Session P21: Computational Methods for Strongly Correlated Systems and Many Body Theory

Spectral weight of the Emery model within different computational schemes

Although the single band Hubbard model captures many important aspects of the phenomenology of the high-temperature superconductors, the three-band Emery model allow the study of additional effects associated with the transfer of charge between the copper and oxygen orbitals and the strong hole repulsion at the oxygen sites. Here we present a comparison of the integrated and angle resolved spectral weight using exact diagonalization, dynamical mean field theory (with a quantum Monte Carlo solver and MaxEnt), and determinantal quantum Monte Carlo in the hole doped regime.   

Equation of state in single-band Hubbard models in two- and three-dimensions

We present results on ground-state energetics of the single-band Hubbard model in two- and three-dimensions with nearest-neighbor hopping and repulsive contact s-wave (on-site) interaction. Our calculations are done with the constrained-path auxiliary-field quantum Monte Carlo method. By incorporating generalized boundary conditions, we reduce finite-size effects due to open and closed shell filling and finite simulation cells. Results are obtained for the kinetic, interaction, and total energies and extrapolated to the thermodynamic limit for a range of interaction strengths ($U/t$) and electron densities.   

Does the $t^{'}{-}t{-}J$ model catch the main features of the cuprates phase diagram?

Using the Green's Function Monte Carlo Technique (GFMC), we investigate the effects of the $t^{'}$ interaction on the phase diagram of the $t{-}J$ model and its possible relevance for the physics of high-temperature superconductors (HTcS). In practice, we consider a very accurate guiding wave function including both magnetic and superconducting order parameters, as well as long-range Jastrow factors, in order to reproduce the correct low-energy spin and charge excitations. The $t^{'}$ interaction induces a suppression of the antiferromagnetic order parameter for hole concentration $\delta \sim 3-4\% $ (for $t^{'}=-0.2t$ and $J/t=0.2$), while the paramagnetic phase is characterized by an incommensurate peak in the spin structure factor. The inclusion of the $t^{'}$ term allows one to strongly suppress superconductivity at small doping, i.e., for $\delta < 6\% $. On the contrary, away from the antiferromagnetic phase, d-wave pairing correlations are enhanced up to the optimally doping region ($\delta \sim 20\%$) Our results then indicate that the $t^{'}{-}t{-}J$ model, though it is a very simple and crude approximation of realistic materials, is able to capture important properties of the HTcS phenomenology   

Fermionic functional renormalization group flows into phases with broken symmetry

We describe how functional renormalization group flows for interacting fermions can be continued into phases with broken symmetries. A symmetry-breaking term in the initial condition for the self-energy prevents a true divergence of the interactions at the critical scale. At the same scale, the anomalous self-energy grows rapidly such that the flow can be followed down to zero scale and all modes can be integrated out. Within simple mean-field models, we demonstrate two versions of this idea: one where the initial symmetry breaking is sent to zero, and another where it is compensated by a counter-term. The latter scheme is capable of detecting symmetry-broken phases separated from the symmetric state by an energy barrier. We discuss generalizations to more realistic models. Refs.: M. Salmhofer et al., Prog. Theor. Phys. 112, 943 (2004); R. Gersch et al., Euro. Phys. J. B 48: 349 (2005); R. Gersch et al., cond-mat/0609520.   

Variational reduced-rensity-matrix theory applied to the hubbard model

The application of variational reduced-density-matrix theory to the Hubbard model will be described. Recent results [Physical Review A 73, 062505 (2006)] demonstrate that computationally efficient N-representability conditions produce accurate ground-state energies and reduced-density-matrices for a wide range of interaction strengths for the one-dimensional lattice. I will discuss various types of N-representability conditions, the relationship between symmetries and reduced-density-matrices, and application of this method to other strongly correlated models. Preliminary results for the two-dimensional Hubbard model will be presented.   

Functional renormalization group and bosonization as a solver for 2D fermionic Hubbard models

The functional renormalization group (fRG) provides an unbiased framework to analyze competing instabilities in two-dimensional electron systems and has been used extensively over the past decade [1]. In order to obtain an equally unbiased tool to interprete the flow, we investigate the combination of a many-patch, one-loop calculation with higher dimensional bosonization [2] of the resulting low-energy action. Subsequently a semi-classical approximation [3] can be used to describe the resulting phases. The spinless Hubbard model on a square lattice with nearest neighbor repulsion is investigated as a test case. [1] M. Salmhofer and C. Honerkamp, Prog. Theor. Phys. 105, 1 (2001). [2] A. Houghton, H.-J. Kwon, J. B. Marston, Adv.Phys. 49, 141 (2000); P. Kopietz, Bosonization of interacting fermions in arbitrary dimensions, (Springer, Berlin, 1997). [3] H.-H. Lin, L. Balents, M. P. A. Fisher, Phys. Rev. B 56, 6569–6593 (1997); J. O. Fjaerestad, J. B. Marston, U. Schollwoeck, Ann. Phys. (N.Y.) 321, 894 (2006).   

Effects of electron-phonon coupling on the d-wave pairing superconducting phase

Recent experimental evidence has shown that the electron-phonon coupling could play a role in the formation of a d-wave pairing superconducting phase. Using a multiscale functional renormalization group (MFRG) technique, we study the effects of electron-phonon coupling in the two-dimensional Hubbard model with a band structure appropriate for the cuprate superconductors. We show that a momentum-independent electron- phonon coupling does not favor d-wave pairing but instead leads to the s-wave pairing and incommensurate density wave ordering.   

Sum-rule Conserving Spectral Functions from the Numerical Renormalization Group

We show how spectral functions for quantum impurity models, i.e. nanosystem embedded in fermionic or bosonic environment, can be calculated very accurately using a complete set of ``discarded'' numerical renormalization group (NRG) eigenstates, recently introduced by Anders and Schiller. The only approximation is to judiciously exploit energy scale separation. Our rigorous derivation avoids both the overcounting ambiguities and the single-shell approximation for the equilibrium density matrix prevalent in current methods including state of the art DM-NRG. The resulting procedure based on the full density matrix of the system (FDM-NRG) ensures that relevant sum rules hold rigorously and spectral features at energies below the temperature can be described accurately.   

New diagrammatic approach to the steady-state transport: nonlinear thermoelectric effects in interacting systems

Steady-state nonequilibrium described by a Gibbsian ensemble $e^{-\beta(H-Y)}$ with the boundary condition operator $Y$ is shown to be equivalent to the Keldysh formulation, through an explicit perturbation calculation of Anderson impurity model. We also show that the diagrammatics can be significantly simplified in the steady-state problems with a single real-time contour, in contrast to the double-contour Keldysh method. We apply this method to a quantum dot system in the Anderson impurity model with finite chemical potential bias and finite temperature gradient across the source-drain leads. We discuss the nonlinear nonequilibrium behavior of the Kondo resonance caused by strong potential and temperature bias.   

Density Matrix Renormalization Group study of magnon bound states of Heisenberg S=1 Y-junctions

Systems of Y-junctions are interesting both from a fundamental viewpoint and because of their potential use in nanoscale devices. Here we present a numerical study of S=1 Heisenberg model Y-junctions using a recently developed Y-junction DMRG algorithm[1]. We will focus on the question of the existence of magnon bound states at the junction, as a function of junction geometry and interaction parameters. \newline \newline [1] Haihui Guo and Steven R. White, Phys. Rev. B 74, 060401 (2006)   

A new warm-up procedure for the density-matrix renormalization group

A density-matrix renormalization group (DMRG) calculation starts with the infinite-system algorithm (the warm-up stage), where the system size $l$ is enlarged by adding new sites in the middle, which is then fed into the finite algorithm where the cut location is moved back and forth to enhance accuracy. Usually a considerable proportion of total calculation time has to be spent on the infinite algorithm, before the finite-size sweeps can be started. This is because at each step the target wavefunction for a different $l$ has to be calculated by some numerical diagonalization technique, and it is more difficult to give a good initial vector than in the finite-size algorithm where $l$ is constant. Here we propose a new infinite algorithm procedure where one value of $l$ is used to provide several blocks with different numbers of sites, which in fact dramatically reduces the overall computational time. This is demonstrated for various models such as the Hubbard model.   

A DMRG study of transport properties and correlations of quantum dots

We study transport through quantum dots using the time-dependent density matrix renormalization group method (tDMRG), recently proposed as a powerful computational tool to investigate transport through interacting nanostructures [1]. Since this technique relies on the numerical solution of finite clusters, we analyze the finite-size dependence of both static properties such as spin and charge fluctuations, spin-spin correlations and the conductance in detail, focusing on the example of one quantum dot. Our study reveals a crucial influence of global quantum numbers of finite clusters such as total spin on the results of tDMRG simulations, reflected in even-odd effects. We further establish a connection between the size of charge fluctuations on the quantum dot and the convergence of tDMRG with system size. Similar substantial even-odd effects exist within the framework of another technique, the embedded cluster approximation method (ECA). For the example of three quantum dots, we show that such even-odd effects strongly affect the spin fluctuations, leading to qualitatively different results for the conductance within ECA. [1] Al-Hassanieh et al., Phys. Rev. B 73, 195304 (2006)   

Quadratic scaling \textit{ab initio} DMRG for strong nondynamic correlation

We have devised a quadratic scaling \textit{ab initio} Density Matrix Renormalization Group (DMRG) algorithm for large, linear systems (such as unbranched polymers and long molecules) [1]. It is particularly suited for the description of strong active-space nondynamic correlation. This new local method (LDMRG) is inherently multireference, compact, variational, size-consistent and size-extensive. The reduced scaling is achieved solely through integral screening and without the artificial construction of correlation domains. Due to the multireference nature of the ansatz, we also do not require restricted localization in the occupied and virtual subspaces. Numerically exact (FCI) correlated energies (in a single-zeta 1-particle basis) up to 1-10$\mu $E$_{h}$ accuracy for systems with up to 100 electrons in 100 active orbitals (i.e. determinant spaces up to dimension 10$^{58})$ are presented. We also demonstrate the performance of the method in the study of the challenging metal-insulator transition in hydrogen-chains. We can now study nondynamic correlation in interesting classes of chemical systems, such as organic (opto-) electronic materials [2], or non-repeating chain-like molecules such as unfolded peptide backbones. [1] Hachmann, Cardoen, Chan, \textit{JCP} 125 (\textbf{2006}), 144101. [2] Hachmann, Dorando, Avil\'{e}s, Chan, \textit{in preparation}.   

Quantum Optimization: Spin Glasses and Wavefunction Annealing

The density matrix renormalization group (DMRG) has been extended in order to analyse the quantum spin glass transition (QSGT) for a random Ising model in a transverse field --$\Gamma$-- on a random graph with fixed connectivity $K=3$. The system is solved easily for a high value of $\Gamma$, and the wavefunction is {\em annealed} decreasing it slowly until the transition is reached. This way, the QSGT has been characterized in detail. A further decrease of $\Gamma$, down to $\Gamma=0$, allows to obtain the solution of the classical minimization problem associated, thus providing a possible alternative route to quantum annealing methods. Reference: J. Rodriguez-Laguna, G.E. Santoro, {\em Quantum Spin Glass Transition: the Ising model on random graphs}, submitted to Phys. Rev. B. ArXiv: {\tt cond-mat/0610661} (2006).