Weyl semimetals Pavel Buividovich Regensburg Simplest model of

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 1 D 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 Ch. SB!!!

Weyl semimetals: realizations Pyrochlore Iridates [Wan et al. ’ 2010] • Strong SO coupling (f-element) • Magnetic ordering Stack of TI’s/OI’s [Burkov, Balents’ 2011] Surface states of TI Spin splitting Tunneling amplitudes Iridium: Rarest/strongest elements Consumption on earth: 3 t/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] • • 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 • Take simplest model of TIs: Wilson-Dirac fermions • Model magnetic doping/parity breaking terms by local terms in the Hamiltonian • Hypercubic symmetry broken by b • 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 2 x 2 hamiltonian Any perturbation (transl. invariant) = just shift of the Weyl point are topologically stable Only “annihilate” with Weyl point of another chirality E. g. Ch. SB 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 [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 • 2 D 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: ü Quark-gluon plasma ü Hadronic matter ü Leptons/neutrinos in Early Universe • Condensed matter physics: ü Weyl semimetals ü 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 d. S/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 1 D minimally doubled fermions • Even number of Weyl points in the BZ • Sum of “chiralities” = 0 1 D 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 -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

“CME” in D=1+1 Fixed cutoff regularization: Shift of integration variable: ZERO UV regularization ambiguity

Dimensional reduction: 2 D axial anomaly Polarization tensor in 2 D: Proper regularization (vector current conserved): [Chen, hep-th/9902199] Final answer: • Value at k 0=0, k 3=0: NOT DEFINED (without IR regulator) • First k 3 → 0, then k 0 → 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 = 1 D 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 Ch. SB (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 • 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? • • • 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 • • •