1. Anomalous transport in parity-breaking Weyl semimetals
Pavel Buividovich
(Regensburg)
CRC 634 Concluding Conference
Darmstadt, 8-12 June 2015

2. Weyl semimetals: “3D graphene” and more
Weyl points survive ChSB!!!

3. Simplest model of Weyl semimetals
Dirac Hamiltonian
with time-reversal/parity-breaking terms
Breaks time-reversal Breaks parity
A lot of intuition from HEP, only recent experiments
Well-studied by now:
Fermi arcs, AHE, Berry flux…

4. Topological stability of Weyl points
Weyl Hamiltonian in momentum space
Full set of operators for 2x2 hamiltonian
Perturbations = just shift of the Weyl point
Weyl point are topologically stable
Berry Flux!!!
Only “annihilate” with
Weyl point of
another chirality

5. Anomalous transport: Hydrodynamics
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

6. 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!!!

7. Chiral Magnetic Effect
???
Lowest Landau level =
1D Weyl fermion
Excess of right-moving particles
Excess of left-moving anti-particles
Directed current along magnetic field
Not surprising – we’ve broken parity

8. Signatures of CME in cond-mat

9. Negative magnetoresistivity
Enhancement of electric conductivity
along magnetic field
Intuitive explanation: no backscattering
for 1D Weyl fermions

10. Chirality pumping and magnetoresistivity
Relaxation time
approximation:
OR: photons with circular polarization
Chiral magnetic wave

11. Negative magnetoresistivity Experimental signature of axial anomaly, Bi1-xSbx , T ~ 4 K

12. Negative magnetoresistivity [ArXiv:1412.6543] ]

13. Negative magnetoresistivity from lattice QCD
NMR in strongly coupled confined phase!!!

14. Non-renormalization of CME: hydrodynamical argument
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]

15. = CME and axial anomaly Expand current-current correlators in μA:
VVA correlators in some special kinematics!!! =
The only scale is µ
k3 >> µ !!!

16. General decomposition of VVA correlator
4 independent form-factors
Only wL is constrained by axial WIs
[M. Knecht et al., hep-ph/ ]

17. Anomalous correlators vs VVA correlator
CME: p = (0,0,0,k3), q=(0,0,0,-k3), µ=1, ν=2, ρ=0
IR SINGULARITY
Regularization: p = k + ε/2, q = -k+ε/2
ε – “momentum” of chiral chemical potential
Time-dependent chemical potential:
No ground state!!!

18. Anomalous correlators vs VVA correlator
Spatially modulated chiral chemical potential
By virtue of Bose symmetry, only w(+)(k2,k2,0)
Transverse form-factor
Not fixed by the anomaly
[PB ]

19. CME and axial anomaly (continued)
In addition to anomaly non-renormalization,
new (perturbative!!!) non-renormalization theorems
[M. Knecht et al., hep-ph/ ]
[A. Vainstein, hep-ph/ ]:
Valid only for massless fermions!!

20. CME and axial anomaly (continued)
Special limit: p2=q2
Six equations for four unknowns… Solution:
Might be subject to corrections due to ChSB!!!

21. CME and inter-fermion interactions
Sources of corrections to CME in WSM:
Spontaneous
chiral symmetry
Breaking
Hydrodynamic/Kinetic arguments invalid with Goldstones!
First principle check with
Overlap fermions
[PB,Kochetkov, in progress]
Radiative QED
corrections
[Miransky,Jensen,
Kovtun,Gursoy ]

22. Effect of interaction: exact chiral symmetry
Continuum Dirac, cutoff regularization,
on-site interactions V [P. B., ]

23. Effect of interactions on CME: Wilson-Dirac lattice fermions
Enhancement of CME due to renormalization of µA
[PB,Puhr,Valgushev, ]

24. Instability of chiral plasmas
μA, QA- not “canonical” charge/chemical potential
Electromagnetic instability of μA
[Fröhlich 2000]
[Ooguri,Oshikawa’12]
[Akamatsu,Yamamoto’13] […]
Chiral kinetic theory (see below)
Classical EM field
Linear response theory
Unstable EM field mode
μA => magnetic helicity
Novel type of “inverse cascade” [ ]

25. Instability of chiral plasmas – simple estimate
Maxwell equations + ohmic conductivity + CME
Energy conservation
Plain wave solution
Dispersion relation
Unstable solutions at large k !!!

26. Real-time simulations: classical statistical field theory approach [Son’93, J. Berges and collaborators]
Full quantum dynamics of fermions
Classical dynamics of electromagnetic fields
Backreaction from fermions onto EM fields
Approximation validity same as kinetic theory
First nontrivial order of expansion in ђ

27. Real-time simulations of chirality pumping [P.B., M.Ulybyshev’15]
Wilson-Dirac fermions with zero bare mass as a lattice model of WSM
Fermi velocity still ~1 (vF << 1 in progress)
Dynamics of fermions is exact, full mode summation (no stochastic estimators)
Technically: ~ 60 Gb / (16x16x32 lattice), MPI
External magnetic field from external source (rather than initial conditions )
Anomaly reproduced up to ~5% error
Energy conservation up to ~2-5%

28. Results from classical statistical field theory

29. Results from classical statistical field theory

30. Initial quantum fluctuations included

31. Initial quantum fluctuations included

32. Initial quantum fluctuations included

33. Conclusions Parity-breaking WSM: dynamical equilibrium
Anomalous transport phenomena: CME, CVE
“Non-dissipative” ground-state transport
CME protected by anomaly
Nontrivial corrections from:
symmetry breaking
radiative QED corrections
BUT: quite small for lattice models
Real-time instability of parity-breaking WSM
Backreaction speeds up chirality decay

34. This work was done with
Maksim Ulybyshev
Semen Valgushev
Matthias Puhr

35. Back-up slides

36. 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

37. 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
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

38. 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

39. 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

40. 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

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

42. 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

43. 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

44. 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

45. 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

46. 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:

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

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

49. 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

50. 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

51. 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)

52. 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

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

54. 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

55. 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

56. 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

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

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

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

60. 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

61. 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

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

63. 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)

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

65. 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!!!

66. 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

67. 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