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Pavel Buividovich (Regensburg)

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TODO: - discuss different kinds of WSM, calculate the spectrum in the presence of mass Fermi arc states - Explain why anomalous response of conductivity is a signature of WSM

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Dirac Hamiltonian with time-reversal/parity-breaking terms Breaks time-reversal Breaks parity

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Axial anomaly = = non-conservation of Weyl fermion number BUT: number of states is fixed on the lattice???

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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

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Enhancement of electric conductivity along magnetic field Intuitive explanation: no backscattering for 1D Weyl fermions

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A lot of confusion in HIC physics Table-top experiments are easier?

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Weyl points survive ChSB!!!

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Pyrochlore Iridates [Wan et al.2010] Strong SO coupling (f-element) Strong SO coupling (f-element) Magnetic ordering Magnetic ordering Stack of TIs/OIs [Burkov,Balents2011] Surface states of TI Spin splitting Iridium: Rarest/strongest elements Consumption on earth: 3t/year Tunneling amplitudes Magnetic doping/TR breaking essential

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How to split energies of Weyl nodes? [Halasz,Balents 2012] Stack of TIs/OIs Stack of TIs/OIs Break inversion by voltage Break inversion by voltage Or break both T/P Or break both T/P Chirality pumping [Parameswaran et al.13] OR: photons with circular polarization

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Take simplest model of TIs: Wilson-Dirac fermions Take simplest model of TIs: Wilson-Dirac fermions Model magnetic doping/parity breaking terms by local terms in the Hamiltonian Model magnetic doping/parity breaking terms by local terms in the Hamiltonian Hypercubic symmetry broken by b Hypercubic symmetry broken by b Vacuum energy is decreased for both b and A Vacuum energy is decreased for both b and A

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Wilson-Dirac with chiral chemical potential: No chiral symmetry No chiral symmetry No unique way to introduce A No unique way to introduce A Save as many symmetries as possible [Yamamoto10] Save as many symmetries as possible [Yamamoto10] Counting Zitterbewegung, not worldline wrapping

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One flavor of Wilson-Dirac fermions One flavor of Wilson-Dirac fermions Instantaneous interactions (relevant for condmat) Instantaneous interactions (relevant for condmat) Time-reversal invariance: no magnetic interactions Time-reversal invariance: no magnetic interactions Kramers degeneracy in spectrum: Complex conjugate pairs Complex conjugate pairs Paired real eigenvalues Paired real eigenvalues External magnetic field causes sign problem! External magnetic field causes sign problem! Determinant is always positive!!! Determinant is always positive!!! Chiral chemical potential: still T-invariance!!! Chiral chemical potential: still T-invariance!!! Simulations possible with Rational HMC Simulations possible with Rational HMC

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Weyl Hamiltonian in momentum space: Full set of operators for 2x2 hamiltonian Any perturbation (transl. invariant) = just shift of the Weyl point = 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:

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Free Weyl Hamiltonian: Unitary matrix of eigenstates: Associated non-Abelian gauge field:

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Classical regime: neglect spin flips = off-diagonal terms in a k Classical action (a p ) 11 looks like a field of Abelian monopole in momentum space Berry flux Berry flux Topological invariant!!! Fermion doubling theorem: In compact Brillouin zone only pairs of monopole/anti-monopole

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What are surface states of a Weyl semimetal? Boundary Brillouin zone Boundary Brillouin zone Projection of the Dirac point Projection of the Dirac point k x (), k y () curve in BBZ k x (), k y () curve in BBZ 2D Bloch Hamiltonian 2D Bloch Hamiltonian Toric BZ Toric BZ Chern-Symons Chern-Symons = total number of Weyl points = total number of Weyl points inside the cylinder inside the cylinder h(, k z ) is a topological Chern insulator h(, k z ) is a topological Chern insulator Zero boundary mode at some Zero boundary mode at some

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Collective motion of chiral fermions High-energy physics: High-energy physics: Quark-gluon plasma Quark-gluon plasma Hadronic matter Hadronic matter Leptons/neutrinos in Early Universe Leptons/neutrinos in Early Universe Condensed matter physics: Condensed matter physics: Weyl semimetals Weyl semimetals Topological insulators Topological insulators

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Classical conservation laws for chiral fermions Energy and momentum Energy and momentum Angular momentum Angular momentum Electric charge No. of left-handed Electric charge No. of left-handed Axial charge No. of right-handed Axial charge No. of right-handed Hydrodynamics: Hydrodynamics: Conservation laws Conservation laws Constitutive relations Constitutive relations Axial charge violates parity New parity-violating transport coefficients

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Lets 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 Strong constraints on parity-violating transport coefficients parity-violating transport coefficients [Son, Surowka 2009] [Son, Surowka 2009] Non-dissipativity of anomalous transport Non-dissipativity of anomalous transport [Banerjee,Jensen,Landsteiner2012] [Banerjee,Jensen,Landsteiner2012]

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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!!!

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1) Weyl semimetals/Top.insulators are crystals 2) Lattice is the only practical non-perturbative regularization of gauge theories First, lets consider axial anomaly on the lattice

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Dimension of Weyl representation: 1 Dimension of Weyl representation: 1 Dimension of Dirac representation: 2 Dimension of Dirac representation: 2 Just one Pauli matrix = 1 Just one Pauli matrix = 1 Weyl Hamiltonian in D=1+1 Three Dirac matrices: Three Dirac matrices: Dirac Hamiltonian:

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Axial anomaly = = non-conservation of Weyl fermion number BUT: number of states is fixed on the lattice???

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DOUBLERS Even number of Weyl points in the BZ Even number of Weyl points in the BZ Sum of chiralities = 0 Sum of chiralities = 0 1D version of Fermion Doubling 1D minimally doubledfermions

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Lets try real two-component fermions Two chiral Dirac fermions Anomaly cancels between doublers Try to remove the doublers by additional terms

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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

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AAAA -A-A-A-A Excess of right-moving particles Excess of right-moving particles Excess of left-moving anti-particles Excess of left-moving anti-particles Directed current Not surprising weve broken parity Effect relevant for nanotubes

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Fixed cutoff regularization: Shift of integration variable: ZERO UV regularization ambiguity

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Polarization tensor in 2D: [Chen,hep-th/9902199] Value at k 0 =0, k 3 =0: NOT DEFINED Value at k 0 =0, k 3 =0: NOT DEFINED (without IR regulator) (without IR regulator) First k 3 0, then k 0 0 First k 3 0, then k 0 0 Otherwise zero Otherwise zero Final answer: Proper regularization (vector current conserved):

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Excess of right-moving particles Excess of right-moving particles Excess of left-moving particles Excess of left-moving particles Directed axial current, separation of chirality Effect relevant for nanotubes AAAA AAAA

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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

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Finite volume: Degeneracy of every level = magnetic flux Additional operators [Wiese,Al-Hasimi, 0807.0630]

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Lowest Landau level = 1D Weyl fermion

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Parallel uniform electric and magnetic fields The anomaly comes only from LLL Higher Landau Levels do not contribute

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Nielsen-Ninomiya picture: Minimally doubled fermions Minimally doubled fermions Two Dirac cones in the Brillouin zone Two Dirac cones in the Brillouin zone For Wilson-Dirac, For Wilson-Dirac, anomaly again stems anomaly again stems from Wilson terms from Wilson terms VALLEYTRONICS VALLEYTRONICS

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CME, Dirac fermions CSE, Dirac fermions AME, Weyl fermions

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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)

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Anomalous current-current correlators: Chiral Separation and Chiral Magnetic Conductivities:

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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 angle Funny Goldstones!!!

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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

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Graphene Nice and simple standard tight-binding model Nice and simple standard tight-binding model Many interesting specific questions Many interesting specific questions Field-theoretic questions (almost) solved Field-theoretic questions (almost) solved Topological insulators Many complicated tight-binding models Many complicated tight-binding models Reduce to several typical examples Reduce to several typical examples Topological classification and universality of boundary states Topological classification and universality of boundary states Stability w.r.t. interactions? Topological Mott insulators? Stability w.r.t. interactions? Topological Mott insulators? Weyl semimetals Many complicated tight-binding models, physics of dirt Many complicated tight-binding models, physics of dirt Simple models capture the essence Simple models capture the essence Non-dissipative anomalous transport Non-dissipative anomalous transport Exotic boundary states Exotic boundary states Topological protection of Weyl points Topological protection of Weyl points