Topological media: quantum liquids, topological insulators & quantum vacuum G. Volovik Landau Institute CC-2010, Chernogolovka, July 26, 2010 1. 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 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 & axions 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 superfluidity 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 Relativistic plasma topological insulator Magnetohydrodynamic Color superfluidity Pair-correlations Photon mass classes of random Turbulence of vortex lines Savvidi vacuum Collective modes Vortex Coulomb matrices propagating vortex front Quark confinement, QCD cosmology plasma velocity independent Intrinsic orbital momentum of quark matter Reynolds number 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: you can't comb the hair on a ball smooth, in Landau theories anti-Grand-Unification Landau view on a many body system many body systems are simple at low energy & temperature equally applied to: weakly excited state of liquid superfluids, can be considered as system solids, of "elementary excitations" & relativistic quantum vacuum Landau, 1941

Universe helium liquids ground state = vacuum vacuum ground state + + elementary elementary particles excitations quasiparticles = elementary particles

why is low energy physics applicable to our vacuum ? characteristic high-energy scale in our vacuum is Planck energy 5 1/2 19 32 EP =(hc /G) ~10 GeV~10 K

high-energy physics is high-energy physics & cosmology extremely ultra-low energy physics belong to ultra-low temperature physics

highest energy in accelerators T of cosmic background radiation 16 Eew ~1 TeV ~ 10 K TCMBR ~ 1 K

−16 E ~ 10 E −32 ew Planck TCMBR ~ 10 EPlanck

cosmology is extremely ultra-low frequency physics cosmological expansion

v(r,t) = H(t) r Hubble law

−60 H ~ 10 EPlanck Hubble parameter

our Universe is extremely close to equilibrium ground state

We should study general properties of equilibrium ground states - quantum vacua Why no freezing at low T? natural masses of elementary particles are of order of characteristic energy scale even at highest temperature the Planck energy we can reach 19 32 16 m ~ EPlanck ~10 GeV~10 K T ~ 1 TeV~10 K everything should be completely frozen out

1016 e−m/T =10 − =0

10−123, 10−1016 another great challenge? main hierarchy puzzle cosmological constant puzzle

4 mquarks , mleptons <<< EPlanck Λ <<<< EPlanck its emergent physics solution: its emergent physics solution:

mquarks = mleptons = 0 Λ = 0

reason: reason: momentum-space topology thermodynamics of quantum vacuum of quantum vacuum Why no freezing at low T?

massless particles & gapless excitations are not frozen out

who protects massless excitations? gapless fermions live near Fermi surface & Fermi point

who protects Fermi surface & Fermi point ? Topology

p we live because z Fermi point is the hedgehog p y protected by topology p x Life protection Fermi point: hedgehog in momentum space

hedgehog is stable: one cannot comb the hair on a ball smooth tools

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

4 Λ <<<< EPlanck mquarks , mleptons <<< EPlanck 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, ... 2. Liquid 3He & 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 µν g (pµ- eAµ - eτ .Wµ)(pν- eAν - eτ .Wν) = 0 analog of Fermi surface Topological stability of Fermi surface Green's function Energy spectrum of G-1= iω − E(p) non-interacting gas of fermionic atoms ω 2 p2 p2 p E(p) = – µ = – F 2m 2m 2m E>0 p (p ) y z empty levels E<0 occupied p levels: p x Fermi sea F Fermi surface E=0 p=p ∆Φ=2π 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 Route to Landau Fermi-liquid

is Fermi surface robust to interaction ?

Sure! Because of topology: ω winding number N=1 cannot change continuously, interaction cannot destroy singularity p (p ) y z then Fermi surface survives in Fermi liquid ? p p x F Landau theory of Fermi liquid is topologically protected & thus is universal ∆Φ=2π Fermi surface: vortex line in p-space all metals have Fermi surface ... G(ω,p)=|G|e iΦ Not only metals. Some superconductore too!

Stability conditions & Fermi surface topologies in a superconductor Gubankova-Schmitt-Wilczek, Phys.Rev. B74 (2006) 064505

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 3. Superfluid 3He-A & Standard Model From Fermi surface to Fermi point

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

p (p ) y z p x where are Dirac particles? Dirac particle - composite object E=cp made of left and right particles

p (p ) y z E p x

p (p ) y z p E=cp x

p (p ) y z 2 2 2 2 p E = c p + m x mixing of left and right particles is secondary effect, which occurs at extremely low temperature

16 Tew ~ 1 TeV~10 K

Fermi (Dirac) points in 3+1 gapless topological matter topologically protected point nodes in: superfluid 3He-A, triplet cold Fermi gases, semi-metal (Abrikosov-Beneslavskii)

Gap node - Fermi 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 Gap node - Fermi 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 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 and gravity emerging near Fermi point Atiyah-Bott-Shapiro construction: linear expansion of Hamiltonian near the nodes in terms of Dirac Γ-matrices

k i . 0 E = vF (p − pF) H = ei Γ (pk − pk) linear expansion near linear expansion near emergent relativity Fermi surface Fermi point

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 Fermi point together with spin, Dirac Γ−matrices, gravity & physical laws: Lorentz & gauge invariance, equivalence principle, etc quantum vacuum as cryo-crystal 4D graphene Michael Creutz JHEP 04 (2008) 017

Fermi (Dirac) points with N3 = +1

Fermi (Dirac) points with N3 = −1 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

dimensional reduction

py

Fully gapped 2+1 system px

~ 1 N3 = ∫ dpxdpy g . (∂p g × ∂p g) 4π x y over the whole 2D momentum space or over 2D Brillouin zone topological insulators & superconductors in 2+1

p-wave 2D superconductor, 3He-A film, HgTe insulator quantum well

p2 − µ c(px + ipy) H = 2m p2 c) (px – ipy) − + µ) 2m

2 2 2 p = px +py

How to extract useful information on energy states from 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 and Quantum Hall effect

2D non-topological 2D topological 2D non-topological insulator insulator insulator or vacuum or vacuum

y ~ y ~ N3 = 0 N3 = 0

current ~ current apply voltage V N3 = 1 −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 e2 ~ σ = N3 xy 4π 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 / semiconductors / 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 semiconductor 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 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.