Discovery of Weyl Fermions, Topological Fermi Arcs and Topological Nodal-Line States of Matter

Princeton University group (click on the picture to view with higher resolution), From left to right: Guang Bian, M. Zahid Hasan (Principal investigator), Nasser Alidoust, Hao Zheng, Daniel S. Sanchez, Suyang Xu and Ilya Belopolski. 

Author: M. Zahid Hasan

Affiliation: Department of Physics, Princeton University, USA

Link to Hasan Research Group: Laboratory for Topological Quantum Matter & Advanced Spectroscopy >>

The eponymous Dirac equation describes the first synthesis of quantum mechanics and special relativity in describing the nature of electron. Its solutions suggest three distinct forms of relativistic particles - the Dirac, Majorana and Weyl fermions [1-3]. In 1929, Hermann Weyl proposed the simplest version of the equation, whose solution predicted massless fermions with a definite chirality or handedness [3]. Weyl’s equation was intended as a model of elementary articles, but in nearly 86 years, no candidate Weyl fermions have ever been established in high-energy experiments. Neutrinos were once thought to be such particles but later found to possess a small mass. Recently, analogs of the fermion particles have been discovered in certain electronic materials that exhibit strong spin-orbit coupling and topological behavior. Just as Dirac fermions emerge as signatures of topological insulators [4], researchers have shown that electronic excitations in semimetals such as tantalum or niobium arsenides (TaAs and NbAs) behave like Weyl fermions [5-7]. And such a behavior is consistent with their topological semimetal bandstructures [8,9].

Past 2Physics article by M. Zahid Hasan:
July 18, 2009: "Topological Insulators : A New State of Quantum Matter"

In 1937 physicist Conyers Herring considered under what conditions electronic bands in solids have the same energy by accident in crystals that lack certain symmetries [10]. Near these accidental band touching points, the low-energy excitations, or electronic quasiparticles can be described by an equation that is essentially identical to the 1929 Weyl equation. In recent times, these touching points have been theoretically studied in the context of topological materials and are referred to as Weyl points and the quasiparticles near them are the emergent Weyl fermions [11]. In these solids, the electrons’ quantum-mechanical wave functions acquire a phase, as though they were moving in a superficial magnetic field that is defined in momentum space. In contrast to a real magnetic field, this fictional field (known as a Berry curvature) admits excitations that behave like magnetic monopoles. These monopoles are topological defects or singularities that locate at the Weyl points. So the real space Weyl points are associated with chiral fermions and in momentum space they behave like magnetic monopoles [11-17]. The fact that Weyl nodes are related to magnetic monopoles suggests they will be found in topological materials that are in the vicinity of a topological phase transition [14,15]. The surface of a topological insulator has a Fermi surface that forms a closed loop in momentum space; in a Weyl semimetal, these loops become non-closed arcs as some symmetry is lifted [11,12]. These Fermi arcs terminate at the location of the bulk Weyl points ensuring their topological nature [12]. Theory had suggested that Weyl semimetals should occur in proximity to topological insulators in which inversion or time-reversal symmetry was broken [12,14,16].

Building on these ideas, researchers, including the Princeton University group, used ab initio calculations to predict candidate materials [8,9] and perform angle-resolved photoemission spectroscopy to detect the Fermi arcs, characteristic of Weyl nodes, on the surface of TaAs and NbAs [5-7]. ARPES is an ideal tool for studying such a topological material as known from the extensive body of works on topological insulators [4]. The ARPES technique involves shooting high-energy photons on a material and measuring the energy, momentum and spin of the ejected electrons both from the surface and the bulk. This allows for the explicit determination of both bulk Weyl nodes and the Fermi-arc surface states (Figure 1).

Figure 1: (click on the image to view with high resolution) Weyl fermion and Fermi arcs (a) Schematic of the band structure of a Weyl fermion semimetal. (b) Correspondence of the bulk Weyl fermions to surface Fermi arc states. (c) ARPES mapping of TaAs Fermi surface. (d) Fermi arc surface states and Weyl nodes on the (001) surface of TaAs. (e) Linear dispersion of Weyl quasi-particles in TaAs. (Adapted form Ref. [5])

In the absence of spin-orbit coupling, the tantalum arsenide material is a nodal-line semimetal in which the bulk Fermi surface is a closed loop in momentum space [8,17,18]. With spin-orbit coupling turned on, the loop-shaped nodal line condenses into discrete Weyl points in momentum space [8]. In this sense the topological nodal-line semimetal can be thought of as a state where the Weyl semimetals originate from by further symmetry breaking (Figure 2). Such a state has been considered in theory previously [17] but it lacked concrete experimental realizations. Very recently, the first example of a topological nodal-line semimetal in the lead tantalum selenide (PbTaSe₂) materials has been reported experimentally [18]. Even though many predictions existed, no concrete experimentally realizable material was found. These findings suggest that Weyl semimetals [5-7] and nodal-line semimetals [17-18] are the first two examples of topological materials that are intrinsically gapless in contrast to topological insulators [4].

Figure 2: (click on the image to view with high resolution) Topological nodal-line semimetals (a) Schematic of a Weyl semimetal and a topological nodal-line semimetal. (b) ARPES mapping and theoretical simulation of (001)-surface band structure of PbTaSe₂ showing the loop-shaped bulk Fermi surface. (c) ARPES spectrum and theoretical band structure along some momentum space directions. (e) Calculated iso-energy band contour showing the nodal line and topological surface states. (Adapted from Ref. [18])

In the 1980s, Nielsen and Ninomiya suggested that exotic effects, like the ABJ (Adler-Bell-Jackiw) chiral anomaly—in which the combination of an applied electric and magnetic fields generates an excess of quasiparticles with a particular chirality—were associated with Weyl fermions and could be observable in 3D crystals [13]. A further correspondence has been established more recently with the increased understanding of materials with band structures that are topologically protected [11-17]. Unusual transport properties that are associated with Weyl fermions, such as a reduction of the electrical resistance in the presence of an applied magnetic field, have already been reported in the TaAs class of materials [19,20] (Figure 3). Weyl materials can also act as a novel platform for topological superconductivity leading to the realization of Weyl-Majorana modes potentially opening a new pathway for investigating qubit possibilities [21]. Weyl particles have also been observed in photonic (bosonic) crystals. In these systems the number of optical modes has an unusual scaling with the volume of the photonic crystal, which may allow for the construction of large-volume single-mode lasers [22]. Development in the last few months seems to suggest that Weyl particles are indeed associated with a number of unexpected quantum phenomena and these findings may lead to applications in next-generation photonics and electronics.

Figure 3: (click on the image to view with high resolution) Signature of the chiral anomaly in the Weyl fermion semimetal TaAs. (a) Magneto-resistance (MR) data of the Weyl semimetal TaAs in the presence of parallel electric and magnetic fields at T = 2 K. The MR decreases as one increases the magnetic field. (b) MR as a function of angle between the electric and the magnetic fields. The negative magneto-resistance is quickly suppressed as one varies the direction of the magnetic ~B field away from that of the electric ~E field. The observed negative MR and its angular dependence serve as the key signature of the chiral anomaly. (c,d) Landau energy spectra of the left- and right-handed Weyl fermions in the presence of parallel electric and magnetic fields. (Adapted from Ref. [20])

References:
[1] Frank Wilczek “Why are there analogies between condensed matter and particle theory?” Physics Today, 51, 11–13 (1998). Abstract.
[2] Palash B. Pal, “Dirac, Majorana and Weyl fermions”, American Journal of Physics, 79, 485–498 (2011). Abstract.
[3] Hermann Weyl, “Elektron und Gravitation. I”, Zeitschrift für Physik, 56, 330 (1929). Abstract.
[4] M. Z. Hasan and C.L. Kane “Topological Insulators”, Review of Modern Physics, 82, 3045 (2010). Abstract.
[5] Su-Yang Xu, Ilya Belopolski, Nasser Alidoust, Madhab Neupane, Guang Bian, Chenglong Zhang, Raman Sankar, Guoqing Chang, Zhujun Yuan, Chi-Cheng Lee, Shin-Ming Huang, Hao Zheng, Jie Ma, Daniel S. Sanchez, BaoKai Wang, Arun Bansil, Fangcheng Chou, Pavel P. Shibayev, Hsin Lin, Shuang Jia, M. Zahid Hasan, “Discovery of a Weyl Fermion Semimetal and Topological Fermi Arcs in TaAs”,  Science, 349, 613 (2015). Abstract.
[6] Su-Yang Xu, Nasser Alidoust, Ilya Belopolski, Zhujun Yuan, Guang Bian, Tay-Rong Chang, Hao Zheng, Vladimir N. Strocov, Daniel S. Sanchez, Guoqing Chang, Chenglong Zhang, Daixiang Mou, Yun Wu, Lunan Huang, Chi-Cheng Lee, Shin-Ming Huang, BaoKai Wang, Arun Bansil, Horng-Tay Jeng, Titus Neupert, Adam Kaminski, Hsin Lin, Shuang Jia, M. Zahid Hasan, “Discovery of a Weyl Fermion state with Fermi arcs in NbAs”, Nature Physics, 11, 748 (2015). Abstract.
[7] B.Q. Lv, H.M. Weng, B.B. Fu, X.P. Wang, H. Miao, J. Ma, P. Richard, X.C. Huang, L.X. Zhao, G.F. Chen, Z. Fang, X. Dai, T. Qian, H. Ding, “Experimental Discovery of Weyl Semimetal TaAs”, Physical Review X, 5, 031013 (2015). Abstract; B.Q. Lv, N. Xu, H.M. Weng, J.Z. Ma, P. Richard, X.C. Huang, L.X. Zhao, G.F. Chen, C.E. Matt, F. Bisti, V.N. Strocov, J. Mesot, Z. Fang, X. Dai, T. Qian, M. Shi, H. Ding, "Observation of Weyl nodes in TaAs", Nature Physics, 11, 724 (2015). Abstract.
[8] Shin-Ming Huang, Su-Yang Xu, Ilya Belopolski, Chi-Cheng Lee, Guoqing Chang, BaoKai Wang, Nasser Alidoust, Guang Bian, Madhab Neupane, Chenglong Zhang, Shuang Jia, Arun Bansil, Hsin Lin, M. Zahid Hasan, “A Weyl Fermion Semimetal with Surface Fermi Arcs in the Transition Metal Monopnictide TaAs Class”, Nature Communications, 6, 7373 (2015). Abstract.
[9] Hongming Weng, Chen Fang, Zhong Fang, B. Andrei Bernevig, Xi Dai, “Weyl Semimetal Phase in Noncentrosymmetric Transition-Metal Monophosphides”, Physical Review X, 5, 011029 (2015). Abstract.
[10] Conyers Herring, “Accidental Degeneracy in the Energy Bands of Crystals”, Physical Review, 52, 365-373 (1937). Abstract.
[11] Ashvin Vishwanath, “Viewpoint: Where the Weyl Things Are”, Physics, 8, 84 (2015). Full Text.
[12] Xiangang Wan, Ari M. Turner, Ashvin Vishwanath, Sergey Y. Savrasov, “Topological Semimetal and Fermi-Arc Surface States in the Electronic Structure of Pyrochlore Iridates”, Physical Review B, 83, 205101 (2011). Abstract.
[13] H. B. Nielsen, Masao Ninomiya, “The Adler-Bell-Jackiw Anomaly and Weyl Fermions in a Crystal”, Physics Letters B, 130, 389 (1983). Abstract.
[14] Shuichi Murakami, “Phase Transition Between the Quantum Spin Hall and Insulator Phases in 3D: Emergence of a Topological Gapless Phase”, New Journal of Physics, 9, 356 (2007). Full Text.
[15] Grigory E. Volovik, "The Universe in a Helium Droplet", Oxford University Press (2003).
[16] A.A. Burkov, Leon Balents, "Weyl semimetal in a Topological Insulator multilayer", Physical Review Letters, 107, 127205 (2011). Abstract.
[17] A. A. Burkov, M. D. Hook, Leon Balents, "Topological Nodal Semimetals", Physical Review B, 84, 235126 (2011). Abstract.
[18] Guang Bian, Tay-Rong Chang, Raman Sankar, Su-Yang Xu, Hao Zheng, Titus Neupert, Ching-Kai Chiu, Shin-Ming Huang, Guoqing Chang, Ilya Belopolski, Daniel S. Sanchez, Madhab Neupane, Nasser Alidoust, Chang Liu, BaoKai Wang, Chi-Cheng Lee, Horng-Tay Jeng, Chenglong Zhang, Zhujun Yuan, Shuang Jia, Arun Bansil, Fangcheng Chou, Hsin Lin, M. Zahid Hasan , "Topological Nodal-Line Fermions in Spin-Orbit Metal PbTaSe₂", Nature Communications", 7:10556 (2016). Abstract.
[19] Xiaochun Huang, Lingxiao Zhao, Yujia Long, Peipei Wang, Dong Chen, Zhanhai Yang, Hui Liang, Mianqi Xue, Hongming Weng, Zhong Fang, Xi Dai, Genfu Chen, "Observation of the chiral anomaly induced negative magneto-resistance in 3D Weyl semi-metal TaAs", Physical Review X, 5, 031023 (2015). Abstract.
[20] Chenglong Zhang, Su-Yang Xu, Ilya Belopolski, Zhujun Yuan, Ziquan Lin, Bingbing Tong, Nasser Alidoust, Chi-Cheng Lee, Shin-Ming Huang, Hsin Lin, Madhab Neupane, Daniel S. Sanchez, Hao Zheng, Guang Bian, Junfeng Wang, Chi Zhang, Titus Neupert, M. Zahid Hasan, Shuang Jia, "Observation of the Adler-Bell-Jackiw chiral anomaly in a Weyl semimetal", arXiv:1503.02630 [cond-mat.mes-hall] (2015). [To appear in Nature Communications].
[21] Anffany Chen, M. Franz, "Superconducting proximity effect and Majorana flat bands in the surface of a Weyl semimetal", arXiv:1601.01727 [cond-mat.supr-con] (2016).
[22] Ling Lu, Zhiyu Wang, Dexin Ye, Lixin Ran, Liang Fu, John D. Joannopoulos, Marin Soljačić, “Experimental Observation of Weyl Points”, Science, 349, 622 (2015). Abstract.

Topological Insulators : A New State of Quantum Matter

M. Zahid Hasan

[This is an invited article based on a series of recent works by the author and his collaborators -- 2Physics.com]

Author: M. Zahid Hasan

Affiliation: Joseph Henry Laboratories of Physics, Department of Physics,
Princeton University, USA

Most quantum states of condensed-matter systems or the fundamental forces are categorized by spontaneously broken symmetries. The remarkable discovery of quantum Hall effects (1980s) revealed that there exists an organizational principle of matter based not on the broken symmetry but only on the topological distinctions in the presence of time-reversal symmetry breaking [1,2]. In the past few years, theoretical developments suggest that new classes of topological states of quantum matter might exist in nature [3,4,5]. Such states are purely topological in nature in the sense that they do not break time-reversal symmetry, and hence can be realized without any applied magnetic field : "Quantum Hall-like effects without magnetic field".

Research Team at Princeton University: [L to R] David Hsieh, Dong Qian, L. Andrew Wray, YuQi Xia

This exotic phase of matter is a subject of intense research because it is predicted to give rise to dissipationless (energy saving) spin currents, quantum entanglements and novel macroscopic behavior that obeys axionic electrodynamics rather than Maxwell's equations [6]. Unlike ordinary quantum phases of matter such as superconductors, magnets or superfluids, topological insulators are not described by a local order parameter associated with a spontaneously broken symmetry but rather by a quantum entanglement of its wave function, dubbed topological order. In a topological insulator this quantum entanglement survives over the macroscopic dimensions of the crystal and leads to surface states that have unusual spin textures.

Topologically ordered phases of matter are extremely rare and are experimentally challenging to identify. The only known example was the quantum Hall effect discovered in the 1980s by von Klitzing (Nobel Prize 1985). It was identified by measuring a quantized magneto-transport in a two-dimensional electron system under a large external magnetic field at very low temperatures, which is characterized by robust conducting states localized along the one-dimensional edges of the sample. Two-dimensional topological insulators, on the other hand, are predicted to exhibit similar edge states even in the absence of a magnetic field because spin-orbit coupling can simulate its effect (Fig.1A) due to the relativistic terms added in a band insulator's Hamiltonian.

Remarkably, three-dimensional topological insulators, an entirely new state of matter with no charge quantum Hall analogue, are also postulated to exist. And its topological order or exotic quantum entanglement is predicted to give rise to unusual conducting two-dimensional surface states (Fig.1B) that have novel spin-selective energy-momentum dispersion relations. Utilizing state-of-the-art angle-resolved photoemission spectroscopy, an international collaboration led by scientists from Princeton University have studied the electronic structure of several bismuth based spin-orbit materials [7,8,9]. By systematic tuning of the incident photon energy, it was possible to isolate surface quantum states from the bulk states, which confirmed that these materials realized a three-dimensional topological insulator phase.

Figure 1. (A) Schematic of the 1D edge states in a 2D topological insulator. The red and blue curves represent the edge current with opposite spin character. (B) Schematic of the 2D surface states in a 3D topological insulator. (C) Most elemental topological Insulators exhibit odd number of Dirac cones on their surface unlike the even numbers observed in graphene. Topological insulator Dirac cones are spin polarized where as Dirac cones in graphene are not.

The remarkable property of the surface states of a 3D topological insulator is that its Fermi surface supports a geometrical quantum entanglement phase, which occurs when the spin-polarized Fermi surface encloses the Kramers' points and on the surface Brillouin zone an odd number of times in total (Fig.2B). ARPES intensity map of the (111) surface states of bulk insulating Bi1-xSbx (Fig.2A) shows that a single Fermi surface encloses . However, determination of the degeneracy of the additional Fermi surface around requires a detailed study of its energy-momentum dispersion. ARPES spectra along the - direction (Fig.2C) reveal that the Fermi surface enclosing is actually composed of two bands, therefore two Fermi surfaces enclose , leading to a total of seven and Fermi surface enclosures.

Figure 2. (A) ARPES surface state (SS) Fermi surface of insulating Bi1-xSbx showing spin polarization directions as indicated by red and blue arrows. (B) Schematic of the SS Fermi surface of a 3D topological insulator. (C) ARPES energy-momentum dispersion of the surface states. The shaded areas denote the bulk bands while the dashed white lines are guides to the eye for surface state dispersions. (D) A single Dirac cone is observed in Bi2Te3.

These results constitute the first direct experimental evidence of a topological insulator in nature which is fully quantum entangled. The observed spin-texture in BiSb is consistent with a magnetic monopole image field beneath the surface. It shows that spin-orbit materials are a new family in which exotic topological order quantum phenomena, such as dissipationless spin currents and axion-like electrodynamics, may be found without the need for an external magnetic field. The results presented in this study also demonstrate a general measurement algorithm of identifying and characterizing topological insulator materials for future research which can be utilized to discover, observe and study other forms of topological order and quantum entanglements in nature. A detailed study of topological order and quantum entanglement can potentially pave the way for fault-tolerant (topological) quantum computing [10].

Figure 3: A new type of quantum matter called a topological insulator contains only half an electron pair (represented by just one Dirac cone in schematic crystal structure at top left), which is observed in the form of a single ring (red) in the center of the electron-map (top right) with electron spin in only one direction. This highly unusual observation shows that if an electron is tagged "red" and then undergoes a full 360-degree revolution about the ring, it does not recover its initial face as an ordinary everyday object would, but instead acquires a different color "blue" (represented by the changing color of the arrows around the ring). This new quantum effect can be the basis for the realization of a rare quantum phase that had been a long-sought key ingredient for developing quantum computers that can be highly fault-tolerant.

References:
[1] K. von Klitzing, G. Dorda, M. Pepper, "New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance", Phys. Rev. Lett. 45, 494-497 (1980). Abstract.
[2] D.C. Tsui, H. Stormer, A.C. Gossard, "Two-dimensional magnetotransport in the extreme quantum limit", Phys. Rev. Lett. 48, 1559-1562 (1982). Abstract.
[3] L. Fu, C. L. Kane and E. J. Mele, "Topological insulators in three dimensions", Physical Review Letters 98, 106803 (2007). Abstract.
[4] J. E. Moore and L. Balents, "Topological invariants of time-reversal-invariant band structures", Physical Review B 75, 121306(R) (2007). Abstract.
[5] S.-C. Zhang, "Topological states of quantum matter", Physics 1, 6 (2008). Abstract.
[6] M. Franz, "High energy physics in a new guise", Physics 1, 36 (2008). Abstract.
[7] D. Hsieh, D. Qian, L. Wray, Y. Xia, Y. S. Hor, R. J. Cava and M. Z. Hasan, "A topological Dirac insulator in a quantum spin Hall phase", Nature 452, 970 (2008). Abstract.
[8] Y. Xia, D. Qian, L. Wray, D. Hsieh, A. Pal, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava and M. Z. Hasan, "Observation of a large-gap topological insulator class with single surface Dirac cone”, Nature Physics 5, 398 (2009). Abstract.
[9] D. Hsieh, Y. Xia, L. Wray, D. Qian, A. Pal, J. H. Dil, J. Osterwalder, F. Meier, G. Bihlmayer, C. L. Kane, Y. S. Hor, R. J. Cava and M. Z. Hasan, "Observation of Unconventional Quantum Spin Textures in Topological Insulators", Science 323, 919 (2009). Abstract.
[10] A. Akhmerov, J. Nilsson, C. Beenakker, “Electrically detected interferometry of Majorana fermions in a topological insulator”, Phys. Rev. Lett. 102, 216404 (2009). Abstract.