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Momentum Space Topology in Standard Model and in Condensed Matter 2469-1 Workshop and Conference on Geometrical Aspects of Quantum States in Condensed Matter 1 - 5 July 2013 Momentum space topology in Standard Model and in condensed matter Grigory Volovik Aalto University and Landau Institute ^Momentum+^ space topology in Standard Model & condensed matter a a Aalto University G. Volovik Landau Institute Geometrical Aspects of Quantum States in Condensed Matter, 1-5 July 2013 1. Gapless & gapped topological media 2. Fermi surface as topological object 3. Fermi points (Weyl, Majorana & Dirac points) & nodal lines * superfluid 3He-A, topological semimetals , cuprate superconductors , graphene vacuum of Standard Model of particle physics in massless phase * gauge fields, gravity, chiral anomaly as emergent phenomena; quantum vacuum as 4D graphene * exotic fermions: quadratic, cubic & quartic dispersion; dispersionless fermions; Horava gravity 4. Flat bands & Fermi arcs in topological matter * surface flat bands: 3He-A, semimetals , cuprate superconductors , graphene , graphite * towards room-temperature superconductivity * 1D flat band in the vortex core *Fermi-arc on the surface of topological matter with Weyl points 5. Fully gapped topological media * superfluid 3He-B, topological insulators , chiral superconductors, quantum spin Hall insulators, vacuum of Standard Model of particle physics in present massive phase, vacua of lattice QCD * Majorana edge states & zero modes on vortices ( planar phase , topological insulator & 3He-B ) 3+1 sources of effective Quantum Field Theories in many-body system & quantum vacuum Lev Landau I think it is safe to say that no one understands Quantum Mechanics Richard Feynman Thermodynamics is the only physical theory of universal content Albert Einstein effective theories Symmetry: conservation laws, translational invariance, of quantum liquids: spontaneously broken symmetry, Grand Unification, ... two-fluid hydrodynamics of superfluid 4He & Fermi liquid theory of liquid 3He missing ingredient Topology: winding number in Landau theories one can’t comb hair on a ball smooth, anti-Grand-Unification Topological classes as defects in momentum space Fermi surface: vortex ring in p-space fully gapped topological matter: p skyrmion in p-space metals, t z normal 3He py p (p ) y z p x px p F 3He-B, topological insulators, quantum-Hall states, 3He-A film, H = + c .p 6\=2/ m vacuum of Standard Model Weyl point - hedgehog in p-space 3He-A, vacuum of SM, topological semimetals (Abrikosov) t Fermi arc on surface py of Weyl materials p , p y z pz / Fermi surface p=p p Dirac strings in p-space terminating on monopole 1 x / pz p=p flat band 2 on vortex in Weyl material Khodel-Shaginyan flat band: /-vortex in p-space EXONHGJHFRUUHVSRQGHQFH WRSRORJ\LQEXONSURWHFWV gapless fermions on surface of fully gapped systems; higher order nodes in nodal topological materials '4XDQWXP+DOOLQVXODWRU +H$ILOP JDSOHVVFKLUDOHGJHVWDWHV *9 'WRSRORJLFDOLQVXODWRU JDSOHVV'LUDFIHUPLRQV 9RONRY3DQNUDWRY VXSHUIOXLG+H% JDSOHVV0DMRUDQDIHUPLRQV 6DORPDD*9 +H$:H\OVHPLPHWDOZLWKSRLQWQRGHV )HUPLDUF QRGDOOLQH RQVXUIDFH 7XWVXPLHWDO JUDSKHQHZLWKSRLQWQRGHV GLVSHUVLRQOHVV'IODWEDQG 5\X+DWVXJDL VHPLPHWDOZLWK)HUPLOLQHV 'IODWEDQGRQWKHVXUIDFH +HLNNLOD*9 EXONGHIHFWFRUUHVSRQGHQFH WRSRORJ\LQEXONSURWHFWVgapless fermions inside topological defect UHODWLYLVWLFVWULQJ IHUPLRQ]HURPRGHVLQFRUH -DFNLZ5RVVL +H$ZLWKSRLQWQRGHV 'IODWEDQGLQWKHFRUH .RSQLQ6DORPDD 'SLSVXSHUFRQGXFWRU 0DMRUDQDIHUPLRQVLQWKHFRUH *9 two major universality classes of gapless fermionic vacua Landau theory of Fermi liquid Standard Model + gravity vacuum with Fermi surface: vacuum with Fermi (Weyl) point: metals, normal 3He 3He-A, planar phase, Weyl semimetal, vacuum of SM gravity emerges from Fermi (Weyl) point μi . g (pμ- eAμ - eo .Wμ)(pi- eAi - eo Wi) = 0 analog of Fermi surface Theory of topological matter: Nielsen, So, Ishikawa, Matsuyama, Haldane, Yakovenko, Horava, Kitaev, Ludwig, Schnyder, Ryu, Furusaki, S-C Zhang, Kane, Liang Fu, ... Fermi surface as topological object Green’s function Energy spectrum of G-1= it< ¡(p) non-interacting gas of fermionic atoms t 2 p2 p2 p ¡(p) = — μ = — F 2m 2m 2m p (p ) ¡ > 0 y z empty levels ¡ < 0 occupied p levels: p x Fermi sea F Fermi surface p=p 6\=2/ ¡ = 0 F Fermi surface: is Fermi surface a domain wall vortex ring in p-space in momentum space? phase of Green’s function G(t,p)=|G|e i\ has winding number N = 1 no! it is a vortex ring Migdal jump, non-Fermi liquids & p-space topology * Singularity at Fermi surface is robust to perturbations: t winding number N=1 cannot change continuously, interaction (perturbative) cannot destroy singularity p (p ) * Typical singularity: Migdal jump y z n(p) p x p F 6\=2/ pF p * Other types of singularity with the same winding number: Fermi surface: Luttinger Fermi liquid, vortex line in p-space marginal Fermi liquid, pseudo-gap ... G(t,p)=|G|e i\ * Example: zeroes in G(t,p) have the same N=1 as poles Z(p,t) zeroes in G(t,p) G(t,p) = Z(p,t) = (t2 + ¡2(p))a it< ¡(p) for a > 1/2 * Important for interacting systems, where quasiparticles are ill defined * Fermi surface exists in superfluids/superconductors, examples: 3He-A in flow & Gubankova-Schmitt-Wilczek, PRB74 (2006) 064505, but Luttinger theorem is not applied quantized vortex in r-space > Fermi surface in p-space homotopy group /1 Topology in r-space Topology in p-space z how is it in p-space ? t y p (p ) y z winding x p number p x F N1 = 1 6\=2/ 6\=2/ vortex ring Fermi surface ^(r)=|^| e i\ G(t,p)=|G|e i\ scalar order parameter Green’s function (propagator) of superfluid & superconductor classes of mapping S1 A U(1) classes of mapping S1 A GL(n,C) manifold of space of broken symmetry vacuum states non-degenerate complex matrices non-topological flat bands due to interaction Khodel-Shaginyan fermion condensate JETP Lett. 51, 553 (1990) GV, JETP Lett. 53, 222 (1991) Nozieres, J. Phys. (Fr.) 2, 443 (1992) E{n(p)} bE{n(p)} = ¡(p)bn(p)ddp = 0 solutions: ¡(p) = 0 or bn(p)=0 ¡(p) n(p) n(p) ¡(p) p p pF p1 flat band p2 bn(p)=0 bn(p)=0 ( )=0 splitting of Fermi surface ( ) ¡(p) = 0 bn p to Fermi ball (flat band) ¡ p = 0 bn(p)=0 S.-S. Lee Non-Fermi liquid from a charged black hole: A critical Fermi ball PRD 79, 086006 (2009) anti-de Sitter/conformal field theory correspondence Flat band as momentum-space dark soliton terminated by half-quantum vortices t p , p y z / p=p p 1 x / half-quantum vortices p=p 2 in 4-momentum space ¡ = 0 Khodel-Shaginyan fermionic condensate (flat band) phase of Green's function changes by / across the "dark soliton" 3. Classes of Fermi points & nodal lines: superfluid 3He-A, Standard Model, semimetals, graphene, cuprate SC, ... surface of 3He-B & topological insulators Weyl point magnetic hedgehog or Berry phase Dirac monopole z pz y py x px again no difference ? hedgehog in r-space right-handed and left-handed hedgehog in p-space massless quarks and leptons (r)=^r are elementary particles (p) = ^p m in Standard Model m Landau CP symmetry close to Fermi point is emergent H = + c m .p right-handed Weyl electron = hedgehog in p-space with spines = spins Topological invariant for right and left elementary particles Berry ‘magnetic’ field Berry phase a(C) = \0 =2/ ^ p Dirac H (p) = monopole C 2p2 Dirac string \0 in (unobservable) momentum space Weyl fermions in 3+1 gapless topological cond-mat topologically protected Weyl points in: topological semi-metal or Weyl metals (Abrikosov-Beneslavskii 1971), 3 He-A (1982), triplet Fermi gases, CoSb3 (arXiv:1204.5905) Gap node - Weyl point (anti-hedgehog) N3 =<1 E N = 1 e dSk g . ( g × g) 3 ijk pi pj p 8/ 1 over 2D surface S p (p ) in 3D p-space y z p x p 2 S N3 =1 2 2 2 2 2 p = px +py +pz Gap node - Weyl point (hedgehog) p2 c(p + ip ) (p) (p) +i (p) 2m < μ x y g3 g1 g2 H = = p2 g (p) <i g (p) <g (p) c(px — ipy ) < μ 1 2 3 ) + ) 2m ) ) emergence of relativistic chiral Weyl fermions near Fermi points original non-relativistic Hamiltonian p2 c(p + ip ) (p) (p) +i (p) 2m < μ x y g3 g1 g2 H = = = o .g(p) p2 g (p) <i g (p) <g (p) c(px — ipy ) < μ 1 2 3 ) + ) 2m ) ) close to nodes, i.e. in low-energy corner left-handed E relativistic chiral fermions emerge particles p 1 ¾¾ p (p ) N3 =<1 y z H = N3 c o . (p<p0) p¾ x E = – cp¾ N3p =1 2 right-handed particles top. invariant determines chirality chirality is emergent ?? in low-energy corner relativistic invariance as well what else is emergent ? bosonic collective modes in two generic fermionic vacua Landau theory of Fermi liquid Standard Model + gravity collective Bose modes: Fermi propagating surface A Fermi oscillation of position point of Fermi point p A p - eA collective Bose modes form effective dynamic of fermionic vacuum: electromagnetic field propagating oscillation of shape propagating of Fermi surface oscillation of slopes 2 2 2 ik E = c p A g pi pk form effective dynamic gravity field Landau, ZhETF 32, 59 (1957) two generic quantum field theories of interacting bosonic & fermionic fields relativistic quantum fields & gravity emerging near Weyl point Atiyah-Bott-Shapiro construction: primary object: linear expansion of Hamiltonian near the nodes in terms of Dirac K-matrices tetrad k a 0 e μ E = vF (p < pF) emergent relativity H = ea K .(pk < pk) a linear expansion near Fermi surface linear expansion near Weyl point secondary object: pz metric p μi ab μ i y g = d ea eb gμi(p - eA - eo .W )(p - eA - eo .W ) = 0 px μ μ μ i i i effective metric: effective effective emergent gravity SU(2) gauge isotopic spin hedgehog in p-space effective field effective electromagnetic electric charge all ingredients of Standard Model :
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