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Physical Review A

Thermal entanglement witness for interacting itinerant fermion systems

André Neves Ribeiro, Cláudio Andrade Macedo

Published Aug 20, 2012• 3 citations
Publisher

Abstract

The Hubbard model describes interacting itinerant fermion systems and is the simplest model capable of describing the essential physics of strongly correlated electron systems. In this model, both spin correlations and charge correlations are important for the entanglement, thus the magnetic susceptibility is not an appropriate thermal entanglement witness. The specific heat reflects spin correlations and charge correlations. We obtain a relation that permits the specific heat to be used directly as a thermal entanglement witness. We calculate, away from half filling, the entanglement critical temperature ${T}_{E}$, below which entanglement is detected, for small linear clusters and the hypercubic lattice in the limit of infinite dimensions by using the exact numerical diagonalization method and the dynamical mean-field theory, respectively. We find, in both cases, that ${T}_{E}$ increases with increasing strength of the on-site Coulomb repulsion $U$.

DOI
10.1103/physreva.86.022320

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The Hubbard model describes interacting itinerant fermion systems and is the simplest model capable of describing the essential physics of strongly correlated electron systems. In this model, both spin correlations and charge correlations are important for the entanglement, thus the magnetic susceptibility is not an appropriate thermal entanglement witness. The specific heat reflects spin correlations and charge correlations. We obtain a relation that permits the specific heat to be used directly as a thermal entanglement witness. We calculate, away from half filling, the entanglement critical temperature TE{T}_{E}TE​, below which entanglement is detected, for small linear clusters and the hypercubic lattice in the limit of infinite dimensions by using the exact numerical diagonalization method and the dynamical mean-field theory, respectively. We find, in both cases, that TE{T}_{E}TE​ increases with increasing strength of the on-site Coulomb repulsion UUU.
We show that the thermal entanglement in a spin system using only magnetic-susceptibility measurements is restricted to the insulator materials. We develop a generalization of the thermal entanglement witness that allows us to get information about the system entanglement with variable local spin lengths that can be used experimentally in conductor or insulator materials. As an application, we study thermal entanglement for the half-filled Hubbard model for linear, square, and cubic clusters. We note that it is the itinerancy of electrons that favors the entanglement. Our results suggest a weak dependence between entanglement and external spin freedom degrees.
The Hubbard model is a longstanding problem in the theory of strongly correlated electrons and a very active one in the experiments with ultracold fermionic atoms. Motivated by current and prospective quantum simulations, we apply a two-dimensional tensor network---an infinite projected entangled pair state---evolved in imaginary time by the neighborhood tensor update algorithm working directly in the thermodynamic limit. With $\mathrm{U}(1)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1)$ symmetry and the bond dimensions up to 29, we generate thermal states down to the temperature of 0.17 times the hopping rate. We obtain results for spin and charge correlators, unaffected by boundary effects. The spin correlators---measurable in prospective ultracold atoms experiments attempting to approach the thermodynamic limit---provide evidence of disruption of the antiferromagnetic background with mobile holes in a slightly doped Hubbard model. The charge correlators reveal the presence of hole-doublon pairs near half filling and signatures of hole-hole repulsion on doping. We also obtain specific heat in the slightly doped regime.
Exchange interactions in spin systems can give rise to quantum entanglement in the ground and thermal states of the systems. We consider a spin tetramer, with spins of magnitude 12\frac{1}{2}21​, in which the spins interact via nearest-neighbor, diagonal, and four-spin interactions of strength J1{J}_{1}J1​, J2{J}_{2}J2​, and KKK, respectively. The ground- and thermal-state entanglement properties of the tetramer are calculated analytically in the various limiting cases. Both bipartite and multipartite entanglements are considered and a signature of the quantum phase transition (QPT), in terms of the entanglement ratio, is identified. The first-order QPT is accompanied by discontinuities in the nearest-neighbor and diagonal concurrences. The magnetic properties of a S=12S=\frac{1}{2}S=21​ antiferromagnetic polyoxovanadate compound V12 are well explained by tetramers, with J2=0{J}_{2}=0J2​=0, K=0K=0K=0, in which the spins interact via the isotropic Heisenberg exchange interaction Hamiltonian. Treating the magnetic susceptibility $\ensuremath{\chi}$ as an entanglement witness (EW), an estimate of the lower bound of the critical entanglement temperature Tc{T}_{c}Tc​ below which entanglement is present in the experimental compound, is determined. Two other cases considered include the symmetric tetramer---i.e., tetrahedron (J1=J2,K=0)({J}_{1}={J}_{2},K=0)(J1​=J2​,K=0)---and the symmetric trimer. In both the cases, there is no entanglement between a pair of spins in the thermal state but multipartite entanglement is present. A second EW based on energy provides an estimate of the entanglement temperature TE{T}_{E}TE​ below which the thermal state is definitely entangled. This EW detects bipartite entanglement in the case of the tetramer describing a square of spins (the case of V12 ) and multipartite entanglement in the cases of the tetrahedron and symmetric trimer.
We have calculated the spin correlation length of the half-filled one-dimensional Hubbard model down to the temperature of T =0.01×(transfer integral) by the Bethe-Ansatz method for the quantum transfer matrix. Its temperature dependence shows the crossover from (2 log π T ) -1 at high temperatures to a T -1 with the log corrections at low temperatures, the low- T behavior being consistent with the prediction by the conformal field theory. When the Coulomb repulsion U is small, we have found at low temperatures another crossover in the proportional constant a from the value for the free electron system to that for the antiferromagnetic Heisenberg model. This crossover is due to the disappearance of charge fluctuation. A similar crossover has been observed for the linear coefficient of specific heat, C / T .
We have calculated the spin correlation length of the half-filled one-dimensional Hubbard model down to the temperature of T =0.01×(transfer integral) by the Bethe-Ansatz method for the quantum transfer matrix. Its temperature dependence shows the crossover from (2 log π T ) -1 at high temperatures to a T -1 with the log corrections at low temperatures, the low- T behavior being consistent with the prediction by the conformal field theory. When the Coulomb repulsion U is small, we have found at low temperatures another crossover in the proportional constant a from the value for the free electron system to that for the antiferromagnetic Heisenberg model. This crossover is due to the disappearance of charge fluctuation. A similar crossover has been observed for the linear coefficient of specific heat, C / T .
We construct entanglement witnesses using fundamental quantum operators of spin models which contain two-particle interactions and have a certain symmetry. By choosing the Hamiltonian as such an operator, our method can be used for detecting entanglement by energy measurement. We apply this method to the Heisenberg model in a cubic lattice with a magnetic field, the XYXYXY model, and other familiar spin systems. Our method provides a temperature bound for separable states for systems in thermal equilibrium. We also study the Bose-Hubbard model and relate its energy minimum for separable states to the minimum obtained from the Gutzwiller ansatz.