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Civil War Book Review

Next generation multi-scale quantum simulations for strongly correlated materials

Shuxiang Yang

Published Aug 22, 2011• 2 citations
Publisher

Abstract

This thesis represents our effort to develop the next generation multi-scale quantum simulation methods suitable for strongly-correlated materials, where complicated phase-diagrams prevail, suggesting complicated underlying physics. We first give a detailed description of the parquet formalism. With its help, different approximate methods can be unified and a hierarchy of approximate methods with different accuracies and computational complexity can thus be designed. Next, we present a numerical solution of the parquet approximation. Results on the Hubbard model are compared to those obtained from Determinant Quantum Monte Carlo (DQMC), FLuctuation EXchange (FLEX), and self-consistent second-order approximation methods. The comparison shows a satisfactory agreement with DQMC and a significant improvement over the FLEX or the self-consistent second-order approximation. The parquet formalism can also be used to analyze the superconducting mechanism of the high-temperature superconductors. The dynamical cluster approximation (DCA) method is used to understand the proximity of the superconducting dome to the quantum critical point in the 2-D Hubbard model. At optimal doping, where Vd is revealed to be featureless, we find a power-law behavior of chi_0d(w=0), replacing the BCS logarithm behavior, and strongly enhanced T_c. After that we propose another multi-scale approach by combining the DCA and the recently introduced dual-fermion formalism. Within this approach, short and long length scale physics is addressed by the DCA cluster calculation, while intermediate length scale physics is addressed diagrammatically using dual fermions. The bare and dressed dual fermionic Green functions scale as O(1/Lc), so perturbation theory on the dual lattice converges very quickly. Lastly, we study the responses to dynamical modulation of the optical lattice potential by analyzing properties of the repulsive fermionic Hubbard model in an optical lattice. We provide numerical evidence showing the modulations by on-site local interaction cannot be ignored, and instead can even strongly contribute to the dynamical behaviors of the system in highly-doped cases.

DOI
10.31390/gradschool_dissertations.2772

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This thesis represents our effort to develop the next generation multi-scale quantum simulation methods suitable for strongly-correlated materials, where complicated phase-diagrams prevail, suggesting complicated underlying physics. We first give a detailed description of the parquet formalism. With its help, different approximate methods can be unified and a hierarchy of approximate methods with different accuracies and computational complexity can thus be designed. Next, we present a numerical solution of the parquet approximation. Results on the Hubbard model are compared to those obtained from Determinant Quantum Monte Carlo (DQMC), FLuctuation EXchange (FLEX), and self-consistent second-order approximation methods. The comparison shows a satisfactory agreement with DQMC and a significant improvement over the FLEX or the self-consistent second-order approximation. The parquet formalism can also be used to analyze the superconducting mechanism of the high-temperature superconductors. The dynamical cluster approximation (DCA) method is used to understand the proximity of the superconducting dome to the quantum critical point in the 2-D Hubbard model. At optimal doping, where Vd is revealed to be featureless, we find a power-law behavior of chi_0d(w=0), replacing the BCS logarithm behavior, and strongly enhanced T_c. After that we propose another multi-scale approach by combining the DCA and the recently introduced dual-fermion formalism. Within this approach, short and long length scale physics is addressed by the DCA cluster calculation, while intermediate length scale physics is addressed diagrammatically using dual fermions. The bare and dressed dual fermionic Green functions scale as O(1/Lc), so perturbation theory on the dual lattice converges very quickly. Lastly, we study the responses to dynamical modulation of the optical lattice potential by analyzing properties of the repulsive fermionic Hubbard model in an optical lattice. We provide numerical evidence showing the modulations by on-site local interaction cannot be ignored, and instead can even strongly contribute to the dynamical behaviors of the system in highly-doped cases.
This thesis presents the development of new numerical methods for the treatment of strongly correlated electron systems based on self-consistent approaches at both the one and the two-particle level such as the parquet formalism. The parquet formalism was solved for the first time on a two-dimensional cluster. When the fully irreducible vertex is approximated by the bare vertex, we obtain the parquet approximation. Its validity was investigated by comparing results that it produces to those of other conserving approximations such as the FLuctuation EXchange (FLEX) approximation or the Second Order Perturbation Theory (SOPT). We found that it provides a significant improvement of FLEX or SOPT and a satisfactory agreement with Quantum Monte Carlo results despite instabilities in the self-consistency at low temperatures and for strong Coulomb interaction. We use the parquet formalism to study the Quantum Critical Point at finite doping in the Hubbard model by decomposing the vertex into its contributions from different channels. We apply this decomposition to the pairing channel and we find that the dominant contribution to the vertex originates in the spin channel even at the quantum critical doping. Furthermore, we explore the divergence of the two parts of the pairing matrix at optimal doping and observe that the irreducible vertex decreases monotonically as the doping is increased while the bare susceptibility exhibits an algebraic divergence at the quantum critical doping supporting the Quantum Critical BCS scenario proposed by She and Zaanen. To circumvent the instabilities in the iteration of the parquet formalism, we explored the dual fermion approach introduced by Rubtsov et al. Here, we extended the formalism to the Dynamical Cluster Approximation, in the process introducing a small parameter in the dual fermions perturbation theory. We demonstrate the quality of the resulting Dual Fermion DCA through a systematic study of the cluster size dependence and of the different perturbative approximations. These efforts represent the initial steps in the development of the Multi-Scale Many Body approach that appropriately treats correlations at different length scales.
We have designed a multiscale approach for strongly correlated systems by combining the dynamical cluster approximation (DCA) and the recently introduced dual fermion formalism. This approach employs an exact mapping from a real lattice to a DCA cluster of linear size Lc{L}_{c}Lc​ embedded in a dual fermion lattice. Short-length-scale physics is addressed by the DCA cluster calculation, while longer-length-scale physics is addressed diagrammatically using dual fermions. The bare and dressed dual fermionic Green functions scale as O(1/Lc)\mathcal{O}(1/{L}_{c})O(1/Lc​), 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 O(1/Lc3)\mathcal{O}(1/{L}_{c}^{3})O(1/Lc3​) with third-order and three-body corrections down by an additional factor of O(1/Lc)\mathcal{O}(1/{L}_{c})O(1/Lc​).
Many technologically important materials display complex behavior due to strong electronic correlations. These include magnets, magnetoresistive materials, lanthanides and actinides and high temperature superconductors. Some of these materials display complex phase diagrams with multiple competing ground states, spin and charge ordering and complex excitations like those associated with spin and charge separation. An unbiased treatment of these phenomena requires a theory able to treat the correlations over all relevant length scales. In part, due to the minus sign problem, numerically exact methods employed to study these systems scale exponentially. Thus, despite great recent progress such as the development of the dynamical mean field approximation and its cluster extensions, there is as yet no unbiased study of these complex phenomena. The Multi-Scale Many-Body (MSMB) formalism and algorithms address these problems by separating the correlations into different length scales and treating each with an appropriate approximation: strong correlations at short length scales are treated by numerically exact Quantum Monte Carlo (QMC) simulations, those at intermediate length scales are treated by Feynman diagrammatic methods, while the weak correlations at long length scales are treated by a (dynamical) mean field approximation with a self-consistently determined effective medium.
This thesis will describe efforts to enhance our ability to simulate the 2D Hubbard model. Chapter 2 provides a background on the main computational techniques used throughout the work, including quantum Monte Carlo (QMC), dynamical mean field theory (DMFT), the dynamical cluster approximation (DCA), the continuous time auxiliary field algorithm (CTAUX), and the Maximum Entropy Method (MEM) for numeric analytic continuation. Chapter 3 presents new work on applying Twisted Boundary Conditions to the DCA framework. This method is applied in a effort to access thermodynamic (i.e. large system) information about the Hubbard model without the immense computational expense required to simulate large lattices directly. Chapter 4 describes efforts to study the extended 2D Hubbard model, which re-introduces non-local interactions between electrons that drive the formation of charge ordered phases. The work includes a thorough analysis of the phase diagram of the model away from half-filling, as well as analysis of the effect of non-local interactions on anti-ferromagnetic fluctuations and competition between the charge order and AFM states. Chapter 5 includes a derivation of the dual fermions diagrammatic expansion and the dual fermions ladder approximation used to compute corrections to single site DMFT calculations. The method is applied to the 2D Hubbard model in order to study the evolution of its spectral function, Fermi surface, and momentum dependent mass renormalization as the temperature, interaction strength, and doping are changed. The great advantage of this method is that we are able to access arbitrary momenta throughout the Brillouin zone, in contrast to the limitation to a few cluster momenta that characterize cluster methods like DCA.
We present a method to treat the two-dimensional (2D) Hubbard model for parameter regimes which are relevant for the physics of the high-Tc{T}_{c}Tc​ superconducting cuprates. Unlike previous attempts to attack this problem, our approach takes into account all fluctuations in different channels on equal footing and is able to treat reasonable large lattice sizes up to $32\ifmmode\times\else\texttimes\fi{}32$. This is achieved by the following three-step procedure: (i) We transform the original problem to a representation (dual fermions) in which all purely local correlation effects from the dynamical mean field theory are already considered in the bare propagator and bare interaction of the new problem. (ii) The strong $1/{(i\ensuremath{\nu})}^{2}$ decay of the bare propagator allows us to integrate out all higher Matsubara frequencies aside from the lowest using low-order diagrams. The new effective action depends only on the two lowest Matsubara frequencies which allows us to (iii) apply the two-particle self-consistent parquet formalism, which takes into account the competition between different low-energy bosonic modes in an unbiased way, on much finer momentum grids than usual. In this way, we were able to map out the phase diagram of the 2D Hubbard model as a function of temperature and doping. Consistently with the experimental evidence for hole-doped cuprates and previous dynamical cluster approximation calculations, we find an antiferromagnetic region at low doping and a superconducting dome at higher doping. Our results also support the role of the van Hove singularity as an important ingredient for the high value of Tc{T}_{c}Tc​ at optimal doping. At small doping, the destruction of antiferromagnetism is accompanied by an increase of charge fluctuations supporting the scenario of a phase-separated state driven by quantum critical fluctuations.
This thesis contains a series of numerical studies of strongly correlated electron systems. In these systems, interesting emergent properties are brought about by strong electron correlations, leading to unusual fluctuations and phase transitions. We start with a review of basic concepts in many-body physics from a field theory point of view, including second quantization and path integral formalisms. We then introduce extensions of the single-particle and two-particle Green’s function formalisms in the singlet superconducting state, which enable us to extract physical information of the systems in the symmetry-broken phase. Next, we proceed to the models and numerical methods we use to study the strongly correlated systems. We introduce a low energy effective model -- the Hubbard model, which contains a subset of the electron degrees of freedom and can be solved numerically using advanced many-body methods. With the dynamical cluster approximation and continuous-time quantum Monte Carlo impurity solver, we are able to study the competing fluctuations in the paramagnetic state and analyze the fluctuations behind superconductivity in the singlet superconducting state. We introduce self-consistent diagrammatic methods used in realistic material calculations in the last part of this thesis, with an outline of group theory concepts that can be used to optimize the simulations. Numerical representations and methods for effectively solving equations in realistic material calculations are discussed afterwards. We briefly review some of the developments in this field, and then introduce spectral methods that are based on mathematical properties of Legendre polynomials for solving both the imaginary- and real-time Dyson equations.