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Physical review. B./Physical review. B

Quantum coherence spectrum and quantum phase transitions

Yanchao Li, Jing Zhang, Hai‐Qing Lin

Published Mar 25, 2020• 27 citations
Publisher

Abstract

We review the internal connection between quantum coherence (QC) and quantum phase transitions (QPTs). We propose two quantities ${L}_{\text{QC}}$ and ${S}_{\text{QC}}$ to detect QPTs of different orders by analyzing the spectrum of QC. Our results show that the two quantities not only can detect different types of QPT but can also exhibit a remarkable robustness against thermal fluctuations. In comparison with other detectors, such as quantum entanglement and QD, ${L}_{\text{QC}}$ and ${S}_{\text{QC}}$ can show more effective and reliable behaviors in characterizing a QPT at zero and finite temperatures. In light of this property and the fundamental role of QC in quantum information processing, QC spectrum-related ${L}_{\text{QC}}$ and ${S}_{\text{QC}}$ have considerable importance in detecting QPTs at a real finite-temperature experimental environment.

DOI
10.1103/physrevb.101.115142

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We review the internal connection between quantum coherence (QC) and quantum phase transitions (QPTs). We propose two quantities LQC{L}_{\text{QC}}LQC​ and SQC{S}_{\text{QC}}SQC​ to detect QPTs of different orders by analyzing the spectrum of QC. Our results show that the two quantities not only can detect different types of QPT but can also exhibit a remarkable robustness against thermal fluctuations. In comparison with other detectors, such as quantum entanglement and QD, LQC{L}_{\text{QC}}LQC​ and SQC{S}_{\text{QC}}SQC​ can show more effective and reliable behaviors in characterizing a QPT at zero and finite temperatures. In light of this property and the fundamental role of QC in quantum information processing, QC spectrum-related LQC{L}_{\text{QC}}LQC​ and SQC{S}_{\text{QC}}SQC​ have considerable importance in detecting QPTs at a real finite-temperature experimental environment.
We propose using maximum coherence (QCmax{\mathrm{QC}}_{\mathrm{max}}QCmax​) to study quantum phase transition (QPT). We investigate several well-known models, such as the XXZ model, ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model, and one-dimensional spin Kitaev model. The results show that QCmax{\mathrm{QC}}_{\mathrm{max}}QCmax​ can be used to detect different types of QPTs, including the Berezinskii-Kosterlitz-Thouless and topological types. In addition, QCmax{\mathrm{QC}}_{\mathrm{max}}QCmax​ is more robust against interferences, such as thermal fluctuations and measurement interferences, than any existing detector. This property enables QCmax{\mathrm{QC}}_{\mathrm{max}}QCmax​ to identify QPTs in experiments, where temperature effects and measurement interferences always exist. Thus, QCmax{\mathrm{QC}}_{\mathrm{max}}QCmax​ might be an ideal tool for studying QPTs because of its measurability, universality, and robustness against interferences.
We study the connections between local quantum coherence (LQC) based on Wigner-Yanase skew information and quantum phase transitions (QPTs). When applied on the one-dimensional Hubbard, XY spin chain with three-spin interaction, and Su-Schrieffer-Heeger models, the LQC and its derivatives are used successfully to detect different types of QPTs in these spin and fermionic systems. Furthermore, the LQC is effective as the quantum discord (QD) in detecting QPTs at finite temperatures, where the entanglement has lost its effectiveness. We also demonstrate that the LQC can exhibit different behaviors in many forms compared with the QD.
We review the main results and ideas showing that quantum correlations at finite temperatures (T), in particular quantum discord, are useful tools in characterizing quantum phase transitions (QPT) that only occur, in principle, at the unattainable absolute zero temperature. We first review some interesting results about the behavior of thermal quantum discord for small spin-1/2 chains and show that they already give us important hints of the infinite chain behavior. We then study in detail and in the thermodynamic limit (infinite chains) the thermal quantum correlations for the XXZ and XY models, where one can clearly appreciate that the behavior of thermal quantum discord at finite T is a useful tool to spotlight the critical point of a QPT.
Aim and Scope of this Book A plethora of systems exhibit phase transitions as the temperature or some other parameter is changed. Examples range from the ice-water phase transition observed in our daily life to the loss of ferromagnetism in iron or to the more sophisticated Mott insulator-superfluid phase transition observed in optical lattices [343]. The last five decades have witnessed a tremendous upsurge in the studies of phase transitions at finite temperature [727, 149, 333, 136, 494, 541, 556]. The success of Landau-Ginzburg theories and the concepts of spontaneous symmetry breaking and the renormalization group [27, 410, 821, 578] in explaining many of the finite temperature phase transitions occurring in nature has been spectacular. In this book, we will consider only a subclass of phase transitions called quantum phase transitions (QPTs) [154, 658, 725, 799, 185, 63, 62, 66, 141, 744] and we will discuss these mainly from the view point of recent studies of information and dynamics. QPTs are zero temperature phase transitions which are driven by quantum fluctuations and are usually associated with a non-analyticity in the ground state energy density of a quantum many-body Hamiltonian. We will focus on continuous QPTs where the order parameter vanishes continuously at the quantum critical point (QCP) at some value of the parameters which characterize the Hamiltonian. We will not discuss first order quantum phase transitions associated with an abrupt change in the order parameter. Usually, a first order phase transition is characterized by a finite discontinuity in the first derivative of the ground state energy density. A continuous QPT is similarly characterized by a finite discontinuity, or divergence, in the second derivative of the ground state energy density, assuming that the first derivative is continuous. This is of course the classical definition; we will later mention some QPTs where the ground state energy density is not necessarily singular.
Pairwise quantum discord (QD) and classical correlation (CC) are studied in the XY\mathit{XY}XY spin chain with three-spin interaction. We analyze their capability in detecting quantum phase transitions (QPTs) at both zero and finite temperatures and find that the pairwise QD of two neighboring spins is more reliable than that of any other distances in identifying QPTs. Both the QD and CC detect quantum critical points associated with first- and higher-order QPTs caused by field and three-spin interactions at finite temperatures. In addition, we find a different finite-size scaling behavior for QD from previous reports for the transverse field Ising case and show some interesting phenomena of QD and entanglement of formation for finite temperatures.
We introduce a coherence susceptibility method, based on the fact that it signals quantum fluctuations, for identifying quantum phase transitions, which are induced by quantum fluctuations. This method requires no prior knowledge of order parameter, and there is no need for careful considerations concerning the choice of a bipartition of the system. It can identify different types of quantum phase transition points exactly. At finite temperatures, where quantum criticality is influenced by thermal fluctuations, our method can pinpoint the temperature frame of quantum criticality, which perfectly coincides with recent experiments.