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arXiv:2406.10178 · Physical review. B./Physical review. B

Finite-temperature detection of quantum critical points: A comparative study

G A P Ribeiro, Gustavo Rigolin

Published Jun 14, 2024• 6 citations
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Abstract

We comparatively study three of the most useful quantum information tools to detect quantum critical points (QCPs) when only finite temperature data are available. We investigate quantitatively how the quantum discord, the quantum-teleportation-based QCP detectors, and the quantum coherence spectrum pinpoint the QCPs of several spin-$1/2$ chains. We work in the thermodynamic limit (infinite number of spins) and with the spin chains in equilibrium with a thermal reservoir at temperature $T$. The models here studied are the XXZ model with and without an external longitudinal magnetic field, the Ising transverse model, and the XY model subjected to an external transverse magnetic field.

DOI
10.1103/physrevb.109.245122

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We comparatively study three of the most useful quantum information tools to detect quantum critical points (QCPs) when only finite temperature data are available. We investigate quantitatively how the quantum discord, the quantum-teleportation-based QCP detectors, and the quantum coherence spectrum pinpoint the QCPs of several spin-1/21/21/2 chains. We work in the thermodynamic limit (infinite number of spins) and with the spin chains in equilibrium with a thermal reservoir at temperature TTT. The models here studied are the XXZ model with and without an external longitudinal magnetic field, the Ising transverse model, and the XY model subjected to an external transverse magnetic field.
We show for the models here investigated that the teleportation-based quantum critical point (QCP) detectors can properly estimate the locations of the QCPs when we are not even close to the thermodynamic limit (infinite spin chains) and when we only have access to finite temperature data. Specifically, by working with spin chains with about ten qubits and in equilibrium with a thermal reservoir at temperature T, we show that it is possible to locate with an error of only a few percents the correct spots of the QCPs for almost all the models studied here. The spin chains we investigate are given by the XXZ model with or without an external longitudinal magnetic field as well as the XX model, the XY model and the Ising model, all of them subjected to an external transverse magnetic field.
In previous work [Phys. Rev. A 107, 052420 (2023)] we showed that the quantum teleportation protocol can be used to detect quantum critical points (QCPs) associated with a couple of different classes of quantum phase transitions, even when the system is away from the absolute zero temperature (T=0T=0T=0). Here, working in the thermodynamic limit (infinite chains), we extend the previous analysis for several other spin-12\frac{1}{2}21​ models. We investigate the usefulness of the quantum teleportation protocol to detect the QCPs of those models when the temperature is either zero or greater than zero. The spin chains we investigate here are described by the XXZXXZXXZ model, the XYXYXY model, and the Ising model, all of them subjected to an external magnetic field. Specifically, we use a pair of nearest-neighbor qubits from an infinite spin chain at thermal equilibrium with a reservoir at temperature TTT as the resource to execute the quantum teleportation protocol. We show that the ability of this pair of qubits to faithfully teleport an external qubit from the chain is dramatically affected as we cross the QCPs related to the aforementioned models. The results here presented together with the ones of aforementioned work suggest that the quantum teleportation protocol is a robust and quite universal tool to detect QCPs even when the system of interest is far from the absolute zero temperature.
We show that the quantum teleportation protocol is a powerful tool to study quantum phase transitions at finite temperatures. We consider a pair of spins from an infinite spin-1/2 chain (XXZXXZXXZ model) in equilibrium with a reservoir at temperature TTT as the resource used by Alice and Bob to implement the teleportation protocol. We show that the efficiency of this pair of spins to teleport a qubit is drastically affected after we cross a quantum critical point (QCP), even for high values of TTT. Also, we show that the present tool is as sharp as quantum discord (QD) to spotlight a QCP, where QD is the best finite TTT QCP detector known to date. Contrary to QD, however, we show that the present tool is easier to compute theoretically and has a direct experimental and operational meaning.
We extend the program initiated by T. Werlang et al. [Phys. Rev. Lett. 105, 095702 (2010)] in several directions. Firstly, we investigate how useful quantum correlations, such as entanglement and quantum discord, are in the detection of critical points of quantum phase transitions when the system is at finite temperatures. For that purpose we study several thermalized spin models in the thermodynamic limit, namely, the XXZ\mathit{XXZ}XXZ model, the XY\mathit{XY}XY model, and the Ising model, all of which with an external magnetic field. We compare the ability of quantum discord, entanglement, and some thermodynamic quantities to spotlight the quantum critical points for several different temperatures. Secondly, for some models we go beyond nearest neighbors and also study the behavior of entanglement and quantum discord for second nearest neighbors around the critical point at finite temperature. Finally, we furnish a more quantitative description of how good all these quantities are in spotlighting critical points of quantum phase transitions at finite TTT, bridging the gap between experimental data and those theoretical descriptions solely based on the unattainable absolute zero assumption.
We show that the teleportation protocol can be efficiently used to detect quantum critical points using finite-temperature data even if all resources needed for its implementation lie within the system under investigation. Contrary to a previous proposal, there is no need to use an external qubit as the input state to be teleported to one of the qubits within the system. Here we use a pair of nearest-neighbor spins from an infinite spin-12\frac{1}{2}21​ chain in equilibrium with a heat bath as the entangled resource of the quantum-teleportation protocol and a third adjacent qubit within the chain itself as the input state to be teleported. For several spin-chain models subjected to an external magnetic field, we show that the efficiency of the teleportation protocol is severely affected as we cross the quantum critical points associated with those spin chains. This abrupt change in efficiency gives us a clear indication of a quantum phase transition.
We examine the ability of quantum discord (QD) and entanglements (concurrence, EoF and negativity) to detect the critical points associated to quantum phase transitions (QPTs) for XY models, i.e., the isotropic XY model with three-spin interactions at zero temperature, and the anisotropic XY model in a transverse magnetic field h at finite temperatures. For the case of zero temperature, we found that both entanglements and QD can spotlight the critical points of QPTs for these two models. Moreover, QD versus distance M exhibits the long-range behavior of quantum correlation for the anisotropic XY model, while entanglement is short-ranged. For the case of finite temperatures, we found that negativity has the same behaviors with concurrence at or near transition points. Moreover, QD for the anisotropic XY model can increase with temperature even in the absence of a magnetic field.