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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