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In this work, we give an analytical derivation of the reduced density matrix between two qubits in a cavity field, which is described by the quantum Rabi model. Using the coherent state method and the generalized hypergeometric functions, we find a complete solution of the 4-by-4 reduced density matrix for each eigen energy by taking the partial trace over the cavity field states. Four kinds of measurements used to describe the quantum entanglement and quantum correlation between the two qubits, i.e., the concurrence, quantum discord, steerable weight, and robustness of coherence, are calculated for the ground state and excited states. For both the weak coupling ( $g\rightarrow 0$ ) and deep-strong coupling ( $g\rightarrow \infty $ ) regimes, quantum entanglement and quantum correlation between the two qubits are vanishing. An appropriate coupling in the deep strong coupling regime between the qubits and the cavity field is required if one wants to establish quantum entanglement between the qubits.

Using numerical renormalization group (NRG) calculation, we construct a dataset with 100K samples, and train six different neural networks for the prediction of spectral functions from density of state (DOS) of the host material. We find that a combination of gated recurrent unit network (GRU) and bidirectional GRU (BiGRU) performances the best among all the six neural networks. The mean absolute error (MAE) of the GRU+BiGRU network can reach 0.052 and 0.043 when this network is evaluated on the original dataset and two other independent datasets. The average time of spectral function predictions from machine learning is on the scale of 10-5 ∼ 10-6 that of traditional impurity solvers for Anderson impurity model. This investigation pave the way for the application of recurrent neural network (RNN) and convolutional neural network (CNN) in the prediction of spectral functions from DOSs in machine learning solvers of magnetic impurity problems.

In this work, we study the magnetic impurity resonance states in monolayer FeSe superconductor for different pairing symmetries. When the magnetic impurity is hybridized to one of the two electronic pockets at M point, we find that there will be two in-gap resonance peaks located symmetrically at the two sides of the Fermi energy. The intensities of the two peaks are comparable for s-wave and d-wave pairings. For the p-wave pairing, the intensity of the positive resonance peak is much smaller than the negative one. When the magnetic impurity is hybridized to the two electronic pockets simultaneously and spin-orbit coupling is considered, the two resonance peaks will be split into four for d-wave pairing. This splitting is absent for s-wave and p-wave pairings. These features can be considered as signatures to distinguish the pairing potential symmetries.

In this work, we study the magnetic-impurity resonance states in superconducting phase of twisted bilayer graphene for different pairing symmetries. Using the two-orbital model proposed by Yuan and Fu in [Phys. Rev. B 98, 045103 (2018)], we find that when the impurity is located at one site of the emergent honeycomb lattice, the spacial distributions of the resonance states will break both the threefold and twofold rotation symmetries of $D_3$ group for pairing symmetries which belong to the irreducible representations of this point group. When the magnetic impurity is located at the center of the emergent honeycomb lattice, the appearance of resonance peak at the position close to the impurity can be considered as a strong evidence of non-$s$-wave pairing., 11 pages, 9 figures

In this work, we investigate the local density of states (LDOS) near a magnetic impurity in single-layer FeSe superconductors. The two-orbital model with spin-orbit coupling proposed in Ref. [{\emph{Phys. Rev. Lett.} \textbf{119}, 267001 (2017)}] is used to describe the FeSe superconductor. In the strong coupling regime, two impurity resonance peaks appear with opposite resonance energies in the LDOS spectral function. For a strong spin-orbit coupling, the superconducting gap in this model is $d$-wave symmetric with nodes, the spatial distributions of the LDOS at the two resonance energies are fourfold symmetric, which reveals typical characteristic of $d$-wave pairing. When the spin-orbit coupling is not strong enough to close the superconducting gap, we find that the spatial distribution of the LDOS at one of the resonance energies manifests $s$-wave symmetry, while the pairing potential preserves $d$-wave symmetry. This result is consistent with previous experimental investigations., 6 pages, 3 figures

We study two-dimensional massive Dirac equation in circular well potential. The energies of bound states are obtained. We demonstrate the Klein paradox of this relativistic wave equation: For large enough potential depth, the bound states disappear from the spectra. Applications to graphene systems are discussed.

In this work, we use the numerical renormalization group (NRG) theory to study the thermodynamics of the two-impurity Anderson model. Two different methods are used to estimate the effect of the Dzyaloshiskii-Moriya (DM) interaction on the variation of the Kondo temperature. When the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction is vanishing, the two different estimations give different tendency. If we use the peak of the specific heat to identify the variation of the Kondo temperature versus the Dzyaloshiskii-Moriya interaction, we get an almost linear function. However, if we use the low temperature universal curve of the impurity entropy, we get a quadratic function. These results indicate that the previous debates about the influence of the spin-orbit coupling on the Kondo temperature may come from the different definitions of the Kondo temperature. When the RKKY interaction is ferromagnetic, there are two stages of the Kondo screening. Both the two estimations demonstrate that the second stage of the Kondo temperature is exponentially dependent on the DM interaction. There results are dramatically different from those calculated via perturbation theory., 6 pages, 3 figures, important references added, All comments are welcome