[Paper Review] The Density Matrix Renormalization Group for Fermion Systems

R. M. Noack, S. R. White+1 more|arXiv (Cornell University)|Apr 30, 1994

Quantum many-body systems4 citations

TL;DR

This paper applies the density matrix renormalization group (DMRG) to interacting fermion systems in more than one dimension, demonstrating its effectiveness for studying strongly correlated electron systems. It reports numerical results for the two-chain Hubbard model, revealing a gapped spin liquid at half-filling and weak algebraic d-wave-like pair field correlations away from half-filling, indicating possible superconducting tendencies.

ABSTRACT

We discuss techniques of the density matrix renormalization group and their application to interacting fermion systems in more than one dimension. We show numerical results for equal--time spin--spin and singlet pair field correlation functions, as well as the spin gap for the Hubbard model on two chains. The system is a gapped spin liquid at half--filling and shows weak algebraic $d$-wave--like pair field correlations away from half--filling.

Motivation & Objective

• To extend the density matrix renormalization group (DMRG) method to interacting fermion systems in more than one dimension.
• To investigate the electronic structure and correlation functions in the two-chain Hubbard model using DMRG.
• To analyze spin and pair field correlations and determine the presence of a spin gap.
• To explore the nature of the ground state, particularly the possibility of d-wave-like superconducting pairing away from half-filling.

Proposed method

• Adapts the DMRG algorithm to handle fermionic degrees of freedom using second quantization and anti-commuting operators.
• Employs a block-decimation approach to iteratively construct the density matrix and project the Hilbert space onto the most relevant states.
• Uses the density matrix to identify and retain the most significant states during the renormalization process, improving convergence and accuracy.
• Performs numerical calculations on a two-chain Hubbard model with periodic boundary conditions to compute equal-time correlation functions.
• Computes spin-spin and singlet pair field correlation functions using the DMRG wavefunction.
• Analyzes the spin gap by examining the energy difference between the singlet and triplet ground states in the two-chain system.

Experimental results

Research questions

• RQ1 Can the DMRG method accurately describe the ground state of interacting fermion systems in two dimensions?
• RQ2 What is the nature of spin and pair field correlations in the two-chain Hubbard model at various fillings?
• RQ3 Does the system exhibit a spin gap, and how does it vary with electron filling?
• RQ4 Are there signatures of d-wave-like pairing correlations in the two-chain Hubbard model away from half-filling?
• RQ5 Is the ground state a gapped spin liquid at half-filling, as suggested by the numerical results?

Key findings

• The two-chain Hubbard model exhibits a gapped spin liquid ground state at half-filling, as indicated by the presence of a spin gap.
• Away from half-filling, the system displays weak algebraic d-wave-like pair field correlations, suggesting potential superconducting pairing tendencies.
• Equal-time spin-spin correlation functions show short-range antiferromagnetic order, consistent with a spin liquid phase.
• Singlet pair field correlation functions decay algebraically with distance, characteristic of d-wave pairing symmetry.
• The spin gap remains finite at half-filling, supporting the stability of the spin liquid phase.
• The DMRG method successfully captures the subtle competition between magnetic order and pairing in the two-chain system.