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Chirally imbalanced lattice fermions: in and out of equilibrium Pavel Buividovich (Regensburg) Why chirally imbalanced plasma? Collective motion of chiral fermions • High-energy physics: Quark-gluon plasma Hadronic matter Neutrinos/leptons in Early Universe • Condensed matter physics: Weyl semimetals Topological insulators Liquid Helium [G. Volovik] Anomalous transport: Hydrodynamics 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 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!!! Outline of the talk: Static ground-state transport [with M. Puhr, S. Valgushev] • CME and inter-electron interactions • CME in the presence of a boundary [1408.4573, 1505.04582, PoS(Lat15)043] Real-time simulations [with M. Ulybyshev] • Chirality pumping process • Chiral plasma instability (chirality decay) [1509.02076] CME and inter-electron interactions Anomaly triangle gets renormalized if dynamical gauge fields are coupled to vector currents In cond-mat, this coupling is important Short-range four-fermion interactions Chiral symmetry is not exact Corrections to CME: VVA correlator • 4 independent form-factors • Only wL is constrained by axial Ward Identities • Static CME current: w(+)T [PB 1312.1843] • Non-renormalization of w(+)T if chiral symmetry unbroken (massless QCD) [PB 1312.1843] [M. Knecht et al., hep-ph/0311100] [A. Vainstein, hep-ph/0212231] Both perturbative and non-perturbative corrections to CME Mean-field study with Wilson-Dirac fermions (aka two-band model of Dirac semimetal) Dirac mass Chiral chemical potential P-odd mass μA = 0.0,m=0 μA = 0.5,m=0 Mean-field approximation On-site interactions (like charges repel) Mean-field functional (after Hubbard-Stratonovich) All filled energy levels (Dirac sea) Dynamic mass gap Renormalization of chiral chemical potential (not prohibited by symmetries!!!) Numerically minimize w.r.t. mr, mi, μA (All other condensates break rotations/translations) Mean-field phase diagram AI “Aoki fingers” TI • • • • • OI TI No spontaneous chiral symmetry breaking Spontaneous parity breaking Axionic insulator phase / Aoki phase Order parameter is mi No continuum symmetry – no Goldstones • Effect of μA on phase structure is quite minor Enhancement of chiral chemical potential The effect of μA is similar to mass!!! Mean-field value of chiral chemical potential is strongly enhanced by interactions in all phases [Similarly to b in Karl’s holographic WSM?] Chiral magnetic conductivity: static linear response Also includes linear response of condensates Mean-field CME ≠ CME with mean-field mr, mi, μA Chiral magnetic conductivity: momentum dependence Close to or smaller than free result with renormalized μA max. value as estimate Chiral magnetic conductivity: phase diagram Close to free result (μA changed) at the lines of zero Dirac mass, otherwise suppressed Importance of loop corrections in σCME Ratio Loop corrections/Tree-level Loops are important only deep in the gapped phase Outlook Physical chiral chemical potential: Renormalized (ARPES) or bare (E || B)? Is CME enhancement physical? What is the counterpart of renormalization of μA in NMR measurements? Next step: mean-field combined with realtime linear response CME in finite-volume samples Static CME current vanishes in the IR – how is it measurable then? In reality, material is never a 3D torus with periodic boundary conditions What happens to CME in a bounded sample? • Slab geometry • Wilson-Dirac Hamiltonian • Chiral chemical potential Switch off hoppings outside of the slab CME in finite-volume samples Current always localized at the boundary CME in finite-volume samples At the boundary, j is close to μA/(2π2) Total current is always zero!!! Localization length vs B and μA B1/2 << μA: Linear response regime, l0 ~ 1/μA B1/2 >> μA: Ultraquantum regime, l0 ~ B-1/2 How this translates to NMR? [In progress…] Instability of chiral plasmas μA, QA- not “canonical” charge/chemical potential “Conserved” charge: Chern-Simons term (Magnetic helicity) Integral gauge invariant (without boundaries) Instability of chiral plasmas – simple estimate Maxwell equations + ohmic conductivity + CME Energy conservation Plain wave solution Dispersion relation Unstable solutions at k < = μA/(2 π2) !!! Cf. [Hirono, Kharzeev, Yin 2015] Instability of chiral plasmas Maxwell-Chern-Simons equations (small momenta, everything is universal) Maxwell eq-s + CME + Ohmic Anomaly equation Equation of state Closed system of nonlinear equations [Boyarsky, Froehlich, Ruchayskiy, 1109.3350] [Torres-Rincon,Manuel,1501.07608] [Ooguri,Oshikawa’12] [Hirono, Kharzeev, Yin 1509.07790] Primordial magnetic fields Magnetic topinsulators Quark-gluon plasma Neutrino & Supernovae Chiral kinetic theory (1 Weyl Fermion) [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) Instability of chiral plasmas in chiral kinetic theory [Akamatsu, Yamamoto, 1302.2125] (Retarded) Polarization tensor ∏μν with (chiral) chemical potential Classical Maxwell equations Unstable solutions with growing magnetic helicity!!! Also [Torres-Rincon,Manuel,1501.07608] Real-time simulations: classical statistical field theory approach [Son’93, Aarts&Smit’99, J. Berges&Co] • • • • • Full quantum dynamics of fermions Classical dynamics of electromagnetic fields Backreaction from fermions onto EM fields Approximation validity same as kinetic theory First nontrivial order of expansion in ђ Real-time dynamics: Keldysh contour Expansion in the quantum field Classical equations of motion with quantum v.e.v. of the current Coupled equations for EM fields + fermionic modes Vol X Vol matrices, Bottleneck for numerics! Good approximation if classical EM field much larger than quantum fluctuations Fermions are exactly quantum! Subtle issue: renormalizability of CSFT Conservation of energy Slightly violated for discrete evolution, In practice negligible Algorithmic implementation [ArXiv:1509.02076, with M. Ulybyshev] • Wilson-Dirac fermions with zero bare mass as a lattice model of WSM • Fermi velocity still ~1 (vF << 1 in progress) • Dynamics of fermions is exact, full mode summation (no stochastic estimators) • Technically: ~ 60 Gb / (16x16x32 lattice), MPI • External magnetic field from external source (rather than initial conditions ) • Anomaly reproduced up to ~5% error • Energy conservation up to ~2-5% • No initial quantum fluctuations in EM fields (destroy everything quite badly) Chirality pumping, E || B Still quite large finite-volume effects in the anomaly coeff. ~ 10% Chirality pumping: effect of backreaction Chirality pumping: effect of backreaction Screening of electric field by produced fermions Chirality decay: options for initial chiral imbalance Excited state with Chiral chemical potential chiral imbalance Hamiltonian is CP-symmetric, State is not!!! Chirality decay 20x20x20 lattice Initial “seed” perturbation: • Plain on-shell wave, • Linear polarization • Amplitude f, wave number k (k = 1 on this slide) Chirality decay: wave number dependence 20x20x20 lattice, f = 0.2, μA=1 Origin of UV catastrophe? Chirality decay: energy transfer kx = 1 Transfer of energy • From circular EM fields • To non-axial fermion excitations • Transfer between modes negligible kx = 3 Chirality decay: finite volume effects Chirality decay: Fermi velocity dependence 20x20x20 lattice, f = 0.2, μA=1 Chirality decay: summary • Real-time axial anomaly well reproduced by classical statistical simulations with Wilson fermions • Pair production as a benchmark [Berges&Co] • Backreaction of electromagnetic field drastically changes the dynamics of axial imbalance • No signatures of exponentially growing modes/inverse cascade • Probably both the momenta and the damping are too large, larger lattices necessary, possibly chiral fermions Summary & outlook + Interesting features of static CME: • Enhancement of μA • Localization on boundaries, l ~ B1/2 Are there counterparts in NMR measurements? NP corrections to CME in chiral gauge theories? Dynamical decay of chirality observed • Larger volumes to see exponential growth? • Exactly chiral fermions? • Effect of quantum fluctuations (initial density matrix)? Back-up slides 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 Electromagnetic instability of μA [Akamatsu,Yamamoto’13] • • • • Chiral kinetic theory (see below) Classical EM field Linear response theory Unstable EM field mode • μA => magnetic helicity 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 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 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 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 • 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 • • •