Session H13: Numerical Methods for Strongly Correlated System: Heisenberg and Density Matrix

Computational Phase Diagrams for Strongly Correlated Quantum Spins

In an extended system of strongly interacting quantum spins, a single spin flip is an example of a microscopic disturbance whose time-evolution is well behaved and given by Heisenberg's equation. At long times most of the disturbance decays exponentially, leaving behind a few excitations whose decay is slower than exponential. These have energies at which the excitation spectrum is singular, separating bands of qualitatively different excitations. We apply the recursion method [Solid State Physics 35, Academic Press, 2l5-94(1980)] to a generalization of Heisenberg's equation for the evolution of an appropriate microscopic disturbance. This produces a continued fraction whose essential singularities are the desired phase boundaries. Calculations for some Heisenberg spin Hamiltonians illustrate this approach.   

Iterative Diagonalization of Inhomogeneous Heisenberg Models

The antiferromagnetic Heisenberg model is one of the most important in describing quantum spins coupled by exchange interactions. Difficulties arise especially in presence of broken symmetry, due for instance to impurities or defects. In these cases, even well-established numerical methods like Lanczos or Monte Carlo, encounter limitations. We propose a numerical method which works even in the absence of translational invariance. We diagonalize the Heisenberg model exploiting the conservation of both the z-component of the total spin and the square of the total spin, a much more complicated procedure but that renders an additional block diagonalization. In essence, the N-site Hamiltonian is built using basis-vectors generated from the direct product of the eigenvectors of the (N-1)-site Hamiltonian and the states of the added N-th spin. The procedure is also applied for the two-leg ladder, an experimental relevant system. Results are shown for ground-state energy and temperature dependent specific heat for chains with local spin impurities or with random distributions of spins 1/2, 1 or 3/2.   

Entanglement perturbation theory for the excitation spectrum in one dimension

A novel many-body method, entanglement perturbation theory, is developed for the excitation spectra in one dimension. Applied to the antiferromagnetic Heisenberg chains with spin one-half and 1, converging and hence exact results are obtained, including known Bethe Ansatz result for spin one-half and DMRG results for spin 1. We have found that the magnons are spread over about 4 lattice sites. An essential ingredient in this theory is the exact, un-renormalized ground state of arbitrary system sizes, which are also calculated by EPT in a simple, general and exact manner.   

A Brief Introduction to the Truncated Eigenfermion Decomposition

We present a computational formalism for the approximate unitary transformation of a many-body fermion Hamiltonian with two-body interactions. This work is a further development of the numerical canonical transformation approach of S. R. White [J. Chem. Phys. 117, 7472 (2002)]. The Hamiltonian can be diagonalized in a basis of \textit{eigenfermion} operators, in which case the eigenstates are all single Slater determinants of eigenfermions. The transformation of two-body interactions generates higher-order interactions that can be approximated by effective two-body interactions using a novel generalization of normal ordering. The error in representating a target eigenstate is minimized by performing the generalized normal ordering with respect to that eigenstate. Numerical results are presented for several test cases, including Hubbard model clusters.   

The Density Matrix Renormalization Group Algorithm for Strongly Correlated Systems: A Generic Implementation

I will present DMRG++, a fully functional generic Density-Matrix Renormalization Group (DMRG) code with sample cases for the Hubbard and Heisenberg model, and for one-dimensional chains and n-leg ladders. My talk will include an overview of the core C++ classes, effective symmetry blocking and parallelization found in DMRG++. I will also explain how to add new strongly correlated electron (SCE) models and geometries with minimal code changes. Even if you are not very familiar with the DMRG or C++, you will be able to understand the main motivations and advantages of generic programming applied to SCE systems.   

A Renormalization Group for Treating 2D Coupled Arrays of Continuum 1D Systems

We study the spectrum of two dimensional coupled arrays of continuum one-dimensional systems by wedding a density matrix renormalization group (DMRG) procedure to a renormalization group improved truncated spectrum approach. To illustrate the methodology we study the spectrum of large arrays of coupled quantum Ising chains. We demonstrate explicitly that the method can treat the various regimes of chains, in particular the three dimensional Ising ordering transition the chains undergo as a function of interchain coupling. We argue that the methodology's success is based on finite size corrections being exponentially small which in turn allows the block DMRG entanglement entropy to be kept to a minimum.   

Low-temperature density matrix renormalization group using regulated polynomial expansion

We propose a new scheme of density matrix renormalization group (DMRG) for low dimensional strongly correlated electron systems at finite temperatures, which is a straightforward extension of the target-state procedure at zero temperature. In order to investigate thermodynamical properties, we employ the target state that is weighted by a Boltzmann factor [1]. Making use of a regulated polynomial expansion [2] and random sampling, we can calculate static and dynamical quantities at finite temperatures. In order to obtain good convergency in high temperature region, we need a large truncation number of the density matrix, while a necessary truncation number is small at low temperatures. The proposed method is, therefore, suitable for lower temperature region. As a demonstration of the method, we show the specific heat and dynamical current-current correlation function of the 1D Hubbard model at half filling. The DMRG results reproduce the exact digitalization results at low temperatures. [1] S. Sota and T. Tohyama, Phys. Rev. B \textbf{78}, 113101 (2008). [2] S. Sota and M. Itoh, J. Phys. Soc. Jpn. \textbf{76}, 054004 (2007).   

Screened Coulomb Interactions of Localized Electrons from First-Principles

We present a recently developed, maximally localized Wannier function approach for calculating the screened Coulomb(U) and exchange (J) interactions of localized electrons in solids. The localized orbitals are constructed using the maximally localized Wannier function approach. The dielectric screening is calculated from first-principles within the random phase approximation. Results for several systems containing strongly localized d electrons will be presented.   

Refinement of a Lanczos-based variational procedure to solve the Holstein model

We propose a slight refinement to the Trugman variational procedure to more efficiently solve the Holstein model with Lanczos methods. The modified Lanczos method converges much more quickly compared with the usual one in the intermediate- and strong- coupling regimes. To get a 6-digit accuracy of ground state energy at intermediate-strong coupling in the adiabatic region (small phonon frequency), only about 5000 basis states are need to be included. We also construct a variational ground state based on the numerical results.   

Plaquette Renormalization Scheme for Tensor Network States

We present a method for contracting a square-lattice tensor network in two dimensions based on auxiliary tensors accomplishing successive truncation (renormalization) of the effective 8-index tensors for $2\times 2$ plaquettes into 4-index tensors. The schheme is variational, and thus the tensors can be optimized by minimizing the energy. Test results for the quantum phase transition of the transverse-field Ising model confirm that even the smallest possible tensors (two values for each tensor index at each renormalization level) produce much better results than the simple product (mean-field) state. We also discuss several extensions of the scheme.   

A Diagrammatic Extension to Dynamical Cluster Approximation based on the Two-Particle Irreducible Vertex at Intermediate Length Scales

We present a non-perturbative multi-scale extension to the Dynamical Cluster Approximation (DCA) based on the two particle irreducible vertex $\Gamma$. The correlations at short length scales are calculated exactly using Quantum Monte Carlo (QMC) on small cluster of size $N_c^{(1)}$, and long length scales are treated at the dynamical mean field level. Intermediate length scales are treated on a second cluster of size $N_c^{(2)}> N_c^{(1)}$ by approximating its two-particle irreducible vertex with that of the smaller cluster, which is calculated by retaining its full momentum and frequency dependence. The resulting self energy of the large cluster is calculated using the Schwinger-Dyson equation. The method is applied to the 2D Hubbard model with cluster sizes $N_c^{(2)}\ge N_c^{(1)}$ and the results are compared with those that are calculated using QMC by increasing the size of the small cluster $N_c^{(1)}$ up to $N_c^{(2)}$.   

Nonlinear conductance oscillation in strong correlation limit of molecular quantum dots near zero bias anomaly

Recent experiments on strong correlation effects in molecular junctions have demonstrated that the interplay of electronic coupling to molecular vibrations and the Coulomb interaction produces intriguing oscillatory structures in the nonlinear conductance near zero bias anomaly at voltages in the energy scale of, presumably, the Kondo temperature. Using the imaginary-time quantum Monte Carlo technique recently developed for strongly correlated nonequilibrium, the nonlinear conductance of the Anderson-Holstein model at finite bias has been calculated. We discuss the mapping between the charge- and spin-Kondo limits and their distinctly different transport physics under finite chemical potential bias. We show that the conductance oscillation emerges at finite bias in the vicinity of the Kondo temperature due to strong electron-vibration coupling. The origin of the oscillation is from the bias-induced strong electron density modes as opposed to direct phonon excitations.   

Role of phonons and of finite temperature on the spectral function of a single hole in a 2D quantum antiferromagnet

We study thermal broadening of the hole spectral function of the two-dimensional $t-J$ and $t-t^{\prime}-t^{\prime\prime}-J$ modela within the non-crossing approximation (NCA) with and without the contribution of optical phonons. We have also studied the range of validity of the NCA by including the role of vertex corrections. The broadening of the quasiparticle peak as well as the transfer of spectral weight as a function of momentum to higher energy string excitations is found to be in reasonably good agreement with experimental angle resolved photo-emission spectroscopy(ARPES) results using a rather large electron-phonon coupling.   

Inelastic Scattering from Local Vibrational Modes

We study a nonuniversal contribution to the dephasing rate of conduction electrons due to local vibrational modes. Bosonization allows us to evaluate the full T-matrix. The inelastic scattering rate is strongly influenced by multiphonon excitations, exhibiting oscillatory behaviour. For higher frequencies, it saturates to a finite, coupling dependent value. In the strong coupling limit, the phonon is almost completely softened, and the inelastic cross section reaches its maximal value. This represents a magnetic field insensitive contribution to the dephasing time in mesoscopic systems, in addition to magnetic impurities.