Momentum Space Topology in Standard Model and in Condensed Matter

2469-1

Workshop and Conference on Geometrical Aspects of Quantum States in Condensed Matter

1 - 5 July 2013

Momentum space topology in Standard Model and in condensed matter

Grigory Volovik Aalto University and Landau Institute

^Momentum+^ space topology in Standard Model & condensed matter a a Aalto University G. Volovik Landau Institute Geometrical Aspects of Quantum States in Condensed Matter, 1-5 July 2013 1. Gapless & gapped topological media 2. Fermi surface as topological object 3. Fermi points (Weyl, Majorana & Dirac points) & nodal lines * superfluid 3He-A, topological semimetals , cuprate superconductors , graphene vacuum of Standard Model of particle physics in massless phase * gauge fields, gravity, chiral anomaly as emergent phenomena; quantum vacuum as 4D graphene * exotic fermions: quadratic, cubic & quartic dispersion; dispersionless fermions; Horava gravity 4. Flat bands & Fermi arcs in topological matter * surface flat bands: 3He-A, semimetals , cuprate superconductors , graphene , graphite * towards room-temperature superconductivity * 1D flat band in the vortex core *Fermi-arc on the surface of topological matter with Weyl points 5. Fully gapped topological media * superfluid 3He-B, topological insulators , chiral superconductors, quantum spin Hall insulators, vacuum of Standard Model of particle physics in present massive phase, vacua of lattice QCD * Majorana edge states & zero modes on vortices ( planar phase , topological insulator & 3He-B ) 3+1 sources of effective Quantum Field Theories in many-body system & quantum vacuum Lev Landau

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

Richard Feynman

Thermodynamics is the only physical theory of universal content

Albert Einstein effective theories Symmetry: conservation laws, translational invariance, of quantum liquids: spontaneously broken symmetry, Grand Unification, ... two-fluid hydrodynamics of superfluid 4He & Fermi liquid theory of liquid 3He

missing ingredient Topology: winding number in Landau theories one can’t comb hair on a ball smooth, anti-Grand-Unification Topological classes as defects in momentum space Fermi surface: vortex ring in p-space fully gapped topological matter: p skyrmion in p-space metals, t z normal 3He py p (p ) y z p x px p F 3He-B, topological insulators, quantum-Hall states, 3He-A film, H = + c .p 6\=2/ m vacuum of Standard Model Weyl point - hedgehog in p-space 3He-A, vacuum of SM, topological semimetals (Abrikosov) t Fermi arc on surface py of Weyl materials p , p y z pz / Fermi surface

p=p p Dirac strings in p-space terminating on monopole 1 x

/ pz p=p flat band 2 on vortex in Weyl material Khodel-Shaginyan flat band: /-vortex in p-space EXONHGJHFRUUHVSRQGHQFH WRSRORJ\LQEXONSURWHFWV gapless fermions on surface of fully gapped systems; higher order nodes in nodal topological materials

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Landau theory of Fermi liquid Standard Model + gravity

vacuum with Fermi surface: vacuum with Fermi (Weyl) point: metals, normal 3He 3He-A, planar phase, Weyl semimetal, vacuum of SM

gravity emerges from Fermi (Weyl) point μi g (pμ- eAμ - eo .Wμ)(pi- eAi - eo .Wi) = 0 analog of Fermi surface

Theory of topological matter: Nielsen, So, Ishikawa, Matsuyama, Haldane, Yakovenko, Horava, Kitaev, Ludwig, Schnyder, Ryu, Furusaki, S-C Zhang, Kane, Liang Fu, ... Fermi surface as topological object Green’s function Energy spectrum of G-1= it< ¡(p) non-interacting gas of fermionic atoms t 2 p2 p2 p ¡(p) = — μ = — F 2m 2m 2m p (p ) ¡ > 0 y z empty levels ¡ < 0 occupied p levels: p x Fermi sea F Fermi surface p=p 6\=2/ ¡ = 0 F Fermi surface: is Fermi surface a domain wall vortex ring in p-space in momentum space? phase of Green’s function G(t,p)=|G|e i\ has winding number N = 1

no! it is a vortex ring Migdal jump, non-Fermi liquids & p-space topology * Singularity at Fermi surface is robust to perturbations: t winding number N=1 cannot change continuously, interaction (perturbative) cannot destroy singularity p (p ) * Typical singularity: Migdal jump y z

n(p) p x p F 6\=2/ pF p * Other types of singularity with the same winding number: Fermi surface: Luttinger Fermi liquid, vortex line in p-space marginal Fermi liquid, pseudo-gap ... G(t,p)=|G|e i\ * Example: zeroes in G(t,p) have the same N=1 as poles Z(p,t) zeroes in G(t,p) G(t,p) = Z(p,t) = (t2 + ¡2(p))a it< ¡(p) for a > 1/2 * Important for interacting systems, where quasiparticles are ill defined * Fermi surface exists in superfluids/superconductors, examples: 3He-A in flow & Gubankova-Schmitt-Wilczek, PRB74 (2006) 064505, but Luttinger theorem is not applied quantized vortex in r-space > Fermi surface in p-space

homotopy group /1 Topology in r-space Topology in p-space z how is it in p-space ? t

y p (p ) y z winding x p number p x F N1 = 1 6\=2/ 6\=2/ vortex ring Fermi surface ^(r)=|^| e i\ G(t,p)=|G|e i\ scalar order parameter Green’s function (propagator) of superfluid & superconductor classes of mapping S1 A U(1) classes of mapping S1 A GL(n,C) manifold of space of broken symmetry vacuum states non-degenerate complex matrices non-topological flat bands due to interaction Khodel-Shaginyan fermion condensate JETP Lett. 51, 553 (1990) GV, JETP Lett. 53, 222 (1991) Nozieres, J. Phys. (Fr.) 2, 443 (1992) E{n(p)} bE{n(p)} = ¡(p)bn(p)ddp = 0 solutions: ¡(p) = 0 or bn(p)=0 ¡(p) n(p) n(p) ¡(p)

p p pF p1 flat band p2

bn(p)=0 bn(p)=0 ( )=0 splitting of Fermi surface ( ) ¡(p) = 0 bn p to Fermi ball (flat band) ¡ p = 0 bn(p)=0 S.-S. Lee Non-Fermi liquid from a charged black hole: A critical Fermi ball PRD 79, 086006 (2009) anti-de Sitter/conformal field theory correspondence Flat band as momentum-space dark soliton terminated by half-quantum vortices

p , p y z / p=p p 1 x / half-quantum vortices p=p 2 in 4-momentum space ¡ = 0 Khodel-Shaginyan fermionic condensate (flat band)

phase of Green's function changes by / across the "dark soliton" 3. Classes of Fermi points & nodal lines: superfluid 3He-A, Standard Model, semimetals, graphene, cuprate SC, ... surface of 3He-B & topological insulators Weyl point magnetic hedgehog or Berry phase Dirac monopole z pz y py

x px again no difference ?

hedgehog in r-space right-handed and left-handed hedgehog in p-space massless quarks and leptons (r)=^r are elementary particles (p) = ^p m in Standard Model m

Landau CP symmetry close to Fermi point is emergent H = + c m .p right-handed Weyl electron = hedgehog in p-space with spines = spins Topological invariant for right and left elementary particles

Berry ‘magnetic’ field Berry phase a(C) = \0 =2/ ^ p Dirac H (p) = monopole C 2p2 Dirac string \0 in (unobservable) momentum space Weyl fermions in 3+1 gapless topological cond-mat topologically protected Weyl points in:

topological semi-metal or Weyl metals (Abrikosov-Beneslavskii 1971), 3 He-A (1982), triplet Fermi gases, CoSb3 (arXiv:1204.5905)

Gap node - Weyl point (anti-hedgehog) N3 =<1 E N = 1 e dSk g . ( g × g) 3 ijk pi pj p 8/ 1 over 2D surface S p (p ) in 3D p-space y z p x p 2 S N3 =1 2 2 2 2 2 p = px +py +pz Gap node - Weyl point (hedgehog)

c(p + ip ) +i

2m < μ x y g3(p) g1(p) g2(p) H = ) = ) p2 c(p — ip ) ) g1(p)

original non-relativistic Hamiltonian

< μ c(px + ipy) g3(p) g1(p) +i g2(p)

2m . H = ) = o g(p) = ) p2 c(p — ip ) ) g1(p)

relativistic invariance as well what else is emergent ? bosonic collective modes in two generic fermionic vacua Landau theory of Fermi liquid Standard Model + gravity

collective Bose modes: Fermi propagating surface A Fermi oscillation of position point of Fermi point p A p - eA collective Bose modes form effective dynamic of fermionic vacuum: electromagnetic field propagating oscillation of shape propagating of Fermi surface oscillation of slopes

2 2 2 ik E = c p A g pi pk

form effective dynamic gravity field

Landau, ZhETF 32, 59 (1957)

two generic quantum field theories of interacting bosonic & fermionic fields relativistic quantum fields & gravity emerging near Weyl point Atiyah-Bott-Shapiro construction: primary object: linear expansion of Hamiltonian near the nodes in terms of Dirac K-matrices tetrad k a 0 e μ E = vF (p < pF) emergent relativity H = ea K .(pk < pk) a linear expansion near Fermi surface linear expansion near Weyl point secondary object: pz metric p μi ab μ i y g = d ea eb gμi(p - eA - eo .W )(p - eA - eo .W ) = 0 px μ μ μ i i i

effective metric: effective effective emergent gravity SU(2) gauge isotopic spin hedgehog in p-space effective field effective electromagnetic electric charge all ingredients of Standard Model : chiral fermions & gauge fields field e = + 1 or <1 emerge in low-energy corner gravity & gauge fields together with spin, Dirac K

for interacting systems Standard Model topological invariant

Topological invariant protected by symmetry

N = 1 e tr dV K G μ G-1 G i G-1G h G-1 Q 2 μih 24/ over S3

G is Green’s function, Q is symmetry operator

GQ =+/< QG

for Standard Model vacuum Q is M2 center group

Q = exp 2/io3

o3 weak isotopic spin of SU(2)

NQ = 16 ng

16 massless Weyl particles in one generation are protected by combined symmetry and topology Weyl point (Dirac monopole) as origin of chiral anomaly example: baryon production from vacuum by hypermagnetic field B

chiral anomaly equation triangle (Adler, Bell, Jackiw) Feynman diagram

• 1 B = NB BY•EY 4/2 Y Y

N = 1 e tr dV BY2 G μ G-1 G i G-1G h G-1 B 2 μih 24/ over S3 Dirac monopole in p-space matrix of baryonic charge matrix of hypecharge Berry ‘magnetic’ field Berry phase a(C) = \0 =2/ • 1 p^ B = B •E B C Y2 2 Y Y Ya aaa H (p) = 4/ C 2p2 Dirac string \0

B a-- baryonic charge Y -- hypercharge a for right C a = +1 earlier indications of axial anomaly in 3He-A from Weyl point: -1 for left anomaly in orbital dynamics & paradox of angular & linear momenta Leggett-Takagi, Ann. Phys. 110, 353 (1978); GV & Mineev, JETP 56, 579 (1982) 6\PPHWU\SURWHFWHGLQYDULDQWVDQGFKLUDODQRPDO\ .E FKDUJH

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spectral flow

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

quasiparticles move from vacuum to the positive energy world, where they are scattered by quasiparticles in bulk and transfer momentum from vortex to normal component this is the source of Kopnin spectral flow force

Bevan, Manninen, Cook, Hook, Hall, Vachaspati & GV Momentum creation by vortices in superfluid 3He as a model of primordial baryogenesis, Nature 386, 689 (1997) ([SHULPHQWDOFKLUDODQRPDO\VSHFWUDOIORZIRUFHRQDYRUWH[

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+HOVLQNL From massless Weyl particles to massive Dirac particles

where are massive Dirac particles? Dirac particle - composite object: E=cp mixture of left and right Weyl particles

p (p ) y z E p x

N=+1 p (p ) y z 16 p E=cp Tew ~ 1 TeV~10 K x N=+1<1=0

p (p ) y z 2 2 2 2 p E = c p + m x

N=<1 fully gapped vacua can be also topologically nontrivial is Dirac vacuum topologically trivial ? (3He-B, topological insulators, ...) p-space analogs of graphene p-space analog of emergence of 2+1 gapless 4D graphene in lattice QCD relativistic fermions in 2D graphene Kaplan 1993, 2011, Creutz JHEP 04 (2008) 017

quantum vacuum as crystal

Weyl/Dirac point N = +1

Weyl/Dirac point N = <1

p-space analog of 3D graphene: superconductor _ - phase emergence of 2+1 relativistic fermions due to topology of graphene nodes

1 -1 N = tr [Ko dl H l H] 4/i

K - symmetry operator, T = +1 T = <1 commuting or anti-commuting with H

close to nodes: T = +1 T = <1 HT = +1 = oxpx + oypy T = +1 T = <1 HT = <1 = oxpx < oypy

K = oz

for real interacting systems the Hamiltonian H(p) is substituted by inverse Green’s function at zero frequency G<1(t=0,p) SU(2) gauge fields emerging near Weyl & Dirac points Atiyah-Bott-Shapiro construction: linear expansion of Hamiltonian near the nodes in terms of Dirac K-matrices

k a H = ea K .(pk < eAk < eo .Wk)

effective tetrad: effective isotopic emergent gravity effective effective spin electromagnetic SU(2) gauge SU(2) gauge field is collective mode field field of vacuum with Weyl point

isotopic spin comes from spin or from band indices

SU(2) field near Dirac points in graphene

HT = +1 = ox(px < Ax < m .Wx) + oy(py < Ay < m .Wy) Summation of topological charges in action exotic fermions: massless fermions with quardatic, cubic & quartic dispersion semi-Dirac fermions 1 -1 N = tr [Ko dl H l H] 4/i bilayer graphene double cuprate layer

N=+1 N=+1 N=0 N=+2 E E E E

p p p p y y y y p p p + p = x + x x x

E2 = (p2/ 2m)2 E=cp E=cp E2 = 2c2p2 + 4m2 nonrelativistic Dirac fermion relativistic massless fermions Dirac fermion with quadratic dispersion re-entrant violation of Lorentz invariance & neutrino physics Klinkhamer & GV arXiv:1109.6624 nonlinear Dirac fermions N=1: Dirac fermions with linear dispersion

0 px + ipy H = ) = m p + m p ) x x y y px — ipy 0

N=2: Dirac fermions with quadratic dispersion

2 Dirac fermions with nonlinear dispersion 0 (px + ipy) H = )

2 ) 0 (px + ipy) (px — ipy) 0 H = ) N ) (px — ipy) 0 N=3: Dirac fermions with cubic dispersion

3 0 (px + ipy) H = ) 3 ) (px — ipy) 0 multiple Fermi point T. Heikkila & GV arXiv:1010.0393 cubic spectrum in trilayer graphene N = 1 + 1 + 1 = 3 N=+1 N=+1 N=+1

E = p3 + += p N= +3 x E = < p3

E=cp E=cp E=cp multilayered graphene spectrum in the outer layers N N = 1 + 1 + 1 + ... E = p E = < pN

what kind of induced gravity emerges near degenerate Fermi point?

route to topological flat band on the surface of 3D material Splitting of Dirac and Weyl points Splitting of quadratic point or trigonal warping

E N = 2 = 1 + 1 + 1 <1 p y p N = +1 y N = +1 N = <1 p x p x

N = +2 N = +1 Horava anisotropic scaling gravity

3 3 3 anisotropic z=3 scaling: x = b x’ , t = b t’ Sgrav = d x dt R b3 b3 b<6 Horava anisotropic z=2 scaling in bilayered graphene

1 -1 N = tr [Ko dl H l H] 4/i

N=+1 N=+1 N=0 N=+2 E E E E

p p p p y y y y p p p + p = x + x x x

E=cp E=cp E2 = 2c2p2 + 4m2 E2 = (p2/ 2m)2

2+1 massless Dirac fermions massive fermions massless Dirac fermions with quadratic dispersion Fermions in 2+1 bylayer graphene single layer zweibein

0 p + ip x y 0 (e1+i e2).(p

double layer

( zweibein )

2 2

0 (px + ipy) 0 [(e1+i e2).(p

anisotropic scaling: x = b x’ , t = b2 t’

Horava-Lifshitz gravity: Horava, Quantum gravity at a Lifshitz point PRD 79, 084008 (2009) Heisenberg-Euler action in massive QED non-linear corrections to Maxwell equations due to vacuum polarization

3 2 2 2 4 2 4 SHE = d x dt [( B < E ) /M + ( B.E) /M ]

2 M is rest energy of electrons B,E << M

What is the Heisenberg-Euler action for relativistic massless QED emerging in condensed matter ? What is the Heisenberg-Euler action for massless QED with exotic Dirac fermions with quadratic and cubic spectrum ?

relation to Horava quantum gravity with anisotropic scaling isotropic QED emerging in 3He-A and single layer graphene

isotropic scaling: x = b x’ , t = b t’ , B = b<2 B’ , E = b<2 E’ , S = S’

3+1 isotropic QED & emerging in Weyl superfluids & semimetals

3 2 2 2 2 SQED = d x dt ( B < E ) ln 1/( B < E ) b3 b b<4 b<4 2 2 imaginary action (Schwinger pair production) at B < E

2+1 isotropic QED emerging in single layer graphene

2 2 2 3/4 SQED = d x dt ( B < E ) b2 b b<3

2 2 imaginary action (Schwinger pair production) at B < E 2+1 anisotropic QED emerging in bylayer graphene

2 2

0 (px + ipy) /2m 0 [(e1+i e2)(p 1 i.e at E > B /m Schwinger pair production 1 -2 2 Im S = / B μ dx exp(< μ f(x) ) 0 2 2 3 μ= m E /B f(x) = x < (1+x)/2 ln(1+x) < (1

2 3 2 pair production mainly occurs at μ > 1 i.e at E > B /m

3 at μ >> 1 f(x) = x /6

4/3 1/3 Schwinger pair production Im S ~ E m

M.I. Katsnelson & G.E. Volovik, Quantum electrodynamics with anisotropic scaling: Heisenberg-Euler action and Schwinger pair production in the bilayer graphene, Pis’ma ZhETF 95, 457 (2012); arXiv:1203.1578 4. Flat bands & Fermi arcs in topological matter

nodal spiral in multilayered graphene nodal lines generates flat band with zero energy nodes in graphene in cuprate superconductors in the top and bottom layers generate flat band on zigzag edge generate flat band on side surface Hekilla, Kopnin, GV Shinsei Ryu

approximate flat band on side surface of graphite formation of nodal spiral in bulk (together with flat band on the surface) by stacking of graphene layers

Hi,j = (mxpx + mypy)bi,j + (m+t+ + m

t+ > t<

N = 1 1

1 -1 N1 = tr [Ko dl H l H] 4/i Emergence of nodal line from gapped branches by stacking graphene layers example of topological bulk-surface correspondense: Nodal spiral generates flat band on the surface projection of spiral on the surface determines boundary of flat band

N = 1 tr K dl H-1 H [ o l ] 4/i C

at each (px,py) except the boundary of circle one has 1D fully gapped state (insulator)

Nout(px,py) = 0 trivial 1D insulator

Nin(px,py) = 1 topological 1D insulator topological insulator has 1D gapless edge state manifold (px,py) of edge states forms flat band N = 1 C N = 0 N = 1 Cout Cin Nodal spiral generates flat band on the surface

projection of nodal spiral on the surface determines boundary of flat band lowest energy states: energy spectrum in bulk bulk states (projection to px , py plane) surface states form the flat band nodal line Modified nodal spiral in rhombohedral graphite: spiral of Fermi surfaces (McClure 1969)

projections on surfaces determine approximate flat band

for conventional graphite: approximate flat band on the lateral surface Nodal lines in graphite tranformed to chain of electron and hole FS

for conventional graphite: approximate flat band on the lateral surface flat flat band band

px = +1 = 0 = +1 = +1 = 0 N N N N N 1 < = +1 = N N ] H l

-1 H

K [ tr i 1 / 4 Flat band on the graphene edge =

N Surface superconductivity in topological semimetals: route to room temperature superconductivity

Extremely high DoS of flat band gives high transition temperature:

normal superconductors: d2p 1 flat band superconductivity: exponentially suppressed 1= g linear dependence 2h2 E(p) transition temperature of Tc on coupling g T = T exp (-1/gi) 2/N < 1 T ¾ gS c F DoS = i(¡) ¾ ¡ c FB interaction DOS N is number of layers coupling flat band area N = 4: i(¡) ¾ ¡<1/2 Kopaev (1970); Kopaev-Rusinov (1987) HYLGHQFHRIURRPWHPSHUDWXUHVXSHUFRQGXFWLYLW\"

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*/DUNLQV<9ODVRY.+ROODQGDU;LY From Weyl point to quantum Hall topological insulators

N = 1 e dSk g . ( g × g) 3 8/ ijk pi pj over 2D surface S in 3D momentum space top. invariant for Weyl point in 3+1 system

hedgehog 2D trivial pz ¾ in p-space N3(pz) = 0 insulator Weyl ¾¾ point N3 = N3(pz p0)

py at each pz one has 2D insulator ¾ 2D topological Hall or fully gapped 2D superfluid N3(pz) = 1 insulator px ~ N (p ) = 1 e dp dp g . ( g × g) 3 z 8/ ijk x y px py over the whole 2D momentum space skyrmion or over 2D Brillouin zone in p-space top. invariant for fully gapped 2+1 system 3D matter with Weyl points: Topologically protected flat band in vortex core vortices in r-space ¾ pz 2D trivial z N3(pz) = 0 insulator at each pz between two values: 2D topological Hall insulator: pF cos h zero energy states E (pz)=0, Weyl in the vortex core flat point (1D flat band 1011.4665) band

¾ 2D topological N3(pz) = 1 QH insulator

< pF cos h

Weyl point ¾ 2D trivial N3(pz) = 0 insulator ~ 1 -1 -1 -1 GV & Yakovenko N3(pz) = 2 tr dpx dpydt G t G G p G G p G 4/ x y (1989) Topologically protected flat band in vortex core of superfluids with Weyl points E (p , Q) z flat band in spectrum of fermions bound to core of 3He-A vortex (Kopnin-Salomaa 1991)

E (pz)

continuous spectrum

bound states flat band of bound states terminates on zeroes Weyl flat band Weyl pz of continuous spectrum point point (i.e on Weyl points) 3He-A with Weyl points: for each |pz | < pF cos h Topologically protected one has 2D topological Hall insulator with Dirac valley (Fermi arc) on surface zero energy edge states E (pz)=0 (Dirac valley PRB 094510 or Fermi arc PRB 205101) pz ¾ 2D trivial N3(pz) = 0 insulator

pF cos h

Weyl point py )=0

2D topological )=0 z

¾ z p p

( N (pz) = 1 3 insulator ( E p E

Fermi arc Fermi arc x

< pF cos h

Weyl point 2D trivial x=xL ¾ x=xR N3(pz) = 0 insulator Edge states at interface between effective two 2+1 topological insulators & Fermi arc y

~ ~ N3 = N< N3 = N+ interface

gapped state x gapped state

E(py, |pz| < pF) left moving empty edge states on the edge of insulator with Index theorem: py 0 ~ number of fermion zero modes N = 1 at interface: 3 one fermion zero mode occupied i = 1 i = N+ < N< GV JETP Lett. 55, 368 (1992) py E < 0 Fermi arc in 2D: p Fermi surface which terminates z on two points: 0 Fermi arc: Fermi surface separates positive and negative energies, but has boundaries

E (py, pz) > 0 Fermi surface ordinary Fermi surface py E (py = 0, |pz| < pF) = 0 has no boundaries E (py, pz) < 0 Fermi surface of bound states pz may leak to higher dimension

Fermi surface of localized states is terminated by projections of Weyl points when localized states merge with continuous spectrum

L spectrum of R spectrum of edge states on edge states on left wall right wall

Tsutsumi, et al PRB 094510 5. Fully gapped topological matter

skyrmions in p-space emergent relativistic fermions as edge states N = 1 e dSk g . ( g × g) 3 8/ ijk pi pj around Fermi point 3+1vacuum with massless fermions

dimensional reduction

Fully gapped 2+1 vacuum

dimensional reduction N~ = 1 e dp dp g . ( g × g) 3 8/ ijk x y px py over the whole 2D momentum space or over 2D Brillouin zone gapless 1+1 edge states

Fully gapped 4+1 vacuum gives 3+1 relativistic fermions (Kaplan, arXiv:1112.0302) topological insulators & gapped superconductors in 2+1 topological insulator = topological gapped superconductor = bulk insulator superconductor with gap in bulk with topologically protected but with topologically protected gapless states on the boundary gapless states on the boundary

3 p-wave 2D superconductor (Sr2RuO4 ?), He-A thin film, CdTe/HgTe/Cd insulator quantum well, planar phase film

who protects gapless states?

p2 < μ c(px + ipy) generic example: 2m p2 = p 2 +p 2 H = ) x y 2 c(p — ip ) p ) x y < 2m + μ

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

Hs = Es Topological invariant in momentum space

c(p + ip ) +i

2m < μ x y g3(p) g1(p) g2(p) H = ) . H = ) = o g(p) p2 c(p — ip ) ) g1(p)

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

unit vector ^g p x g (px,py) sweeps unit sphere ~ N3 (μ > 0) = 1 quantum phase transition: from topological to non-topologicval insulator/superconductor

c(p + ip ) +i

2m < μ x y g3(p) g1(p) g2(p) H = ) . = ) = o g(p) p2 c(p — ip ) ) g1(p)

μ = 0 intermediate state at μ = 0 must be gapless quantum phase transition

¾ 6N = 0 is origin of fermion zero modes 3 ¾ at the interface between states with different N3 p-space invariant in terms of Green’s function & topological QPT

~ 1 2 μ -1 i -1 h -1 N3= 2 e tr d p dt G G G G G G 24/ μih GV & Yakovenko ~ J. Phys. CM 1, 5263 (1989) N3 = 6 quantum phase transitions in thin 3He-A film ~ N3 = 4

transition between plateaus ~ must occur via gapless state! N3 = 2 gap gap ~ gap a N3 = 0 film thickness a1 a2 a3 QPT from trivial plateau-plateau QPT to topological state between topological states topological quantum phase transitions

transitions between ground states (vacua) of the same symmetry, but different topology in momentum space QPT interrupted example: QPT between gapless & gapped matter by thermodynamic transitions no symmetry change T (temperature) T along the path no change of symmetry along the path line of 1-st order transition different asymptotes qc q n <6/T when T => 0 T e T 2-nd order transition qc q - parameter of system topological topological semi-metal insulator quantum phase transition at q q=qc broken symmetry T line of other topological QPT: 1-st order transition Lifshitz transition, line of transtion between topological and nontopological superfluids, 2-nd order transition plateau transitions, confinement-deconfinement transition, ... q interface between two 2+1 topological insulators or gapped superfluids

~ ~ N3 = N< N3 = N+ interface x

gapless interface gapped state gapped state

* domain wall in 2D chiral superconductors: px - component

2 p < ip p +ip

p x y x y

< μ c(px + i py tanh x ) ~ ~ 2m = 1 N = +1 H = ) N < 3 p2 3 c(p — i p tanh x ) ) x x y < 2m + μ 0

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

~ ~ N3 = N< N3 = N+ interface gapped state gapped state

E(py) left moving edge states empty py 0 GV JETP Lett. 55, 368 (1992) occupied

Index theorem: number of fermion zero modes at interface:

i = N+ < N<

empty

)

( E

x insulator

or vacuum 0 =

2D non-topological 3 ~

occupied

edge states edge right moving right

= 0 =

current right right insulator

1 +J 2D topological

= y

3 left left

~ J N

= =

current y J

0 current current =

3 ~

N y insulator

or vacuum p

2D non-topological

occupied

edge states edge

)

left moving left

(

0 empty Edge states and currents

V 0

+ )

x y insulator

or vacuum (

deficit E = 2D non-topological ~

of right moving states moving right of

edge states edge right moving right

current

V =

insulator ~ 1

y right right 2D topological =

2 ~ /

N +J e

left left

current = J

= =

J V

< 0 = ~

current current y insulator

or vacuum p V

2D non-topological

edge states edge

left moving left

)

y p

apply voltage

(

extra number extra of left moving states moving left of Edge states & intrinsic QHE: topological invariant determines Hall quantization Intrinsic quantum Hall effect & momentum-space invariant

e 2 ~ μih 2 S = N e d x dt A F A - electromagnetic field CS 16/ 3 μ ih μ

p-space invariant r-space invariant

2 ~ electric current J = bS / b A = e N E x CS x 3 y 4/ film of topological quantum liquid ^ z ~ ^ N y , E 3

^ x , J 8 quantized intrinsic Hall conductivity 6 (without external magnetic field)

2 ~ 4 m = e N xy 3 4/ 2

GV & Yakovenko a J. Phys. CM 1, 5263 (1989) film thickness general Chern-Simons terms & momentum-space invariant (interplay of r-space and p-space topologies)

~ S = 1 N eμih d2x dt A I F J CS 16/ 3IJ μ ih r-space invariant p-space invariant protected by symmetry dimensional reduction of chiral anomaly in 3+1

~ 1 e tr [ d2p dt Q Q G μ G-1 G i G-1G h G-1] N = μih I J 3IJ 24/2 I Q - charge interacting with gauge field A I μ gauge fields can be real, artificial or auxiliary Q=e for electromagnetic field A μ ^ z z z Q=m for effective spin-rotation field A ( A = aH ) z μ 0 ^ id/dt

z 1 z spin current J = (aN dH /dy + N Ey) x 4/ ss se

spin-spin QHE spin-charge QHE

2D singlet superconductor: s-wave: N = 0 spin/spin N ss m = ss p + ip : N = 2 xy 4/ x y ss dxx-yy + idxy : Nss = 4 film of planar phase of superfluid 3He

spin/charge N m = se GV & Yakovenko xy 4/ J. Phys. CM 1, 5263 (1989) spin quantum Hall effect: planar phase film of 3He & 2D topological insulator

c(p + i p m ) 2m < μ x y z H = ) p2 c(p — i p m ) ) x y z < 2m + μ ~ 1 e tr [ d2p dt G μ G-1 G i G-1G h G-1] = 0 N = μih 3 24/2

~ 1 2 μ -1 i -1 h -1 e tr [ d p dt m G G G G G G ] N = μih z se 24/2 ~ + ~ < N3 = +1 N3 = <1 ~ ~ + ~ < ~ ~+ ~ < N = = 3 N3 + N 3 = 0 Nse N3 < N 3 = 2 spin quantum Hall effect z 1 spin current J = N Ey spin-charge QHE x 4/ se

spin/charge GV & Yakovenko Nse N = 2 m = se J. Phys. CM 1, 5263 (1989) xy 4/ Intrinsic spin-current quantum Hall effect & edge state z 1 z spin current J = (aN dH /dy + N Ey) x 4/ ss se

spin-charge QHE

y y

N = 0 N = 0 se N = 2 se se x

E(py)

right moving left moving spin down spin down electric current is zero spin current is nonzero 3D topological superfluids / insulators / semiconductors / vacua gapless topologically fully gapped topologically nontrivial vacua nontrivial vacua

3He-A, 3He-B, Standard Model Standard Model above electroweak transition, below electroweak transition, semimetals, topological insulators, 4D graphene triplet & singlet (cryocrystalline vacuum) chiral superconductor, ...

Bi2Te3 Present vacuum as semiconductor or insulator

3 conduction bands E (p) of d-quarks electric charge q=-1/3

3 conduction bands of u-quarks, q=+2/3 conduction electron band, q=-1

neutrino band, q=0 p neutrino band, q=0

valence electron band, q=-1

3 valence u-quark bands q=+2/3 dielectric and magnetic properties of vacuum (running coupling constants) 3 valence d-quark bands q=-1/3 Quantum vacuum: Dirac sea electric charge of quantum vacuum Q= Y qa = N [-1 + 3×(-1/3) + 3×(+2/3) ] = 0 a fully gapped 3+1 topological matter 3 superfluid He-B, topological insulator Bi2Te3 , present vacuum of Standard Model

* Standard Model vacuum as topological insulator

Topological invariant protected by symmetry

N = 1 e tr dV Q G μ G-1 G i G-1G h G-1 Q 2 μih 24/ over 3D momentum space

G is Green’s function at t=0, Q is symmetry operator GQ =+/< QG

Standard Model vacuum: Q=a5 Ga5 =< a5G

NQ = 8ng

8 massive Dirac particles in one generation topological superfluid 3He-B Ho2 =< o2H p2

— μ c m.p K = o 2m* B p2 2

N = = — μ o +c m.p o ( 3 B 1 p2 (2m* ) (c m.p — + μ B 2m*

1/m* non-topological superfluid topological 3He-B

NQ = 0 NQ = +2 Dirac vacuum 0 μ 1/m* NQ = <1 NQ = +1 = 0

Dirac Dirac — S cBm.p

NQ = <2 NQ = 0 N = (c m.p + ( topological superfluid non-topological superfluid B S

GV JETP Lett. 90, 587 (2009) Boundary of 3D gapped topological superfluid x,y

vacuum 3He-B p2 — μ+U(z) .

2m* cBm p

N = N = +2 p2 ( NQ = 0 Q (c m.p — + μ—U(z) B 2m* z 0

spectrum of Majorana fermion zero modes μ—U(z) H = c ^z . m x p = c (m p

0 z NQ = 0 fermion zero modes on Dirac wall

Dirac wall — S(z) cm.p N = ( (cm.p + S(z) Dirac vacuum Dirac vacuum

N = <1 N = +1 Volkov-Pankratov, Q Q 2D massless fermions in inverted contacts JETP Lett. 42, 178 (1985) S < 0 S > 0 z nodal line 0 S chiral fermions

NQ = +1 Dirac point z at the wall 0

NQ = <1 Dirac point

in Bi2Te3 Dirac point is below FS: Bi2Te3 nodal line on surface of topological insulator Majorana fermions: edge states p2 on the boundary of 3D gapped topological matter — μ+U(z) .

2m* cm p

N = p2 ( (cm.p — + μ—U(z) * boundary of topological superfluid 3He-B 2m* x,y μ—U(z) vacuum 3He-B Majorana fermions on wall N = +2 NQ = 0 NQ = +2 Q

0 z 0 NQ = 0

spectrum of fermion zero modes

* Dirac domain wall Hzm = c (mxpy

z — S(z) cm.p

0 2D massless fermions N = ( in inverted contacts N = <1 cm.p Q ( + S(z) JETP Lett. 42, 178 (1985) Majorana fermions on interface in topological superfluid 3He-B cx p2 — μ

2m* mxcxpx+mycypy+mzczpz

N = ( N = +2 p2 N = <2 Q (m c p +m c p +m c p — + μ Q x x x y y y z z z 2m* cy 0

N = +2 NQ = +2 NQ = <2 NQ = < 2 Q

phase diagram domain wall

one of 3 "speeds of light" changes sign across wall cz N = +2 Majorana Q fermions spectrum of fermion zero modes z Hzm = c (mxpy

NQ = < 2 Zero energy states in the core of vortices in topological superfluids

vortices in fully gapped 3+1 system

fermion zero modes in vortex core Bound states of fermions on cosmic strings and vortices Spectrum of quarks in core of electroweak cosmic string

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

E (pz , Q) E (pz , Q) Q=2

Q=1

pz Q=0 pz

Q=-1

Q=-2

E(pz) = - cpz for d quarks E(pz) = cpz for u quark asymmetric branches cross zero energy

Index theorem: Number of asymmetric branches = N Jackiw & Rossi N is vortex winding number Nucl. Phys. B190, 681 (1981) Bound states of fermions on vortex in s-wave superconductor Caroli, de Gennes & J. Matricon, Phys. Lett. 9 (1964) 307 NQ = 0

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

E(Q,pz) = < Q t0 (pz)

= 2/ E << Q=-7/2 t0 6 F 6 Q=-5/2 Q=-3/2 Q=-1/2 1/2 3/2 5/2 7/2 Q pz

asymmetric branch as function of Q

Angular momentum Q is half-odd integer no true fermion zero modes: in s-wave superconductor no asymmetric branch as function of pz

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

Number of asymmetric Q-branches = 2N is the existence of fermion zero modes N is vortex winding number related to topology in bulk?

GV JETP Lett. 57, 244 (1993) fermions zero modes on symmetric vortex in 3He-B topological 3He-B at μ > 0 : NQ = 2 E (Q, pz = 0 ) E (pz , Q) E (pz , Q)

Q=4 Q=4 Q=3 Q=3 Q=2 Q=2 Q=1 Q=1

Q=0 p Q=0 1 234 Q z pz

Q=-1 Q=-1 Q=-2 Q=-2 Q=-3 Q=-3 Q=-4 Q=-4

EQ=-Qt0 helicity + helicity < 2 t0=6 /EF << 6 gapless fermions on Q=0 branch form Q is integer for p-wave superfluid 3He-B 1D Fermi-liquid

Misirpashaev & GV Fermion zero modes in symmetric vortices in superfluid 3He, Physica B 210, 338 (1995) fermions zero modes on symmetric vortex in 3He-B topological 3He-B at μ > 0 : NQ = 2 E (Q, pz = 0 ) E (pz , Q)

Q=4 Q=3 Q=2 Q=1

Q=0 p 1 234 Q z Q=-1 Q=-2 Q=-3 Q=-4

EQ=-Qt0 gapless fermions on Q=0 branch form

2 t0=6 /EF << 6 1D Fermi-liquid

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

Misirpashaev & GV Fermion zero modes in symmetric vortices in superfluid 3He, Physica B 210, 338 (1995) topological quantum phase transition in bulk & in vortex core 1/m* non-topological superfluid topological superfluid 3He-B

NQ = 0 NQ = +2 μ

Q=4 Q=3 Q=2 Q=1 pz E (pz , Q) Q=0 pz Q=-1 Q=-2 Q=-3 Q=-4

μ < 0 μ > 0 vs vs

NQ = 0 NQ = 2 superfluid 3He-B as non-relativistic limit of relativistic triplet superconductor

c_.p+ `M < μR a56

N = relativistic triplet superconductor ( ( a56 < c_.p < `M + μR

cp << M μ << M

p2 — μ c m.p

2m B 3

N = superfluid He-B p2 ( (c m.p — + μ B 2m

2 cB = c 6 /M m = M / c

2 2 2 (μ + M) = μR + 6 phase diagram of topological states of relativistic triplet superconductor

3He-B p2 — μ .

2m cBm p

N = p2 ( (c m.p — + μ B 2m

c_.p+ `M < μR a56

N = ( ( a56 < c_.p < `M + μR energy spectrum in relativistic triplet superconductor

222 μ =M <6 |μ |<μ* R R R |μ |>μ* R R gapless spectrum at topological quantum phase transition soft quantum phase transition: Higgs transition in p-space

c_.p+ `M < μR a56

N = ( a56 < c_.p < `M + μR ( spectrum of non-relativistic 3He-B

2 2 μ<0 μ=0 μ<6 /M μ>6 /M soft quantum phase transition: Higgs transition in p-space

gapless spectrum at topological quantum phase p2 transition 1/m* — μ .

2m* cBm p

N = p2 ( (cBm.p — + μ NQ = 0 NQ = +2 2m*

0 μ NQ = <1 NQ = +1 Dirac Dirac

NQ = <2 NQ = 0 fermion zero modes in relativistic triplet superconductor

c_.p+ `M < μR a56

N = ( a56 < c_.p < `M + μR (

pz pz

vortices in topological superconductors have fermion zero modes

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

1 tr d3p dt dq G G-1 G G-1G G-1G G-1G G-1] N = [ t q px py pz 5 4/3i

for vortices in Dirac vacuum N = N winding number 5

E (pz) pz

N invariant was introduced by Golterman, Jansen & Kaplan for lattice fermions 5 Phys. Lett. B 301 (1993) 219 see also M.A. Zubkov & GV Momentum space topological invariants for the 4D relativistic vacua with mass gap Nucl. Phys. B 860 (2012) 295 RELAXATION IN THE VORTEX STATE T/T 0 0.15 0.16c 0.17 0.175 0.6 rad/s P = 0.5 bar 1.6 rad/s 1.2 rad/s

-1 0.6 , s

τ 0.2 rad/s = 0 0.4

0.2 Relaxation rate 1/ Relaxation rate

0 0 0.1 0.2 0.3 0.4 0.5 ThermometerThermometer fork width fork [∝ width,exp(− Hz/T )], Hz

Relaxation rate: 1/τ = 1/τ0() + C() exp(−/T ) BROKEN SYMMETRY OF VORTEX CORES IN 3 He-B

P, bar Solid 30 A B 20

10 Normal

0 012T,mK

A A 1.0 1.0

0.8 0.8

0.6 0.6

0.4 0.4 ∼ few ξ Broken symmetry core Axisymmetric core

Ikkala, Hakonen, Bunkov, Krusius et al 1982- Salomaa, Volovik, Thuneberg et al DAMPING OF SPIN PRECESSION VIA VORTEX CORES

Torque from precessing magnetic moment puts vortex core in twisting motion (oscillations / precession) ⇓ M Transitions between the core-bound fermion states are triggered and the core gets overheated ⇓ Dissipation

(Kopnin and Volovik, 1998)

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