Journal of the Korean Physical Society

Strong Odd Bond - Weak Even Bond Picture in the Half-Filled One-Dimensional Hubbard Model

Myung-Hoon Chung

Published Apr 1, 2019• 2 citations

Abstract

We study the strong odd bond - weak even bond picture by measuring the bond energy and the half-chain entanglement entropy in the one-dimensional Hubbard model. By using the infinite density matrix renormalization group with matrix product operator, we obtain the ground state of the matrix product state. As a function of the bond dimension, the bond energy and the half-chain entanglement entropy support the strong odd bond - weak even bond picture. From the expectation values of the occupation number, we numerically observe that the odd - even effect appears only for the half-filling case with a finite bond dimension. Our results confirm that the matrix product operator method is successful in exploring itinerant fermionic systems.

DOI: 10.3938/jkps.74.674

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Journal of the Korean Physical Society

Strong Odd Bond - Weak Even Bond Picture in the Half-Filled One-Dimensional Hubbard Model

Myung-Hoon Chung

Published Apr 1, 2019• 2 citations

Publisher

Abstract

DOI: 10.3938/jkps.74.674

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arXiv:1901.09151Physical review. B./Physical review. BMay 24, 201916 citations

The one-dimensional

\mathrm{SU}(n)

Hubbard model is investigated numerically for

n=2

, 3, 4, and 5 at half-filling and

1∕n

filling using the density-matrix renormalization-group method. The energy gaps and various quantum information entropies are calculated. In the half-filled case, finite spin and charge gaps are found for arbitrary positive

U

n>2

. Furthermore, it is shown that the transition to the gapped phase at

{U}_{c}=0

is of Kosterlitz-Thouless type and is accompanied by a bond dimerization both for even and odd

n

. In the

1∕n

-filled case, the transition has similar features as the metal-insulator transition in the half-filled SU(2) Hubbard model. The charge gap opens exponentially slowly for

U>{U}_{c}=0

, the spin sector remains gapless, and the ground state is nondimerized.

U

and system size. In both the one- and two-dimensional cases, we find that the subsystem entropy scales proportionally to

{U}^{2}

for weak

U

and proportionally to volume. Nevertheless, the optimized momentum-space DMRG can provide variationally accurate results for the two-dimensional Hubbard model at weak coupling for moderate system sizes.

i

and

j

refer to the lattice sites and $\ensuremath{\sigma}$ to the spin. The interaction-energy functional $W[{\ensuremath{\gamma}}_{ij\ensuremath{\sigma}}]$ is calculated exactly for representative finite periodic systems by performing exact Lanczos diagonalizations. The relationship between $W[{\ensuremath{\gamma}}_{ij\ensuremath{\sigma}}]$ and the entropy $S[{\ensuremath{\eta}}_{\mathbit{k}\ensuremath{\sigma}}]$ of independent fermions with natural-orbital occupations ${\ensuremath{\eta}}_{\mathbit{k}\ensuremath{\sigma}}$ is analyzed. A simple approximation to the interaction energy of the half-filled Hubbard model is proposed, which takes the form $W=W(S[{\ensuremath{\eta}}_{\mathbit{k}\ensuremath{\sigma}}])$. Using this functional we derive the ground-state energy, kinetic energy, average number of double occupations, charge distribution, magnetic susceptibility, and field-induced spin polarization in one-, two-, and three-dimensional periodic lattices. The limit of infinite dimensions is also explored. The accuracy of the method is assessed by comparison with available exact numerical or analytical results. Goals, limitations, and possible extensions of the domain of applicability of the functional are discussed.