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Topological media: quantum liquids, topological insulators & quantum vacuum
G. Volovik
Landau Institute
CC-2010, Chernogolovka, July 26, 2010

1. Introduction: Universe as object of ultralow temperature physics * quantum vacuum as topological medium * effective quantum field theories emerging at low T 2. Fermi surface as topological object *normal 3He, metals, 3. Dirac (Fermi) points as topological objects * superfluid 3He-A, semimetals, cuprate superconductors,

4. Fully gapped topological media * superfluid 3He-B, topological * * *

* * *

graphene, vacuum of Standard Model of particle physics in massless phase topological invariants for gapless 2D and 3D topological matter QED, QCD and gravity as emergent phenomena quantum vacuum as cryo-crystal, -phase of superfluid 3He

insulators, chiral superconductors, vacuum of Standard Model of particle physics in present massive phase topological invariants for gapped 2D and 3D topological matter edge states & Majorana fermions ( planar phase of 3He & surface of 3He-B )

intrinsic QHE & spin-QHE role of momentum-space topology

5. Conclusion * important

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


3+1 sources of effective Quantum Field Theories in many-body system & quantum vacuum
I think it is safe to say that no one understands

Lev Landau

Quantum Mechanics
Richard Feynman

Thermodynamics

is the only physical theory of universal content Albert Einstein

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

effective theories of quantum liquids: two-fluid hydrodynamics of superfluid 4He & Fermi liquid theory of liquid 3He missing ingredient in Landau theories

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


Landau view on a many body system
many body systems are simple at low energy & temperature weakly excited state of liquid can be considered as system of "elementary excitations" Landau, 1941
helium liquids Universe

equally applied to: superfluids, solids, & relativistic quantum vacuum

ground state = vacuum
ground state + elementary excitations vacuum + elementary particles

quasiparticles = elementary particles

why is low energy physics applicable to our vacuum ?


characteristic high-energy scale in our vacuum is Planck energy

EP =(hc5/G)
high-energy physics is extremely ultra-low energy physics
highest energy in accelerators Eew ~1 TeV ~ 1016K

1/2

~10

19

GeV~1032K

high-energy physics & cosmology belong to ultra-low temperature physics
T of cosmic background radiation

T T
cosmological expansion

CMBR

~1K
-32

Eew ~ 10

-16

E

Planck

CMBR

~ 10

E

Planck

cosmology is extremely ultra-low frequency physics

v(r,t) = H(t) r
H ~ 10
-60

Hubble law Hubble parameter

E

Planck

our Universe is extremely close to equilibrium ground state We should study general properties of equilibrium ground states - quantum vacua


Why no freezing at low T?
natural masses of elementary particles are of order of characteristic energy scale the Planck energy

even at highest temperature we can reach

m~E

Planck

~10

19

GeV~1032K
16

T ~ 1 TeV~1016K
everything should be completely frozen out

e

-m/T

=10

-10

=0

10

-123

, 10

-10

16

another great challenge?


main hierarchy puzzle m ,m <<< E

cosmological constant puzzle <<<< E
4 Planck

quarks

leptons

Planck

its emergent physics solution: m =m =0

its emergent physics solution: =0

quarks

leptons

reason: momentum-space topology of quantum vacuum

reason: thermodynamics of quantum vacuum


Why no freezing at low T?

massless particles & gapless excitations are not frozen out

who protects massless excitations?


gapless fermions live near Fermi surface & Fermi point who protects Fermi surface & Fermi point ?
we live because Fermi point is the hedgehog protected by topology

Topology
p
z

p

y

p

x

Life protection
Fermi point: hedgehog in momentum space

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


tools Topology in momentum space

Thermodynamics
responsible for properties of vacuum energy problems of cosmological constant: perfect equilibrium Lorentz invariant vacuum has

responsible for properties of fermionic and bosonic quantum fields in the background of quantum vacuum Fermi point in momentum space protected by topology is a source of massless Weyl fermions, gauge fields & gravity

=0

;

perturbed vacuum has nonzero on order of perturbation



<<<< E

4 Planck

m

quarks

,m

leptons

<<< E

Planck


Quantum vacuum as topological substance: universality classes physics at low T is determined by gapless excitations

universality classes of gapless vacua

topology is robust to deformations: nodes in spectrum survive interaction

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

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

two major universality classes of gapless fermionic vacua

Landau theory of Fermi liquid

Standard Model + gravity

vacuum with Fermi surface

vacuum with Fermi point

gravity emerges from

Fermi point
analog of

gµ(pµ- eAµ - e .Wµ)(p- eA - e .W) = 0

Fermi surface


Topological stability of Fermi surface
Energy spectrum of non-interacting gas of fermionic atoms p2 p2 p2 E(p) = 2m ­ µ = ­F 2m 2m E>0 E<0 empty levels
occupied levels: Fermi sea

Green's function G-1= i - E(p)

p (p )
y z

p p=p
F

p
F

x

Fermi surface E=0

=2 Fermi surface: vortex ring in p-space phase of Green's function G(,p)=|G|e i

is Fermi surface a domain wall in momentum space?

has winding number N = 1 no! it is a vortex ring


Route to Landau Fermi-liquid
is Fermi surface robust to interaction ?

Sure! Because of topology: winding number N=1 cannot change continuously, interaction cannot destroy singularity



p (p )
y z

then Fermi surface survives in Fermi liquid ?

p Landau theory of Fermi liquid is topologically protected & thus is universal
all metals have Fermi surface ... Not only metals. Some superconductore too!
Stability conditions & Fermi surface topologies in a superconductor Gubankova-Schmitt-Wilczek, Phys.Rev. B74 (2006) 064505

p
F

x

=2 Fermi surface: vortex line in p-space

G(,p)=|G|e i




quantized vortex in r-space Fermi surface in p-space homotopy group Topology in r-space
z how is it in p-space ? y x =2 vortex ring
1

Topology in p-space
p (p )
y z

winding number N1 = 1
Fermi surface

p =2

p
F

x

(r)=|| e i
scalar order parameter of superfluid & superconductor

G(,p)=|G|e i
Green's function (propagator)

classes of mapping S1 U(1)
manifold of broken symmetry vacuum states

classes of mapping S1 GL(n,C)
space of non-degenerate complex matrices

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

magnetic hedgehog vs right-handed electron
z y p
z

p

y

x

p

x

again no difference ?

hedgehog in r-space

(r)=^ r

right-handed and left-handed massless quarks and leptons are elementary particles in Standard Model Landau CP symmetry is emergent

hedgehog in p-space

(p) = ^ p
close to Fermi point

H = + c .p

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


E=cp p (p )
y z

p

x


where are Dirac particles? E=cp p (p )
y z

Dirac particle - composite object made of left and right particles

E

p

x

p (p )
y z

E=cp p (p )
y z

p

x

p

E2 = c2p2 + m2 mixing of left and right particles is secondary effect, which occurs at extremely low temperature

x

T

ew

~ 1 TeV~1016K

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

Gap node - Fermi point (anti-hedgehog)

N

3

= 1e 8

ijk



dS g . (pi g â pj g)

k

N3 =-1
p
1

E p (p )
y z

over 2D surface S in 3D p-space

p
p
2

x

N3 =1
Gap node - Fermi point (hedgehog)

S

2

H=

)

p2 2m - µ

c(px + ipy)

c(px ­ ipy)

-

p +µ 2m
2

))
=

g3(p)

g1(p) +i g2(p)

g1(p) -i g2(p)

-g3(p)

)


emergence of relativistic QFT near Fermi (Dirac) point
original non-relativistic Hamiltonian p2 c(px + ipy) 2m - µ H= = 2 c(px ­ ipy) - p + µ 2m close to nodes, i.e. in low-energy corner relativistic chiral fermions emerge

)

))

g3(p)

g1(p) +i g2(p)

g1(p) -i g2(p)
left-handed particles
1 N3 =-1

-g3(p)

)

= .g(p)
E p (p )
y z

p

H = N 3 c .p E = ± cp

p

x

N3 =1 p
2

right-handed particles

chirality is emergent ?? what else is emergent ?

top. invariant determines chirality in low-energy corner relativistic invariance as well


bosonic collective modes in two generic fermionic vacua Landau theory of Fermi liquid
Fermi surface

Standard Model + gravity
collective Bose modes: propagating oscillation of position of Fermi point A

Fermi point

p p - eA
collective Bose modes of fermionic vacuum: propagating oscillation of shape of Fermi surface form effective dynamic electromagnetic field propagating oscillation of slopes E2 = c2p2 gikpi p
k

form effective dynamic gravity field Landau, ZhETF 32, 59 (1957)

two generic quantum field theories of interacting bosonic & fermionic fields


relativistic quantum fields and gravity emerging near Fermi point
Atiyah-Bott-Shapiro construction: linear expansion of Hamiltonian near the nodes in terms of Dirac -matrices

E = vF (p - pF)
linear expansion near Fermi surface emergent relativity

0 H = eik i .(pk - pk)

linear expansion near Fermi point

p

z

p

y

gµ(pµ- eAµ - e .Wµ)(p- eA - e .W) = 0
p

x

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

hedgehog in

p

-space

all ingredients of Standard Model : chiral fermions & gauge fields emerge in low-energy corner of together with spin, Dirac -matrices, gravity & physical laws: Lorentz & gauge invariance, equivalence principle, etc

gravity & gauge fields are collective modes vacua with Fermi point


quantum vacuum as cryo-crystal

4D graphene Michael Creutz JHEP 04 (2008) 017

Fermi (Dirac) points with N3 = +1 Fermi (Dirac) points with N3 = -1


4. From Fermi point to intrinsic QHE & topological insulators N
3

=

1e 8

ijk



dSk g . (pi g â pj g)

over 2D surface S in 3D momentum space

3+1vacuum with Fermi point

dimensional reduction
p
y

Fully gapped 2+1 system
p
x

~ N

3

=

1 4



dpxdpy g . (px g â py g)

over the whole 2D momentum space or over 2D Brillouin zone


topological insulators & superconductors in 2+1
p-wave 2D superconductor, 3He-A film, HgTe insulator quantum well

H=

)

p2 2m - µ

c(px + ipy) p2 - +µ 2m
2

c(px ­ ipy)

)

p2 = px2 +py

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

H = E


Topological invariant in momentum space

H=

)

p2 2m - µ

c(px + ipy)

c(px ­ ipy)

-

p2 +µ 2m
2

)

p
y

H=

)

g3(p)

g1(p) +i g2(p)

g1(p) -i g2(p)

-g3(p)

)

= .g(p)

p2 = px2 +py

fully gapped 2D state at µ = 0

N

~

3

=

1 4

d2p g . (px g â py g)

GV, JETP 67, 1804 (1988)

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

unit vector
p
x

^ g

g (px,py)

sweeps unit sphere ~ N3 (µ > 0) = 1

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

)

p2 2m - µ

c(px + ipy)

c(px ­ ipy)

-

p +µ 2m
2

))
=

g3(p)

g1(p) +i g2(p)
~

g1(p) -i g2(p)

-g3(p) N3
~

)

= .g(p)

Topological invariant in momentum space ~ N3 = 1 d2p g . (px g â py g)

4



trivial insulator

N3 = 1
topological insulator µ

N3 = 0

~

intermediate state at µ = 0 must be gapless

µ=0
quantum phase transition

at the interface between states with different N3

N3 = 0 is origin of fermion zero modes

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

N3=

~

1 24

2

e

µ

tr d2p d G µ G-1 G G-1G G N3 = 6
~

-1 GV & Yakovenko J. Phys. CM 1, 5263 (1989)

quantum phase transitions in thin 3He-A film ~ N3 = 4
~ transition between plateaus must occur via gapless state!

N3 = 2 gap N3 = 0 a1 QPT from trivial to topological state
~

gap

gap

a film thickness

a2 a3 plateau-plateau QPT between topological states


topological quantum phase transitions
transitions between but

ground states (vacua) of the same symmetry, different topology in momentum space
QPT interrupted by thermodynamic transitions

example: QPT between gapless & gapped matter T (temperature)

T
no change of symmetry along the path different asymptotes when T => 0

no symmetry change along the path

line of 1-st order transition

Tn
topological semi-metal

q

e -/T
c topological insulator

q
T

q c 2-nd order transition

q

- parameter of system

quantum phase transition at

q=qc
T

q
broken symmetry line of 1-st order transition line of 2-nd order transition

other topological QPT: Lifshitz transition, transtion between topological and nontopological superfluids, plateau transitions, confinement-deconfinement transition, ...

q


Zero energy states on surface of topological insulators & superfluids
p

y

Fully gapped 2+1 system

p

x

~ N

3

=

1 4



dpxdpy g . (px g â py g)

Fully gapped 3+1 system

Majorana fermions on the surface and in the vortex cores


interface between two 2+1 topological insulators or gapped superfluids
y
interface

~ N3 = N-

~ N3 = N+
x

gapless interface gapped state

gapped state

* domain wall in 2D chiral superconductors:

px - component px - ipy ~ N3= -1
0

H=

)

p2 2m - µ

c(px + i py tanh x ) p2 - +µ 2m

c(px ­ i py tanh x )

)

px+ipy ~ N3 = +1
x

py - component

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

interface

~ N3 = N-
gapped state

~ N3 = N+
gapped state

E(py)
empty 0 occupied left moving edge states

p

y

GV JETP Lett. 55, 368 (1992)

Index theorem: number of fermion zero modes at interface:

= N+ - N-


Edge states and currents

2D non-topological insulator or vacuum

2D topological insulator y
current

y
current

2D non-topological insulator or vacuum

~ N3 = 0
empty

~ N3 = 1

~ N3 = 0
x

E(py)
left moving edge states 0 right moving edge states

E(py)

empty

p

y

0 occupied occupied

p

y

current Jy = Jleft +Jright = 0


Edge states and Quantum Hall effect
2D non-topological insulator or vacuum 2D topological insulator y
current

2D non-topological insulator or vacuum
y

apply voltage V

-V/2

~ N3 = 1

current

~ N3 = 0

~ N3 = 0
x

V/2

E(py)-V/2

E(py)+V/2
right moving edge states
y

0

p

V/2
0

p

y

-V/2

left moving edge states current Jy = Jleft +Jright = xyEy



xy

~ e2 N = 4

3

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

spin current J

z x

=

1 (N dHz/dy + N E ) ss se y 4

spin-spin QHE 2D singlet superconductor:

spin-charge QHE



spin/spin xy

= Nss 4

s-wave: px + ipy: dxx-yy + idxy :

Nss = 0 Nss = 2 Nss = 4

film of planar phase of superfluid 3He



spin/charge xy

= Nse 4

GV & Yakovenko J. Phys. CM 1, 5263 (1989)


planar phase film of 3He & 2D topological insulator

H= ~ N3 = 1e 2 24

)

p2 2m - µ c(px ­ i py z)

c(px + i py z ) p2 - +µ 2m

)
-1

µ

tr

[ d2p d G µ G-1 G G-1G G-1] = 0 [ d2p d
z G G
µ -1

~ Nse =

1e 242

µ

tr

G G-1G G
~+ N3 = +1

]
~- N3 = -1

~ ~+ ~- N3 = N3 + N3 = 0 spin quantum Hall effect spin current J = x
z

~ ~+ ~ - Nse = N3 - N3 = 2

1NE 4 se

y

spin-charge QHE
GV & Yakovenko J. Phys. CM 1, 5263 (1989)



spin/charge xy

= Nse 4

N =2
se


Intrinsic spin-current quantum Hall effect & edge state spin current J = x
z

1 (N dHz/dy + N E ) ss se y 4
spin-charge QHE
y
current

y
current

N =0
se
spin

N =2
se
spin

N =0
se

-V/2 E(py)-V/2
left moving spin up

V/2 E(py)+V/2 V/2

x

right moving spin up

p -V/2

y



spin/charge xy

= Nse 4

p

y

right moving spin down electric current is zero spin current is nonzero

left moving spin down


3D topological superfluids / insulators / semiconductors / vacua
gapless topologically nontrivial vacua fully gapped topologically nontrivial vacua

3He-A, Standard Model above electroweak transition, semimetals, 4D graphene (cryocrystalline vacuum)

3He-B, Standard Model below electroweak transition, topological insulators, triplet & singlet chiral superconductor, ...

Bi2Te3


Present vacuum as semiconductor or insulator
3 conduction bands of d-quarks electric charge q=-1/3 3 conduction bands of u-quarks, q=+2/3 conduction electron band, q=-1 neutrino band, q=0 neutrino band, q=0 valence electron band, q=-1 3 valence u-quark bands q=+2/3 3 valence d-quark bands q=-1/3
dielectric and magnetic properties of vacuum (running coupling constants)

E (p)

p

Quantum vacuum: Dirac sea
electric charge of quantum vacuum Q= qa = N [-1 + 3â(-1/3) + 3â(+2/3) ] = 0
a


fully gapped 3+1 topological matter
superfluid 3He-B, topological insulator Bi2Te3 , present vacuum of Standard Model * Standard Model vacuum as topological insulator Topological invariant protected by symmetry

N =

1e 242

µ

tr dV G µ G-1 G G-1G G
over 3D momentum space

-1

G is Green's function at =0, is symmetry operator G =+/- G
Standard Model vacuum: =5

G 5 = - 5G

N = 8n

g

8 massive Dirac particles in one generation


topological superfluid 3He-B

=

(

p2 ­µ 2m*

H 2 = - 2H
=

cB.p
p2 2m*

cB.p ­

+µ

(

(

p2 ­ µ 3 +cB.p 1 2m*

)

K = 2

1/m* non-topological superfluid N = 0 N = -1 Dirac N = - 2 topological superfluid N = 0 non-topological superfluid 0 topological 3He-B N = +2 N = +1 Dirac = µ

Dirac vacuum 1/m* = 0

(

­ cB.p

cB.p +

(

GV JETP Lett. 90, 587 (2009)


Boundary of 3D gapped topological superfluid
x,y = vacuum
3

He-B

(

p2 ­ µ+U(z) 2m*

cB.p

cB.p

p2 ­ + µ­U(z) 2m*

(

N = 0 0

N = +2 z
Majorana particle = Majorana anti-particle 1/2 of fermion:

b = b+

µ­U(z)
Majorana fermions on wall
0

spectrum of Majorana zero modes

N = +2 z

Hzm = cB ^ . x p = cB (xpy-ypx) z
helical fermions

N = 0


fermion zero modes on Dirac wall
Dirac wall = Dirac vacuum N = -1 Dirac vacuum N = +1

(

­ (z) c.p c.p

+ (z)

(
nodal line

Volkov-Pankratov, 2D massless fermions in inverted contacts JETP Lett. 42, 178 (1985)

<0 0
chiral fermions

>0 z

N = +1
z
0

Dirac point at the wall Dirac point

N = -1

in Bi2Te3 Dirac point is below FS: nodal line on surface of topological insulator

Bi2Te3


Majorana fermions: edge states
on the boundary of 3D gapped topological matter
=

* boundary of topological superfluid 3He-B
x,y vacuum
3

(

p2 ­ µ+U(z) 2m*

c.p

c.p

­

p2 + µ­U(z) 2m*

(

He-B
Majorana fermions on wall

µ­U(z)

N = 0 0

N = +2 z

N = +2
0

N = 0

spectrum of fermion zero modes

* Dirac domain wall
chiral fermion

Hzm = c (xpy-ypx)
helical fermions

N = +1
z
0

N = -1

=

(

­ (z) c.p c.p

+ (z)

(

Volkov-Pankratov, 2D massless fermions in inverted contacts JETP Lett. 42, 178 (1985)


Majorana fermions on interface in topological superfluid 3He-B

cx
N = +2

=

(

p2 ­µ 2m*

xcxpx+ycypy+zczpz
p2 ­ +µ 2m*

xcxpx+ycypy+zczpz

(

N = -2 0

cy
N = -2

N = - 2

N = +2

N = +2

domain wall

phase diagram

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

cz
Majorana fermions

N = +2

spectrum of fermion zero modes
0

z

Hzm = c (xpy-ypx)

N = - 2


Conclusion
Momentum-space topology determines:

universality classes of quantum vacua effective field theories in these quantum vacua topological quantum phase transitions (Lifshitz, plateau, etc.) quantization of Hall and spin-Hall conductivity topological Chern-Simons & Wess-Zumino terms quantum statistics of topological objects spectrum of edge states & fermion zero modes on walls & quantum vortices chiral anomaly & vortex dynamics, etc.