lattice studies of topologically nontrivial non-abelian gauge field configurations in an external...
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![Page 1: Lattice studies of topologically nontrivial non-Abelian gauge field configurations in an external magnetic field in an external magnetic field P. V. Buividovich](https://reader036.vdocuments.mx/reader036/viewer/2022062314/56649f1c5503460f94c32e91/html5/thumbnails/1.jpg)
Lattice studies of topologically nontrivial non-Abelian gauge field configurations
in an external magnetic field
P. V. Buividovich (Regensburg University)
Workshop on QCD in strong magnetic fields 12-16 November 2012, Trento, Italy
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• Chiral Magnetic Effect: charge separation in an external magnetic field due to chirality fluctuations
• Chirality fluctuations: reflect the fluctuations of the topology of non-Abelian gauge fields
• In real QCD: instanton tunneling (zero temperature) or sphaleron transitions
Introduction
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Describing CME in Euclidean space
CME is a dynamical phenomena, Euclidean QFT (and lattice) rather describe stationary states.
Background fields?• Induce static chirality imbalance due to
chiral chemical potential [Kharzeev, Fukushima, Yamamoto,…]
• Consider topologically nontrivial classical solutions (instantons, calorons, dyons)
• Consider correlators of observables with local chirality [ArXiv:0907.0494]
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Describing CME in Euclidean space
Charge separation in stationary states?
Observables?
• Electric currents (allowed by torus topology)
• Spin parts of magnetic and electric dipole moments (but is it charge separation?)
• Global dipole moment prohibited by torus topology
• Local density of electric charge
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Charge separation in the instanton background
Instanton ~ point in Euclidean space
Tensors from which the Lorentz-invariant current can be constructed:• Distance to the instanton center • Electromagnetic field strength tensor • Dual field strength tensor
All such expressions average to zero
No global current
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Charge separation in the instanton background
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Dirac spectrum for instanton with magnetic field
Motivated by recent work [ArXiv:1112.0532 Basar, Dunne, Kharzeev]
• What is the Dirac eigenspectrum for instanton in magnetic field?
• Are there additional zero modes with different chiralities?
• How does the structure of the eigenmodes change?
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Dirac spectrum for instanton with magnetic field
Overlap Dirac operator, 164 lattice, ρ = 5.0 – “Large Instanton Limit”
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IPR and localization of Dirac eigenmodes
• IPR, inverse participation ratio:
Localization on a single point: IPR = 1, Uniform spread: IPR = 1/V
• ρ(x) is the eigenmode density:
• Geometric extent of the eigenmodes:
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IPR of low-lying Dirac eigenmodes
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Geometric extent of the zero mode
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Geometric extent of low-lying modes
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Geometric structure of low-lying modes
Mode Number =>
Mag
netic
field
=
>
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IPR and localization of Dirac eigenmodes
• There are no additional zero modes• Zero modes are extended in the
direction of the magnetic field • Zero modes become more localized in
transverse directions• Overall IPR only weakly depends on the
magnetic field• Geometric parameters of higher modes
weakly depend on magnetic fields
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Charge separation at finite temperature: caloron background [work with F. Bruckmann]
• Caloron: A generalization of the instanton for one compact (time) direction = Finite T
[Harrington, Kraan, Lee]• Trivial/nontrivial holonomy: stable at
high/low temperatures• Strongly localized solution
Action density Zero mode density
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Charge separation at finite temperature: caloron background [work with F. Bruckmann]
• Pair of BPS monopole/anti-monopole, separated by some distance.
• Explicit breaking of parity invariance• Only axial symmetry• not prohibited by symmetries• Can calorons be relevant for the
description of the CME/Charge separation?
Numerical study of current densities in the caloron background
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Electric current definition for chiral fermions
Fermion propagating in a fixed gauge field configuration
Overlap lattice Dirac operator [ArXiv:hep-lat/9707022, Neuberger]
Dw is a local Wilson-Dirac operator, sign() is nonlocal
Current is not exactly conserved
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Net electric current along the magnetic field
• 163x4 lattice• monopole/anti-monopole distance = 8
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Current density profile along the caloron axis
B = 0.12 || caloron axis
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Current density profile along the caloron axis
B = 0.24 || caloron axis
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Transverse profile of the current density
B = 0.24 || caloron axis
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Net electric charge
• 163x4 lattice• monopole/anti-monopole distance = 8
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Net electric charge vs. magnetic field
B || caloron axis
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Charge density profile along the caloron axis
B = 0.12 || caloron axis
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B = 0.24 || caloron axis
Charge density profile along the caloron axis
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B = 0.12 || caloron axis
Transverse charge density profile@monopole
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B = 0.12 || caloron axis
Transverse charge density profile@midpoint
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B = 0.12 || caloron axis
Transverse charge density profile@anti-monopole
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Some physical estimates
• Fix lattice spacing from • ρ = 0.33 fm – characteristic caloron
size, from [ArXiv:hep-ph/0607315, Gerhold, Ilgenfritz, Mueller-Preussker] a ≈ 0.10 fm (163x4)
• Consider a dilute gas of calorons/anticalorons
• Concentration n ~ 1 fm-4 [ArXiv:hep-ph/0607315]
• Charge fluctuations Averaging over 4D orientations
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Some physical estimates
• 4D volume:
Fireball volume Collision duration
Au-Au, b = 4 fm, [ArXiv:0907.1396,
Skokov, Illarionov, Toneev]
⟨ ⟨𝑸𝟐 ⟩ ⟩ 𝟎 .𝟎𝟏Compare with in [ArXiv:0711.0950, Kharzeev, McLerran, Warringa]
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Conclusions
Instanton• Charge separation not allowed by
symmetries• Magnetic-field-induced fluctuations of
electric dipole moment• No additional zero modes in the
magnetic fieldCaloron
• Charge generation if B || caloron axis• Charge and current distributions are
strongly localized• Reasonable estimates for charge
fluctuations