The insertion of a magnetic impurity in a superconductor induces a first order quantum phase transition as the coupling to the electronic spin density increases. As the transition is crossed, a discontinuity is exhibited by various quantities, such as the total spin density, the total gap function, and the gap function at the impurity location. The location of other quantum phase transitions have been detected by singularities in entanglement measures of the system. In this work we show that the single-site and two-site von Neumann entropies, the mutual information, and the Meyer-Wallach measure show discontinuities at the quantum phase transition. The negativity is less sensitive to the transition. We study in detail these quantities as a function of spin coupling and distance from the impurity center.
The insertion of a magnetic impurity in a superconductor induces a first order quantum phase transition as the coupling to the electronic spin density increases. As the transition is crossed, a discontinuity is exhibited by various quantities, such as the total spin density, the total gap function, and the gap function at the impurity location. The location of other quantum phase transitions have been detected by singularities in entanglement measures of the system. In this work we show that the single-site and two-site von Neumann entropies, the mutual information, and the Meyer-Wallach measure show discontinuities at the quantum phase transition. The negativity is less sensitive to the transition. We study in detail these quantities as a function of spin coupling and distance from the impurity center.
We introduce a partial state fidelity approach to quantum phase transitions. We consider a superconducting lattice with a magnetic impurity inserted at its center, and look at the fidelity between partial (either one-site or two-site) quantum states. In the vicinity of the point of the quantum phase transition, we observe a sudden drop of the fidelity between two one-site partial states corresponding either to the impurity location or its close vicinity. This enables us to identify the on-site magnetization as the order parameter for the phase transition studied. In the case of two-site states, the fidelity reveals the transition point as long as one of the two electron sites is located at the impurity, while the other lies elsewhere in the lattice. We also determine the Uhlmann mixed state geometric phase, recently introduced in the study of the structural change of the system state eigenvectors in the vicinity of the lines of thermal phase transitions, and find it to be trivial, both for one- and two-site partial states, except when an electron site is at the impurity. This means that the system partial state eigenvectors do not contribute significantly to the enhanced state distinguishability around the point of this quantum phase transition. Finally, we use the fidelity to analyze the total amount of correlations contained within a composite system, showing that, even for the smallest one-site states, it features an abrupt quantitative change in the vicinity of the point of the quantum phase transition.
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 consider a spin ladder model which is known to have matrix product states as exact ground states with spin liquid characteristics. The model has two critical-point transitions at the parameter values u=0 and $\ensuremath{\infty}$. We study the variation of entanglement and fidelity measures in the ground states as a function of u and specially look for signatures of quantum phase transitions at u=0 and $\ensuremath{\infty}$. The two different entanglement measures used are S(i) (the single-site von Neumann entropy) and S(i,j) (the two-body entanglement). At the quantum critical point (QCP) $u=\ensuremath{\infty}$, the entanglement measure E[=S(i),S(i,j)] vanishes but remains nonzero at the other QCP u=0. The first and second derivatives of E with respect to the parameter u and the entanglement length associated with S(i,j) are further calculated to identify special features, if any, near the QCPs. We further determine the GS fidelity F and a quantity ln∣D∣ related to the second derivative of F and show that these quantities calculated for finite-sized systems are good indicators of QPTs occurring in the infinite system.
Topological insulators and topological superconductors display various topological phases that are characterized by different Chern numbers or by gapless edge states. In this work we show that various quantum information methods such as the von Neumann entropy, entanglement spectrum, fidelity, and fidelity spectrum may be used to detect and distinguish topological phases and their transitions. As an example we consider a two-dimensional p-wave superconductor, with Rashba spin-orbit coupling and a Zeeman term. The nature of the phases and their changes are clarified by the eigenvectors of the k-space reduced density matrix. We show that in the topologically nontrivial phases the highest weight eigenvector is fully aligned with the triplet pairing state. A signature of the various phase transitions between two points on the parameter space is encoded in the k-space fidelity operator.
Bipartite quantum entanglement of electron spins of a Cooper pair in a dirty s-wave superconductor is investigated at absolute zero temperature in terms of the exchange interaction and potential scattering as well as the relative distance between the two electrons. In the case of transition-metal impurities, we utilize the T-matrix approach to obtain the relevant Green's functions. We employ the two-electron spin-space density matrix, which is associated with the perturbed Green's functions. We show that a two-spin state can be described by the Werner state for dirty superconductors as well as for clean ones. We find that both the first and second quantum phase transitions (QPTs) are generated by the competition between potential scattering and other interactions. Meanwhile, it is not possible to determine the allowable value for each interaction individually. In the case of rare-earth impurities, concurrence, as a measure of bipartite entanglement, is discussed in terms of the relative distance and collision times for all finite and infinite Debye frequencies. The results confirm that quantum correlation and QPTs can be tuned by impurities.
Abstract By considering transition-metal (Shiba-Rusinov model) and rare-earth metal impurities (Abrikosov-Gorkov theory) effect on a many-body system, i.e., a BCS s-wave superconductor, quantum bipartite entanglement of two electrons of the Cooper pairs in terms of the exchange interaction, J, the potential scattering, V(playing an important role, unexpectedly), and the distance of two electron spins of the Cooper pair is calculated at zero temperature by using two-electron spin-space density matrix (Werner state). In transition-metal case, we find new quantum phase transitions (QPTs). The changes of J, which causes to have localized excited state, V and the pairing interaction (via energy gap) lead to the displacement of the QPTs (interactions act in the same direction, however sometimes the pairing interaction causes the competition with other interactions), regardless of their effects on the value of concurrence. Determining the allowable values of all interactions by itself is not possible, due to the smallness of the perturbed Green’s functions (appearing in the density matrix). For non-magnetic and magnetic (rare-earth) impurity cases, concurrence versus the distance and collision times is discussed for all finite and infinite Debye frequency. The quantum correlation, instability of the system and what's more important QPT can be tuned by the impurity.