Topological media: quantum liquids, topological insulators & quantum vacuum

Transcription

1 Topological media: quantum liquids, topological insulators & quantum vacuum Landau Institute G. Volovik CC-2010, Chernogolovka, July 26, Introduction: Universe as object of ultralow temperature physics * quantum vacuum as topological medium * effective quantum field theories emerging at low T 2. Fermi surface as topological object *normal 3He, metals, 3. Dirac (Fermi points as topological objects * superfluid 3He-A, semimetals, cuprate superconductors, graphene, vacuum of Standard Model of particle physics in massless phase * topological invariants for gapless 2D and 3D topological matter * QED, QCD and gravity as emergent phenomena * quantum vacuum as cryo-crystal, α-phase of superfluid 3He 4. Fully gapped topological media * superfluid 3He-B, topological insulators, chiral superconductors, vacuum of Standard Model of particle physics in present massive phase * topological invariants for gapped 2D and 3D topological matter * edge states & Majorana fermions ( planar phase of 3He & surface of 3He-B * intrinsic QHE & spin-qhe 5. Conclusion * important role of momentum-space topology

2 Torsion & spinning strings, torsion instanton Fermion zero modes on strings & walls Antigravitating (negative-mass string Gravitational Aharonov-Bohm effect Domain wall terminating on string String terminating on domain wall Monopoles on string & Boojums Witten superconducting string Soft core string, Q-balls Kibble mechanism Dark matter detector Primordial magnetic field Baryogenesis by textures Cosmological & Newton constants dark energy dark matter Effective gravity Bi-metric & conformal gravity Graviton, dilaton Spin connection Rotating vacuum Vacuum dynamics conformal anomaly & strings Inflation Branes matter creation vacuum gravity black holes ergoregion, event horizon Hawking & Unruh effects black hole entropy Vacuum instability Superfluidity of neutron star vortices, glitches shear flow instability neutron stars Z &W strings skyrmions Alice string Pion string cosmology Gravity QCD quark matter Quark condensate Nambu--Jona-Lasinio Vaks--Larkin Color superfluidity Savvidi vacuum cosmic strings 3 He nuclear physics physical vacuum High Energy Physics Plasma Physics Nuclei vs 3He droplet Shell model Pair-correlations Collective modes Quark confinement, QCD cosmology Intrinsic orbital momentum of quark matter Emergence & effective theories Weyl & Majorana fermions Vacuum polarization, screening - antiscreening, running coupling Symmetry breaking (anisotropy of vacuum Parity violation -- chiral fermions Vacuum instability in strong fields, pair production Casimir force, quantum friction Fermionic charge of vacuum Higgs fields & gauge bosons Momentum-space topology Hierarchy problem, Supersymmetry high-t & chiral superconductivity Condensed Matter Phenome nology phase transitions Neutrino oscillations Chiral anomaly & axions Spin & isospin, String theory CPT-violation, GUT low dimensional systems Bose condensates hydrodynamics disorder random anisotropy unity of physics 3 He Grand Unification Gap nodes Low -T scaling mixed state Broken time reversal 1/2-vortex, vortex dynamics Films: FQHE, Statistics & charge of skyrmions & vortices Edge states; spintronics 1D fermions in vortex core Critical fluctuations Mixture of condensates Vector & spinor condensates BEC of quasipartcles, magnon BEC & laser meron, skyrmion, 1/2 vortex General; relativistic; spin superfluidity multi-fluid rotating superfluid Shear flow instability quantum phase transitions & momentum-space topology Larkin- Imry-Ma Relativistic plasma topological insulator Magnetohydrodynamic Photon mass classes of random Turbulence of vortex lines Vortex Coulomb matrices propagating vortex front plasma velocity independent Reynolds number

3 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 Symmetry: conservation laws, translational invariance, spontaneously broken symmetry, Grand Unification,... effective theories of quantum liquids: two-fluid hydrodynamics of superfluid 4 He & Fermi liquid theory of liquid 3 He Topology: you can't comb the hair on a ball smooth, anti-grand-unification missing ingredient in Landau theories

4 Landau view on a many body system many body systems are simple at low energy & temperature weakly excited state of liquid can be considered as system of "elementary excitations" Landau, 1941 equally applied to: superfluids, solids, & relativistic quantum vacuum helium liquids ground state + elementary excitations ground state = vacuum quasiparticles = elementary particles Universe vacuum + elementary particles why is low energy physics applicable to our vacuum?

5 characteristic high-energy scale in our vacuum is Planck energy E P =(hc 5 /G 1/ GeV10 32 K high-energy physics is extremely ultra-low energy physics highest energy in accelerators E ew 1 TeV K high-energy physics & cosmology belong to ultra-low temperature physics T of cosmic background radiation T CMBR 1 K E ew E Planck T CMBR E Planck cosmology is extremely ultra-low frequency physics cosmological expansion v(r,t = H(t r Hubble law H E Planck Hubble parameter our Universe is extremely close to equilibrium ground state We should study general properties of equilibrium ground states - quantum vacua

6 Why no freezing at low T? natural masses of elementary particles are of order of characteristic energy scale the Planck energy m E Planck 1019 GeV10 32 K even at highest temperature we can reach T 1 TeV1016 K everything should be completely frozen out e m/t =10 = , another great challenge?

7 main hierarchy puzzle cosmological constant puzzle m quarks, m leptons <<< E Planck 4 Λ <<<< E Planck its emergent physics solution: its emergent physics solution: m quarks = m leptons = 0 Λ = 0 reason: momentum-space topology of quantum vacuum reason: thermodynamics of quantum vacuum

8 Why no freezing at low T? massless particles & gapless excitations are not frozen out who protects massless excitations?

9 gapless fermions live near Fermi surface & Fermi point who protects Fermi surface & Fermi point? Topology we live because Fermi point is the hedgehog protected by topology p z p y Life protection p x Fermi point: hedgehog in momentum space hedgehog is stable: one cannot comb the hair on a ball smooth

10 tools Thermodynamics Topology in momentum space responsible for properties of vacuum energy problems of cosmological constant: perfect equilibrium Lorentz invariant vacuum has Λ = 0; perturbed vacuum has nonzero Λ on order of perturbation Λ <<<< E 4 Planck responsible for properties of fermionic and bosonic quantum fields in the background of quantum vacuum Fermi point in momentum space protected by topology is a source of massless Weyl fermions, gauge fields & gravity m quarks, m leptons <<< E Planck

11 Quantum vacuum as topological substance: universality classes physics at low T is determined by gapless excitations universality classes of gapless vacua topology is robust to deformations: nodes in spectrum survive interaction Horava, Kitaev, Ludwig, Schnyder, Ryu, Furusaki,...

12 2. Liquid 3 He & effective theory of vacuum with Fermi surface two major universality classes of gapless fermionic vacua Landau theory of Fermi liquid Standard Model + gravity vacuum with Fermi surface vacuum with Fermi point gravity emerges from Fermi point analog of Fermi surface g µν (p µ - ea µ - eτ.w µ (p ν - ea ν - eτ.w ν = 0

13 Topological stability of Fermi surface Energy spectrum of non-interacting gas of fermionic atoms E(p = E>0 empty levels Fermi surface E=0 p2 2m µ = E<0 p2 occupied levels: Fermi sea 2m p F 2m p=p F is Fermi surface a domain wall in momentum space? 2 Green's function G -1 = iω E(p ω Φ=2π Fermi surface: vortex ring in p-space p y (p z pf phase of Green's function G(ω,p= G e iφ has winding number N = 1 p x no! it is a vortex ring

14 Route to Landau Fermi-liquid is Fermi surface robust to interaction? Sure! Because of topology: winding number N=1 cannot change continuously, interaction cannot destroy singularity then Fermi surface survives in Fermi liquid? ω p y (p z pf p x Landau theory of Fermi liquid is topologically protected & thus is universal all metals have Fermi surface... Not only metals. Some superconductore too! Φ=2π Fermi surface: vortex line in p-space G(ω,p= G e iφ Stability conditions & Fermi surface topologies in a superconductor Gubankova-Schmitt-Wilczek, Phys.Rev. B74 (

15 quantized vortex in r-space Fermi surface in p-space Topology in r-space z homotopy group π 1 how is it in p-space? Topology in p-space ω y p y (p z vortex ring Φ=2π Ψ(r= Ψ e iφ scalar order parameter of superfluid & superconductor classes of mapping S 1 U(1 manifold of broken symmetry vacuum states x winding number N 1 = 1 Fermi surface Φ=2π G(ω,p= G e iφ pf Green's function (propagator p x classes of mapping S 1 GL(n,C space of non-degenerate complex matrices

16 3. Superfluid 3 He-A & Standard Model From Fermi surface to Fermi point z magnetic hedgehog vs right-handed electron y p z p y x p x again no difference? hedgehog in r-space σ(r=r ^ right-handed and left-handed massless quarks and leptons are elementary particles in Standard Model hedgehog in p-space ^ σ(p = p close to Fermi point Landau CP symmetry is emergent H = + c σ.p right-handed electron = hedgehog in p-space with spines = spins

17 E=cp p y (p z p x

18 where are Dirac particles? E=cp Dirac particle - composite object made of left and right particles p y (p z E p x p y (p z E=cp p x p y (p z p x E 2 = c 2 p 2 + m 2 mixing of left and right particles is secondary effect, which occurs at extremely low temperature T ew 1 TeV10 16 K

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20 Fermi (Dirac points in 3+1 gapless topological matter topologically protected point nodes in: superfluid 3 He-A, triplet cold Fermi gases, semi-metal (Abrikosov-Beneslavskii N 3 = 1 eijk ds k g. ( p i 8π g pj g over 2D surface S in 3D p-space Gap node - Fermi point (anti-hedgehog N 3 = 1 p 1 E p y (p z N 3 =1 p 2 Gap node - Fermi point (hedgehog S 2 p x H = p2 2m µ c(p x ip y c(p x + ip y p2 + µ 2m g 3 (p g 1 (p +i g 2 (p = g 1 (p i g 2 (p g 3 (p

21 emergence of relativistic QFT near Fermi (Dirac point original non-relativistic Hamiltonian p2 c(p x + ip y H = 2m µ p2 c(p x ip y + µ 2m close to nodes, i.e. in low-energy corner relativistic chiral fermions emerge H = N 3 c τ.p E = ± cp chirality is emergent?? g 3 (p g 1 (p +i g 2 (p = g 1 (p i g 2 (p g 3 (p left-handed particles p 1 N 3 = 1 N 3 =1 p 2 right-handed particles top. invariant determines chirality in low-energy corner = τ.g(p E p y (p z p x what else is emergent? relativistic invariance as well

22 bosonic collective modes in two generic fermionic vacua Landau theory of Fermi liquid Standard Model + gravity collective Bose modes: Fermi surface collective Bose modes of fermionic vacuum: propagating oscillation of shape of Fermi surface Fermi point propagating oscillation of position of Fermi point p p - ea form effective dynamic electromagnetic field propagating oscillation of slopes A E 2 = c 2 p 2 g ik p i p k form effective dynamic gravity field Landau, ZhETF 32, 59 (1957 two generic quantum field theories of interacting bosonic & fermionic fields

23 relativistic quantum fields and gravity emerging near Fermi point Atiyah-Bott-Shapiro construction: linear expansion of Hamiltonian near the nodes in terms of Dirac Γ-matrices E = v F (p p F linear expansion near Fermi surface emergent relativity H = e k i Γ i 0.(p k p k linear expansion near Fermi point p z p y g µν (p µ - ea µ - eτ.w µ (p ν - ea ν - eτ.w ν = 0 hedgehog in p-space p x effective metric: emergent gravity effective electromagnetic field all ingredients of Standard Model : chiral fermions & gauge fields emerge in low-energy corner together with spin, Dirac Γ matrices, gravity & physical laws: Lorentz & gauge invariance, equivalence principle, etc effective SU(2 gauge field effective isotopic spin effective electric charge e = + 1 or 1 gravity & gauge fields are collective modes of vacua with Fermi point

24 quantum vacuum as cryo-crystal 4D graphene Michael Creutz JHEP 04 ( Fermi (Dirac points with Fermi (Dirac points with N 3 = +1 N 3 = 1

25 4. From Fermi point to intrinsic QHE & topological insulators N 3 = 1 eijk ds k g. ( p i 8π g pj g over 2D surface S in 3D momentum space 3+1vacuum with Fermi point dimensional reduction p y Fully gapped 2+1 system p x N 3 = 1 dp x dp y g. ( p x 4π g py g over the whole 2D momentum space or over 2D Brillouin zone

26 topological insulators & superconductors in 2+1 p-wave 2D superconductor, 3 He-A film, HgTe insulator quantum well H = p2 2m µ c(p x ip y c(p x + ip y p2 + µ 2m p 2 = p x 2 +py 2 How to extract useful information on energy states from Hamiltonian without solving equation Hψ = Eψ

27 Topological invariant in momentum space H = p2 2m µ c(p x ip y c(p x + ip y p2 + µ 2m g 3 (p g 1 (p +i g 2 (p H = g 1 (p i g 2 (p g 3 (p = τ.g(p p 2 = p x 2 +py 2 N 3 = 1 d 2 p g. ( p x 4π g py g fully gapped 2D state at µ = 0 GV, JETP 67, 1804 (1988 Skyrmion (coreless vortex in momentum space at µ > 0 p y p x unit vector g (p x,p y g^ sweeps unit sphere N 3 (µ > 0 = 1

28 H = quantum phase transition: from topological to non-topologicval insulator/superconductor Topological invariant in momentum space N 3 = p2 2m µ c(p x ip y c(p x + ip y p2 + µ 2m 1 d 2 p g. ( p x 4π g py g g 3 (p g 1 (p +i g 2 (p = g 1 (p i g 2 (p g 3 (p trivial insulator N 3 = 0 N 3 N 3 = 1 topological insulator = τ.g(p µ intermediate state at µ = 0 must be gapless µ = 0 quantum phase transition N 3 = 0 is origin of fermion zero modes at the interface between states with different N 3

29 p-space invariant in terms of Green's function & topological QPT N 3 = 1 24π 2 e µνλ tr d 2 p dω G µ G -1 G ν G -1 G λ G -1 quantum phase transitions in thin 3 He-A film N 3 = 4 N 3 = 6 GV & Yakovenko J. Phys. CM 1, 5263 (1989 N 3 = 2 transition between plateaus must occur via gapless state! N 3 = 0 QPT from trivial to topological state gap gap a 1 a 2 a 3 gap plateau-plateau QPT between topological states a film thickness

30 example: QPT between gapless & gapped matter T (temperature topological semi-metal q c topological quantum phase transitions transitions between ground states (vacua of the same symmetry, but different topology in momentum space no change of symmetry along the path q - parameter of system quantum phase transition at q=q c different asymptotes when T => T n q e /T 0 c T topological insulator other topological QPT: Lifshitz transition, transtion between topological and nontopological superfluids, plateau transitions, confinement-deconfinement transition,... T T QPT interrupted by thermodynamic transitions no symmetry change along the path line of 1-st order transition broken symmetry q 2-nd order transition q q line of 1-st order transition line of 2-nd order transition

31 Zero energy states on surface of topological insulators & superfluids p y Fully gapped 2+1 system p x N 3 = 1 dp x dp y g. ( p x 4π g py g Fully gapped 3+1 system Majorana fermions on the surface and in the vortex cores

32 interface between two 2+1 topological insulators or gapped superfluids y N 3 = N interface N 3 = N + x gapped state gapless interface gapped state * domain wall in 2D chiral superconductors: H = p2 2m c(p x i p y tanh x µ c(p x + i p y tanh x p2 + µ 2m p x - component p x ip y N 3 = 1 0 p x +ip y N 3 = +1 x p y - component

33 Edge states at interface between two 2+1 topological insulators or gapped superfluids y N 3 = N interface N 3 = N + gapped state gapped state empty E(p y 0 left moving edge states occupied p y GV JETP Lett. 55, 368 (1992 Index theorem: number of fermion zero modes at interface: ν = N + N

34 Edge states and currents empty E(p y 2D non-topological insulator or vacuum left moving edge states current y 2D topological insulator N 3 = 0 N 3 = 0 N 3 = 1 current y right moving edge states 2D non-topological insulator or vacuum x E(p y empty 0 p y 0 p y occupied occupied current J y = J left +J right = 0

35 Edge states and Quantum Hall effect 2D non-topological insulator or vacuum 2D topological insulator 2D non-topological insulator or vacuum apply voltage V N 3 = 0 V/2 current E(p y V/2 E(p y +V/2 y N 3 = 1 V/2 current y N 3 = 0 x V/2 0 p y left moving edge states right moving edge states V/2 0 p y current J y = J left +J right = σ xy E y σ xy = e 2 N 3 4π

36 Intrinsic spin-current quantum Hall effect & momentum-space invariant spin current J x z = 1 4π (γn ss dh z /dy + N se E y spin-spin QHE spin-charge QHE 2D singlet superconductor: spin/spin σ xy = N ss 4π s-wave: N ss = 0 p x + ip y : N ss = 2 d xx-yy + id xy : N ss = 4 film of planar phase of superfluid 3 He spin/charge σ xy = N se 4π GV & Yakovenko J. Phys. CM 1, 5263 (1989

37 planar phase film of 3He & 2D topological insulator p2 c(p x + i p y σ z H = 2m µ p2 c(p x i p y σ z + µ 2m N 3 = 1 eµνλ tr [ d 2 p dω G µ G -1 G ν G -1 G λ G -1 ] = 0 24π 2 N se = 1 eµνλ tr [ d 2 p dω σ z G µ G -1 G ν G -1 G λ G -1 ] 24π 2 N 3 = N N = 0 N + 3 = +1 N 3 = 1 + N N se = + 3 = N 3 2 spin quantum Hall effect spin current J x z = 1 Nse E y 4π spin-charge QHE spin/charge σ xy = N se 4π N se = 2 GV & Yakovenko J. Phys. CM 1, 5263 (1989

38 Intrinsic spin-current quantum Hall effect & edge state spin current J x z = 1 (γn 4π ss dh z /dy + N se E y spin-charge QHE y y N se = 0 V/2 spin current N se = 2 spin current V/2 N se = 0 x E(p y V/2 E(p y +V/2 right moving spin up left moving spin up V/2 p y spin/charge σ xy = N se 4π V/2 p y right moving spin down electric current is zero spin current is nonzero left moving spin down

39 3D topological superfluids / insulators / semiconductors / vacua gapless topologically nontrivial vacua fully gapped topologically nontrivial vacua 3He-A, Standard Model above electroweak transition, semimetals, 4D graphene (cryocrystalline vacuum 3He-B, Standard Model below electroweak transition, topological insulators, triplet & singlet chiral superconductor,... Bi 2 Te 3

40 Present vacuum as semiconductor or insulator 3 conduction bands of d-quarks electric charge q=-1/3 E (p 3 conduction bands of u-quarks, q=+2/3 conduction electron band, q=-1 neutrino band, q=0 neutrino band, q=0 p valence electron band, q=-1 3 valence u-quark bands q=+2/3 3 valence d-quark bands q=-1/3 Quantum vacuum: Dirac sea electric charge of quantum vacuum Q= Σ q a = N [ (-1/3 + 3 (+2/3 ] = 0 a dielectric and magnetic properties of vacuum (running coupling constants

41 fully gapped 3+1 topological matter superfluid 3 He-B, topological insulator Bi 2 Te 3, present vacuum of Standard Model * Standard Model vacuum as topological insulator Topological invariant protected by symmetry N Κ = 1 24π 2 e µνλ tr dv Κ G µ G -1 G ν G -1 G λ G -1 over 3D momentum space G is Green's function at ω=0, Κ is symmetry operator GΚ =+/ ΚG Standard Model vacuum: Κ=γ 5 Gγ 5 = γ 5 G N Κ = 8n g 8 massive Dirac particles in one generation

42 topological superfluid 3 He-B ( Η = p2 c B σ.p 2m* µ p2 c B σ.p 2m* + µ ( ( ( p2 2m* µ = τ 3 +c B σ.p τ 1 Hτ 2 = τ 2 H K = τ 2 1/m* non-topological superfluid topological 3 He-B N Κ = 0 N Κ = +2 0 N Κ = 1 N Κ = +1 µ Dirac vacuum 1/m* = 0 Dirac N Κ = 2 topological superfluid N Κ = 0 Dirac non-topological superfluid Η = ( Μ c B σ.p c B σ.p + Μ GV JETP Lett. 90, 587 (2009

43 Boundary of 3D gapped topological superfluid vacuum x,y 3 He-B Η = ( p2 c B σ.p 2m* µ+u(z p2 c B σ.p 2m* + µ U(z ( N Κ = 0 N Κ = +2 Majorana fermions on wall 0 µ U(z N Κ = +2 z Majorana particle = Majorana anti-particle 1/2 of fermion: b = b + spectrum of Majorana zero modes H zm = c ^ B z. σ x p = c B (σ x p y σ y p x N Κ = 0 0 z helical fermions

44 fermion zero modes on Dirac wall Dirac wall Dirac vacuum Dirac vacuum Η = ( Μ(z cσ.p cσ.p + Μ(z ( N Κ = 1 Μ < 0 N Κ = +1 Μ > 0 Volkov-Pankratov, 2D massless fermions in inverted contacts JETP Lett. 42, 178 (1985 chiral fermions N Κ = 1 Μ 0 N Κ = +1 z 0 Dirac point at the wall z Dirac point nodal line in Bi 2 Te 3 Dirac point is below FS: nodal line on surface of topological insulator Bi 2 Te 3

45 Majorana fermions: edge states on the boundary of 3D gapped topological matter * boundary of topological superfluid 3 He-B vacuum N Κ = 0 x,y 3 He-B N Κ = +2 Η = ( Majorana fermions on wall p2 cσ.p 2m* µ+u(z p2 cσ.p 2m* + µ U(z µ U(z N Κ = +2 ( 0 z N Κ = 0 0 spectrum of fermion zero modes * Dirac domain wall H zm = c (σ x p y σ y p x chiral fermion N Κ = 1 Μ 0 N Κ = +1 z Η = ( Μ(z cσ.p helical fermions cσ.p + Μ(z ( Volkov-Pankratov, 2D massless fermions in inverted contacts JETP Lett. 42, 178 (1985

46 Majorana fermions Η = p2 on interface in topological superfluid 3 He-B 2m* µ σ x c x p x +σ y c y p y +σ z c z p z p2 (σ x c x p x +σ y c y p y +σ z c z p z 2m* + µ ( N Κ = 2 c x N Κ = +2 0 c y N Κ = 2 N Κ = +2 N Κ = +2 N Κ = 2 domain wall phase diagram one of 3 "speeds of light" changes sign across wall Majorana fermions c z N Κ = +2 spectrum of fermion zero modes N Κ = 2 0 z H zm = c (σ x p y σ y p x

47 Conclusion Momentum-space topology determines: universality classes of quantum vacua effective field theories in these quantum vacua topological quantum phase transitions (Lifshitz, plateau, etc. quantization of Hall and spin-hall conductivity topological Chern-Simons & Wess-Zumino terms quantum statistics of topological objects spectrum of edge states & fermion zero modes on walls & quantum vortices chiral anomaly & vortex dynamics, etc.