1. Chirally imbalanced lattice fermions: in and out of equilibrium
Pavel Buividovich
(Regensburg)

2. Why chirally imbalanced plasma?
Collective motion of chiral fermions
High-energy physics:
Quark-gluon plasma
Hadronic matter
Neutrinos/leptons in Early Universe
Condensed matter physics:
Weyl semimetals
Topological insulators
Liquid Helium [G. Volovik]

3. 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

4. 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!!!

5. Outline of the talk: Static ground-state transport
[with M. Puhr, S. Valgushev]
CME and inter-electron interactions
CME in the presence of a boundary
[ , , PoS(Lat15)043]
Real-time simulations
[with M. Ulybyshev]
Chirality pumping process
Chiral plasma instability (chirality decay)
[ ]

6. CME and inter-electron interactions
Anomaly triangle gets renormalized
if dynamical gauge fields are coupled
to vector currents
In cond-mat, this coupling is important
Short-range four-fermion interactions
Chiral symmetry is not exact

7. Corrections to CME: VVA correlator
4 independent form-factors
Only wL is constrained by axial Ward Identities
Static CME current: w(+)T [PB ]
Non-renormalization of w(+)T if chiral symmetry
unbroken (massless QCD) [PB ]
[M. Knecht et al., hep-ph/ ]
[A. Vainstein, hep-ph/ ]
Both perturbative and non-perturbative
corrections to CME

8. Chiral chemical potential
Mean-field study with Wilson-Dirac fermions (aka two-band model of Dirac semimetal)
Chiral chemical potential
Dirac mass
P-odd mass
μA = 0.0,m=0
μA = 0.5,m=0

9. Mean-field approximation
On-site interactions
(like charges repel)
Mean-field functional (after Hubbard-Stratonovich)
Dynamic
mass gap
Renormalization
of chiral
chemical potential
(not prohibited
by symmetries!!!)
All filled energy levels
(Dirac sea)
Numerically minimize w.r.t. mr, mi, μA
(All other condensates break rotations/translations)

10. Mean-field phase diagramAI
“Aoki fingers” TI OI TI
No spontaneous chiral symmetry breaking
Spontaneous parity breaking
Axionic insulator phase / Aoki phase
Order parameter is mi
No continuum symmetry – no Goldstones
Effect of μA on phase structure is quite minor

11. Enhancement of chiral chemical potential
The effect of μA
is similar to mass!!!
Mean-field value of chiral chemical potential is strongly enhanced by interactions in all phases
[Similarly to b in Karl’s holographic WSM?]

12. Chiral magnetic conductivity: static linear response
Also includes linear response of condensates
Mean-field CME ≠ CME with mean-field mr, mi, μA

13. Chiral magnetic conductivity: momentum dependence
Close to or smaller than free result
with renormalized μA max. value as estimate

14. Chiral magnetic conductivity: phase diagram
Close to free result (μA changed)
at the lines of zero Dirac mass, otherwise suppressed

15. Importance of loop corrections in σCME
Ratio Loop corrections/Tree-level
Loops are important only deep in the gapped phase

16. Outlook
Physical chiral chemical potential:
Renormalized (ARPES) or bare (E || B)?
Is CME enhancement physical?
What is the counterpart of renormalization
of μA in NMR measurements?
Next step: mean-field combined with real-time linear response

17. CME in finite-volume samples
Static CME current vanishes in the IR –
how is it measurable then?
In reality, material is never a
3D torus with periodic boundary conditions
What happens to CME in a bounded sample?
Slab geometry
Wilson-Dirac Hamiltonian
Chiral chemical potential
Switch off hoppings
outside of the slab

18. CME in finite-volume samples
Current always localized at the boundary

19. CME in finite-volume samples
At the boundary, j is close to μA/(2π2)
Total current is always zero!!!

20. Localization length vs B and μA
B1/2 << μA: Linear response regime, l0 ~ 1/μA
B1/2 >> μA: Ultraquantum regime, l0 ~ B-1/2
How this translates to NMR? [In progress…]

21. Instability of chiral plasmas
μA, QA- not “canonical” charge/chemical potential
“Conserved” charge:
Chern-Simons term
(Magnetic helicity)
Integral gauge invariant
(without boundaries)

22. Instability of chiral plasmas – simple estimate
Maxwell equations + ohmic conductivity + CME
Energy conservation
Plain wave solution
Dispersion relation
Unstable solutions at k <  = μA/(2 π2) !!!
Cf. [Hirono, Kharzeev, Yin 2015]

23. Closed system of nonlinear equations
Instability of chiral plasmas Maxwell-Chern-Simons equations (small momenta, everything is universal)
Maxwell eq-s + CME + Ohmic
Anomaly equation
Equation of state
Closed system of nonlinear equations
[Boyarsky, Froehlich, Ruchayskiy, ]
[Torres-Rincon,Manuel, ] [Ooguri,Oshikawa’12]
[Hirono, Kharzeev, Yin ]
Primordial magnetic fields
Magnetic topinsulators
Quark-gluon plasma
Neutrino & Supernovae

24. Chiral kinetic theory (1 Weyl Fermion) [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)

25. (Retarded) Polarization tensor ∏μν with (chiral) chemical potential
Instability of chiral plasmas in chiral kinetic theory [Akamatsu, Yamamoto, ]
(Retarded) Polarization tensor ∏μν with (chiral) chemical potential
Classical Maxwell equations
Unstable solutions
with growing
magnetic helicity!!!
Also [Torres-Rincon,Manuel, ]

26. Real-time simulations: classical statistical field theory approach [Son’93, Aarts&Smit’99, J. Berges&Co]
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 dynamics: Keldysh contour

28. Expansion in the quantum field

29. Classical equations of motion with quantum v.e.v. of the current
Coupled equations for EM fields + fermionic modes
Vol X Vol matrices,
Bottleneck for numerics!
Good approximation if classical EM field much larger
than quantum fluctuations
Fermions are exactly quantum!
Subtle issue: renormalizability of CSFT

30. Conservation of energy for discrete evolution, In practice negligible
Slightly violated
for discrete evolution,
In practice negligible

31. Algorithmic implementation [ArXiv:1509.02076, with M. Ulybyshev]
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%
No initial quantum fluctuations in EM fields
(destroy everything quite badly)

32. Chirality pumping, E || B
Still quite large
finite-volume effects
in the anomaly coeff.
~ 10%

33. Chirality pumping: effect of backreaction

34. Chirality pumping: effect of backreaction
Screening of electric field by produced fermions

35. Chirality decay: options for initial chiral imbalance
Excited state with
chiral imbalance
Chiral chemical potential
Hamiltonian is
CP-symmetric,
State is not!!!

36. 20x20x20 lattice Chirality decay Initial “seed” perturbation:
Plain on-shell wave,
Linear polarization
Amplitude f, wave number k (k = 1 on this slide)

37. Chirality decay: wave number dependence Origin of UV catastrophe?
20x20x20 lattice, f = 0.2, μA=1
Origin of UV catastrophe?

38. Chirality decay: energy transfer
kx = kx = 3
Transfer of energy
From circular EM fields
To non-axial fermion
excitations
Transfer between modes
negligible

39. Chirality decay: finite volume effects

40. Chirality decay: Fermi velocity dependence
20x20x20 lattice, f = 0.2, μA=1

41. Chirality decay: summary
Real-time axial anomaly well reproduced by classical statistical simulations with Wilson fermions
Pair production as a benchmark [Berges&Co]
Backreaction of electromagnetic field drastically changes the dynamics of axial imbalance
No signatures of exponentially growing modes/inverse cascade
Probably both the momenta and the damping are too large, larger lattices necessary, possibly chiral fermions

42. Summary & outlook + Interesting features of static CME:
Enhancement of μA
Localization on boundaries, l ~ B1/2
Are there counterparts in NMR measurements?
NP corrections to CME in chiral gauge theories?
Dynamical decay of chirality observed
Larger volumes to see exponential growth?
Exactly chiral fermions?
Effect of quantum fluctuations (initial density matrix)?

43. Back-up slides

44. 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

45. 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

46. 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

47. 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

48. 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

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

50. 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

51. 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

52. 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

53. 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

54. 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:

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

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

57. 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

58. 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

59. 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)

60. 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

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

62. 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

63. 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

64. 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

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

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

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

68. 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

69. 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

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

71. 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)

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

73. 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!!!

74. 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

75. 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