paritybreaking Weyl semimetals Pavel Buividovich Regensburg CRC 634 Concluding Conference Darmstadt 812 June 2015 Weyl semimetals 3D graphene and more Weyl points survive ID: 310504
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Slide1
Anomalous transport in parity-breaking Weyl semimetals
Pavel Buividovich(Regensburg)CRC 634 Concluding ConferenceDarmstadt, 8-12 June 2015Slide2
Weyl semimetals: “3D graphene” and more
Weyl points survive ChSB!!!Slide3
Simplest model of Weyl semimetalsDirac Hamiltonian with time-reversal/parity-breaking terms
Breaks time-reversal Breaks parity
Well-studied by now:
Fermi arcs, AHE, Berry flux…
A lot of intuition from HEP, only recent experimentsSlide4
Full set of operators for 2x2 hamiltonianPerturbations = just shift of the Weyl point
Weyl point are topologically stable Berry Flux!!!
Only “annihilate” with
Weyl
point of
another chirality
Topological stability of
Weyl
points
Weyl
Hamiltonian in momentum spaceSlide5
Anomalous transport: HydrodynamicsClassical conservation laws for chiral fermionsEnergy and momentum Angular momentumElectric charge No. of
left-handedAxial charge No. of right-handed
Hydrodynamics:
Conservation laws
Constitutive relations
Axial charge violates
parity
New parity-violating
transport coefficientsSlide6
Anomalous transport: CME, CSE, CVEChiral 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!!!Slide7
Chiral Magnetic Effect
μA-μA
Excess of right-moving particles
Excess of left-moving anti-particles
Directed current along magnetic field
Not surprising – we’ve broken
parity
Lowest Landau level =
1D
Weyl
fermion
???Slide8
Signatures of CME in cond-matSlide9
Negative magnetoresistivity
Enhancement of electric conductivity
along magnetic field
Intuitive explanation: no backscattering
for 1D
Weyl
fermionsSlide10
Chirality pumping and magnetoresistivity
OR: photons with circular polarizationChiral magnetic wave
Relaxation time
approximation:Slide11
Negative magnetoresistivityExperimental signature of axial anomaly, Bi1-xSb
x , T ~ 4 K Slide12
Negative magnetoresistivity [ArXiv:1412.6543]]Slide13
Negative magnetoresistivity from lattice QCD
NMR
in strongly coupled
confined phase
!!!Slide14
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]Slide15
CME and axial anomalyExpand current-current correlators
in μA: VVA correlators
in some special kinematics!!!
The only scale is µ
k3 >> µ
!!!
=Slide16
General decomposition of VVA correlator 4 independent form-factors
Only wL is constrained by axial WIs [M. Knecht
et al.
,
hep
-ph/0311100]Slide17
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!!!Slide18
Anomalous correlators vs VVA correlator
Spatially modulated chiral chemical potentialBy virtue of Bose symmetry, only w
(+)
(k
2
,k
2
,0)
Transverse form-factor
Not fixed by the anomaly
[PB 1312.1843]Slide19
CME and axial anomaly (continued)In addition to anomaly non-renormalization,new (perturbative
!!!) non-renormalization theorems[M. Knecht et al.
, hep-ph
/0311100]
[A.
Vainstein
,
hep-ph
/0212231]
:
Valid only for
massless
fermions
!!Slide20
CME and axial anomaly (continued)Special limit: p
2=q2Six equations for four unknowns… Solution:
Might be subject to
corrections
due to
ChSB
!!!Slide21
CME and inter-fermion interactions
Sources of corrections to CME in WSM: Spontaneous chiral symmetryBreaking
Hydrodynamic/Kinetic arguments invalid with
Goldstones!
First principle check with
Overlap fermions
[
PB,Kochetkov
, in progress]
Radiative
QED
corrections
[
Miransky,Jensen
,
Kovtun,Gursoy
2014-2015]Slide22
Effect of interaction: exact chiral symmetry
Continuum Dirac, cutoff regularization, on-site interactions V
[P. B., 1408.4573]Slide23
Effect of interactions on CME:Wilson-Dirac lattice fermions
Enhancement of CME due to renormalization of
µA
[PB,Puhr,Valgushev,1505.
04582
]Slide24
μA, QA- not “canonical” charge/chemical potential
Electromagnetic instability of μA
[Frö
hlich
2000]
[Ooguri,Oshikawa’12]
[
Akamatsu,Yamamoto’13
] […]
Chiral kinetic theory (see below)
Classical EM field
Linear response theory
Unstable EM field mode
μ
A
=> magnetic helicity
Novel type of “inverse cascade”
[1504.04854]
Instability of chiral plasmasSlide25
Instability of chiral plasmas – simple estimateMaxwell equations + ohmic conductivity + CME
Energy conservationPlain wave solution Dispersion relation
Unstable solutions at
large k !!!Slide26
Real-time simulations:classical statistical field theory approach[Son’93, J. Berges and collaborators]
Full quantum dynamics of fermionsClassical dynamics of electromagnetic fields
Backreaction from fermions onto EM fields
Approximation validity same as kinetic theory
First nontrivial order of expansion in
ђSlide27
Real-time simulations of chirality pumping[P.B., M.Ulybyshev’15]
Wilson-Dirac fermions with zero bare mass as a lattice model of WSMFermi 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%Slide28
Results from classical statistical field theorySlide29
Results from classical statistical field theorySlide30
Initial quantum fluctuations includedSlide31
Initial quantum fluctuations includedSlide32
Initial quantum fluctuations includedSlide33
ConclusionsParity-breaking WSM: dynamical equilibriumAnomalous transport phenomena: CME
, CVE“Non-dissipative” ground-state transportCME 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
decaySlide34
This work was done withMaksim
UlybyshevMatthias PuhrSemen
ValgushevSlide35
Back-up slidesSlide36
Weyl semimetals: realizationsPyrochlore Iridates
[Wan et al.’2010]Strong SO coupling (f-element)Magnetic orderingStack of TI’s/OI’s[Burkov,Balents’2011]
Surface states of TI Spin splitting
Iridium:
Rarest/strongest elements
Consumption on earth: 3t/year
Tunneling amplitudes
Magnetic doping/TR breaking essentialSlide37
Weyl semimetals with μAHow 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
μ
A
[Akamatsu,Yamamoto’13]
Chiral kinetic theory (see below)
Classical EM field
Linear response theory
Unstable EM field mode
μ
A
=> magnetic
helicitySlide38
Lattice model of WSMTake simplest model of TIs: Wilson-Dirac fermionsModel magnetic doping/parity breaking terms by local terms in the Hamiltonian
Hypercubic symmetry broken by b
Vacuum energy is decreased for both
b
and
μ
ASlide39
Weyl semimetals: no sign problem!Wilson-Dirac with
chiral chemical potential:No chiral symmetry
No
unique
way
to
introduce
μ
A
Save
as
many
symmetries
as
possible
[Yamamoto‘10]
Counting
Zitterbewegung,not worldline wrappingSlide40
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 HMCSlide41
Weyl points as monopoles in momentum spaceFree
Weyl Hamiltonian:
Unitary matrix of
eigenstates
:
Associated non-Abelian gauge field:Slide42
Weyl points as monopoles in momentum spaceClassical regime: neglect spin flips = off-diagonal terms in a
kClassical 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-monopoleSlide43
Fermi arcs[Wan,Turner,Vishwanath,Savrasov’2010]What are surface states of a Weyl semimetal?
Boundary Brillouin zoneProjection of the Dirac pointkx
(θ
),
k
y
(
θ
)
– curve in BBZ
2D Bloch Hamiltonian
Toric
BZ
Chern
-Symons
= total number of
Weyl
points
inside the cylinder
h(
θ
,
k
z
) is a topological
Chern insulator Zero boundary mode at some θSlide44
Why anomalous transport?Collective motion of chiral fermionsHigh-energy physics:Quark-gluon plasmaHadronic
matterLeptons/neutrinos in Early UniverseCondensed matter physics:Weyl semimetalsTopological insulatorsSlide45
Why anomalous transport on the lattice?1) Weyl semimetals/
Top.insulators are crystals2) Lattice is the only practical non-perturbative
regularization of gauge theories
First, let’s consider
axial anomaly
on the latticeSlide46
Dimension of Weyl representation: 1Dimension of Dirac representation: 2Just one
“Pauli matrix” = 1Weyl Hamiltonian in D=1+1 Three Dirac matrices:
Dirac Hamiltonian:
Warm-up: Dirac fermions in D=1+1 Slide47
Warm-up: anomaly in D=1+1 Slide48
Axial anomaly on the latticeAxial anomaly = = non-conservation of Weyl fermion number
BUT: number of states is fixed on the lattice???Slide49
Anomaly on the (1+1)D lattice
DOUBLERS
Even number of
Weyl
points in the BZ
Sum of “
chiralities
” = 0
1D version of
Fermion Doubling
1D minimally
doubled
fermionsSlide50
Anomaly on the (1+1)D latticeLet’s try “real” two-component fermions
Two chiral “Dirac” fermionsAnomaly cancels between doublers
Try to remove the doublers
by additional termsSlide51
Anomaly on the (1+1)D lattice
A)B)C)
D)
A)
B)
D
)
C)
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…
(1+1)D Wilson fermionsSlide52
Now, finally, transport: “CME” in D=1+1
μA
-
μ
A
Excess of right-moving particles
Excess of left-moving anti-particles
Directed current
Not surprising – we’ve broken
parity
Effect relevant for nanotubes Slide53
“CME” in D=1+1
Fixed cutoff regularization:
Shift of integration
variable:
ZERO
UV regularization
ambiguitySlide54
Dimensional reduction: 2D axial anomaly
Polarization tensor in 2D: [Chen,hep-th/9902199]
Value at
k
0
=0, k
3
=0
:
NOT DEFINED
(without IR regulator)
First
k
3
→
0
, then
k
0
→
0
Otherwise zero
Final answer:
Proper regularization (vector current conserved): Slide55
Excess of right-moving particlesExcess of left-moving particlesDirected axial current, separation of chiralityEffect relevant for nanotubes
“CSE” in D=1+1
μ
A
μ
ASlide56
“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 densitySlide57
Going to higher dimensions: Landau levels for Weyl fermionsSlide58
Going to higher dimensions: Landau levels for Weyl fermions
Finite volume:
Degeneracy of every level = magnetic flux
Additional operators
[
Wiese,Al-Hasimi
, 0807.0630] Slide59
LLL, the Lowest Landau Level
Lowest Landau level = 1D Weyl fermionSlide60
Anomaly in (3+1)D from (1+1)DParallel uniform electric and magnetic fieldsThe anomaly comes only from LLL
Higher Landau Levels do not
contributeSlide61
Anomaly on (3+1)D lattice Nielsen-Ninomiya picture:Minimally doubled fermions
Two Dirac cones in the Brillouin zoneFor Wilson-Dirac, anomaly again stems from Wilson terms
VALLE
YTRONICSSlide62
Anomalous transport in (3+1)D from (1+1)D CME, Dirac fermions
CSE, Dirac fermions“AME”, Weyl fermions
Slide63
Chiral kinetic theory [Stephanov,Son]
Classical action and
equations of motion with gauge fields
Streaming equations in phase space
More consistent
is the Wigner
formalism
Anomaly = injection of particles at zero
momentum (level crossing)Slide64
CME and CSE in linear response theory
Anomalous current-current correlators: Chiral Separation and Chiral Magnetic Conductivities: Slide65
Chiral symmetry breaking in WSMMean-field free energy
Partition functionFor 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!!!Slide66
Electromagnetic response of WSMAnomaly: chiral rotation has nonzero Jacobian in
E and BAdditional term in the actionSpatial 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 Slide67
SummaryGrapheneNice and simple “standard tight-binding model”Many interesting specific questionsField-theoretic questions (almost) solved
Topological insulatorsMany complicated tight-binding modelsReduce to several typical examplesTopological 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