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Weyl semimetals
Pavel Buividovich
(Regensburg)
Simplest model of Weyl semimetals
Dirac Hamiltonian
with time-reversal/parity-breaking terms
Breaks time-reversal
Breaks parity
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???
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
Nielsen-Ninomiya and Dirac/Weyl
semimetals
Enhancement of electric conductivity
along magnetic field
Intuitive explanation: no backscattering
for 1D Weyl fermions
Nielsen-Ninomiya and Dirac/Weyl semimetals
Field-theory motivation
A lot of confusion in HIC physics…
Table-top experiments are easier?
Weyl semimetals
Weyl points survive ChSB!!!
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
Chirality pumping
[Parameswaran et al.’13]
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
OR: photons with • μ => magnetic helicity
A
circular polarization
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
•
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
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:
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
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
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]
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!!!
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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