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Finite-entanglement scaling properties of the one-dimensional fermion Hubbard model

Min-Chul Cha

Published Aug 21, 2019• 0 citations
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We consider the Ising phase of the antiferromagnetic XXZ Heisenberg chain on a finite-size lattice with N sites. We compute the large-NNN behavior of the spin stiffness, obtaining the correlation length $\ensuremath{\xi}.$ We use our results to discuss the scaling behavior of the metal-insulator transitions in one-dimensional systems, taking into account the mapping between the XXZ Heisenberg chain and the spinless fermion model, and known results for the Hubbard model. We study the scaling properties of both the Hubbard model and the XXZ Heisenberg chain by solving numerically the Bethe ansatz equations. We find that for some range of values of $\ensuremath{\xi}/N,$ the scaling behavior may be observed for the Hubbard model but not for the XXZ Heisenberg chain. We show how $\ensuremath{\xi}$ can be obtained from the scaling properties of the spin stiffness for small system sizes. This method can be applied to models having not an exact solution, illuminating their transport properties.
We study the effects of limited entanglement in the one-dimensional Hubbard model by representing the ground states in the form of matrix product states. Finite-entanglement scaling behavior over a wide range is observed at half-filling. The critical exponents characterizing the length scale in terms of the size of the matrices used are obtained, confirming the theoretical prediction that the values of the exponents are solely determined by the central charge. The entanglement spectrum shows that a global double degeneracy occurs in the ground states with a charge gap. We also find that the Mott transition, tuned by changing the chemical potential, always occurs through a first-order transition, and the metallic phase has a few conducting states, including the states with a mean-field nature close to the critical point, as expected in variational matrix product states with a finite amount of entanglement.