1. Weyl semimetals
Pavel Buividovich
(Regensburg)

2. Weyl semimetals
TODO:
- discuss different kinds of WSM, calculate the spectrum in the presence of mass
Fermi arc states
- Explain why anomalous response of conductivity is a signature of WSM

3. Simplest model of Weyl semimetals
Dirac Hamiltonian
with time-reversal/parity-breaking terms
Breaks time-reversal Breaks parity

4. Nielsen, Ninomiya and Dirac/Weyl semimetals
Axial anomaly on the lattice?
Axial anomaly =
= non-conservation of Weyl fermion number
BUT: number of states is fixed on the lattice???

5. Nielsen, Ninomiya and Dirac/Weyl semimetals
Weyl points separated
in momentum space
In compact BZ, equal number of right/left handed Weyl points
Axial anomaly = flow of charges from/to left/right Weyl point

6. Nielsen-Ninomiya and Dirac/Weyl semimetals
Enhancement of electric conductivity
along magnetic field
Intuitive explanation: no backscattering
for 1D Weyl fermions

7. Nielsen-Ninomiya and Dirac/Weyl semimetals

8. Field-theory motivation
A lot of confusion in HIC physics…
Table-top experiments are easier?

9. Weyl points survive ChSB!!!
Weyl semimetals
Weyl points survive ChSB!!!

10. Weyl semimetals: realizations
Pyrochlore Iridates
[Wan et al.’2010]
Strong SO coupling (f-element)
Magnetic ordering
Stack of TI’s/OI’s
[Burkov,Balents’2011]
Surface states of TI Spin splitting
Tunneling amplitudes
Iridium:
Rarest/strongest elements
Consumption on earth: 3t/year
Magnetic doping/TR breaking essential

11. Weyl semimetals with μA
How to split energies of Weyl nodes?
[Halasz,Balents ’2012]
Stack of TI’s/OI’s
Break inversion by voltage
Or break both T/P
Chirality pumping
[Parameswaran et al.’13]
Electromagnetic instability of μA
[Akamatsu,Yamamoto’13]
Chiral kinetic theory (see below)
Classical EM field
Linear response theory
Unstable EM field mode
μA => magnetic helicity
OR: photons with circular polarization

12. Lattice model of WSM Take simplest model of TIs: Wilson-Dirac fermions
Model magnetic doping/parity breaking terms by local terms in the Hamiltonian
Hypercubic symmetry broken by b
Vacuum energy is decreased for both b and μA

13. Weyl semimetals: no sign problem!
Wilson-Dirac with chiral chemical potential:
No chiral symmetry
No unique way to introduce μA
Save as many symmetries as possible [Yamamoto‘10]
Counting Zitterbewegung,
not worldline wrapping

14. Weyl semimetals+μA : no sign problem!
One flavor of Wilson-Dirac fermions
Instantaneous interactions (relevant for condmat)
Time-reversal invariance: no magnetic interactions
Kramers degeneracy in spectrum:
Complex conjugate pairs
Paired real eigenvalues
External magnetic field causes sign problem!
Determinant is always positive!!!
Chiral chemical potential: still T-invariance!!!
Simulations possible with Rational HMC

15. Topological stability of Weyl points
Weyl Hamiltonian in momentum space:
Full set of operators for 2x2 hamiltonian
Any perturbation (transl. invariant)
= just shift of the Weyl point
Weyl point are topologically stable
Only “annihilate” with Weyl point of another chirality
E.g. ChSB by mass term:

16. Weyl points as monopoles in momentum space
Free Weyl Hamiltonian:
Unitary matrix of eigenstates:
Associated non-Abelian gauge field:

17. Weyl points as monopoles in momentum space
Classical regime: neglect spin flips =
off-diagonal terms in ak
Classical action
(ap)11 looks like a field of Abelian monopole in momentum space
Berry flux
Topological invariant!!!
Fermion doubling theorem:
In compact Brillouin zone
only pairs of
monopole/anti-monopole

18. Fermi arcs [Wan,Turner,Vishwanath,Savrasov’2010]
What are surface states of a Weyl semimetal?
Boundary Brillouin zone
Projection of the Dirac point
kx(θ), ky(θ) – curve in BBZ
2D Bloch Hamiltonian
Toric BZ
Chern-Symons
= total number of Weyl points
inside the cylinder
h(θ, kz) is a topological Chern insulator
Zero boundary mode at some θ

19. Why anomalous transport?
Collective motion of chiral fermions
High-energy physics:
Quark-gluon plasma
Hadronic matter
Leptons/neutrinos in Early Universe
Condensed matter physics:
Weyl semimetals
Topological insulators

20. Hydrodynamic approach
Classical conservation laws for chiral fermions
Energy and momentum
Angular momentum
Electric charge No. of left-handed
Axial charge No. of right-handed
Hydrodynamics:
Conservation laws
Constitutive relations
Axial charge violates parity
New parity-violating
transport coefficients

21. Hydrodynamic approach
Let’s try to incorporate
Quantum Anomaly into Classical Hydrodynamics
Now require positivity of entropy production…
BUT: anomaly term
can lead to any sign of dS/dt!!!
Strong constraints on
parity-violating transport coefficients
[Son, Surowka ‘ 2009]
Non-dissipativity of anomalous transport
[Banerjee,Jensen,Landsteiner’2012]

22. Anomalous transport: CME, CSE, CVE
Chiral Magnetic Effect
[Kharzeev, Warringa,
Fukushima]
Chiral Separation Effect
[Son, Zhitnitsky]
Chiral Vortical Effect
[Erdmenger et al.,
Teryaev, Banerjee et al.]
Flow vorticity
Origin in
quantum anomaly!!!

23. Why anomalous transport on the lattice?
1) Weyl semimetals/Top.insulators are crystals
2) Lattice is the only practical non-perturbative
regularization of gauge theories
First, let’s consider
axial anomaly
on the lattice

24. Warm-up: Dirac fermions in D=1+1
Dimension of Weyl representation: 1
Dimension of Dirac representation: 2
Just one “Pauli matrix” = 1
Weyl Hamiltonian in D=1+1
Three Dirac matrices:
Dirac Hamiltonian:

25. Warm-up: anomaly in D=1+1

26. Axial anomaly on the lattice
= non-conservation of Weyl fermion number
BUT: number of states is fixed on the lattice???

27. Anomaly on the (1+1)D lattice
1D minimally
doubled
fermions
DOUBLERS
Even number of Weyl points in the BZ
Sum of “chiralities” = 0
1D version of Fermion Doubling

28. Anomaly on the (1+1)D lattice
Let’s try “real” two-component fermions
Two chiral “Dirac” fermions
Anomaly cancels between doublers
Try to remove the doublers by additional terms

29. Anomaly on the (1+1)D lattice
(1+1)D Wilson fermions
In A) and B):
In C) and D): A) B) C) D) B)
Maximal mixing of chirality at BZ boundaries!!!
Now anomaly comes from the Wilson term
+ All kinds of nasty renormalizations… A) B) D) C)

30. Now, finally, transport: “CME” in D=1+1
Excess of right-moving particles
Excess of left-moving anti-particles
Directed current
Not surprising – we’ve broken parity
Effect relevant for nanotubes

31. “CME” in D=1+1 Fixed cutoff regularization: Shift of integration
variable: ZERO
UV regularization
ambiguity

32. Dimensional reduction: 2D axial anomaly
Polarization tensor in 2D:
Proper regularization (vector current conserved):
[Chen,hep-th/ ]
Final answer:
Value at k0=0, k3=0: NOT DEFINED
(without IR regulator)
First k3 → 0, then k0 → 0
Otherwise zero

33. Directed axial current, separation of chirality
“CSE” in D=1+1 μA μA
Excess of right-moving particles
Excess of left-moving particles
Directed axial current, separation of chirality
Effect relevant for nanotubes

34. Energy flux = momentum density
“AME” or “CVE” for D=1+1
Single (1+1)D Weyl fermion at finite temperature T
Energy flux = momentum density
(1+1)D Weyl fermions, thermally excited states:
constant energy flux/momentum density

35. Going to higher dimensions: Landau levels for Weyl fermions

36. Going to higher dimensions: Landau levels for Weyl fermions
Finite volume:
Degeneracy of every level = magnetic flux
Additional operators [Wiese,Al-Hasimi, ]

37. LLL, the Lowest Landau Level
Lowest Landau level = 1D Weyl fermion

38. Anomaly in (3+1)D from (1+1)D
Parallel uniform electric and magnetic fields
The anomaly comes only from LLL
Higher Landau Levels do not
contribute

39. Anomaly on (3+1)D lattice
Nielsen-Ninomiya picture:
Minimally doubled fermions
Two Dirac cones in the Brillouin zone
For Wilson-Dirac,
anomaly again stems
from Wilson terms
VALLEYTRONICS

40. Anomalous transport in (3+1)D from (1+1)D
CME, Dirac fermions
CSE, Dirac fermions
“AME”, Weyl fermions

41. Chiral kinetic theory [Stephanov,Son]
Classical action and
equations of motion with gauge fields
More consistent
is the Wigner
formalism
Streaming equations in phase space
Anomaly = injection of particles at zero
momentum (level crossing)

42. CME and CSE in linear response theory
Anomalous current-current correlators:
Chiral Separation and Chiral Magnetic Conductivities:

43. Chiral symmetry breaking in WSM
Mean-field free energy
Partition function
For ChSB (Dirac fermions)
Unitary transformation of SP Hamiltonian
Vacuum energy and Hubbard action are not changed
b = spatially rotating condensate = space-dependent θ angle
Funny Goldstones!!!

44. Electromagnetic response of WSM
Anomaly: chiral rotation has nonzero Jacobian in E and B
Additional term in the action
Spatial shift of Weyl points:
Anomalous Hall Effect:
Energy shift of Weyl points
But: WHAT HAPPENS IN GROUND STATE (PERIODIC EUCLIDE???)
Chiral magnetic effect
In covariant form

45. Topological insulators
Summary
Graphene
Nice and simple “standard tight-binding model”
Many interesting specific questions
Field-theoretic questions (almost) solved
Topological insulators
Many complicated tight-binding models
Reduce to several typical examples
Topological classification and universality of boundary states
Stability w.r.t. interactions? Topological Mott insulators?
Weyl semimetals
Many complicated tight-binding models, “physics of dirt”
Simple models capture the essence
Non-dissipative anomalous transport
Exotic boundary states
Topological protection of Weyl points