Akzyanov, R S

Khokhlov D.A., Akzyanov R.S., Kapranov A.V.

Abstract We study the effects of magnetisation on the properties of the doped topological insulator of the Bi 2 Se 3 family with nematic superconductivity. We found that the direction of the in-plane magnetisation fixes the direction of the nematicity in the system. The chiral state is more favourable than the nematic state for large values of out-of-plane magnetisation. Overall, the critical temperature of the nematic superconductivity is robust against magnetisation. We explore in detail the spectrum of the system with the pinned direction of the nematic order parameter Δ y . Without magnetisation, there is a full gap in the spectrum because of finite hexagonal warping. At an out-of-plane mz or orthogonal in-plane mx magnetisation that is strong enough, the spectrum is closed at the nodal points that are split by the magnetisation. Flat Majorana surface states connect such split bulk nodal points. Parallel magnetisation my lifts the nodal points and opens a full gap in the spectrum. We discuss relevant experiments and propose experimental verifications of our theory.

Akzyanov R.S., Rakhmanov A.L.

We study the influence of the antiferromagnetic order on the surface states of topological insulators.We derive an effective Hamiltonian for these states, taking into account the spatial structure of the antiferromagnetic order.We obtain a typical (gapless) Dirac Hamiltonian for the surface states when the surface of the sample is not perturbed.Gapless spectrum is protected by the combination of time-reversal and half-translation symmetries.However, a shift in the chemical potential of the surface layer opens a gap in the spectrum away from the Fermi energy.Such a gap occurs only in systems with finite antiferromagnetic order.We observe that the system topology remains unchanged even for large values of the disorder.We calculate the spectrum using the tight-binding model with different boundary conditions.In this case we get a gap in the spectrum of the surface states.This discrepancy arises due to the violation of the combined time-reversal symmetry.We compare our results with experiments and density functional theory calculations.

Akzyanov R.S.

In this study, we investigate the effect of random point disorder on the surface states of a topological insulator with out-of-plane magnetization. We consider the disorder within a high-order Born approximation. The Born series converges to the one branch of the self-consistent Born approximation (SCBA) solution at low disorder. As the disorder strength increases, the Born series converges to another SCBA solution with the finite density of states within the magnetization-induced gap. Further increase of the disorder strength leads to a divergence of the Born series, showing the limits of the applicability of the Born approximation. We find that the convergence properties of this Born series are closely related to the properties of the logistic map, which is known as a prototypical model of chaos. We also calculate the longitudinal and Hall conductivities within the Kubo formulas at zero temperature with the vertex corrections for the velocity operator. Vertex corrections are important for describing transport properties in the strong disorder regime. In the case of a strong disorder, the longitudinal conductivity is weakly dependent on the disorder strength, while the Hall conductivity decreases with increasing disorder.

Kapranov A.V., Akzyanov R.S., Rakhmanov A.L.

In the topological superconductor with the nematic superconductivity in the ${E}_{u}$ representation, it is possible to have different types of vortices. One is associated with the vorticity in the particle-hole space and corresponds to the Abrikosov vortex. Another type corresponds to the vorticity in the spin space and is called the spin vortex. We study the interaction of the Abrikosov vortex with the spin vortices. We derive the free energy of the sample with the Abrikosov and the strain-induced spin vortices using the Ginzburg-Landau approach for the two-component superconducting order parameter. We calculate the critical strain at which the spin vortex is formed. We show that the spin vortex and the Abrikosov vortex attract each other, and as a result, they have a common core. We apply Bogoliubov--de Gennes equations to study electronic states in a combined vortex structure. We show that no zero-energy states (Majorana fermions) are localized near the common vortex core of the Abrikosov vortex and the spin vortex of any type. Possible experimental realization is discussed.

Akzyanov R.S., Rakhmanov A.L.

We study the physics of the Josephson effect in the odd-parity nematic superconductors within the Ginzburg-Landau approach. The order parameter is a two-component vector that transforms as a coordinate vector in the $(x,y)$ plane under rotation and its direction is usually refer to as the nematicity direction. A nontrivial interplay between the nematicity and crystallographic axes of the superconductors that form the junction makes the Josephson effect quite unusual. We derive current-phase relations for different configurations of the junction, crystallographic axes of the sample, and the nematicity direction in the superconductors. We obtain that the Meissner kernel in the considered samples has off-diagonal components and the transverse phase difference across the junction can induce a Josephson current that flows along the contact. We show that such an anomalous Josephson Hall effect can be observed without any magnetization. We calculate the magnetic field dependence of the maximum current through the junction. We find that the period of the Fraunhofer oscillations of the maximum Josephson current depends on the geometry of the junction, the direction of the magnetic field, and the nematicity directions. The obtained results can be generalized to other superconductors with nondiagonal Meissner kernels.

Khokhlov D.A., Akzyanov R.S., Rakhmanov A.L.

Shapiro D.S., Remizov S.V., Lebedev A.V., Babukhin D.V., Akzyanov R.S., Zhukov A.A., Bork L.V.

Dmitriy S. Shapiro1,2,3,∗ Sergey V. Remizov, Andrey V. Lebedev, Danila V. Babukhin, Ramil S. Akzyanov, Andrey A. Zhukov, and Leonid V. Bork Dukhov Research Institute of Automatics (VNIIA), 127055 Moscow, Russia National University of Science and Technology MISiS, 119049 Moscow, Russia Institute for Quantum Materials and Technologies, Karlsruhe Institute of Technology, 76021 Karlsruhe, Germany V. A. Kotel’nikov Institute of Radio Engineering and Electronics, Russian Academy of Sciences, Moscow 125009, Russia Department of Physics, National Research University Higher School of Economics, Moscow 101000, Russia Moscow Institute of Physics and Technology, 141700, Institutskii Per. 9, Dolgoprudny, Moscow Distr., Russia and Institute for Theoretical and Applied Electrodynamics, Russian Academy of Sciences, 125412 Moscow, Russia

Bannikov M.I., Akzyanov R.S., Zhurbina N.K., Khaldeev S.I., Selivanov Y.G., Zavyalov V.V., Rakhmanov A.L., Kuntsevich A.Y.

M.I. Bannikov, 2 R.S. Akzyanov, 4, 5 N.K. Zhurbina, S.I. Khaldeev, 2 Yu.G. Selivanov, V.V. Zavyalov, 2 A. L. Rakhmanov, 5 and A.Yu. Kuntsevich 2 P.N. Lebedev Physical Institute, Russian Academy of Sciences, Moscow 119991, Russia National Research University Higher School of Economics, Moscow 101000, Russia Dukhov Research Institute of Automatics, Moscow, 127055 Russia Moscow Institute of Physics and Technology, Dolgoprudny, Moscow Region, 141700 Russia Institute for Theoretical and Applied Electrodynamics, Russian Academy of Sciences, Moscow, 125412 Russia P.L. Kapitza Institute of Physical Problems, Moscow, Russia

Akzyanov R.S.

We study the effects of the Lifshitz transition from closed to open Fermi surface in dirty topological insulators with the nematic superconductivity near the critical temperature. We solve linearized Gor’kov equations and find that the nematic superconductor with an open Fermi surface has a lower critical temperature and more susceptible to the disorder than the superconductor with the closed Fermi surface. We propose that correspondence between the critical temperature and stability against the disorder is the general feature of the superconductivity. We investigate the effects of the Lifshitz transition on the competition between superconducting phases in a topological insulator. Open Fermi surface is beneficial for the nematic order parameter ∆ 4 in competition with orbital-triplet ∆ 2 and disfavors nematic state over the s-wave order parameter. We study Meissner currents in both clean and dirty limits. We found that transition from closed to open Fermi surface increases anisotropy of Meissner currents. Finite disorder suppresses superconducting density stronger than critical temperature. We compare our results with the existing experimental data.

Khokhlov D.A., Akzyanov R.S.

We investigate the response of a doped topological insulator Bi2Se3 with spin-triplet nematic superconductivity to external magnetization. We calculate the Zeeman part of magnetic susceptibility for nematic and chiral superconducting phases near Tc in Ginzburg-Landau formalism. Superconducting order parameter from Eu representation has non-trivial coupling with the transversal Zeeman field that results in a paramagnetic response to a magnetization. The topology of a Fermi surface has a strong influence on magnetic susceptibility. Lifshitz transition from closed to open Fermi surface eventually leads to phase transition from the nematic to chiral phase. At the transition point, magnetic susceptibility diverges. Also, we study the effects of the electron-electron interaction on the competition between nematic and chiral phases. We found that in a real system, electron-electron interaction can drive nematic to chiral phase only in the vicinity of the phase transition. We compare our results with the existing experimental data.

Akzyanov R.S., Rakhmanov A.L.

Using the Ginzburg-Landau approach we show that the strain of the nematic superconductor can generate a specific (nematic) vorticity. In the case of doped topological insulators that vorticity forms a spin vortex. We find two types of topologically different spin vortices that either enhance (type I) or suppress (type II) superconductivity far from the vortex core. We apply Bogoliubov-de Gennes equations to study electronic states in the nematic superconductor with spin vortices. We find that in the case of the vortex of type I, zero-energy states are localized near the vortex core. These states can be identified as Majorana Kramer’s pairs. In the case of the vortex of type II, there are no localized zero-energy states. Thus, we establish a non-trivial connection between the strain and Majorana fermions in the doped topological insulators with nematic superconductivity.

Khokhlov D.A., Akzyanov R.S.

We theoretically investigate quasiparticle interference in superconducting topological insulators with the nematic order parameter. This order parameter spontaneously breaks the rotational symmetry of the crystal. Such rotational symmetry breaking is visible in the quasiparticle interference picture both in coordinate and momentum spaces. For a small bias voltages quasiparticle interference incommensurate with the crystal symmetry and shows nematic behavior. If the bias voltage is comparable with the value of the order parameter interference picture is similar to the interference picture of the normal state. Interference patterns are sensitive to the orientation of the nematicity. We compare our results with the existing experimental data.

Stolyarov V.S., Sheina V.A., Khokhlov D.A., Vlaic S., Pons S., Aubin H., Akzyanov R.S., Vasenko A.S., Menshchikova T.V., Chulkov E.V., Golubov A.A., Cren T., Roditchev D.

Inelastic interactions of quantum systems with the environment usually wash coherent effects out. In the case of Friedel oscillations, the presence of disorder leads to a fast decay of the oscillation amplitude. Here we show both experimentally and theoretically that in three-dimensional topological insulator Bi2Te3 there is a nesting-induced splitting of coherent scattering vectors which follows a peculiar evolution in energy. The effect becomes experimentally observable when the lifetime of quasiparticles shortens due to disorder. The amplitude of the splitting allows an evaluation of the lifetime of the electrons. A similar phenomenon should be observed in any system with a well-defined scattering vector regardless of its topological properties.

Akzyanov R.S.

Abstract We study the spin conductivity of the bulk states of three-dimensional topological insulators within Kubo formalism. Spin Hall effect is the generation of the spin current that is perpendicular to the applied voltage. In the case of a three-dimensional topological insulator, applied voltage along x direction generates transverse spin currents along y and z directions with comparable values. We found that finite non-universal value of the spin conductivity exists in the gapped region due to the inversion of bands. Contribution to the spin conductivity from the vertex corrections enhances the spin conductivity from the filled states. These findings explain large spin conductivity that has been observed in topological insulators.

Akzyanov R.S., Kapranov A.V., Rakhmanov A.L.

We show that a spontaneous strain and spontaneous magnetization can arise in the doped topological insulators with a two-component superconducting vector order parameter. The details of the effects crucially depend on the symmetry of the superconducting order parameter, whether it is nematic or chiral. The transition from the nematic state to the chiral one can be performed by application of a magnetic field while the transition from the chiral state to the nematic one is tuned by the external strain. These transitions associated with a jump of the magnetic susceptibility and mechanical stiffness. Possible experimental observations of the predicted effects are discussed.

Apinyan V., Kopeć T.

Nagorkin V., Schimmel S., Gebauer P., Isaeva A., Baumann D., Koitzsch A., Büchner B., Hess C.

We investigated the electronic properties of the topological insulator Bi2Te3 by scanning tunneling microscopy and spectroscopy at low temperature. We obtained high-resolution quasiparticle interference data of the topological surface Dirac electrons at different energies. Spin-selective joint density of states calculations were performed for surface and bulk electronic states to interpret the observed quasiparticle interference data. The topological properties of our crystals are demonstrated by the absence of backscattering along with the linear energy dispersion of the dominant scattering vector. In addition, we detect nondispersive scattering modes which we associate with bulk-surface scattering and, thus, allow an approximate identification of the bulk energy gap range based on our quasiparticle interference data. Measurements of differential conductance maps in magnetic fields up to 15 T have been carried out, but no strong modifications could be observed. Published by the American Physical Society 2025

Zhang Z.B.

De Leo M., Borgna J.P., Huenchul C.

Karabassov T.

Kamenskaya T.A., Eliseyev I.A., Davydov V.Y., Kuntsevich A.Y.

We have modified the metal-assisted transfer technique to obtain large-area, few-layer flakes from transition metal dichalcogenide bulk crystals by introducing an initial stage—exfoliation of the bulk crystal onto an intermediate substrate, specifically a silicon wafer coated with polyvinyl alcohol. Following this, we thermally evaporate silver onto the sample and transfer the top layers of the crystal along with the silver layer to the target substrate. This technique allows the production of visually non-corrugated single- and few-layer flakes with high yield. A direct comparison of the micro-Raman and micro-photoluminescence spectra of flakes exfoliated using our method with the spectra of those exfoliated from the scotch tape reveals differences in their properties. We identify signatures of deformations in the flakes exfoliated from the intermediate substrate, indicating the presence of static friction between the substrate and the flake. Our findings, thus, suggest a useful method to induce intrinsic deformation in 2D materials.

Rechciński R., Khindanov A., Pikulin D.I., Liao J., Rokhinson L.P., Chen Y.P., Lutchyn R.M., Väyrynen J.I.

Khokhlov D.A., Akzyanov R.S., Kapranov A.V.

Abstract We study the effects of magnetisation on the properties of the doped topological insulator of the Bi 2 Se 3 family with nematic superconductivity. We found that the direction of the in-plane magnetisation fixes the direction of the nematicity in the system. The chiral state is more favourable than the nematic state for large values of out-of-plane magnetisation. Overall, the critical temperature of the nematic superconductivity is robust against magnetisation. We explore in detail the spectrum of the system with the pinned direction of the nematic order parameter Δ y . Without magnetisation, there is a full gap in the spectrum because of finite hexagonal warping. At an out-of-plane mz or orthogonal in-plane mx magnetisation that is strong enough, the spectrum is closed at the nodal points that are split by the magnetisation. Flat Majorana surface states connect such split bulk nodal points. Parallel magnetisation my lifts the nodal points and opens a full gap in the spectrum. We discuss relevant experiments and propose experimental verifications of our theory.

Akzyanov R.S., Rakhmanov A.L.

We study the influence of the antiferromagnetic order on the surface states of topological insulators.We derive an effective Hamiltonian for these states, taking into account the spatial structure of the antiferromagnetic order.We obtain a typical (gapless) Dirac Hamiltonian for the surface states when the surface of the sample is not perturbed.Gapless spectrum is protected by the combination of time-reversal and half-translation symmetries.However, a shift in the chemical potential of the surface layer opens a gap in the spectrum away from the Fermi energy.Such a gap occurs only in systems with finite antiferromagnetic order.We observe that the system topology remains unchanged even for large values of the disorder.We calculate the spectrum using the tight-binding model with different boundary conditions.In this case we get a gap in the spectrum of the surface states.This discrepancy arises due to the violation of the combined time-reversal symmetry.We compare our results with experiments and density functional theory calculations.

Smylie M.P., Islam Z., Glatz A., Gu G.D., Schneeloch J.A., Zhong R.D., Rosenkranz S., Kwok W.-., Welp U.

Eaton A., Mukherjee D., Fertig H.A.

Mel’nikov A.S., Samokhvalov A.V.

Starting from the Bogolubov–de Gennes equations for superconductor with an arbitrary anisotropic Fermi surface we derive the Andreev-type theory accounting for the quasiparticle trajectory interference and corresponding nonquasiclassical quantization effects. The resulting Andreev equations are applied for the analysis of the subgap quasiparticle states localized at the pinned vortex cores. The normal reflection of electrons and holes at the defects is shown to result in the qualitative transformation of the subgap spectra and formation of new types of bound quasiparticle states. We focus on the effect of the Fermi surface anisotropy on the quasiparticle spectral properties including opening of soft and hard minigaps in the spectrum and spatial behavior of the local density of states around pinned vortex cores.

Smith M., Quito V.L., Burkov A.A., Orth P.P., Martin I.

We present a theory for thin films of the Dirac semimetal ${\mathrm{Cd}}_{3}{\mathrm{As}}_{2}$ in the presence of magnetic fields. We show that, above a critical thickness, specific subbands $n$ of thin film ${\mathrm{Cd}}_{3}{\mathrm{As}}_{2}$ are in a quantum spin Hall insulator regime and study their response to in- and out-of-plane magnetic fields. We find that sufficiently large in-plane Zeeman fields drive the system toward a 2D Dirac semimetal regime, provided the field is directed perpendicular to a high-symmetry mirror plane. For other directions, we find the Dirac points to be weakly gapped. We further investigate how the system responds to finite out-of-plane field components, both starting from the quantum spin Hall regime at small in-plane fields and from the 2D Dirac semimetal regimes at larger in-plane fields, addressing recent experimental observations in A. C. Lygo et al. [Phys. Rev. Lett. 130, 046201 (2023)] and B. Guo et al. [Phys. Rev. Lett. 131, 046601 (2023)].

Sboychakov A.O., Rozhkov A.V., Rakhmanov A.L.

We argue that doped twisted bilayer graphene with magical twist angle can become superconducting. In our theoretical scenario, the superconductivity coexists with the spin-density-wave--like ordering. Numerical mean-field analysis demonstrates that the spin-density-wave order, which is much stronger than the superconductivity, leaves parts of the Fermi surface ungapped. This Fermi surface serves as a host for the superconductivity. Since the magnetic texture at finite doping breaks the point group of the twisted bilayer graphene, the stabilized superconducting order parameter is nematic. We also explore the possibility of a purely Coulomb-based mechanism of superconductivity in the studied system. The screened Coulomb interaction is calculated within the random phase approximation. It is shown that near the half-filling the renormalized Coulomb repulsion indeed induces the superconducting state, with the order parameter possessing two nodes on the Fermi surface. We estimate the superconducting transition temperature, which turns out to be very low. The implications of our proposal are discussed.

Pan H., Liu Z., Hou D., Gao Y., Niu Q.

The Onsager's reciprocal relation in spin transport is proved from the thermodynamic point of view and further confirmed by deriving the spin Hall effect and its inverse using the semiclassical theory. The intrinsic part of the spin conductivity is shown to be a tensor of rank 2 instead of rank 3, to be only of Hall type, and to not dissipate heat, resulting in a planar spin Hall effect governed by a spin-repulsion vector with clear geometric origin.

Estyunina T.P., Shikin A.M., Estyunin D.A., Eryzhenkov A.V., Klimovskikh I.I., Bokai K.A., Golyashov V.A., Kokh K.A., Tereshchenko O.E., Kumar S., Shimada K., Tarasov A.V.

One of the approaches to manipulate MnBi2Te4 properties is the magnetic dilution, which inevitably affects the interplay of magnetism and band topology in the system. In this work, we carried out angle-resolved photoemission spectroscopy (ARPES) measurements and density functional theory (DFT) calculations for analysing changes in the electronic structure of Mn1−xGexBi2Te4 that occur under parameter x variation. We consider two ways of Mn/Ge substitution: (i) bulk doping of the whole system; (ii) surface doping of the first septuple layer. For the case (i), the experimental results reveal a decrease in the value of the bulk band gap, which should be reversed by an increase when the Ge concentration reaches a certain value. Ab-initio calculations show that at Ge concentrations above 50%, there is an absence of the bulk band inversion of the Te pz and Bi pz contributions at the Γ-point with significant spatial redistribution of the states at the band gap edges into the bulk, suggesting topological phase transition in the system. For case (ii) of the vertical heterostructure Mn1−xGexBi2Te4/MnBi2Te4, it was shown that an increase of Ge concentration in the first septuple layer leads to effective modulation of the Dirac gap in the absence of significant topological surface states of spatial redistribution. The results obtained indicate that surface doping compares favorably compared to bulk doping as a method for the Dirac gap value modulation.

Gao A., Liu Y., Qiu J., Ghosh B., V. Trevisan T., Onishi Y., Hu C., Qian T., Tien H., Chen S., Huang M., Bérubé D., Li H., Tzschaschel C., Dinh T., et. al.

Quantum geometry in condensed-matter physics has two components: the real part quantum metric and the imaginary part Berry curvature. Whereas the effects of Berry curvature have been observed through phenomena such as the quantum Hall effect in two-dimensional electron gases and the anomalous Hall effect (AHE) in ferromagnets, the quantum metric has rarely been explored. Here, we report a nonlinear Hall effect induced by the quantum metric dipole by interfacing even-layered MnBi 2 Te 4 with black phosphorus. The quantum metric nonlinear Hall effect switches direction upon reversing the antiferromagnetic (AFM) spins and exhibits distinct scaling that is independent of the scattering time. Our results open the door to discovering quantum metric responses predicted theoretically and pave the way for applications that bridge nonlinear electronics with AFM spintronics.

Wang N., Kaplan D., Zhang Z., Holder T., Cao N., Wang A., Zhou X., Zhou F., Jiang Z., Zhang C., Ru S., Cai H., Watanabe K., Taniguchi T., Yan B., et. al.

The Berry curvature and quantum metric are the imaginary part and real part, respectively, of the quantum geometric tensor, which characterizes the topology of quantum states1. The Berry curvature is known to generate a number of important transport phenomena, such as the quantum Hall effect and the anomalous Hall effect2,3; however, the consequences of the quantum metric have rarely been probed by transport measurements. Here we report the observation of quantum-metric-induced nonlinear transport, including both a nonlinear anomalous Hall effect and a diode-like non-reciprocal longitudinal response, in thin films of a topological antiferromagnet, MnBi2Te4. Our observations reveal that the transverse and longitudinal nonlinear conductivities reverse signs when reversing the antiferromagnetic order, diminish above the Néel temperature and are insensitive to disorder scattering, thus verifying their origin in the band-structure topology. They also flip signs between electron- and hole-doped regions, in agreement with theoretical calculations. Our work provides a means to probe the quantum metric through nonlinear transport and to design magnetic nonlinear devices. Quantum-metric-induced nonlinear transport, including the nonlinear anomalous Hall effect and a diode-like response, is observed in thin films of a topological antiferromagnet, providing a means to design magnetic nonlinear devices.

Hecker M., Willa R., Schmalian J., Fernandes R.M.

Electronically ordered states that break multiple symmetries can melt in multiple stages, similarly to liquid crystals. In a partially melted phase, known as vestigial phase, a bilinear made out of combinations of the multiple components of the primary order parameter condenses. Multicomponent superconductors are thus natural candidates for vestigial order since they break both the $\text{U}(1)$-gauge and also time-reversal or lattice symmetries. Here, we use group theory to classify all possible real-valued and complex-valued bilinears of a generic two-component superconductor on a tetragonal or hexagonal lattice. While the more widely investigated real-valued bilinears correspond to vestigial nematic or ferromagnetic order, the little explored complex-valued bilinears correspond to a vestigial charge-$4e$ condensate, which itself can have an underlying $s$-wave, ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave, or ${d}_{xy}$-wave symmetry. To properly describe the fluctuating regime of the superconducting Ginzburg-Landau action and thus access these competing vestigial phases, we employ both a large-$N$ and a variational method. We show that while vestigial order can be understood as a weak-coupling effect in the large-$N$ approach, it is akin to a moderate-coupling effect in the variational method. Despite these distinctions, both methods yield similar results in wide regions of the parameter space spanned by the quartic Landau coefficients. Specifically, we find that the nematic and ferromagnetic phases are the leading vestigial instabilities, whereas the various types of charge-$4e$ order are attractive albeit subleading vestigial channels. The only exception is for the hexagonal case, in which the nematic and $s$-wave charge-$4e$ vestigial states are degenerate. We discuss the limitations of our approach, as well as the implications of our results for the realization of exotic charge-$4e$ states in material candidates.

Akzyanov R.S.

In this study, we investigate the effect of random point disorder on the surface states of a topological insulator with out-of-plane magnetization. We consider the disorder within a high-order Born approximation. The Born series converges to the one branch of the self-consistent Born approximation (SCBA) solution at low disorder. As the disorder strength increases, the Born series converges to another SCBA solution with the finite density of states within the magnetization-induced gap. Further increase of the disorder strength leads to a divergence of the Born series, showing the limits of the applicability of the Born approximation. We find that the convergence properties of this Born series are closely related to the properties of the logistic map, which is known as a prototypical model of chaos. We also calculate the longitudinal and Hall conductivities within the Kubo formulas at zero temperature with the vertex corrections for the velocity operator. Vertex corrections are important for describing transport properties in the strong disorder regime. In the case of a strong disorder, the longitudinal conductivity is weakly dependent on the disorder strength, while the Hall conductivity decreases with increasing disorder.

Cao T., Shao D., Huang K., Gurung G., Tsymbal E.Y.

Kapranov A.V., Akzyanov R.S., Rakhmanov A.L.

In the topological superconductor with the nematic superconductivity in the ${E}_{u}$ representation, it is possible to have different types of vortices. One is associated with the vorticity in the particle-hole space and corresponds to the Abrikosov vortex. Another type corresponds to the vorticity in the spin space and is called the spin vortex. We study the interaction of the Abrikosov vortex with the spin vortices. We derive the free energy of the sample with the Abrikosov and the strain-induced spin vortices using the Ginzburg-Landau approach for the two-component superconducting order parameter. We calculate the critical strain at which the spin vortex is formed. We show that the spin vortex and the Abrikosov vortex attract each other, and as a result, they have a common core. We apply Bogoliubov--de Gennes equations to study electronic states in a combined vortex structure. We show that no zero-energy states (Majorana fermions) are localized near the common vortex core of the Abrikosov vortex and the spin vortex of any type. Possible experimental realization is discussed.

Yokoyama M., Nishigaki H., Ogawa S., Nita S., Shiokawa H., Matano K., Zheng G.

Electronic nematicity, a consequence of rotational symmetry breaking, is an emergent phenomenon in various new materials. In order to fully utilize the functions of these materials, ability of tuning them through a knob, the nematic director, is desired. Here we report a successful manipulation of the nematic director, the vector order parameter (d vector), in the spin-triplet superconducting state of ${\mathrm{Cu}}_{x}{\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}$ by magnetic fields. At $H=0.5\phantom{\rule{0.16em}{0ex}}\mathrm{T}$, the ac susceptibility related to the upper critical field shows a twofold symmetry in the basal plane. At $H=1.5\phantom{\rule{0.16em}{0ex}}\mathrm{T}$, however, the susceptibility shows a sixfold symmetry, which has never been reported before in any superconductor. These results indicate that the $\mathbf{d}$ vector initially pinned to a certain direction is unlocked by a threshold field to respect the trigonal crystal symmetry. We further reveal that the superconducting gap in different crystals converges to ${p}_{x}$ symmetry at high fields, although it differs at low fields.

Tan H., Yan B.

Many experiments observed a metallic behavior at zero magnetic fields (antiferromagnetic phase, AFM) in ${\mathrm{MnBi}}_{2}{\mathrm{Te}}_{4}$ thin film transport, which coincides with gapless surface states observed by angle-resolved photoemission spectroscopy, while it can become a Chern insulator at field larger than 6 T (ferromagnetic phase, FM). Thus, the zero-field surface magnetism was once speculated to be different from the bulk AFM phase. However, recent magnetic force microscopy refutes this assumption by detecting persistent AFM order on the surface. In this Letter, we propose a mechanism related to surface defects that can rationalize these contradicting observations in different experiments. We find that co-antisites (exchanging Mn and Bi atoms in the surface van der Waals layer) can strongly suppress the magnetic gap down to several meV in the AFM phase without violating the magnetic order but preserve the magnetic gap in the FM phase. The different gap sizes between AFM and FM phases are caused by the exchange interaction cancellation or collaboration of the top two van der Waals layers manifested by defect-induced surface charge redistribution among the top two van der Waals layers. This theory can be validated by the position- and field-dependent gap in future surface spectroscopy measurements. Our work suggests suppressing related defects in samples to realize the quantum anomalous Hall insulator or axion insulator at zero fields.

How P.T., Yip S.K.

A two-component superconductor may hypothetically support a vestigial order phase above its superconducting transition temperature, with rotational or time-reversal symmetry spontaneously broken while remaining nonsuperconducting. This has been suggested as an explanation for the observed normal state nematicity of the nematic superconductor ${M}_{x}{\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}$. We examine the condition for this vestigial order to occur within Ginzburg-Landau theory with order parameter fluctuations, on both the nematic and chiral sides of the theory. Contrary to prior theoretical results, we rule out a large portion of parameter space for possible vestigial order. We argue that very extreme anisotropy is one prerequisite for the formation of a stable vestigial phase via this mechanism, which is likely not met in real materials.

Qiu J., Tzschaschel C., Ahn J., Gao A., Li H., Zhang X., Ghosh B., Hu C., Wang Y., Liu Y., Bérubé D., Dinh T., Gong Z., Lien S., Ho S., et. al.

Using circularly polarized light to control quantum matter is a highly intriguing topic in physics, chemistry and biology. Previous studies have demonstrated helicity-dependent optical control of chirality and magnetization, with important implications in asymmetric synthesis in chemistry; homochirality in biomolecules; and ferromagnetic spintronics. We report the surprising observation of helicity-dependent optical control of fully compensated antiferromagnetic order in two-dimensional even-layered MnBi2Te4, a topological axion insulator with neither chirality nor magnetization. To understand this control, we study an antiferromagnetic circular dichroism, which appears only in reflection but is absent in transmission. We show that the optical control and circular dichroism both arise from the optical axion electrodynamics. Our axion induction provides the possibility to optically control a family of $${{{\mathcal{PT}}}}$$ -symmetric antiferromagnets ( $${{{\mathcal{P}}}}$$ , inversion; $${{{\mathcal{T}}}}$$ , time-reversal) such as Cr2O3, even-layered CrI3 and possibly the pseudo-gap state in cuprates. In MnBi2Te4, this further opens the door for optical writing of a dissipationless circuit formed by topological edge states. The authors demonstrate control of antiferromagnetic order using helical light.

Chang C., Liu C., MacDonald A.H.

The quantum Hall effect, discovered by von Klitzing more than 40 years ago, requires strong magnetic fields for its realization. More recently it was found that the effect can also be realized in zero magnetic field as a result of spontaneous time-reversal symmetry breaking. This Colloquium discusses the physics underlying this quantum anomalous Hall effect, the materials it is observed in, and potential applications.

Li S., Liu T., Liu C., Wang Y., Lu H., Xie X.C.

Abstract Topological materials, which feature robust surface and/or edge states, have now been a research focus in condensed matter physics. They represent a new class of materials exhibiting nontrivial topological phases, and provide a platform for exploring exotic transport phenomena, such as the quantum anomalous Hall effect and the quantum spin Hall effect. Recently, magnetic topological materials have attracted considerable interests due to the possibility to study the interplay between topological and magnetic orders. In particular, the quantum anomalous Hall and axion insulator phases can be realized in topological insulators with magnetic order. MnBi2Te4, as the first intrinsic antiferromagnetic topological insulator discovered, allows the examination of existing theoretical predictions; it has been extensively studied, and many new discoveries have been made. Here we review the progress made in MnBi2Te4 from both experimental and theoretical aspects. The bulk crystal and magnetic structures are surveyed first, followed by a review of theoretical calculations and experimental probes on the band structure and surface states, and a discussion of various exotic phases that can be realized in MnBi2Te4. The properties of MnBi2Te4 thin films and the corresponding transport studies are then reviewed, with an emphasis on the edge state transport. Possible future research directions in this field are also discussed.

Chen P., Yao Q., Xu J., Sun Q., Grutter A.J., Quarterman P., Balakrishnan P.P., Kinane C.J., Caruana A.J., Langridge S., Li A., Achinuq B., Heppell E., Ji Y., Liu S., et. al.

The intrinsic magnetic topological insulator MnBi2Te4 (MBT) provides a platform for the creation of exotic quantum phenomena. Novel properties can be created by modification of the MnBi2Te4 framework, but the design of stable magnetic structures remains challenging. Here we report ferromagnet-intercalated MnBi2Te4 superlattices with tunable magnetic exchange interactions. Using molecular beam epitaxy, we intercalate ferromagnetic MnTe layers into MnBi2Te4 to create [(MBT)(MnTe)m]N superlattices and examine their magnetic interaction properties using polarized neutron reflectometry and magnetoresistance measurements. Incorporation of the ferromagnetic spacer tunes the antiferromagnetic interlayer coupling of the MnBi2Te4 layers through the exchange-spring effect at MnBi2Te4/MnTe hetero-interfaces. The MnTe thickness can be used to modulate the relative strengths of the ferromagnetic and antiferromagnetic order, and the superlattice periodicity can tailor the spin configurations of the synthesized multilayers. The magnetic exchange interaction of MnBi2Te4—an intrinsic magnetic topological insulator—can be tuned by intercalating ferromagnetic layers of MnTe.

Akzyanov R.S., Rakhmanov A.L.

We study the physics of the Josephson effect in the odd-parity nematic superconductors within the Ginzburg-Landau approach. The order parameter is a two-component vector that transforms as a coordinate vector in the $(x,y)$ plane under rotation and its direction is usually refer to as the nematicity direction. A nontrivial interplay between the nematicity and crystallographic axes of the superconductors that form the junction makes the Josephson effect quite unusual. We derive current-phase relations for different configurations of the junction, crystallographic axes of the sample, and the nematicity direction in the superconductors. We obtain that the Meissner kernel in the considered samples has off-diagonal components and the transverse phase difference across the junction can induce a Josephson current that flows along the contact. We show that such an anomalous Josephson Hall effect can be observed without any magnetization. We calculate the magnetic field dependence of the maximum current through the junction. We find that the period of the Fraunhofer oscillations of the maximum Josephson current depends on the geometry of the junction, the direction of the magnetic field, and the nematicity directions. The obtained results can be generalized to other superconductors with nondiagonal Meissner kernels.