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arXiv:1508.07477 · Physical Review B

Entanglement structure of the Hubbard model in momentum space

G. Ehlers, Jenő Sólyom, Örs Legeza, R. M. Noack

Published Dec 8, 2015• 34 citations
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Abstract

We study the properties of the ground states of the one- and two-dimensional Hubbard models at half-filling and moderate doping using entanglement-based measures, which we calculate numerically using the momentum-space density matrix renormalization group (DMRG). In particular, we investigate quantities such as the single-site entropy and two-site mutual information of single-particle momentum states as well as the behavior of the bipartite subsystem entropy for partitions in momentum space. The distribution of these quantities in momentum space gives insight into the fundamental nature of the ground state, insight that can be used to make contact with weak-coupling-based analytic approaches and to optimize numerical methods, the momentum-space DMRG in particular. We study the site and subsystem entropies as a function of interaction strength $U$ 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.

DOI
10.1103/physrevb.92.235116

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The performance of the density matrix renormalization group (DMRG) is strongly influenced by the choice of the local basis of the underlying physical lattice. We demonstrate that, for the two-dimensional Hubbard model, the hybrid--real-momentum-space formulation of the DMRG is computationally more efficient than the standard real-space formulation. In particular, we show that the computational cost for fixed bond dimension of the hybrid-space DMRG is approximately independent of the width of the lattice, in contrast to the real-space DMRG, for which it is proportional to the width squared. We apply the hybrid-space algorithm to calculate the ground state of the doped two-dimensional Hubbard model on cylinders of width four and six sites; at n=0.875n=0.875n=0.875 filling, the ground state exhibits a striped charge-density distribution with a wavelength of eight sites for both U/t=4.0U/t=4.0U/t=4.0 and 8.08.08.0. We find that the strength of the charge ordering depends on U/tU/tU/t and on the boundary conditions. Furthermore, we investigate the magnetic ordering as well as the decay of the static spin, charge, and pair-field correlation functions.
In order to optimize the ordering of the lattice sites in the momentum space and quantum chemistry versions of the density-matrix renormalization group (DMRG) method we have studied the separability and entanglement of the target state for the one-dimensional Hubbard model and various molecules. By analyzing the behavior of von Neumann entropy we have found criteria that help to fasten convergence. An initialization procedure has been developed which maximizes the Kullback-Leibler entropy and extends the active space in a dynamical fashion. The dynamically extended active space procedure reduces significantly the effective system size during the first half-sweep and accelerates the speed of convergence of momentum space DMRG and quantum chemistry DMRG to a great extent. The effect of lattice site ordering on the number of block states to be kept during the RG procedure is also investigated.
In order to optimize the ordering of the lattice sites in the momentum space and quantum chemistry versions of the density matrix renormalization group (DMRG) method we have studied the separability and entanglement of the target state for the 1-D Hubbard model and various molecules. By analyzing the behavior of von Neumann and Neumann-Renyi entropies we have found criteria that help to fasten convergence. A new initialization procedure has been developed which maximizes the Kullback-Leibler entropy and extends the active space (AS) in a dynamical fashion. The dynamically extended active space (DEAS) procedure reduces significantly the effective system size during the first half sweep and accelerates the speed of convergence of momentum space DMRG and quantum chemistry DMRG to a great extent. The effect of lattice site ordering on the number of block states to be kept during the RG procedure is also investigated.
A momentum-space approach of the density-matrix renormalization-group (DMRG) method is developed. Ground-state energies of the Hubbard model are evaluated using this method and compared with exact diagonalization as well as quantum Monte Carlo results. It is shown that the momentum-space DMRG is a very useful numerical tool for studying the Hubbard model and other fundamental models of interacting electrons in two dimensions. For the Hubbard model in two dimensions, the momentum-space DMRG method reproduces accurately the exact diagonalization results of ground-state energies on a 4\ifmmode\times\else\texttimes\fi{}4 lattice and yields new upper bounds of ground-state energies on an 8\ifmmode\times\else\texttimes\fi{}8 lattice.
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.