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arXiv:cond-mat/9801228

Density-matrix renormalization study of the Hubbard model on a Bethe lattice

Marie-Bernadette Lepetit, Maixent Cousy, G. M. Pastor

Published Jan 22, 1998• 0 citations
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Abstract

The half-filled Hubbard model on the Bethe lattice with coordination number $z=3$ is studied using the density-matrix renormalization group (DMRG) method. Ground-state properties such as the energy $E$, average local magnetization $$, its fluctuations $ - < \hat S_z>^2$ and various spin correlation functions $ - < S_z(i)> < S_z(j)>$ are determined as a function of the Coulomb interaction strength $U/t$. The calculated local magnetic moments $$ increase monotonically with increasing Coulomb repulsion $U/t$ forming an antiferromagnetic spin-density-wave state which matches the two sublattices of the bipartite Bethe lattice. At large $U/t$, $$ is strongly reduced with respect to the saturation value 1/2 due to exchange fluctuations between nearest neighbors (NN) spins ($|< S_z(i)>|\simeq 0.35$ for $U/t\to +\infty$). $ - < S_z(i)>^2$ shows a maximum for $U/t=2.4$--2.9 which results from the interplay between the usual increase of $$ with increasing $U/t$ and the formation of important permanent moments $$ at large $U/t$. 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 $U/t$. 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.

DOI
10.48550/arxiv.cond-mat/9801228

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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 $ - < \hat S_z>^2$ and various spin correlation functions $ - < S_z(i)> < S_z(j)>$ are determined as a function of the Coulomb interaction strength $U/t$. 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 ($|< S_z(i)>|\simeq 0.35$ for $U/t\to +\infty$). $ - < S_z(i)>^2$ shows a maximum for $U/t=2.4$--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.
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.
Two-dimensional density-matrix renormalization group method is employed to examine the ground-state phase diagram of the Hubbard model on the triangular lattice at half-filling. The calculation reveals two discontinuities in the double occupancy with increasing the repulsive Hubbard interaction UUU at ${U}_{\mathrm{c}1}\ensuremath{\sim}7.8t$ and ${U}_{\mathrm{c}2}\ensuremath{\sim}9.9t$ (ttt being the hopping integral), indicating that there are three phases separated by first-order transitions. The absence of any singularity in physical quantities for $0\ensuremath{\le}U<{U}_{\mathrm{c}1}$ implies a metallic phase in this regime. For U>Uc2U>{U}_{\mathrm{c}2}U>Uc2​, the local spin density induced by an applied pinning magnetic field exhibits a three sublattice feature, which is compatible with the ${120}^{\ensuremath{\circ}}$ N\'eel-ordered state realized in the limit of $U\ensuremath{\rightarrow}\ensuremath{\infty}$. For Uc1<U<Uc2{U}_{\mathrm{c}1}<U<{U}_{\mathrm{c}2}Uc1​<U<Uc2​, a response to the applied pinning magnetic field is comparable to that in the metallic phase with a relatively large spin correlation length, but showing neither valence bond nor chiral magnetic order, which therefore resembles gapless spin liquid. However, the spin structure factor for the intermediate phase exhibits the maximum at the K\mathrm{K}K and ${\mathrm{K}}^{\ensuremath{'}}$ points in the momentum space, which is not compatible to spin liquid with a large spinon Fermi surface. The calculation also finds that the pairing correlation function monotonically decreases with increasing UUU and thus the superconductivity is unlikely in the intermediate phase.
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.
An efficient density matrix renormalization group (DMRG) algorithm is presented and applied to Y junctions, systems with three arms of nnn sites that meet at a central site. The accuracy is comparable to DMRG of chains. As in chains, new sites are always bonded to the most recently added sites and the superblock Hamiltonian contains only new or once renormalized operators. Junctions of up to $N=3n+1\ensuremath{\approx}500$ sites are studied with antiferromagnetic (AF) Heisenberg exchange JJJ between nearest-neighbor spins SSS or electron transfer ttt between nearest neighbors in half-filled Hubbard models. Exchange or electron transfer is exclusively between sites in two sublattices with ${N}_{A}\ensuremath{\ne}{N}_{B}$. The ground state (GS) and spin densities ${\ensuremath{\rho}}_{r}=\ensuremath{\langle}{S}_{r}^{z}\ensuremath{\rangle}$ at site rrr are quite different for junctions with S=1/2S=1/2S=1/2, 1, 3/2, and 2. The GS has finite total spin SG=2S(S){S}_{G}=2S(S)SG​=2S(S) for even (odd) NNN and for MG=SG{M}_{G}={S}_{G}MG​=SG​ in the SG{S}_{G}SG​ spin manifold, ${\ensuremath{\rho}}_{r}>0(<0)$ at sites of the larger (smaller) sublattice. S=1/2S=1/2S=1/2 junctions have delocalized states and decreasing spin densities with increasing NNN. S=1S=1S=1 junctions have four localized Sz=1/2{S}_{z}=1/2Sz​=1/2 states at the end of each arm and centered on the junction, consistent with localized states in S=1S=1S=1 chains with finite Haldane gap. The GS of S=3/2S=3/2S=3/2 or 2 junctions of up to 500 spins is a spin density wave with increased amplitude at the ends of arms or near the junction. Quantum fluctuations completely suppress AF order in S=1/2S=1/2S=1/2 or 1 junctions, as well as in half-filled Hubbard junctions, but reduce rather than suppress AF order in S=3/2S=3/2S=3/2 or 2 junctions.
Magnetic properties of the two-dimensional Hubbard-model are inferred from results of Monte Carlo simulations. Lattice sizes up to 8\ifmmode\times\else\texttimes\fi{}8 and temperatures down to T=t20T=\frac{t}{20}T=20t​ (t=hoppingt=\mathrm{hopping}t=hopping) were studied. The half-filled system is found to exhibit antiferromagnetic long-range order for all values of the Coulomb repulsion UUU. The low-temperature magnetic properties are found to be well described by spin-wave theory with renormalized local moment and spin-wave velocity. Numerical evidence presented suggests that when doped the system loses the long-range order immediately away from half filling.
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