Advanced Angle-Resolved Photoemission Spectroscopy (ARPES), STM and Quantum Topology Probing Bulk-Boundary correspondence (essence of topology) |
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Spectrocopic methods are capable of decisively proving and probing topological states of matter. The most prominent examples are in the field of A novel experimental approach to topological quantum phenomena: Traditionally spectroscopic methods have been used to characterize electronic behavior in quantum matter whereas initial discoveries originated from transport methods. Works in 3D topological insulators suggest that spectroscopic methods such as Science 323, 919 (2009), Nature 452, 970 (2008), Nature Physics (2009), Nature 460, 1106 (2009), Science 332, 560 (2011), Nature Physics (2019), Nature 562, 91–95 (2018), Nature 460, 1101 (2009) and Nature (2019), Science 365, 1278 (2019)). Previously topological quantum phenomena (quantum Hall like effects) were being probed mainly with transport methods pioneered by von Klitzing (1980). Following our demonstration of application of spin-ARPES (Science 2009, Nature 2009), there are world-wide efforts to apply this technique and its derivatives to probe and study novel topological quantum phenomena in condensed matter systems.
Traditionally spectroscopic methods have been used to characterize electronic or spin behavior in quantum matter whereas initial discoveries originated from non-spectroscopic methods. My work focuses on the theme which I often like to call spectroscopy for discovering new states of quantum matter. Weyl Semimetal discovery methods ( US Patent App. 10/214,797 )
Weyl Semimetal discovery methods ( US Patent App. 15/352,279 ) Approved.
Three dimensional topological insulators (3D-TI) (originally called "Topological Insulators" to distinguish them from 2D quantum Hall type effects and insulators) are the first example of topological order in the bulk solids (there is no genuine quantum Hall effect in three dimensions). They feature a "protected" metallic Dirac-like surface state (2DEG or planar "topological metal") where electron's spin and momentum are locked to each other and possess half the degrees of freedom present in an ordinary electron Fermi gas. Strong spin-orbit coupling leads to an insulating bulk and the surface states are protected by time reversal symmetry and belong to the Z2 class (see theory by Kane-Mele'05a, Kane-Mele'05b leading to 3DTI Fu-Kane-Mele'07, Moore-Balents'07; precursor theory of TR-breaking 2D topological insulator by Haldane in 1988 (T-breaking 2D topological insulators, Chern insulators) building up on Thouless et.al., TKNN theory of topological invariants). "Topological Magnets" (APS invited Talk)
In experiments, 3D Topological Insulators are an example of non-quantum-Hall-like topological matter experimentally discovered and reported (2007) around the same time, in parallel, as the spin Hall edge-states in Hg(Cd)Te, in 2007. Both the spin quantum Hall effect (Wurzburg-STANFORD team, Science 2007) and 3D Topological Insulator Surface States (my team, Nature 452, 970 (2008), submitted in 2007, "Search&Discovery" Physics Today and KITP 2007) were reported the same year 2007 (a few months apart) using two independent and unrelated experimental methods. Experimentally, these two are unrelated. The spin quantum Hall effect (QSHE) can be thought of as two copies of well-known IQH (integer quantum Hall) states put together in two dimensions. Since IQH state is a 2D topological insulator, spin quantum Hall effect is also a 2D topological insulator but time-reversal invariant (protected by Z2 invariant). On the other hand, the 3D Topological Insulators are a new and distinct state of matter which cannot be reduced to multiple copies of IQH and there is no spin Hall effect in 3D (the term "Topological Insulators" was originally used exclusively for the novel and unprecedented 3D state since there is no spin Hall like effect there). The 3D state is thus an example of non-quantum-Hall-like topological matter and the first realization of topologically ordered bulk solid in nature (Topological Insulators, Physics World 2011). Experimentally, 3D TI did not arise from quantum spin Hall effect. Additionally, transport measurements cannot provide a proof of Z2 topology either (Physics World 2011). All of the 2D topological insulator examples (IQH, FQH, QSH) including the fractional one (FQH) involving Coulomb interaction are understood in the standard picture of quantized electron orbits in a spin-independent or spin-dependent magnetic field, the 3D topological insulator defies such description and is a novel type of topological order which cannot be reduced to multiple copies of quantum-Hall-like states. In fact, the 3D topological insulator exists not only in zero magnetic field, they also differ from the 2D variety in three very important aspects: 1) they possess topologically protected 2D metallic surfaces (a new type of 2DEG) rather than the 1D edges, 2) they can work at room temperature (300K and beyond, largegap topological insulators) rather than cryogenic (mK) temperatures required for the QSH effects and,3) they occur in standard bulk semiconductors rather than at buried interfaces of ultraclean semiconductor heterostructures thus tolerate stronger disorder than the IQH-like states. The non-quantum-Hall-like (novel) character and the extremely rich physics (surface 2DEG, helical fermion gas) of the 3D state has led to a world-wide research interest in this topic in general. Also the novel experimental approach in studying topological quantum phenomena demonstrated by us are now being used by many other groups world-wide studying many other topological materials and phenomena. Why Topological Surface States (TIs) are so exciting? Topological Insulators (3DTI) is new and unprecedented and cannot be reduced to multiple copies of quantum Hall or spin Hall like states. Most topological states of matter are realized in two or lower dimensions (quantum Hall states, quantum spin Hall effect, non-Fermi liquid chains and wires, quantum spin-liquids etc.). Unlike all others, neither strong electron-electron interactions (necessary for quantum spin liquids), high magnetic fields and low temperatures (necessary for quantum Hall states), nor low dimensionality (needed for quantum Hall states, spin quantum Hall states (QSHE), and non-Fermi liquid spin chains) are needed for the 3D topological insulator. The theoretical and experimental discovery of the 3D TIs - the first example of topological order in bulk solids - has generated much experimental and theoretical efforts to understand and utilize all aspects of these quantum phenomena and the materials that exhibit them. One of the major challenges in going from quantum Hall-like 2D states to 3D topological insulators is to develop new experimental approaches/methods to precisely probe this novel form of topological-order since the standard tools and settings that work for IQH-state also work for QSH states. The method to probe 2D topological-order is exclusively with charge transport (pioneered by Von Klitzing in the 1980s), which either measures quantized transverse conductance plateaus in IQH systems or longitudinal conductance in quantum spin Hall (QSH) systems. In a 3D topological insulator, the boundary itself supports a two dimensional electron gas (2DEG) and transport is not (Z2) topologically quantized hence cannot directly probe the topological invariants ?o or the topological quantum numbers analogous to the Chern numbers of the IQH systems. This is unrelated to the fact that the present materials have some extrinsic or residual/impurity conductivity in their naturally grown bulk. In this paper, we review the birth of momentum- and spin-resolved spectroscopy as a new experimental approach and as a directly boundary sensitive method to study and prove topological-order in three-dimensions via the direct measurements of the topological invariants ?o that are associated with the Z2 topology of the spin-orbit band structure and opposite parity band inversions, which led to the experimental discovery of the first 3D topological insulator in Bi-based semiconductors. Topological Surface States (A New Type of 2D Electron System) & Topological Insulators: Experimentally demonstrated, a three dimensional topological insulator (3D-TI) features a protected two-dimensional electron gas on its surface. The high magnetic fields, low temperatures or low dimensionality are not necessary for retaining the topological protection or topological order of a macroscopic 2DEG on the surface of a topological insulator. Nature 452, 970 (2008), submitted in 2007, also KITP 2007 "Search&Discovery" Physics Today Experimental Discovery : Topological Surface States - A New Type of 2D Electron Systems Measurements of topological invariants {vo}: Transport measurements that have been the key to probe topological order in conventional quantum Hall like systems cannot (even in theory!) be used to measure the topological quantum numbers of Z2 TIs (Fu-Kane's {vo}). We have shown a technique/method for the direct measurement of Z2 topological quantum numbers {vo} for the first time: Science 323, 919 (2009), later further expanded: Science 332, 560 (2011) see below for details. Topological (Z2) Order is more directly manifested in the spin and momentum correlated motion of electrons on the surface. This leads to a new type of 2DEG where the electron's spin and linear momentum are one-to-one locked. Such a 2DEG only carries half of the total degrees of freedom of a conventional 2DEG and in the vicinity of the Kramers' point takes the form of a half Dirac gas: Nature 460, 1101 (2009) and further details in related materials Phys. Rev. Lett. (2009) News in Physics Today Consequence of Z2 topological order: Spin-Momentum locking and pi-Berry's phase lead to the absence of elastic backscattering on the surfaces. By combining spin-ARPES and tunneling (collaboration), we have demonstrated such absence of elastic backscattering: Nature 460, 1106 (2009) (STM+Spin-ARPES), Berry's phase was shown at Science 323, 919 (2009) Discovery of the next generation and "room temperature topological insulators": the Bi2Se3 class: The advantages of large band gap and simple spin-polarized Dirac cone topology of these spin-orbit insulators led to our observation of topological quantum phenomena at room temperatures without magnetic fields and without high purity semiconductors Discovery of Single-Dirac-Cone TSS Bi2Se3 as a TI class: N&V NatPhys (2009) KITP Proc. 2008 "The Hydrogen Atom of Topological Insulator" BiSb (2007) to Bi2Se3 KITP (2008) Nature Physics 5, 398 (2009) submitted in 2008 and (see above Nature (2009)) Phys. Rev. Lett. 103, 146401 (2009) further work on Bi2Te3 and Sb2Te3 in comparison with Bi2Se3 Phys. Rev. Lett. 105, 036404 (2010) Bi2Se3 related TIs such as spin-orbit BiTlSe2 class of materials Given that the topological insulators are standard bulk semiconductors and their topological characteristics can survive to high temperatures, their novel properties could lead to many exciting applications. An exciting progress along this line is that the Bi2Se3 class of 3D TIs can be turned in to superconductors to form the host material for Majorana Fermions (Nature Physics 6, 855 (2010)). Also both integer and fractional quantum Hall effects have been reported in this Bi2Se3 class of materials by other groups indicating the high mobility of the topological surface state. These developments have unleashed a world-wide experimental effort to understand all aspects of electrical and spin properties leading to a nearly graphene-like revolution in physics (Physics World 2011). Magnetic Symmetry Breaking: Topological protection and degree of robustness: The Dirac cone materials are probed via the modification of surface potential: How robust the topological properties of a Topological Insulator surface are investigated (Nature Physics 7, 32 (2011)). This paper reported preliminary results regarding magntism, for a full detail see, Magnetic Topological Insulators : The effect of time-reversal symmetry leads to unconventional spin textures on the surface of a topological insulator. Hedgehog spin texture and Berry's phase tuning in a magnetic topological insulator was observed in our experiments; Nature Physics 7, 032 (2011) Coulomb/disorder/magnetic perturbation effects. Nature Physics 8, 616 (2012) for Magnetic symmetry breaking, for details see Phys. Rev. B (2012) A novel experimental approach to topological quantum phenomena: Traditionally spectroscopic methods have been used to characterize electronic behavior in quantum matter whereas initial discoveries originated from transport methods. Our works in 3D topological insulators suggest that spectroscopic methods such as ARPES can be utilized to discover novel topological quantum phenomena (Science 323, 919 (2009), Nature 452, 970 (2008), Nature 460, 1106 (2009), Science 332, 560 (2011)). Previously topological quantum phenomena (quantum Hall like effects) were being probed mainly with transport methods pioneered by von Klitzing (1980). Following our demonstration of application of spin-ARPES, there are world-wide efforts to apply this technique and its derivatives to probe and study novel topological quantum phenomena in condensed matter systems ["Search&Discovery" Physics Today 2009; Physics World 2011]. Superconducting Symmetry Breaking: Superconducting Topo. Insulators and Topological-Order: Superconducting topological insulators can serve as Majorana platforms. Observation of topological order in a superconducting doped topological insulator demonstrates a platform for realizing Majorana fermions. Some of the doped topological insulators that superconduct at low temperatures may also turn out to be topological superconductors proposed in theory based on our experimental observations and spin-ARPES results Nature Physics 6, 855 (2010) and additional details in Phys. Rev. B (2012) Bulk Insulating Topological Insulators: Recently, highly bulk insulating topological insulators have also been realized where Dirac surface states contribute more than 70% of the total conduction channel (Preprint at Xiong et.al., arXiv:1101.1315v1 on Bi-based 3D TIs identified by us, Preprint at S.-Y. Xu et.al., arXiv:1007.5111v1 (2010) which led to our work reported at Phys. Rev. B (2012)). Also see works by other groups Phys. Rev. B (2010) [ our eariler ARPES on Bi2Te2Se at arXiv (2010)] Phys. Rev. B (2012) and Phys. Rev. B (2013) BTS family of topological insulators; arXiv (2010) Nature Commun. 04, 2991 (2013) (see the TKI section below for details) Adiabatic Continuation Method for Predicting TI & Topological Phase Transition: Working with Hsin Lin and others we demonstrated that first-principles based adiabatic continuation approach is a very powerful and efficient tool for constructing topological phase diagrams and locating non-trivial topological insulator materials. Applied to real materials our results demonstrated the efficacy of adiabatic continuation as a useful tool for exploring topologically nontrivial alloying systems and for identifying new topological insulators even when the underlying lattice does not possess inversion symmetry, and the approaches based on parity analysis of Fu-Kane are not viable. Nature Materials 9, 546 (2010) and more recently at Phys. Rev. B (2013) Topological Phase transition and Texture inversion: It is believed that a trivial insulator can be twisted into a topological state by modulating the spin-orbit interaction driving the system through a topological quantum phase transition. We reported the observation of such a phase transition in a tunable spin-orbit system where the topological state formation is visualized. In the topological state, vortex-like polarization states are observed to exhibit 3D vectorial textures, which collectively feature a chirality transition as the spin-momentum locked electrons on the surface go through the zero carrier density point. Such phase transition and texture inversion can be the physical basis for observing fractional charge (±e/2) and other fractional topological phenomena. Science 332, 560 (2011) and in a related system, see Phys. Rev. Lett. 109, 186403 (2012) Topological Crystalline Insulator (TCI) Phase: Z2 topological insulator protected by time-reversal symmetry is realized via spin-orbit interaction-driven band inversion. The topological phase in the Bi1?xSbx system is due to an odd number of band inversions. We experimentally investigated the possibility of a mirror symmetry-protected topological crystalline insulator phase in the Pb1?xSnxTe class of materials that has been theoretically predicted (by Fu et.al.,) to exist in its end compound SnTe. Our observation of the spin-polarized Dirac surface states in the inverted Pb1?xSnxTe and their absence in the non-inverted compounds related via a topological phase transition provide the experimental groundwork for opening the research on novel topological order in quantum devices. Nature Commun. 03, 1192 (2012) TCI-phase and BI to TCI Phase Transition Science 341, 1496 (2013) Dirac node formation and mass acquisition in a topological crystalline insulator (TCI)-phase, in collaboration with scanning tunneling spectroscopy (Madhavan) group at Boston and on orbital-texture physics, see Nature Physics (in press) (2014) "3D Graphene", Topological 3D Dirac Semimetals: Symmetry-broken three-dimensional (3D) topological Dirac semimetal systems with strong spin-orbit coupling can host many exotic Hall-like phenomena and Weyl fermion quantum transport. Using high-resolution angle-resolved photoemission spectroscopy, we performed systematic electronic structure studies on Cd 3 As 2 , which has been predicted to be the parent material, from which many unusual topological phases can be derived. We observe a highly linear bulk band crossing to form a 3D dispersive Dirac cone projected at the Brillouin zone centre by studying the (001)-cleaved surface. Remarkably, an unusually high in-plane Fermi velocity up to 1.5 x 10^ 6 m/s is observed in our samples, where the mobility is known up to 40,000 cm 2/ V.s, suggesting that Cd3As2 can be a promising candidate as an anisotropic-hypercone (three-dimensional) high spin-orbit analogue of 3D graphene. Our discovery of the Dirac-like bulk topological semimetal phase in Cd3As2 opens the door for exploring higher dimensional spin-orbit Dirac physics in a real material. The 3D Dirac semimetals can be realized from topo. phase transition. Near the critical point a 3D Dirac version of Graphene is realized which was probed in our 2011 Science paper (more recently elaborated at our Nat.Com. paper): Science 332, 560 (2011) and more recently in other materials such as Cd3As2 (see below) Nature Commun. 05, 3786 (2014) topological 3D Dirac Semimetal Cd3As2 Preprint on Na3Bi (2013) topological 3D Dirac Semimetal Na3Bi Ultrafast Time-resolved response of Topological Insulators: The advent of topological insulators has made it possible to realize two-dimensional spin polarized gases of relativistic fermions with unprecedented properties in condensed matter. Their photoconductive control with ultrafast light pulses is of interest in optoelectronics. In collaboration with M. Marsi group (Paris) we probed the interplay of surface and bulk transient carrier dynamics in a photoexcited topological insulator Bi2Se3 and related materials. Currently, we are probing other exotic aspects of ultrafast response of bulk insulating TI materials (preprint) and correlated electron systems. Nature Commun. 05, 3786 (2014) Relativistic nanoscale Schottky barrier (with M. Marsi group) Preprint (2014) Exotic ultrafast response of TIs (our recent work) Nanoscale Ultra-Thin-Films/MBE of Topological Insulators and related spin-orbit films: Understanding the spin behaviour of boundary modes in ultrathin topological insulator films is critically essential for the design and fabrication of functional nanodevices. Using spin-resolved photoemission spectroscopy with p-polarized light in topological insulator Bi2Se3 thin films, we reported tunnelling-dependent evolution of spin configuration in topological insulator thin films across the metal-to-insulator transition and separately in magnetically doped thin-films. Nature Physics 8, 616 (2012) Magnetic Thin Films Nature Commun. 05, 3786 (2014) Tunelling and spin-texture evolution in films for potential devices. Nature Commun. (in press) (2014) Spin-orbit physics in molydiselenide films for potential devices. Topological Kondo or Mixed-Valence Correlated Electron Systems: Topological States can also arise in correlated electron systems such as Kondo or Mixed-Valence Insulators. By combining low-temperature and high energy-momentum resolution of the laser-based ARPES technique, we probed the surface electronic structure of the anomalous conductivity regime (sub 6K) in SmB6. We observe that the bulk bands exhibit a Kondo gap of 15 meV and identify in-gap low-lying states within a 4 meV window of the Fermi level on the (001)-surface of this material. These states disappear as temperature is raised above 15K in correspondence with the complete disappearance of the 2D conductivity channels in SmB6. Our bulk and surface measurements carried out in the transport-anomaly-temperature regime (T 10K) are consistent with the first-principle predicted Fermi surface topology of a topological Kondo insulator phase in this material. Nature Commun. 04, 2991 (2013) Preprint on YbB6 (2014) An expt. algorithm for topo Kondo & mixed-val. insulators (2013) Physics World 2011 First five experimental papers on 3DTI (Topological Insulators)
Nature 452, 970 (2008); D.Hsieh, D.Qian, Y.Xia et.al., [April, '08] Submt.(2007) Science 323, 919 (2009); D.Hsieh, Y.Xia, L.A.Wray et al., [February, '09] Submt.(2008) Nature Physics 5, 398 (2009); Y.Xia, D.Qian, L.A.Wray, D.Hsieh et al., [May '09] Submt.(2008) Extended version at Nature 460, 1101 (2009); D.Hsieh, Y.Xia, D.Qian et.al., Submt.(2009) Phys.Rev.B 79, 195208 (2009); Y.Hor, A.Richardella, Y.Xia, D.Hsieh et.al., [May '09] Submt.(2009) Science 325,178 (2009); Y.L.Chen, J.Analytis, . S.-C. Zhang et al., [Stanford] [June '09] Submt.(Mar. 2009) |
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**Nature News (Homepage, July 2017)**: ..Topology Reshaping Physics

Ideas borrowed from graphene (Dirac semimetal) and topological insulators (in 3D) allow one to generalize to talk about a Dirac Semimetal in three dimensions and there can be many different types of 2D and 3D Dirac Semimetals with their experimental realizations in non-Bi or Bi-based spin-orbit materials :

For recent developments in topological materials see talks at

**KITP program on topological materials..**

and Inst. for Advanced Study (IAS, Princeton) School on Topological Matter

**Reading List**FOCUS issue on "

**Topological Semimetals**" by Nature Materials (Collection of Articles)

Magnetic metals/semimetals with Dirac fermions:

**PhysicsWorld (2018)**: "Physicists find new ‘control knob’ for the quantum topological world" (

**Nature 2018**)

**"Wily Weyl"**: PhysicsWorld (2018)

"By considering the topology of chiral crystals, a new type of massless fermion,

connected with giant arc-like surface states, are predicted.

Such Kramers–Weyl fermions should manifest themselves

in a wide variety of chiral materials."

__News at Nature Materials 2018__Dirac-Weyl Semimetals in 2D and 3DGraphene is an example of Dirac semimetal. Another example of Dirac Semimetal in two dimensions is the surface states of topological insulators while Fermi level is tuned to lie at the Dirac point inside the bulk gap of the semiconductor (see, "A tunable topological insulator in the spin helical Dirac transport regime";
Hsieh, Xia, Wray et.al., Nature 460, 1101 (2009)).
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The recently discovered three-dimensional or bulk topological insulators are expected to exhibit exotic quantum phenomena. It is believed that a trivial insulator can be twisted into a topological state by modulating the spin-orbit interaction or the crystal lattice, driving the system through a topological quantum phase transition. By directly measuring the topological quantum numbers and invariants, we report the observation of a phase transition in a tunable spin-orbit system, BiTl(S |
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3D Dirac semimetals appear at the critical point between a conventional insulator and a 3D topological insulator (see papers below) Topological Phase Transition and Texture Inversion in a Tunable Topological Insulator. S.-Y. Xu, Y. Xia, L. A. Wray et al.;Science 332, 560 (2011) |
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Topology of the electronic structure of a crystal is manifested in its surface states. In topological insulators B |
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Symmetry-broken three-dimensional (3D) topological Dirac semimetal systems with strong spin-orbit coupling can host many exotic Hall-like phenomena and Weyl fermion quantum transport. |
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Observation of Fermi Arc Surface States in a Topological Metal. Published in S.-Y. Xu, C. Liu, S. Kushwaha, et al., Science 347, 294 (2014). |
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Ultraquantum magnetoresistance in single-crystalline Ag2Se. Published in C. Zhang, H. Li, T.-R. Chang, et al., arXiv:1502.02324 (2015). |
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Observation of a bulk 3D Dirac multiplet, Lifshitz transition, and nestled spin states in Na _{3}Bi. Preprint by S.-Y. Xu, C. Liu, S. K. Kushwaha, et al., arXiv:1312.7624 (2013). |
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Surface Versus Bulk Dirac States Tuning in a Three-Dimensional Topological Dirac Semimetal. Published in M. Neupane, S.-Y. Xu, N. Alidoust, et al., Phys. Rev. B 91, 241114(R) (2015). |
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A strongly robust type II topological Weyl fermion semimetal state in Ta3S2
Guoqing Chang, Su-Yang Xu, Daniel S. Sanchez, Shin-Ming Huang, Chi-Cheng Lee, Tay-Rong Chang, Guang Bian, Hao Zheng, Ilya Belopolski, Nasser Alidoust, Horng-Tay Jeng, Arun Bansil, Hsin Lin, and M. Zahid Hasan Science Adv. e1600295 (2016) |
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Type-II Symmetry Protected Topological Dirac Semimetals
Tay-Rong Chang, Su-Yang Xu, Daniel S. Sanchez, Wei-Feng Tsai, Shin-Ming Huang, Guoqing Chang, Chuang-Han Hsu, Guang Bian, Ilya Belopolski, Zhi-Ming Yu, Shengyuan A. Yang, Titus Neupert, Horng-Tay Jeng, Hsin Lin, and M. Zahid Hasan Physical Review Letters 119, 026404 (2017) |
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Mirror-protected Dirac fermions on Weyl Semimetal Surfaces (2017) | ||

(Theoretical prediction and experimental observation of) "Mirror Protected Dirac Fermions on a Weyl Semimetal NbP Surface" Hao Zheng, Guoqing Chang, Shin-Ming Huang, Cheng Guo, Xiao Zhang, Songtian Zhang, Jiaxin Yin, Su-Yang Xu, Ilya Belopolski, Nasser Alidoust, Daniel S. Sanchez, Guang Bian, Tay-Rong Chang, Titus Neupert, Horng-Tay Jeng, Shuang Jia, Hsin Lin, and M. Zahid Hasan Phys. Rev. Lett. 119, 196403 (2017) |
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Topological Hopf-link semimetals (2017) | ||

"Topological Hopf-link semimetals" G. Chang, S.-Y. Xu et.al., "Topological Hopf and Chain Link Semimetal States and Their Application to Co2MnGa" Phys. Rev. Lett. 119, 156401 (2017) |
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