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arXiv:quant-ph/0602048 · Physical Review A

Single-site entanglement of fermions at a quantum phase transition

Daniel Larsson, Henrik Johannesson

Published Apr 25, 2006• 72 citations
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Abstract

We show that the single-site entanglement of a generic spin-1/2 fermionic lattice system can be used as a reliable marker of a finite-order quantum phase transition, given certain provisos. We discuss the information contained in the single-site entanglement measure, and provide illustrations from the Mott--Hubbard metal-insulator transitions of the one-dimensional (1D) Hubbard model, and the (1D) Hubbard model with long-range hopping.

DOI
10.1103/physreva.73.042320

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We show that the single-site entanglement of a generic spin-1/2 fermionic lattice system can be used as a reliable marker of a finite-order quantum phase transition, given certain provisos. We discuss the information contained in the single-site entanglement measure, and provide illustrations from the Mott--Hubbard metal-insulator transitions of the one-dimensional (1D) Hubbard model, and the (1D) Hubbard model with long-range hopping.
The study of entanglement properties of quantum critical many-particle systems has become a subject of intense interest. While the basic features of entanglement scaling for critical spin-1/2 systems (coupled qubits) are by now fairly well understood, entanglement properties of critical fermions (or bosons) are less well studied. In an effort to contribute to this problem, we have analyzed the single-site entanglement of a generic spin-1/2 lattice fermion system and found that (under certain provisos) this measure can be used as a reliable marker to identify and partly characterize a quantum critical point. We illustrate our findings by exact analytical results for the single-site entanglement at the magnetic and Mott-Hubbard transitions of the 1D Hubbard model and at the Mott-Hubbard transitions of the 1D Hubbard model with long-range hopping.
Entanglement has been widely investigated in condensed-matter systems since they are considered good candidates for developing quantum technologies. Additionally, entanglement is a powerful tool to explore quantum phase transitions in strongly correlated systems, with the von Neumann entropy being considered a proper measure of quantum entanglement for pure bipartite systems. For lattice systems, in particular, the single-site entanglement quantifies how much information about the quantum state of the remaining sites can be obtained by a measurement at a single site. Here, we use quantum Monte Carlo calculations to obtain the average single-site entanglement for the two-dimensional Hubbard model in different geometries, probing the effects of varying temperature and interaction strength. We find that the average single-site entanglement signals the quantum phase transitions in such systems, allowing us to identify and characterize signatures of quantum phase transitions even at finite temperatures. We also analyze the relation between entanglement and magnetic susceptibility: in all the geometries considered, we find regimes in which the quantities are roughly linearly connected, revealing their link to the Mott transition.
"We studied the one-dimensional Bose-Hubbard model, by means of the Density Matrix Renormalization Group (DMRG). The U parameter, which contains the Coulombian repulsion information, is fixed to U = 1 and it is performed a swept over the hopping parameter t. The entanglement is investigated by implementing the one-site, two-site and block von Neumann entropy looking for which of these measures give account of the quantum phase transition from Mott insulator to superfluid. The one-site and two-site entropy do not give information of the phase transition of the system. In contrast, the block entropy shows, in the Mott insulator phase a behavior that saturates rapidly to a finite value and in the superfluid phase shows a logarithmic behavior that agrees with the mean field theory. From the derivative of the block entropy average as a function of t we found the phase transition takes place at tc =0.15."
The role of two-point and multipartite entanglement at quantum phase transitions (QPTs) in correlated electron systems is investigated. We consider a bond-charge extended Hubbard model exactly solvable in one dimension which displays various QPTs---with two (qubit) as well as more (qudit) on-site degrees of freedom involved. The analysis is carried out by means of appropriate measures of bipartite/multipartite quantum correlations. It is found that all transitions ascribed to two-point correlations are characterized by an entanglement range which diverges at the transition points. The exponent coincides with that of the correlation length at the transitions. We introduce the correlation ratio, namely, the ratio of quantum mutual information and single-site entanglement. We show that at T=0T=0T=0, it captures the relative role of two-point and multipartite quantum correlations at transition points, generalizing to qudit systems the entanglement ratio. Moreover, a finite value of quantum mutual information between infinitely distant sites is seen to quantify the presence of off-diagonal long-range order induced by multipartite entanglement.
We study the critical behavior of the single-site entanglement entropy S at the Mott metal-insulator transition in infinite-dimensional Hubbard model. For this model, the entanglement between a single site and rest of the lattice can be evaluated exactly, using the dynamical mean-field theory (DMFT). Both the numerical solution using exact diagonalization and the analytical one using two-site DMFT gives S - Sc∝ α log2[(1/2 - Dc)/Dc](U - Uc), with Dcthe double occupancy at Ucand α < 0 being different on two sides of the transition.