The Universe in a helium droplet:from semi-metals & topological insulators to particle physics & cosmology

Landau InstituteG. Volovik

Zelenogorsk, July 12, 2010

3. Dirac (Fermi) points as topological objects2. Fermi surface as topological object

5. Thermodynamics & dynamics of quantum vacuum

4. Fully gapped topological media

* semimetals, cuprate superconductors, graphene, superfluid 3He-A, vacuum of Standard Model of particle physics in massless phase* topological invariants for gapless 2D and 3D topological matter* emergent physical laws near Dirac point (Fermi point)

* superfluid 3He-B, topological insulators, chiral superconductors, vacuum of Standard Model of particle physics in present massive phase

* intrinsic QHE & spin-QHE, Chern-Simons action, chiral anomaly, vortex dynamics, ...

* topological invariants for gapped 2D and 3D topological matter* edge states, fermion zero modes on vortices & Majorana fermions

* vacuum energy & cosmological constant* cosmology as relaxation to equilibrium

1. Introduction: helium liquids & Universe

* thermodynamics of quantum vacuum & cosmological constant* quantum vacuum as topological medium* effective quantum field theories

3He

unity of physics

Condensed Matter

lowdimensional

systems

Bosecondensates

high-T & chiral super-

conductivity

blackholes

vacuumgravity

cosmic strings

physicalvacuum

neutronstars nuclear

physics

hydrodynamics

disorderphase

transitions

High EnergyPhysics

PlasmaPhysics

PhenomenologyQCD

Gravity

cosmologyFilms: FQHE,Statistics & charge ofskyrmions & vorticesEdge states; spintronics1D fermions in vortex coreCritical fluctuations

Superfluidity of neutron star vortices, glitches

shear flow instability

Nuclei vs 3He droplet Shell model Pair-correlations Collective modes

quarkmatter

Quark condensate Nambu--Jona-Lasinio Vaks--Larkin Color superfluidity Savvidi vacuum Quark confinement, QCD cosmology Intrinsic orbital momentum of quark matter

General; relativistic;spin superfluiditymulti-fluid rotating superfluid Shear flow instability Magnetohydrodynamic Turbulence of vortex lines propagating vortex frontvelocity independent Reynolds number

Relativistic plasmaPhoton massVortex Coulomb plasma

Mixture of condensates Vector & spinor condensatesBEC of quasipartcles, magnon BEC & lasermeron, skyrmion, 1/2 vortex

Kibble mechanism Dark matter detector Primordial magnetic fieldBaryogenesis by textures & strings Inflation Branes matter creation

Torsion & spinning strings, torsion instantonFermion zero modes on strings & walls

Antigravitating (negative-mass) stringGravitational Aharonov-Bohm effect

Domain wall terminating on stringString terminating on domain wall

Monopoles on string & BoojumsWitten superconducting string

Soft core string, Q-balls Z &W strings

skyrmionsAlice stringPion string

Cosmological &Newton constants

dark energydark matter

Effective gravityBi-metric &

conformal gravityGraviton, dilatonSpin connection

Rotating vacuumVacuum dynamics

conformal anomaly

Emergence & effective theoriesVacuum polarization, screening - antiscreening, running couplingSymmetry breaking (anisotropy of vacuum)Parity violation -- chiral fermionsVacuum instability in strong fields, pair productionCasimir force, quantum frictionFermionic charge of vacuumHiggs fields & gauge bosonsMomentum-space topologyHierarchy problem, Supersymmetry Neutrino oscillations Chiral anomaly & axions Spin & isospin, String theory CPT-violation, GUT

Gap nodes Low -T scaling mixed stateBroken time reversal1/2-vortex, vortex dynamics

ergoregion, event horizonHawking & Unruh effects

black hole entropy Vacuum instability

quantum phase transitions& momentum-space topology

random anisotropy Larkin- Imry-Ma topological insulator classes of random matrices

3He Grand Unification

From Landau Fermi liquid & two-fluid hydrodynamicsto physics of quantum vacuum & cosmology

Einstein general relativityLandau two-fluid hydrodynamics

Landau theory of Fermi liquid as effective theory of fermionic vacuum

Standard Model + gravitysuperfluid 3He-A

planar phase & 3He-B Dirac vacuum as topological insulator

* liquid 3He and Universe

* extension of Landau theory from Fermi surface to Fermi point

* from Fermi point to QHE & topological insulators

quantum vacuum as self-sustained system cosmology as approach to equilibrium

* from helium liquids to dynamics of quantum vacuum & cosmology

* superfluid 4He and Universe

Helium liquids & the Universe

Topology: you can't comb the hair on a ball smooth, anti-Grand-Unification

Thermodynamics is the only physical theory of universal content

I think it is safe to say that no one understands Quantum Mechanics

Richard Feynman

Albert Einstein

3+1 sources of effective Quantum Field Theories

Symmetry: conservation laws, translational invariance, spontaneously broken symmetry, Grand Unification, ...

missing ingredientin Landau theories

effective theoriesof quantum liquids:

two-fluid hydrodynamicsof superfluid 4He

& Fermi liquid theory ofliquid 3He

Lev Landau

many body systems are simple at low energy & temperatureLandau view on a many body system

ground state+

elementaryexcitations

Universehelium liquids

vacuum+

elementaryparticles

quasiparticles = elementary particles

ground state = vacuum

Landau, 1941

weakly excited state of liquid can be considered as systemof "elementary excitations"

equally applied to:superfluids,

solids,&

relativistic quantum vacuum

why is low energy physicsapplicable to our vacuum ?

classical low-energy propertyof quantum vacuum

classical low-energy propertyof quantum liquids

Landau equations & Einstein equationsare effective theories

describing dynamics ofmetric field + matter (quasiparticles)

Landau equations Einstein equations

1. superfluid 4He & effective theories of hydrodynamic type

Einstein general relativityLandau two-fluid hydrodynamics

N.G. Berloff & P.H. RobertsNonlocal condensate models of superfluid helium

J. Phys. A32, 5611 (1999)

∆ ∼ h2ρ2/3m−5/3

Landau quasiparticles in superfluid 4He: phonons & rotons

this is vortex ringof minimal size

Landau estimateof vortex gap is correct !

this is energy of vortex ring

of minimal size

Landau 1941 roton:quasiparticles of vortex spectrum

Landau gap in vortex spectrum

phononlong waveexcitation Landau 1946 roton

short wave quasiparticle

two types of rotons in Landau theory in original 1941 theory: quantum of vortex motion

in modified 1946 theory: short wavelength quasiparticle

Effective metric in Landau two-fluid hydrodynamics

ds2 = − c2dt2 + (dr − vsdt)2 ds2 = gµυdxµdxν

gµυpµpν = 0

g00 = −1 g0 i = −vsi gij = c2δij−vs

ivsj

c speed of sound

vs superfluid velocity

Doppler shifted phonon spectrum in moving superfluid

reference frame for phonon is draggedby moving liquid

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

(E − p.vs)2 − c2 p2 = 0

E − p.vs= cp

pν =(−E, p)

review:Barcelo, Liberati & Visser,

Analogue GravityLiving Rev. Rel. 8 (2005) 12

move p.vs to the left

effective metric

E = cp + p.vs

take square

Landau critical velocity = black hole horizon

superfluid 4He Gravity

acoustic horizon black hole horizon

v2(r) = 2GM = c2 rh____ __r r

v2(r) = c2 rh__r

vs(r)

superfluidvacuum

gravity v(r)

vs= c black hole horizon at g00= 0

where flow velocity(velocity of local frame in GR)reaches Landau critical velocity

v(rh) = vLandau = c

v(rh)=c

r = rh

Painleve-Gulstrand metric

Schwarzschild metric

(Unruh, 1981)

Hawking radiation is phonon/photon creationabove Landau critical velocity

ds2 = - dt2 (c2-v2) + 2 v dr dt + dr2 + r2dΩ2

g00 g0r

ds2 = - dt2 (c2-v2) + dr2 /(c2-v2) + r2dΩ2

g00 grr

acoustic gravity general relativity

Superfluids Universe

ds2 = gµυdxµdxν= 0

geometry of space timefor matter

geometry of effective space timefor quasiparticles (phonons)

geodesics for phonons

Landau two-fluid equations

equationfor normalcomponent

equationsfor superfluidcomponent

equationfor matter

Einstein equations of GR

geodesics for photons

dynamic equations

for metric field gµυ

metric theories of gravity

gµυ

ρ + .(ρvs + PMatter) = 0 .vs + (µ + vs

2/2) = 0 . (Rµν - gµνR/2) = Tµν

8πG

1 Matter

1/2 of GR Tµν = 0 ;ν Matter

is gravity fundamental ?

as hydrodynamics ?

ρ + .(ρvs + PMatter) = 0 .vs + (µ + vs

2/2) = 0 .

classical generalrelativity

quantum vacuumsuperfluid helium

underlying microscopicquantum systemat high energy

emergentlow-energy

effective theory

classical 2-fluidhydrodynamics

it may emerge as classical outputof underlying quantum system

(Rµν - gµνR/2) - Λgµν = Tµν Matter

8πG

1

message from: liquid helium to: gravity

ν Tµν = 0 Matter

many body systems are simple at low energy & temperatureapplication of Landau view on a many body system to Universe

ground state+

elementaryexcitations

Universehelium liquids

vacuum+

elementaryparticles

Landau, 1941

weakly excited state of liquid can be considered as systemof "elementary excitations"

weakly excited state of Universeshould be also simple

but why is low energy physicsapplicable to our Universe ?

Eew ~ 10−16EPlanck

H ~ 10−60EPlanck

high-energy physics isextremely ultra-low energy physics

high-energy physics & cosmologybelong to ultra-low temperature physics

cosmology is extremely ultra-low frequency physics

our Universe is extremely close to equilibrium ground state

We should study general properties of equilibrium ground states - quantum vacua

highest energy in accelerators

Eew ~1 TeV ~ 1016K

TCMBR ~ 10−32EPlanck

T of cosmic background radiation

TCMBR ~ 1 K

EP =(hc5/G)1/2~1019 GeV~1032K

characteristic high-energy scale in our vacuum is Planck energy

cosmological expansion

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

Hubble parameter

Hubble law

even at highest temperaturewe can reach

e−m/T =10 =0−1016

Why no freezing at low T?

m ~ EPlanck ~1019 GeV~1032K

natural masses of elementary particlesare of order of characteristic energy scale

the Planck energy

10−123, 10−1016

another great challenge?

everything should be completely frozen out

T ~ 1 TeV~1016K

main hierarchy puzzle

mquarks , mleptons <<< EPlanck

cosmological constant puzzle

Λ <<<< EPlanck

Λ = 0mquarks = mleptons = 0

its emergent physics solution: its emergent physics solution:

reason:momentum-space topology

of quantum vacuum

reason:thermodynamics

of quantum vacuum

4

Why no freezing at low T?

massless particles & gapless excitationsare not frozen out

who protects massless excitations?

we live becauseFermi point is the hedgehog

protected by topology

px

py

pz

Fermi point:hedgehog in momentum space

gapless fermions livenear

Fermi surface & Fermi pointTopology

no life withouttopology ?

who protects massless fermions?

hedgehog is stable:one cannot comb the hair

on a ball smooth

Life protection

Topology in momentum spaceThermodynamics

responsible for propertiesof vacuum energy

problems of cosmological constant:perfect equilibrium Lorentz invariant vacuum

has Λ = 0;

perturbed vacuum has nonzero Λon order of perturbation

responsible for properties offermionic and bosonic quantum fields

in the background of quantum vacuum

Fermi point in momentum spaceprotected by topology is a source of

massless Weyl fermions, gauge fields & gravity

tools

mquarks , mleptons <<< EPlanckΛ <<<< EPlanck4

Horava, Kitaev, Ludwig, Schnyder, Ryu, Furusaki, ...

Quantum vacuum as topological substance: universality classes

universality classes of gapless vacua

physics at low T is determined by gapless excitations

topology is robustto deformations:

nodes in spectrumsurvive interaction

vacuum with Fermi surface vacuum with Fermi point

two major universality classes of fermionic vacua

gµν(pµ- eAµ - eτ .Wµ)(pν- eAν - eτ .Wν) = 0gravity emerges from

Fermi pointanalog of

Fermi surface

2. Liquid 3He & effective theory of vacuum with Fermi surface

Landau theory of Fermi liquid Standard Model + gravity

Topological stability of Fermi surface

Fermi surface:vortex ring in p-space

∆Φ=2π

pxp

F

py

(pz)

ω

Fermi surfaceE=0

Energy spectrum ofnon-interacting gas of fermionic atoms

E<0occupied

levels:Fermi sea

p=pF

phase of Green's function

Green's function

E>0empty levels

G(ω,p)=|G|e iΦ

G-1= iω − E(p)

has winding number N = 1

E(p) = p2

2m– µ = p

2

2m – pF

2

2m

no!it is a vortex ring

is Fermi surface a domain wall

in momentum space?

Route to Landau Fermi-liquid

Fermi surface:vortex line in p-space

∆Φ=2π

pxp

F

py

(pz)

ωSure! Because of topology:

winding number N=1 cannot change continuously,interaction cannot destroy singularity

G(ω,p)=|G|e iΦ

is Fermi surface robust to interaction ?

then Fermi surface survives in Fermi liquid ?

all metals have Fermi surface ...

Stability conditions & Fermi surface topologies in a superconductor

Gubankova-Schmitt-Wilczek, Phys.Rev. B74 (2006) 064505

Not only metals.

Some superconductore too!

Landau theory of Fermi liquidis topologically protected & thus is universal

Topology in r-space

quantized vortex in r-space ≡ Fermi surface in p-space

windingnumberN1 = 1

classes of mapping S1 → U(1) classes of mapping S1 → GL(n,C)

Topology in p-space

vortex ring

scalar order parameterof superfluid & superconductor

Green's function (propagator)

∆Φ=2π

Ψ(r)=|Ψ| e iΦ G(ω,p)=|G|e iΦ

y

x

z

Fermi surface

∆Φ=2π

pxp

F

py

(pz)

ωhow is it in p-space ?

space ofnon-degenerate complex matrices

manifold ofbroken symmetry vacuum states

homotopy group π1

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

magnetic hedgehog vs right-handed electron

hedgehog in r-space

z

x

y

px

py

σ(r)=r ^ ^ σ(p) = p

H = + c σ .p

pz

hedgehog in p-space

right-handed electron = hedgehog in p-space with spines = spins

close to Fermi point

again no difference ?

Landau CP symmetryis emergent

right-handed and left-handedmassless quarks and leptons

are elementary particlesin Standard Model

p

x

E=cp

py

(pz)

where are Dirac particles?

p

x

E=cp

py

(pz)

p

x

E=cp

py

(pz)

Dirac particle - composite objectmade of left and right particles

py

(pz)

px

E

E2 = c2p2 + m2

mixing of left and right particlesis secondary effect, which occurs

at extremely low temperature

Tew ~ 1 TeV~1016K

N3 =−1

over 2D surface Sin 3D p-space

8πN3 = 1 e

ijk ∫ dSk g . (∂pi g × ∂pj g)

2 p

1 p

N3 =1

Gap node - Fermi point(anti-hedgehog)

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

p

x

E

py

(pz)

H =c(px + ipy)

c(px – ipy)

p2

2m

p2

2m)) g3(p) g1(p) +i g2(p)

g1(p) −i g2(p) −g3(p) = ))− µ

+ µ−

S2

Gap node - Fermi point(hedgehog)

left-handedparticles

right-handedparticles

H = N3 c τ .p

E = ± cp

close to nodes, i.e. in low-energy cornerrelativistic chiral fermions emerge

2 p

1 p

H =c(px + ipy)

c(px – ipy)

p2

2m

p2

2m)) g3(p) g1(p) +i g2(p)

g1(p) −i g2(p) −g3(p) = ))− µ

+ µ− = τ .g(p)

emergence of relativistic QFT near Fermi (Dirac) point

original non-relativistic Hamiltonian

chirality is emergent ??

what else is emergent ?relativistic invariance as well

p

x

E

py

(pz)

top. invariant determines chiralityin low-energy corner

N3 =−1

N3 =1

two generic quantum field theories of interacting bosonic & fermionic fields

bosonic collective modes in two generic fermionic vacua

Landau theory of Fermi liquid Standard Model + gravity

collective Bose modes:

propagatingoscillation of position

of Fermi pointp → p - eA

form effective dynamicelectromagnetic field

propagatingoscillation of slopes

E2 = c2p2 → gikpi pk

form effective dynamicgravity field

AFermipoint

collective Bose modesof fermionic vacuum:

propagatingoscillation of shape

of Fermi surface

Fermisurface

Landau, ZhETF 32, 59 (1957)

relativistic quantum fields and gravity emerging near Fermi point

px

py

pz

hedgehog in p-space

effective metric:emergent gravity

emergent relativitylinear expansion nearFermi surface

linear expansion nearFermi point

effectiveSU(2) gauge

field

effectiveisotopic spin

gµν(pµ- eAµ - eτ .Wµ)(pν- eAν - eτ .Wν) = 0

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

Atiyah-Bott-Shapiro construction:linear expansion of Hamiltonian near the nodes in terms of Dirac Γ-matrices

H = eik Γi .(pk − pk) 0

E = vF (p − pF)

effectiveelectromagnetic

field

effectiveelectric charge

e = + 1 or −1

gravity & gauge fieldsare collective modes

of vacua with Fermi point

emergent Landautwo-fluid hydrodynamics

emergent general covariance& general relativity

EPlanck >> ELorentz EPlanck << ELorentz

ELorentz << EPlanck

ELorentz ~ 10−3 EPlanck

ELorentz >> EPlanck

ELorentz > 109 EPlanck

crossover from Landau 2-fluid hydrodynamics to Einstein general relativitythey 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

Universe3He-A with Fermi point

dimensional reduction

3+1vacuum with Fermi point

Fully gapped 2+1 system

4. From Fermi point to intrinsic QHE & topological insulators

over 2D surface Sin 3D momentum space

over the whole 2D momentum spaceor over 2D Brillouin zone

8πN3 = 1 e

ijk ∫ dSk g . (∂pi g × ∂pj g)

4πN3 = 1 ∫ dpxdpy g . (∂px g × ∂py g) ~

py

px

topological insulators & superconductors in 2+1

How to extract useful information on energy states from Hamiltonianwithout solving equation

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

H =c(px + ipy)

c(px – ipy)

p2

2m

p2

2m)) − µ

+ µ−

p2 = px2 +py

2

Hψ = Eψ

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

Topological invariant in momentum space

N3 (µ > 0) = 1~

4πN3 = 1 ∫ d2p g . (∂px g × ∂py g) ~

g (px,py)

unit vector

sweeps unit sphere

py

px

g

fully gapped 2D state at µ = 0

g3(p) g1(p) +i g2(p)

g1(p) −i g2(p) −g3(p) H = ))H =

c(px + ipy)

c(px – ipy)

p2

2m

p2

2m)) − µ

+ µ− = τ .g(p)

p2 = px2 +py

2

GV, JETP 67, 1804 (1988)

quantum phase transition:from topological to non-topologicval insulator/superconductor

Topological invariant in momentum space

intermediate state at µ = 0 must be gapless

4πN3 = 1 ∫ d2p g . (∂px g × ∂py g) ~

H =c(px + ipy)

c(px – ipy)

p2

2m

p2

2m)) g3(p) g1(p) +i g2(p)

g1(p) −i g2(p) −g3(p) = ))− µ

+ µ− = τ .g(p)

N3 = 1 ~

N3~

N3 = 0 ~

µ

quantum phase transition

µ = 0

∆N3 = 0 is origin of fermion zero modesat the interface between states with different N3

∼∼

topologicalinsulator

trivialinsulator

p-space invariant in terms of Green's function & topological QPT

film thickness

gapgap

gap

quantum phase transitionsin thin 3He-A film

plateau-plateau QPTbetween topological states

QPT from trivialto topological state

N3 = 2 ~

N3 = 0 ~

N3 = 4 ~

N3 = 6 ~

a

a1 a2 a3

24π2N3= 1 e

µνλ tr ∫ d2p dω G ∂µ G-1 G ∂ν G-1G ∂λ G-1~

transition between plateausmust occur via gapless state!

GV & YakovenkoJ. Phys. CM 1, 5263 (1989)

q - parameter of systemqc

quantum phase transition at q=qc

other topological QPT:Lifshitz transition,

transtion between topological and nontopological superfluids,plateau transitions,

confinement-deconfinement transition, ...

example: QPT between gapless & gapped matter

topological quantum phase transitions

transitions between ground states (vacua) of the same symmetry,

but different topology in momentum space

no symmetry changealong the path

broken symmetry

different asymptotes

when T => 0

T (temperature)

T n e −∆/Tqc

Tno change of symmetry

along the path

topologicalsemi-metal

topologicalinsulator

QPT interruptedby thermodynamic transitions

q

q

line of1-st order transition

2-nd order transitionT

q

line of1-st order transition

line of2-nd order transition

T

Fully gapped 3+1 system

Majorana fermions on the surfaceand in the vortex cores

Fully gapped 2+1 system

Zero energy states on surface of topological insulators & superfluids

4πN3 = 1 ∫ dpxdpy g . (∂px g × ∂py g) ~

py

px

py - component

x

y

interface between two 2+1 topological insulators or gapped superfluids

H =c(px + i py tanh x )

c(px – i py tanh x )

p2

2m

p2

2m)) − µ

+ µ−

px+ipypx − ipy

x 0

N3 = +1 N3= −1 ~~

px - component

gapped state

gaplessinterface

gapped state

inter

faceN3 = N−

~N3 = N+ ~

* domain wall in 2D chiral superconductors:

Edge states at interface between two 2+1 topological insulators or gapped superfluids

0 py

E(py)

left movingedge states

occupied

empty

Index theorem:number of fermion zero modes

at interface:

ν = N+ − N−

gapped stategapped state

y

inter

faceN3 = N−

~N3 = N+ ~

GV JETP Lett. 55, 368 (1992)

0 py

E(py)x

y y

curre

nt

curre

nt

N3 = 0 ~ N3 = 0

~

Edge states and currents

2D topologicalinsulator

2D non-topologicalinsulator

or vacuum

2D non-topologicalinsulator

or vacuum

current Jy = Jleft +Jright = 0

left movingedge states

occupied

emptyempty

occupied

0 py

E(py)

right movingedge states

N3 = 1 ~

0 py

E(py)−V/2 E(py)+V/2

Edge states and Quantum Hall effect

2D topologicalinsulator

2D non-topologicalinsulator

or vacuum

2D non-topologicalinsulator

or vacuum

current Jy = Jleft +Jright = σxyEy

left movingedge states

0 py

V/2

V/2

−V/2

−V/2

right movingedge states

apply voltage V

4πσ

xy = e

2 N3

~

x

y y

curre

nt

curre

ntN3 = 0 ~

N3 = 0 ~

N3 = 1 ~

N

a

2

4

6

8

film of topological quantum liquid

Intrinsic quantum Hall effect & momentum-space invariant

4πσ

xy = e

2 N

film thickness

quantized intrinsic Hall conductivity(without external magnetic field)

p-space invariant r-space invariant

4π = e

2 N3

Ey electric current Jx = δS

CS / δA

x

z

x , J ^

y , E^

Aµ- electromagnetic field

16πS

CS = N

eµνλ ∫ d2x dt A

µ F

νλ

e2

~

~

~

~

GV & YakovenkoJ. Phys. CM 1, 5263 (1989)

3

3

3

general Chern-Simons terms & momentum-space invariant(interplay of r-space and p-space topologies)

16πS

CS = 1 N

3IJ eµνλ ∫ d2x dt A

µI F

νλJ

ΚI

- charge interacting with gauge field Aµ

I

Κ=e for electromagnetic field Aµ

r-space invariantp-space invariant protected by symmetry

24π2 1 e

µνλtr [

∫d2p dω Κ

I Κ

J G ∂µ G-1 G ∂ν G-1G ∂λ G-1]N

3IJ =

gauge fields can bereal, artificial or auxiliary

Κ=σz for effective spin-rotation field Aµ

z ( A0z= γH z) ^

id/dt −γσ.H = id/dt − σ.A0

applied Pauli magnetic field plays the role of components of effective SU(2) gauge field Aµ

i

^ ^

~

~

Intrinsic spin-current quantum Hall effect & momentum-space invariant

16πS

CS = 1 N

3IJ eµνλ ∫ d2x dt A

µI F

νλJ

4π = 1 (γN

ss dHz/dy + N

se Ey) spin current J

xz = δS

CS / δA

xz

spin-spin QHE spin-charge QHE

2D singlet superconductor:

film of planar phase of superfluid 3He

s-wave: Nss = 0px + ipy: Nss = 2dxx-yy + idxy : Nss = 44π

σxy

= Nss spin/spin

4πσ

xy = Nse

spin/charge GV & YakovenkoJ. Phys. CM 1, 5263 (1989)

~

planar phase film of 3He & 2D topological insulator

spin quantum Hall effect

4π = 1 N

se Ey spin current J

xz = δS

CS / δA

xz

spin-charge QHE

4πσ

xy = Nse

spin/charge GV & YakovenkoJ. Phys. CM 1, 5263 (1989)

H =c(px + i py σz )

c(px – i py σz)

p2

2m

p2

2m)) − µ

+ µ− H =

c(σx px + σy py)

c(σx px + σy py)

p2

2m

p2

2m)) − µ

+ µ−

N3 = −1~ −

N3 = +1~ +

N3 = ~ −

N3 = ~

N3 + 0~ +

N3 = ~ −

N3Κ = ~

N3 − 2~+

Nse

= N3Κ = 2~

24π2 1 e

µνλtr [

∫d2p dω Κ G ∂µ G-1 G ∂ν G-1G ∂λ G-1]N

3K = Κ = τ3σz Κ = σz

~

24π2 1 e

µνλtr [

∫d2p dω G ∂µ G-1 G ∂ν G-1G ∂λ G-1] = 0 N

3 =

~

Intrinsic spin-current quantum Hall effect & edge state

4π = 1 (γN

ss dHz/dy + N

se Ey) spin current J

xz

spin-charge QHE

4πσ

xy = Nse

spin/charge

electric current is zerospin current is nonzero

V/2−V/2 x

y y

curre

nt

curre

ntN

se= 2

Nse

= 0Nse

= 0

spin

curre

nt

spin

curre

nt

py

E(py)−V/2 E(py)+V/2

left movingspin up

right movingspin down

py

V/2

−V/2 right movingspin up

left movingspin down

3D topological superfluids/insulators/semiconductors

gapless topologicallynontrivial vacua

3He-A, Standard Model above electroweak transition,

semimetals

3He-B, Standard Model below electroweak transition,

topological insulators (Be2Se3 , ...),triplet & singlet color/chiral superconductors, ...

fully gapped topologicallynontrivial vacua

E (p)

p

conduction electron band, q=-1

3 conduction bandsof d-quarks

electric charge q=-1/3

3 conduction bandsof u-quarks, q=+2/3

3 valence u-quark bandsq=+2/3

3 valence d-quark bands q=-1/3

valence electron band, q=-1

Quantum vacuum: Dirac sea

electric charge of quantum vacuum Q= Σ qa = N [-1 + 3×(-1/3) + 3×(+2/3) ] = 0

a

Present vacuum as semiconductor or insulator

neutrino band, q=0

neutrino band, q=0

dielectric and magneticproperties of vacuum(running coupling constants)

* Standard Model vacuum as topological insulator

superfluid 3He-B, topological insulator Bi2Te3 , present vacuum of Standard Model

Standard Model vacuum:

Topological invariant protected by symmetry

8 massive Dirac particles in one generation

24π2NΚ= 1 e

µνλ tr ∫ dV Κ G ∂µ G-1 G ∂ν G-1G ∂λ G-1

over 3D momentum space

GΚ =+/− ΚG

Κ=γ5 Gγ5 =− γ5G

G is Green's function at ω=0, Κ is symmetry operator

fully gapped 3+1 topological matter

NΚ = 8ng

topological superfluid 3He-B

topological 3He-B

topological superfluid

non-topological superfluid

non-topological superfluid

µ

Dirac Dirac

NΚ = +2 NΚ = 0

NΚ = 0 NΚ = −2

1/m*

0NΚ = −1 NΚ = +1

p2

2m*– µ

p2

2m*+ µ–

cBσ.p

cBσ.p( (Η =

– Μ

+ Μ

cBσ.p

cBσ.p( (Η =

1/m* = 0

GV JETP Lett. 90, 587 (2009)

K = τ2

Dirac vacuum

p2

2m*– µ( )= τ3 +cBσ.p τ1

Hτ2 =− τ2H

Boundary of 3D gapped topological superfluid

vacuum

z

z

NΚ = 0

x,y

0

NΚ = 0

NΚ = +2

p2

2m*– µ+U(z)

p2

2m*+ µ–U(z)–

cBσ.p

cBσ.p( (Η =

0

µ–U(z)

Majoranafermionson wall

3He-B

NΚ = +2

helical fermions

spectrum of Majorana fermion zero modes

Hzm = cB z . σ x p = cB (σxpy−σypx)^

fermion zero modes on Dirac wall

– Μ(z)

+ Μ(z)

cσ.p

cσ.p( (Η =

NΚ = −1

NΚ = +1

z 0

Μ

Volkov-Pankratov,2D massless fermionsin inverted contactsJETP Lett. 42, 178 (1985)

chiralfermions

chiralfermions

z

NΚ = +1

Dirac vacuum

0

Dirac vacuum

Dirac wall

NΚ = −1

Μ > 0Μ < 0

Majorana fermions: edge stateson the boundary of 3D gapped topological matter

3He-Bvacuum

z

NΚ = +2 NΚ = 0

x,y

0NΚ = 0

NΚ = +2

p2

2m*– µ+U(z)

p2

2m*+ µ–U(z)–

cσ.p

cσ.p( (Η =

– Μ(z)

+ Μ(z)

cσ.p

cσ.p( (Η =

0

µ–U(z)

NΚ = −1

NΚ = +1

z 0

Μ

* boundary of topological superfluid 3He-B

* Dirac domain wall

helical fermions

Volkov-Pankratov,2D massless fermionsin inverted contactsJETP Lett. 42, 178 (1985)

Majoranafermionson wall

chiralfermion

spectrum of fermion zero modes

Hzm = c (σxpy−σypx)

Majorana fermions

NΚ = − 2

NΚ = +2

p2

2m*– µ

p2

2m*+ µ–

σxcxpx+σycypy+σzczpz

σxcxpx+σycypy+σzczpz( (Η =

z 0

on interface in topological superfluid 3He-B

one of 3 "speeds of light" changes sign across wall

spectrum of fermion zero modesMajoranafermions

domain wallphase diagram

Hzm = c (σxpy−σypx)

cx

cz

NΚ = − 2 NΚ = +2

cy

NΚ = +2

NΚ = −2

NΚ = −2

NΚ = +2

0

Lancaster experiments:probing edge states of 3He-B with vibrating wire

E

2pFv∆−pFv

vc = ∆/3pF

v <vc

v

2v

v >vc

E

2pFv∆−pFv

spectrum of surface states

top of theoccupied band

bottom of thefree band

continuous spectrum

vortices in fully gapped 3+1 system

fermion zero modes in vortex core

Zero energy states in the core of vortices in topological superfluids

E (pz , Q)

pz

Q=2

Q=1

Q=0

Q=-2

Q=-1

E (pz , Q)

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

pz

E(pz) = - cpz for d quarks

Spectrum of quarks in core of electroweak cosmic string

asymmetric branches cross zero energy

Bound states of fermions on cosmic strings and vortices

E(pz) = cpz for u quark

Number of asymmetric branches = NN is vortex winding number

Index theorem: Jackiw & Rossi Nucl. Phys. B190, 681 (1981)

E (pz , Q)

pz

Q=-3/2Q=-5/2

Q=-1/2

Q=-7/2

E (Q, pz = 0 )

3/21/2 5/2 7/2Q

E(Q,pz) = − Q ω0 (pz)

Number of asymmetric Q-branches = 2NN is vortex winding number

is the existence of fermion zero modesrelated to topology in bulk?

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

no true fermion zero modes: no asymmetric branch as function of pz

asymmetricbranch

as function of Q

Bound states of fermions on vortex in s-wave superconductor

Angular momentum Q is half-odd integerin s-wave superconductor

ω0 = ∆2/ EF << ∆

Caroli, de Gennes & J. Matricon, Phys. Lett. 9 (1964) 307

GV JETP Lett. 57, 244 (1993)

NΚ = 0

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

pz

Q=2Q=3

Q=1

Q=0

Q=4

Q=-3Q=-2

Q=-4

Q=-1

topological 3He-B at µ > 0 : NΚ = 2

Q1 2 3 4

E (pz , Q)

pz

Q=2Q=3

Q=1

Q=0

Q=4

Q=-3Q=-2

Q=-4

Q=-1

EQ=-Qω0

ω0=∆2/EF << ∆

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

gapless fermions on Q=0 branch form

1D Fermi-liquid

fermions zero modes on symmetric vortex in 3He-B

helicity + helicity −

Misirpashaev & GV Fermion zero modes in symmetric vortices in superfluid 3He,

Physica B 210, 338 (1995)

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

pz

Q=2Q=3

Q=1

Q=0

Q=4

Q=-3Q=-2

Q=-4

Q=-1

Q1 2 3 4

EQ=-Qω0

ω0=∆2/EF << ∆

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

gapless fermions on Q=0 branch form

1D Fermi-liquid

Misirpashaev & GV Fermion zero modes in symmetric vortices in superfluid 3He,

Physica B 210, 338 (1995)

topological 3He-B at µ > 0 : NΚ = 2

fermions zero modes on symmetric vortex in 3He-B

pz

Q=2Q=3

Q=1

Q=0

Q=4

Q=-3Q=-2

Q=-4

Q=-1

E (pz , Q)pz

topological quantum phase transition in bulk & in vortex core

vs

NΚ = 2

vs

NΚ = 0

µ > 0µ < 0

topological superfluid 3He-Bnon-topological superfluid

µ

NΚ = +2 NΚ = 0

1/m*

p2

2m– µ

p2

2m+ µ–

cBσ.p

cBσ.p( (Η =

γ5∆

γ5∆ − cα.p − βM + µR

cα.p+ βM − µR( (Η =

superfluid 3He-B as non-relativistic limit of relativistic triplet superconductor

superfluid 3He-B

relativistic triplet superconductor

cB = c ∆ /M

cp << Mµ << M

(µ + M)2 = µR + ∆2

m = M / c2

2

p2

2m– µ

p2

2m+ µ–

cBσ.p

cBσ.p( (Η =

γ5∆

γ5∆ − cα.p − βM + µR

cα.p+ βM − µR( (Η =

phase diagram of topological states of relativistic triplet superconductor

3He-B

γ5∆

γ5∆ − cα.p − βM + µR

cα.p+ βM − µR ((Η =

energy spectrum in relativistic triplet superconductor

gapless spectrumat topologicalquantum phasetransition

soft quantum phasetransition:Higgs transitionin p-space

R |µ

R|<µ*µ

R2 2 2=M −∆

R* |µ

R|>µ

spectrum of non-relativistic 3He-B

µ

Dirac Dirac

NΚ = +2 NΚ = 0

NΚ = 0 NΚ = −2

1/m*

0NΚ = −1 NΚ = +1

p2

2m*– µ

p2

2m*+ µ–

cBσ.p

cBσ.p( (Η =

µ<∆2/M µ<0 µ=0 µ>∆2/M

gapless spectrumat topological

quantum phasetransition

soft quantum phasetransition:

Higgs transitionin p-space

γ5∆

γ5∆ − cα.p − βM + µR

cα.p+ βM − µR ((Η =

fermion zero modes in relativistic triplet superconductor

vortices intopological superconductorshave fermion zero modes

pzpz

generalized index theorem ?

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

for vortices in Dirac vacuum

winding number

4π3i 1 tr [

∫d3p dω dφ

G ∂ω G-1 G ∂φ G

-1G ∂px G-1G ∂py G

-1G ∂pz G-1]N

5 =

N5

= N

E (pz)

pz

T (mK)H (T)p

(b

ar)

1

1

2

0.5

A

A

A

B

B

B NORMAL

NORMAL

A1

symmetry breaking phase transitions

G=SO(3)xSO(3)xU(1)

SO(3)xSO(3)xU(1) SO(3)

SO(3)

homotopy group

π1(G/H)=π1(U(1)xSO(3))=ZxZ2

vortices in superfluid 3He-B

1 + 1 = 2

N1 = 0, 1, 2, 3 ... and ν = 0, 1

N1 = 1 , ν =0mass vortex

-- rotation matrix

N1 = 0 , ν =1spin vortexN1 = ν = 1

spin-mass vortex

1 + 1 = 0

winding numbers

counterflowregion

soliton

cluster ofmass

vortices(N1=2, ν=0)

spin–massvortex

doubly quantizedvortex as pair

of spin–mass vorticesconfined by soliton

boundaryof

rotatingcontainer

soliton

vs

(N1=1, ν=–1) ≡ (N1=1, ν=1)

∆αβ ~ eiΦ σαβR i kik

R ki

most symmetricvortex core

vortex with ferromagnetic A-phase core

non-axisymmetric vortex

pair of half-quantum vortices

breaking of parity

breaking of axialsymmetry

B-phaseorder parameter

B-phaseorder parameter A-phase

order parameter

ξ

ρ

ρ

N =1/2 N = 1 N =1/2

∆B

∆B

∆A

symmetry breaking in the 3He-B vortex core

Phase diagram of first ordervortex-core transition in 3He-B

Pekola, Simola, Hakonen, Krusius, et al., PRL 53, 584 (1984)

Solid

Temperature (mK)0 1 2 3

0

10

20

30

40

Pres

sure

(ba

r)

3He-BNormalFermiliquid

non-axisymmetricvortex core

vortexwith

A-phasecore

3He-A

most symmetric 3He-B vortex

experiment: magnetic core superconductivity along the core

Witten superconducting cosmic string & v-vortex

v-vortex:breaking of parity

additional breaking of symmetry in the core

B-phaseorder parameter

ξ

ρ

∆B

B-phaseorder parameterA-phase

orderparameter

ρ

∆B

∆A

Higgs field

second Higgs field

ρ

H

HA

most symmetric cosmic string

Witten string:breaking of electromagnetic symmetry

Higgs field

ξ = 1/mHiggs ξ

ρ

H

Hakonen, Krusius, Salomaa, et al., PRL 51, 1362 (1983)

E. Witten, Nucl. Phys. B 249, 557 (1985) Salomaa & Volovik, PRL 51, 2040 (1983)

broken parity in the 3He-B vortex core

B-phaseorder parameterA-phase

orderparameter

∆B

∆A

B-phaseorder parameter

A-phaseorder

parameter

∆B

∆A

electric polarization in the core

paritytransformation

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

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

pz

Q=2Q=3

Q=1

Q=0

Q=4

Q=-3Q=-2

Q=-4

Q=-1

zero energy states in symmetric N=1 3He-A vortex

Q1 2 3 4

Fermions on Q=0 branch form

flat Fermi-band

(fermionic condensate or Khodel state)

flat band

Kopnin & Salomaa, PRB 44, 9667 (1991)

FMagnus + FIordanskii + FKopnin+ FStokes = 0

forces on vortices:three nondissipative forces + friction force acting on a vortex line

Iordanskii force

Gravitational Aharonov-Bohm

effect

Kopnin force

Axial anomaly

Magnus–Joukowski lifting force in classical hydrodynamics

vortexvelocity

vL

heat bathvelocity

vn

momentum transfer from negativeenergy states in the core to heat bath

analog of baryogenesis

FMagnus = κ × ρ(vL - vs)

FIordanskii = κ × ρn(vs - vn) FKopnin = κ × C(T)(vn - vL)

FStokes = −γ (vL - vn)

momentum transfer between vortex and superfluid vacuum

Stokes friction force

Aharonov-Bohm scatteringof quasiparticles on a vortex

vacuumvelocity

vs

vs

l field vs

x

y

z

Momentogenesis by N=2 vortex-skyrmion

vortex-skyrmion with N=2m=2

circulation quanta

m = (1/4π) ∫∫ dx dy ( l • ( ∂l /∂x × ∂l /∂y) ) = 1

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

F = ∫ d3r P = (1/2π2) ∫ d3r (B•E ) pF l = (1/2π2) h pF3

∫ d3r (∇× l • dl /dt ) l

= 2π h (1/3π2) pF3 z × (vn − vL)

•

^

l=l(r-vt)

q

chiral anomaly equation

(Adler, Bell, Jackiw)

applied to 3He-A applied to Standard Model

spectral flow produces

Chiral anomaly: matter-antimatter asymmetry of Universe (baryon asymmetry) and Kopnin force

momentum from vacuumof fermion zero modes

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

baryons from vacuum

spec

tral f

low

B = (1/4π2) BY•EY Σ B C Y2•

a a a aP = (1/4π2) B•E Σ P C e2•

a

a

a

a a a

B = Σ B n • •

a aP = Σ P n • •

a a

aC = +1 for right

-1 for left

B -- baryonic charge

Y -- hypercharge a

a

P -- momentum (fermionic charge)

e -- effective electric charge

aC = +1 for right

-1 for left

0.4 0.6 0.8 1

–1

0

T / Tc

Magnus + Iordanskii +Kopnin (spectral-flow)

ω0τ ‹‹ 1Kopnin spectral flowforce almost compensatescontributions ofMagnus + Iordanskii

Magnus + Iordanskii forces

turbulent vortex flow

superfluid Reynolds number

laminar vortex flow

ω0τ ›› 1Magnus force

friction force

nondissipativeforces

Manchester experiment(red circles)

Experimental and theoretical forces on a vortex

theory(solid line)

d||

d⊥–1

Reα(T)=d||

1– d⊥ ≈ ω0τ

Reα(Tc) = 0 Reα(0)= ∞ Reα(T ~0.6 Tc) ~ 1

turbulent and laminar vortex evolution

0 4 s 8 s 24 s

44 s 68 s 128 s 160 s

ΩT=0.8 Tc

T=0.4 Tc

0

Reα < 1

Reα > 1

regularturbulent

TURBULENT VORTEX GENERATION

0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75

1.1

1.2

1.3

1.4

1.5

Ωc

(ra

d/s)

T/Tc

P = 29.0 bar

169 events640 events

T/Tc

TTon = 0.590 Tcσ = 0.033 cT

0.4 0.5 0.6 0.7

frac

tion

of e

vent

s

0

1

0.5

0.4 1 2 5q(T)

[Finne et al. Nature 424, 1022 (2003)]

Conclusion to topological medium part

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.

Momentum-space topology determines:

quantum vacuum as self-sustained system

dynamics of cosmological `constant'

5. From helium liquids to dynamics of Lorentz invariant vacuum

recipe to cook Universe

Dark Energy 70%

Dark Matter 30%

Baryonic Matter 4%

Visible Matter 0,4%

cosmological constant Λ is possible candidate for dark energy

Λ= εDark Energy

dark energy problem

dark matter forms clusters like ordinary matter

Dark and Dark!What is the difference?

∗

εMatter/εcrit

observational cosmology:dark energy vs dark matter

luminositydistances

Knop et al 2003

Acoustic peakSpergel et al 2003

of GalaxiesAllen et al 2002

εDarkEnergy

εcrit

εcrit = 3H2/8πG = 10−29g/cm3

H - Hubble parameter

Cosmological Term

* Original Einstein equations

* 1917: Einstein added the cosmological term

& obtained static solution: Universe as 3D sphere

cosmological constant

(Rµν - gµνR/2) - Λgµν = Tµν Matter

8πG

1

Λ = 0.5 εMatter = 1/(8πGR2)

x12 + x2

2 + x32 + x4

2 = R2

(Rµν - gµνR/2) = Tµν Matter

8πG

1

radius of 3D sphere energy density of matter

it is finitebut no boundaries ?!

matter is a source of gravity field

perfect Universe !

but it is curved!I would prefer to live in flat Universe

Estimation of cosmological constant

Λ = 0.5 εMatter = 1/(8πGR2)

the first and the last one: after 90 years nobody could improve it

order of magnitude is OK, why blunder?

this was correct estimation of Λ

compare with observed Λ = 2.3 εMatter

when I was discussing cosmological problems withEinstein, he remarked that the introduction of the

cosmological term was the biggest blunder of his life

-- George Gamow, My World Line, 1970

arguments against Λ !

* 1917: de Sitter found stationary solutionof Einstein equations without matter

* 1924: Hubble : Universe is not stationary* 1929: Hubble : recession of galaxies

What?empty space gravitates !?

(Rµν - gµνR/2) - Λgµν = 0 8πG

1

R= exp (Ht) H2 = 3/(8πΛG)

Away with curvature !and with cosmological term !

1923: expanding version of de Sitter Universe:

Wait !

Wenn schon keine quasi-statische Welt,dann fort mit dem kosmologischen Glied.

A. Einstein H. Weyl, 23 Mai 1923

the question arises whether it is possibleto represent the observed facts

without introducing a curvature at all.

Einstein & de Sitter, PNAS 18 (1932) 213

no source of gravity field !

Epoch of quantum mechanics: Λ as vacuum energy

(Rµν - gµνR/2) - Λgµν = Tµν Matter

8πG

1

(Rµν - gµνR/2) = Λgµν + Tµν Matter

8πG

1

physical vacuum as source of gravity

En = hν(n + 1/2)

n = 0 : zero point energyof quantum fluctuations

you do not believe Einstein?energy must gravitate !

quantum field is set of oscillators

move Λ to the right

zero point energy has weight ?

it is filled withquantum oscillations

vacuum is not empty ?

Latin: vacuus empty

Equation of state of quantum vacuum

what is vacuum?

pumping the vacuum by piston

`empty space'

`empty space'

Λ = εvac = −pvac

Evac= εvacV

pvac = − dE/dV= − εvac

ε = −p

ε < 0

p =−ε > 0vacuum

pressureof vacuum

energy densityof vacuum

ε > 0

p =−ε < 0applied external forceF = εA, negative pressure

applied external forceF = εA, positive pressure

vacuum

right !

vacuum ismedium with equation of state

(Rµν - gµνR/2) = Tµν + Tµν MatterVacuum

8πG

1

VacuumTµν = Λgµν

εzero point + εDirac = (1/2)Σ E(k) - Σ E(k) = bosons

(νb - νf)ck4 Planckfermions

zero-point energy numberof bosonic

fields

Planck scale

numberof Dirac

fields

energy of Dirac vacuum

supersymmetry:symmetry between fermions and bosons

νb = νf

εzero point ~ 10120Λupper limit

maybe Λ = 0 ?!

photonic vacuum:n(k)=0

vacuum is too heavy !

there is no supersymmetry below TeV

Σk (n(k) + 1/2) ck weight

of photon vacuum

Dirac vacuum

How heavy is aether?

k

x

E = ck

ky

(kz)

occupiednegative energy

levels

weightof Dirac vacuum

Luminiferous aether (photons)

Λ in supernova era

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

distant supernovae: accelerating Universe(Perlmutter et al., Riess et al.)

you are right,

but Λ is not zero

*

Universe is flat !

*

Λ = εvac = εDark Energy = 70%

εDark Matter = 30%

εVisible Matter = 0,4%

Kepler's Supernova 1604from ` De Stella Nova in Pede Serpentarii '

What can condensed matter physicist say on Λ ?

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

Why condensed matter ??!

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

0−1

= 0

magic word: regularization

wisdom of particle physicist:

problems:

* Why is vacuum not extremely heavy?

* Why is vacuum as heavy as (dark) matter ?

* Why is vacuum gravitating? Why is Λ non-zero?it is easier to accept that Λ=0than 123 orders of magnitude smaller

Λobservation = εDark Energy ~ 2−3 εDM ~ 10−47 GeV4

Cosmological constant paradox

Λobservation ~ 10−123ΛTheorytoo bad for theory

Λtheory = εzero point energy ~ EPlanck ~ 1076 GeV4 4

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

Λexp ~ 2-3 εDark Matter ~ 10−123Λbare

0−1

= 0*magic word: regularization

*Polyakov conjecture: dynamical screeneng of Λ by infrared fluctuations of metric

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

wisdom of particle physicist:

A.M. PolyakovPhase transitions and the Universe, UFN 136, 538 (1982)

De Sitter space and eternity, Nucl. Phys. B 797, 199 (2008)

V. Gurarie, Nonequilibrium dynamics of weakly and strongly paired superconductors: 0905.4498A.F. Volkov & S.M. Kogan, JETP 38, 1018 (1974)

Barankov & Levitov, ...

Λbare ~ εzero point

what is natural value of cosmological constant ?

Λ = 0Λ = EPlanck4

time dependent cosmological constant

could be in early Universe should be in old Universe

Λ ~ 0Λ ∼ EPlanck4

how to describe quantum vacuum & vacuum energy Λ ?

* quantum vacuum is Lorentz-invariant

* 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

charge density nis not Lorentz invariant

q must be Lorentz invariant

does such q exist ?

L n = γ(n + v.j)

L q = q

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

wvac = −1

Hawking suggested to introduce specialfield which describes the vacuum onlyHawking, Phys. Lett. B 134, 403 (1984)

relativistic invariant conserved charges q

qαβ = q gαβ

∇α qαβ

= 0

∇α qα

= 0 qα = ?

∇α qαβµν

= 0

q αβµν = q eαβµν

possible

impossible

Duff & van NieuwenhuizenPhys. Lett. B 94, 179 (1980)

examples of vacuum variable q

Fκλµν = q (-g)1/2 eκλµν 24

1 q2 = − Fκλµν F κλµν

Fκλµν = ∇[κ Aλµν]

12

q<GαβGµν>= (gαµgβν−gανgβµ) q = <GαβGαβ> <Gαβ> = 0

24

q<GαβGµν>= (-g)1/2 eαβµν q = <GαβGαβ>

∇µuν = q gµν

4-form field

gluon condensates in QCD

topologicalcharge density

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

~

thermodynamics in flat spacethe same as in cond-mat

conservedcharge Q

thermodynamicpotential

Lagrange multiplieror chemical potential µ

equilibrium vacuum

Q = dV q

Ω =E − µQ = dV (ε (q)− µq)

pressure P = − dE/dV= −ε + q dε/dq

dε/dq = µ

dΩ/dq = 0

equilibrium self-sustained vacuum

dε/dq = µ

ε - q dε/dq = −P = 0

E = V ε(Q/V)

vacuum energy & cosmological constant

pressure

energyof equilibriumself-sustained vacuum

cosmologicalconstant

equation of state

vacuum variablein equilibriumself-sustained vacuum

P = −ε + q dε/dq= -ΩΛ = Ω =ε − µ q

P = −Ωcosmologicalconstantin equilibriumself-sustainedvacuumself-tuning:

two Planck-scale quantitiescancel each other

in equilibrium self-sustained vacuum

Λ = ε − µq = 0

equilibrium self-sustained vacuum

dε/dq = µ

ε - q dε/dq = −P = 0

4ε (q) ~ EPlanck

4 EPlanck

4 EPlanck

2 q ~ µ ~ EPlanck

dynamics of q in flat space

action

whatever is the origin of q the motion equation for q is the same

integration constant µ in dynamics becomes chemical potential in thermodynamics

4-form field Fκλµν as an example of conserved charge q in relativistic vacuum

Fκλµν = ∇[κ Aλµν] ∇κ (F κλµν q−1dε/dq) = 0

S= dV dt ε (q)

Maxwell equation

solution

motionequation

∇κ (dε/dq) = 0

∇κ (dε/dq) = 0

dε/dq = µ

24

1 q2 = − Fκλµν F κλµν

F κλµν = q eκλµν

dynamics of q in curved space 4-form field and chiral condensate

action

S = d 4x (-g)1/2 [ ε (q) + K(q)R ] + Smatter

cosmological term

gravitational coupling K(q) is determined by vacuumand thus depends on vacuum variable q

dε/dq + R dK/dq = µ

Einsteintensor

integrationconstant

matter

K(Rgµν− 2Rµν) + (ε − µq)gµν − 2( ∇µ∇ν − gµν∇λ∇λ) K = Tµν

motionequation

q becomes dynamical only when K depends on q

Einsteinequations

∇µ T µν = 0= ε gµν

dε/dq = µ q = const

Λ = ε − µq (Rgµν− 2Rµν) + Λ gµν = Tµν

motionequation

originalEinstein

equations

case of K=const restores original Einstein equations

16πG

1

16πG

1K = G - Newton constant

Λ - cosmological constant

Minkowski solution

cosmological term

dε/dq = µ

Λ = ε(q) − µq = 0

dε/dq + R dK/dq = µ

Einsteintensor

matter

K(Rgµν− 2Rµν) + (ε − µq)gµν − 2( ∇µ∇ν − gµν∇λ∇λ) K = Tµν

Maxwellequations

Minkowskivacuumsolution

Einsteinequations

vacuum energy in action: ε (q) ~ 4

EPlanck

thermodynamic vacuum energy: ε − µq = 0

∇µ T µν = 0

R = 0

Model vacuum energy

dε/dq = µ

ε − µq = 0

q = q0Minkowskivacuumsolution

vacuumcompressibility

vacuumstability

emptyspace

q = 0

self-sustainedvacuumq = q0

µ = µ0 = − 3χq0

1

2χ

1

q02

q2

3q04

q4 ε (q) = ( ) − +

V

1

dP

dVχ = −

χ

1 = (q2 d2ε/dq2 )q=q0

> 0

ε(q)

q q q0 q0

Ω(q) = ε(q) − µ0q

qmin

pressureof vacuum compressibility of vacuum

energy densityof vacuum

Minkowski vacuum (q-independent properties)

Λ = Ωvac = −Pvac Pvac = − dE/dV= − Ωvac

χvac = −(1/V) dV/dPpressure fluctuations

<(∆Pvac)2> = T/(Vχvac)

<(∆Λ)2> = <(∆P)2>

natural value of Λdetermined by macroscopic

physics

natural value of χvacdetermined by microscopic

physics

χvac ~ E Planck

−4

volume of Universeis large:

V > TCMB/(Λ2χvac)

V > 1028 Vhor

Λ = 0 1/χvac = q2 d2εvac /dq2

dynamics of q in curved space: relaxation of Λ at fixed µ=µ0

Λ(q) = ε(q) − µ0q

dε/dq + R dK/dq = µ0

K(Rgµν− 2Rµν) + gµν Λ(q) − 2( ∇µ∇ν − gµν∇λ∇λ) K = Tµν

Maxwellequations

dynamicsolution

similar to scalar field with mass M ~ EPlanckA.A. Starobinsky, Phys. Lett. B 91, 99 (1980)

Einsteinequations

ω ~ EPlanck

matter

q(t) − q0 ~ q0t

sin ωtΛ(t) ~ ω2

t2sin2ωt

H(t) = a(t)

a(t) .

3t

2 (1− cos ωt)=

Relaxation of Λ (generic q-independent properties)

natural solution of the main cosmological problem ?

Λ relaxes from natural Planck scale valueto natural zero value

cosmological "constant" Hubble parameter Newton "constant"

ω ~ EPlanck

Λ(t) ~ ω2 t2

sin2ωtG(t) = GN (1 + )

ωt

sin ωtH(t) =

a(t)

a(t) .

3t

2 (1− cos ωt)=

<Λ(tPlanck)> ~ EPlanck 4

Λ(t = ∞) = 0

present value of Λ

ω ~ EPlanck Λ(t) ~ ω2 t2

sin2ωt

<Λ(tPlanck)> ~ EPlanck 4

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

dynamics of Λ:from Planck to present value

coincides with present value of dark energysomething to do with coincidence problem ?

dynamics of Λ in cosmology

ω ~ EPlanck ω = 2∆

Λ(t) ~ ω2 t2

sin2ωt

dynamics of ∆ in superconductor

δ|∆(t)|2 ~ ω3/2 t1/2

sin ωt

ε(t) − εvac ~ ω t

sin2ωt

final states:

equilibrium vacuum with Λ = 0

ground state of superconductor

t = 0 t = + ∞intial states:

nonequilibrium vacuum with Λ ∼ EPlanck

superconductor with nonequilibrium gap ∆

4

F.R. Klinkhamer & G.E. Volovik Dynamic vacuum variable &equilibrium approach in cosmology PRD 78, 063528 (2008)Self-tuning vacuum variable &cosmological constant, PRD 77, 085015 (2008)

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

V. Gurarie, Nonequilibrium dynamics of weakly and strongly paired superconductors: 0905.4498A.F. Volkov & S.M. Kogan, JETP 38, 1018 (1974)

Barankov & Levitov, ...

reversibility of the process

ω ~ EPlanck ω = 2∆

Λ(t) ~ ω2 t2

sin2ωtδ|∆(t)|2 ~ ω3/2

t1/2

sin ωt

reversible energy transferfrom coherent degree of freedom (vacuum)

to particles (Landau damping)

reversible energy transferfrom vacuum to gravity

inverse process in contracting Universe ?

t = 0 t = + ∞t = − ∞

Minkowski vacuum as attractorallow µ to relax (Dolgov model)

both µ & q relax to equilibrium values µ0 & q0cosmological constant Λ relaxes to zero

t

q(t)

q0

F. Klinkhamer & GVTowards 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

* vacuum energy has Planck scale value in equilibrium

but this energy is not gravitating

4 ε(q)~ EPlanck

* thermodynamic energy of equilibrium vacuum

* gravitating energy is thermodynamic vacuum energy

Ω(q0) = ε(q0) − q0 dε/dq0 = 0

Ω(q) = ε − q dε/dq Tµν = Λgµν = Ω(q)gµν

Tµν = Λgµν = ε(q)gµν