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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