quasiparticle interactions - namely QPI) of CO and Co on Cu(111) surface. Magnetic Co adatoms are expected to generate a spin-orbit split in dispersion in QPI(q) space, the existence of which is confirmed by the k-space angle-resolved photo-emission spectroscopy (ARPES) of Cu(111) surface in the recent years. Hence the previously observed electron-phonon kink and spin-orbit splitting of the dispersion, as well as the scattering properties of CO molecules and Co adatoms, should also be observable in QPI space via FT-STS of Cu(111), and compatible with previous studies on similar systems. We are using a low temperature (4.2 K) commercial  Scanning Tunneling Microscope (CREATEC STM) that operates using Nanonis electronic controllers and software which allows us to perform FT-STS as well as topological imaging.   

Deconfined criticality in "easy-plane" SU($N$) anti-ferromagnets

Motivated by evidence for deconfined criticality in SU($N$) anti-ferromagnets, we investigate the phase diagram of these models in the case where the SU($N$) symmetry is reduced to rotations about the diagonal generators ("easy-plane" symmetry). We carry out extensive numerical simulations using quantum Monte Carlo, revealing a first-order magnetic to valence bond solid phase transition that becomes a continuous deconfined transition at large $N$. We support our numerical data by performing epsilon expansions of the easy-plane deformed $CP^{N-1}$ field theory near both the upper and lower critical dimensions. This renormalization group analysis shows that the symmetric deconfined fixed point is unstable in the presence of easy-plane anisotropy, resulting in a runaway flow for intermediate values of $N$ and a flow towards a stable easy-plane deconfined fixed point at large $N$, which is consistent with the critical behavior of our lattice models.   

Magnetic transitions and quantum criticality in the three-dimensional Hubbard model

We analyze the (quantum) critical properties of the simplest model for electronic correlations, the Hubbard model, in three spatial dimensions by means of the dynamical mean field theory (DMFT, including all local correlations) and the dynamical vertex approximation (D$\Gamma$A, including non-local correlations on all length scales). Both in the half-filled/unfrustrated [1] and in the hole-doped system [2] the transition temperature is significantly lowered by including non-local fluctuations.\\\\ In the latter case, however, the magnetic order becomes incommensurate, eventually leading to a complete suppression of the order and giving rise to a magnetic quantum critical point (QCP) at zero temperature [2]. We analyze the (quantum) critical properties of this QCP (e.g. critical exponents) and relate our findings to the standard theory of quantum criticality in metals, the Hertz-Millis-Moriya theory. \\\\ $[1]$ G. Rohringer, A. Toschi, A. A. Katanin, and K. Held, Phys. Rev. Lett. {\bf 107}, 256402 (2011).\\ $[2]$ T. Sch\"afer, A. A. Katanin, K. Held, and A. Toschi, in preparation.   

A Multiorbital DMFT Analysis of Electron-Hole Asymmetry in the Dynamic Hubbard Model

The dynamic Hubbard model (DHM) improves on the description of strongly correlated electron systems provided by the conventional single-band Hubbard model through additional electronic degrees of freedom, namely a second, higher energy orbital and associated hybridization parameters for interorbital transitions. The additional orbital in the DHM provides a more realistic modeling of electronic orbital "relaxation" in real lattices. One result of orbital relaxation is a clear electron-hole asymmetry, absent in the single-band case. We have employed the computational technique of dynamical mean field theory, generalized to the two-orbital case, to study this asymmetry with respect to varying system parameters, including both intersite and intrasite orbital hybridization as well as the role played by Mott physics. Our results stand in good agreement with previous exact diagonalization studies of the DHM.   

Interlaced coarse-graining for the dynamical cluster approximation

The negative sign problem remains a challenging limiting factor in quantum Monte Carlo simulations of strongly correlated fermionic many-body systems. The dynamical cluster approximation (DCA) makes this problem less severe by coarse-graining the momentum space to map the bulk lattice to a cluster embedded in a dynamical mean-field host. Here, we introduce a new form of an interlaced coarse-graining and compare it with the traditional coarse-graining. We show that it leads to more controlled results with weaker cluster shape and smoother cluster size dependence, which with increasing cluster size converge to the results obtained using the standard coarse-graining. In addition, the new coarse-graining reduces the severity of the fermionic sign problem. Therefore, it enables calculations on much larger clusters and can allow the evaluation of the exact infinite cluster size result via finite size scaling. To demonstrate this, we study the hole-doped two-dimensional Hubbard model and show that the interlaced coarse-graining in combination with the DCA$^+$ algorithm permits the determination of the superconducting $T_c$ on cluster sizes, for which the results can be fitted with the Kosterlitz-Thouless scaling law.   

Diagrammatic Monte Carlo sampling of the dual-fermion expansion for the Hubbard model

The dual-fermion approach provides a formally exact prescription for calculating the properties of a correlated electron system in terms of a diagrammatic expansion around dynamical mean-field theory (DMFT). The approach can address the full range of interactions, is exact in both the weak- and strong-coupling limits, and naturally incorporates long-range correlations beyond the reach of current cluster extensions to DMFT. Practical implementations have so far been limited to leading-order or ladder-type approximations to the expansion. In this work we compute the dual-fermion expansion for the Hubbard model to higher orders by means of a diagrammatic Monte Carlo algorithm which stochastically samples all diagram topologies. This approach allows a systematic check for the convergence of the series and hence provides a route towards a fully controlled treatment of correlated electrons.   

Hubbard operator density functional theory for Fermionic lattice models

We formulate an effective action as a functional of Hubbard operator densities whose stationary point delivers all local static information of the interacting lattice model. Using the variational principle, we get a self-consistent equation for Hubbard operator densities. The computational cost of our approach is set by diagonalizing the local Fock space. We apply our method to the one and two band Hubbard model (including crystal field and on-site exchange) in infinite dimensions where the exact solution is known. Excellent agreement is obtained for the one-band model. In the two-band model, good agreement is obtained in the metallic region of the phase diagram in addition to the metal-insulator transition. While our approach does not address frequency dependent observables, it has a negligible computational cost as compared to dynamical mean field theory and could be highly applicable in the context total energies of strongly correlated materials and molecules.   

Towards an ab-initio treatment of nonlocal electronic correlations with dynamical vertex approximation

Recently, approaches such as the dynamical vertex approximation (D$\Gamma$A) [1] or the dual-fermion method [2] have been developed. These diagrammatic approaches are going beyond dynamical mean field theory (DMFT) by including nonlocal electronic correlations on all length scales as well as the local DMFT correlations. Here we present our efforts to extend the D$\Gamma$A methodology to ab-initio materials calculations (ab-initio D$\Gamma$A) [3]. Our approach is a unifying framework which includes both GW and DMFT-type of diagrams, but also important nonlocal correlations beyond, e.g. nonlocal spin fluctuations. In our multi-band implementation we are using a worm sampling technique [4] within continuous-time quantum Monte Carlo in the hybridization expansion to obtain the DMFT vertex, from which we construct the reducible vertex function using the two particle-hole ladders. As a first application we show results for transition metal oxides. [1] A. Toschi, A. A. Katanin, and K. Held, Physical Review B 75, 045118 (2007). [2] A. N. Rubtsov, M. I. Katsnelson, A. I. Lichtenstein, Physical Review B 77, 033101 (2008). [3] A. Toschi et al., Annalen der Physik 523, 698 (2011) [4] P. Gunacker et al., Physical Review B 92, 155102 (2015)   

Extremely Correlated Fermi Liquid theory: Imposing the hole density sum-rule

The analytical theory of extremely strongly correlated Fermi liquids (ECFL) for the large U models, when applied to Cuprate superconductors in the nodal direction, provides ARPES spectral line shapes that are very close to experiments. Approximate lowest order calculations within this formalism also closely reproduce the spectral line shapes for the single impurity Anderson model found using the Numerical Renormalization Group. Similarly excellent comparison is possible with the Dynamical Mean Field Theory self energy for the Hubbard model in high dimensions. However these calculations yields too large an energy scale for frequency dependence, in the proximity of integer (or Mott) filling. We show that the theory permits the imposition of sum rules for hole density, rather than the electron density used earlier, on the Greens functions of the theory. The numerical results of these variants are presented, and compared to the earlier calculations. The new results go a long way towards resolving the energy scale problem, while retaining the excellence of line shapes.   

Space-time separation of electronic correlations

While second-order phase transitions always cause strong nonlocal fluctuations, their effect on spectral properties crucially depends on the dimensionality. First, we show that for the important case of three dimensions the electron self-energy is well separable into a local dynamical part and static nonlocal contributions$[1]$. In particular, using the dynamical vertex approximation for the doped 3D Hubbard model, we demonstrate that the quasiparticle weight remains essentially momentum independent, despite overall large nonlocal corrections to the self-energy when approaching the spin-ordered state. This generalizes earlier empirical findings of this property in the iron pnictides$[2]$ and transition metal oxides$[3]$ based on Hedin's {\it GW} approximation. With this insight, we here propose a "space-time-separated" scheme for many-body perturbation theory that is up to ten times more efficient than current implementations. Finally, we discuss limits of the space-time separation of correlation effects by studying the crossover from three to two dimensions.\\[0.25cm] $[1]$ T. Sch{\"a}fer, A. Toschi, JMT. PRB 91, 121107(R) (2015)\\ $[2]$ JMT, M. van Schilfgaarde, G. Kotliar. PRL 109, 237010 (2012)\\ $[3]$ JMT, M. Casula, T. Miyake, S. Biermann PRB 90, 165138 (2014)   

Computation of ab initio energy savings due to magnetic interactions

The double-exchange mechanism [1] is the traditional explanation for antiferromagnetic coupling between magnetic ions. In this theory, the energy savings within the context of a hopping model is derived from kinetic energy terms. However, the connection to ab initio energy savings to our knowledge has not been studied using an explicitly correlated theory that can obtain quantitative accuracy. Our study focuses on determining, from ab initio calculations, whether the origin of interactions in magnetic systems is explainable through conventional arguments. To this end, we investigate the contributions (kinetic, ionic, electron-electron) to the total energy of the (Mn-O-Mn)$^{\mathrm{+2}}$system using quantum Monte Carlo techniques. We will report on progress in elucidating the connection between the ab initio energy savings and the effective model energy savings that result in an antiferromagnetic interaction in this system. [1] Zener. Phys. Rev. , 403 (1951)