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In this work, we apply the advantage distillation method to improve the performance of a practical twin-field quantum key distribution system under collective attack. Compared with the previous analysis result given by Maeda, Sasaki and Koashi [Nature Communication 10, 3140 (2019)], the maximal transmission distance obtained by our analysis method will be increased from 420 km to 470 km. By increasing the loss-independent misalignment error to 12%, the previous analysis method can not overcome the rate-distance bound. However, our analysis method can still overcome the rate-distance bound when the misalignment error is 16%. More surprisingly, we prove that twin-field quantum key distribution can generate positive secure key even if the misalignment error is close to 50%, thus our analysis method can significantly improve the performance of a practical twin-field quantum key distribution system.

Quantum key distribution (QKD) allows two remote parties to share information-theoretic secret keys. Many QKD protocols assume the phase of encoding state can be continuous randomized from 0 to 2π, which, however, may be questionable in the experiment. This is particularly the case in the recently proposed twin-field (TF) QKD, which has received a lot of attention since it can increase the key rate significantly and even beat some theoretical rate-loss limits. As an intuitive solution, one may introduce discrete-phase randomization instead of continuous randomization. However, a security proof for a QKD protocol with discrete-phase randomization in the finite-key region is still missing. Here, we develop a technique based on conjugate measurement and quantum state distinguishment to analyze the security in this case. Our results show that TF-QKD with a reasonable number of discrete random phases, e.g., 8 phases from {0,π/4,π/2,…,7π/4}, can achieve satisfactory performance. On the other hand, we find the finite-size effects become more notable than before, which implies that more pulses should be emit in this case. More importantly, as a the first proof for TF-QKD with discrete-phase randomization in the finite-key region, our method is also applicable in other QKD protocols.

The Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction has been extensively explored in isotropic Dirac systems with linear dispersion, which typically follows an exponent decaying rate with the impurity distance R , i.e., J ∝ 1/ R ^d (1/ R ^2 ^d ^−1 ) in d -dimensional systems at finite (zero) Fermi energy. This fast decay makes it rather difficult to be detected and limits its application in spintronics. Here, we theoretically investigate the influence of anisotropic dispersion on the RKKY interaction, and find that the introduction of non-relativistic dispersion in semi-Dirac semimetals (S-DSMs) can significantly prolong the decay of the RKKY interaction and can remarkably enhance the Dzyaloshinskii–Moriya interaction around the relativistic direction. The underlying physics is attributed to the highly increased density of states in the linear-momentum direction as a result of the interplay of relativistic and non-relativistic electrons. Furthermore, we propose a general formula to determine the decaying rate of the RKKY interaction, extending the typical formula for isotropic DSMs. Our results suggest that the S-DSM materials are a powerful platform to detect and control the magnetic exchange interaction, superior to extensively adopted isotropic Dirac systems.

Phase-matching quantum key distribution (PM-QKD) provides a promising solution to surpass the fundamental rate–distance bound without quantum repeaters. In this paper, we insert an additional advantage distillation (AD) step after quantum communication to improve the performance of PM-QKD. Simulation results show that, by splitting the raw key into blocks of only a few bits so as to identify highly correlated bit pairs, the AD method can tolerate high system misalignment errors and improve the secret key rate and transmission distance significantly, which is very promising in current PM-QKD systems.

Abstract We investigate the thermoelectric effect on a topological insulator surface with particular interest in impurity-induced resonant states. To clarify the role of the resonant states, we calculate the dc and ac conductivities and the thermoelectric coefficients along the longitudinal direction within the full Born approximation. It is found that at low temperatures, the impurity resonant state with strong energy de-pendence can lead to a zero-energy peak in the dc conductivity, whose height is sensitively dependent on the strength of scattering potential, and even can reverse the sign of the thermopower, implying the switching from n- to p-type carriers. Also, we exhibit the thermoelectric signatures for the filling process of a magnetic band gap by the resonant state. We further study the impurity effect on the dynamic optical conductivity, and find that the resonant state also generates an optical conductivity peak at the absorption edge for the interband transition. These results provide new perspectives for understanding the doping effect on topological insulator materials.

When an electron is incident from a Weyl material to an insulator and totally reflected, it suffers a reflection group delay and a reflection shift (Goos–Hänchen and/or Imbert–Fedorov shifts). We found the group delay is negative for half of the incident states. The negative group delay does not mean the electron is bounced back before its injection, but is an effective acceleration of the electron near the interface induced by self-interference. The reflection shift orients circulating the points at which the surface-bulk state transition occurs. The reflection shift and the group delay cause velocity correction of the bound states in the Weyl material sandwiched by two insulators. The velocity correction features induced by the negative group delay were verified by a tight-binding calculation, in which the concept of group delay is not used.

By generalizing the Landau quantization Boltzmann equation to a finite-size system, we investigate the magnetotransport in disordered Weyl semimetals (WSMs) for both the ballistic and diffusive regimes. It is found that, in the diffusive limit, the chiral chemical potential would drive extra electrical current to flow through the chiral channels in the n=0 Landau levels, leading to emergence of the positive longitudinal magnetoconductivity (LMC). The positive LMC will exhibit the angular narrowing phenomenon if ℏω_{c}≫E_{F}sqrt[l_{a}/l_{e}], where ω_{c}, E_{F}, and l_{a(e)} denote, respectively, the cyclotron frequency, the Fermi energy, and the intravalley (intervalley) relaxation length. The anomalous magnetic field dependence of the angular narrowing, i.e., the width and height of the positive LMC peak could increase or decrease with increasing the magnetic field, can be attributable to quantum oscillations of the chiral anomaly. In addition, the quantum oscillations are sensitive to the temperature and impurity scattering, which may also cause some other anomalous properties for the LMC. Our findings are helpful to understand the anomalous behaviors of the positive LMC in WSMs.

If an electron beam hits onto the interface of a Weyl-node-mismatch junction, a shift of the beam center on the interface happens when the beam is reflected or transmitted, where the junction consists of two materials of the same Weyl semimetal and one of them is rotated with respect to the other by an angle. We calculate the longitudinal and transverse shift components (the Goos–Hänchen and Imbert–Fedorov shifts). The reflection shift for total reflection cases is much more remarkable than the shift for transmitted cases. There exists a semi-vortex structure of the reflection shift on the in-plane k -space. The vortex is induced by the touch between bulk bands and interface bands. The formation of such interface bands is explained by the pulley-group model, in which the Weyl cones serve as wheels and the surface and interface bands act as ropes. A surface rope connects wheels of opposite chiralities, and an interface rope links the wheels for the two side materials of the same chirality.

We studied the transport properties of the Weyl-node-mismatch junction, beside which the Weyl nodes are mismatch so that their projections on the interface plane cannot overlap. When electrons of one valley are injected onto the junction, the refraction is not coplanar with the injection-reflection plane. The non-coplanar deviation angle is valley dependent, and so if the injection consists of both valley components, the birefraction will take place. When electrons coming from a narrow rod transit across the junction into the bulk region on the other side, the electrons will be diffracted in the unconfined region. The diffraction is dispersed near the rod-bulk interface and is laterally deflected to the direction along the line connecting the Weyl node projections of incident and transmitted sides.

We theoretically investigate the Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction between magnetic impurities distributed in the vicinity of the surface of a Weyl semimetal. Contrary to previous studies, we further take into account the influence of interplay of the Fermi arc and bulk states, and interface refection. It is shown that the RKKY pattern is significantly mediated by the Fermi-arc surface state along with the interface reflection. The Fermi-arc surface state mediates the RKKY interaction by interfering with the bulk states. The resulting interference contribution in the short-range impurity distance R is comparable in magnitude to the bulk-band contribution and even dominates the latter near the surface. It either enhances or weakens the bulk contribution, depending on the relative orientation of impurities and Fermi energy. More importantly, for the long-range impurity distance the interference term dominates in that it can prolong the decay rate from the original bulk R ^−5 -law to R ^−2 ( R ^−3 ) for finite (zero) Fermi energy. The interface reflection not only enhances the magnitude of the RKKY interaction and changes its anisotropy from the original XXZ to XYZ or Ising spin model, but also generates extra twisted RKKY terms parallel to the line connecting Weyl nodes, lacked in the scenario without the interface effect. They originate from the interaction between the impurity and the mirror image of the other impurity. We further analyze in detail the spatial anisotropy of the decay rate and beating pattern. These findings provide a deeper insight into surface magnetic interaction mediated by Weyl fermions.

We investigate the Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction between two magnetic impurities placed on a phosphorene surface based on the tight-binding model. It is found that the RKKY interaction exhibits strong anisotropy along different lattice directions and is sensitive to the deformed direction. The RKKY interaction is largest for impurities distributed in the armchair direction while the strain effect is strongest for the deformation exerted along the zigzag direction. Applied linear strain can increase the RKKY magnitude nonlinearly, and observably prolong the decay rate from the exponent to the $1/{R}^{2}$ law with the impurity distance. Most interestingly, near the strain-induced closing point of the bandgap, we find that the RKKY interaction is no longer simply ferromagnetic or antiferromagnetic, but presents oscillatory behavior, exhibiting the transition from ferromagnetism to antiferromagnetism. This originates from the combination effect of negative energy in the conduction band due to modification of the Fermi surface and the narrowing bandgap, and importantly both of them are reached simultaneously just by tuning the strain. Therefore, the strain effect proves to be an alternative approach to engineering impurity interactions in phosphorene materials.

Phosphorene (a monolayer of black phosphorus) is a large gap semiconductor with high mobility and has great application potential. Numerical calculations reveal that phosphorene is a topologically trivial material and can only host edge bands on specified edges such as the zigzag edge. A linearly polarized irradiation on the phosphorene lattice results in the dynamic gaps in the quasi-energy spectrum. We found that the irradiation polarized in the zigzag direction induces new edge bands within the dynamic gaps on any type of edge (zigzag, armchair, or other bearded edge). We proposed a new gauge independent quantity, $\delta +g$ , to account for the appearence of universal edge bands, where δ is the detune and g is the light induced valence-conduction band transition element. The number of edge bands in the dynamic gaps is reflected by the winding number of it.

We investigate the influence of in-plane spin-exchange interactions on a topological insulator (TI) surface doped with nanomagnets under the second perturbation theory. We propose a novel self-filling mechanism of the surface-state band gap. It is found that when the out-of-plane exchange coupling favors an energy gap around the Dirac point, the in-plane component tends to suppress the induced gap, and even fill it completely. Our theory is based on the spin-flipping inelastic scattering, which creates a complex structure of self-energy, effectively modifying the band gap by renormalizing the magnetic moment and chemical potential. We explicitly analyze the filling effect in the electronic dispersion relation and density of states for different scenarios set by systemic parameters. This self-filling effect induced by spin-exchange coupling itself opens new perspectives for understanding of various magnetically doping phenomena on the TI materials and is expected to mediate the controversy concerning the magnetically doping induced gap.

The interaction effect between the surface states of a topological insulator (TI) and a STM-coupled Anderson impurity is studied by using equations of motion of the Green’s functions. Remarkably, we show that when a coupling between the Anderson impurity and the STM tip is included, the tunneling resonance and the Kondo peak can be tuned to be exactly at the Dirac point, by adjusting the impurity level and Fermi energy, such that the local density of states at the Dirac point is significantly enhanced. This is in contrast to the case of a STM-decoupled Anderson impurity, where both resonances are always fully suppressed at the Dirac point. Our finding suggests a pathway to experimentally control the fundamental properties of the electrons on a TI surface.