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

Chiral chemical potential Mean-field study with Wilson-Dirac fermions (aka two-band model of Dirac semimetal) Chiral chemical potential Dirac mass 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) Dynamic mass gap Renormalization of chiral chemical potential (not prohibited by symmetries!!!) All filled energy levels (Dirac sea) 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 real-time 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]

Closed system of nonlinear equations 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)

(Retarded) Polarization tensor ∏μν with (chiral) chemical potential 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 for discrete evolution, In practice negligible 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 imbalance Chiral chemical potential Hamiltonian is CP-symmetric, State is not!!!

20x20x20 lattice Chirality decay Initial “seed” perturbation: Plain on-shell wave, Linear polarization Amplitude f, wave number k (k = 1 on this slide)

Chirality decay: wave number dependence Origin of UV catastrophe? 20x20x20 lattice, f = 0.2, μA=1 Origin of UV catastrophe?

Chirality decay: energy transfer kx = 1 kx = 3 Transfer of energy From circular EM fields To non-axial fermion excitations Transfer between modes negligible

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

Anomaly on the (1+1)D lattice 1D minimally doubled fermions DOUBLERS 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 In A) and B): In C) and D): A) B) C) 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 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

Directed axial current, separation of chirality “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

Energy flux = momentum density “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

Topological insulators 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