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arXiv:quant-ph/0405130 · Journal of Physics A Mathematical and General

Entanglement and off-diagonal long-range order of an η-pairing state

Heng Fan, Seth Lloyd

Published May 25, 2005• 25 citations
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Abstract

Off-diagonal long-range order (ODLRO) which is believed to be one characteristic of superconductivity is a quantum phenomenon not describable in classical mechanical terms. The quantum state constructed by η-pairing demonstrates ODLRO. Entanglement is a key concept of the quantum information processing and has no classical counterpart. We study the entanglement property of the η-pairing quantum state by concurrence and entropy which are two measures of the entanglement. We show that the concurrence of entanglement between one-site and the rest sites is exactly the correlation function of the ODLRO for the η-pairing state in the thermodynamic limit. So, when the η-pairing state is entangled, it demonstrates ODLRO and is thus in the superconducting phase, if it is a separable state, there is no ODLRO. In the thermodynamical limit, the entanglement between the M -site and other sites of the η-pairing state does not vanish. Other types of ODLRO of the η-pairing state are presented. We show that the behaviour of the ODLRO correlation functions is equivalent to that of the entanglement of the η-pairing state. The scaling of the entropy of the entanglement for the η-pairing state is studied.

DOI
10.1088/0305-4470/38/23/014

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Off-diagonal long-range order (ODLRO) which is believed to be one characteristic of superconductivity is a quantum phenomenon not describable in classical mechanical terms. The quantum state constructed by η-pairing demonstrates ODLRO. Entanglement is a key concept of the quantum information processing and has no classical counterpart. We study the entanglement property of the η-pairing quantum state by concurrence and entropy which are two measures of the entanglement. We show that the concurrence of entanglement between one-site and the rest sites is exactly the correlation function of the ODLRO for the η-pairing state in the thermodynamic limit. So, when the η-pairing state is entangled, it demonstrates ODLRO and is thus in the superconducting phase, if it is a separable state, there is no ODLRO. In the thermodynamical limit, the entanglement between the M -site and other sites of the η-pairing state does not vanish. Other types of ODLRO of the η-pairing state are presented. We show that the behaviour of the ODLRO correlation functions is equivalent to that of the entanglement of the η-pairing state. The scaling of the entropy of the entanglement for the η-pairing state is studied.
In this paper, we intend to show that macroscopic entanglement is possible at high temperatures. We have analysed multipartite entanglement produced by the η-pairing mechanism, which features strongly in the fermionic lattice models of high T c superconductivity. This problem is shown to be equivalent to calculating multipartite entanglement in totally symmetric states of qubits. It is demonstrated that we can conclusively calculate the relative entropy of entanglement within any subset of qubits in the overall symmetric state. Three main results are then presented. First, the condition for superconductivity, namely existence of the off-diagonal long-range order (ODLRO), is dependent not on two-site entanglement but just classical correlations as the sites become more and more distant. Secondly, the entanglement that does survive in the thermodynamical limit is the entanglement of the total lattice and, at half-filling, it scales with the log of the number of sites. It is this entanglement that will exist at temperatures below the superconducting critical temperature, which can currently be as high as 160 K. Finally, it is proved that a complete mixture of symmetric states does not contain any entanglement in the macroscopic limit. On the other hand, a mixture of symmetric states possesses the same two qubit entanglement features as the pure states involved, in the sense that the mixing does not destroy entanglement for a finite number of qubits, albeit it does decrease it. Furthermore, maximal mixing of symmetric states does not destroy ODLRO and classical correlations. We discuss generalizations to the subsystems of any dimensionality (i.e. higher than spin-half).
The quantum states built with the $\mathrm{\ensuremath{\eta}}$ pairing mechanism, i.e., $\mathrm{\ensuremath{\eta}}$ pairing states, were first introduced in the context of high-temperature superconductivity where they were recognized as important example of states allowing for off-diagonal long-range order (ODLRO). In this paper we describe the structure of the correlations present in these states when considered in their momentum representation and we explore the relations between the quantum bipartite and multipartite correlations exhibited in kkk space and the direct lattice superconducting correlations. In particular, we show how the negativity between paired momentum modes is directly related to the ODLRO. Moreover, we investigate the dependence of the block entanglement on the choice of the modes forming the block and on the ODLRO; consequently we determine the multipartite content of the entanglement through the evaluation of the generalized Meyer-Wallach measure in the direct and reciprocal lattice. The determination of the persistency of entanglement shows how the network of correlations depicted exhibits a self-similar structure which is robust with respect to ``local'' measurements. Finally, we recognize how a relation between the momentum-space quantum correlations and the ODLRO can be established even in the case of BCS states.
We show that in the translation invariant case and in the antiferromagnetic phase, the reduced density matrix ρ2 has no off-diagonal long-range order of on-site electron pairs for the single-band Hubbard model on a cubic lattice away from half filling at finite temperatures both for the positive coupling and for the negative coupling. In these cases the model can not give a mechanism for the superconductivity caused by the condensation of on-site electron pairs and the nearestneighbor electron pairs. PACS numbers: 71.10.+x, 74.20.-z, 05.30.-d. 1 It was proposed the possibility that the Hubbard model can give a mechanism to characterize high temperature superconductivity[1]. It has been shown that the η pairing states for the single band Hubbard model with constant hopping matrix element for nearest neighbor sites have ODLRO[2]. It has also been shown that in the ground state the model with negative coupling has off-diagonal long-range order (ODLRO) on some bipartite lattice[3]. However, as the temperature T> 0 whether the reduced density matrix ρ2 has ODLRO is inconclusive. The purpose of this work is to study the ODLRO of ρ2 in the single-band Hubbard model with constant hopping matrix element for nearest neighbor pairs on a cubic lattice at finite temperatures. In the translation invariant case we obtain an equation ( (10) in the following) which is satisfied by the long-range correlation functions of on-site electron pairs. From which we find that there is no ODLRO of on-site electron pairs. Then we consider the case with an antiferromagnetic background. In this case we will show that there is still no ODLRO of on-site electron pairs. The Hamiltonian of the model is H = −t ∑ (a † rar ′ + b † rb ∑
We discuss new possibilities for off-diagonal long range order (ODLRO) in spin chains involving operators which add or delete sites from the chain. For the one-dimensional, periodic Heisenberg, inverse square exchange, and XY\mathrm{XY}XY models we give numerical evidence for the hidden ODLRO conjectured by Anderson. A connection to the singlet pair correlations in one-dimensional models of interacting electrons is made and briefly discussed.