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```Dirac and Weyl fermions
in condensed matter systems: an introduction
Fa Wang ( 王垡 )
ICQM, Peking University

2015/12/6@PKU
Preamble: Dirac/Weyl fermions
●
●
Dirac equation: reconciliation of
special relativity
and quantum mechanics
–
Positive and “negative” energy solutions (anti-particles).
–
Lorentz invariance.
Weyl equation: w/ Dirac mass m=0, “chirality”
is conserved, project Dirac equation (4x4) onto
subspaces, it becomes two decoupled Weyl (2x2) equations
2015/12/6@PKU
Outline
●
Recap of solid state physics
●
2D/3D “Dirac/Weyl semimetals” in CMP
●
–
2D materials: graphene, TI surface
–
3D materials: Dirac/Weyl semimetals
Phenomena related to Dirac/Weyl fermions in CMP
–
“monopole” of Berry curvature, surface “Fermi arc”
–
Landau levels and “chiral anomaly”
2015/12/6@PKU
Recap: electrons in crystals
●
●
Ref.: any solid state physics textbook
Electrons in solid state systems should in principle be
treated as non-relativistic fermions
–
●
Energy scale ~ 10eV, much smaller than electron rest mass
Electron bands: as 0th approx., assume the nuclei form static
periodic lattice, and electrons move in the periodic potential.
–
Bloch theorem: electron eigenstates are products of planewave and a periodic Bloch function u,
for lattice translation vector R.
–
Electron energy eigenvalues
are periodic in k-space,
for reciprocal lattice vector G.
2015/12/6@PKU
Recap: electrons in crystals
●
Electron bands(cont'd)
–
Usually just give
in the 1st Brillouin
zone(BZ), the “unit cell” in k-space.
–
Depending on whether the Fermi energy
is inside a band or in band gap, the system
can be metal or insulator.
–
“effective mass” m* is different from free
electron,
if m*<0, it is hole-like carrier.
–
Effective Hamiltonian at band degeneracy
points nodes may be Dirac/Weyl-like [Herring, PhysRev'37]
2015/12/6@PKU
Recap: electrons in crystals
●
Effective Hamiltonian: for bands n...(n+m-1),
–
Usually use k-indep. Bloch basis,
w/ hermitian “Hamiltonian”
–
m bands are degenerate [H(k) prop. to identity matrix] :
in general k needs to satisfy (m2-1) conditions
–
Symmetries may reduce the number of conditions
–
Without symmetry, only m=2 and k in 3D can have generic
point solution of band degeneracy (usually 3D Weyl points).
–
Usually expand H(k) around the degenerate points (“nodes”)
to linear order in k: 3D Dirac/Weyl are
vi are (anisotropic) velocities.
2015/12/6@PKU
“Trivial” realization of Dirac fermions
in solid state systems: 1D metal
●
For a 1D (spinless) metal with two “Fermi points” at ±kF,
low energy electrons are described by “right/left mover”
under the 1D Dirac equation
–
Mass term
(back-scattering)
is usually forbidden by lattice
translation symmetry (conservation
of momentum mod G)
–
Electron interactions are important.
Free fermion description is generically not valid in 1D.
2015/12/6@PKU
Outline
●
Recap of solid state physics
●
2D/3D “Dirac/Weyl semimetals” in CMP
●
–
2D materials: graphene, TI surface
–
3D materials: Dirac/Weyl semimetals
Phenomena related to Dirac/Weyl fermions in CMP
–
“monopole” of Berry curvature, surface “Fermi arc”
–
Landau levels and “chiral anomaly”
2015/12/6@PKU
2D Dirac fermions: graphene
●
Ref.: Geim et al. RevModPhys'09.
●
2D honeycomb lattice of carbon atoms
●
States close to Fermi level are pz orbitals
Y.Yao et al Phys.Rev.B'07
●
Brillouin zone
Two spin-degenerate bands (4 bands) at BZ corners (K&K')
touch at Fermi level and disperse linearly
2015/12/6@PKU
2D Dirac fermions: graphene
●
Under basis of pz orbitals
effective Hamiltonians around K and K'=-K are
–
This is determined by the crystal & time-reversal symmetries
–
The Dirac mass is due to spin-orbit coupling and is tiny.
–
Breaking A/B sublatt. symmetry can induce mass term
–
Many phenomena related to Dirac fermions
have been predicted/observed in graphene.
●
for normal incidence on a potential barrier
2015/12/6@PKU
2D Dirac/Weyl fermions:
surface states of 3D topological insulators(TI)
●
Ref.: Qi&Zhang, RevModPhys'11
●
Many 3D TI materials: BiSb alloy, Bi2(Se,Te)3,
●
Simplest model: massive Dirac fermions (4 bands) in 3D,
–
Trivial/non-trivial TI: M0&M2 are of same/different sign.
–
Surface: mass domain wall, M0 changes sign.
–
Surface state: Jackiw-Rebbi soliton on mass domain wall.
e.g. TI(M0<0) in z<0 and vacuum(M0>0) in z>0, two surface
states satisfy
and
and
follow 2x2 gapless Dirac Hamiltonian (half of graphene)
2015/12/6@PKU
2D Dirac/Weyl fermions:
surface states of 3D topological insulators(TI)
●
Time-reversal(TR) symmetry protects
the gaplessness of TI surface states.
–
Zeeman field can open gap on
quantum anomalous Hall effect:
each surface contribute
e2/2h Hall conductance
[observed by CZChang et al. Science'13]
2015/12/6@PKU
YLChen et al. Science'10
2D Dirac/Weyl fermions:
surface states of 3D topological insulators(TI)
●
Topological field theory description of 3D TI
–
Couple bulk 3D TI to EM field, integrate out fermions, the
action contains a “topological θ-term”
–
θ is analogous to the “axion field”. Due to TR symmetry,
θ=0(trivial) or θ=π (TI) mod 2π.
–
Surface preserves TR, but θ cannot smoothly change from
trivial to TI without breaking TR. To reconcile these,
fermions on surface should be gapless.
–
On surface, θ jumps by ±π, it seems to produce a ChernSimons term for surface action (quantum anomalous Hall)
2015/12/6@PKU
3D Dirac fermions: Dirac semimetals
●
4-band degeneracy point in k-space with linear dispersions.
–
●
Special crystal symmetry is needed to forbid the Dirac mass.
Example: Cd3As2.
–
Two Dirac points at ±kD
on 4-fold rotation axis.
–
Dirac mass is “forbidden”
by C4v symmetry:
4 states are two pairs of
2dim'l irreducible rep. of C4v;
energy difference of two
pairs change sign along Γ-Z.
2015/12/6@PKU
Neupane et al. NatCommun'14
3D Weyl fermions: Weyl semimetals
●
●
2-band degeneracy point in k-space with linear dispersions.
–
NO symmetry needed to protect this degeneracy.
–
Such Weyl nodes must appear in pairs of opposite chirality.
–
Weyl nodes of opposite chirality can be mutually gapped out
if they meet in k-space. Break inversion and/or TR symmetry
to separate them. [Burkov&Balents, PhysRevLett'11]
–
Weyl nodes are difficult to locate
(usually not on high symmetry lines)
Example: TaAs.
no inversion symmetry,
12 pairs of Weyl nodes.
2015/12/6@PKU
Outline
●
Recap of solid state physics
●
2D/3D “Dirac/Weyl semimetals” in CMP
●
–
2D materials: graphene, TI surface
–
3D materials: Dirac/Weyl semimetals
Phenomena related to Dirac/Weyl fermions in CMP
–
“monopole” of Berry curvature, surface “Fermi arc”
–
Landau levels and “chiral anomaly”
2015/12/6@PKU
Prerequisite: Berry curvature in k-space
●
For a band with Bloch function
its (Abelian) Berry
connection
Berry curvature
–
Analogue of vector potential & mag. field in k-space.
Semiclassical equations of motion of electron on this band:
red term: anomalous velocity[Sundaram&Niu, PhysRevB'99]
–
The Chern number over a closed surface in k-space,
is an integer, if the band is non-degenerate on this surface.
–
Intrinsic anomalous Hall conductivity for 2D system
Fully occupied band w/ Chern
number shows quantum anomalous Hall effect.
2015/12/6@PKU
2D “Chern insulator” and chiral edge states
●
●
Fully occupied bands with nonzero total Chern number:
“Chern insulator(CI)” w/ quantum anomalous Hall effect.
Simplest model: massive Dirac fermion (2 bands) in 2D
–
Occupied band,
has nonzero Chern#
if M0&M2 are of different sign (non-trivial).
–
Define unit vector
skyrmion number of n,
–
Edge: CI(M0<0) in x<0 and vacuum(M0>0) in x>0.
Jackiw-Rebbi soliton: edge states satisfy
and
and are 1D chiral fermion
2015/12/6@PKU
Chern# =
Weyl node as “monopole” of Berry curvature
●
For a Weyl point at kW ,
the Berry curvature of occupied band,
is
–
Total Berry flux thru. surface enclosing Weyl node(s) is
–
Weyl node~“monopole” in k-space, “topologically” stable.
–
due to periodicity in k-space,
the total Berry
flux thru. surface of entire 3D BZ must be zero,
therefore sum of all Weyl points' chirality in
3D BZ must be zero: Weyl points appear in
pairs of opposite chirality.
Balents, Physics'11
2015/12/6@PKU
Surface “Fermi arc” from Weyl nodes
●
For two Weyl nodes of opposite chirality at kx=±kW ,ky=kz=0.
–
The Chern# of |kx|<kW and |kx|>kW k-planes differ by 1.
Suppose |kx|<kW k-planes have nonzero Chern#.
–
With real space surface at z=0, kx,y are still conserved.It is a
2D system in yz-plane with edge at z=0. For |kx|<kW there
will be chiral edge states
–
The Fermi surface on z=0 real space surface (2D kxky-system)
is a non-closed “arc”
connecting the projection
of Weyl nodes on kxky-plane.
2015/12/6@PKU
Turner&Vishwanath arXiv:1301.0330
Surface “Fermi arc” from Weyl nodes
●
Surface “Fermi arc” is now used as the “smoking gun”
signature of Weyl semimetals:
–
Claimed ARPES observations for TaAs
[SYXu et al. Science'15, BQLv et al. PhysRevX'15]
2015/12/6@PKU
SYXu et al. Science'15
Landau levels(LLs) from Dirac/Weyl fermions
●
For uniform mag. field B=Bz,
define
Weyl Hamiltonian becomes
–
Energy eigenvalues are
and
w/ eigenstate
–
In 2D (kz=0): unevenly spaced
Landau level energy
–
Single Weyl point produces a
chiral 0th Landau level.
for n>0,
STM observation of Landau levels in graphene,
Miller et al. Science'09
LLs from 3D Weyl points of
opposite chirality
2015/12/6@PKU
“Chiral anomaly”: 1D metal
●
Consider 1D spinless metal, the low energy theory is
–
It seems that right/left-mover number
are separately conserved
–
However under an “electric field” F,
–
Semiclassical picture:
occupied states in k-space “move to the right”,
particles are transferred from left-mover to right-mover
2015/12/6@PKU
Chiral anomaly in 3D Weyl semimetals
●
semimetal,
the right/left-handed Weyl fermion number are not
conserved under EM field,
–
Semiclassical picture:
1D chiral anomaly on n=0 LL,
LL degeneracy
–
If there are back-scattering,
magnetoresistance(MR)
[DTSon&Spivak, PhysRevB'13]
–
This effect also applies to gapless Dirac semimetal.
2015/12/6@PKU
3D (massless) Dirac fermions in ZrTe5.
●
●
“Chiral anomaly”
observed in MR
Optical transition between
Landau levels observed
[RYChen, ZGChen, XYSong,
Schneeloch, GDGu, FWang,
NLWang, PhysRevLett'15]
Q.Li et al. arXiv:1412.6543
RYChen et al. PhysRevLett'15: magneto-optical reflectance,
peaks indicate transitions between nth and (n+1)th Landau levels,
2015/12/6@PKU
Summary
●
Dirac/Weyl fermions emerge in 2D/3D solid state systems
as special band degeneracy points
–
●
Many high-energy physics phenomena related to
Dirac/Weyl fermions have analogy in CMP.
–
●
Topological surface chiral fermions ~ Jackiw-Rebbi solition.