[Paper Review] DMRG and the Two Dimensional t-J Model

Ian P. McCulloch, A. R. Bishop+1 more|arXiv (Cornell University)|Mar 5, 2001

Quantum many-body systems2 references4 citations

TL;DR

This paper applies the non-Abelian Density Matrix Renormalization Group (DMRG) algorithm to the two-dimensional t-J model, enabling more accurate ground state calculations by preserving SU(2) spin symmetry. The study finds a stable striped ground state with constant hole density per stripe (0.5 < d < 1) at J/t ≈ 0.35, indicating stripes are distinct from phase separation and likely arise from nanoscale phase separation with antiphase domain walls.

ABSTRACT

We describe in detail the application of the recent non-Abelian Density Matrix Renormalization Group (DMRG) algorithm to the two dimensional t-J model. This extension of the DMRG algorithm allows us to keep the equivalent of twice as many basis states as the conventional DMRG algorithm for the same amount of computational effort, which permits a deeper understanding of the nature of the ground state.

Motivation & Objective

• To investigate the nature of the ground state in the two-dimensional t-J model, particularly the existence and stability of stripe phases.
• To test whether stripes represent a true ground state or are artifacts of phase separation in the t-J model.
• To apply and validate the non-Abelian DMRG algorithm for systems with SU(2) spin symmetry, improving accuracy over conventional DMRG.
• To examine finite-size effects and boundary-induced hole localization in striped versus phase-separated regimes.
• To clarify the role of antiphase boundaries in stripe formation and their connection to nanoscale phase separation.

Proposed method

• The non-Abelian DMRG algorithm is used, which generalizes conventional DMRG by preserving SU(2) spin symmetry and allowing block states to transform under irreducible representations of SU(2).
• The Hamiltonian is formulated in a basis labeled by particle number N and total spin j, corresponding to the global symmetry group U(1) ⊗ SU(2).
• The t-J Hamiltonian is defined on a no-double-occupancy subspace, with kinetic hopping term (-t) and spin exchange term (J) on nearest-neighbor sites.
• The algorithm tracks the density matrix of reduced density matrices in the block basis, retaining states with highest eigenvalues to optimize entanglement truncation.
• Calculations are performed on square lattices up to 24×6 with open boundary conditions and half-periodic boundary conditions in the y-direction to stabilize stripe order.
• The method compares results in the SU(2) basis with those from the conventional S^z basis to assess the impact of symmetry preservation.

Experimental results

Research questions

• RQ1 Does the two-dimensional t-J model exhibit a striped ground state distinct from phase separation, particularly at J/t ≈ 0.35?
• RQ2 How does the non-Abelian DMRG algorithm improve the accuracy and stability of ground state calculations compared to conventional DMRG for spin-ful systems?
• RQ3 What is the hole density per stripe in the ground state, and does it remain constant across different system sizes?
• RQ4 How do open boundaries influence hole distribution, and what does this imply about finite-size effects in stripe formation?
• RQ5 Are antiphase boundaries a consequence of stripe order, or do they represent a deeper mechanism such as nanoscale phase separation?

Key findings

• The non-Abelian DMRG algorithm successfully identifies a striped ground state in the 2D t-J model at J/t ≈ 0.35, with three stripes observed on a 24×6 lattice.
• The hole density per stripe is found to be in the range 0.5 < d < 1, suggesting a constant hole density per stripe independent of system size.
• In the striped regime, holes are repelled by open boundaries, whereas in phase-separated regimes, holes are attracted to the boundaries, indicating a fundamental difference between the two phases.
• The presence of antiphase boundaries is confirmed as a key signature of stripe order, supporting the idea that stripes are not merely phase separation but a distinct quantum phase.
• The results suggest that stripes may arise from nanoscale phase separation with periodic ferron (ferromagnetic) droplets forming antiphase domain walls.
• Finite-size effects are significant, particularly due to boundary conditions, and the critical J/t for stripe formation may be sensitive to system geometry and boundary type.