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Chirally imbalanced
lattice fermions:
in and out of equilibrium
Pavel Buividovich
(Regensburg)
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]
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
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!!!
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
[1408.4573, 1505.04582, PoS(Lat15)043]
Real-time simulations
[with M. Ulybyshev]
• Chirality pumping process
• Chiral plasma instability (chirality decay)
[1509.02076]
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
Corrections to CME: VVA correlator
• 4 independent form-factors
• Only wL is constrained by axial Ward Identities
• Static CME current: w(+)T [PB 1312.1843]
• Non-renormalization of w(+)T if chiral symmetry
unbroken (massless QCD) [PB 1312.1843]
[M. Knecht et al., hep-ph/0311100]
[A. Vainstein, hep-ph/0212231]
Both perturbative and non-perturbative
corrections to CME
Mean-field study with Wilson-Dirac fermions
(aka two-band model of Dirac semimetal)
Dirac mass
Chiral chemical potential
P-odd mass
μA = 0.0,m=0
μA = 0.5,m=0
Mean-field approximation
On-site interactions
(like charges repel)
Mean-field functional (after Hubbard-Stratonovich)
All filled
energy levels
(Dirac sea)
Dynamic
mass gap
Renormalization
of chiral
chemical potential
(not prohibited
by symmetries!!!)
Numerically minimize w.r.t. mr, mi, μA
(All other condensates break rotations/translations)
Mean-field phase diagram
AI
“Aoki fingers”
TI
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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
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?]
Chiral magnetic conductivity:
static linear response
Also includes linear response of condensates
Mean-field CME ≠ CME with mean-field mr, mi, μA
Chiral magnetic conductivity:
momentum dependence
Close to or smaller than free result
with renormalized μA
max. value as estimate
Chiral magnetic conductivity: phase diagram
Close to free result (μA changed)
at the lines of zero Dirac mass, otherwise suppressed
Importance of loop corrections in σCME
Ratio Loop corrections/Tree-level
Loops are important only deep in the gapped phase
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 realtime linear response
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
CME in finite-volume samples
Current always localized at the boundary
CME in finite-volume samples
At the boundary, j is close to μA/(2π2)
Total current is always zero!!!
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…]
Instability of chiral plasmas
μA, QA- not “canonical” charge/chemical potential
“Conserved” charge:
Chern-Simons term
(Magnetic helicity)
Integral gauge invariant
(without boundaries)
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]
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, 1109.3350]
[Torres-Rincon,Manuel,1501.07608] [Ooguri,Oshikawa’12]
[Hirono, Kharzeev, Yin 1509.07790]
Primordial magnetic fields
Magnetic topinsulators
Quark-gluon plasma
Neutrino & Supernovae
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)
Instability of chiral plasmas in chiral kinetic
theory [Akamatsu, Yamamoto, 1302.2125]
(Retarded) Polarization tensor ∏μν with
(chiral) chemical potential
Classical Maxwell equations
Unstable solutions
with growing
magnetic helicity!!!
Also [Torres-Rincon,Manuel,1501.07608]
Real-time simulations:
classical statistical field theory approach
[Son’93, Aarts&Smit’99, J. Berges&Co]
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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
Real-time dynamics: Keldysh contour
Expansion in the quantum field
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
Conservation of energy
Slightly violated
for discrete evolution,
In practice negligible
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)
Chirality pumping, E || B
Still quite large
finite-volume effects
in the anomaly coeff.
~ 10%
Chirality pumping: effect of backreaction
Chirality pumping: effect of backreaction
Screening of electric field by produced fermions
Chirality decay:
options for initial chiral imbalance
Excited state with
Chiral chemical potential
chiral imbalance
Hamiltonian is
CP-symmetric,
State is not!!!
Chirality decay
20x20x20
lattice
Initial “seed” perturbation:
• Plain on-shell wave,
• Linear polarization
• Amplitude f, wave number k (k = 1 on this slide)
Chirality decay: wave number dependence
20x20x20 lattice, f = 0.2, μA=1
Origin of UV catastrophe?
Chirality decay: energy transfer
kx = 1
Transfer of energy
• From circular EM fields
• To non-axial fermion
excitations
• Transfer between modes
negligible
kx = 3
Chirality decay: finite volume effects
Chirality decay: Fermi velocity dependence
20x20x20 lattice, f = 0.2, μA=1
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
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)?
Back-up slides
Weyl semimetals: realizations
Pyrochlore Iridates
Stack of TI’s/OI’s
[Wan et al.’2010]
• Strong SO coupling (f-element)
• Magnetic ordering
[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
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]
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Chiral kinetic theory (see below)
Classical EM field
Linear response theory
Unstable EM field mode
• μA => magnetic helicity
Lattice model of WSM
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Take simplest model of TIs: Wilson-Dirac fermions
Model magnetic doping/parity breaking terms by local
terms in the Hamiltonian
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Hypercubic symmetry broken by b
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Vacuum energy is decreased for both b and μA
Weyl semimetals: no sign problem!
Wilson-Dirac with chiral chemical potential:
• No chiral symmetry
• No unique way to introduce μA
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Save as many symmetries as possible
[Yamamoto‘10]
Counting Zitterbewegung,
not worldline wrapping
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
Weyl points as monopoles in
momentum space
Free Weyl Hamiltonian:
Unitary matrix of eigenstates:
Associated non-Abelian gauge
field:
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
Fermi arcs
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[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
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
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
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:
Warm-up: anomaly in D=1+1
Axial anomaly on the lattice
Axial anomaly =
= non-conservation of Weyl fermion number
BUT: number of states is fixed on the lattice???
Anomaly on the (1+1)D lattice
DOUBLERS
1D minimally
doubled
fermions
• Even number of Weyl points in the BZ
• Sum of “chiralities” = 0
1D version of Fermion Doubling
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
Anomaly on the (1+1)D lattice
(1+1)D Wilson fermions
A)
B)
C)
D)
In A) and B):
In C) and 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)
Now, finally, transport:
“CME” in D=1+1
μA
-μA
• Excess of right-moving particles
• Excess of left-moving anti-particles
Directed current
Not surprising – we’ve broken parity
Effect relevant for nanotubes
“CME” in D=1+1
Fixed cutoff regularization:
Shift of integration
variable: ZERO
UV regularization
ambiguity
Dimensional reduction: 2D axial anomaly
Polarization tensor in 2D:
Proper regularization (vector current conserved):
[Chen,hep-th/9902199]
Final answer:
• Value at k0=0, k3=0: NOT DEFINED
(without IR regulator)
• First k3 → 0, then k0 → 0
• Otherwise zero
“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
“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
Going to higher dimensions:
Landau levels for Weyl fermions
Going to higher dimensions:
Landau levels for Weyl fermions
Finite volume:
Degeneracy of every level = magnetic flux
Additional operators [Wiese,Al-Hasimi, 0807.0630]
LLL, the Lowest Landau Level
Lowest Landau level = 1D Weyl fermion
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
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
Anomalous transport in (3+1)D
from (1+1)D
CME, Dirac fermions
CSE, Dirac fermions
“AME”, Weyl fermions
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)
CME and CSE in linear response theory
Anomalous current-current correlators:
Chiral Separation and Chiral Magnetic Conductivities:
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!!!
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
Summary
Graphene
• Nice and simple “standard tight-binding model”
• Many interesting specific questions
• Field-theoretic questions (almost) solved
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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?
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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
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