Anomalous transport on the lattice
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Anomalous transport on the lattice. Pavel Buividovich (Regensburg). Chiral Magnetic Effect [ Kharzeev , Warringa , Fukushima]. Chiral Separation Effect [Son, Zhitnitsky ]. Chiral Vortical Effect [ Erdmenger et al. , Banerjee et al. ]. Anomalous transport: CME, CSE, CVE.

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Anomalous transport on the lattice

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Anomalous transport on the lattice



Chiral Magnetic Effect

[Kharzeev, Warringa,


Chiral Separation Effect

[Son, Zhitnitsky]

Chiral Vortical Effect

[Erdmengeret al.,

Banerjee et al.]

Anomalous transport: CME, CSE, CVE

  • Dissipative transport

  • (conductivity, viscosity)

  • No ground state

  • T-noninvariant(but CP)

  • Spectral function = anti-Hermitean part of retarded correlator

  • Work is performed

  • Dissipation of energy

  • First k → 0, then w → 0

  • Anomalous transport

  • (CME, CSE, CVE)

  • Ground state

  • T-invariant (but not CP!!!)

  • Spectral function = Hermitean part of retarded correlator

  • No work is performed

  • No dissipation of energy

  • First w → 0, then k → 0

T-invariance and absence of dissipation

  • Folklore on CME & CSE:

  • Transport coefficients are RELATED to anomalies (axial and gravitational)

  • and thus protected from:

    • perturbative corrections

    • IR effects (mass etc.)

  • Check these statements as applied to the lattice.

  • What is measurable?

  • How should one measure?

  • What should one measure?

  • Does it make sense at all? If

Anomalous transport: CME, CSE, CVE

CME with overlap fermions

ρ = 1.0, m = 0.05

CME with overlap fermions

ρ = 1.4, m = 0.01

CME with overlap fermions

ρ = 1.4, m = 0.05

Staggered fermions [G. Endrodi]

Bulk definition of μ5 !!! Around 20% deviation

CLAIM: constant magnetic field in finite volume

is NOT a small perturbation

“triangle diagram” argument invalid

(Flux is quantized, 0 → 1 is not a perturbation, just like an instanton number)

CME: “Background field” method

More advanced argument:

in a finite volume

Solution: hide extra flux in the delta-function

Fermions don’t note this singularity if

Flux quantization!

  • Partition function of Dirac fermions in a finiteEuclidean box

  • Anti-periodic BC in time direction, periodic BC in spatial directions

  • Gauge field A3=θ – source for the current

  • Magnetic field in XY plane

  • Chiral chemical potential μ5in the bulk

CME in constant field: analytics

Dirac operator:

Creation/annihilation operators in magnetic field:

Now go to the Landau-level basis:

CME in constant field: analytics


zero modes

Higher Landau levels

Dirac operator in the basis of LLL states:

Vector current:

Closer look at CME: LLL dominance

Prefactor comes from LL degeneracy

Only LLL contribution is nonzero!!!

Polarization tensor in 2D:

Proper regularization (vector current conserved):


Final answer:

Dimensional reduction: 2D axial anomaly

  • Value at k0=0, k3=0: NOT DEFINED

  • (without IR regulator)

  • First k3→ 0, then k0→ 0

  • Otherwise zero

μ5is not a physical quantity, just Lagrange multiplier

Chirality n5 is (in principle) observable

Express everything in terms of n5

To linear order in μ5:

Singularities of Π33cancel !!!

Note: no non-renormalization for two loops or higher and no dimensional reduction due to 4D gluons!!!

Chirality n5vsμ5

Anomalous current-current correlators:

CME and CSE in linear response theory

Chiral Separation and Chiral Magnetic Conductivities:

Integrate out time-like loop momentum

Relation to canonical formalism

Chiral Separation: finite-temperature regularization

Virtual photon hits out fermions

out of Fermi sphere…

Chiral Separation Conductivity: analytic result



of propagators

and polarization tensor

in terms of

chiral states

Chiral Magnetic Conductivity: finite-temperature regularization






Chiral Magnetic Conductivity: finite-temperature regularization

  • Individual contributions of chiral states are divergent

  • The total is finite and coincides with CSE (upon μV→ μA)

  • Unusual role of the Dirac mass

  • non-Fermi-liquid behavior?

Pauli-Villars regularization

(not chiral, but simple and works well)

Chiral Magnetic Conductivity: still regularization

  • Pauli-Villars: zero at small momenta

  • -μA k3/(2 π2) at large momenta

  • Pauli-Villars is not chiral (but anomaly is reproduced!!!)

  • Might not be a suitable regularization

  • Individual regularization in each chiral sector?

  • Let us try overlap fermions

  • Advanced development of PV regularization

  • = Infinite tower of PV regulators [Frolov, Slavnov, Neuberger, 1993]

  • Suitable to define Weyl fermions non-perturbatively

Chiral Magnetic Conductivity: regularization

Lattice chiral symmetry: Lüscher transformations

Overlap fermions

These factors encode the anomaly

(nontrivial Jacobian)

First consider coupling to axial gauge field:

Assume local invariance under

modified chiral transformations

[Kikukawa, Yamada, hep-lat/9808026]:

Dirac operator with axial gauge fields


(Integrable) equation for Dov!!!

We require that Lüscher transformations generate

gauge transformations of Aμ(x)

Overlap fermions with axial gauge fields and chiral chemical potential

Linear equations for “projected” overlap or


Dirac operator with axial gauge field

Overlap fermions with axial gauge fields and chiral chemical potential: solutions

Dirac operator with chiral chemical potential

  • Only implicit definition (so far)

  • Hermitean kernel (at zero μV) = Sign() of what???

  • Potentially, no sign problem in simulations

Current-current correlators on the lattice

Consistent currents!Naturaldefinition:charge transfer

Axial vertex:

In terms of kernel spectrum + derivatives of kernel:

Numerically impossible for arbitrary background…

Krylov subspace methods? Work in progress…

Derivatives of overlap

Derivatives of GW/inverse GW projection:

Chiral Separation Conductivity with overlap

Good agreement with continuum expression

Chiral Separation Conductivity with overlap

Small momenta/large size: continuum result

Chiral Magnetic Conductivity with overlap

At small momenta, agreement with PV regularization

Chiral Magnetic Conductivity with overlap

As lattice size increases, PV result is approached

Let us try “slightly’’ non-conserved current

Role of conserved current

Perfect agreement with “naive” unregularizedresult

Role of anomaly

Replace Dov with projected operator

Lüscher transformations Usual chiral rotations

Is it necessary to use overlap? Try Wilson-Dirac

Agreement with PV, but large artifacts

They might be crucial in interacting theories

Expand anomalous correlators in μV or μA:

CME, CSE and axial anomaly

VVA correlator in some special kinematics!!!

At fixed k3, the only scale is µ

Use asymptotic expressions for k3 >> µ !!!

Ward Identities fix large-momentum behavior

CME, CSE and axial anomaly: IR vs UV

  • CME: -1 x classical result, CSE: zero!!!

General decomposition of VVA correlator

  • 4 independent form-factors

  • Only wL is constrained by axial WIs

[M. Knechtet al., hep-ph/0311100]

CSE: p = (0,0,0,k3), q=0, µ=2, ν=0, ρ=1 ZERO

CME: p = (0,0,0,k3), q=(0,0,0,-k3), µ=1, ν=2, ρ=0


Regularization: p = k + ε/2, q = -k+ε/2

ε – “momentum” of chiral chemical potential

Time-dependent chemical potential:

Anomalous correlatorsvs VVA correlator

Spatially modulated chiral chemical potential

Anomalous correlatorsvs VVA correlator

By virtue of Bose symmetry, only w(+)(k2,k2,0)

Transverse form-factor

Not fixed by the anomaly

In addition to anomaly non-renormalization,

new (perturbative!!!) non-renormalization theorems

[M. Knechtet al., hep-ph/0311100]

[A. Vainstein, hep-ph/0212231]:

CME and axial anomaly (continued)

Valid only for massless QCD!!!

Special limit: p2=q2

Six equations for four unknowns… Solution:

CME and axial anomaly (continued)

Might be subject to NP corrections due to ChSB!!!

Linear response of currents to “slow” rotation:

Chiral Vortical Effect

In terms of correlators

Subject to

PT corrections!!!

  • A naïve method [Yamamoto, 1303.6292]:

  • Analytic continuation of rotating frame metric

  • Lattice simulations with distorted lattice

  • Physical interpretation is unclear!!!

  • By virtue of Hopf theorem:

  • only vortex-anti-vortex pairs allowed on torus!!!

  • More advanced method

  • [Landsteiner, Chernodub & ITEP Lattice, 1303.6266]:

  • Axial magnetic field = source for axial current

  • T0y = Energy flow along axial m.f.

  • Measure energy flow in the background axial

  • magnetic field

Lattice studies of CVE

  • First lattice calculations: ~ 0.05 x theory prediction

  • Non-conserved energy-momentum tensor

  • Constant axial field not quite a linear response

  • For CME: completely wrong results!!!...

  • Is this also the case for CVE?

  • Check by a direct calculation, use free overlap

Lattice studies of CVE

  • No holomorphic structure

  • No Landau Levels

  • Only the Lowest LL exists

  • Solution for finite volume?

Constant axial magnetic field

Conserved lattice energy-momentum tensor:

not known

How the situation can be improved, probably?

Momentum from shifted BC [H.Meyer, 1011.2727]

ChiralVortical Effect from shifted boundary conditions

We can get total conserved momentum


Momentum density


Energy flow

  • Anomalous transport: very UV and IR sensitive

  • Non-renormalizability only in special kinematics

  • Proofs of exactness: some nontrivial assumptions

  • Fermi-surface

  • Quasi-particle excitations

  • Finite screening length

  • Hydro flow of chiral particles is a strange object

  • Field theory: NP corrections might appear, ChSB

  • Even Dirac massdestroys the usual Fermi surface

  • What happens to “chiral” Fermi surface and to anomalous transport in confinement regime?

  • What are the consequences of inexact ch.symm?


Backup slides

Chemical potential for conserved charge (e.g. Q):


gauge transform

In the action

Via boundary conditions

For anomalous charge:

Chemical potential for anomalous charges

General gauge transform

BUT the current is not conserved!!!

Chern-Simons current

Topological charge density

  • Simplest method: introduce sources in the action

  • Constant magnetic field

  • Constant μ5 [Yamamoto, 1105.0385]

  • Constant axial magnetic field [ITEP Lattice, 1303.6266]

  • Rotating lattice??? [Yamamoto, 1303.6292]

  • “Advanced” method:

  • Measure spatial correlators

  • No analytic continuation necessary

  • Just Fourier transforms

  • BUT: More noise!!!

  • Conserved currents/

  • Energy-momentum tensor

  • not known for overlap

CME and CVE: lattice studies

Dimensional reduction with overlap

First Lx,Ly→∞ at fixed Lz, Lt, Φ !!!

  • L3 →∞, Lt fixed: ZERO (full derivative)

  • Result depends on the ratio Lt/Lz

IR sensitivity: aspect ratio etc.

2D axial anomaly:


polarization tensor:


polarization tensor:

Importance of conserved current

Dirac eigenmodesin axial magnetic field

  • Landau levels for vector magnetic field:

  • Rotational symmetry

  • Flux-conserving singularity not visible

  • Dirac modes in axial magnetic field:

  • Rotational symmetry broken

  • Wave functions are localized on the boundary

  • (where gauge field is singular)

  • “Conservation of complexity”:

  • Constant axial magnetic field in finite volume

  • is pathological

Dirac eigenmodesin axial magnetic field

  • CME is related to anomaly (at least) perturbativelyin massless QCD

  • Probably not the case at nonzero mass

  • Nonperturbative contributions could be important (confinement phase)?

  • Interesting to test on the lattice

  • Relation valid in linear response approximation

  • Hydrodynamics!!!

Six equations for four unknowns… Solution:

CME and axial anomaly (continued)

Fermi surface singularityAlmost correct, but what is at small p3???

Full phase space is available only at |p|>2|kF|

  • Measure spatial correlators+ Fourier transform

  • External magnetic field: limit k0 →0 required after k3 →0, analytic continuation???

  • External fields/chemical potential are not compatible with perturbativediagrammatics

  • Static field limit not well defined

  • Result depends on IR regulators


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