Topological semimetals: from to flat band

^ ^ G. Volovik † Aalto University Landau Institute a a Frascati, 7 Settembre, 2011 1. Gapless & gapped topological media 2. as topological object 3. Fermi points (Weyl, Majorana & Dirac points) & nodal lines * superfluid 3He-A, topological semimetals , cuprate superconductors , vacuum of Standard Model of particle physics in massless phase * QED, QCD and gravity as emergent phenomena; quantum vacuum as 4D graphene * exotic : quadratic, cubic & quartic dispersion; dispersionless fermions 4. Flat bands on surface of topological matter * superfluid 3He-A, semimetals , cuprate superconductors , graphene , * towards room-temperature superconductivity 5. 1D flat band in the vortex core and Fermi-arc on the surface of topological matter with Weyl points Heikkilä, Kopnin, GV arXiv:1012.0905, 1011.4185, 1011.4665, 1103.2033 6. Supplemented material: Fully gapped topological media * superfluid 3He-B, topological insulators , chiral superconductors, vacuum of Standard Model of particle physics in present massive phase * edge states & Majorana fermions ( planar phase , & 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 : 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 classes of topological matter as momentum-space objects Fermi surface: vortex ring in p-space fully gapped topological matter: p skyrmion in p-space , ω z normal 3He py p (p ) y z p x px p F 3He-B, topological insulators, 3He-A film, H = + c .p ∆Φ=2π σ vacuum of Standard Model Weyl point - hedgehog in p-space 3He-A, vacuum of SM, topological semimetals ω Fermi arc on py 3He-A surface p , p y z pz π Fermi surface

p=p p Dirac strings in p-space terminating on monopole 1 x

π pz p=p flat band 2 on 3He-A vortex flat band (Khodel state): π-vortex in p-space topological correspondence: topology in bulk protects gapless fermions on the surface or in vortex core bulk-surface correspondence:

2D Quantum Hall insulator & 3He-A film chiral edge states

3D topological insulator Dirac fermions on surface

superfluid 3He-B Majorana fermions on surface

superfluid 3He-A, Fermi arc on surface

graphene dispersionless 1D flat band on surface

semimetal with Fermi lines 2D flat band on the surface bulk-vortex correspondence: relativistic string / superfluid 3He-B 1D zero modes (Jackiw-Rossi) superfluid 3He-A 1D flat band in the core (Kopnin-Salomaa) New topological object in momentum space: flat band with zero energy

Flat band on the surface of topological matter with nodal lines 2. 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 (Weyl) point: metals, normal 3He 3He-A, planar phase, Weyl semimetal, vacuum of SM

gravity emerges from Fermi (Weyl) point µν g (pµ- eAµ - eτ .Wµ)(pν- eAν - eτ .Wν) = 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, ... Topological stability of Fermi surface Green's function Energy spectrum of G-1= iω − ε(p) non-interacting gas of fermionic atoms ω 2 p2 p2 p ε(p) = – µ = – F 2m 2m 2m p (p ) ε > 0 y z empty levels ε < 0 occupied p levels: p x Fermi sea F Fermi surface p=p ∆Φ=2π ε = 0 F Fermi surface: is Fermi surface a domain wall vortex ring in p-space in momentum space? phase of Green's function G(ω,p)=|G|e iΦ 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: ω 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 ∆Φ=2π 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(ω,p)=|G|e iΦ * Example: zeroes in G(ω,p) have the same N=1 as poles Z(p,ω) zeroes in G(ω,p) G(ω,p) = Z(p,ω) = (ω2 + ε2(p))γ iω − ε(p) for γ > 1/2 * Important for interacting systems, where 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 π1 Topology in r-space Topology in p-space z how is it in p-space ? ω

y p (p ) y z winding x p number p x F N1 = 1 ∆Φ=2π ∆Φ=2π vortex ring Fermi surface Ψ(r)=|Ψ| e iΦ G(ω,p)=|G|e iΦ scalar order parameter Green's function (propagator) of superfluid & superconductor classes of mapping S1 → U(1) classes of mapping S1 → 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)} δE{n(p)} = ∫ε(p)δn(p)ddp = 0 solutions: ε(p) = 0 or δn(p)=0

ε(p) n(p) n(p) ε(p)

p p pF p1 flat band p2

δn(p)=0 δn(p)=0 ( )=0 splitting of Fermi surface ( ) ε(p) = 0 δn p to Fermi ball (flat band) ε p = 0 δn(p)=0

S.-S. Lee Non-Fermi liquid from a charged black hole: A critical Fermi ball PRD 79, 086006 (2009) Flat band as momentum-space dark soliton terminated by half-quantum vortices

ω

p , p y z π p=p p 1 x π half-quantum vortices p=p 2 in 4-momentum space ε = 0 Khodel-Shaginyan fermionic condensate (flat band)

phase of Green's function changes by π 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

magnetic hedgehog vs Weyl point 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 σ in Standard Model σ

Landau CP symmetry close to is emergent H = + c σ .p right-handed Weyl = hedgehog in p-space with spines = spins E=cp

p (p ) y z p x Chiral Weyl fermions in Standard Model

for interacting systems Standard Model topological invariant

Topological invariant protected by symmetry

N = 1 e tr dV K G ∂µ G-1 G ∂ν G-1G ∂λ G-1 Κ 2 µνλ ∫ 24π over S3

G is Green's function, Κ is symmetry operator GΚ =+/− ΚG

for Standard Model vacuum Κ = exp 2πiτ3 weak isotopic spin

NΚ = 16 ng

16 massless Weyl particles in one generation are protected by combined symmetry and topology From massless Weyl particles to massive Dirac particles

where are massive Dirac particles? Dirac particle - composite object: E=cp mixture of left and right Weyl particles

p (p ) y z E p x

N=+1 p (p ) y z 16 p E=cp Tew ~ 1 TeV~10 K x N=+1−1=0

p (p ) y z 2 2 2 2 p E = c p + m x

N=−1 fully gapped vacua can be also topologically nontrivial is Dirac vacuum topologically trivial ? (3He-B, topological insulators, ...) Weyl fermions in 3+1 gapless topological cond-mat topologically protected Weyl points in:

topological semi- (Abrikosov-Beneslavskii 1971), 3He-A (1982), triplet Fermi gases

Gap node - Weyl point (anti-hedgehog) N = 1 k 3 − E N = 1 e dS g . (∂ g × ∂ g) 3 ijk ∫ pi pj p 8π 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 c(p – ip ) g1(p) −i g2(p) −g3(p) ) x y − + µ) ) ) 2m emergence of relativistic QFT near Fermi (Dirac) point

original non-relativistic Hamiltonian p2 c(p + ip ) (p) (p) +i (p) 2m − µ x y g3 g1 g2 H = = = τ .g(p) p2 c(p – ip ) g1(p) −i g2(p) −g3(p) ) x y − + µ) ) ) 2m close to nodes, i.e. in low-energy corner left-handed E relativistic chiral fermions emerge particles p 1 p (p ) N3 =−1 y z H = N3 c τ .p p x E = ± cp N3p =1 2 right-handed particles top. invariant determines 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 → 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 → 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 Γ-matrices tetrad µ k a 0 ea E = vF (p − pF) emergent relativity H = ea Γ .(pk − pk) secondary object: linear expansion near linear expansion near metric Fermi surface Weyl point µν ab µ ν g = η ea eb

pz µν g (pµ- eAµ - eτ .Wµ)(pν- eAν - eτ .Wν) = 0 py

effective metric: effective effective px emergent gravity SU(2) gauge isotopic spin effective field effective electromagnetic electric charge field e = + 1 or −1 hedgehog in p-space

all ingredients of Standard Model : gravity & gauge fields chiral fermions & gauge fields are collective modes emerge in low-energy corner of vacua with Weyl point together with spin, Dirac Γ−matrices, gravity & physical laws: Lorentz & gauge invariance, equivalence principle, etc crossover from Landau 2-fluid hydrodynamics to Einstein general relativity they represent two different limits of hydrodynamic type equations

equations for gµν depend on hierarchy of ultraviolet cut-off's: Planck energy scale EPlanck vs Lorentz violating scale ELorentz

EPlanck >> ELorentz EPlanck << ELorentz emergent Landau emergent general covariance two-fluid hydrodynamics & general relativity

3He-A with Fermi point Universe

ELorentz << EPlanck ELorentz >> EPlanck −3 9 ELorentz ~ 10 EPlanck ELorentz > 10 EPlanck quantum vacuum as crystal 4D graphene Michael Creutz JHEP 04 (2008) 017

Fermi (Dirac) points with N3 = +1

Fermi (Dirac) points with N3 = −1 topology of graphene nodes

1 -1 N = tr [Ko∫ dl H ∂l H] 4πi Ν = +1 Ν = −1 K - symmetry operator, commuting or anti-commuting with H Ν = +1 close to nodes: Ν = −1

HΝ = +1 = τxpx + τypy Ν = +1 Ν = −1

HΝ = −1 = τxpx − τypy

K = τz exotic fermions: bilayer graphene massless fermions with quardatic dispersion double cuprate layer semi-Dirac fermions surface of top. insulator fermions with cubic and quartic dispersion neutrino physics

1 -1 N = tr [ K o∫ dl H ∂l H] 4πi

N=+1 N=+1 N=0 N=+2 E E E E

p p p p y y y y p p p + p = x + x x x

E=cp E=cp E2 = 2c2p2 + 4m2 E2 = (p2/ 2m)2

Dirac fermions massive fermions massless fermions with quadratic dispersion multiple Fermi point T. Heikkilä & GV arXiv:1010.0393 cubic spectrum in trilayer graphene N = 1 + 1 + 1 = 3 N=+1 N=+1 N=+1

E = p3 + += p N= +3 x E = − p3

E=cp E=cp E=cp

multilayered graphene spectrum in the outer layers N N = 1 + 1 + 1 + ... E = p E = − pN

what kind of induced gravity emerges near degenerate Fermi point?

route to topological flat band on the surface of 3D material Flat bands in topological matter

nodal spiral in multilayered graphene nodal lines generates flat band with zero energy nodes in graphene in cuprate superconductors in the top and bottom layers generate flat band on zigzag edge generate flat band on side surface Hekilla, Kopnin, GV Shinsei Ryu

approximate flat band on side surface of graphite formation of nodal spiral in bulk (together with flat band on the surface) by stacking of graphene layers

Hi,j = (σxpx + σypy)δi,j + (σ+t+ + σ−t−)δi,j+1

t+ > t−

N = 1 1

1 -1 N1 = tr [ K o∫ dl H ∂l H] 4πi Emergence of nodal line from gapped branches by stacking graphene layers Nodal spiral generates flat band on the surface

projection of spiral on the surface determines boundary of flat band

N = 1 tr K dl H-1 ∂ H 1 [ o∫ l ] 4πi C

at each (px,py) except the boundary of circle one has 1D gapped state (insulator) N = 0 trivial 1D insulator outside N = 1 topological 1D insulator inside at each (px,py) inside the circle one has 1D gapless edge state N1 = 1 this is flat band C

Coutside Cinside Nodal spiral generates flat band on the surface

projection of nodal spiral on the surface determines boundary of flat band lowest energy states: energy spectrum in bulk bulk states (projection to px , py plane) surface states form the flat band nodal line Helicity of nodal spiral

t+ < t− t+ > t− Modified nodal spiral in rhombohedral graphite: spiral of Fermi surfaces (McClure 1969)

projections on surfaces determine approximate flat band Nodal lines in graphite tranformed to chain of electron and hole FS

for conventional graphite: approximate flat band on the lateral surface Gapless topological matter with protected flat band on surface or in vortex core

non-topological flat bands due to interaction E(p) Khodel-Shaginyan fermion condensate JETP Lett. 51, 553 (1990) flat band GV, JETP Lett. 53, 222 (1991) p Nozieres, J. Phys. (Fr.) 2, 443 (1992) p1 p2 splitting of Fermi surface to flat band flat band in soliton 2 2 2 H = τ3 (px +pz −pF )/2m + τ1c(z)pz 2 2 nodes at pz = 0 and px = pF

1 -1 N = tr [ K o∫ dl H ∂l H] 4πi

soliton spectrum in soliton c E( px)

flat band 0 z -pF pF flat flat band band

px = +1 = 0 = +1 = +1 = 0 N N N N N 1 − = = +1 N N ] H l ∂

-1 H

dl

∫ o K

[

tr i 1 π 4

=

Flat band on the graphene edge graphene the on band Flat N Surface superconductivity in topological semimetals: route to room temperature superconductivity

Extremely high DOS of flat band gives high transition temperature:

normal superconductors: d2p 1 flat band superconductivity: exponentially suppressed 1= g ∫ linear dependence 2h2 E(p) transition temperature of Tc on coupling g

Tc = TF exp (-1/gν) "Recent studies of the correlations between the internal Tc ∼ gSFB microstructure of the samples and the transport properties suggest that superconductivity might be localized at the interaction DOS interfaces between crystalline graphite regions interaction of different orientations, running parallel to the flat band graphene planes." PRB. 78, 134516 (2008) area multiple Fermi point

Kathryn Moler: possible 2D superconductivityof twin boundaries From Weyl point to quantum Hall topological insulators 1 k N3 = e ∫ dS g . (∂p g × ∂p g) 8π ijk i j over 2D surface S in 3D momentum space top. invariant for Weyl point in 3+1 system

hedgehog 2D trivial pz ∼ in p-space N3(pz) = 0 insulator Weyl ∼∼ point N3 = N3(pz p0)

py at each pz one has 2D insulator ∼ 2D topological Hall or fully gapped 2D superfluid N3(pz) = 1 insulator px ~ 1 . N3 (pz) = ∫ dpxdpy g (∂px g × ∂py g) over4 theπ whole 2D momentum space skyrmion or over 2D Brillouin zone in p-space top. invariant for fully gapped 2+1 system 3D matter with Weyl points: Topologically protected flat band in vortex core vortices in r-space ∼ pz 2D trivial z N3(pz) = 0 insulator at each pz between two values: 2D topological Hall insulator: pF cos λ zero energy states E (pz)=0, Weyl in the vortex core flat point (1D flat band 1011.4665) band

∼ 2D topological N3(pz) = 1 QH insulator

− pF cos λ

Weyl point ∼ 2D trivial N3(pz) = 0 insulator ~ 1 -1 -1 -1 GV & Yakovenko N3(pz) = 2 tr ∫ dpx dpydω G ∂ω G G ∂p G G ∂p G 4π x y (1989) Topologically protected flat band in vortex core of superfluids with Weyl points E (p , Q) z flat band in spectrum of fermions bound to core of 3He-A vortex (Kopnin-Salomaa 1991)

pz

E (pz)

continuous spectrum

bound states flat band of bound states terminates on zeroes Weyl flat band Weyl pz of continuous spectrum point point (i.e on Weyl points) 3He-A with Weyl points: for each |pz | < pF cos λ Topologically protected one has 2D topological Hall insulator with Dirac valley (Fermi arc) on surface zero energy edge states E (pz)=0 (Dirac valley PRB 094510 or Fermi arc PRB 205101)

pz ∼ 2D trivial N3(pz) = 0 insulator

pF cos λ

Weyl point py )=0

2D topological )=0 z

∼ z p p

( N (pz) = 1 3 insulator ( E p E

Fermi arc Fermi arc x

− pF cos λ

Weyl point 2D trivial x=xL ∼ x=xR N3(pz) = 0 insulator Edge states at interface between effective two 2+1 topological insulators & Fermi arc y

~ ~ N3 = N− N3 = N+ interface gapped state x gapped state

E(py, |pz| < pF) left moving empty edge states on the edge of insulator with Index theorem: py 0 ~ number of fermion zero modes N = 1 at interface: 3 one fermion zero mode occupied ν = 1 ν = N+ − N− GV JETP Lett. 55, 368 (1992) py E < 0 Fermi arc in 2D: p Fermi surface which terminates z on two points: −pF E = 0 pF projections of Weyl points E > 0 Fermi arc: Fermi surface separates positive and negative energies, but has boundaries

E (py, pz) > 0 Fermi surface ordinary Fermi surface py E (py = 0, |pz| < pF) = 0 has no boundaries E (py, pz) < 0 Fermi surface of bound states pz may leak to higher dimension

Fermi surface of localized states is terminated by projections of Weyl points when localized states merge with continuous spectrum

L spectrum of R spectrum of edge states on edge states on left wall right wall

Tsutsumi, et al PRB 094510 Horava anisotropic scaling gravity

3 3 3 anisotropic z=3 scaling: x = b x' , t = b t' Sgrav = ∫ d x dt R b3 b3 b−6 Horava anisotropic scaling in bilayered graphene

1 -1 N = tr [ K o∫ dl H ∂l H] 4πi

N=+1 N=+1 N=0 N=+2 E E E E

p p p p y y y y p p p + p = x + x x x

E=cp E=cp E2 = 2c2p2 + 4m2 E2 = (p2/ 2m)2

2+1 massless Dirac fermions massive fermions massless Dirac fermions with quadratic dispersion relativistic quantum fields and gravity emerging near Weyl point Atiyah-Bott-Shapiro construction: linear expansion of Hamiltonian near the nodes in terms of Dirac Γ-matrices

k i 0 H = ei Γ .(pk − pk) linear expansion near Weyl point

effective tetrad: emergent Γ−matrices µν emergent gravity g (pµ- eAµ - eτ .Wµ)(pν- eAν - eτ .Wν) = 0

pz effective metric: effective effective emergent gravity SU(2) gauge isotopic spin py effective field effective electromagnetic electric charge px field e = + 1 or −1

what gravity & gauge fields emerge in vacua with quadratic hedgehog in p-space Dirac point in bilayer graphene ? 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 Fermions in 2+1 bylayer graphene single layer zweibein 0 px + ipy 0 (e (p) +i e ).(p −A) H = 1 2 = σx px + σy py = ) px – ipy 0 (e (p) − i e ).(p −A) ) 0 ( 1 2 ) double layer zweibein 2 2 0 (px + ipy) 0 [(e1(p) +i e2).(p −A)] H = = 2 () px – ipy) ( (p) i ).(p A) 2 0 ) [ e − e − ] ) 1 2 0 )

anisotropic scaling: x = b x' , t = b2 t' 2+1 anisotropic QED emerging in bylayer graphene

2 2 0 (px + ipy) 0 [(e1(p) +i e2)(p −A)] H = = 2 () px – ipy) ( (p) i )(p A) 2 0 ) [ e − e − ] ) 1 2 0 ) anisotropic scaling: x = b x' , t = b2 t' , B = b−2 B' , E = b−3 E' , S = S'

2 2 4/3 −1 SQED = ∫ d x dt ( B − E − E E ) 4 b2 b2 b−4 b− b−4 3+1 isotropic QED emerging in Weyl semimetal 2 2 isotropic scaling: x = b x' , t = b t' , B = b− B' , E = b− E' , S = S' 3 2 2 SQED = ∫ d x dt ( B − E ) b3 b b−4 b−4 2+1 isotropic QED emerging in single layer graphene 2 2 2 3/4 SQED = ∫ d x dt ( B − E ) b2 b b−3 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

& vortex dynamics, etc.

flat band & room-temperature superconductivity

superfuid phases 3He serve as primer for topological matter: quantum vacuum of Standard Model, topological superconductors & topological insulators, etc.

we need: low T, high H, miniaturization, atomically smooth surface, nano-detectors, ... and fabrication of samples with room-temperature surface superconductivity 4. From Fermi point to intrinsic QHE & topological insulators 1 k N3 = e ∫ dS g . (∂p g × ∂p g) 8π ijk i j over 2D surface S in 3D momentum space

3+1vacuum with Fermi point hedgehog in p-space dimensional reduction

Fully gapped 2+1 system

~ 1 skyrmion N3 = ∫ dpxdpy g . (∂p g × ∂p g) 4π x y in p-space over the whole 2D momentum space or over 2D Brillouin zone topological insulators & gapped superconductors in 2+1 topological insulator = topological gapped superconductor = bulk insulator superconductor with gap in bulk with topologically protected but with topologically protected gapless states on the boundary gapless states on the boundary

3 p-wave 2D superconductor (Sr2RuO4 ?), He-A thin film, CdTe/HgTe/Cd insulator quantum well, planar phase film

who protects gapless states?

p2 − µ c(px + ipy) generic example: 2m 2 2 2 H = p = px +py p2 c) (px – ipy) − + µ) 2m

How to extract useful information on energy states from this Hamiltonian without solving equation

Hψ = Eψ Topological invariant in momentum space

p2 c(p + ip ) (p) (p) +i (p) 2m − µ x y g3 g1 g2 H = H = = τ .g(p) p2 c(p – ip ) g1(p) −i g2(p) −g3(p) ) x y − + µ) ) ) 2m p2 = p 2 +p 2 x y fully gapped 2D state at µ = 0

~ 1 2 N3 = ∫ d p g . (∂p g × ∂p g) GV, JETP 67, 1804 (1988) 4π x y

Skyrmion (coreless vortex) in momentum space at µ > 0 py

unit vector ^g p x g (px,py) sweeps unit sphere ~ N3 (µ > 0) = 1 quantum phase transition: from topological to non-topologicval insulator/superconductor

p2 c(p + ip ) (p) (p) +i (p) 2m − µ x y g3 g1 g2 H = = = τ .g(p) p2 c(p – ip ) g1(p) −i g2(p) −g3(p) ) x y − + µ) ) ) 2m ~ N3 Topological invariant in momentum space ~ ~ N3 = 1 N = 1 d2p g . (∂ g ∂ g) trivial 3 ∫ px × py insulator 4π ~ topological insulator N3 = 0 µ

µ = 0 intermediate state at µ = 0 must be gapless quantum phase transition

∼ ∆N = 0 is origin of fermion zero modes 3 ∼ at the interface between states with different N3 p-space invariant in terms of Green's function & topological QPT

~ 1 2 µ -1 ν -1 λ -1 N3= 2 e tr ∫ d p dω G ∂ G G ∂ G G ∂ G 24π µνλ GV & Yakovenko ~ J. Phys. CM 1, 5263 (1989) N3 = 6 quantum phase transitions in thin 3He-A film ~ N3 = 4

transition between plateaus ~ must occur via gapless state! N3 = 2 gap gap ~ gap a N3 = 0 film thickness a1 a2 a3 QPT from trivial plateau-plateau QPT to topological state between topological states topological quantum phase transitions

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

py

Fully gapped 2+1 system px

~ 1 N3 = ∫ dpxdpy g . (∂p g × ∂p g) 4π x y

Fully gapped 3+1 system

Majorana fermions on the surface and in the vortex cores interface between two 2+1 topological insulators or gapped superfluids

y

~ ~ N3 = N− N3 = N+ interface x

gapless interface gapped state gapped state

* domain wall in 2D chiral superconductors: px - component

p2 p − ip p +ip c(p + i p tanh x ) x y x y 2m − µ x y ~ ~ H = N3= −1 N3 = +1 p2 ) c(px – i py tanh x ) − + µ ) 0 x 2m

py - component Edge states at interface between two 2+1 topological insulators or gapped superfluids y

~ ~ N3 = N− N3 = N+ interface gapped state gapped state

E(py) left moving edge states empty py 0 GV JETP Lett. 55, 368 (1992) occupied

Index theorem: number of fermion zero modes at interface:

ν = N+ − N− Edge states and currents

2D non-topological 2D topological 2D non-topological insulator insulator insulator or vacuum or vacuum y y ~ ~ N = 0 = current 3 N3 0 ~ current N3 = 1 x E(py) empty E(py) empty left moving right moving edge states edge states py 0 py 0

occupied occupied

current Jy = Jleft +Jright = 0 Edge states & intrinsic QHE: topological invariant determines Hall quantization

2D topological 2D non-topological insulator 2D non-topological insulator insulator or vacuum or vacuum y ~ y ~ N = 0 N = 0 current ~ apply voltage V N = 1 current −V/2 V/2 x

E(py)−V/2 E(py)+V/2

right moving edge states V/2 py p 0 0 y −V/2 left moving edge states current Jy = Jleft +Jright = σxyEy extra number deficit of left moving states 2 ~ σ = e N of right moving states xy 4π Intrinsic quantum Hall effect & momentum-space invariant

e 2 ~ µνλ 2 S = N e ∫ d x dt A F A - electromagnetic field CS 16π 3 µ νλ µ

p-space invariant r-space invariant

2 ~ electric current J = δS / δ A = e N E x CS x 3 y 4π film of topological quantum liquid ^ z ~ ^ N y , E 3

^ x , J 8 quantized intrinsic Hall conductivity 6 (without external magnetic field)

2 ~ 4 σ = e N xy 3 4π 2

GV & Yakovenko a J. Phys. CM 1, 5263 (1989) film thickness general Chern-Simons terms & momentum-space invariant (interplay of r-space and p-space topologies)

~ S = 1 N eµνλ ∫ d2x dt A I F J CS 16π 3IJ µ νλ r-space invariant p-space invariant protected by symmetry

~ 1 e tr [ d2p dω Κ Κ G ∂µ G-1 G ∂ν G-1G ∂λ G-1] N = µνλ ∫ I J 3IJ 24π2 I Κ - charge interacting with gauge field A I µ gauge fields can be real, artificial or auxiliary Κ=e for electromagnetic field A µ ^ z z z Κ=σ for effective spin-rotation field A ( A = γH ) z µ 0 ^ id/dt −γσ.H = id/dt − σ^.A 0 i applied Pauli magnetic field plays the role of components of effective SU(2) gauge field A µ Intrinsic spin-current quantum Hall effect & momentum-space invariant

z 1 z spin current J = (γN dH /dy + N Ey) x 4π ss se

spin-spin QHE spin-charge QHE

2D singlet superconductor: s-wave: N = 0 spin/spin N ss σ = ss p + ip : N = 2 xy 4π x y ss dxx-yy + idxy : Nss = 4

film of planar phase of superfluid 3He

spin/charge N σ = se GV & Yakovenko xy 4π J. Phys. CM 1, 5263 (1989) planar phase film of 3He & 2D topological insulator p2 − µ c(px + i py σz ) H = 2m p2 ) c(px – i py σz) − + µ ) 2m ~ 1 e tr [ d2p dω G ∂µ G-1 G ∂ν G-1G ∂λ G-1] = 0 N = µνλ ∫ 3 24π2

~ 1 2 µ -1 ν -1 λ -1 e tr [ d p dω σ G ∂ G G ∂ G G ∂ G ] N = µνλ ∫ z se 24π2 ~ + ~ − N3 = +1 N3 = −1 ~ ~ + ~ − ~ ~+ ~ − N = = 3 N3 + N 3 = 0 Nse N3 − N 3 = 2 spin quantum Hall effect z 1 spin current J = N Ey spin-charge QHE x 4π se

spin/charge GV & Yakovenko Nse N = 2 σ = se J. Phys. CM 1, 5263 (1989) xy 4π Intrinsic spin-current quantum Hall effect & edge state z 1 z spin current J = (γN dH /dy + N Ey) x 4π ss se

spin-charge QHE

y y

N = 0 N = 0 se N = 2 se se x −V/2 spin current spin currentV/2

E(py)−V/2 E(py)+V/2 right moving spin up left moving V/2 spin up spin/charge Nse py σ = py xy −V/2 4π

right moving left moving spin down spin down electric current is zero spin current is nonzero 3D topological superfluids / insulators / / vacua gapless topologically fully gapped topologically nontrivial vacua nontrivial vacua

3He-A, 3He-B, Standard Model Standard Model above electroweak transition, below electroweak transition, semimetals, topological insulators, 4D graphene triplet & singlet (cryocrystalline vacuum) chiral superconductor, ...

Bi2Te3 Present vacuum as or insulator

3 conduction bands E (p) of d-quarks electric charge q=-1/3

3 conduction bands of u-quarks, q=+2/3 conduction electron band, q=-1

neutrino band, q=0 p neutrino band, q=0

valence electron band, q=-1

3 valence u-quark bands q=+2/3 dielectric and magnetic properties of vacuum (running coupling constants) 3 valence d-quark bands q=-1/3 Quantum vacuum: Dirac sea electric charge of quantum vacuum Q= Σ qa = N [-1 + 3×(-1/3) + 3×(+2/3) ] = 0 a fully gapped 3+1 topological matter 3 superfluid He-B, topological insulator Bi2Te3 , present vacuum of Standard Model

* Standard Model vacuum as topological insulator

Topological invariant protected by symmetry

N = 1 e tr dV Κ G ∂µ G-1 G ∂ν G-1G ∂λ G-1 Κ 2 µνλ ∫ 24π over 3D momentum space

G is Green's function at ω=0, Κ is symmetry operator GΚ =+/− ΚG

Standard Model vacuum: Κ=γ5 Gγ5 =− γ5G

NΚ = 8ng

8 massive Dirac particles in one generation topological superfluid 3He-B Hτ2 =− τ2H p2 – µ c σ.p K = 2m* B 2 τ2 p Η = = – µ τ3 +cBσ.p τ1 p2 2m* c σ.p – + µ ( ) ( B 2m* ( 1/m* non-topological superfluid topological 3He-B

NΚ = 0 NΚ = +2 Dirac vacuum 0 µ 1/m* NΚ = −1 NΚ = +1 = 0

Dirac Dirac – Μ cBσ.p NΚ = −2 NΚ = 0 Η = c σ.p + topological superfluid non-topological superfluid ( B Μ ( GV JETP Lett. 90, 587 (2009) Boundary of 3D gapped topological superfluid p2 – µ+U(z) . x,y 2m* cBσ p Η = p2 vacuum 3 c σ.p – + µ–U(z) He-B ( B 2m* (

N = +2 NΚ = 0 Κ

z 0 Majorana particle = Majorana anti-particle 1/2 of fermion: b = b+ µ–U(z) spectrum of Majorana zero modes Majorana ^ fermions Hzm = cB z . σ x p = cB (σxpy−σypx) on wall NΚ = +2 helical fermions 0 z NΚ = 0

fermion zero modes on Dirac wall

Dirac wall – Μ(z) cσ.p Η = cσ.p + Μ(z) ( Dirac vacuum Dirac vacuum (

N = −1 N = +1 Volkov-Pankratov, Κ Κ 2D massless fermions in inverted contacts JETP Lett. 42, 178 (1985) Μ < 0 Μ > 0 z nodal line 0 Μ chiral fermions

NΚ = +1 Dirac point z at the wall 0

NΚ = −1 Dirac point

in Bi2Te3 Dirac point is below FS: Bi2Te3 nodal line on surface of topological insulator Majorana fermions: edge states p2 on the boundary of 3D gapped topological matter – µ+U(z) . 2m* cσ p Η = p2 cσ.p – + µ–U(z) * boundary of topological superfluid 3He-B ( 2m* ( x,y µ–U(z) vacuum 3He-B Majorana fermions on wall N = +2 NΚ = 0 NΚ = +2 Κ

0 z 0 NΚ = 0

spectrum of fermion zero modes

* Dirac domain wall Hzm = c (σxpy−σypx) helical fermions Μ chiral fermion NΚ = +1 Volkov-Pankratov, z – Μ(z) cσ.p 0 2D massless fermions Η = in inverted contacts N = −1 cσ.p Κ ( + Μ(z) ( JETP Lett. 42, 178 (1985) Majorana fermions on interface in topological superfluid 3He-B cx p2 – µ 2m* σxcxpx+σycypy+σzczpz Η = N = +2 p2 N = −2 Κ σ c p +σ c p +σ c p – + µ Κ ( x x x y y y z z z 2m* ( cy 0

N = +2 NΚ = +2 NΚ = −2 NΚ = − 2 Κ

phase diagram domain wall

one of 3 "speeds of light" changes sign across wall cz N = +2 Majorana Κ fermions spectrum of fermion zero modes z Hzm = c (σxpy−σypx) 0

NΚ = − 2 Zero energy states in the core of vortices in topological superfluids

vortices in fully gapped 3+1 system

fermion zero modes in vortex core Bound states of fermions on cosmic strings and vortices Spectrum of quarks in core of electroweak cosmic string

quantum numbers: Q - angular momentum & pz - linear momentum

E (pz , Q) E (pz , Q) Q=2

Q=1

pz Q=0 pz

Q=-1

Q=-2

E(pz) = - cpz for d quarks E(pz) = cpz for u quark asymmetric branches cross zero energy

Index theorem: Number of asymmetric branches = N Jackiw & Rossi N is vortex winding number Nucl. Phys. B190, 681 (1981) Bound states of fermions on vortex in s-wave superconductor Caroli, de Gennes & J. Matricon, Phys. Lett. 9 (1964) 307 NΚ = 0

E (Q, p = 0 ) z E (pz , Q)

E(Q,pz) = − Q ω0 (pz)

= 2/ E << Q=-7/2 ω0 ∆ F ∆ Q=-5/2 Q=-3/2 Q=-1/2 1/2 3/2 5/2 7/2 Q pz

asymmetric branch as function of Q

Angular momentum Q is half-odd integer no true fermion zero modes: in s-wave superconductor no asymmetric branch as function of pz

Index theorem for approximate fermion zero modes: Index theorem for true fermion zero modes?

Number of asymmetric Q-branches = 2N is the existence of fermion zero modes N is vortex winding number related to topology in bulk?

GV JETP Lett. 57, 244 (1993) fermions zero modes on symmetric vortex in 3He-B 3 topological He-B at µ > 0 : NΚ = 2 E (Q, pz = 0 ) E (pz , Q) E (pz , Q)

Q=4 Q=4 Q=3 Q=3 Q=2 Q=2 Q=1 Q=1

Q=0 p Q=0 1 234 Q z pz

Q=-1 Q=-1 Q=-2 Q=-2 Q=-3 Q=-3 Q=-4 Q=-4

EQ=-Qω0 helicity + helicity − 2 ω0=∆ /EF << ∆ gapless fermions on Q=0 branch form Q is integer for p-wave superfluid 3He-B 1D Fermi-liquid

Misirpashaev & GV Fermion zero modes in symmetric vortices in superfluid 3He, Physica B 210, 338 (1995) fermions zero modes on symmetric vortex in 3He-B 3 topological He-B at µ > 0 : NΚ = 2 E (Q, pz = 0 ) E (pz , Q)

Q=4 Q=3 Q=2 Q=1

Q=0 p 1 234 Q z Q=-1 Q=-2 Q=-3 Q=-4

EQ=-Qω0 gapless fermions on Q=0 branch form

2 ω0=∆ /EF << ∆ 1D Fermi-liquid

Q is integer for p-wave superfluid 3He-B

Misirpashaev & GV Fermion zero modes in symmetric vortices in superfluid 3He, Physica B 210, 338 (1995) topological quantum phase transition in bulk & in vortex core 1/m* non-topological superfluid topological superfluid 3He-B

NΚ = 0 NΚ = +2 µ

Q=4 Q=3 Q=2 Q=1 pz E (pz , Q) Q=0 pz Q=-1 Q=-2 Q=-3 Q=-4

µ < 0 µ > 0 vs vs

NΚ = 0 NΚ = 2 superfluid 3He-B as non-relativistic limit of relativistic triplet superconductor

cα.p+ βM − µR γ5∆ Η = relativistic triplet superconductor γ ∆ . ( 5 − cα p − βM + µR (

cp << M µ << M

p2 – µ c σ.p 2m B 3 Η = superfluid He-B p2 c σ.p – + µ ( B 2m (

2 cB = c ∆ /M m = M / c

2 2 2 (µ + M) = µR + ∆ phase diagram of topological states of relativistic triplet superconductor

3He-B p2 – µ . 2m cBσ p Η = p2 c σ.p – + µ ( B 2m ( cα.p+ βM − µR γ5∆ Η = γ ∆ . ( 5 − cα p − βM + µR ( energy spectrum in relativistic triplet superconductor

222 µ =M −∆ |µ |<µ* R R R |µ |>µ* R R gapless spectrum at topological quantum phase transition soft quantum phase transition: Higgs transition in p-space

cα.p+ βM − µR γ5∆ Η = γ ∆ . ( 5 − cα p − βM + µR ( spectrum of non-relativistic 3He-B

2 2 µ<0 µ=0 µ<∆ /M µ>∆ /M soft quantum phase transition: Higgs transition in p-space

gapless spectrum at topological quantum phase p2 transition 1/m* – µ . 2m* cBσ p Η = p2 cBσ.p – + µ NΚ = 0 NΚ = +2 ( 2m* ( 0 µ NΚ = −1 NΚ = +1 Dirac Dirac

NΚ = −2 NΚ = 0 fermion zero modes in relativistic triplet superconductor

cα.p+ βM − µR γ5∆ Η = γ ∆ . ( 5 − cα p − βM + µR (

pz pz

vortices in topological superconductors have fermion zero modes

generalized index theorem ? possible index theorem for fermion zero modes on vortices (interplay of r-space and p-space topologies)

1 tr d3p dω dφ G ∂ G-1 G ∂ G-1G ∂ G-1G ∂ G-1G ∂ G-1 N = [ ∫ ω φ px py pz ] 5 4π3i

for vortices in Dirac vacuum N = N winding number 5

E (pz)

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