[Paper Review] Density Matrix Embedding Theory and Strongly Correlated Lattice Systems

Bo-Xiao Zheng|arXiv (Cornell University)|Mar 27, 2018

Physics of Superconductivity and Magnetism186 references3 citations

TL;DR

This paper introduces an advanced formulation of Density Matrix Embedding Theory (DMET) with broken symmetry and improved impurity solvers to study strongly correlated lattice systems, particularly the Hubbard and three-band Hubbard models. It achieves a controlled, comprehensive ground state phase diagram, confirming d-wave superconductivity and inhomogeneous orders, and identifies a highly compressible vertical stripe phase at 1/8 doping, revealing limitations of the one-band model and extending DMET to finite temperature via superoperator formalism.

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).

Motivation & Objective

• To develop a symmetry-broken DMET formulation to study high-temperature superconductivity and competing orders in cuprate models.
• To extend DMET's applicability by integrating advanced impurity solvers beyond exact diagonalization, including DMRG, AFQMC, and active-space quantum chemistry methods.
• To perform finite-size scaling analysis to ensure numerical reliability in phase diagram calculations.
• To investigate the strong-coupling, underdoped regime of the Hubbard model and determine the true ground state at 1/8 doping.
• To explore beyond-the-one-band model physics using the three-band Hubbard model and first-principles downfolded Hamiltonians.

Proposed method

• Formulate DMET with broken spin and particle-number symmetries to access superconducting and inhomogeneous phases.
• Implement approximate impurity solvers—DMRG, auxiliary-field quantum Monte Carlo, and active-space methods—to treat larger fragments than exact diagonalization allows.
• Use a hybrid chemical potential optimization scheme with weighted averaging and parabolic/linear extrapolation to accelerate convergence to target electron density.
• Apply the Davidson algorithm for efficient diagonalization of large Hamiltonian matrices in impurity solver calculations.
• Develop BitGen, a code generator for fermionic algebras, to automate tensor contractions and code generation for numerical implementations.
• Extend DMET to finite temperature using a superoperator representation of density matrices, enabling thermal correlation studies.

Experimental results

Research questions

• RQ1 What is the true ground state of the Hubbard model in the strong-coupling, underdoped regime, particularly at 1/8 doping?
• RQ2 Does d-wave superconductivity emerge in the Hubbard model on a square lattice with controlled numerical uncertainties?
• RQ3 How do finite-size effects influence the phase diagram of the Hubbard model, and can they be systematically extrapolated?
• RQ4 To what extent does the one-band Hubbard model capture the essential physics of cuprate superconductors?
• RQ5 Can the three-band Hubbard model and first-principles downfolded Hamiltonians explain the observed inhomogeneous and superconducting orders beyond the one-band model?

Key findings

• The ground state of the Hubbard model at 1/8 doping is a highly compressible, filled vertical stripe phase in the coupling regime relevant to cuprates.
• A comprehensive ground state phase diagram was computed with well-controlled numerical uncertainties, confirming the existence of d-wave superconductivity and various inhomogeneous orders.
• The one-band Hubbard model is found to have limitations in capturing the full physics of cuprates, particularly in the strong-coupling, underdoped regime.
• The three-band Hubbard model and downfolded cuprate Hamiltonians reveal additional physics beyond the one-band model, suggesting the need for multi-orbital descriptions.
• The finite-temperature extension of DMET using superoperator formalism enables the study of thermal properties in strongly correlated systems, including cuprates.
• The chemical potential optimization algorithm with adaptive weighting and extrapolation achieves convergence in fewer than four DMET iterations, ensuring numerical efficiency.