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Stripes in a Three-Chain Hubbard Ladder: a Comparison of Density-Matrix Renormalization Group and Constrained-Path Monte Carlo Results

J. Bonča, James E. Gubernatis, M. Guerrero, Eric Jeckelmann, Steven R. White

Published Mar 1, 1998• 0 citations
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Abstract

Using both the density-matrix renormalization group method and the constrained-path quantum Monte Carlo method, we studied the ground-state energies and the spin and hole densities of a 12x3 Hubbard model with open boundary conditions and six holes doped away from half-filling. Results obtained with these two methods agree well in the small and intermediate U regimes. For U/t{>=}6 we find a ground-state with charge inhomogeneities consistent with stripes. (c) 2000 The American Physical Society.

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arXiv:cond-mat/9805116•Physical review. B, Condensed matter•Oct 1, 1998•52 citations
Using both the density-matrix renormalization group method and the constrained-path quantum Monte Carlo method, we studied the ground-state energies and the spin and hole densities of a 12x3 Hubbard model with open boundary conditions and six holes doped away from half-filling. Results obtained with these two methods agree well in the small and intermediate U regimes. For U/t{>=}6 we find a ground-state with charge inhomogeneities consistent with stripes. (c) 2000 The American Physical Society.
Using both the density-matrix renormalization group method and the constrained-path quantum Monte Carlo method, we studied the ground-state energies and the spin and hole densities of a $12\ifmmode\times\else\texttimes\fi{}3$ Hubbard model with open boundary conditions and six holes doped away from half-filling. Results obtained with these two methods agree well in the small and intermediate U regimes. For U/t> 6U/t>~6U/t> 6 we find a ground-state with charge inhomogeneities consistent with stripes.
Using both the density-matrix renormalization group method and the constrained-path quantum Monte Carlo method, we have studied the ground-state energies and the spin and hole densities of a 12×312 \times 312×3 Hubbard model with open boundary conditions and 6 holes doped away from half-filling. Results obtained with these two methods agree well in the small and intermediate UUU regimes. For U/t≥6U/t \geq 6U/t≥6 we find a ground-state with stripes.
We investigate the formation of stripes in $7r\ifmmode\times\else\texttimes\fi{}6$ Hubbard ladders with 4r4r4r holes doped away from half filling using the density-matrix renormalization group (DMRG) method. A parallelized code allows us to keep enough density-matrix eigenstates (up to m=8000m=8000m=8000) and to study sufficiently large systems (with up to 7r=217r=217r=21 rungs) to extrapolate the stripe amplitude to the limits of vanishing DMRG truncation error and infinitely long ladders. Our work gives strong evidence that stripes exist in the ground state for strong coupling (U=12t)(U=12t)(U=12t) but that the structures found in the hole density at weaker coupling (U=3t)(U=3t)(U=3t) are an artifact of the DMRG approach.
The performance of the density matrix renormalization group (DMRG) is strongly influenced by the choice of the local basis of the underlying physical lattice. We demonstrate that, for the two-dimensional Hubbard model, the hybrid--real-momentum-space formulation of the DMRG is computationally more efficient than the standard real-space formulation. In particular, we show that the computational cost for fixed bond dimension of the hybrid-space DMRG is approximately independent of the width of the lattice, in contrast to the real-space DMRG, for which it is proportional to the width squared. We apply the hybrid-space algorithm to calculate the ground state of the doped two-dimensional Hubbard model on cylinders of width four and six sites; at n=0.875n=0.875n=0.875 filling, the ground state exhibits a striped charge-density distribution with a wavelength of eight sites for both U/t=4.0U/t=4.0U/t=4.0 and 8.08.08.0. We find that the strength of the charge ordering depends on U/tU/tU/t and on the boundary conditions. Furthermore, we investigate the magnetic ordering as well as the decay of the static spin, charge, and pair-field correlation functions.
We use the density-matrix renormalization group to calculate the phase diagram of the one-dimensional extended Hubbard chain at half filling. We find that for one specific U,U,U, the critical value of Vc{V}_{c}Vc​ for the charge-density-wave versus spin-density-wave phase transition is smaller than that predicted from the Monte Carlo simulation. Our results are consistent with the strong- and weak-coupling limit results, whereas the previous results from the Monte Carlo simulations deviate from the weak-coupling limit result. As UUU increases, we find that the first-order phase transition develops gradually. The tricritical point is estimated to occur at Uc=4−−6,{U}_{c}=4--6,Uc​=4−−6, not Uc=3{U}_{c}=3Uc​=3 as previously predicted from the Monte Carlo simulation. Due to the topological difference, we observe that with the periodic boundary condition, the phase transition is more abrupt for finite chains of length N=4MN=4MN=4M than for chains of N=4M+2,N=4M+2,N=4M+2, where MMM is an integer, for the same UUU and V.V.V. The difference will diminish in the thermodynamic limit.
Using density matrix renormalization group calculations, we compare results obtained for the $t\ensuremath{-}J,$ one-band Hubbard, and three-band Hubbard models of a two-leg CuO ladder. Spin and charge gaps, pair binding energies, and effective pair hoppings are calculated for a wide range of parameters. All three models have an insulating state at a filling corresponding to one hole per Cu site. For physically relevant parameters their spin gaps are similar in size but they exhibit quite different charge gaps. We find that the binding energy of a pair of doped holes is of the order of the undoped ladder spin gap for all three models. The main difference between the models is the size of the effective pair hopping, which is significantly larger in the three-band model with parameters appropriate for CuO materials than in the other two models.