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arXiv:1803.10259

Density Matrix Embedding Theory and Strongly Correlated Lattice Systems

Bo-Xiao Zheng

Published Mar 27, 2018• 4 citations
arXivHTMLPDF

Abstract

This thesis describes the development of the density matrix embedding theory (DMET) and its applications to lattice strongly correlated electron problems, including a review of DMET theory and algorithms (Ch 2), investigation of finite size scaling (Ch 3), Applications to high-temperature superconductivity (Ch 4-6), a framework for finite-temperature DMET (Ch 7).

DOI
10.48550/arxiv.1803.10259

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This thesis describes the development of the density matrix embedding theory (DMET) and its applications to lattice strongly correlated electron problems, including a review of DMET theory and algorithms (Ch 2), investigation of finite size scaling (Ch 3), Applications to high-temperature superconductivity (Ch 4-6), a framework for finite-temperature DMET (Ch 7).
This thesis describes several topics related to finite temperature studies of strongly correlated systems: finite temperature density matrix embedding theory (FT-DMET), finite temperature metal-insulator transition, and quantum algorithms including quantum imaginary time evolution (QITE), quantum Lanczos (QLanczos), and quantum minimally entangled typical thermal states (QMETTS) algorithms. While the absolute zero temperature is not reachable, studies of physical and chemical problems at finite temperatures, especially at low temperature, is essential for understanding the quantum behaviors of materials in realistic conditions. Here we define low temperature as the temperature regime where the quantum effect is not largely dissipated due to thermal fluctuation. Treatment of systems at low temperature is specially difficult compared to both high temperature - where classical approximation can be applied - and zero temperature where only the ground state is required to describe the system of interest. FT-DMET is a wavefunction-based embedding scheme which can handle finite temperature simulations of a variety of strongly correlated problems. The "high-level in low-level" framework enables FT-DMET to tackle large bulk sizes and capture the majority of the entanglement at the same time. FT-DMET formulations and implementation details for both model systems and ab initio problems are provided in Chapter 2 and Chapter 3. Metal-insulator transition is a common but important phase transition in many strongly correlated materials. The widely accepted scheme to distinguish an insulator from a metal is band structure theory based on a single-particle picture. However, insulating phases caused by disorder or strong correlation cannot be explained merely with the band structure. In Chapter 4, we demonstrate that electron locality/mobility is a more general criteria to detect metal-insulator transition. We further introduce complex polarization as the order parameter to reflect the electron locality/mobility and provide a formalism based on thermofield theory to evaluate the complex polarization at finite temperature. Quantum algorithms are designed to perform simulations on a quantum device. The infrastructure of a quantum processing unit (QPU) utilizes the superposition property of quantum bits (qubits), and thus can potentially outplay the classical simulations in computational scaling for certain problems. In Chapter 5, we introduce the QITE algorithm, which can be applied to quantum simulations of both ground state and finite temperature problems. We further introduce a subspace method, QLanczos algorithm, and a a finite temperature quantum algorithm, QMETTS, where QITE is used as a building block for the two algorithms. We demonstrate above quantum algorithms with simulations on both classical computers and quantum computers.
Density matrix embedding theory (DMET) is a quantum embedding theory for strongly correlated systems. From a computational perspective, one bottleneck in DMET is the optimization of the correlation potential to achieve self-consistency, especially for heterogeneous systems of large size. We propose a new method, called projected DMET (p-DMET), which achieves self-consistency without needing to optimize the correlation potential. We demonstrate the performance of p-DMET on the two-dimensional Hubbard model.
This thesis is divided into three chapters. In the first chapter we outline a simple and numerically inexpensive approach to describe the spectral features of the single-impurity Anderson model. The method combines aspects of the density matrix embedding theory (DMET) approach with a spectral broadening approach inspired by those used in numerical renormalization group (NRG) methods. At zero temperature for a wide range of U , the spectral function produced by this approach is found to be in good agreement with general expectations as well as more advanced and complex numerical methods such as DMRG-based schemes. The theory developed here is simply transferable to more complex impurity problems The second chapter outlines the density matrix embedding methodology in the context of electronic structure applications. We formulate analytical gradients for energies obtained from DMET focusing on two scenarios: RHF-in-RHF embedding and FCI-in-RHF embedding. The former involves solving the small embedded system at Restricted Hartree-Fock (RHF) level of theory. This serves to check the validity of the formulas by reproducing the RHF results on the full system for energies and gradients. The latter scenario employs full configuration interaction (FCI) as a high level solver for the small embedded system. Our results show that only Hellmann-Feynman terms, which involve derivatives of one- and two-electron terms in the atomic orbital basis, are required to calculate energy gradients in both cases. We applied our methodology to the problem of H 10 ring dissociation where the analytical gradients matched those obtained numerically. The gradient formulation is applicable to geometry optimization of strongly correlated molecules and solids. It can also be used in ab initio molecular dynamics where forces on nuclei are obtained from DMET energy gradients. In the final chapter we focus on the study of finite-temperature equilibrium properties of quantum systems in continuous space. We formulate an expansion of the partition function in continuous-time and use Monte Carlo to sample terms in the resulting infinite series. Such a strategy has been highly successful in quantum lattice models but has found scant application in off-lattice systems. The Monte Carlo estimate of the average energy of quantum particles in continuous space subject to simple model potentials is found to converge with low statistical error to the exact solutions even when very high perturbation order is required. We outline two ways in which the algorithm can be applied to more complex problems. First, by drawing an analogy between the formulation in continuous-time with the discretized, Trotter factorized version of standard path integral Monte Carlo (PIMC). This allows one to use the suite of standard PIMC moves to carry out the position sampling required to obtain the weight of each time configuration. Finally, we propose an alternate route by fitting the many-particle potential with multidimensional Gaussians which provides an analytical form for the position integrals.
We extend our density matrix embedding theory (DMET) [Phys. Rev. Lett.2012, 109, 186404] from lattice models to the full chemical Hamiltonian. DMET allows the many-body embedding of arbitrary fragments of a quantum system, even when such fragments are open systems and strongly coupled to their environment (e.g., by covalent bonds). In DMET, empirical approaches to strong coupling, such as link atoms or boundary regions, are replaced by a small, rigorous quantum bath designed to reproduce the entanglement between a fragment and its environment. We describe the theory and demonstrate its feasibility in strongly correlated hydrogen ring and grid models; these are not only beyond the scope of traditional embeddings but even challenge conventional quantum chemistry methods themselves. We find that DMET correctly describes the notoriously difficult symmetric dissociation of a 4 × 3 hydrogen atom grid, even when the treated fragments are as small as single hydrogen atoms. We expect that DMET will open up new ways of treating complex strongly coupled, strongly correlated systems in terms of their individual fragments.
We describe an efficient quantum embedding framework for realistic ab initio density matrix embedding theory (DMET) calculations in solids. We discuss in detail the choice of orbitals and mapping to a lattice, treatment of the virtual space and bath truncation, and the lattice-to-embedded integral transformation. We apply DMET in this ab initio framework to a hexagonal boron nitride monolayer, crystalline silicon, and nickel monoxide in the antiferromagnetic phase, using large embedded clusters with up to 300 embedding orbitals. We demonstrate our formulation of ab initio DMET in the computation of ground-state properties such as the total energy, equation of state, magnetic moment, and correlation functions.
Density Matrix Embedding Theory (DMET) presents a novel approach in capturing the physics of strongly correlated systems. It essentially describes finite fragments in the presence of environment while explicitly allowing quantum entanglement between both. DMET has been remarkably successful in studying systems ranging from molecules to solids and the paradigmatic Hubbard model. In this talk, we will present extensions of DMET to systems that are allowed to break SU(2) spin symmetry as well as U(1) particle number symmetry. This will enable us to study a range of broken symmetry phases such as topological superconductivity and, more interestingly, their interplay with strong correlations.