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### Anomalous transport: CME, CSE, CVE

### T-invariance and absence of dissipation

### Anomalous transport: CME, CSE, CVE

### CME with overlap fermions

### CME with overlap fermions

### CME with overlap fermions

### CME: “Background field” method

### CME in constant field: analytics

### CME in constant field: analytics

### Closer look at CME: LLL dominance

### Dimensional reduction: 2D axial anomaly

### Chirality n5vsμ5

### CME and CSE in linear response theory

### Chiral Separation: finite-temperature regularization

### Chiral Magnetic Conductivity: finite-temperature regularization

### Chiral Magnetic Conductivity: finite-temperature regularization

### Chiral Magnetic Conductivity: still regularization

### Chiral Magnetic Conductivity: regularization

### Overlap fermions

### Dirac operator with axial gauge fields

### Overlap fermions with axial gauge fields and chiral chemical potential

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

### Current-current correlators on the lattice

### Derivatives of overlap

### Chiral Separation Conductivity with overlap

### Chiral Separation Conductivity with overlap

### Chiral Magnetic Conductivity with overlap

### Chiral Magnetic Conductivity with overlap

### Role of conserved current

### Is it necessary to use overlap? Try Wilson-Dirac

### CME, CSE and axial anomaly

### CME, CSE and axial anomaly: IR vs UV

### General decomposition of VVA correlator

### Anomalous correlatorsvs VVA correlator

### Anomalous correlatorsvs VVA correlator

### CME and axial anomaly (continued)

### CME and axial anomaly (continued)

### Chiral Vortical Effect

### Lattice studies of CVE

### Lattice studies of CVE

### Constant axial magnetic field

### ChiralVortical Effect from shifted boundary conditions

### Conclusions

### Chemical potential for anomalous charges

### CME and CVE: lattice studies

### Dimensional reduction with overlap

### IR sensitivity: aspect ratio etc.

### Dirac eigenmodesin axial magnetic field

### CME and axial anomaly (continued)

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

### Conclusions

[Kharzeev, Warringa,

Fukushima]

Chiral Separation Effect

[Son, Zhitnitsky]

Chiral Vortical Effect

[Erdmengeret al.,

Banerjee et al.]

- (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

- 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

ρ = 1.0, m = 0.05

ρ = 1.4, m = 0.01

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)

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

Dirac operator:

Creation/annihilation operators in magnetic field:

Now go to the Landau-level basis:

(topological)

zero modes

Higher Landau levels

Dirac operator in the basis of LLL states:

Vector current:

Prefactor comes from LL degeneracy

Only LLL contribution is nonzero!!!

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

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!!!

Anomalous current-current correlators:

Chiral Separation and Chiral Magnetic Conductivities:

Integrate out time-like loop momentum

Relation to canonical formalism

Virtual photon hits out fermions

out of Fermi sphere…

of propagators

and polarization tensor

in terms of

chiral states

s,s’

–

chiralities

regularization

required!!!

- 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?

(not chiral, but simple and works well)

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

Lattice chiral symmetry: Lüscher transformations

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]:

Require

(Integrable) equation for Dov!!!

We require that Lüscher transformations generate

gauge transformations of Aμ(x)

Linear equations for “projected” overlap or

propagator

Dirac operator with axial gauge field

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

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 GW/inverse GW projection:

Good agreement with continuum expression

Small momenta/large size: continuum result

At small momenta, agreement with PV regularization

As lattice size increases, PV result is approached

Let us try “slightly’’ non-conserved current

Perfect agreement with “naive” unregularizedresult

Agreement with PV, but large artifacts

They might be crucial in interacting theories

Expand anomalous correlators in μV or μA:

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

- 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

IR SINGULARITY

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

ε – “momentum” of chiral chemical potential

Time-dependent chemical potential:

Spatially modulated chiral chemical potential

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]:

Valid only for massless QCD!!!

Six equations for four unknowns… Solution:

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

Linear response of currents to “slow” rotation:

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

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

- No Landau Levels
- Only the Lowest LL exists
- Solution for finite volume?

Conserved lattice energy-momentum tensor:

not known

How the situation can be improved, probably?

Momentum from shifted BC [H.Meyer, 1011.2727]

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?

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

Non-compact

gauge transform

In the action

Via boundary conditions

For anomalous charge:

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

First Lx,Ly→∞ at fixed Lz, Lt, Φ !!!

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

- Result depends on the ratio Lt/Lz

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

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:

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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