Introduction to Quantum Materials Leon Balents, KITP
QS3 School, June 11, 2018 Quantum Materials • What are they? Materials where electrons are doing interesting quantum things • The plan: • Lecture 1: Concepts in Quantum Materials • Lecture 2: Survey of actual materials Themes of modern QMs
• Order • Topology • Entanglement • Correlations • Dynamics Order: symmetry
• Symmetry: a way to organize matter • A symmetry is some operation that leaves a system (i.e. a material) invariant (unchanged
• In physics, we usually mean it leaves the Hamiltonian invariant
U
• Why symmetry? • It is persistent: it only changes through a phase transition • It has numerous implications: • Quantum numbers and degeneracies • Conservation laws • Brings powerful mathematics of group theory • The set of all symmetries of a system form its symmetry group. Materials with different symmetry groups are in different phases Ising model
• A canonical example z z x z z
H = J hx
• Basic symmetries of our world: • space-time (Lorentz/Poincare) symmetry • spatial isotropy and translations • time reversal • Charge/particle number conservation • Approximate symmetries (sometimes) • spin-rotation • various internal quantum numbers • These things are broken down to varying degrees in different QMs RuCl3 LiNaSO4 Tourmaline Proustite Li2ZrF6 Coquimbite Portlandite Fluocerite-(La)
(#172) P63 (#173) P6 (#174) P6/m (#175) P63/m (#176) P622 (#177) P6122 (#178) P6522 (#179) Cystallography tetracosakis(µ2-Methoxo)- catena-[2,2'-(biphenyl-4,4'- dodecakis(µ2-proline)-dodeca- iron(iii) dodecaperchlorate AgF )(H2O)4 •NephCrystaleline structure:LiNaCO3 diyldiimino)di 230benzene crystallographic-1,3,5-triol] Fluorapatite space groups3 LaBTB m2 (#187) P6c2 (#188) P62m (#189) P62c (#190) P6/mmm (#191)P6/mcc (#192) P63/mcm (#193) P63/mmc (#194) • Classifies the arrangements of atoms (which break the symmetries of free space)
BaTi(Si O ) Na O KCaF(CO3) • Wallpaper3 9 groups2 2 inSrBe 2d3O4 (c.f. 2dAlB2 materials)Beryl ZrI3 Graphite
3 (#203) Im3 (#204) Pa3 (#205) Ia3 (#206) P432 (#207) P4232 (#208) F432 (#209) F4132 (#210) I432 (#211) • Basic input to many things • Phonons, elasticity, band structure...
Dodecasil Na1-x•WO3 LOADSPyrite of extremelyYttria BIF useful-9-Cu stuffBe3 Pon2 BilbaoPCN-20 Te(OH)6 NiHg c (#219) I43d (#220)crystallographicPm3m (#221) Pn3n (#22 server...2) Pm3n (#223) Pn3m (#224) Fm3m (#225) Fm3c (#226) Fd3m (#227) • Structural phase transition = change of space group.
Katoite Mn3B7O13I hydrogarnet ZIF-71-RHO Co-Squarate V6SnSi (NH4)[(Mo12O36)(AsO4)Mo(MoO)] NaCl LTA Spinel J B Goodenough tilting of the CuO, octahedra-which aretetragonal 3. Electron correlatlon energies (c/a =- 1)"along a [ 1lo] axis to transform the structure fromthe tetragonal symmetry of figure l(a)to the Substitution of a larger Ba2+ or Sr2+ ion for La3+ in orthorhombic symmetry of figure l(b) below IT; = 530 K La,CuO, not only expands the mean A-0 bond length, [4]. Bending of the Cu-0-Cu bond angle from 180" butalso oxidises the CuO,planes; removal of anti- Chemical and structural relationships in high-T, materials permits a matching of the sizes of the basal-plane lattice bondingelectrons from theplane shortens the Cu-0 parameters of the two intergrowth layers without com- bondlength. Consequently the tolerance factor t pression of the Cu-0 bond length. increasesLa,CuO, with and y Nd,CuO,;in the system the former La,-,Sr,CuO,, is a parent com-so I; plane Cu-0 bonds whereas a reduction would add anti- decreasespound for with p-type increasing superconductivity, y in thephase the latterdiagram for n-typeof bonding c~,*~-,,~electrons and thus expand the in-plane figuresuperconductivity. 3. Moreover, as both t and the oxidation state of Cu-0 bonds against the internal compressive force. As 2.4. The T'-tetragonal structure the CuO,Stoichiometric planes increases, La,CuO, the drivinghasthe tetragonal force(T) for insert- a result, La,CuO, is readily oxidised, but attempts to structure of figure l(a) at temperatures T > IT;; aco- reduce it have failed. A t < 1 at high temperatures allows theinsertion of ingoperative excess oxygen tilting decreases,of the CuO, and octahedra stoichiometric at lower oxygen tem- excess interstitial oxygen into the La202+dlayers. concentrationsperatures (T are< IT;) readilyyields the achieved orthorhombic in air (0) in structurethe com- However, a Coulomb repulsion between the interstitial positionalof figure range l(b) [3,0.07 41. < The y < phase 0.15 changewhereas, at forT =x IT;< 0.07,is a 2.2. Oxidation and c-axis oxide ions forces a strong relaxation of the it istypical necessary second-order, to annealsoft-mode in transition.N, to avoidIn insertion both of Even with electronordering, a t < 1 occurs at tem- c-axis oxygen neighbouring an interstitial, so onlya excessstructures, oxygen. (Cu0,)'- Annealing planes in (or 0, sheets allows Textension < IT;) alternate of the peratures high enough (T > 400 "C) for the insertion of small concentrationcan be accommodated before a rangewith of twooxygen (Lao)' stoichiometry layershaving tothe yrocksalt = 0.27;structure. for y 0.27, > interstitial oxygen between adjacent La0 planes;the Thisconfiguration has two immediate consequences phase transition occurs within the La20,+d layers. The a high oxygen pressureappears to be necessary to interstitial oxygen atoms occupy the sites coordinated that are common to all the p-type copper oxide super- layer transforms from a rocksalt to a fluorite configu- prevent oxygen loss [14,151. Oxygen-deficient by four La andfour c-axis oxygen A similar insertion conductors: (i) the layers are alternately charged posi- [S]. ration by adisplacement of the c-axis oxygen to the La, -,S~,CUO,-~has oxygen vacancies at the c-axis of interstitial oxygen occurs in La,NiO, The inter- tively and negatively, which creates an interlayer [7]. interstitial sites; in the absence of any c-axis oxygen, the sites. nal electric field parallel to the c axis stabilises an inter- internal electric field that shifts the energies within one oxygen is coordinated by only four Ln3+ ions (figure 2). A strikingfeature of figure 3 is the appearance of stitial oxide ion 0,-,so holes are introduced into the layer relative to those in the other; and (ii) stabilisation Such a displacement is possible because of the stability CuO, planes. The interstitial oxide ionsexpand the high-T,of an superconductivityintergrowth structure in requires only a anarrow matchingcomposi- of the La,02+a layers and shorten the Cu-0 bond lengths in of Cu2+ ions in square-coplanar coordination. In this tionalbond range lengths between across thean antiferromagneticinterface of the two and layers. a metal- The the CuO, planes, thus increasing the effective value of T'-tetragonal structure, the excess oxygen occupies the lic significancephase within of whattheinternal appears electricstructurally fieldis to beillustrated a single the tolerance factor t and suppressing the driving force c-axis sites; the normal and interstitial sites of the T- rangebelow of bySr asolidtrapping solution. of mobileMoreover, holesthe from the Nee1 CuO, tem- for further oxidation. Lowering the temperature to re- and T-tetragonal structures are interchanged. The fluo- peratureplanes inTN superconductors of antiferromagnetic with La,CuO,+, excess or disordered is establish the driving force reduces the 02--ion mobil- rite La,O, layer has a larger a axis, which increases the extremelyoxygen sensitivein the YBa,Cu,O,+, to the oxidation structure state and of theby CuO, the ity, so L~,CUO,+~prepared in 1 atm air has a 6 S 0.02 varied influence of Pr on thesuperconductivity. Here effective tolerance factor t. In fact, the expansion tends planes. A TN = 326 K has been observed in La,CuO, in the bulk. Near the surface of theparticles or we emphasise the structurakhemical consequences of to place the CuO, planesunder tension, so the [16], but the introduction of excess oxygen commonly grains, however, a 6 > 0.02 may introduce filamentary bond length mismatch. Cu-0-Cubond angles remain 180" to lowest tem- lowers TN to about 240 K. In La,-,Sr,CuO,, the long- superconductivity [8, 91. Under 23 kbarand a high A measure of the bond length matching across the peratures. Bending of the Cu-0-Cu bond in this phase range antiferromagnetic order disappears near y = 0.02, oxygen pressure, the entire antiferromagnetic semicon- interlayer interface for the tetragonal structure of figure doesnot appear to be anoption; thebond angles a low-temperature spin-glass transition occurring in a ductor La,CuO, is transformed into the superconduc- l(a) is the tolerance factor remain 180" to Ln = Gd, and other phases are formed narrow transitional range 0.02 < y < 0.05 between the tor La,CuO,.,, [lo] ; under 300 bar oxygen pressure, on substitution of smaller rare earth ions. Significantly, antiferromagneticand superconductivet = (A-o)/$(B-o) compositional (3) La,CuO,,,, contains in the bulk both the semicon- the T'-tetragonal phases can be doped n-type, but not ranges.where These thebond unusual lengthsfeatures A-0 and signal EO achange have tradi-from ductor and the superconductor phases [l 1, 121. p-type. Theaddition of antibondingelectrons to the strongtionally electron beencorrelation taken to be thesum energies of the in empiricallytheanti- 180" Cu-0-Cubonds relieves the tensile stress; ferromagneticdetermined,room-temperaturecompositions to weakionic radii,electron i.e.corre- (RLa removal of antibondingelectrons would increase this lation+ R,)energies and (Rcu in the + R,)metallic in La,CuO,. compositions; But this it tradition, therefore 2.3. Symmetry lowering stress. The maximum n-type doping with Ce substitut- becomesdevised necessaryby Goldschmitt to ask to how predict the magnitudewhich of the ABX,of the As t decreases with temperature below those tem- ions in Ln, -,Ce,CuO, decreases with the size of the copperoxides magnetic and fluorides moments would pc" haveare changing thecubic-perovskite with y. peratures where oxygen insertion is possible, the structure, deflects attention from the significance of the Ln3+ion; it is y = 0.25, 0.20 and 0.15 for Ln = BeforeStructural turningto theexperimental answerphase to this La,CuO, transitionstructure allows acooperative, nearly rigid La,,,,Nd,~,, ,Nd and Gdrespectively [13]. question,quite different it is useful thermal to expansionreview briefly coefficients how the of the electron A-X and B-X bondlengths. A t 2: 1 at the temperature of formation would result in a t < 1 at room temperature because the softer A-X bondshave larger thermal expansion coefficients thanthe harder B-X bonds. In the layer structure of La,CuO,,a t < 1 places the CuO, planes of the tetragonal phase under compression and the (Lao), layers under tension. This bond length mismatch is relieved in three successive steps.Talk about order parameters
2.1. Electron ordering
The Cu2+ : 3d9 configuration contains one hole in the 3d shell. Ordering of the hole into the 3d,2-,2 orbital creates four shorter Cu-0 bonds in the CuO, plane and two longer Cu-0 bonds to the apical oxygen on the c Sr content x axis. Theanomalously large c/a ratio of T-tetragonal FlgureLa,CuO, 3. Phase was diagram recognised for theearly system [S] to L~,-,S~,CUO,-~ be a manifesta- Figure 2. Structure of tetragonal (T') Nd,CuO, p = y - 26. Adapted from [15]. (a) Ibl tion of this electron ordering. Clearly an oxidation that Figure 1. Structures of (a)tetragonal (T) and (b) removes moreantibonding electrons from the CuO, orthorhombic (0)La,CuO,. The arrows indicate the planes-whether g* or n*-shortens furtherthe in- direction of the tilting of the CuO, octahedra. 28 tetragonal orthorhombic
27 “Order parameters” ~ soft phonon modes Magnetism
• Fundamental symmetry is time-reversal
S
Ce Sb Incr~shg field HII 001 ttt
FIG, 3. Magnetic phase diagram of CeSb for an in- creasing field applied along a [001] direction. The dashed lines correspond s'(». q) to a change of the domain
distribution. An arrow & or & represents the mag- netization up or down along the z axis of a (001) plane. S2 (5 ~ demeln Kz) Magnetism s, ( =",) S2( .q)(
I 0 10 15 20 T(K)
which do not correspond to that of an usual meta- dicates a first-order transition. The phase S' dis- magnet. appears at II=39 koe and in higher field the sys- Broken T-reversal systems are very rich:tem becomes completely saturated. Figure 5 gives • the field dependence of the intensity maximum of A. Low-temperature zone the [20k] superlattice reflections together with the At 4.2 K, when the magnetic field increases, the [111]integrated intensity. The critical fields show 1651 magnetic spacephase S,groups(k=0.572=47) remains stable up to a field only a weak temperature dependence, however they of 21 kQe. At higher field a new phase S' appears are affected by hysteresis effects which are quite characterized by a propagation vector k = —,'. At important for the S-S' phase change (about 9 kOe). 20.9 the coexistence of the S and S' in- in with Weyl fermions in magnetic Kondo system CeSb koe phases These results are good agreement magne- C Guo et al. Decreasing field H// 001J 2 ffff
F(» c)
FIG. 4. Magnetic phase diagram of CeSb for a de- creasing field applied '(» along a [001] direction. An Q) arrow ) or ) representsthe ff magnetization up or down I along the g-axis of a (001) plane.
S2 (single domain kz) (»2 ff ff ) , ll !l. ), . i. , l CeSb "T(K) Rossat-Mignod et al, 1977
Fig. 1 a Possible routes for achieving Weyl fermions. b–e Crystal structure and characterisation of CeSb. a Schematic diagram for three means of achieving Weyl fermions. Route I is via the breaking of inversion symmetry, route II is via breaking time reversal symmetry by applying a magnetic field, and route III is via breaking time reversal symmetry in the magnetic state. b Crystal structure of CeSb. c Dc magnetic susceptibility of CeSb in an applied field of 0.1 T measured upon warming after zero-field cooling (ZFC) and field cooling (FC) showing a magnetic transition at around 17 K, and another one around 8 K to the antiferro–ferromagnetic (AFF) phase where the ZFC and FC curves split. d Temperature dependence of the electrical resistivity of CeSb and LaSb. e Magnetic contribution to the resistivity of CeSb obtained by subtracting the data for LaSb, plotted on a logarithmic temperature scale. The straight line indicates the logarithmic increase of the resistivity with decreasing temperature below about 80 K, suggesting the presence of Kondo scattering dropping abruptly, due to the onset of magnetic order. The several low temperatures. Below 10 K, there is an anomaly around transition corresponds to entering the antiferromagnetic phase 4.3 T, which indicates the transition from an AFF phase to a field- with paramagnetic layers, where there is no net magnetisation induced FM state. At higher temperatures in the magnetic state, and the zero-field cooling (ZFC) and field cooling (FC) curves do the transition to the FM phase is pushed to higher fields which is not split. At around 8 K, there is a significant difference between consistent with previous reports.30 While the data at 12 and 10 K the ZFC and FC data, which signals the onset of the also show a positive magnetoresistance, at 6 K the enhancement antiferro–ferromagnetic (AFF) phase where all the Ce moments is significantly greater. The magnetoresistance continues to order ferromagnetically within the layer.30 The electrical resistivity become stronger with decreasing temperature, namely by a (ρ(T)) of both CeSb and LaSb is shown in Fig. 1d. While LaSb factor of ≈520 at 0.3 K and 9 T with no indication of saturation, behaves as a simple metal, ρ(T) of CeSb is significantly enhanced. similar to the isostructural non-magnetic compounds LaSb and Upon cooling there is a decrease of ρ(T) from 300 to 80 K below LaBi.36–38 Such a large positive magnetoresistance when the which it increases reaching a maximum at 35 K. Figure 1e shows current and field are perpendicular has been found in materials the magnetic contribution to ρ(T) demonstrating that below 80 K proposed to display a chiral anomaly.13, 18, 19, 39, 40 there is a logarithmic increase of the magnetic resistivity, likely We also measured the magnetoresistance upon varying the due to incoherent Kondo scattering, followed by a decrease below angle θ between the current I and applied field B. As shown in 35 K, presumably arising from the onset of coherence. Fig. 2c, at 2 K the magnetoresistance undergoes a significant decrease as θ is reduced, becoming negative near 0°. It can be Angular-dependence of the magnetoresistance seen more clearly in Fig. 3a that when the current and applied fi A particularly important signature of Weyl fermions is a negative eld are parallel, a negative magnetoresistance appears above the magnetoresistance when the applied field is parallel to the current transition from the AFF to the FM state. As the temperature is increased, the decrease of the resistivity with field becomes less direction, which arises due to a population imbalance between rapid and starts to be observed at a higher field. At temperatures Weyl fermions of different chiralities induced by a magnetic field, 34, 35 below 10 K, the negative magnetoresistance is very sensitive to resulting in a net current. We therefore measured the the alignment of magnetic field and current and is destroyed by a magnetoresistance of CeSb as a function of angle and tempera- slight deviation from 0°, as shown by the 2 and 6 K data in Fig. 3b. fi ture. Figure 2a displays ρ(T) measured in various applied elds Similar features were also observed on measurements of another fi along [001], perpendicular to the current direction. In zero eld, ρ sample where the current direction was rotated by an angle of (T) continues to decrease down to the lowest temperature. Upon about 20° compared to the one in Fig. 3, indicating that the applying a magnetic field there is a minimum of the resistivity behaviour is reproducible when the current direction with respect before it significantly increases at low temperatures, indicating a to the crystal axes is changed (see Supplementary Information). very large positive magnetoresistance, before flattening at the In the AFF state, the magnetoresistance remains positive and lowest temperatures. The magnetoresistance is shown in Fig. 2b at shows a sharp drop at the transition to the FM state. This step-like npj Quantum Materials (2017) Published in partnership with Nanjing University Superconductivity
• Charge conservation symmetry is spontaneously broken in a SC • Order parameter ~ “pair wavefunction” = (k)✏
• Several distinct superconducting phases
M. Norman, 2011 Twisted bilayer graphene
• RESEARCHRecentARTICLE results (Y. Cao et al, 2018)
could be unconventional?
Figure 3 | Magnetic field response of the superconducting states in Fig. 1). (c) Differential resistance dVxx/dI versus dc bias current I for MA-TBG. (a-b) Four-probe resistance as a function of density n and different B⊥ values, measured for device M2. (d) Rxx-T curves for different perpendicular magnetic field B⊥ in device M1 and M2 respectively. Apart B⊥ values, measured for device M1. (e) Perpendicular and parallel critical from the similar dome structures around half-filling as in Fig. 2b-c, magnetic field versus temperature for device M1 (50% normal state there are notably oscillatory features near the boundary between the resistance). The fitting curves are plotted according to the corresponding superconducting phase and the correlated insulator phase. These formulas in Ginzburg-Landau theory for a 2D superconductor. oscillations can be understood as phase-coherent transport through Measurements in (a-c) are all taken at 70 mK. inhomogeneous regions in the device (see Methods and Extended Data
ACCELERATED ARTICLE PREVIEW
8 | NA TURE | VOL 000 | 00 MONTH 2018 © 2018 Macmillan Publishers Limited, part of Springer Nature. All rights reserved. J. A. Mydosh and P.M. Oppeneer: Colloquium: Hidden order, superconductivity, ... 1303 More subtleresistivity T exhibits order stronger decreases as the purity of the starting Uð Þ material is increased. A characteristic plot of T for the two a and c directions in the body-centered ð Þ tetragonal (bct) unit cell of URu2Si2 (see below) is shown in Fig. 4. There is a negative temperature coefficient d=dT at high temperatures,3644 followed byJ.A. a Mydosh maximum and P.M. Oppeneer at 75 K, sig- Classic example: nalinghidden the onset of lattice order coherence, and then a dramatic drop • Table 2. Summary of analytic theories and models proposed to explain the HO, with an emphasis to low temperatureson the recent contributions. and superconductivity. For proposals of specific multipolar Presently magnetic one order canon the U ions, see find resistivityTable ratios3. of 500 or more in the best of today’s Barzykin and Gorkov (1995) three-spin correlations [45] samples. TheKasuya explanation (1997) of the high-temperatureuraniumresistivity dimerisation [46] (T * 100 K)Ikeda reaching and Ohashi (1998)500 cm is open:d either-spin density a wavestrong [47] Okuno and Miyake (1998) CEF and quantum fluctuations [48] Kondo-likeChandra scattering et al. (2002) of incoherent, atomicorbital U spins currents [49 takes] Viroszek et al. (2002) unconv. spin density wave [50] place or, sinceMineev the and Zhitomirsky resistivity (2005) is above the Joffe-Regelstaggered spin density limit wave [51] k ‘ 1 (theVarma product and Zhu (2006) of the Fermi momentumhelicity and (Pomeranchuk) the mean order [52] F ’ Elgazzar et al. (2009) dynamical symmetry breaking [53] free path),Kotetes variable et al. (2010) range hopping occurs.chiral Asd-density the wave local [54] Dubi and Balatsky (2011) hybridization wave [55] U spins disappearPepin et al. (2011) with the onset of coherencemodulated when spin liquid the [56] Fujimoto (2011) spin nematic order [57] temperatureRiseborough is lowered et al. (2012) and the heavy-fermionunconv. state spin-orbital is cre- density wave [58] ated, the spinDas (2012) (fluctuation) scattering is removedspin-orbital density and wave a [59] Chandra et al. (2013) hastatic order [60] coherent low-carrierHsu and Chakravarty state (2013) without significantsinglet-triplet scatteringd-density is wave [61] formed. The superconducting transition temperature Tc varies Table 3. Summary of proposals for a specific multipolar magnetic ordering on the uranium ion to significantlyexplain (between the HO, with 0.8 an and emphasis 1.5 on K)the recent with contributions. sample Note quality that different and symmetries are purity and appearspossible for high-rank to coexist multipoles, thereforeon a some microscopic kind of multipoles scaleappear more with than once. the HO withoutNieuwenhuys disturbing (1987) it (Broholm et al., 1987dipole; (2Isaacs1)order[62] Santini and Amoretti (1994) quadrupolar (22)order[63] et al., 1990).Kiss and Fazekas (2005) octupolar (23)order[64] FIG. 2. Specific heat as a function of temperature for URu2Si2. Hanzawa and Watanabe (2005) octupolar order [65] 2 Hanzawa (2007) incommensurate octupole [66] Top: C=T vs T ; bottom: C=T vs T with the superconducting 4 URu2Si2, Palstra Haule and Kotliar (2009) hexadecapolar (2 )order[67] transition also shown. Note the large extrapolated specific-heat Cricchio et al. (2009) dotriacontapolar (25)order[68] 2 Harima et al. (2010) antiferro quadrupolar order [69] coefficient of 180 mJ=mole K . From Palstra et al., 1985. Thalmeier and Takimoto (2011) E(1, 1)-type quadrupole [70] et al, 1985 Kusunose and Harima (2011) antiferro hexadecapole [71] Ikeda et al. (2012) E−-type dotriacontapole [72] Rau and Kee (2012) E-type dotriacontapole [73] The magnetic response is strongly Ising like, as there is a Ressouche et al. (2012) dotriacontapolar order [16] magnetic signal only along c which begins to deviate from a Full explanation, critique and possible validation of theories listed in Tables 2 and 3 are local-moment Curie-Weiss dependence below 150 K. The beyond the scope of this review. We will only briefly discuss a relevant theory according maximum at 60 K indicates the coherence temperature fromto its relation toMydosh the given experiment. Finally,+ Openeer, we treat the debate over the localized2014 description that relies on the experimental realization of a strong Ising-like anisotropy TÃ and the formation of a heavy Fermi liquid. The HO in URu2Si2 that even occurs in dilute Th1 x Ux Ru2Si2.ThehugeIsinganisotropyof − transition is hardly seen but it corresponds to the intersection quasiparticles in URu2Si2 has been emphasized as a hallmark of localized 5f character [13,60]. Notwithstanding, here we will illustrate that an itinerant DFT description can also of the drop with the plateau below 20 K [cf. Pfleiderer, explain the Ising anisotropy. Continuing along this line, we examine several recent itinerant Mydosh, and Vojta (2006)]. Clearly the susceptibility of models. URu2Si2 is not that of a conventional bulk antiferromagnet. Nevertheless, there are several uranium compounds that do show a similar Ising-like behavior, for example, URhAl and UCo2Si2 [see, e.g., Sechovsky´ and Havela (1998) and Mihalik et al. (2006)]. Although the HO transition always occurs at 17.5 K and is robust, not dependent on sample quality, the low-temperature properties are indeed sample dependent. In particular, the
FIG. 3. Susceptibility of URu2Si2 with applied field (2 T) along FIG. 4. Top: Overview of the resistivity along the a and c axes. the a and c axes. Note the deviation from the Curie-Weiss law Bottom: Expanded view of the low-temperature resistivity illustrat- ( 3:5 =U; 65 K) along the c axis below 150 K. ing the HO transition (T 17:5K) and the superconducting one eff B CW o ¼ From¼Palstra et al., 1985¼À. (T 0:8K). From Palstra, Menovsky, and Mydosh, 1986. c ¼
Rev. Mod. Phys., Vol. 83, No. 4, October–December 2011 Pseudo-gap
charge order loop currents T-reversal breaking pair density wave stripes nematic RVB ... competing/intertwined/ hidden/fluctuating orders?? ElectronREVIEW ARTICLE nematicNATURE PHYSICS DOI: 10.1038/NPHYS2877
symmetry is just tetragonal to orthorhombic, but seems to be that electrons, not lattice, are driving the order
Figure 1 | Schematic phase diagramschematic of hole-doped and electron-doped of Fe122 iron pnictides ofmaterials the BaFe2As2 family. Thefrom blue area denotes stripe-type orthorhombic magnetism, the red area denotes nematic/orthorhombic paramagnetic order, and the yellow area superconductivity. The green area 56 corresponds to a magnetically ordered state that preserves tetragonal (C4) symmetry, as observed recently . The dark red region denotes a regime with strong nematic fluctuations. Dotted lines illustrate theFernandez magnetic and nematic transition et linesal,2014 inside the superconducting dome 73. Second-order (first-order) transitions are denoted by solid (dashed) lines. The insets show the temperature-dependence of the nematic (') and magnetic (M) order parameters in dierent regions of the phase diagram: region (I) corresponds to simultaneous first-order magnetic and nematic transitions; region (II) corresponds to split second-order nematic and first-order magnetic transitions; and region (III) corresponds to split second-order transitions. At x 0 the = transitions are split, like in region (II), but their separation is small 5,6.
1 class points to a di erent origin of the nematic phase (see Fig. 2 for Because the transition is driven by 1, the coe cient 1 , which, a schematic representation): above Tnem, corresponds to the inverse susceptibility of this order parameter, vanishes at T T and becomes negative for T < T , Structural distortion: the lattice parameters a and b along the x = nem nem • whereas 2 and 3 remain finite and positive (although fluctuations and y directions become di erent 5. Such an order is normally of 2 and 3 may shift Tnem slightly). For T < Tnem, 1 orders associated with a phonon-driven structural transition. 1 1/2 on its own: 1 ( 1 /b) . If the coupling constants ij in Charge/orbital order: the occupations nxz and nyz (and on- h i =± • 15 equation (1) were zero, the other two fields 2 and 3 would not site energies) of the dxz and dyz Fe-orbitals become di erent . order, but once the are finite, a non-zero 1 instantly induces The appearance of such an order is normally associated with ij h i finite values of the secondary order parameters 2 12 2 1 , divergent charge fluctuations. h i = h i 3 13 3 1 . As a consequence, there is only one nematic Spin order: the static spin susceptibility mag(q) becomes h i = h i • transition temperature at which all three i become non-zero (for di erent along the qx and qy directions of the Brillouin zone h i 7 example, the lattice symmetry is broken at the same temperature before a conventional SDW state develops . The appearance at which electronic nematic order emerges). Thus, it is not possible of such an order is normally associated with divergent to determine which causes the instability by looking solely at equi- quadrupole magnetic fluctuations. librium order parameters. A further experimental complication is The fact that these three order parameters are non-zero in the the presence of nematic twin domains below Tnem, which e ectively averages 1 to zero. This problem can be circumvented by apply- nematic phase leads to a dilemma, which can be best characterized h i as a physics realization of the ‘chicken and egg problem’: all three ing a small detwinning uniaxial stress 13,14, which acts as a conjugate types of order (structural, orbital and spin-driven nematic) must field to 1 and breaks the tetragonal symmetry at all temperatures, be present no matter which drives the nematic instability. This similar to an external magnetic field applied on a ferromagnet. follows from the fact that bi-linear combinations of the order One way to select the primary order is to carefully study parameters that break the same symmetry (in our case, the fluctuations in the symmetry-unbroken phase at T >Tnem. Because tetragonal symmetry of the system) are invariant under symmetry the primary order parameter 1 acts as an external field for the transformations and must therefore appear in the Landau free secondary order parameters, 2 and 3, fluctuations of the former energy. Suppose that one of the three order parameters is the renormalize the susceptibilities of the latter. For small 1 1 (i 2,3) we have primary one — that is, its fluctuations drive the nematic instability. i i ⌧ = Let us call it 1 and the other two 2 and 3, respectively. The free 2 2 2 2 1 2 1 , 3 3 1 3 1 (2) energy has the generic form ˜ ⇡ + 12 ˜ ⇡ + 13 2 1 1 2 b 4 where 1 1 is the susceptibility of the primary field. The F [ 1, 2, 3] 12 1 2 = h i = 2 1 1 + 4 1 + renormalized susceptibilities of the secondary fields do diverge at the nematic transition, however, for small enough 12 and 13, 2 1 1 ˜ 1 2 1 2 and 3 begin to grow only in the immediate vicinity of Tnem, where 2 2 13 1 3 3 3 (1) ˜ +2 + + 2 +··· 1 is already large. If one can measure the three susceptibilities
98 NATURE PHYSICS VOL 10 FEBRUARY 2014 www.nature.com/naturephysics | | | Order questions
• What is the hidden order in URu2Si2? • Is the pseudo-gap region of the cuprates a consequence of some hidden order? • Are there mechanisms for unifying different orders in QMs - i.e. can you give a sharp meaning to intertwinement etc.? Topology
Mathematical study of objects that can and cannot be smoothly transformed into one another Topology
In physics, “knotted” states can become new phases of matter with robust properties Topological invariant
W=0 W=1 W=-1 W=-2
1 dz W [ ]= Z Winding number C 2⇡i z 2 IC 1 z (s) = ds 0 2⇡i z(s) Z Topology of order
• Spontaneously broken symmetry leads to topological defects: configurations of order parameter that cannot be smoothly “unwound”
XY/U(1) order: vortices discrete order: domain walls
The topological theory of defects in ordered ~acedia*& N. D. Mermin
Laboratory of Atomic and Solid State Physics Cornell University, Ithaca, New York 14853
Aspects of the theory of homotopy groups are described in a mathematical style closer to that of condensed matter physics than that of topology. The aim is to make more readily accessible to physicists the recent applications of homotopy theory to the study of defects in ordered media. Although many physical examples are woven into the development of the subject, the focus is on Inathematical pedagogy rather than on a systematic review of applications. vector order: skyrmions/hedgehogs
CONTENTS B. Group-theoretic description of the order- 609 parameter space I. Introduction 592 C. Examples of order-parameter spaces. as 611 II. Ordered Media and Defects 594 coset spaces 611 A. Examples of ordered media 594 1. Planar spins 612 Planar spins 594 2. Ordinary spins 612 2. Ordinary spins 594 3. Nematic s 612 3. Nematic liquid crystals 594 4. Biaxial nematics 612 4. Biaxial nematics 595 5. Super fluid helium-3 5. Super fluid helium-3 595 Properties of the fundamental group of B. A simple example of the topology of a topological group defects: planar spins in two dixnensions V. The Fundamental Group of the Order- C. An even simpler illustration: ordinary parameter Space spins in the plane 600 A. The fundamental theorexn on the III. The Fundamental Group 600 fund amental group A. The fundamental group at a point 601 1. The correspondence between loops in 1. Loops at x 601 coset space and the connected 2. Homotopies at x 601 components of the isotropy subgroup 3. The product of two loops 602 2. Proof of the isomorphism between 616 4. The product of homotopy clas es r, (G/II) and a/II, of loops 602 B. Computing the fundamental group of the 617 5. The fundamental group at x, 7t~(A, x) 602 order-parameter space 617 6. A few simple examples 603 Planar spins 617 a. The circle 603 2. Ordinary spins 617 b. The surface of a sphere 603 3. Nematic s 618 c. The figure-eight space 604 4. Biaxial nematics 619 B. The fundamental group of a connected space 604 5. Superfluid helium-3 619 1. The isomorphism between fundamental VI. Media with Non-Abelian Fundamental Groups groups based at different points 604 A. More on the nature of line defects in 620 Uniqueness (or lack of uniqueness) of biaxial nematics the path isomorphisms between based B. The combination of defects in the non- fund amental groups 605 Abelian case C. The fundamental group and the classes of C. The entanglement of line defects freely homotopic loops 606 VII. The Second Homotopy Group and the D. The fundamental group and the Cl.assification of Point Defects in Three 625 combination of line defects ' 607 Dimensions 625 IV. Group-theoretic Structure of the Order- A. The second homotopy group at x, z2(R, x) 626 parameter Space 608 B. The second homotopy group, 'Jt 2(B) A. Continuous groups 608 C. The fundamental theorem on the second homotopy group of coset spaces D. The action of jr~ on 7t~. classes of freely homotopio spheres in G/H *A preliminary version of this paper appeared as Technical E. Examples of point defects 631 Beport No. 3045 of the Materials Science Center of Cornell 1. Planar splns 631 University. It has been substantially revised at the University 2. Ordinary spins 631 of Sussex, with the partial financial, support of the Science Be- 3. Nematics 631 search Council of Great Britain and with the generous help and 4. Biaxial nematic s 632 hospitality of A. J. Leggett and D. F. Brewer. The revision 5. Super fluid helium-3 632 has also been supported in part by the National Science Founda- VIII. Ordered Media with Broken Translational 632 tion under Grant No. DMB 77-18329 and through the Materials Symmetry in the Uniform State Science Center of Cornell University, Technical Beport No. A. The naive generalization 4021. 1. Order-parameter space fA complementary review of homotopy, from the point of view 2. Defects and homotopy groups of a mathematician and field theorist, by Louis Michel is sched- 3. Computing the homotopy groups uled to appear in a future issue of Beviews of Modern Physics. B. Critique of the naive generalization
Reviews of Modern Physics, Vol. 51, No. 3, July 1979 Copyright 0& 1979 American Physical Society Topological band theory
555
AUGUST 1, 1937 PH YSI CAL REVIEW VOLUM E 52
The Structure of Electronic Excitation Levels in Insulating Crystals Uber die Quantenmechanik der Elek~ronen in Kristallgittern. GREGORY H. WANNIER Princeton University, Princeton, ¹mJersey* Von Felix Bloeh in Leipzig. (Received May 13, 1937} 5lit 2 Abbildungen. (Eingegangen am 10. August 1928.) In this article, a method is devised to study the energy spectrum for an excited electron con- Die Bewegung eines Elektrons im Gitter wird untersucht, indem wir uns dieses figuration in an ideal crystal. The con6guration studied consists of a single excited electron durch ein zun~chst streng dreifaeh periodisches Kraftfeld schematisieren. Unter taken out of a full band of N electrons. The multiplicity of the state is N'. It is shown that Hinzunahme der Fermischen Statistik auf die Elektronen gestattet unser Modell because of the Coulomb attraction between the electron and its hole states are split off from the bottom of the excited Bloch band; for these states the electron¹cannot"escape its hole Aussagen fiber den von ihnen herrfihrenden AnteiI der spezi[ischen W~rme des Kristalls. Ferner wird gezeigt, dal] die Berficksichtigung der thermischen Gitter- completely. The analogy of these levels to the spectrum of an atom or molecule is worked out schwingungen Gr51]enordnung and Temperaturabh~ingigkeit der elektrischen Leit- quantitatively. The bottom of the 8loch band appears as "ionization potential" and the Bloch [~ihigkeit yon Metallen in qualitativer Ubereinstimmung mit der Erfahrung ergibt. band itself as the continuum above this threshold energy.
Einleitung. Die Elekfronentheorie der MetalLe hat seit einiger Zeit For~schritte zu verzeichnen, die in der Anwendung quantentheo- ~OR several years, there have been two com- (3) We shall derive some general results and retischer Prinziplen auf das Elektronengas begriindet sind. Zun~chst hat peting pictures in use to describe the be- discuss their consequences. Pauli* unter der Annahme, da~ die Metatlelektronen sieh vSllig frel im havior of electrons in crystals. The one adopted Gitter bewegen kOnnen und der Fermischen** Statistik gehorchen, den in most theoretical calculations and especially 1. BASIC %AVE FUNCTIONS AND ENERGY MATRIX successful for metals describes each electron by a temperaturunabh~nglgen Paramagnetlsmus der Alkalien zu erkl~ren ver- . running wave, but Frenkel has shown that in It would no doubt be more satisfactory for moeht. Die elektrischen undE thermischen Eigenschaften des Elektronen- insulating crystals, to discuss the Hamiltonian many cases the more elementary atomic picture gases sind dann yon Sommerfeld, Houston und Eckar~*** n~her atomic functions rather func- may be the better approximation. ' This ap- using than Bloch tions. But this line of has been hampered untersucht worden. Die Tatsache freier Leitungse]ektronen wird yon parent contradiction has been removed by attack ihnen als gegeben betrachtet und ihre Wechselwlrkung mi~ dem Gitter Slater and Shockley, ' who showed with a simpli- by the fact that atomic functions are not orthogonal. build or- nur dureh eine zunachst ph~inomenologlsch elngefiihrte, dann yon fied model that the two types of states actually %e can, however, up thogonal functions all the H o u st o n**** strenger begrfindete freie Weg]ange mitbe1~ickslchtigt. coexist in a crystal. It is the purpose of the having advantages of atomic ones by starting out from a Bloch Sehliel]lich hat Heisenberg-~ gezeigt, daL] im anderen Grenzfa]l, wo zu- present paper to treat this question in a quantita- approximation. Let us assume then that a Bloch n~chst die Elektronen an die Ionen ]m Gitter gebunden gedacht und erst tive way, starting out from the actual Hamil- or Fock method has given us functions b(k, x) of in nachster N~herung die Austauschvorgange unter ihnen berficksichtigt tonian of the system. shall restrict ourselves in this article to energy W(k). Then the required functions are werden, das fiir den Ferromagnetismus entseheidende intermolekulare Feld %e insulators containing one electron in the lowest seine Erklgrung finder. a(x —n) =1/(X)& exp —ik„n)b(k„, ' excited state, and we shall study the energy P L x), (1) Hier soll ein Zwisehenstandpunkt zwischen den beiden oben er- spectrum of this single configuration, neglecting w~hnten Behandlungsweisen elngenommen werden, insofern, als tier Aus- perturbations arising from other configurations. where N is the number of cells in the crystal and tausch der Elektronen unberiicksichtigt bleibt, sie dagegen nicht einfach As to the method we shall proceed in the fol- the k's are as usual determined by some bound- lowing way: ary condition. Formula to set of func- * W. PauJi, ZS. f. Phys. 41, 81, 1927. (1) We shall construct orthogonal "atomic" (1) applies any Bloch ** E. Fermi, ebenda 36, 902, 1926. wave functions and express the energy matrix in tions, but it might be interesting to get some *** A. Sommerfeld, W. V. Houston, C. Eckart, ebenda 47, 1, 1928.k this vector insight into the structure of thexa' s. For this system. **** W.V. Houston, ebenda 48, 449, 1928. (2) The energy matrix contains many terms purpose let us 6rst make the ad koc assumption "i" W. Heisenberg, ebenda 49, 619, 1928. having the periodicity of the lattice and a few (valid for free electrons) that b is of the form whichThouless,have not; we shall develop1984a method which takes them both into account. b(k„x) =exp Cik„x] b(x),
* I want to express my thanks to Princeton University where the periodic factor b(x) is independent of k. for the grant of its Swiss-American Exchange Fellowship Then we find explicitly: for the year 1936—37. ' J. Frenkel, Phys. Rev. 1'7, 17 (1931); Physik. Zeits. Sowjetunion 9, 158 (1936); Physik. Zeits, Sowjetunion 8, 'The unit of length adopted in this article is the ele- 185 (1935). mentary translation in the direction of each of the crystal ~ J. C. Slater and W. S. Shockley, Phys. Rev. SG, 705 axes. In some deductions the crystal is assumed to be (1936). simple cubic, but this could easily be removed. Chern number
ik r • Bloch states n(r)=e · un,k(r) ~ = i u ~ u • Berry gauge field An h n|rk| ni ~ = ~ ~ Bn rk ⇥ An • Net Berry flux gives Chern number 1 2 z qn = d k n Z 2⇡ B 2 Z Phase transitions
• There is no smooth way to go from one topological sector to another IQHE
• Hall conductance measures Chern number
h ⇢ = xy ne2 ⇢xx peak signifies gapless quantum critical state
• Passage from one Chern number to another is a quantum phase transition (with no order parameter!) IQHE
• The boundary between states with different Chern number is gapless
q=0 q=1
chiral edge states cannot backscatter IQHE
• The boundary between states with different Chern number is gapless
q=0 q=1
Bulk-boundary correspondence:
Chern Number = NR-NL IQHE
• Edge states are “half” of the low energy
excitations of a 1DEG large distance E
L R
k Halperin, 1982 IQHE
• Edge states are “half” of the low energy
excitations of a 1DEG large distance E
L R
k Halperin, 1982 General rule: surface state of a d-dimensional TI cannot be realized in a d-1 dimensional system Summary - Chern Insulators • Winding of the one-electron wavefunction over the Brillouin zone • Quantified by a topological invariant: the Chern number, which coincides with Hall conductance • States with different values of the topological invariant are different phases • Tuning from one to another requires a quantum phase transition • The interface between two different values has gapless “protected” edge states, which are “anomalous”: they could not exist in an isolated 1-dimensional system Time-reversal symmetry
• Would be nice to have topological quantization in materials in “natural conditions” • The Berry curvature is odd under time- reversal ( k)= (k) B B • This implies the Chern number vanishes • For decades it was believed this meant electronic states are topologically trivial with TRS Z2 TIs
2d: Kane, Mele (2005); Bernevig, Hughes, Zhang (2006) 3d: L. Fu, C. Kane, E. Mele (2007); J. Moore, LB (2007)
• Even with TR symmetry, a different type of TI is possible (with spin-orbit coupling) • 2d: “QSHE” • Roughly understood as opposite IQHE’s for up and down electrons Z2 TIs
2d: Kane, Mele (2005); Bernevig, Hughes, Zhang (2006) 3d: J. Moore, LB (2007); L. Fu, C. Kane, E. Mele (2007) • Q = parity of band crossings between TRI momenta at the surface
8
Symmetry d E Conduction Band Conventional ν=0 AZ ⇥⌅⇧12345678 Insulator A 0000 Z 0 Z 0 Z 0 Z (a) (b) EF AIII 001Z 0 Z 0 Z 0 Z 0 AI 100000Z 0 Z2 Z2 Z Quantum spin BDI 111Z 000Z 0 Z2 Z2 ν=1 Hall insulator Valence Band D 010Z2 Z 000Z 0 Z2 −π/a k0 −π/a DIII 11 1Z2 Z2 Z 000Z 0 AII 10 0 0 Z2 Z2 Z 000Z FIG. 5 Edge states in the quantum spin Hall insulator. (a) CII 1 11Z 0 Z2 Z2 Z 000 shows the interface between a QSHIFull and zone an ordinary for insula-TI C 0 “trivial”10 0 Z 0 Z2 Z2 Z “topological”00 tor, and (b) shows the edge state dispersion in the graphene model, in which up and down spins propagate in opposite CI 1 11 00Z 0 Z2 Z2 Z 0 directions.
TABLE I Periodic table of topological insulators and super- conductors. The 10 symmetry classes are labeled using the (2001) corresponds to the d = 4 entry in class A or AII. notation of Altland and Zirnbauer (1997) (AZ) and are spec- There are also other entries in physical dimensions that ified by presence or absence of symmetry ⇥, particle-hole have yet to be filled by realistic systems. The search is symmetry ⌅ and chiral symmetryT ⇧ = ⌅⇥. 1and0denotes the presence and absence of symmetry, with± 1 specifying on to discover such phases. the value of ⇥2 and ⌅2. As a function of symmetry± and space dimensionality, d,thetopologicalclassifications(Z, Z2 and 0) show a regular pattern that repeats when d d +8. III. QUANTUM SPIN HALL INSULATOR ! The 2D topological insulator is known as a quantum 3. Periodic table spin Hall insulator. This state was originally theorized to exist in graphene (Kane and Mele, 2005a) and in 2D Topological insulators and superconductors fit to- semiconductor systems with a uniform strain gradient gether into a rich and elegant mathematical structure (Bernevig and Zhang, 2006). It was subsequently pre- that generalizes the notions of topological band theory dicted to exist (Bernevig, Hughes and Zhang, 2006), and described above (Schnyder, et al., 2008; Kitaev, 2009; was then observed (K¨onig, et al., 2007), in HgCdTe quan- Schnyder, et al., 2009; Ryu, et al., 2010). The classes tum well structures. In section III.A we will introduce of equivalent Hamiltonians are determined by specifying the physics of this state in the model graphene system the symmetry class and the dimensionality. The symme- and describe its novel edge states. Section III.B will re- try class depends on the presence or absence of sym- view the experiments, which have also been the subject metry (8) with ⇥2 = 1 and/or particle-hole symmetryT of the review article by K¨onig, et al. (2008). (15) with ⌅2 = 1. There± are 10 distinct classes, which are closely related± to the Altland and Zirnbauer (1997) classification of random matrices. The topological clas- A. Model system: graphene sifications, given by Z, Z2 or 0, show a regular pattern as a function of symmetry class and dimensionality and In section II.B.2 we argued that the degeneracy at the can be arranged into the periodic table of topological in- Dirac point in graphene is protected by inversion and sulators and superconductors shown in Table I. symmetry. That argument ignored the spin of the The quantum Hall state (Class A, no symmetry; d = electrons.T The spin orbit interaction allows a new mass 2 2), the Z2 topological insulators (Class AII, ⇥ = 1; term in (3) that respects all of graphene’s symmetries. In d =2, 3) and the Z2 and Z topological superconductors the simplest picture, the intrinsic spin orbit interaction 2 (Class D, ⌅ = 1; d =1, 2) described above are each commutes with the electron spin Sz, so the Hamiltonian entries in the periodic table. There are also other non decouples into two independent Hamiltonians for the up trivial entries describing di↵erent topological supercon- and down spins. The resulting theory is simply two copies ducting and superfluid phases. Each non trivial phase is the Haldane (1988) model with opposite signs of the Hall predicted, via the bulk-boundary correspondence to have conductivity for up and down spins. This does not violate gapless boundary states. One notable example is super- symmetry because time reversal flips both the spin and 3 T fluid He B (Volovik, 2003; Roy, 2008; Schnyder, et al., xy. In an applied electric field, the up and down spins 2008; Nagato, Higashitani and Nagai, 2009; Qi, et al., have Hall currents that flow in opposite directions. The 2009; Volovik, 2009), in (Class DIII, ⇥2 = 1, ⌅2 = +1; Hall conductivity is thus zero, but there is a quantized s d = 3) which has a Z classification, along with gapless 2D spin Hall conductivity, defined by Jx" Jx# = xyEy with s Majorana fermion modes on its surface. A generalization xy = e/2⇡ – a quantum spin Hall e↵ect. Related ideas of the quantum Hall state introduced by Zhang and Hu were mentioned in earlier work on the planar state of J. Moore, LB (2007) Z2 TIs L. Fu, C. Kane, E. Mele (2007)
• In 3d, there are four Z2 parities • 3 “weak” parities - describe layered 2d TIs • 1 “strong” parity - describes intrinsically 3d physics 8
Symmetry d E Conduction Band Conventional ν=0 AZ ⇥⌅⇧12345678 Insulator odd number A 0000 Z 0 Z 0 Z 0 Z (a) (b) EF of Dirac AIII 001Z 0 Z 0 Z 0 Z 0 AI 100000Z 0 Z2 Z2 Z cones at any Quantum spin BDI 111Z 000Z 0 Z2 Z2 ν=1 Hall insulator Valence Band surface D 010Z2 Z 000Z 0 Z2 −π/a k0 −π/a DIII 11 1Z2 Z2 Z 000Z 0 AII 10 0 0 Z2 Z2 Z 000Z FIG. 5 Edge states in the quantum spin Hall insulator. (a) CII 1 11Z 0 Z2 Z2 Z 000 shows the interface between a QSHI and an ordinary insula- C 0 10 0 Z 0 Z2 Z2 Z 00 tor, and (b) shows the edge state dispersion in the graphene model, in which up and down spins propagate in opposite CI 1 11 00Z 0 Z2 Z2 Z 0 directions.
TABLE I Periodic table of topological insulators and super- conductors. The 10 symmetry classes are labeled using the (2001) corresponds to the d = 4 entry in class A or AII. notation of Altland and Zirnbauer (1997) (AZ) and are spec- There are also other entries in physical dimensions that ified by presence or absence of symmetry ⇥, particle-hole have yet to be filled by realistic systems. The search is symmetry ⌅ and chiral symmetryT ⇧ = ⌅⇥. 1and0denotes the presence and absence of symmetry, with± 1 specifying on to discover such phases. the value of ⇥2 and ⌅2. As a function of symmetry± and space dimensionality, d,thetopologicalclassifications(Z, Z2 and 0) show a regular pattern that repeats when d d +8. III. QUANTUM SPIN HALL INSULATOR ! The 2D topological insulator is known as a quantum 3. Periodic table spin Hall insulator. This state was originally theorized to exist in graphene (Kane and Mele, 2005a) and in 2D Topological insulators and superconductors fit to- semiconductor systems with a uniform strain gradient gether into a rich and elegant mathematical structure (Bernevig and Zhang, 2006). It was subsequently pre- that generalizes the notions of topological band theory dicted to exist (Bernevig, Hughes and Zhang, 2006), and described above (Schnyder, et al., 2008; Kitaev, 2009; was then observed (K¨onig, et al., 2007), in HgCdTe quan- Schnyder, et al., 2009; Ryu, et al., 2010). The classes tum well structures. In section III.A we will introduce of equivalent Hamiltonians are determined by specifying the physics of this state in the model graphene system the symmetry class and the dimensionality. The symme- and describe its novel edge states. Section III.B will re- try class depends on the presence or absence of sym- view the experiments, which have also been the subject metry (8) with ⇥2 = 1 and/or particle-hole symmetryT of the review article by K¨onig, et al. (2008). (15) with ⌅2 = 1. There± are 10 distinct classes, which are closely related± to the Altland and Zirnbauer (1997) classification of random matrices. The topological clas- A. Model system: graphene sifications, given by Z, Z2 or 0, show a regular pattern as a function of symmetry class and dimensionality and In section II.B.2 we argued that the degeneracy at the can be arranged into the periodic table of topological in- Dirac point in graphene is protected by inversion and sulators and superconductors shown in Table I. symmetry. That argument ignored the spin of the The quantum Hall state (Class A, no symmetry; d = electrons.T The spin orbit interaction allows a new mass 2 2), the Z2 topological insulators (Class AII, ⇥ = 1; term in (3) that respects all of graphene’s symmetries. In d =2, 3) and the Z2 and Z topological superconductors the simplest picture, the intrinsic spin orbit interaction 2 (Class D, ⌅ = 1; d =1, 2) described above are each commutes with the electron spin Sz, so the Hamiltonian entries in the periodic table. There are also other non decouples into two independent Hamiltonians for the up trivial entries describing di↵erent topological supercon- and down spins. The resulting theory is simply two copies ducting and superfluid phases. Each non trivial phase is the Haldane (1988) model with opposite signs of the Hall predicted, via the bulk-boundary correspondence to have conductivity for up and down spins. This does not violate gapless boundary states. One notable example is super- symmetry because time reversal flips both the spin and 3 T fluid He B (Volovik, 2003; Roy, 2008; Schnyder, et al., xy. In an applied electric field, the up and down spins 2008; Nagato, Higashitani and Nagai, 2009; Qi, et al., have Hall currents that flow in opposite directions. The 2009; Volovik, 2009), in (Class DIII, ⇥2 = 1, ⌅2 = +1; Hall conductivity is thus zero, but there is a quantized s d = 3) which has a Z classification, along with gapless 2D spin Hall conductivity, defined by Jx" Jx# = xyEy with s Majorana fermion modes on its surface. A generalization xy = e/2⇡ – a quantum spin Hall e↵ect. Related ideas of the quantum Hall state introduced by Zhang and Hu were mentioned in earlier work on the planar state of Band inversion
• Different classes of bands cannot be smoothly deformed into one another
Bi2Se3
no SOC w/ SOC Band inversion
• Different classes of bands cannot be smoothly deformed into one another
Bi2Se3
normal insulator topological insulator Single Dirac cone
• 1/4 Graphene • 2d Dirac mass breaks TR • Cannot be found in any 2d model
= ⇡ =0 Single Dirac cone
• 1/4 Graphene • 2d Dirac mass breaks TR • Cannot be found in any 2d model
states decay into bulk = ⇡ not allowed Single Dirac cone
• 1/4 Graphene • 2d Dirac mass breaks TR • Cannot be found in any 2d model B If time reversal is broken, the Dirac cone is gapped and TI trivial topol. inversion becomes equivalent to trivial insulator TIs Broken with TRS TRS
integer Chern Z2 invariant number = Hall = magneto- conductance electric polarizability
chiral edge helical edge/ states surface states Topological superconductors Periodic table of topological insulators and superconductors
Complete and exhaustive classification of topological insulators and SCs
“periodic“Periodic table” of table”topological insulators of / superconductors TIs+TSCs
Name T C S=CT d=1 d=2 d=3 IQHE A 0 0 0 - d - Chern AIII 0 0 1 - Z2 TI polyacetylen insulator AI +1 0 0 - - - BDI +1 +1 1 d - - chiral p-wave D 0 +1 0 2 d - TRI top. triplet SC (He3 B) DIII -1 +1 1 2 2 QSHE Z2 3D Z2 top. insulator chiral AII -1 0 0 - d 2 d 2 Topo. CII -1 -1 1 - 2 chiral d-wave topo. SC C 0 -1 0 - d - SC CI +1 -1 1 - - TRI top. singlet SC
He3B new topological insulators / superconductors in one, two, and three dimensions Dimensional hierarchy relatesA. Schnyder topological phases in different symmetry classes and differentet al, dimensions 2008 free fermion Ryu, Schnyder, Furusaki, Ludwig, arXiv:0912.2157 (to appear in NJP) 14 TIs Kitaev, 2009 2
implementation of the ideas introduced here would constitute a critical step towards this ultimate goal.
I. MAJORANA FERMIONS IN ‘SPINLESS’ p-WAVE KitaevSUPERCONDUCTING Chain WIRES We begin by discussing the physics of a single wire. Valu- able intuition can be garnered from Kitaev’s toy model for a • 1d spinless,spinlessp-wave superconducting“p-wave”N-site superconductor chain23: N N 1 i H = µ c† c (tc† c + e c c + h.c.) x x x x+1 | | x x+1 x=1 x=1 2 X X (1) where cx is a spinless fermion operator and µ, t>0, and FIG. 1: (a) Pictorial representation of the ground state of Eq. (1) in implementation of the ideas introduced here would constitute• Topological ei respectively for denote |μ the|< chemical 2 t potential, tunneling the limit µ =0, t = . Each spinless fermion in the chain is | | | | a critical step towards this ultimate goal. strength, and pairing potential. The bulk- and end-state struc- decomposed in terms of two Majorana fermions A,x and B,x. Ma- ture becomes particularly transparent in the special case23 joranas B,x and A,x+1 combine to form an ordinary, finite energy A,1 B,1 A,2 B,2 B,3 A,N B,N µ =0, t = . Here it is usefulA,3 to express fermion, leaving two zero-energy end Majoranas A,1 and B,N as | | shown23. (b) A spin-orbit-coupled semiconducting wire deposited on an s-wave superconductor can be driven into a topological supercon- I. MAJORANA FERMIONS IN ‘SPINLESS’ p-WAVE 1 i ducting state exhibiting such end Majorana modes by applying an cx = e 2 ( B,x + i A,x), (2) SUPERCONDUCTING WIRES 2 external magnetic field1,2. (c) Band structure of the semiconducting wire when B =0(dashed lines) and B =0(solid lines). When µ lies in the band gap generated by the field,6 pairing inherited from the We begin by discussing the physics of a single wire. Valu- with ↵,x = ↵† ,x Majorana fermion operators satisfying , =2 . These expressions expose the proximate superconductor drives the wire into the topological state. able intuition can be garnered from Kitaev’s toy model for a { ↵,xtogether↵0,x0 } ↵↵ 0thesexx0 two “Majorana” spinless, p-wave superconducting N-site chain23: defining characteristics of Majorana fermions—they are their own antiparticle and constitute ‘half’ of an ordinary fermion. fermions form one two-level systemsimplest Hamiltonian describing such a wire reads In this limit the Hamiltonian can be written as N N 1 2 2 i ~ @x H = µ c† cx (tc† cx+1 + e cxcx+1 + h.c.) N 1 = dx † µ i~uˆe @x x x | | H x 2m · x=1 x=1 H = it . (3) Z ✓ X X B,x A,x+1 (1) x=1 gµBBz z i' X x +( e x x + h.c.) . (4) where cx is a spinless fermion operator and µ, t>0, and FIG. 1: (a) Pictorial representation of the ground state of Eq. (1) in 2 | | # " i ◆ e respectively denote the chemical potential, tunneling theConsequently, limit µ =0 , B,xt =and .A,x Each+1 spinlesscombine fermion to form in an the ordi- chain is | | The operator ↵x corresponds to electrons with spin ↵, effec- | | decomposednary fermion in termsd =( of two Majorana+ i fermions)/2 which A,x and costs B,x en-. Ma- strength, and pairing potential. The bulk- and end-state struc- x A,x+1 B,x tive mass m, and chemical potential µ. (We suppress the spin 23 joranasergy 2 t,B,x reflectingand A,x the+1 wire’scombine bulk to form gap. an Conspicuously ordinary, finite ab- energy ture becomes particularly transparent in the special case indices except in the pairing term.) In the third term, u denotes fermion,sent from leavingH, however, two zero-energy are endand Majoranas , which A,1 representand B,N as µ =0, t = . Here it is useful to express A,1 B,N the (Dresselhaus31 and/or Rashba32) spin-orbit strength, and | | shownend-Majorana23. (b) A spin-orbit-coupled modes. These can semiconducting be combined into wire an deposited ordi- on =( x, y, z) is a vector of Pauli matrices. This coupling annarys-wave (though superconductor highly non-local) can be driven zero-energy into a topologicalfermion dend supercon-= 1 favors aligning spins along or against the unit vector ˆe , which i 2 ducting( + statei exhibiting)/2. Thus such there end Majorana are two degenerate modes by ground applying an cx = e ( B,x + i A,x), (2) A,1 B,N we assume lies in the (x, y) plane. The fourth term represents 2 external magnetic field1,2. (c) Band structure of the semiconducting states 0 and 1 = dend† 0 , where dend 0 =0, which serve the Zeeman coupling due to the magnetic field B < 0. Note wire when| i B =0| i (dashed| lines)i and B =0| i (solid lines). When µ z as topologically protected qubit states. Figure 1(a) illustrates that spin-orbit enhancement can lead to33 g 2. Finally, the lies in the band gap generated by the field,6 pairing inherited from the with ↵,x = ↵† ,x Majorana fermion operators satisfying this physics pictorially. last term reflects the spin-singlet pairing inherited from the ↵,x, ↵ ,x =2 ↵↵ xx . These expressions expose the proximateAway from superconductor this special drives limit the the wire Majorana into the end topological states no state. { 0 0 } 0 0 s-wave superconductor via the proximity effect. defining characteristics of Majorana fermions—they are their longer retain this simple form, but survive provided the bulk To understand the physics of Eq. (4), consider first Bz = own antiparticle and constitute ‘half’ of an ordinary fermion. gap remains finite23. This occurs when µ < 2t, where a =0. The dashed lines in Fig. 1(c) illustrate the band simplestpartially Hamiltonian filled band pairs. describing The bulk such gap closes a| wire| when readsµ =2t, In this limit the Hamiltonian can be written as | | structure here—clearly no ‘spinless’ regime is possible. In- and for larger µ a topologically2 2 trivial superconducting state troducing a magnetic field generates a band gap Bz at | | ~ @x / | | N 1 without end= Majoranasdx † emerges. Hereµ pairingi~u occursˆe @ inx either zero momentum as the solid line in Fig. 1(c) depicts. When H x 2m · H = it . (3) a fully occupiedZ or vacant✓ band. µ lies inside of this gap the system exhibits only a single pair B,x A,x+1 x=1 Realizinggµ Kitaev’sBBz topologicalz superconductingi' state exper- of Fermi points as desired. Turning on which is weak com- X x +( e x x + h.c.) . (4) imentally requires2 a system which| is| effectively# " spinless— pared to the gap then effectively p-wave pairs fermions in the i.e. ◆ p k k Consequently, and combine to form an ordi- , exhibits one set of Fermi points—and -wave pairs at the lower band with momentum and , driving the wire into B,x A,x+1 TheFermi operator energy. Bothcorresponds criteria can to be electrons satisfied inwith a spin-orbit- spin ↵, effec-Kitaev’s topological phase1,2. [Singlet pairing in Eq. (4) gen- nary fermion d =( + i )/2 which costs en- ↵x x A,x+1 B,x tivecoupled mass semiconductingm, and chemical wire potential depositedµ. on (We an s suppress-wave super- the spinerates p-wave pairing because spin-orbit coupling favors op- ergy 2t, reflecting the wire’s bulk gap. Conspicuously ab- 1,2 indicesconductor except by applying in the pairing a magnetic term.) field In the[see third Fig. term, 1(b)].u Thedenotesposite spins for k and k states in the lower band.] Quan- sent from H, however, are and , which represent A,1 B,N the (Dresselhaus31 and/or Rashba32) spin-orbit strength, and end-Majorana modes. These can be combined into an ordi- =( x, y, z) is a vector of Pauli matrices. This coupling nary (though highly non-local) zero-energy fermion d = end favors aligning spins along or against the unit vector ˆe , which ( + i )/2. Thus there are two degenerate ground A,1 B,N we assume lies in the (x, y) plane. The fourth term represents states 0 and 1 = d† 0 , where dend 0 =0, which serve | i | i end| i | i the Zeeman coupling due to the magnetic field Bz < 0. Note as topologically protected qubit states. Figure 1(a) illustrates that spin-orbit enhancement can lead to33 g 2. Finally, the this physics pictorially. last term reflects the spin-singlet pairing inherited from the Away from this special limit the Majorana end states no s-wave superconductor via the proximity effect. longer retain this simple form, but survive provided the bulk To understand the physics of Eq. (4), consider first Bz = gap remains finite23. This occurs when µ < 2t, where a =0. The dashed lines in Fig. 1(c) illustrate the band partially filled band pairs. The bulk gap closes| | when µ =2t, | | structure here—clearly no ‘spinless’ regime is possible. In- and for larger µ a topologically trivial superconducting state B | | troducing a magnetic field generates a band gap z at without end Majoranas emerges. Here pairing occurs in either zero momentum as the solid line in Fig. 1(c) depicts./ | When| a fully occupied or vacant band. µ lies inside of this gap the system exhibits only a single pair Realizing Kitaev’s topological superconducting state exper- of Fermi points as desired. Turning on which is weak com- imentally requires a system which is effectively spinless— pared to the gap then effectively p-wave pairs fermions in the i.e., exhibits one set of Fermi points—and p-wave pairs at the lower band with momentum k and k, driving the wire into Fermi energy. Both criteria can be satisfied in a spin-orbit- Kitaev’s topological phase1,2. [Singlet pairing in Eq. (4) gen- coupled semiconducting wire deposited on an s-wave super- erates p-wave pairing because spin-orbit coupling favors op- conductor by applying a magnetic field1,2 [see Fig. 1(b)]. The posite spins for k and k states in the lower band.] Quan- Topological semi-metals
• Another class of topological states are semi-metals, in which band touching is protected by topology • Graphene: 2d Dirac fermion • TaAs: 3d Weyl fermion
• Cd3Se2: 3d Dirac fermion
• Mn3Sn: 3d magnetic Weyl fermions • many more! • In these systems the touching points are like “topological defects”: singularities of Berry curvature • Near the touching points, electrons have unusual dynamics: lots of potential for interesting physics! ACCIDENTAL D EGEN ERAC Y 373
For crystals with an inversion center, contacts X+(k„,i), E (k„ i), of odd and' of even eigen- of equivalent manifolds M'(k), 3f'(k) may occur functions fs, ' be counted which have energies at all points k of an endless curve, or of a number E'(k,) ~& E'(k„). Now the quantity of such curves, in k-space. These contact curves cannot be destroyed or broken by any infini- tesimal change in the potential U which pre- serves the inversional symmetry. It is vanishingly is an integer, and according to whether this improbable for such curves to lie in planes of integer is odd or even the number of circuits of symmetry in the B-Z; however a contact curve the second type along which contact between the may pass through a symmetry axis at a point bands i and j occurs must be odd or even. Since where necessary degeneracy or contact of any crystal with an inversion center can be made inequivalent manifolds occurs. by an infinitesimal change in the form of U into Suppose that for a crystal with an inversion one whose space group is merely its translation center a contact of inequivalent manifolds group plus the inversion, this implies certain 3E'(k), M'(k) occurs at a point k on a sym- restrictions on the numbers of contact curves metry axis, and suppose that m'(k) and m'(k) which may occur for crystals of higher sym- are each one-dimensional. Then if the vector g metry. Prediction of the existence of curves of (proportional in the Hartree case to (P„', iVPq, & ))'contact of equivalent manifolds may therefore does not vanish, a curve of contactACCIDENTAL DEGENERACYmust pass be365 possible from a knowledge merely of the basis vectors for a real representation of the is just like the theory of electronic wave functions space group of the crystal, and that the normal and their energies: frequency can be plotted as a through k. This curvemodes maybelonging to aberepresentationa curvewhich is functionofWeylof wavecontactvector, and sticking togetherenergiesof semimetalof the different M'(k, ) at the eight irreducible in the field of real numbers, even two or more of these frequency bands will occur though reducible in the complex field, must all at wave vectors k where G' has multidimensional of equivalent manifoldshave the sameoffrequency.the 7 Thustypemathematicallyjustrepresentationsdescribed,or where case (b) or case points(c), as k„. the theory of normal modes and their frequencies defined above, occurs. It is a pleasure for me to express my thanks to or it may be a curve ~ Cf.ofE. Wigner,contactGott. Nachr. (1930), p. 133.of inequivalentProfessor E. Wigner, who suggested this problem. For a crystal without an inversion center, the manifolds in a planeAUGUSTof15,symmetry.1937 PHYSICALNaturallyREVIEW if VOLUMenergyE 52 separation 8E(k+x) in the neigborhood there is no such symmetryAccidentalplaneDegeneracy ininthe EnergytheBands spaceof Crystals of a point k where contact of equivalent mani- CONYERS HERRING Princeton University, Princeton, Net Jersey group, the former alternative must(Received hold.June 16, 1937) folds occurs may be expected to be of the order The circumstances are investigated under which two wave functions occurring in the Hartree or I'ock solution for a crystal can have the same reduced wave vector and the same energy, It — For a crystal whose spaceis found that coincidencegroupof the energies of waveconsistsfunctions with the same symmetryonlyproperties, of ~ as ~ +0, for all directions of x. as well as those with different symmetries, is often to be expected. Some qualitative features are derived of the way in which energy varies with wave vector near wave vectors for which degeneracy occurs. All these results, like those of the preceding paper, should be applicable of its translation group alsoplusto the frequency anspectrum ofinversion,the normal modes of vibration of a crystal.three For a crystal with an inversion center, the "N previous papers, by Bouckaert, Smoluchow- The analysis necessary to answer these ques- k' types of contact curves- - ski, and Wigner,may' and by the author,occur,' certain tionswhichis rather tedious. Despitearethis and theenergyfact separation 8E(k') at a point near a properties of the wave functions and energy that it may not be of practical significance to values of an electron moving in the periodic field bother about too fine details in an approximate most easily describedof a crystalwhenwere derived.energyThese properties wereis theory,consideredthe discussion to be given below maycurvebe of contact of equivalent manifolds may be the properties necessitated by the symmetry of of value in forming pictures of the energy band the crystal and by the reality of the Hamiltonian. structures of metals, especially of multivalent k' as a trebly periodic Thefunctiontwo questions to be discussedofin thiswavepaper ones. Invectorparticular, it is hopedinthat the completeexpected to be of the order of the distance of are: determination of energy as a function of wave (1) In the solution of Hartree's or Fock's vector by interpolation from the results of cal- the infinite reciprocalequations forlatticea crystal to what extentspace.may one culationsTheof the Wigner-Seitz-Slaterfirst type willfrombe the curve. expect to encounter accidental coincidences in facilitated and made more reliable. The results energy between two one-electron wave functions of this paper also apply, as did those of I, to the type is a simple closedwith the samecircuitwave vector? Bywhich"accidental" frequencyis spectrumdistinctof the normal modes of All kinds of contacts of equivalent manifolds coincidences are to be understood coincidences vibration of a crystal; however numerical cal- not necessitated by the symmetry and reality of culation of these frequencies has not yet ad- from the circuit obtainedthe Hamiltonian.from it by thevanced inversionas far as has the calculation of electronicexcept the ones described above are vanishingly (2) If the energies of two or more bands bands. 3 coincide at wave vector k, whether accidentally The notation to be used is the same as in I. k~ —k. The second typeor for reasons isof symmetrya simpleand reality, how mayclosedIn addition, thecircuitsymbol LM', 3P] will be improbable.intro- In particular, the occurrence of the energies of these bands be expected to vary duced to represent the subspace of Hilbert space with wave vector in the neighborhood of k? spanned together by any two linear manifolds of wave functions M' and M'. which either coincides' Bouckaert,withSmoluchowski, and theWigner, Phys. inverseRev. 50, circuit of equivalent manifolds ' isolated points of contact 58' (1936), hereafter referred to as BSW. Calculations for a simple cubic lattice have been made or can be brought intoPrecedingcoincidencepaper, hereafter referred to as I. withby M. Blackman,it Proc.byRoy. Soc.2xA159, 416 (1937).for crystals with an inversion center is vanish- times a translation of the reciprocal lattice. The ingly improbable. third type is a curve extending periodically to infinity. Now consider any energy band i, and I should like to express my gratitude to Pro- the band j next above it. For each of the eight fessor E. Wigner for his interest in this work, and distinct points k„(r= 1 to 8) of the B-Z whose to Dr. L. P. Bouckaert and Dr. R. Smoluchowski G~" contain the inversion let the numbers for some interesting discussions. ACCIDENTAL D EGEN ERAC Y 373
For crystals with an inversion center, contacts X+(k„,i), E (k„ i), of odd and' of even eigen- of equivalent manifolds M'(k), 3f'(k) may occur functions fs, ' be counted which have energies at all points k of an endless curve, or of a number E'(k,) ~& E'(k„). Now the quantity of such curves, in k-space. These contact curves cannot be destroyed or broken by any infini- tesimal change in the potential U which pre- serves the inversional symmetry. It is vanishingly is an integer, and according to whether this improbable for such curves to lie in planes of integer is odd or even the number of circuits of symmetry in the B-Z; however a contact curve the second type along which contact between the may pass through a symmetry axis at a point bands i and j occurs must be odd or even. Since where necessary degeneracy or contact of any crystal with an inversion center can be made inequivalent manifolds occurs. by an infinitesimal change in the form of U into Suppose that for a crystal with an inversion one whose space group is merely its translation center a contact of inequivalent manifolds group plus the inversion, this implies certain 3E'(k), M'(k) occurs at a point k on a sym- restrictions on the numbers of contact curves metry axis, and suppose that m'(k) and m'(k) which may occur for crystals of higher sym- are each one-dimensional. Then if the vector g metry. Prediction of the existence of curves of (proportional in the Hartree case to (P„', iVPq, & ))'contact of equivalent manifolds may therefore does not vanish, a curve of contactACCIDENTAL DEGENERACYmust pass be365 possible from a knowledge merely of the basis vectors for a real representation of the is just like the theory of electronic wave functions space group of the crystal, and that the normal and their energies: frequency can be plotted as a through k. This curvemodes maybelonging to aberepresentationa curvewhich is functionofWeylof wavecontactvector, and sticking togetherenergiesof semimetalof the different M'(k, ) at the eight irreducible in the field of real numbers, even two or more of these frequency bands will occur though reducible in the complex field, must all at wave vectors k where G' has multidimensional of equivalent manifoldshave the sameoffrequency.the 7 Thustypemathematicallyjustrepresentationsdescribed,or where case (b) or case points(c), as k„. the theory of normal modes and their frequencies defined above, occurs. Bandstructure HgTe It is a pleasure for me to express my thanks to or it may be a curve ~ Cf.ofE. Wigner,contactGott. Nachr. (1930), p. 133.of inequivalentProfessor E. Wigner, who suggested this problem. For a crystal without an inversion center, the B.A Bernevig, T.L. Hughes, S.C. Zhang, Science 314, 1757 (2006) manifolds in a planeAUGUSTof15,symmetry.1937 PHYSICALNaturallyREVIEW if VOLUMenergyE 52 separation 8E(k+x) in the neigborhood there is no such symmetryAccidentalplaneDegeneracy ininthe EnergytheBands spaceof Crystals of a point k where contact Eof 4.0nmequivalent 6.2 mani-nm 7.0 nm CONYERS HERRING Princeton University, Princeton, Net Jersey group, the former alternative must(Received hold.June 16, 1937) folds occurs may be expected to be of the order The circumstances are investigated under which two wave functions occurring in the Hartree k or I'ock solution for a crystal can have the same reduced wave vector and the same energy, It — For a crystal whose spaceis found that coincidencegroupof the energies of waveconsistsfunctions with the same symmetryonlyproperties, of ~ as ~ +0, for all directions of x. as well as those with different symmetries, is often to be expected. Some qualitative features are derived of the way in which energy varies with wave vector near wave vectors for which degeneracy occurs. All these results, like those of the preceding paper, should be applicable of its translation group alsoplusto the frequency anspectrum ofinversion,the normal modes of vibration of a crystal.three For a crystal with an inversion center, the E1 H1 "N previous papers, by Bouckaert, Smoluchow- The analysis necessary to answer these ques- k' - ' normal inverted types of contact curves- ski, and Wigner,mayand by the author,occur,' certain tionswhichis rather tedious. Despitearethis and theenergyfact separation 8E(k') at a point near a properties of the wave functions and energy that it may not be of practical significance to gap gap values of an electron moving in the periodic field bother about too fine details in an approximate ~ E1 H1 most easily describedof a crystalwhenwere derived.energyThese properties wereis theory,consideredthe discussion to be given below maycurvebe of contactH =of equivalentv~ kmanifolds may be the properties necessitated by the symmetry of of value in forming pictures of the energy band the crystal and by the reality of the Hamiltonian. structures of metals, especially of multivalent k' as a trebly periodic Thefunctiontwo questions to be discussedofin thiswavepaper ones. Invectorparticular, it is hopedinthat the completeexpected to be of the order of the distance of are: determination of energy as a function of wave · (1) In the solution of Hartree's or Fock's vector by interpolation from the results of cal- the infinite reciprocalequations forlatticea crystal to what extentspace.may one culationsTheof the Wigner-Seitz-Slaterfirst type willfrombe the curve. expect to encounter accidental coincidences in facilitated and made more reliable. The results energy between two one-electron wave functions of this paper also apply, as did thoseAof I,two-componentto the spinor in three type is a simple closedwith the samecircuitwave vector? Bywhich"accidental" frequencyis spectrumdistinctof the normal modes of All kinds of contacts of equivalent manifolds coincidences are to be understood coincidences vibration of a crystal; however numerical cal- not necessitated by the symmetry and reality of culation of these frequencies has not yet ad- from the circuit obtainedthe Hamiltonian.from it by thevanced inversionas far as has the calculation of electronicexcept the ones described above are vanishingly (2) If the energies of two or more bands bands. 3 dimensions: “half” of a Dirac fermion. coincide at wave vector k, whether accidentally The notation to be used is the same as in I. k~ —k. The second typeor for reasons isof symmetrya simpleand reality, how mayclosedIn addition, thecircuitsymbol LM', 3P] will be improbable.intro- In particular, the occurrence of the energies of these bands be expected to vary duced to represent the subspace of Hilbert space with wave vector in the neighborhood of k? spanned together by any two linear manifolds of wave functions M' and M'. which either coincides' Bouckaert,withSmoluchowski, and theWigner, Phys. inverseRev. 50, circuitWeylisolated fermionspoints of havecontact aof equivalentchiralitymanifolds and ' 58' (1936), hereafter referred to as BSW. Calculations for a simple cubic lattice have been made or can be brought intoPrecedingcoincidencepaper, hereafter referred to as I. withby M. Blackman,it Proc.byRoy. Soc.2xA159, 416 (1937).for crystals with an inversion center is vanish- times a translation of the(Diracreciprocal semimetalslattice. also exist)The ingly improbable.must be massless third type is a curve extending periodically to infinity. Now consider any energy band i, and I should like to express my gratitude to Pro- the band j next above it. For each of the eight fessor E. Wigner for his interest in this work, and distinct points k„(r= 1 to 8) of the B-Z whose to Dr. L. P. Bouckaert and Dr. R. Smoluchowski G~" contain the inversion let the numbers for some interesting discussions. ACCIDENTAL D EGEN ERAC Y 373
For crystals with an inversion center, contacts X+(k„,i), E (k„ i), of odd and' of even eigen- of equivalent manifolds M'(k), 3f'(k) may occur functions fs, ' be counted which have energies at all points k of an endless curve, or of a number E'(k,) ~& E'(k„). Now the quantity of such curves, in k-space. These contact curves cannot be destroyed or broken by any infini- tesimal change in the potential U which pre- serves the inversional symmetry. It is vanishingly is an integer, and according to whether this improbable for such curves to lie in planes of integer is odd or even the number of circuits of symmetry in the B-Z; however a contact curve the second type along which contact between the may pass through a symmetry axis at a point bands i and j occurs must be odd or even. Since where necessary degeneracy or contact of any crystal with an inversion center can be made inequivalent manifolds occurs. by an infinitesimal change in the form of U into Suppose that for a crystal with an inversion one whose space group is merely its translation center a contact of inequivalent manifolds group plus the inversion, this implies certain 3E'(k), M'(k) occurs at a point k on a sym- restrictions on the numbers of contact curves metry axis, and suppose that m'(k) and m'(k) which may occur for crystals of higher sym- are each one-dimensional. Then if the vector g metry. Prediction of the existence of curves of (proportional in the Hartree case to (P„', iVPq, & ))'contact of equivalent manifolds may therefore does not vanish, a curve of contactACCIDENTAL DEGENERACYmust pass be365 possible from a knowledge merely of the basis vectors for a real representation of the is just like the theory of electronic wave functions space group of the crystal, and that the normal and their energies: frequency can be plotted as a through k. This curvemodes maybelonging to aberepresentationa curvewhich is functionofWeylof wavecontactvector, and sticking togetherenergiesof semimetalof the different M'(k, ) at the eight irreducible in the field of real numbers, even two or more of these frequency bands will occur though reducible in the complex field, must all at wave vectors k where G' has multidimensional of equivalent manifoldshave the sameoffrequency.the 7 Thustypemathematicallyjustrepresentationsdescribed,or where case (b) or case points(c), as k„. the theory of normal modes and their frequencies defined above, occurs. It is a pleasure for me to express my thanks to or it may be a curve ~ Cf.ofE. Wigner,contactGott. Nachr. (1930), p. 133.of inequivalentProfessor E. Wigner, who suggested this problem. For a crystal without an inversion center, the manifolds in a planeAUGUSTof15,symmetry.1937 PHYSICALNaturallyREVIEW if VOLUMenergyE 52 separation 8E(k+x) in the neigborhood there is no such symmetryAccidentalplaneDegeneracy ininthe EnergytheBands spaceof Crystals of a point k where contact of equivalent mani- CONYERS HERRING Princeton University, Princeton, Net Jersey group, the former alternative must(Received hold.June 16, 1937) folds occurs may be expected to be of the order The circumstances are investigated under which two wave functions occurring in the Hartree or I'ock solution for a crystal can have the same reduced wave vector and the same energy, It — For a crystal whose spaceis found that coincidencegroupof the energies of waveconsistsfunctions with the same symmetryonlyproperties, of ~ as ~ +0, for all directions of x. as well as those with different symmetries, is often to be expected. Some qualitative features are derived of the way in which energy varies with wave vector near wave vectors for which degeneracy occurs. All these results, like those of the preceding paper, should be applicable of its translation group alsoplusto the frequency anspectrum ofinversion,the normal modes of vibration of a crystal.three For a crystal with an inversion center, the "N previous papers, by Bouckaert, Smoluchow- The analysis necessary to answer these ques- k' types of contact curves- - ski, and Wigner,may' and by the author,occur,' certain tionswhichis rather tedious. Despitearethis and theenergyfact separation 8E(k') at a point near a properties of the wave functions and energy that it may not be of practical significance to values of an electron moving in the periodic field bother about too fine details in an approximate most easily describedof a crystalwhenwere derived.energyThese properties wereis theory,consideredthe discussion to be given below maycurvebe of contact of equivalent manifolds may be the properties necessitated by the symmetry of of value in forming pictures of the energy band the crystal and by the reality of the Hamiltonian. structures of metals, especially of multivalent k' as a trebly periodic Thefunctiontwo questions to be discussedofin thiswavepaper ones. Invectorparticular, it is hopedinthat the completeexpected to be of the order of the distance of are: determinationEitherof energy as a function inversionof wave or time-reversal (1) In the solution of Hartree's or Fock's vector by interpolation from the results of cal- the infinite reciprocalequations forlatticea crystal to what extentspace.may one culationsTheof the Wigner-Seitz-Slaterfirst type willfrombe the curve. expect to encounter accidental coincidences in facilitated and made more reliable. The results energy between two one-electron wave functions of this paper also apply, as did those of I, to the type is a simple closedwith the samecircuitwave vector? Bywhich"accidental" frequencyis spectrumdistinctof(orthe normal both)modes of All kinds mustof becontacts brokenof equivalent manifolds coincidences are to be understood coincidences vibration of a crystal; however numerical cal- not necessitated by the symmetry and reality of culation of these frequencies has not yet ad- from the circuit obtainedthe Hamiltonian.from it by thevanced inversionas far as has the calculation of electronicexcept the ones described above are vanishingly (2) If the energies of two or more bands bands. 3 coincide at wave vector k, whether accidentally The notation to be used is the same as in I. k~ —k. The second typeor for reasons isof symmetrya simpleand reality, how mayclosedIn addition, thecircuitsymbol LM', 3P] will be improbable.intro- In particular, the occurrence of the energies of these bands be expected to vary duced to represent the subspace of Hilbert space with wave vector in the neighborhood of k? spanned together by any two linear manifolds of wave functions M' and M'. which either coincides' Bouckaert,withSmoluchowski, and theWigner, Phys. inverseRev. 50, circuit of equivalent manifolds ' isolated points of contact 58' (1936), hereafter referred to as BSW. Calculations for a simple cubic lattice have been made or can be brought intoPrecedingcoincidencepaper, hereafter referred to as I. withby M. Blackman,it Proc.byRoy. Soc.2xA159, 416 (1937).for crystals with an inversion center is vanish- times a translation of the reciprocal lattice. The ingly improbable. third type is a curve extending periodically to infinity. Now consider any energy band i, and I should like to express my gratitude to Pro- the band j next above it. For each of the eight fessor E. Wigner for his interest in this work, and distinct points k„(r= 1 to 8) of the B-Z whose to Dr. L. P. Bouckaert and Dr. R. Smoluchowski G~" contain the inversion let the numbers for some interesting discussions. ACCIDENTAL D EGEN ERAC Y 373
For crystals with an inversion center, contacts X+(k„,i), E (k„ i), of odd and' of even eigen- of equivalent manifolds M'(k), 3f'(k) may occur functions fs, ' be counted which have energies at all points k of an endless curve, or of a number E'(k,) ~& E'(k„). Now the quantity of such curves, in k-space. These contact curves cannot be destroyed or broken by any infini- tesimal change in the potential U which pre- serves the inversional symmetry. It is vanishingly is an integer, and according to whether this improbable for such curves to lie in planes of integer is odd or even the number of circuits of symmetry in the B-Z; however a contact curve the second type along which contact between the may pass through a symmetry axis at a point bands i and j occurs must be odd or even. Since where necessary degeneracy or contact of any crystal with an inversion center can be made inequivalent manifolds occurs. by an infinitesimal change in the form of U into Suppose that for a crystal with an inversion one whose space group is merely its translation center a contact of inequivalent manifolds group plus the inversion, this implies certain 3E'(k), M'(k) occurs at a point k on a sym- restrictions on the numbers of contact curves metry axis, and suppose that m'(k) and m'(k) which may occur for crystals of higher sym- are each one-dimensional. Then if the vector g metry. Prediction of the existence of curves of (proportional in the Hartree case to (P„', iVPq, & ))'contact of equivalent manifolds may therefore does not vanish, a curve of contactACCIDENTAL DEGENERACYmust pass be365 possible from a knowledge merely of the basis vectors for a real representation of the is just like the theory of electronic wave functions space group of the crystal, and that the normal and their energies: frequency can be plotted as a through k. This curvemodes maybelonging to aberepresentationa curvewhich is functionofWeylof wavecontactvector, and sticking togetherenergiesof semimetalof the different M'(k, ) at the eight irreducible in the field of real numbers, even two or more of these frequency bands will occur though reducible in the complex field, must all at wave vectors k where G' has multidimensional of equivalent manifoldshave the sameoffrequency.the 7 Thustypemathematicallyjustrepresentationsdescribed,or where case (b) or case points(c), as k„. the theory of normal modes and their frequencies defined above, occurs. It is a pleasure for me to express my thanks to or it may be a curve ~ Cf.ofE. Wigner,contactGott. Nachr. (1930), p. 133.of inequivalentProfessor E. Wigner, who suggested this problem. For a crystal without an inversion center, the manifolds in a planeAUGUSTof15,symmetry.1937 PHYSICALNaturallyREVIEW if VOLUMenergyE 52 separation 8E(k+x) in the neigborhood there is no such symmetryAccidentalplaneDegeneracy ininthe EnergytheBands spaceof Crystals of a point k where contact of equivalent mani- CONYERS HERRING Princeton University, Princeton, Net Jersey group, the former alternative must(Received hold.June 16, 1937) folds occurs may be expected to be of the order The circumstances are investigated under which two wave functions occurring in the Hartree or I'ockS.solution Murakami,for 2007 a crystal can have the same reduced wave vector and the same energy, It — For a crystal whose spaceis found that coincidencegroupof the energies of waveconsistsfunctions with the same symmetryonlyproperties, of ~ as ~ +0, for all directions of x. as wellX.as thoseWanwith different et alsymmetries,, 2011is often to be expected. Some qualitative features are derived of the way in which energy varies with wave vector near wave vectors for which degeneracy occurs. All these results, like those of the preceding paper, should be applicable of its translation group alsoplustoA.the frequencyBurkov+LB,anspectrum ofinversion,the normal 2011modes of vibration of a crystal.three For a crystal with an inversion center, the "N previous papers, by Bouckaert, Smoluchow- The analysis necessary to answer these ques- k' types of contact curves- - ski, and Wigner,may' and by the author,occur,' certain tionswhichis rather tedious. Despitearethis and theenergyfact separation 8E(k') at a point near a properties of the wave functions and that it not be of significance energy may practical to 1.0 values of an electron moving in the periodic field bother about too fine details in an approximate most easily describedof a crystalwhenwere derived.energyThese properties wereis theory,consideredthe discussion to be given below maycurvebe of contact of equivalent manifolds may be the properties necessitated by the symmetry of of value in forming pictures of the energy band the crystal and by the reality of the Hamiltonian. structures of metals, especially of multivalent k' as a trebly periodic Thefunctiontwo questions to be discussedofin thiswavepaper ones. Invectorparticular, it is hopedinthat the completeexpected to be of the order of the distance of are: determination of energy as a function of wave 0.5 Weyl points are (1) In the solution of Hartree's or Fock's vector by interpolation from the results of cal- the infinite reciprocalequations forlatticea crystal to what extentspace.may one culationsTheof the Wigner-Seitz-Slaterfirst type willfrombe the curve. expect to encounter accidental coincidences in facilitated and made more reliable. The results “monopoles” of energy between two one-electron wave functions of this paper also apply, as did those of I, to the with "accidental" type is a simple closedthe samecircuitwave vector? Bywhich frequencyis spectrumdistinctof the normal modes of All kinds0.0 of contacts of equivalent manifolds coincidences are to be understood coincidences vibration of a crystal; however numerical cal- not necessitated by the symmetry and reality of culation of these frequencies has not yet ad- Berry curvature: from the circuit obtainedthe Hamiltonian.from it by thevanced inversionas far as has the calculation of electronicexcept the ones described above are vanishingly (2) If the energies of two or more bands bands. 3 coincide at wave vector k, whether accidentally The notation to be used is the same as in I. -0.5 topology in k- k~ —k. The second typeor for reasons isof symmetrya simpleand reality, how mayclosedIn addition, thecircuitsymbol LM', 3P] will be improbable.intro- In particular, the occurrence of the energies of these bands be expected to vary duced to represent the subspace of Hilbert space with wave vector in the neighborhood of k? spanned together by any two linear manifolds of wave functions M' and M'. which either coincides' Bouckaert,withSmoluchowski, and theWigner, Phys. inverseRev. 50, circuit of equivalent space!manifolds ' isolated points of contact 58' (1936), hereafter referred to as BSW. Calculations for a simple cubic lattice have been made Preceding paper, hereafter referred to as I. by M. Blackman, Proc. Roy. Soc. A159, 416 (1937). -1.0 or can be brought into coincidence with it by 2x for crystals-1.0 with-0.5 an0.0 inversion0.5 1.0 center is vanish- times a translation of the reciprocal lattice. The ingly improbable.Review: A.M. Turner, A. Vishwanath, arXiv:1301.0330 third type is a curve extending periodically to infinity. Now consider any energy band i, and I should like to express my gratitude to Pro- the band j next above it. For each of the eight fessor E. Wigner for his interest in this work, and distinct points k„(r= 1 to 8) of the B-Z whose to Dr. L. P. Bouckaert and Dr. R. Smoluchowski G~" contain the inversion let the numbers for some interesting discussions. ACCIDENTAL D EGEN ERAC Y 373
For crystals with an inversion center, contacts X+(k„,i), E (k„ i), of odd and' of even eigen- of equivalent manifolds M'(k), 3f'(k) may occur functions fs, ' be counted which have energies at all points k of an endless curve, or of a number E'(k,) ~& E'(k„). Now the quantity of such curves, in k-space. These contact curves cannot be destroyed or broken by any infini- tesimal change in the potential U which pre- serves the inversional symmetry. It is vanishingly is an integer, and according to whether this improbable for such curves to lie in planes of integer is odd or even the number of circuits of symmetry in the B-Z; however a contact curve the second type along which contact between the may pass through a symmetry axis at a point bands i and j occurs must be odd or even. Since where necessary degeneracy or contact of any crystal with an inversion center can be made inequivalent manifolds occurs. by an infinitesimal change in the form of U into Suppose that for a crystal with an inversion one whose space group is merely its translation center a contact of inequivalent manifolds group plus the inversion, this implies certain 3E'(k), M'(k) occurs at a point k on a sym- restrictions on the numbers of contact curves metry axis, and suppose that m'(k) and m'(k) which may occur for crystals of higher sym- are each one-dimensional. Then if the vector g metry. Prediction of the existence of curves of (proportional in the Hartree case to (P„', iVPq, & ))'contact of equivalent manifolds may therefore does not vanish, a curve of contactACCIDENTAL DEGENERACYmust pass be365 possible from a knowledge merely of the basis vectors for a real representation of the is just like the theory of electronic wave functions space group of the crystal, and that the normal and their energies: frequency can be plotted as a through k. This curvemodes maybelonging to aberepresentationa curvewhich is functionofWeylof wavecontactvector, and sticking togetherenergiesof semimetalof the different M'(k, ) at the eight irreducible in the field of real numbers, even two or more of these frequency bands will occur though reducible in the complex field, must all at wave vectors k where G' has multidimensional of equivalent manifoldshave the sameoffrequency.the 7 Thustypemathematicallyjustrepresentationsdescribed,or where case (b) or case points(c), as k„. the theory of normal modes and their frequencies defined above, occurs. It is a pleasure for me to express my thanks to or it may be a curve ~ Cf.ofE. Wigner,contactGott. Nachr. (1930), p. 133.of inequivalentProfessor E. Wigner, who suggested this problem. For a crystal without an inversion center, the manifolds in a planeAUGUSTof15,symmetry.1937 PHYSICALNaturallyREVIEW if VOLUMenergyE 52 separation 8E(k+x) in the neigborhood there is no such symmetryAccidentalplaneDegeneracy ininthe EnergytheBands spaceof Crystals of a point k where contact of equivalent mani- CONYERS HERRING Princeton University, Princeton, Net Jersey group, the former alternative must(Received hold.June 16, 1937) folds occurs may be expected to be of the order The circumstances are investigated under which two wave functions occurring in the Hartree or I'ockS.solution Murakami,for 2007 a crystal can have the same reduced wave vector and the same energy, It — For a crystal whose spaceis found that coincidencegroupof the energies of waveconsistsfunctions with the same symmetryonlyproperties, of ~ as ~ +0, for all directions of x. as wellX.as thoseWanwith different et alsymmetries,, 2011is often to be expected. Some qualitative features are derived of the way in which energy varies with wave vector near wave vectors for which degeneracy occurs. All these results, like those of the preceding paper, should be applicable of its translation group alsoplustoA.the frequencyBurkov+LB,anspectrum ofinversion,the normal 2011modes of vibration of a crystal.three For a crystal with an inversion center, the "N previous papers, by Bouckaert, Smoluchow- The analysis necessary to answer these ques- k' types of contact curves- - ski, and Wigner,may' and by the author,occur,' certain tionswhichis rather tedious. Despitearethis and theenergyfact separation 8E(k') at a point near a properties of the wave functions and energy that it may not be of practical significance to • Striking properties: values of an electron moving in the periodic field bother about too fine details in an approximate most easily describedof a crystalwhenwere derived.energyThese properties wereis theory,consideredthe discussion to be given below maycurvebe of contact of equivalent manifolds may be the properties necessitated by the symmetry of of value in forming pictures of the energy band the crystal and by the reality of the Hamiltonian. structures of metals, especially of multivalent k' as a trebly periodic Thefunctiontwo questions to be discussedofin thiswavepaper ones. Invectorparticular, it is hopedinthat the completeexpected• Anomalousto be of the order Hallof effectthe distance of are: determination of energy as a function of wave (1) In the solution of Hartree's or Fock's vector by interpolation from the results of cal- the infinite reciprocalequations forlatticea crystal to what extentspace.may one culationsTheof the Wigner-Seitz-Slaterfirst type willfrombe the curve. expect to encounter accidental coincidences in facilitated and made more reliable. The results energy between two one-electron wave functions of this paper also apply, as did those of I, to the • ABJ “anomaly”: strong type is a simple closedwith the samecircuitwave vector? Bywhich"accidental" frequencyis spectrumdistinctof the normal modes of All kinds of contacts of equivalent manifolds coincidences are to be understood coincidences vibration of a crystal; however numerical cal- not necessitated by the symmetry and reality of culation of these frequencies has not yet ad- from the circuit obtainedthe Hamiltonian.from it by thevanced inversionas far as has the calculation of electronicexcept the ones described above are vanishingly (2) If the energies of two or more bands bands. 3 negative MR for I ∥ B coincide at wave vector k, whether accidentally The notation to be used is the same as in I. k~ —k. The second typeor for reasons isof symmetrya simpleand reality, how mayclosedIn addition, thecircuitsymbol LM', 3P] will be improbable.intro- In particular, the occurrence of the energies of these bands be expected to vary duced to represent the subspace of Hilbert space with wave vector in the neighborhood of k? spanned together by any two linear manifolds ' of wave functions M' and M'. • Surface Fermi arcs which either coincidesBouckaert,withSmoluchowski, and theWigner, Phys. inverseRev. 50, circuit of equivalent manifolds ' isolated points of contact 58' (1936), hereafter referred to as BSW. Calculations for a simple cubic lattice have been made or can be brought intoPrecedingcoincidencepaper, hereafter referred to as I. withby M. Blackman,it Proc.byRoy. Soc.2xA159, 416 (1937).for crystals with an inversion center is vanish- times a translation of the reciprocal lattice. The ingly improbable.Review: A.M. Turner, A. Vishwanath, arXiv:1301.0330 third type is a curve extending periodically to infinity. Now consider any energy band i, and I should like to express my gratitude to Pro- the band j next above it. For each of the eight fessor E. Wigner for his interest in this work, and distinct points k„(r= 1 to 8) of the B-Z whose to Dr. L. P. Bouckaert and Dr. R. Smoluchowski G~" contain the inversion let the numbers for some interesting discussions. LETTER RESEARCH
a 4 324 b 8 Figure 2 | Magnetic field dependence of the AHE 200 K in Mn3Sn. a, Field dependence of the Hall resistivity ρH (left axis) and the longitudinal Mn Sn Mn Sn H 3 100 K 3 resistivity ρ (right axis) at 300 K in the magnetic 2 322 4 300 K field B [2110] with the electric current I [0110]. T = 300 K 350 K b, Field dependence of the Hall resistivity ρH at (
μΩ 400 K various temperatures in B [0110] with I [0001]. cm) cm)
0 320 cm) 0 c, d, The Hall conductivity σH versus B measured μΩ μΩ
( ( , H H in B [2110] [0110] and [0001] obtained at 300 K (c) and 100 K (d). e, Magnetization dependence I || [0110] B || [0110] –2 318 –4 of ρH at 300 K. f, Field dependence of AF = − − at 300 K. The arrows in ρH ρH R0sBRμ0M I || [0001] B || [2110] the hexagon at lower left in a and b indicate the field and current directions in the hexagonal –4 316 –8 Mn–1 –0.53 0Sn 0.5 1 magnetic–1 –0.5 00.5 1 Weyllattice of Mn3Sn. NATURE PHYSICS DOI: 10.1038/NPHYS4181 B (T) B (T) LETTERS cd40 150 102 B || [0110] L10-FePt (ref. 7) L1 -FePd (ref. 7) Mn Sn I || [0001] 0 100 3 L10-MnGa (ref. 7) 101 D022-Mn2Ga (ref. 7) 20 Co/Ni films (ref. 7) T = 100 K B || [2110] Nd2Mo2O7 (ref. 30)
) ) 50 B || [0001] 100
–1 –1 I || [0110] Fe (ref. 37) )
I || [0110] − 1 Co (ref. 37) cm cm 0 0−1 Fe3O4 (B < 0.8 T) (ref. 38) –1 –1 10
Ω Ω MnGe (140 K, B > 2 T) (ref. 39) | ( µ V K ji ( (
| S MnGe (100 K, B > 5 T) (ref. 39) H H B || [0110] B || [0001] MnGe (20 K, B < 14 T) (ref. 39) I || [0001] –5010−2 Mn3Sn I || [0110] Pt/Fe multilayer N = 1 (B < 5 T) (ref. 40) –20 Pt/Fe multilayer N = 4 (B < 5 T) (ref. 40) Pt/Fe multilayer N = 7 (B < 5 T) (ref. 40) T = 300 K –10010−3 Pt/Fe multilayer N = 9 (B < 5 T) (ref. 40) B || [2110], I || [0110] Mn3.06Sn0.94 Mn Sn –40 –15010−4 3.09 0.91 –1 –0.5 0 0.5 1 –110−4 –0.510−3 10−002 10−1 .5 100 1 101 B (T) B (T)M (T) ef6 6 Figure 4 | Magnetization dependence of the spontaneous Nernst eect for ferromagnetic metals and Mn3Sn. Full logarithmic plot of the anomalous B || [0110], B || [0110], I || [0001] anomalousI || [0001]Hall effect,Nernst signal Sji versus the magnetizationanomalousM for a variety Nernst of ferromagnetic effect, metals and Mn 3Sn measured at various temperatures and fields (Methods). | | Mn SnIt shows the general trend for4 ferromagnets that Sji increases withMnMSn. The shaded region indicates the linear relation Sji Qs µ0M, with Qs ranging 4 3 | | 3 | |=| | | | from 0.05µVK 1 T 1 to 1µVK 1 T 1. The Nernst signal data points obtained at various temperatures for Sample 1 (Mn Sn ) for B 0110 and Nakatsuji et al, 2015 ⇠ ⇠ Ikhlas et al, 2017 3.06 0.94 k[ ¯ ] for Sample 2 (Mn Sn ) (blue filled circles) for B 0110 (green filled circle) do not follow the relation, and reach almost the same value as the largest T = 300 K 3.09 0.91 B || [2110] k[ ¯ ] T = 300 K 2 B || [2110] among ferromagnetic metals2 with three orders of magnitude smaller M. I || [0110] I || [0110] cm) cm) 11,32 Received 6 June 2016; accepted 23 May 2017; 0 several Weyl points nearby0 the Fermi level . Our study further μΩ μΩ ( published online 24 July 2017
highlights the complementary roles of anomalous Hall conductivity (
B || [0001] H
B || [0001] AF H ↵
ji and transverse thermoelectric conductivity ji in revealing the I || [0110] –2 I || [0110]Effects comparabletopological character of band–2 structure. to or exceedingReferences Finally, from the viewpoint of application for thermoelectric 1. Smith, A. W. The transverse thermomagnetic e ect in nickel and cobalt. power generation, ANE could be useful as it facilitates the Phys. Rev. 33, 295–306 (1911). –4 fabrication of a module–4 structurally much simpler than the 2. Kondorskii, E. I. & Vasileva, R. Degree of localization of magnetic electron and ferromagneticconventional metals one using the Seebeck at e roomect6. The orthogonal temperaturethe Nernst–Ettingshausen e ect in ferromagnetic metals. Sov. Phys. JEPT 18, orientation of the voltage output to the thermal heat flow (Fig. 1a) 277–278 (1964). –6 –6 3. Lee, W.-L., Watauchi, S., Miller, V. L., Cava, R. J. & Ong, N. P. Anomalous Hall –20 –10 0102enables0 a lateral series connection–1 of –0.5 a single kind00 of ferromagnet.5 1 heat current and Nernst e ect in the CuCr2Se4 x Brx ferromagnet. Phys. Rev. B (T) M (m B per f.u.) with alternating magnetization direction (Fig. 1a inset). This Lett. 93, 226601 (2004). simplifies a thermopile structure to e ciently cover the surface of 4. Miyasato, T. et al. Crossover behavior of the anomalous Hall e ect and a heat source (Fig. 1a inset),2 in comparison with the conventional anomalous Nernst e ect in itinerant ferromagnets. Phys.10 Rev.,12 Lett. 99, Correspondingly, the Hall conductivity,thermoelectric σH = − module ρH/ρ using, for thein-plane Seebeck e in-planeect, which anisotropy consists energy086602 (2007). up to the fourth-order term , which is fields along both [2110] and [0110] ofshows a pillar a structure large jump of alternating and narrow p- and n-typeconsistent semiconductors. with the observed5. Pu, Y., Chiba, small D., Matsukura, coercivity. F., Ohno, This H. & further Shi, J. Mott indicates relation for anomalous that Hall and Nernst e ects in Ga1 x Mnx As ferromagnetic semiconductors. hysteresis (Fig. 2c, d). For instance, withTo increase B || [0110] power, σ density,H has large a thermopile values should by rotating ideally cover the the net ferromagneticPhys. Rev. Lett. 101, moment,117208 (2008). one may switch the staggered −1 −1 entire surface of a heat source,−1 and− therefore,1 a micro-fabricated 6. Sakuraba, Y. et al. Anomalous Nernst e ect in L1100-FePt/MnGa,12 thermopiles for near zero field, ~20 Ω cm at 300thermopile K and arraynearly has 100 to be Ω arranged cm as densely at asmoment possible. However,direction ofnew the thermoelectric triangular applications. spin Appl.structure Phys. Express 6,.033003 This (2013). switch 100 K. This is again quite large for anas longAFM as and a ferromagnet comparable is used, to theirthose inherent should stray be fields the may origin7. of Hasegawa, the sign K. et change al. Material of dependence the Hall of anomalouseffect, as Nernst we e discussect in values found in ferromagnetic metalsperturb3,17. On magnetization the other hand, direction the of Hall neighbouring below. modules, On heating, and thisperpendicularly ferromagnetic magnetized component ordered-alloy thin vanishes films. Appl. at Phys. the Lett. Néel106, limit the integration density. 252405 (2015). conductivity for B || [0001] (c axis) shows no hysteresis but only a temperature of 430 8.K, Xiao,above D., Yao,which Y., Fang, the Z. &hysteresis Niu, Q. Berry-phase disappears e ect in anomalous in both the Our discovery of a new class of material that produces almost thermoelectric transport. Phys. Rev. Lett. 97, 026603 (2006). linear field dependence. no stray fields but exhibits a large ANE isT and highly B importantdependence9. of Xiao, the D., magnetization Chang, M.-C. & Niu, Q. (Fig. Berry phase3a and e ects its on inset). electronic properties. The magnetization curve M(B) showsfor the anisotropic application towardhysteresis thermoelectric similar powerTo generation, reveal the and temperatureRev. Mod. Phys.evolution82, 1959–2007 of the (2010). spontaneous component 10. Nakatsuji, S., Kiyohara, N. & Higo, T. Large anomalous Hall e ect in a to that found for the Hall effect. Forit example, should allow M us versus to design B || a[0110] thermopile at withof the a muchAHE, denser both the zero-field Hall resistivity ρH(B = 0) and the zero- integration of thermoelectric modules to e ciently cover a heat non-collinear antiferromagnet at room temperature. Nature 527, temperatures between 100 K and 400 K shows a clear hysteresis, indi- field longitudinal resistivity212–215 (2015).ρ(B = 0) were measured after cooling sam- source than the ferromagnetic counterparts. While the observed 11. Yang, H. et al. Topological Weyl semimetals in the chiral antiferromagnetic cating that a weak ferromagnetic momentvalues in (4–7 this workmμBwould per formula be still far unit from theples size in necessary a magnetic for field of BFC = 7 T from 400 K down to 5 K and materials Mn3Ge and Mn3Sn. New J. Phys. 19, 015008 (2017). (f.u.)) changes its direction with coercivityapplication, of only our a study few indicateshundred that oersted the magnetic subsequently Weyl metals such setting12. B Bauer, to 0 G. at E. 5 W., K Saitoh, (Methods). E. & van Wees, Figure B. J. Spin 4a caloritronics. shows theNat. Mater.tem-11, as Mn3Sn should be particularly useful to obtain a larger ANE by 391–399 (2012). (Fig. 3a). Whereas the in-plane M is almost isotropic and has a narrow perature dependence13. Uchida, K.ofet al the. Observation zero-field of the spin Seebeck Hall e ect. conductivityNature 455, enhancing the Berry curvature at EF. Further studies to develop the 2 hysteresis, the magnetization shows technologyonly a linear for application dependence such on as thin-film B for growthσH(B and= 0) coercivity = − ρH(B =778–781 0)/ρ (2008).(B = 0) obtained after the above field- 14. Slachter, A., Bakker, F. L., Adam, J.-P. & van Wees, B. J. Thermally driven B || [0001] at all the temperatures measuredcontrol of suchbetween magnets 100 will K be and important 450 K to buildcooling a thermoelectric (FC) procedurespin using injection three from a different ferromagnet intoconfigurations a non-magnetic metal. of theNat. Phys.mag-6, power generator. (Fig. 3b). The similar anisotropic and hysteretic behaviours found in netic field (BFC) and electric879–882 (2010). current (I) directions. Here, σzx stands for both ρ (B) and M(B) indicate that the existence of the small and soft the Hall conductivity15. Huang,obtained S. Y., Wang, after W. G.,the Lee, FC S. F., procedure Kwo, J. & Chien, in C. L.B Intrinsic || [0110] H Methods spin-dependent thermal transport. Phys. Rev. Lett. 107, 216604FC (2011). ferromagnetic component allows us to switch the sign of the Hall effect. with I || [0001], and16. σyz Nagaosa, for B N.,FC Sinova, || [2110] J., Onoda, and S., I MacDonald, || [0110] A.. Both H. & Ong, show N. P. Anomalous large Methods, including statements of data availability and any Hall e ect. Rev. Mod. Phys. 82, 1539–1592 (2010). −1 Indeed, previous neutron diffractionassociated measurements accession codes and and theoretical references, are availablevalues at in thelow temperatures,17. Bruno, P. in andHandbook in ofparticular, Magnetism and | Advancedσzx| exceeds Magnetic Materials100 ΩVol. 1 −1 analyses clarified that the inverse onlinetriangular version spin of this structure paper. has no cm at T < 80 K. Both(eds |σ Kronmuller,zx| and |σ H.yz &| Parkins,decrease S.) Ch. on 22, heating 540–558 (John but Wiley, still 2007). retain
NATURE PHYSICS | VOL 13 | NOVEMBER 2017 | www.nature.com/naturephysics 12 NOVEMBER 2015 | VOL 527 | NATURE | 2131089 © 2015 Macmillan Publishers© 2017 Macmillan Limited. Publishers All rights Limited, reserved part of Springer Nature. All rights reserved. Topological band theory
• Lots more topology if you include all symmetries and atomic structure: “TCIs”
ARTICLE doi:10.1038/nature23268 ARTICLE DOI: 10.1038/s41467-017-00133-2 OPEN Symmetry-based indicators of band topology in the 230 space groups Topological quantum chemistry Barry Bradlyn1*, L. Elcoro2*, Jennifer Cano1*, M. G. Vergniory 3,4,5*, Zhijun Wang 6*, C. Felser7, M. I. Aroyo2 & B. Andrei Bernevig3,6,8,9 Hoi Chun Po1,2, Ashvin Vishwanath1,2 & Haruki Watanabe3
Since the discovery of topological insulators and semimetals, there has been much research into predicting and The interplay between symmetry and topology leads to a rich variety of electronic topological experimentally discovering distinct classes of these materials, in which the topology of electronic states leads to robust phases, protecting states such as the topological insulators and Dirac semimetals. Previous surface states and electromagnetic responses. This apparent success, however, masks a fundamental shortcoming: results, like the Fu-Kane parity criterion for inversion-symmetric topological insulators, topological insulators represent only a few hundred of the 200,000 stoichiometric compounds in material databases. demonstrate that symmetry labels can sometimes unambiguously indicate underlying band However, it is unclear whether this low number is indicative of the esoteric nature of topological insulators or of a topology. Here we develop a systematic approach to expose all such symmetry-based fundamental problem with the current approaches to finding them. Here we propose a complete electronic band theory, indicators of band topology in all the 230 space groups. This is achieved by first developing which builds on the conventional band theory of electrons, highlighting the link between the topology and local chemical an efficient way to represent band structures in terms of elementary basis states, and then bonding. This theory of topological quantum chemistry provides a description of the universal (across materials), global isolating the topological ones by removing the subset of atomic insulators, defined by the properties of all possible band structures and (weakly correlated) materials, consisting of a graph-theoretic description of existence of localized symmetric Wannier functions. Aside from encompassing all earlier momentum (reciprocal) space and a complementary group-theoretic description in real space. For all 230 crystal symmetry results on such indicators, including in particular the notion of filling-enforced quantum band groups, we classify the possible band structures that arise from local atomic orbitals, and show which are topologically non- insulators, our theory identifies symmetry settings with previously hidden forms of band trivial. Our electronic band theory sheds new light on known topological insulators, and can be used to predict many more. topology, and can be applied to the search for topological materials.
For the past century, chemists and physicists have advocated funda- which provides a good description nearby the high-symmetry ki points. mentally different perspectives on materials: chemists have adopted However, to determine the global band structure, the different k · p an intuitive ‘local’ viewpoint of hybridization, ionic chemical bond- expansions about each ki need to be patched together. Group theory ing and finite-range interactions, whereas physicists have described places constraints—‘compatibility relations’—on how this can be done. materials through band structures or Fermi surfaces in a non-local, Each solution to these compatibility relations gives groups of bands momentum-space picture. These two descriptions seem disjoint, espe- with different connectivities, corresponding to different physically cially with the advent of topological insulators, which are exclusively realizable phases of matter (trivial or topological). We solve all of the understood in terms of the non-trivial topology of Bloch Hamiltonians compatibility relations for all 230 space groups by mapping connectivity throughout the Brillouin zone in momentum space. Despite the in band theory to the graph-theoretic problem of constructing multi- apparent success in predicting some (mostly time-reversal-invariant) partite graphs. Classifying the allowed connectivities of energy bands topological insulators, conventional band theory is ill-suited to their becomes a combinatorial problem of graph enumeration: we present a natural treatment. Given the paucity of known topological insulators fully tractable, algorithmic solution. (fewer than 400 materials out of the 200,000 existent in crystal struc- Second, we develop the tools to compute the way in which the real- ture databases), one may ask whether topo logical materials are truly space orbitals in a material determine the symmetry character of the so rare, or if this reflects a failing of the conventional theory. electronic bands. Given only the Wyckoff positions and the orbital The topological properties of energy bands are intrinsically global symmetry (s, p, d and so on) of the elements or orbitals in a material, in momentum space. The duality between real and momentum (direct we derive the symmetry character of all of the energy bands at all points
1 Department of Physics, University of California, Berkeley, CA 94720, USA. 2 Department of Physics, Harvard University, Cambridge, MA 02138, USA. and reciprocal) space suggests that the properties of bands that are in the Brillouin zone. We do this by extending the notion of a band 3 Department of Applied Physics, University of Tokyo, Tokyo 113-8656, Japan. Correspondence and requests for materials should be addressed to non-local in momentum space will manifest locally in real space. Here representation, first introduced in refs 2 and 3, to the physically relevant A.V. (email: [email protected]) we unify the real- and momentum-space descriptions of solids, to case of materials with spin–orbit coupling (SOC) and/or time-reversal NATURE COMMUNICATIONS | 8: 50 | DOI: 10.1038/s41467-017-00133-2 | www.nature.com/naturecommunications 1 provide a powerful, complete and predictive band theory. Our symmetry. A band representation consists of all energy bands (and procedure provides a complete understanding of the structure of energy Bloch functions) that arise from localized orbitals respecting the crystal bands in a material, and links the topological properties of the material symmetry (and possibly time-reversal symmetry). The set of band to the chemical orbitals at the Fermi level. It is therefore a theory of representations is strictly smaller than the set of groups of bands topological quantum chemistry. obtained from our graph theory4. We identify a special subset We develop the complete theory of topological bands in two main of ‘elementary’ band representations (EBRs)2,5, elaborated on in steps. First, we compile all of the possible ways energy bands in a solid Supplementary Information, which correspond to the smallest sets can be connected throughout the Brillouin zone to obtain all realizable of bands that can be derived from localized atomic-like Wannier band structures in all non-magnetic space groups. Crystal symme- functions6. The 10,403 different EBRs for all of the space groups, tries place strong constraints on the allowed connections of bands. At Wyckoff positions and orbitals are presented in ref. 7. high-symmetry points (ki) in the Brillouin zone, Bloch functions are If the number of electrons is a fraction of the number of connected classified by irreducible representations of the symmetry group of ki, bands (connectivity) that form an EBR, then the system is a symmetry- which also determine the degeneracy. Away from these high-symmetry enforced semimetal. The EBR method allows us to easily identify points, fewer symmetry constraints exist and some degeneracies are candidate semimetallic materials. We find that the largest possible lowered. This result is central to the k · p approach1 to band structure, number of connected bands in an EBR is 24 and hence the smallest
1Princeton Center for Theoretical Science, Princeton University, Princeton, New Jersey 08544, USA. 2Department of Condensed Matter Physics, University of the Basque Country UPV/EHU, Apartado 644, 48080 Bilbao, Spain. 3Donostia International Physics Center, P. Manuel de Lardizabal 4, 20018 Donostia-San Sebastián, Spain. 4Department of Applied Physics II, University of the Basque Country UPV/EHU, Apartado 644, 48080 Bilbao, Spain. 5Max Planck Institute for Solid State Research, Heisenbergstrasse 1, 70569 Stuttgart, Germany. 6Department of Physics, Princeton University, Princeton, New Jersey 08544, USA. 7Max Planck Institute for Chemical Physics of Solids, 01187 Dresden, Germany. 8Laboratoire Pierre Aigrain, Ecole Normale Supérieure-PSL Research University, CNRS, Université Pierre et Marie Curie-Sorbonne Universités, Université Paris Diderot-Sorbonne Paris Cité, 24 rue Lhomond, 75231 Paris Cedex 05, France. 9Sorbonne Universités, UPMC Université Paris 06, UMR 7589, LPTHE, F-75005 Paris, France. * These authors contributed equally to this work.
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