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arXiv:1111.1442 · Physical Review B

Density matrix renormalization group algorithm for Bethe lattices of spin- 12 or spin-1 sites with Heisenberg antiferromagnetic exchange

Manoranjan Kumar, S. Ramasesha, Z. G. Soos

Published Apr 6, 2012• 19 citations
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Abstract

A density matrix renormalization group (DMRG) algorithm is presented for the Bethe lattice with connectivity $Z=3$ and antiferromagnetic exchange between nearest-neighbor spins $s=1/2$ or 1 sites in successive generations $g$. The algorithm is accurate for $s=1$ sites. The ground states are magnetic with spin $S(g)={2}^{g}s$, staggered magnetization that persists for large $g>20$, and short-range spin correlation functions that decrease exponentially. A finite energy gap to $S>S(g)$ leads to a magnetization plateau in the extended lattice. Closely similar DMRG results for $s$ $=$ 1/2 and 1 are interpreted in terms of an analytical three-site model.

DOI
10.1103/physrevb.85.134415

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A density matrix renormalization group (DMRG) algorithm is presented for the Bethe lattice with connectivity Z=3Z=3Z=3 and antiferromagnetic exchange between nearest-neighbor spins s=1/2s=1/2s=1/2 or 1 sites in successive generations ggg. The algorithm is accurate for s=1s=1s=1 sites. The ground states are magnetic with spin S(g)=2gsS(g)={2}^{g}sS(g)=2gs, staggered magnetization that persists for large g>20g>20g>20, and short-range spin correlation functions that decrease exponentially. A finite energy gap to S>S(g)S>S(g)S>S(g) leads to a magnetization plateau in the extended lattice. Closely similar DMRG results for sss === 1/2 and 1 are interpreted in terms of an analytical three-site model.
The half-filled Hubbard model on the Bethe lattice with coordination number z=3z=3z=3 is studied using the density-matrix renormalization group (DMRG) method. Ground-state properties such as the energy EEE, average local magnetization $ $, its fluctuations $ - ^2$ and various spin correlation functions $ - $ are determined as a function of the Coulomb interaction strength U/tU/tU/t. The calculated local magnetic moments $ $ increase monotonically with increasing Coulomb repulsion U/tU/tU/t forming an antiferromagnetic spin-density-wave state which matches the two sublattices of the bipartite Bethe lattice. At large U/tU/tU/t, $ $ is strongly reduced with respect to the saturation value 1/2 due to exchange fluctuations between nearest neighbors (NN) spins (∣∣≃0.35| |\simeq 0.35∣∣≃0.35 for U/t→+∞U/t\to +\inftyU/t→+∞). $ - ^2$ shows a maximum for U/t=2.4U/t=2.4U/t=2.4--2.9 which results from the interplay between the usual increase of $ $ with increasing U/tU/tU/t and the formation of important permanent moments $ $ at large U/tU/tU/t. NN sites show antiferromagnetic spin correlations which increase with increasing Coulomb repulsion. In contrast next NN sites are very weakly correlated over the whole range of U/tU/tU/t. The accuracy of the DMRG results is discussed by comparison with tight-binding exact results, independent DMRG calculations for the Heisenberg model and simple first-order perturbation estimates.
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A modified density matrix renormalization group (DMRG) algorithm is applied to the zigzag spin-12\frac{1}{2}21​ chain with frustrated antiferromagnetic exchange J1{J}_{1}J1​ and J2{J}_{2}J2​ between first and second neighbors. The modified algorithm yields accurate results up to ${J}_{2}/{J}_{1}\ensuremath{\approx}4$ for the magnetic gap $\ensuremath{\Delta}$ to the lowest triplet state, the amplitude BBB of the bond order wave phase, the wavelength $\ensuremath{\lambda}$ of the spiral phase, and the spin correlation length $\ensuremath{\xi}$. The J2/J1{J}_{2}/{J}_{1}J2​/J1​ dependences of $\ensuremath{\Delta}$, BBB, $\ensuremath{\lambda}$, and $\ensuremath{\xi}$ provide multiple comparisons to field theories of the zigzag chain. The twist angle of the spiral phase and the spin structure factor yield additional comparisons between DMRG and field theory. Attention is given to the numerical accuracy required to obtain exponentially small gaps or exponentially long correlations near a quantum phase transition.
The density matrix renormalization group (DMRG) method generates the low-energy states of linear systems of NNN sites with a few degrees of freedom at each site by starting with a small system and adding sites step by step while keeping constant the dimension of the truncated Hilbert space. DMRG algorithms are adapted to open chains with inversion symmetry at the central site, to cyclic chains and to weakly coupled chains. Physical properties rather than energy accuracy is the motivation. The algorithms are applied to the edge states of linear Heisenberg antiferromagnets with spin S≥1S \ge 1S≥1 and to the quantum phases of a frustrated spin-1/2 chain with exchange between first and second neighbors. The algorithms are found to be accurate for extended Hubbard and related 1D models with charge and spin degrees of freedom.
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