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The in a helium droplet: from semi-metals & topological insulators to & cosmology G. Volovik Zelenogorsk, July 12, 2010 Landau Institute 1. Introduction: helium liquids & Universe * effective quantum field theories * quantum vacuum as topological medium * thermodynamics of quantum vacuum & 2. Fermi surface as topological object 3. Dirac (Fermi) points as topological objects * semimetals, cuprate superconductors, graphene, superfluid 3He-A, vacuum of of particle physics in massless phase * topological invariants for gapless 2D and 3D topological * emergent physical laws near Dirac point (Fermi point) 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, zero modes on vortices & Majorana * intrinsic QHE & spin-QHE, Chern-Simons action, chiral anomaly, vortex dynamics, ... 5. Thermodynamics & dynamics of quantum vacuum * vacuum energy & cosmological constant * cosmology as relaxation to equilibrium Torsion & spinning strings, torsion instanton Emergence & effective theories Fermion zero modes on strings & walls Vacuum polarization, screening - antiscreening, running coupling Antigravitating (negative-mass) string 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 3He Grand Unification Soft core string, Q-balls Momentum-space topology Kibble mechanism Z &W strings Hierarchy problem, 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, mixed state Cosmological & & strings CPT-violation, GUT Broken time reversal Newton constants high-T & chiral High Energy 1/2-vortex, vortex dynamics cosmology super- Films: FQHE, dark matter matter Physics conductivity Statistics & charge of Effective creation skyrmions & vortices Bi-metric & low vacuum Edge states; spintronics conformal gravity Condensed gravity Gravity 3 dimensional 1D fermions in vortex core , 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 Helium liquids & the Universe From Landau Fermi liquid & two-fluid hydrodynamics to physics of quantum vacuum & cosmology * superfluid 4He and Universe

Landau two-fluid hydrodynamics Einstein

* liquid 3He and Universe

Landau theory of Fermi liquid as effective theory of fermionic vacuum

* extension of Landau theory from Fermi surface to Fermi point

superfluid 3He-A Standard Model + gravity

* from Fermi point to QHE & topological insulators

planar phase & 3He-B Dirac vacuum as topological insulator * from helium liquids to dynamics of quantum vacuum & cosmology quantum vacuum as self-sustained system cosmology as approach to equilibrium 3+1 sources of effective Quantum Field Theories 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 ? 1. superfluid 4He & effective theories of hydrodynamic type

Landau two-fluid hydrodynamics Einstein general relativity

classical low-energy property classical low-energy property of quantum liquids of quantum vacuum

Landau equations Einstein equations

Landau equations & Einstein equations are effective theories describing dynamics of metric field + matter (quasiparticles) Landau quasiparticles in superfluid 4He: phonons & rotons two types of rotons in Landau theory in original 1941 theory: quantum of vortex motion in modified 1946 theory: short wavelength quasiparticle

Landau 1941 roton: Landau gap in vortex spectrum quasiparticles of vortex spectrum ∆ ∼ h2ρ2/3m−5/3

this is energy this is vortex ring of vortex ring of minimal size of minimal size phonon long wave Landau estimate excitation Landau 1946 roton of vortex gap is correct ! short wave quasiparticle

N.G. Berloff & P.H. Roberts Nonlocal condensate models of superfluid helium J. Phys. A32, 5611 (1999) Effective metric in Landau two-fluid hydrodynamics

Doppler shifted phonon spectrum in moving superfluid review: Barcelo, Liberati & Visser, E = cp + p.vs c speed of sound Analogue Gravity Living Rev. Rel. 8 (2005) 12 vs superfluid velocity move p.vs to the left

E − p.vs= cp take square

2 2 2 µυ p =(−E, p) (E − p.vs) − c p = 0 g pµpν = 0 ν 00 0 i i ij 2 ij i j g = −1 g = −vs g = c δ −vs vs effective metric

inverse metric gµυ determines effective in which phonons move along geodesic curves

2 2 2 2 2 µ ν ds = − c dt + (dr − vsdt) ds = gµυdx dx reference frame for phonon is dragged by moving liquid Landau critical velocity = black hole horizon Painleve-Gulstrand metric superfluid 4He Gravity 2 2 2 2 2 2 2 ds = - dt (c -v ) + 2 v dr dt + dr + r dΩ g 00 g0r acoustic horizon black hole horizon 2 2 r 2 2____GM 2 __rh v (r) = c __h ds2 = - dt2 (c2-v2) + dr2 /(c2-v2) + r2dΩ2 v (r) = = c r r r gravity v(r) vs(r) g00 grr

horizon at g00= 0 vs= c where flow velocity black hole r = rh (velocity of local frame in GR) v(r )=c reaches Landau critical velocity h

superfluid v(rh) = vLandau = c vacuum (Unruh, 1981)

Hawking radiation is phonon/photon creation above Landau critical velocity Superfluids Universe

acoustic gravity metric theories of gravity general relativity

geometry of effective space time g geometry of space time for quasiparticles (phonons) µυ for matter

2 µ ν geodesics for photons geodesics for phonons ds = gµυdx dx = 0

Landau two-fluid equations Einstein equations of GR . Matter dynamic equations ρ + .(ρvs + P ) = 0 Matter for metric field gµυ 1 (Rµν - gµνR/2) = Tµν . 2 vs + (µ + vs /2) = 0 8πG equations equation equation for superfluid for normal µν T;ν Matter = 0 for matter 1/2 of GR component component message from: liquid helium to: gravity

is gravity fundamental ? it may emerge as classical output of underlying quantum system

as hydrodynamics ?

underlying microscopic superfluid helium quantum system at high energy quantum vacuum

classical 2-fluid emergent classical general hydrodynamics low-energy relativity effective theory

. Matter ρ + .(ρvs + P ) = 0 1 Matter . 2 (Rµν - gµνR/2) - Λgµν = Tµν vs + (µ + vs /2) = 0 8πG Tµν = 0 ν Matter application of Landau view on a many body system to Universe many body systems are simple at low energy & temperature weakly excited state of liquid weakly excited state of Universe can be considered as system should be also simple of "elementary excitations"

Landau, 1941

helium liquids Universe

ground state vacuum + + elementary elementary excitations particles

but why is low energy physics applicable to our Universe ? 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? who protects massless fermions? gapless fermions live near Topology Fermi surface & Fermi point

no life without 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 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 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 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 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 ~ S = 1 N eµνλ ∫ d2x dt A I F J CS 16π 3IJ µ νλ z z 1 z spin current J = δS / δA = (γN dH /dy + N Ey) x CS 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 2 p − µ c(σx px + σy py) − µ c(px + i py σz ) H = 2m H = 2m p2 2 p ) c(σx px + σy py) − + µ c(px – i py σ ) − + µ ) ) z ) 2m 2m ~ 1 e tr [ d2p dω G ∂µ G-1 G ∂ν G-1G ∂λ G-1] = 0 N = µνλ ∫ 3 24π2 ~ Κ = τ3σz 1 e tr [ d2p dω Κ G ∂µ G-1 G ∂ν G-1G ∂λ G-1] Κ = σz N = µνλ ∫ 3K 24π2 ~ + ~ − N3 = +1 N3 = −1 ~ ~ + ~ − ~ ~+ ~ − N = = 3 N3 + N 3 = 0 N3Κ N3 − N 3 = 2 spin quantum Hall effect z z 1 spin current J = δS / δA = N Ey spin-charge QHE x CS x 4π se

spin/charge ~ GV & Yakovenko Nse N = = 2 σ = se N3Κ 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 current N = 2 se se current x −V/2 spin current V/2 spin current

left moving E(py)−V/2 E(p )+V/2 y spin down left moving V/2 spin up spin/charge Nse py σ = py V/2 xy − 4π right moving right moving spin up spin down electric current is zero spin current is nonzero 3D topological superfluids/insulators/semiconductors

gapless topologically fully gapped topologically nontrivial vacua nontrivial vacua

3He-A, 3He-B, Standard Model above electroweak transition, Standard Model below electroweak transition, semimetals topological insulators (Be2Se3 , ...), triplet & singlet color/chiral superconductors, ... 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 x,y

vacuum 3He-B p2 – µ+U(z) . 2m* cBσ p Η = N = +2 p2 NΚ = 0 Κ c σ.p – + µ–U(z) ( B 2m* ( z 0

spectrum of zero modes µ–U(z) H = c ^z . σ x p = c (σ p −σ p ) Majorana zm B B x y y x fermions on wall NΚ = +2 helical fermions

0 z NΚ = 0 fermion zero modes on Dirac wall Dirac wall

Dirac vacuum Dirac vacuum – Μ(z) cσ.p N = −1 N = +1 Κ Κ Η = ( cσ.p + Μ(z) ( Μ < 0 Μ > 0 z 0

Μ chiral fermions chiral NΚ = +1 fermions z 0

NΚ = −1 Volkov-Pankratov, 2D massless fermions in inverted contacts JETP Lett. 42, 178 (1985) 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 Lancaster experiments: probing edge states of 3He-B with vibrating wire v

spectrum of surface states continuous spectrum 2v E

∆−pFv v

vc = ∆/3pF

E

v >vc 2pFv ∆−pFv 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 vortices in superfluid 3He-B

symmetry breaking phase transitions homotopy group

SO(3)xSO(3)xU(1) SO(3) π1(G/H)=π1(U(1)xSO(3))=ZxZ2 winding numbers A1 NORMAL N1 = 0, 1, 2, 3 ... and ν = 0, 1 A 1 + 1 = 2 1 + 1 = 0 A B N1 = 1 , ν =0 N1 = 0 , ν =1 mass vortex spin vortex A N1 = ν = 1 spin-mass vortex B NORMAL 1 B G=SO(3)xSO(3)xU(1) 2 (bar) p doubly quantized H (T)0.5 spin–mass cluster of vortex as pair boundaryT (mK) of spin–mass vortices SO(3)1 vortex mass of vortices confined by soliton rotating (N1=2, ν=0) container soliton i iΦ k ki soliton ∆αβ ~ e σαβR vs ki R -- rotation matrix counterflow region

(N1=1, ν=–1) ≡ (N1=1, ν=1) symmetry breaking in the 3He-B vortex core Phase diagram of first order ∆ ξ B vortex-core transition in 3He-B B-phase 40 most symmetric order parameter Solid vortex core 3He-A 30 vortex ρ with non-axisymmetric A-phase vortex core core breaking of parity 20 Normal 3He-B Pressure (bar) Fermi ∆B 10 liquid B-phase order parameter A-phase order parameter 0 0123 ∆A Temperature (mK) ρ Pekola, Simola, Hakonen, Krusius, et al., vortex with ferromagnetic A-phase core PRL 53, 584 (1984)

non-axisymmetric vortex

breaking of axial N =1/2 N =1/2 N = 1 symmetry pair of half-quantum vortices Witten superconducting cosmic string & v-vortex most symmetric 3He-B vortex most symmetric cosmic string ξ = 1/m ξ ξ ∆B Higgs H B-phase Higgs field order parameter

ρ ρ

additional breaking of symmetry in the core v-vortex: Witten string: breaking of parity breaking of electromagnetic symmetry

H ∆B B-phase Higgs field

A-phase order parameter HA ∆A order second Higgs field parameter ρ ρ

Salomaa & Volovik, PRL 51, 2040 (1983) E. Witten, Nucl. Phys. B 249, 557 (1985) experiment: magnetic core superconductivity along the core Hakonen, Krusius, Salomaa, et al., PRL 51, 1362 (1983) broken parity in the 3He-B vortex core

parity transformation ∆B ∆B B-phase B-phase A-phase order parameter order parameter order parameter ∆A

A-phase order ∆ parameter A

electric polarization in the core Bound states of fermions on v-vortex in 3He-B

M. A. Silaev, Spectrum of bound fermion states on vortices in 3He-B, JETP Lett. 2009 flat band zero energy states in symmetric N=1 3He-A vortex

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

Q=4 Q=3 Q=2 Q=1 Fermions on Q=0 branch form

Q=0 flat Fermi-band 1234 Q pz

Q=-1 (fermionic condensate or Khodel state) Q=-2 Q=-3 Q=-4

Kopnin & Salomaa, PRB 44, 9667 (1991) forces on vortices: three nondissipative forces + friction force acting on a vortex line

heat bath Iordanskii force velocity Kopnin force Gravitational vn Aharonov-Bohm Axial anomaly effect

FIordanskii = κ × ρn(vs - vn) FKopnin = κ × C(T)(vn - vL) momentum transfer from negative Aharonov-Bohm scattering energy states in the core to heat bath of quasiparticles on a vortex analog of baryogenesis vacuum vortex velocity velocity vs FMagnus = κ × ρ(vL - vs) vL momentum transfer between vortex and superfluid vacuum Magnus–Joukowski lifting force in classical hydrodynamics

Stokes friction force FMagnus + FIordanskii + FKopnin+ FStokes = 0 FStokes = −γ (vL - vn) Momentogenesis by N=2 vortex-skyrmion

m = (1/4π) ∫∫ dx dy ( l • ( ∂l /∂x × ∂l /∂y) ) = 1 vortex-skyrmion with N=2m=2 v l field s circulation quanta

z y l=l(r-vt) x vs

Momentum transfer from vacuum to the heat bath (matter) gives extra topological force on skyrmion (spectral-flow force)

3 • 2 3 2 3 3 F = ∫ d r P = (1/2π ) ∫ d r (B•E ) pF l = (1/2π ) h pF ∫ d r (∇× l • dl /dt ) l 2 3 ^ = 2π h (1/3π ) pF z × (vn − vL) Chiral anomaly: matter-antimatter asymmetry of Universe ( asymmetry) and Kopnin force

q

spectral flow

momentum from vacuum from vacuum of fermion zero modes spectral flow produces • • P = P n• B = B n• Σ a a Σ aa momentum (fermionic charge) B -- baryonic charge Pa -- a Y -- hypercharge ea -- effective electric charge a • • 2 2 P = (1/4π2) B•E P C e2 chiral B = (1/4π ) B •E Σ B C Y Σa aaa Y Y a aaa 3 anomaly applied to He-A equation applied to Standard Model for right (Adler, Bell, Jackiw) C = +1 for right Ca = +1 a -1 for left -1 for left

quasiparticles move from vacuum to the positive energy world, where they are scattered by quasiparticles in bulk and transfer momentum from vortex to normal component

this is the source of Kopnin spectral flow force Experimental and theoretical forces on a vortex

friction force theory (solid line) Manchester experiment (red circles) d||

0 0.4 0.6 0.8 1 T / Tc nondissipative ω0τ ‹‹ 1 forces Kopnin spectral flow d⊥–1 force almost compensates –1 contributions of Magnus + Iordanskii + Magnus + Iordanskii Kopnin (spectral-flow) ω0τ ›› 1 Magnus force Magnus + Iordanskii forces

1– d superfluid Reynolds number ⊥ Reα(T)= ≈ ω0τ d|| Re (0)= Reα(Tc) = 0 Reα(T ~0.6 Tc) ~ 1 α ∞ laminar vortex flow turbulent vortex flow turbulent and laminar vortex evolution

0 4 s 8 s 24 s Ω T=0.8 Tc Reα < 1

0 44 s 68 s 128 s 160 s

T T =0.4 c Reα > 1 TURBULENT VORTEX GENERATION turbulent regular q(T) 0.4 1 2 5

1.5

1.4

1.3 (rad/s) c Ω 1.2

1.1 P = 29.0 bar 10.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 T/Tc

0.5 Ton = 0.590 Tc 169 events σT = 0.033 Tc 640 events

fraction of events 0 [Finne et al. 0.4 0.5 0.6 0.7 T/Tc Nature 424, 1022 (2003)] Conclusion to topological medium part

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. 5. From helium liquids to dynamics of Lorentz invariant vacuum

quantum vacuum as self-sustained system

dynamics of cosmological `constant' dark energy problem

recipe to cook Universe

Dark Energy 70%

Dark Matter 30% Baryonic Matter 4%

Visible Matter 0,4%

Dark and Dark! What is the difference?

dark matter forms clusters like ordinary matter

cosmological constant Λ is possible candidate for dark energy

Λ= εDark Energy observational cosmology: dark energy vs dark matter

2 −29 3 εcrit = 3H /8πG = 10 g/cm H - Hubble parameter

luminosity distances Knop et al 2003

εDarkEnergy ∗ Acoustic peak εcrit Spergel et al 2003

of Galaxies Allen et al 2002

εMatter/εcrit Cosmological Term

* Original Einstein equations

1 Matter matter is a source of gravity field (Rµν - gµνR/2) = Tµν 8πG

* 1917: Einstein added the cosmological term 2 2 2 2 2 x1 + x2 + x3 + x4 = R 1 Matter (Rµν - gµνR/2) - Λgµν = Tµν 8πG it is finite cosmological constant but no boundaries ?!

& obtained static solution: Universe as 3D sphere

2 Λ = 0.5 εMatter = 1/(8πGR ) perfect Universe ! energy density of matter radius of 3D sphere

but it is curved! I would prefer to live in flat Universe Estimation of cosmological constant when I was discussing cosmological problems with Einstein, he remarked that the introduction of the cosmological term was the biggest blunder of his life -- George Gamow, My World Line, 1970

2 Λ = 0.5 εMatter = 1/(8πGR )

compare with observed Λ = 2.3 εMatter

order of magnitude is OK, why blunder? this was correct estimation of Λ

the first and the last one: after 90 years nobody could improve it arguments against Λ !

* 1917: de Sitter found stationary solution of Einstein equations without matter 1 (Rµν - gµνR/2) - Λgµν = 0 8πG no source of gravity field !

What? 1923: expanding version of de Sitter Universe: empty space gravitates !? R= exp (Ht) H2 = 3/(8πΛG)

* 1924: Hubble : Universe is not stationary Wenn schon keine quasi-statische Welt, * 1929: Hubble : recession of galaxies dann fort mit dem kosmologischen Glied. A. Einstein H. Weyl, 23 Mai 1923 Away with curvature ! and with cosmological term ! the question arises whether it is possible to represent the observed facts Wait ! without introducing a curvature at all. Einstein & de Sitter, PNAS 18 (1932) 213 Epoch of quantum mechanics: Λ as vacuum energy

1 Matter (Rµν - gµνR/2) - Λgµν = Tµν 8πG move Λ to the right 1 Matter (Rµν - gµνR/2) = Λgµν + Tµν 8πG physical vacuum as source of gravity quantum field is set of oscillators

vacuum is not empty ? En = hν(n + 1/2) Latin: vacuus empty n = 0 : zero point energy of quantum fluctuations it is filled with quantum oscillations

zero point energy has weight ?

you do not believe Einstein? energy must gravitate ! Equation of state of quantum vacuum 1 Vacuum Matter what is vacuum? (Rµν - gµνR/2) = Tµν + Tµν 8πG vacuum is Vacuum medium with equation of state Tµν = Λgµν ε = −p Λ = εvac = −pvac right ! energy density pressure of vacuum of vacuum pumping the vacuum by piston

ε > 0 `empty space' vacuum applied external force p =−ε < 0 F = εA, negative pressure Evac= εvacV

pvac = − dE/dV= − εvac ε < 0 `empty space' vacuum applied external force p =−ε > 0 F = εA, positive pressure How heavy is aether?

Luminiferous aether (photons) Dirac vacuum E = ck Σk (n(k) + 1/2) ck photonic vacuum: weight of photon vacuum weight k (k ) n(k)=0 of Dirac vacuum y z k ε + ε = (1/2)Σ E(k) - Σ E(k) = ( ν - ν )ck4 x zero point Dirac bosons fermions b f Planck

zero-point energy number of bosonic Planck scale occupied negative energy energy of Dirac vacuum fields number levels of Dirac fields 120 εzero point ~ 10 Λupper limit vacuum is too heavy !

maybe Λ = 0 ?!

supersymmetry: there is no supersymmetry below TeV symmetry between fermions and bosons

νb = νf Λ in supernova era Kepler's Supernova 1604 distant supernovae: accelerating Universe from ` De Stella Nova in Pede Serpentarii ' (Perlmutter et al., Riess et al.)

−123 Λexp = 2-3 εDark Matter = 10 εzero point

Λ = εvac = εDark Energy = 70% εDark Matter = 30% εVisible Matter = 0,4%

* *

Universe is flat !

you are right, but Λ is not zero −123 Λexp = 2-3 εDark Matter = 10 εzero point

it is easier to accept that Λ = 0 than 123 orders smaller

magic word: regularization 1 wisdom of particle physicist: − = 0 0

What can condensed matter physicist say on Λ ?

Why condensed matter ??! Cosmological constant paradox

−47 4 Λobservation = εDark Energy ~ 2−3 εDM ~ 10 GeV

4 76 4 Λtheory = εzero point energy ~ EPlanck ~ 10 GeV

−123 too bad for theory Λobservation ~ 10 ΛTheory

problems: * Why is vacuum not extremely heavy?

it is easier to accept that Λ=0 * Why is vacuum gravitating? Why is Λ non-zero? than 123 orders of magnitude smaller * Why is vacuum as heavy as (dark) matter ? −123 Λexp ~ 2-3 εDark Matter ~ 10 Λbare Λbare ~ εzero point

*it is easier to accept that Λ = 0 than 123 orders smaller 1 *magic word: regularization − = 0 wisdom of particle physicist: 0

*Polyakov conjecture: dynamical screeneng of Λ by infrared fluctuations of metric A.M. Polyakov Phase transitions and the Universe, UFN 136, 538 (1982) De Sitter space and eternity, Nucl. Phys. B 797, 199 (2008)

*Dynamical evolution of Λ similar to that of gap ∆ in superconductors after kick

V. Gurarie, Nonequilibrium dynamics of weakly and strongly paired superconductors: 0905.4498 A.F. Volkov & S.M. Kogan, JETP 38, 1018 (1974) Barankov & Levitov, ... what is natural value of cosmological constant ?

4 Λ = EPlanck Λ = 0 time dependent cosmological constant

4 Λ ∼ EPlanck Λ ~ 0 could be in early Universe should be in old Universe how to describe quantum vacuum & vacuum energy Λ ?

* quantum vacuum has equation of state w=−1 Λ = εvac = wvac Pvac

* quantum vacuum is Lorentz-invariant wvac = −1 * quantum vacuum is a self-sustained medium, which may exist in the absence of environment * for that, vacuum must be described by conserved charge q q is analog of particle density n in liquids

q must be Lorentz invariant L q = q

charge density n is not Lorentz invariant L n = γ(n + v.j)

Hawking suggested to introduce special field which describes the vacuum only does such q exist ? Hawking, Phys. Lett. B 134, 403 (1984) relativistic invariant conserved charges q

possible

αβ αβµν ∇α q = 0 ∇α q = 0

αβ αβ αβµν αβµν q = q g q = q e Duff & van Nieuwenhuizen Phys. Lett. B 94, 179 (1980)

impossible

α α ∇α q = 0 q = ? examples of vacuum variable q

4-form field Fκλµν = ∇[κ Aλµν]

1 1/2 2 κλµν Fκλµν = q (-g) eκλµν q = − Fκλµν F 24 condensates in QCD

q αβ = (gαµgβν−gανgβµ) q = = 0 12 αβ q 1/2 ~ αβ topological = (-g) eαβµν q = 24 αβ charge density

Einstein-aether theory (T. Jacobson, A. Dolgov)

∇µuν = q gµν thermodynamics in flat space the same as in cond-mat

conserved Q = dV q charge Q thermodynamic Lagrange multiplier Ω =E − µQ = dV (ε (q)− µq) potential or chemical potential µ

pressure P = − dE/dV= −ε + q dε/dq E = V ε(Q/V)

dΩ/dq = 0 equilibrium vacuum equilibrium self-sustained vacuum dε/dq = µ dε/dq = µ ε - q dε/dq = −P = 0 vacuum energy & cosmological constant equilibrium self-sustained vacuum vacuum variable dε/dq = µ 2 q ~ µ ~ EPlanck in equilibrium ε - q dε/dq = −P = 0 self-sustained vacuum energy 4 of equilibrium ε (q) ~ EPlanck self-sustained vacuum pressure P = −ε + q dε/dq= -Ω cosmological Λ = Ω =ε − µ q constant P = −Ω cosmological equation of state constant Λ = ε − µq = 0 in equilibrium self-sustained self-tuning: vacuum two Planck-scale quantities 4 4 cancel each other E E in equilibrium self-sustained vacuum Planck Planck dynamics of q in flat space whatever is the origin of q the motion equation for q is the same

action S= dV dt ε (q) motion equation ∇κ (dε/dq) = 0 solution dε/dq = µ integration constant µ in dynamics becomes chemical potential in thermodynamics

4-form field Fκλµν as an example of conserved charge q in relativistic vacuum 1 2 κλµν Maxwell equation q = − Fκλµν F 24

Fκλµν = ∇[κ Aλµν] κλµν −1 ∇κ (F q dε/dq) = 0 ∇κ (dε/dq) = 0 F κλµν = q eκλµν dynamics of q in 4-form field and chiral condensate action

4 1/2 S = d x (-g) [ ε (q) + K(q)R ] + Smatter gravitational coupling K(q) is determined by vacuum and thus depends on vacuum variable q

q becomes dynamical only when K depends on q motion dε/dq + R dK/dq = µ integration equation constant

Einstein K(Rg − 2R ) + (ε − µq)g − 2( ∇ ∇ − g ∇λ∇ ) K = T equations µν µν µν µ ν µν λ µν

Einstein cosmological matter ∇ T µν = 0 tensor term = ε gµν µ case of K=const restores original Einstein equations

1 K = G - Newton constant 16πG

motion equation dε/dq = µ q = const 1 original (Rg − 2R ) + Λ g = T Λ = ε − µq Einstein 16πG µν µν µν µν equations Λ - cosmological constant Minkowski solution

Maxwell equations dε/dq + R dK/dq = µ

Einstein K(Rg − 2R ) + (ε − µq)g − 2( ∇ ∇ − g ∇λ∇ ) K = T equations µν µν µν µ ν µν λ µν

Einstein cosmological matter tensor term µν ∇µ T = 0

Minkowski d /dq 4 vacuum R = 0 ε = µ vacuum energy in action: ε (q) ~ EPlanck solution Λ = ε(q) − µq = 0 thermodynamic vacuum energy: ε − µq = 0 1 q2 q4 Model vacuum energy ε (q) = − + ( 2 4 ) 2χ q0 3q0

Minkowski dε/dq = µ q = q0 vacuum 1 solution ε − µq = 0 µ = µ0 = − 3χq0

ε(q) Ω(q) = ε(q) − µ0q

q q q 0 min q 0 q empty self-sustained space vacuum q = 0 q = q0

1 dV 1 vacuum q2 d2 /dq2 vacuum χ = − = ( ε )q=q0 > 0 compressibility V dP χ stability Minkowski vacuum (q-independent properties)

2 Λ = Ωvac = −Pvac Pvac = − dE/dV= − Ωvac <(∆Pvac) > = T/(Vχvac) 2 2 energy density pressure χvac = −(1/V) dV/dP <(∆Λ) > = <(∆P) > of vacuum of vacuum compressibility of vacuum pressure fluctuations

natural value of Λ natural value of χvac volume of Universe determined by macroscopic determined by microscopic is large: physics physics 1/χ = q2 d2ε /dq2 2 Λ = 0 vac vac V > TCMB/(Λ χvac) −4 χvac ~ E 28 Planck V > 10 Vhor dynamics of q in curved space: relaxation of Λ at fixed µ=µ0

Maxwell equations dε/dq + R dK/dq = µ0

Einstein λ matter K Rg − 2R + g Λ(q) − 2 ∇ ∇ − g ∇ ∇ K = T equations ( µν µν) µν ( µ ν µν λ) µν

Λ(q) = ε(q) − µ0q dynamic solution 2 . sin ωt 2 sin ωt a(t) 2 q(t) − q0 ~ q0 Λ(t) ~ ω H(t) = = (1− cos ωt) t t2 a(t) 3t

ω ~ EPlanck

similar to with mass M ~ EPlanck A.A. Starobinsky, Phys. Lett. B 91, 99 (1980) Relaxation of Λ (generic q-independent properties) 2 . 2 sin ωt a(t) 2 sin ωt Λ(t) ~ ω H(t) 2 = = (1− cos ωt) G(t) = GN (1 + ) t a(t) 3t ωt cosmological "constant" Hubble parameter Newton "constant" ω ~ EPlanck

4 <Λ(tPlanck)> ~ EPlanck

Λ(t = ∞) = 0

natural solution of the main cosmological problem ? Λ relaxes from natural Planck scale value to natural zero value present value of Λ dynamics of Λ: from Planck to present value

2 2 sin ωt Λ(t) ~ ω ω ~ EPlanck t2

4 <Λ(tPlanck)> ~ EPlanck

2 2 −120 4 <Λ(tpresent)> ~ EPlanck / tpresent ~ 10 EPlanck

coincides with present value of dark energy something to do with coincidence problem ? Dynamical evolution of Λ similar to that of gap ∆ in superconductors after kick dynamics of Λ in cosmology dynamics of ∆ in superconductor

sin2ωt sin ωt Λ(t) ~ ω2 δ|∆(t)|2 ~ ω3/2 t2 t1/2

ω ~ EPlanck ω = 2∆

F.R. Klinkhamer & G.E. Volovik Dynamic vacuum variable & equilibrium approach in cosmology t = + PRD 78, 063528 (2008) t = 0 ∞ Self-tuning vacuum variable & cosmological constant, intial states: final states: PRD 77, 085015 (2008) 4 nonequilibrium vacuum with Λ ∼ EPlanck equilibrium vacuum with Λ = 0

superconductor with nonequilibrium gap ∆ ground state of superconductor sin2ωt ε(t) − εvac ~ ω t V. Gurarie, Nonequilibrium dynamics of weakly and strongly paired superconductors: 0905.4498 A.F. Volkov & S.M. Kogan, JETP 38, 1018 (1974) Barankov & Levitov, ... reversibility of the process

sin2ωt sin ωt Λ(t) ~ ω2 δ|∆(t)|2 ~ ω3/2 t2 t1/2

ω ~ EPlanck ω = 2∆

reversible energy transfer reversible energy transfer from vacuum to gravity from coherent degree of freedom (vacuum) to particles (Landau damping)

inverse process in contracting Universe ? t = − ∞ t = 0 t = + ∞ Minkowski vacuum as attractor allow µ to relax (Dolgov model)

q(t)

q0

t both µ & q relax to equilibrium values µ0 & q0 cosmological constant Λ relaxes to zero

F. Klinkhamer & GV Towards a solution of the cosmological constant problem JETP Lett. 91, 259 (2009) properties of relativistic quantum vacuum as a self-sustained system

* quantum vacuum is characterized by conserved charge q

q has Planck scale value in equilibrium 4 ε(q)~ EPlanck * vacuum energy has Planck scale value in equilibrium but this energy is not gravitating Tµν = Λgµν = ε(q)gµν * gravitating energy is thermodynamic vacuum energy Tµν = Λgµν = Ω(q)gµν Ω(q) = ε − q dε/dq

* thermodynamic energy of equilibrium vacuum Ω(q0) = ε(q0) − q0 dε/dq0 = 0