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International Journal of Modern Physics C

NUMERICAL RENORMALIZATION GROUP METHOD IN WEAKLY COUPLED QUANTUM SPIN CHAINS: COMPARISON WITH EXACT DIAGONALIZATION

Jose Vicente Alvarez, S. Moukouri

Published Jun 1, 2005• 6 citations
Publisher

Abstract

We show that numerical quasi-one-dimensional renormalization group allows accurate study of weakly coupled spin chains with modest computational effort. We perform a systematic comparison with exact diagonalization results in two- and three-leg spin ladders with a transverse Hamiltonian that can involve frustration. Due to the variational nature of the algorithm, the accuracy can be arbitrarily improved enlarging the basis of eigenstates in the transverse direction. We observe that the precision of the algorithm is directly correlated to the binding of the chains. We also show that the method performs especially well in magnetically frustrated systems.

DOI
10.1142/s0129183105007522

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We show that numerical quasi-one-dimensional renormalization group allows accurate study of weakly coupled spin chains with modest computational effort. We perform a systematic comparison with exact diagonalization results in two- and three-leg spin ladders with a transverse Hamiltonian that can involve frustration. Due to the variational nature of the algorithm, the accuracy can be arbitrarily improved enlarging the basis of eigenstates in the transverse direction. We observe that the precision of the algorithm is directly correlated to the binding of the chains. We also show that the method performs especially well in magnetically frustrated systems.
We show that numerical quasi-one-dimensional renormalization group allows accurate study of weakly coupled chains with modest computational effort. We perform a systematic comparison with exact diagonalization results in two and three-leg spin ladders with a transverse Hamiltonian that can involve frustration. Due to the variational nature of the algorithm, the accuracy can be arbitrarily improved enlarging the basis of eigenstates of the density matrix defined in the transverse direction. We observe that the precision of the algorithm is directly correlated to the binding of the chains. We also show that the method performs especially well in frustrated systems.
The Kato-Bloch perturbation formalism is used to present a density-matrix renormalization-group method for strongly anisotropic two-dimensional systems. This method is used to study Heisenberg chains weakly coupled by the transverse couplings ${J}_{\ensuremath{\perp}}$ and Jd{J}_{d}Jd​ (along the diagonals). An extensive comparison of the renormalization group and quantum Monte Carlo results for parameters where the simulations by the latter method are possible shows a very good agreement between the two methods. It is found, by analyzing ground state energies and spin-spin correlation functions, that there is a transition between two ordered magnetic states. When ${J}_{d}∕{J}_{\ensuremath{\perp}}\ensuremath{\lesssim}0.5$, the ground state displays a N\'eel order. When ${J}_{d}∕{J}_{\ensuremath{\perp}}\ensuremath{\gtrsim}0.5$, a collinear magnetic ground state in which interchain spin correlations are ferromagnetic becomes stable. In the vicinity of the transition point, ${J}_{d}∕{J}_{\ensuremath{\perp}}\ensuremath{\approx}0.5$, the ground state is disordered. But, the nature of this disordered ground state is unclear. While the numerical data seem to show that the chains are disconnected, the possibility of a genuine disordered two-dimensional state, hidden by finite size effects, cannot be excluded.
I apply a two-step density-matrix renormalization group method to the anisotropic two-dimensional Hubbard model. As a prelude to this study, I compare the numerical results to the exact one for the tight-binding model. I find a ground-state energy which agrees with the exact value up to four digits for systems as large as $24\ifmmode\times\else\texttimes\fi{}25$. I then apply the method to the interacting case. I find that for strong Hubbard interaction, the ground state is dominated by magnetic correlations. These correlations are robust even in the presence of strong frustration. Interchain pair tunneling is negligible in the singlet and triplet channels and is not enhanced by frustration. For weak Hubbard couplings, due to interchain interaction, interchain nonlocal singlet pair tunneling is enhanced and magnetic correlations are strongly reduced. This suggests a possible superconductive ground state. The pairing mechanism is reminiscent of the Kohn-Luttinger mechanism.
A formulation of numerical real-space renormalization groups for quantum many-body problems is presented and several algorithms utilizing this formulation are outlined. The methods are presented and demonstrated using S=1/2 and S=1 Heisenberg chains as test cases. The key idea of the formulation is that rather than keep the lowest-lying eigenstates of the Hamiltonian in forming a new effective Hamiltonian of a block of sites, one should keep the most significant eigenstates of the block density matrix, obtained from diagonalizing the Hamiltonian of a larger section of the lattice which includes the block. This approach is much more accurate than the standard approach; for example, energies for the S=1 Heisenberg chain can be obtained to an accuracy of at least ${10}^{\mathrm{\ensuremath{-}}9}$. The method can be applied to almost any one-dimensional quantum lattice system, and can provide a wide variety of static properties.
A method is proposed to improve the accuracy of approximate techniques for strongly correlated electrons that use reduced Hilbert spaces. As a first step, the method involves a change of basis that incorporates exactly part of the short-distance interactions. The Hamiltonian is rewritten in new variables that better represent the physics of the problem under study. A Hilbert-space expansion performed in the new basis follows. The method is successfully tested using both the Heisenberg model and the $t\ensuremath{-}J$ model with holes on two-leg ladders and chains, including estimations for ground-state energies, static correlations, and spectra of excited states. An important feature of this technique is its ability to calculate dynamical responses on clusters larger than those that can be studied using exact diagonalization. The method is applied to the analysis of the dynamical spin structure factor $S(\mathbf{q},\ensuremath{\omega})$ on clusters with $2\ifmmode\times\else\texttimes\fi{}16$ sites and 0 and 2 holes. Our results confirm previous studies [M. Troyer, H. Tsunetsugu, and T. M. Rice, Phys. Rev. B 53, 251 (1996)] that suggested that the state of the lowest energy in the spin-1 2-hole subspace corresponds to the bound state of a hole pair and a spin triplet. Implications of this result for neutron-scattering experiments both on ladders and planes are discussed.
Novel randomness-induced disordered ground states in two-dimensional (2D) quantum spin systems have been attracting much interest. For quantitative analysis of such random quantum spin systems, one of the most promising numerical approaches is the tensor-network strong-disorder renormalization group (tSDRG), which was basically established for one-dimensional (1D) systems. In this paper, we propose a possible improvement of its algorithm toward 2D random spin systems, focusing on a generating process of the tree network structure of tensors, and precisely examine their performances for the random antiferromagnetic Heisenberg model not only on the 1D chain but also on the square- and triangular-lattices. On the basis of comparison with the exact numerical results up to 36 site systems, we demonstrate that accuracy of the optimal tSDRG algorithm is significantly improved for the 1D and 2D systems in the strong-randomness regime.