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A DMRG study of the half-filled Hubbard model on a Bethe lattice: a signature of the Mott-Hubbard transition

Marie-Bernadette Lepetit, Maixent Cousy

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 $ - ^2$ and various spin correlation functions $ - $ 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 ($| |\simeq 0.35$ for $U/t\to +\infty$). $ - ^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.

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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.

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Journal of Physics Conference Series•Aug 12, 2013•2 citations
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 - ^2−2 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 - ^2−2 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.
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.
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.
We investigate the ground-state phase diagram of the half-filled one-dimensional Hubbard model with next-nearest-neighbor hopping using the density-matrix renormalization-group technique as well as an unrestricted Hartree-Fock approximation. We find commensurate and incommensurate disordered magnetic insulating phases and a spin-gapped metallic phase in addition to the one-dimensional Heisenberg phase. At large on-site Coulomb repulsion U, we make contact with the phase diagram of the frustrated Heisenberg chain, which has spin-gapped phases for sufficiently large frustration. For weak U, sufficiently large next-nearest-neighbor hopping t2{t}_{2}t2​ leads to a band structure with four Fermi points rather than two, producing a spin-gapped metallic phase. As U is increased in this regime, the system undergoes a Mott-Hubbard transition to a frustrated antiferromagnetic insulator.
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 attractive Hubbard model is investigated in the framework of lattice density-functional theory (LDFT). The ground-state energy E=T+WE=T+WE=T+W is regarded as a functional of the single-particle density matrix ${\ensuremath{\gamma}}_{ij}$ with respect to the lattice sites, where $T[\ensuremath{\gamma}]$ represents the kinetic and crystal-field energies and $W[\ensuremath{\gamma}]$ the interaction energy. Aside from the exactly known functional $T[\ensuremath{\gamma}]$, we propose a simple scaling approximation to $W[\ensuremath{\gamma}]$, which is based on exact analytic results for the attractive Hubbard dimer and on a scaling hypothesis within the domain of representability of $\ensuremath{\gamma}$. As applications, we consider one-, two-, and three-dimensional finite and extended bipartite lattices having homogeneous or alternating onsite energy levels. In addition, the Bethe lattice is investigated as a function of coordination number. Results are given for the kinetic, Coulomb, and total energies, as well as for the density distribution ${\ensuremath{\gamma}}_{ii}$, nearest-neighbor bond order ${\ensuremath{\gamma}}_{ij}$, and pairing energy $\ensuremath{\Delta}{E}_{p}$, as a function of the interaction strength ∣U∣/t|U|/t∣U∣/t, onsite potential $\ensuremath{\varepsilon}/t$, and band filling n=Ne/Nan={N}_{e}/{N}_{a}n=Ne​/Na​. Remarkable even-odd and super-even oscillations of $\ensuremath{\Delta}{E}_{p}$ are observed in finite rings as a function of band filling. Comparison with exact Lanczos diagonalizations and density-matrix renormalization-group calculations shows that LDFT yields a very good quantitative description of the properties of the model in the complete parameter range, thus providing a significant improvement over the mean-field approaches. Goals and limitations of the method are discussed.
The half-filled Hubbard model on the honeycomb lattice is investigated by numerically exact large-scale quantum Monte Carlo simulations for lattice sizes up to 2592 sites. By performing careful finite-size scaling for the spin-spin correlation functions, calculated with a high degree of accuracy, we find that the ground state is antiferromagnetically long-range ordered at U / t = 4, where U is the on-site Hubbard interaction and t is the nearest neighbor hopping. Our result is in sharp contrast to a recent report [Meng et al. , Nature 464, 847 (2010)], where instead strong evidence of spin liquid behavior is found.