pavel buividovich (regensburg) crc 634 concluding conference darmstadt, 8-12 june 2015
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Anomalous transport in parity-breaking Weyl semimetalsPavel Buividovich(Regensburg)CRC 634 Concluding ConferenceDarmstadt, 8-12 June 2015
Weyl semimetals: 3D graphene and more
Weyl points survive ChSB!!!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 fluxA lot of intuition from HEP, only recent experiments
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 chiralityTopological stability of Weyl pointsWeyl Hamiltonian in momentum space
Anomalous transport: HydrodynamicsClassical conservation laws for chiral fermionsEnergy and momentum Angular momentumElectric charge No. of left-handedAxial charge No. of right-handed
Hydrodynamics:Conservation lawsConstitutive relations
Axial charge violates parity
New parity-violating transport coefficients
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!!!Chiral Magnetic Effect
A-A
Excess of right-moving particlesExcess of left-moving anti-particles
Directed current along magnetic field Not surprising weve broken parity
Lowest Landau level = 1D Weyl fermion
???Signatures of CME in cond-mat
Negative magnetoresistivity
Enhancement of electric conductivity along magnetic fieldIntuitive explanation: no backscatteringfor 1D Weyl fermionsChirality pumping and magnetoresistivity
OR: photons with circular polarizationChiral magnetic wave
Relaxation timeapproximation:
Negative magnetoresistivityExperimental signature of axial anomaly, Bi1-xSbx , T ~ 4 K
Negative magnetoresistivity [ArXiv:1412.6543]]
Negative magnetoresistivity from lattice QCD
NMR in strongly coupled confined phase!!!Non-renormalization of CME: hydrodynamical argument
Lets try to incorporate Quantum Anomaly into Classical Hydrodynamics
Now require positivity of entropy productionBUT: anomaly termcan lead to any sign of dS/dt!!!
Strong constraints on parity-violating transport coefficients [Son, Surowka 2009]Non-dissipativity of anomalous transport [Banerjee,Jensen,Landsteiner2012]
CME and axial anomalyExpand current-current correlators in A:
VVA correlators in some special kinematics!!!
The only scale is k3 >> !!!
=General decomposition of VVA correlator
4 independent form-factors Only wL is constrained by axial WIs
[M. Knecht et al., hep-ph/0311100]Anomalous correlators vs VVA correlatorCME: p = (0,0,0,k3), q=(0,0,0,-k3), =1, =2, =0
IR SINGULARITYRegularization: p = k + /2, q = -k+/2 momentum of chiral chemical potential
Time-dependent chemical potential:
No ground state!!!Anomalous correlators vs VVA correlatorSpatially modulated chiral chemical potential
By virtue of Bose symmetry, only w(+)(k2,k2,0)
Transverse form-factorNot fixed by the anomaly[PB 1312.1843]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!!CME and axial anomaly (continued)
Special limit: p2=q2Six equations for four unknowns Solution:
Might be subject to corrections due to ChSB!!!CME and inter-fermion interactions Sources of corrections to CME in WSM: Spontaneous chiral symmetryBreaking
Hydrodynamic/Kinetic arguments invalid with Goldstones!First principle check withOverlap fermions [PB,Kochetkov, in progress]Radiative QEDcorrections[Miransky,Jensen,Kovtun,Gursoy 2014-2015]
Effect of interaction: exact chiral symmetry
Continuum Dirac, cutoff regularization, on-site interactions V [P. B., 1408.4573]Effect of interactions on CME:Wilson-Dirac lattice fermions
Enhancement of CME due to renormalization of A [PB,Puhr,Valgushev,1505. 04582]
A, QA- not canonical charge/chemical potential
Electromagnetic instability of A [Frhlich 2000] [Ooguri,Oshikawa12] [Akamatsu,Yamamoto13] []Chiral kinetic theory (see below)Classical EM fieldLinear response theoryUnstable EM field modeA => magnetic helicityNovel type of inverse cascade [1504.04854]Instability of chiral plasmas
Instability of chiral plasmas simple estimateMaxwell equations + ohmic conductivity + CME
Energy conservation
Plain wave solution
Dispersion relation
Unstable solutions at large k !!!
Real-time simulations:classical statistical field theory approach[Son93, J. Berges and collaborators] Full quantum dynamics of fermionsClassical dynamics of electromagnetic fieldsBackreaction from fermions onto EM fieldsApproximation validity same as kinetic theoryFirst nontrivial order of expansion in
Real-time simulations of chirality pumping[P.B., M.Ulybyshev15]
Wilson-Dirac fermions with zero bare mass as a lattice model of WSMFermi velocity still ~1 (vF magnetic helicity
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 A
Weyl semimetals: no sign problem!Wilson-Dirac with chiral chemical potential:No chiral symmetryNo unique way to introduce A Save as many symmetries as possible [Yamamoto10]
Counting Zitterbewegung,not worldline wrapping
Weyl semimetals+A : 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 pairsPaired real eigenvaluesExternal 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 spaceClassical regime: neglect spin flips = off-diagonal terms in akClassical action
(ap)11 looks like a field of Abelian monopole in momentum space Berry flux Topological invariant!!!
Fermion doubling theorem:In compact Brillouin zoneonly pairs of monopole/anti-monopole
Fermi arcs[Wan,Turner,Vishwanath,Savrasov2010]What are surface states of a Weyl semimetal?Boundary Brillouin zoneProjection of the Dirac pointkx(), ky() curve in BBZ
2D Bloch HamiltonianToric BZChern-Symons = total number of Weyl points inside the cylinderh(, kz) is a topological Chern insulator
Zero boundary mode at some
Why anomalous transport?Collective motion of chiral fermionsHigh-energy physics:Quark-gluon plasmaHadronic matterLeptons/neutrinos in Early UniverseCondensed matter physics:Weyl semimetalsTopological insulators
Why anomalous transport on the lattice?1) Weyl semimetals/Top.insulators are crystals
2) Lattice is the only practical non-perturbativeregularization of gauge theories
First, lets consider axial anomaly on the latticeDimension 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
Warm-up: anomaly in D=1+1
Axial anomaly on the latticeAxial anomaly = = non-conservation of Weyl fermion numberBUT: number of states is fixed on the lattice???
Anomaly on the (1+1)D lattice
DOUBLERSEven number of Weyl points in the BZSum of chiralities = 01D version of Fermion Doubling
1D minimallydoubled fermionsAnomaly on the (1+1)D latticeLets try real two-component fermions
Two chiral Dirac fermionsAnomaly cancels between doublers
Try to remove the doublers by additional terms
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 fermions
Now, finally, transport: CME in D=1+1
A-A
Excess of right-moving particlesExcess of left-moving anti-particles
Directed current Not surprising weve broken parity
Effect relevant for nanotubes
CME in D=1+1
Fixed cutoff regularization:
Shift of integrationvariable: ZERO
UV regularizationambiguityDimensional reduction: 2D axial anomaly
Polarization tensor in 2D:
[Chen,hep-th/9902199] Value at k0=0, k3=0: NOT DEFINED (without IR regulator)First k3 0, then k0 0Otherwise zero
Final answer: Proper regularization (vector current conserved):
Excess of right-moving particlesExcess of left-moving particles
Directed axial current, separation of chirality
Effect relevant for nanotubes
CSE in D=1+1AA
AME or CVE for D=1+1 Single (1+1)D Weyl fermion at finite temperature TEnergy flux = momentum density
(1+1)D Weyl fermions, thermally excited states:constant energy flux/momentum densityGoing to higher dimensions: Landau levels for Weyl fermions
Going to higher dimensions: Landau levels for Weyl fermions
Finite volume:Degeneracy of every level = magnetic fluxAdditional 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)DParallel uniform electric and magnetic fieldsThe anomaly comes only from LLL
Higher Landau Levels do notcontributeAnomaly on (3+1)D lattice Nielsen-Ninomiya picture:Minimally doubled fermionsTwo Dirac cones in the Brillouin zoneFor 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 fieldsStreaming equations in phase spaceMore consistentis the WignerformalismAnomaly = 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 WSMMean-field free energy
Partition function
For ChSB (Dirac fermions)
Unitary transformation of SP HamiltonianVacuum energy and Hubbard action are not changedb = spatially rotating condensate = space-dependent angleFunny Goldstones!!!Electromagnetic response of WSMAnomaly: chiral rotation has nonzero Jacobian in E and BAdditional 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
SummaryGrapheneNice and simple standard tight-binding modelMany interesting specific questionsField-theoretic questions (almost) solved
Topological insulatorsMany complicated tight-binding modelsReduce to several typical examplesTopological classification and universality of boundary statesStability w.r.t. interactions? Topological Mott insulators?
Weyl semimetalsMany complicated tight-binding models, physics of dirtSimple models capture the essenceNon-dissipative anomalous transportExotic boundary statesTopological protection of Weyl points