topological insulators, topological …topological...topological insulators, topological...

Topological insulators, topologicalsuperconductors and Weyl fermionsemimetals: discoveries, perspectives andoutlooks

M Zahid Hasan1,2, Su-Yang Xu1 and Guang Bian1

1 Laboratory for Topological Quantum Matter and Spectroscopy: B7 Jadwin Hall, Department of Physics,Princeton University, Princeton, NJ 08544, USA2 Princeton Center for Complex Materials, Princeton Institute for the Science and Technology of Materials,Princeton University, Princeton, NJ 08544, USA

E-mail: [email protected]

Received 12 October 2014Accepted for publication 18 June 2015Published 3 September 2015

AbstractUnlike string theory, topological physics in lower dimensional condensed matter systems is anexperimental reality since the bulk-boundary correspondence can be probed experimentally inlower dimensions. In addition, recent experimental discoveries of non-quantum-Hall-liketopological insulators, topological superconductors, Weyl semimetals and other topologicalstates of matter also signal a clear departure from the quantum-Hall-effect-like transportparadigm that has dominated the field since the 1980s. It is these new forms of matter thatenabled realizations of topological-Dirac, Weyl cones, helical-Cooper-pairs, Fermi-arc-quasiparticles and other emergent phenomena in fine-tuned photoemission (ARPES)experiments since ARPES experiments directly allow the study of bulk-boundary (topological)correspondence. In this proceeding we provide a brief overview of the key experiments anddiscuss our perspectives regarding the new research frontiers enabled by these experiments.Taken collectively, we argue in favor of the emergence of topological-condensed-matter-physics in laboratory experiments for which a variety of theoretical concepts over the last 80years paved the way.

Keywords: topological insulators, topological superconductors, quantum Hall effects

(Some figures may appear in colour only in the online journal)

The Bi-based topological materials constitute the first exam-ple of topological state existing in bulk solids, which is alsothe first example of the non-quantum-Hall (NQH)-like topo-logical state (NQH-TS) or insulator (NQH-TI) in nature [120]. Another example of NQH-TS is a Weyl semimetal(WSM) which can only exist in three dimensions. Unlike thewell-known and well-studied 2D quantum Hall effect, NQH-TSs cannot be described by a single topological invariant likea Chern number for the quantum Hall effect; instead, theyinvolve a collection of more general topological invariants.For example, a 3D TI is described by four invariants or aWSM is characterized by the monopole-antimonopole

separations in momentum space (whereas quantum spin Hall(QSH) effect state or 2D TI is by a single invariant).Experimentally, NQH-TSs are rather extreme yet can berealized at room temperatures without magnetic fields and canbe turned into exotic magnets, Dirac semimetals, Kondoinsulators, and topological superconductors (TSC), leading toworld-wide intense research activities since 2007 [1114].Their NQH like nature is the key to their distinction fromconventional quantum-Hall-like measurement and probemethodology. Quantum-Hall-like states can be identified viatransport methods while the NQH-TSs often do not havequantized charge transport that would directly reveal the

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topological invariant. The NQH-TSs not only can exist inzero magnetic field but also differ from the quantum Hall orspin Hall effects in several important aspects: (1) they aredescribed by multiple topological invariants, (2) they usuallypossess metallic 2D surfaces arising from topologically pro-tected surface states (TSS) or Fermi arcs rather than 1D edgemodes, (3) they are protected even at room temperaturesrather than cryogenic temperatures unlike 2D QSH effectsthat have been experimentally realized so far, (4) they arehighly resistant to disorder because they occur in bulksemiconductors rather than at buried interfaces of hetero-structures, (5) they can be turned into superconductors orWSM via alloying, existing only in 3D. The 2D surfaceelectron states that arise on the boundaries of NQH-TSs shownovel and unprecedented properties that cannot be achieved inthe bulk of any isolated 2D or 3D system (see figure 1). TheseNQH-TI (NQH-like) phases were first identified by angle-and spin- resolved photoemission spectroscopies (spin-ARPES) [13, 2933]. This class of experimental techniqueshas been productive in identifying a number of closely-relatedtopological states and novel phenomena including topologicalcrystalline insulators (TCI), topological Kondo insulators,topological quantum phase transition (TQPT), topologicalDirac semimetals, magnetic TIs, TSC, and WSM. In thisproceeding, we first briefly review the discovery of theintrinsic NHQ-TIs and related materials and related TSC thatare not QSH-like and discuss the key results that demonstrate

their defining properties as predicted in various theories. Thenwe discuss the experimental discovery of Weyl Fermionsemimetals and topological Fermi arc states of matter andpresent our perspectives and outlooks regarding the newphysics and future directions of the field.

Figure 1. Non-quantum-Hall like phases of topological insulators exist in higher dimensions. In lower dimensions topological insulators arequantum Hall like and exhibit transport quantization in multiples of e2/h. NQH character of 3DTIs are detailed in [13]. Images are adaptedfrom [13, 8].

Figure 2. A 2007 online report of Dirac fermions and topologicalsurface states in topological materials Bi(Sn)Sb signalling non-quantum-Hall-like behavior (no quantized charge transport in 3DTIbased on the topological invariants that define their topology). Seereference [13, 8] for details.

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Phys. Scr. T164 (2015) 014001 M Z Hasan et al

1. The first topological insulator state in Bi1xSbx

The first NQH-TI was experimentally observed in the bis-muth-antimonide semiconductors Bi x1 (Sn)ySbx and Bi x1Sbx, see figures 2 and 3. The non-trivial topological naturedescribed by the invariant structure ( ; )0 1 2 3 of this systemwas demonstrated by directly measuring the bulk-boundarycorrespondence of the electronic structure, as reported in[29, 30]. The key observations include (1) the surface statebands cross the Fermi level an odd number times between the

two Kramers ( and M ) points in the surface Brillouin zone;(2) the surface states connect across the bulk energy gap fromthe bulk conduction band to the bulk valence band; (3) spin-resolved ARPES measurements showed that surface states aresingly degenerate. In particular, the surface state Fermi sur-face that encloses the point exhibits a helical Dirac spintexture, where the direction of the spin polarization of anelectron on this Fermi surface is locked to its momentumdirection. These systematic ARPES data shown in figure 3demonstrate that Bi-Sb semiconductors in their inverted

Figure 3. The first non-quantum-Hall-like topological insulator: the topological gapless surface states in bulk insulating Bi-Sbsemiconductors with a Berry phase. (a), The surface Fermi surface and surface state band dispersion second derivative image of Bi0.9 Sb0.1along M . The Fermi crossings of the surface state are denoted by yellow circles. (b), The resistivity curves of Bi and Bi0.9 Sb0.1. (c),Schematic dependence of bulk band energies of Bi x1 Sbx as a function of x (based on band calculations). (d), Fermi surface map of thesurface states. (e), Schematic of the BiSb surface state observed in panel (d). White lines show scan directions 1 and 2. (f)(i), Surfaceband dispersion along direction 1 and 2 (panels (f,h)) and the corresponding energy-distribution curves (panels (g,i)). (Adapted fromHsieh et al 2008 Nature 452 970. Some panels are modified from figures in reference [13, 29]).

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regime are 3D TIs. A few comments are in order: (1) theseexperiments on 3D TIs are not extensions of QSH effects (2Dtopological insulators described by a single invariant as ininteger quantum Hall or spin Hall effect states in HgTe, forexample). In 2007 two different types of experiments(transport and ARPES) independently reported the QSHeffect and topological insulator Dirac surface states, respec-tively. These two papers are a few months apart and bothwere completed and submitted in 2007 [2, 20, 21]. Moreimportantly, since transport in Z2 topological insulators arenot strictly quantized, the key is to measure the bulk-bound-ary correspondence (not charge transport) providing the firstproof of topology. This was done only via the ARPES mea-surements that for the first time demonstrated spin-momentumhelical locking of the gaplessness of the electron states on theboundary, while the bulk bands are gapped and inverted via atopological phase transition [2, 20, 29, 30]. Clearly, it is thebulk-boundary correspondence that was the first to prove thetopology [13, 8, 20, 29, 30].

We further note that: (1) it is clear that one of the majorchallenges in going from quantum-Hall-like 2D states toNHQ-like topological phases is to utilize different experi-mental methods to precisely probe a novel form of symmetryprotected topological state since the standard tools and set-tings that work for IQHE-state and QSH states but not forNHQ-TSs. The method to probe 2D topological-order isexclusively with charge transport, which measures quantizedconductance in IQHE systems and QSH systems. In a 3Dtopological insulator, the boundary itself supports a twodimensional electron gas (2DEG) and conductance is not (Z2)topologically quantized. Therefore, transport cannot directlyprobe the topological invariants ( ; )0 1 2 3 that differ fromthe Chern numbers of the IQH systems [1, 2, 8, 20, 29, 30].(2) The Bi-based materials have been extensively studied inthe past for their enhanced thermoelectric properties but thatextensive body of work did not actually report any evidenceof Dirac cones, spin-momentum locking or bulk-boundary(topological) correspondence revealing their topologicalinvariant structure ( ; )0 1 2 3 . These aspects are detailed in[3] and references therein. Their topological nature was firstreported in [13, 2931].

2. BiSb to (Bi/Sb)2(Se/Te)3 family of bulk insulatingtopological insulators

2.1. Research on topological insulator Bi1x Sbx led to thediscovery of Bi2Se3 class as topological insulators

Shortly after the discovery of Bi x1 Sbx, the search began fora 3D TI with a larger band gap and a single Dirac surfacestate, in part because the band gap of Bi x1 Sbx is small formaking devices. The Bi x1 Sbx class possesses an invariantstructure of ( 1; 1, 1, 1)0 1 2 3 = = = = . A second gen-eration of 3D TI materials, especially Bi2Se3, which wasdiscovered through this search, offered the potential fortopologically protected behavior at room temperatures.Starting in 2008, work by the Princeton group used spin-

ARPES measurements and first-principles theory and calcu-lations to study the surface band structure of Bi2Se3 andreported observation of the characteristic signature of a TI inthe form of a single spin-polarized Dirac cone (figure 4)[31, 33] by extending their earlier works on Bi-based mate-rials (see figures 2 and 3) suggesting a topological invariantstructure of ( 1; 0, 0, 0)0 1 2 3 = = = = [2, 20]. Aboutthe same time, concurrent first-principles calculations by theStanford group [32] used electronic structure methods toshow that Bi2Se3 is just one of several new large band gapTIs. Princeton experimental results and theoretical predictionsand Stanford theoretical predictions appeared on the sameissue of the journal [31, 32]. All these Bi-based materialswere identified using ARPES methods [3335]. The Bi2Se3surface state was observed to be a nearly ideal single Diraccone near the Dirac node or the Kramers point [31, 33].Another advantage is that Bi2Se3 is stoichiometric so can beprepared at higher level of purity. Finally and perhaps mostimportantly for many subsequent experiments, Bi2Se3 has alarge band gap of around 0.3 eV (3600 K). This indicates thatthe TI state in Bi2Se3 is robust at room temperature, asdemonstrated in [33] (see also figure 4).

2.2. Spin-momentum helical locking

The main ARPES and spin-ARPES results that demonstratedthe single Dirac cone TI states in Bi2Se3 and Bi2Te3 werereported in [31, 33], see figure 4. The key results are sum-marized as follows: (1) the Fermi surface of both Bi x2CaxSe3 and Bi2Te3 consists of a circular contour thatencloses the time-reversal invariant point (figure 4); (2)the surface states form a 2D Dirac cone, which spansacross the bulk band gap; (3) the surface states are spin-polarized and spin-momentum locked due to topology, andthe direction of the effective spin polarization is uniquelylocked to its linear momentum (figures 4(e) and (f)). Suchhelical locking patterns in momentum-space amount to a Berrys phase on the surface with an invariant structure of( 1; 0, 0, 0)0 1 2 3 = = = = . These data serve as a directobservation of the spin helical Dirac fermions on the surface.

2.3. Topological distinction from a Bloch band insulator

A topological insulator differs from a conventional bandinsulator (semiconductor) in that it features a non-trivialtopological invariant in its bulk electronic wavefunctionspace. A TI can be tuned from a conventional insulator bygoing through an adiabatic band inversion process in the bulk.Such a quantum phase transition from a conventional bandinsulator to a TI, involving a change of the bulk topologicalinvariant, is defined as a topological (quantum) phase tran-sition. This fully tunable topological phase transition con-taining the essential physics was first demonstrated in a seriesof ARPES and spin-resolved ARPES measurements on theBi-based material class BiTl(S1 Se)2 [42] as shown infigure 5. Once the materials are tuned below the transition, thegapless helical locking patterns in momentum-spaceamounting to a Berrys phase on the surface are destroyed

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and the surface becomes insulating. These results show thatspin-momentum locking is tied to the topological phasetransition; therefore, it is the bulk that drives it on the surface.This is a direct demonstration of bulk-boundary correspon-dence or topological nature of the surface states.

2.4. Protection via time-reversal symmetry

The Dirac point on the surface states is protected by time-reversal symmetry. Thus non-magnetic disorder cannot opena gap and elastic backscattering is forbidden on the surface.This aspect of topological insulator surface can be seendirectly from the spin-resolved surface band structure (see,

figure 6). This behavior was also confirmed via the quasi-particle interference measurements [86, 87]. However, acheck of the topological character would be to look for a gapor spectral weight suppression by breaking time-reversalsymmetry. This has been demonstrated by doping magneticimpurities Mn into topological insulator Bi2Se3 films.Figure 7 shows the out-of-plane spin polarization (Pz) mea-surements of the TSS in the vicinity of the Dirac point gap ofa Mn(2.5%)-Bi2Se3 sample. The observed hedgehog-like spintexture clearly breaks time-reversal symmetry. Such an exoticspin groundstate [44] plays an important role in realizing thequantized anomalous Hall effect and the axion electro-dynamics in magnetic topological insulators.

Figure 4. Observation of helical Dirac fermions with spin-momentum locking and Berry Phase. (ad), ARPES Fermi surface map of the(111) surface of Bi2 CaSe3 and Bi2Te3. Red arrows denote the direction of the spin polarization around the Fermi surface. The shadedregions in (c) and (d) are the first-principles calculated projections of the bulk bands. (e), Measured y component of spin-polarization alongthe M direction at E 20B = meV, in which only surface states are present. Inset shows a schematic of the cut direction carried out inthe pz-coupling polarization mode. (f), Measured x (red triangles) and z (black circles) components of the spin-polarization along the M direction at EB = 20 meV. (g), Spin-resolved spectra obtained from the y component of the spin polarization data. (h), Fitted values of thespin polarization vector (P) demonstrate the helical spin texture of the Dirac cone. (i), Angle-integrated intensity shows a linear energydependence, consistent with the expected density of states of a Dirac cone. (j), A schematic illustration, based on the data, of the surface stateDirac cone of a topological insulator. (k) and (l), ARPES dispersion maps of Bi2Se3 Dirac surface states at T = 300 and 10 K, respectively.The topological invariant structure of this system is shown to be as ( 1; 0, 0, 0)0 1 2 3 = = = = . (Adapted with modifications from Hsiehet al 2009 Nature 460 1101 [33] and 2008 Nature 452 970 [1, 3, 8, 29]).

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2.5. Protection via space-group symmetry

The experimentally discovered bismuth-based TIs such as Bi-Sb and Bi2Se3 are all Z2 topological insulators. Such a Z2topological insulator state is protected by time-reversal sym-metry, characterized by the 4 topological invariants( ; )0 1 2 3 , and has an odd number of Dirac surface states. Anatural question is that whether one can find new classes of

topological insulators that arise from other discrete symme-tries and are governed by different topological invariants.This motivation led to the prediction of the TCI state [112]. Ina TCI, the space group symmetries of a crystal replace the roleof time-reversal symmetry in a Z2 TI. It was discovered thatthe properties of TCIs differed fundamentally from those ofZ2 TIs because the point group symmetries played a crucialrole [112]. Such a distinction leads to many exotic properties,

Figure 5. Topological (quantum) phase transition and the change of Berry phase in BiTl(S1 Se)2.(a), High resolution ARPES dispersionmappings along a pair of time-reversal invariant points or Kramers points ( and M ). (b), Compositional evolution of band structuremeasured over a wide energy and momentum range. (c), The phase diagram showing the topological phase transition by tuning the chemicalcomposition of BiTl(S1 Se)2, which leads to a change of Berry phase from to 2 on the surface. (Adapted from S-Y Xu et al 2011Science 332 560 [1, 8, 42]).

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such as higher order (nonlinear) surface band crossings,topological states without spinorbit coupling, and crystallinesymmetry protected topological superconductivity or Cherncurrents [112, 115118]. Although theory has proposed TCIstates protected by a number of crystalline symmetriesincluding the mirror symmetry and the four-fold (C4) or six-fold (C6) rotational symmetries [112, 119, 120], the mirrorsymmetry case attracted the most interest since a real materialcandidate, Pb x1 SnxTe(Se), was quickly identified by first-principles band structure calculations [113]. Figure 8 showsARPES results including spin-resolution that demonstrate themirror-protected TCI state in Pb x1 SnxTe [46]. ConcurrentARPES works (without spin-resolution) by two other groupsreported the same TCI state in related materials, Pb x1 SnxSeand SnTe [45, 47]. Specifically, as shown in figure 8, the keyobservations include (1) the presence of two surface stateDirac cones near each X point (figures 8(f)(h); (2) a total offour (an even number of) Dirac cone surface states in thesurface BZ; (3) none of the surface state Fermi surface con-tours enclose any of the Kramers points, in agreement withthe fact that Pb x1 SnxTe(Se) is not a Z2 TI; (4) all surfacestate Dirac points are located along the X X mirrorline momentum space directions, revealing protection fromthe crystalline mirror symmetries; (5) for the two Dirac conesnear an X point, spin-resolved measurements show four dis-tinct spin polarizations with the configuration of , , , ,as one sweeps along the mirror line X direction inmomentum space at an given energy. The observation of aneven number of Dirac cones along the mirror lines

experimentally demonstrates the TCI state in Pb x1 SnxTe,and the measured surface state spin texture [46] reveals thetopological invariant, the mirror Chern number n 2M = , ofthe TCI state of study.

In general, as these experiments reveal, the power ofARPES stems from its ability to directly measure the topo-logical invariants ( ; )0 1 2 3 that define a topological state ofmatter especially due to the fact that many of these noveltopological states are not expected to exhibit charge transportquantization even in theory; thus, traditional transport meth-ods like that of quantum-Hall-like measurements do not apply(since they do not directly couple to ( ; )0 1 2 3 ). These factsbecome more evidently clear in the case of WSM which wewill show later in this proceeding. ARPES has been the onlydecisive experimental method for discovering their existencein materials.

2.6. Bulk insulating topological materials and quantum Halleffects on the surface of a topological insulator

Although it was shown that the chemical potential of TIssurface states can be tuned by the method of surface chemicaladsorption (see figure 9) [33], it is important to note that thismethod only tunes the chemical potential near the surface ofthe sample. This was one of the issues that prevented theprotected surface states in 3D TIs from being used to realizemany of the predicted novel phenomena in the context ofapplications [17, 110, 111, 114], especially the interfacetopological superconductivity via the proximity effect.

Figure 6. Absence of elastic backscattering in topological surface states with a Berry phase. (a), Spin-momentum-locked character of theDirac states as revealed by spin-ARPES. (b), The absence of back scattering is a direct consequence of the non-trivial spin texture. (c), Thechiral spin-momentum-locked spin texture of the upper and lower parts of the Dirac surface states. (d), The spin-selected quasiparticleinterference pattern resulting from the the chiral spin texture of the topological surface states. Charge and Hall transport of intrinsic bulkinsulating Bi-based TIs are shown in figure 9 (e), The topological order and chiral spin texture of the Dirac surface states. The topologicalinvariant structure of this system can be written as ( 1; 0, 0, 0)0 1 2 3 = = = = . (Adapted with modifications from S-Y Xu et al 2011Science 332 560 [1, 8, 42]).

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This shortcoming, which has nothing to do with theirtopological character or its experimental proof, was removedby the discovery of the highly bulk-insulating intrinsic 3D TImaterial, BiSbTeSe2 [79]. Remarkably, BiSbTeSe2 exhibitssurface-dominated conduction in electrical transport evenclose to room temperature. At low temperatures and highmagnetic fields perpendicular to the top and bottom surfaces,well-developed integer quantized Hall plateaux wereobserved, where the two parallel surfaces each contributed ahalf-integer e h2 quantized Hall conductance, accompaniedby vanishing longitudinal resistance (figure 9). This markedthe clear observation of quantum Hall effects realized on thesurface of a Bi-based TI. It is well-known that a high electronmobility is required in order to observe quantum Hall effectsin any 2DEG. In the case of TI, the 2DEG is the TSS. There isan additional requirement that the bulk of the TI needs to be

highly insulating so that only surface states contribute to thetransport. Therefore, the observation of quantum Hall con-ductance in the TSS provides a demonstration that some ofthe Bi-based TI samples such as BiSbTeSe2 has the ultra-lowbulk conductivity and high surface mobility. They areintrinsic bulk insulators for all practical purposes. These twoproperties are important for realizing many of the finer phe-nomena and applications that have been proposed in recenttopological theories [17, 110, 111, 114]. Perhaps, theimportant thing here is the fact that bulk insulating (intrinsic)topological insulators not only exist but also their surfacetransport can be isolated. Similarly, Bi-based materials canalso be used to design and fabricate 2D topological insulatorsfor device implementations (figure 10) [48, 67]. Transportmeasurements as such, however, do not reveal the topologicalinvariants ( ; , , )0 1 2 3 that define the state, since by

Figure 7. Berry phase tuning and hedgehog spin texture in a magnetized topological insulator. (a), Magnetization measurements usingmagnetic circular dichroism shows out-of-plane ferromagnetic character of the Mn-Bi2Se3 MBE film surface through the observed hystereticresponse. The inset shows the ARPES observed gap at the Dirac point in the Mn(2.5%)-Bi2Se3 film sample. (b), Spin-integrated ARPESdispersion map on a representative piece of Mn(2.5%)-Bi2Se3 film sample using 9 eV photons. The blue arrows represent the spin textureconfiguration in close vicinity of the gap revealed by the spin-resolved measurements. (c), Measured out-of-plane spin polarization as afunction of binding energy at different momentum values. The momentum value of each spin polarization curve is noted on the top. The polarangles () of the spin polarization vectors obtained from these measurements are also noted. The 90 polar angle observed at point suggeststhat the spin vector at is along the vertical direction. The spin behavior at and its surrounding momentum space reveals a hedgehog-likespin configuration for each Dirac band separated by the gap. Inset shows a schematic of the revealed hedgehog-like spin texture. (d), Thetime-reversal breaking spin texture features a singular hedgehog-like configuration when the chemical potential is tuned to lie within themagnetic gap. (Adapted from S-Y Xu et al 2012 Nat. Phys. 8 616 [44]).

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Figure 8.Mirror symmetry protected topological surface states in the topological crystalline insulator. (a), Pb x1 SnxTe(Se) crystalizes in theNaCl structure. (b), The bulk Brillouin zone (BZ) of Pb x1 SnxTe. The mirror planes are shown using green and light brown colors. Thesemirror planes project onto the (001) crystal surface as the X X mirror lines. (ce), First-principles band structure calculation on the(001) surface of SnTe. Dirac surface states are found along the X X mirror line momentum space direction. (f,g), ARPES dispersioncut and the corresponc measured along the X X mirror line direction. (h), Constant energy contour maps at different binding energies.Two surface state Dirac cones are observed on opposite sides of the X point along the X X mirror line direction. i, ARPES dispersionmap of Pb0.6 Sn Te0.4 along the mirror line direction. The white dotted lines show the binding energies chosen for spin-resolvedmeasurements SR-Cut 1 at E 0.06B = eV and SR-Cut 2 E 0.70B = eV respectively. Inset: measured spin polarization are shown by the greenand blue arrows on top of the ARPES constant energy contour at binding energy E 0.06B = eV for SR-Cut1. (j, k), Measured in-plane spin-resolved intensity (j) and in-plane spin polarization (k) of the surface states (SR-Cut 1) near the Fermi level at E 0.06B = eV. (l, m) Measuredin-plane spin-resolved intensity (l) and in-plane spin polarization (m) of the bulk valence bands (SR-Cut 2) at high binding energy E 0.70B =eV. (Adapted with modification from S-Y Xu et al 2012 Nat. Commun. 3 1192 [1, 8, 46]).

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5T = 0.35 KB = 0T

t

BSTS nb(Vbg)

Vbg

Vbg (V)

Vbg (V) Vbg (V)

Vbg (V)

b(Vbg)

nt

4

3

2

Rxx

(k

)

Rxx

(k

) Rxy (k)

1

0-100

1.5

1.0

0.5

0.0-100 -60

0.2

0.0k y (

-1)

EB(e

V)

-0.2

-0.2

0

-0.2

-0.4

0.0 0.2 -0.2 0.0 0.2 -0.2 0.0 0.2

Low

High

-80 -40 -20 0 20

-1

0

1

2

xy = 1/2 +

xy bottom (e

2/h)

-2

430

-3/2 -1/2 1/2 3/2 vb = 5/2

25

20

B (T

)15

10

5

0-100 -80 -60 -40 -20

00

-25

-50-75

Vbg

(V)

B = 31T

1v

253

1-1-3

3-1

0 20

0.01000

0.02605

0.06787

0.17680

0.46060

1.20000

-80 -60 -40 -20 0

xx

20

100

80

60

40

20

0-100 -80 -60 -40 -20 0 20

-75

-50

-25

0

251

23

B = 31 TRxxRxy

Surface state quantum Hall transport

(B = 31 T)xy-xy

(B = 31 T)(B = -31 T)

2

3

1/2 + 5/2

1/2 + 3/2

1/2 + 1/2

1/2 - 1/2

1/2 - 3/2

0

-1

v = 1

3

(Vbg - V

D )/V

xx (e

2 / h) x

x (e2

/ h)

Insulating Topological Insulators(EF in bulk gap)

a b

c d

Figure 9. Surface state quantum Hall effect (transport) in the intrinsic bulk insulating topological insulator BiSbTeSe2 and surface gatingprocess in a related compound. (a), Representative electric field effect curve, showing the resistance Rxx measured as a function of Vbg. (b),

Longitudinal resistance (Rxx) and Hall resistance (Rxy) versus backgate voltage (Vbg) at 0.35 K. (c), Extracted 2D longitudinal and Hall

conductivities in units of e h2 . (d), A 2D color plot showing Rxx as a function of magnetic field B and backgate voltage Vbg measured at

0.35 K. (Adapted from D Hsieh et al 2009 Nature 460 1101 [33]; Xu et al 2014 Nat. Phys. [79]). (e)(g), ARPES surface Fermi surface mapsof Bi2 CaSe3 at different NO2 dosages. (h)(j), High resolution ARPES surface band dispersions through after an NO2 dosage of 0 L, 0.01L, and 0.1 L, respectively.

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looking at the transport data alone it is not possible to discernwhether the quantum-Hall-like effect is any different frominteger QHE known from the 1980s. We emphasize this pointsince many authors have misunderstood this important point.

The experimental results taken collectively noted so farreveal that 3D TIs are a new form of matter and cannot bereduced to multiple copies of quantum Hall or spin Hall likestates. Most topological states of matter known before arerealized in two or lower dimensions (quantum Hall states,QSH effect, non-Fermi liquid chains, quantum spin-liquidswith topological order etc). Unlike all others, neither strongelectron correlations (necessary for quantum spin liquids),strong magnetic fields and low temperatures (necessary forquantum Hall states), nor low dimensionality (needed forquantum Hall states, spin quantum Hall states, and non-Fermiliquid spin chains) are needed for the 3D topological insu-lator. The theoretical and experimental discovery of the 3DTIs the first example of symmetry protected topologicalstate in bulk solids has thus generated much experimentaland theoretical efforts to understand and utilize all aspects of

these quantum phenomena and the materials that exhibit them[1], as demonstrated in figures 11 and 12.

3. Topological superconductors

While the general concept behind topological super-conductivity predates topological insulators [121124], recentconsiderations of topological band theory and related invar-iant structures can be used to topologically classify super-conductors that are direct analogs of topological insulators(for reviews, see [3, 7, 125, 126]). The topological insulatoranalogs of TSC with helical pairings (as opposed to chiralstates) can be experimentally designed from topologicalinsulators via the proximity effect. Such a helical TSC istopologically distinct from the the chiral or time-reversalbroken systems [3, 17, 36, 74, 75, 80, 114, 118, 127143].

Superconductivity is a collective phenomenon whereelectrons at the Fermi level cannot exist as single particles butare attracted to each other, forming Cooper pairing and

Figure 10. Bi-based 2D topological insulator (2D TI) (a), schematic of stacking 2D spin-orbit atomic layers (2D topological insulators) toform 3D bulk topological insulator with chiral topological surface states by tuning the interlayer interactions. (b), The ARPES spectra takenfrom Bi2Se3 thin films with thickness from 1QL to 7QL.(c), The EDC curves of the spectra shown in (b). The central EDC curvesdemonstrate the formation of topological bands as the film thickness increases. Adapted with modification from reference [67].

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condensation. This causes an energy gap, the superconductinggap, in the electronic single-particle spectrum. In a topolo-gical insulator, the bulk electronic structure has an insulatingband gap, whereas the surface shows protected Dirac electronstates due to the non-trivial topology. It has been proposedthat one can use a similar picture to qualitatively understand aTSC. A TSC has a superconducting gap in the bulk but showsprotected metallic states on its boundaries or surfaces. How-ever, unlike a topological insulator where the surface statesconsist of electrons, the surface states in a TSC are made upof Majorana fermions. A Majorana fermion is a fermionicparticle that is its own antiparticle. It was originally proposedin high energy physics as a way to understand neutrinos buthas not been conclusively observed as any fundamental par-ticle. On the other hand, it is believed that Majorana boundstates arising in condensed matter systems can play a pivotalrole in building fault-tolerant qubits. Moreover, it has beenpredicted that a certain quantum phase transition associatedwith a TSC can realize condensed-matter supersymmetry[114]. TSC thus provide a rare and exciting platform to rea-lize Majorana fermions and supersymmetry physics in similarcondensed matter settings.

With the advent of topological insulators, it was pro-posed in theory by Fu and Kane that a TSC can be realized by

inducing superconductivity in the spin-helical Dirac electronstates on the surface of a topological insulator [17]. The firststep towards the realization of this proposal requires a cleardemonstration of helical-Cooper pairing. Helical-Cooperpairing is defined as the superconducting Bose condensationof a spin-momentum locked Dirac electron gas. For example,bulk superconducting states were recently observed in Cu-intercalated Bi2Se3 (CuxBi2Se3) [132]. Although it is foundby ARPES that CuxBi2Se3 retains the Dirac surface state [36](see figure 13), no superconducting gap has been conclusivelyobserved in the Dirac surface states, perhaps due to the factthat in the current materials superconducting volume fractionis really low [132]. Hence, for a while, a direct experimentaldemonstration of helical-Cooper pairing remained elusive.

Recently, the helical Cooper pairing in a spin-momentumlocked Dirac electron gas was directly demonstrated by uti-lizing spin- and momentum-resolved photoemission spectro-scopy with sufficiently high resolution and at sufficiently lowtemperature [80]. This was achieved through the observationof the momentum-resolved Bogoliubov quasi-particle spec-trum of a topological insulator (Bi2Se3) in proximity to asuperconducting substrate [80]. This is the first time that atopological superconducting gap has been observed inexperiments, which therefore demonstrates a 2D TSC at the

Figure 11. Opening developments of the field. Bi-based materials have opened the field for experimental studies of novel topologicalphenomena covering topics such as hedgehog magnetism, anomalous quantum Hall effects, topological superconductivity, Dirac semimetals,topological heavy-fermion insulators, topological quantum phase transition and many more. See a comprehensive review in reference [1, 8].Intrinsic bulk insulating Bi-based TIs are shown in figure 9.

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interface between an s-wave superconductor and a TI. Thereis consensus in the field that previous works using othertechniques led to inconclusive results, highlighting the degreeof experimental challenges involved in this topic. It is themomentum and spin resolution of the measurements (lackingin transport or STM) that made it possible to isolate bulkversus surface superconductivity induced in the proximitychannel figure 14. A schematic layout of the heterostructuresample that consists of the s-wave superconductor and the TIfilm Bi2Se3 along with a TEM image of this interface isshown in figures 14(a) and (b). Figure 14(e) shows the low-temperature and high-resolution ARPES measurements,which reveal the existence of a superconducting gap in theDirac surface states with a clear surface-bulk contrastachieved through momentum-space imaging of the states. Themagnitude of the superconducting gap is nearly isotropicaround the Fermi surface (figure 14 (f)), which is consistentwith the time-reversal invariant helical nature of the surface

state superconductivity as expected theoretically[3, 7, 17, 127]. It is worth noting that although helical Cooperpairing can also occur in other 2D systems with spin-polar-ized bands such as a Rashba 2DEG, a key distinction is that ina Rashba 2DEG there are two channels (an even number) forhelical Cooper pairing per Brillouin zone (the helical Cooperpairing within the inner and the outer Fermi surfaces,respectively, see figure 14(h)). In contrast, in the TI/SC sys-tem, there is only one channel (an odd number) for helicalCooper pairing (the single channel provided by the Diracsurface states, figure 14(g)). The odd number of helicalCooper pairing channels, as demonstrated here, guaranteesthe existence of Majorana boundstates [17, 114, 127131].Figure 14(i) presents model calculation analysis, whichdirectly reveals the p pix y superconducting order parameterin the Dirac surface states, supporting the helical Cooperpairing nature, as shown in figure 15.

Figure 12. Opening experimental developments of the field based on the number of follow-up papers. Bi-based materials have opened thefield for experimental studies of many novel topological phenomena. In 2007 two different types of experiments independently reportedquantum spin Hall 2D effect and topological insulator Dirac surface states. These two papers are a few months apart and both completed andsubmitted in 2007 [2, 20, 21]. It is the bulk-boundary correspondence (via ARPES measurements) in TI that proved its topology for the firsttime [13, 8, 20, 29, 30] and cp the field for further research. Examples of intrinsic bulk insulating Bi-based TIs are shown in figure 9.

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One immediate future direction enabled by the demon-stration of a 2D TSC is the detection of Majorana fermions.This can be achieved by studying the magnetic vortices.Majorana bound states are expected there because each vortexeffectively creates a 0D non-topological defect in a 2D TSC.Equally interestingly, theory predicts the condensed-mattersupersymmetry at the critical point of the TQPT associatedwith the 2D TSC. This is conceptually shown in figures 16(c)and (d), depicting how magnetic doping or an external in-plane magnetic field can drive the system to the critical pointbetween a helical superconductivity state and a normal Diracelectron gas state. In addition, the chemical potential is tunedto the Dirac point, for example, via surface NO2 adsorption(figure 16(b)). Under this condition, theory predicts that TSS(a fermionic excitation) and the fluctuations of super-conducting order (a bosonic excitation) satisfy the super-symmetry relationship, and therefore, strikingly, possess thesame Fermi/Dirac velocity and same lifetime or self-energy [114].

4. Weyl fermion semimetals with topologicalFermi arcs

A third example of NQH-like topological state of matter is aWSM. A WSM is a gapless topologically protected statewhose low-energy bulk excitations are Weyl fermions, whilethe surface exhibits non-closed Fermi arcs [11, 145, 148, 149].A WSM exists without any connection to any quantum Hall-like effect in any sense and, like intrinsic (4-invariant) 3Dtopological insulators, it is strictly 3D with no true 2D analog

of a topological Weyl fermion system. While the distinctionmay be a bit subtle in the case of 3D TIs since there are weak-3D TIs (which are QSH like) and strong-3D TIs (NQH-likethat we focused on here) but for the WSM it is far moreclear. In 1929, H. Weyl noted that the Dirac equation takes asimple form if the mass term is set to zero [144]:

cpi L = , with being the conventional Paulimatrices. Such a 2 2 fermion, the Weyl fermion, is asso-ciated with a definite chirality. Around the same time, theproblem of accidental degeneracies in quantum systems wassolved by von Neumann and Wigner. In 1937, C Herringfused these concepts by considering degeneracies in electro-nic bands that arise in a lattice. Specifically, one can realize aband crossing by tuning three momenta in a 3D lattice, andthe dispersion near these touching points can be written as:H v p p p( )F x x y y z z = + + where p now is the momentumdeviation from the degeneracy point. These degeneracy pointsare called the Weyl nodes. Despite these ancient papers,very little progress in finding Weyl fermion nodes in crystallattices had been made. The discoveries of topologicalinsulators provided fresh perspectives into this problem[3, 7, 11, 14, 42, 81, 144162]. It was realized that a crystalwhose low energy excitations satisfy the Weyl equation, aWSM, is a topological phase of matter even though it doesnot have an energy gap as in a topological insulator. Thus, theWSM state broadens the classification of topological phasesof matter beyond insulators and provides a clear example ofNQH-like state of matter. The WSM cannot be reduced tomultiple copies of integer Hall-like states. The Weyl fermionsat zero energy correspond to points of bulk band degeneracy,the Weyl nodes, which are strictly 3D. The chiral charge

Figure 13. Bulk superconductivity and Dirac surface states in doped Bi2Se3. (a), Bulk and surface electrons are non-degenerate at the Fermilevel. VB, valence band; CB, conduction band. (b), A phase diagram compares the measured superconducting topology with preliminaryexpectations based on cases in which (A) each Cu atom donates one doped electron, (B) the experimental chemical potential is applied togeneralized-gradient-approximation band structure and (C) the experimental chemical potential is applied to the band structure of undopedBi2Se3. (cf), Electronic states expected below Tc for even-parity superconductivity (c, e) and an example of odd-parity topologicalsuperconductivity (d, f). (e) and (f) show states within 5 meV of the Fermi level. Both cases allow the superconducting state to host non-Abelian particles such as Majorana fermions. (Adapted from Wray et al 2010 Nat. Phys. [1, 8, 36]).

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associated with each node, which can be understood as asource or a sink of Berry flux, serves as the topologicalinvariant of this new topological phase of matter. The non-trivial topological nature guarantees gapless surface states onthe boundary of a bulk sample. These surface states take theform of Fermi arcs connecting the projection of bulk Weyl

points in the surface Brillouin zone [11, 148]. In contrast totopological insulators where only the surface states areinteresting [3, 7], a WSM features unusual band structure bothin the bulk and on the surface [11]. This opens up unpar-alleled research opportunities, where both bulk and surfacesensitive experimental probes can measure the topological

Figure 14. Spectroscopically-resolved proximity-induced topological superconductivity. (a), A schematic layout of ultra-thin Bi2Se3 filmsepitaxially grown on the (0001) surface of single crystalline s-wave superconductor (SC) 2 H-NbSe2 (T 7.2c = K) using the MBE technique.(b), High resolution transmission electron microscopy (TEM) measurements of the Bi2Se3/NbSe2 interface at 200 keV electron energy. Anatomically abrupt transition from NbSe2 layered structure to the layered quintuple layer structure of Bi2Se3 is resolved, showing a goodatomically flat interface crystal quality. (c), Momentum-integrated ARPES intensity curves over a wide binding energy window (core-levelspectra) taken on a representative 3QL Bi2Se3 ( 3 nm) film grown on NbSe2 before and after removing the amorphous selenium cappinglayer (decapping). (d), Fermi surface map taken at an incident photon energy of 50 eV. The white dotted lines indicate the momentum-spacecut-directions chosen to study the surface state superconducting gap as a function of Fermi surface angle around the surface state Fermisurface. (e), Symmetrized and normalized ARPES spectra along 1 through 5 respectively (red) and their surface gap fittings. (f), Fermisurface angle dependence of the estimated superconducting gap around the surface state Fermi surface. (g), Illustrations for helical-Cooperpairing in a spin-momentum locked helical electron gas and (h), The conventional s-wave Cooper pairing in an ordinary superconductor. (i),Model calculation results of a topological insulator film in proximity to an s-wave superconductor show the calculated total superconductingpairing amplitude and its decomposed singlet ( S

) and triplet ( T

) components on the top surface of a 4-unit-cell thick TI interfaced with an

s-wave superconductor. (Adapted with modification from Xu et al 2014 Nat. Phys. [1, 8, 80]).

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Figure 15. Surface topological superconducting gap fitted with p pix y form of the order parameter. (a), APRES spectrum taken from a 4 QLBi2Se3/NbSe2 scface states. (b), Fitting to the symmetrized ARPES EDC curves of bulk and surface states showing the superconductingcoherence peak. The surface coherence peaks can be fitted with a p pix y form of the order-parameter. (c), Schematic of the helical Cooperpairing. (Partially adapted with modifications from Xu et al 2014 Nat. Phys. [80]).

Figure 16. Interface transport and conditions for emergent supersymmetry. (a), Point-contact transport (dI /dV versus bias voltage) as afunction of temperature. (b), Measured surface state dispersion upon in situ NO2 surface adsorption on the surface of a 7QL Bi2Se3/NbSe2sample using incident photon energy of 55 eV at temperature of 20 K. The NO2 dosage in the unit of Langmuir (1L 1 10 6= torr s) isnoted on the top-right corners of the panels, respectively. The white dotted lines in the last panel are guides to the eye. (c, d), Theoreticallyproposed [114] unusual criticality related to supersymmetry phenomena can be realized in the topological insulator/s-wave superconductorinterface as the chemical potential is tuned to the surface state Dirac point and an in-plane magnetization drives the system to the critical pointbetween superconducting and normal Dirac metal phases. Fermions and bosons are expected to feature the same band velocity and quasi-paticle lifetime [114], where fermion velocity is estimated to be around 5 10 m s5 1 from the ARPES result. (e), Schematic of Majoranabound state at the vortex core in a topological superconductor. (Adapted from Xu et al 2014 Nat. Phys. [1, 8, 80]).

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Figure 17. Weyl semimetal state in TaAs. (a), Boby-centred tetragonal structure of TaAs, shown as stacked TaAs layers. An electricpolarization is induced due to dimples in the TaAs lattice. The right panels show top-down views at different vertical positions, emphasizingthe crystal structure is composed of square lattices that are shifted with respect to one another. (b), In the absence of spinorbit coupling, thereare two line nodes on the k 0x = mirror plane, Mx, and two line nodes on the k 0y = mirror plane, My. In the presence of spinorbit coupling,each line node vaporizes into six Weyl points. The Weyl points are denoted by small circles. Black and white show the opposite chiralcharges of the Weyl points. (c), The calculated (001) surface states on the top surface of TaAs.

Figure 18. Experimental discovery (ARPES experiments) of the Weyl Fermion semimetal state. (a), Vacuum ultra-violate ARPES Fermisurface map of TaAs using incident photon energy of 90 eV, which reveals the surface state Fermi surface consisting of Fermi arcs. (b), High-resolution ARPES Fermi surface map of the crescent-shaped double Fermi arcs. (c), Soft x-ray ARPES Fermi surface of TaAs using incidentphoton energy of 650 eV, which reveals the bulk Weyl nodes. (d), Energy dispersion map of the linearly dispersive bulk Weyl cone. (e),Schematic illustration of the co-propagating property of the crescent-shaped double Fermi arcs. (f), Schematic illustration of the Weylsemimetal state that possess two single Weyl nodes with the opposite (1) chiral charges. At the surface, a Fermi arc exists which consists ofthe projections of the two Weyl nodes.

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nature and detect novel quantum phenomena. Specifically, theWeyl fermions in the bulk provide a condensed-matter rea-lization of the chiral anomaly, giving rise to a negativemagnetoresistance under parallel electric and magnetic fields,novel optical conductivity, non-local transport and local non-conservation of ordinary current predicted in theories[147, 152155]. At the same time, the Fermi arc surface statesare predicted to show novel quantum oscillations in magneto-transport, as well as unusual quantum interference effects intunneling spectroscopy. The realization of Weyl fermions in acondensed matter system, the novel topological classification,and the unusual transport phenomena associated with both thebulk Weyl points and Fermi arc surface states have attractedworldwide research interest [3, 7, 11, 14, 42, 81, 83, 85, 144162]. Weyl fermions have long been known in quantum fieldtheory, but have not been observed as a fundamental particlein nature [144146]. However, realizing the WSM state inreal materials has also been proven to be difficult. Earlyproposals relied on time reversal breaking, predicting theirexistence in magnetic phases such as in pyrochlore or mag-netic multilayers [148150]. However, the case of Weyl fer-mions in these materials remained experimentallyunconfirmed; therefore, for many years, a WSM has not be

found in experiments and has become a much-sought-outtreasure of condensed matter physics.

First-principles band structure calculation studies [81]seem to suggest that TaAs could be an inversion symmetrybreaking WSM (figure 17). This is promising since here it isnot necessary to control the size of the magnetic domains[83, 148150] or to fine-tune the chemical composition as inan alloy [150, 151]. In the absence of spinorbit interactions,the conduction and valence bands intersect on 4 closed loopsthat remain on the kx = 0 and ky = 0 mirror planes. Addingspin-orbit coupling leads to a gapping of these intersections,but the bands then intersect at points with opposite chiralities,slightly displaced from these planes. On the (001) surface ofthe TaAs, the 24 Weyl nodes project onto 16 points, leadingto 8 projected Weyl nodes near the surface BZ edges X and Ypoints with a projected chiral charge of 1 and 8 projectedWeyl nodes near the midpoints of X or Y lines witha projected chiral charge of 2. This results in the existence ofFermi arc surface states that connect the projected Weyl nodeson the (001) surface as seen in the surface calculation shownin figure 17(c). Preliminary angle-resolved photoemissionexperiments on TaAs [82] provided strong evidence thatdemonstrate TaAs as the first WSM. Both the Fermi arc

Figure 19. Topological condensed matter physics. Topological variants of metal, insulator, magnet and superconductor have beendemonstrated via ARPES experiments [13, 163] (based on presentation at Nobel symposium on new forms of matter 2014 [163]).

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surface states [83] and the bulk Weyl cones and Weyl nodeswere observed in ARPES measurements. Figure 18(a) showsthe surface state Fermi surface measured using vacuumultraviolet (low-energy) ARPES. The ARPES data shows apair of Fermi arcs near the midpoint of each X or Y line. Each pair of Fermi arcs has a crescent-like shape. Themomentum space locations where the two arcs terminate intoeach other correspond to the projections of the Weyl nodeswith projected chiral charges of 2. In figure 18(c), the bulkband Fermi surface, which consists of discrete Fermi points,the Weyl nodes, is revealed by soft X-ray ARPES measure-ments. The linear dispersion of the bulk Weyl cones is clearlyseen in the energy dispersion measurements shown infigure 18(d). The simultaneous observation of both surfacestate Fermi arcs and bulk Weyl cones sugest the existence of aWSM state in this compound.

Observation of a WSM state can pave the way for thedetection of chiral anomaly in solids. The chiral anomaly,commonly known as the AdlerBellJackiw anomaly, is animportant concept in quantum field theory, playing a key rolein the standard model of particle physics. The number ofWeyl fermions with a given chirality, which is expected to beconserved classically, is not conserved in quantum field the-ory once quantum fluctuations are considered. In a WSMcrystal, the Weyl cones with opposite chiralities are separatedin momentum space. Therefore, one can imagine that elec-trons can be pumped from one Weyl cone to the other withthe opposite chirality that is separated in momentum space, inthe presence of parallel magnetic and electric fields, whichviolates the conservation of the particle number with a givenchirality, giving rise to a novel analog of the chiral anomaly ina condensed matter system [11, 147, 152155]. Apart fromthe elegant analogy between condensed matter and highenergy physics, the chiral anomaly can potentially be used indevices. Recent theories have predicted its potential applica-tions in designing valleytronic devices [155]. Another futureprospect is the possibility for studying heterostructures con-sisting of an s-wave superconductor and a WSM, where thesuperconducting proximity effect can be also topologicallynon-trivial [162] which perhaps could be even more exoticthan that in the case of the 3DTI/SC interfaces we discussedearlier in this proceeding.

Unlike string theories in high energy physics, theexperimental discoveries of novel topological phases ofmatter, especially, the NQH-like topological insulators, TSCwith helical pairings, WSM with topological Fermi arcs, andrelated topological states of matter, reveal a clear paradigmshift in experimental approaches and methodologies to proveand discover new topological states of matter. It is these newforms of matter demonstrated via ARPES methodology ofproving bulk-boundary (topological) correspondence thatenabled the experimental realizations of topological-Dirac,Weyl cones, helical-Cooper-pairs, Fermi-arc-quasiparticlesand many more emergent phenomena. These results, takencollectively as summarized in figure 19, suggest the emer-gence of topological-condensed-matter-physics in labora-tory experiments for which a variety of theoretical conceptsover the last 80 years paved the way.

Acknowledgments

The authors acknowledge N Alidoust, A Bansil, I Belopolski,G Bian, G Cao, R J Cava, T-R Chang, Y P Chen, T-C Chiang,F-C Chou, Y Chuang, J H Dil, A V Fedorov, Z Fisk, L Fu, DGrauer, D Hsieh, Y Hor, S Jia, C L Kane, H Lin, C Liu, MNeupane, J Osterwalder, D Qian, A Richardella, N Samarth,D Sanchez, R Sankar, P P Shibayev, L A Wray, Y-Q Xia andH Zheng for collaborations and U.S. DOE DE-FG-02-05ER46200, No. AC03-76SF00098, and No. DE-FG02-07ER46352 for support. M Z H acknowledges visiting-sci-entist support from Lawrence Berkeley National Laboratoryand additional support from the A P Sloan Foundation (NewYork) and Gordon and Betty Moore foundation (California)through the grant number GBMF4547.

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