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Vortex liquid in electromagnetically
superconducting vacuum due to strong
magnetic field: numerical lattice results
V. V. Braguta, P. V. Buividovich, M. N. Chernodub,
A. Yu. Kotov and M. I. Polikarpov
Based on: arXiv:1008.1055, arXiv:1101.0117,
arXiv:1111.4401, arXiv:1104.3767,
arXiv:1104.4404 + ongoing work
Superconductivity
Type I
Type II
I. Any superconductor has zero electrical DC resistance
II. Any superconductor is an enemy of the magnetic field:
1) weak magnetic fields are expelled by
all superconductors (the Meissner effect)
2) strong enough magnetic field always kills superconductivity
The claim:
In a background of strong enough magnetic field
the vacuum becomes a superconductor.
The superconductivity emerges in empty space.
Literally, “nothing becomes a superconductor”.
M.Ch., arXiv:1008.1055, arXiv:1101.0117
The claim seemingly contradicts textbooks which state that:
1. Superconductor is a material (= a form of matter, not an empty space)
2. Weak magnetic fields are suppressed by superconductivity
3. Strong magnetic fields destroy superconductivity
General features
Some features of the superconducting state of vacuum:
1. spontaneously emerges above the critical magnetic field
or
Bc ≃ 1016 Tesla = 1020 Gauss
2
2
2
eBc≃ mρ ≃ 31 mπ ≃ 0.6 GeV
2. usual Meissner effect does not exist
3. perfect conductor (= zero DC resistance) in one spatial
dimension (along the axis of the magnetic field)
4. insulator in other (perpendicular) directions
Too strong critical magnetic field?
eBc≃ mρ ≃ 31 mπ ≃ 0.6 GeV
2
2
2
Over-critical magnetic fields (of the strength B ~ 2...3 Bc)
may be generated in ultraperipheral heavy-ion collisions
(duration is short, however – detailed calculations are required)
LHC
eBc
LHC
eBc
RHIC
ultraperipheral
W. T. Deng and X. G. Huang,
Phys.Rev. C85 (2012) 044907
A. Bzdak and V. Skokov,
Phys.Lett. B710 (2012) 171
+ Vladimir Skokov,
private communication.
Approaches to the problem:
0. General arguments;
1. Effective bosonic model for electrodynamics of ρ mesons
based on vector meson dominance
[M.Ch., arXiv:1008.1055]
2. Effective fermionic model (the Nambu-Jona-Lasinio model)
[M.Ch., arXiv:1101.0117]
3. Nonperturbative effective models based on gauge/gravity
duality (AdS/CFT)
[Erdmenger, Kerner, Strydom (Munich, Germany), arXiv:1106.4551]
[Callebaut, Dudal, Verschelde (Gent U., Belgium), arXiv:1105.2217]
5. Numerical simulation of vacuum
ITEP Lattice Group, Moscow, arXiv:1104.3767 + (this talk)
Pairing of quarks in strong magnetic field
Well-known “magnetic catalysis”:
S.P. Klevansky and R. H. Lemmer ('89); H. Suganuma and T. Tatsumi ('91);
V. P. Gusynin, V. A. Miransky and I. A. Shovkovy ('94, '95, '96,...)
attractive channel: spin-0 flavor-diagonal states
enhances chiral
symmetry breaking
0
B
This talk:

attractive channel: spin-1 flavor-offdiagonal states (quantum numbers of ρ mesons)
electrically charged
condensates: leads
to electromagnetic
superconductivity
0
B
Bc
Key players: ρ mesons and vacuum
- ρ mesons:
• electrically charged (q= ±e) and neutral (q=0) particles
• spin: s=1, vector particles
• quark contents: ρ+ =ud, ρ– =du, ρ0 =(uu-dd)/21/2
• mass: mρ=775.5 MeV
• lifetime: τρ=1.35 fm/c (very short: size of the ρ meson is 0.5 fm)
- vacuum: QED+QCD, zero tempertature and density
Naïve qualitative picture of quark pairing in the
electrically charged vector channel: ρ mesons
- Energy of a relativistic particle in the external magnetic field Bext:
momentum along
the magnetic field axis
nonnegative integer number
projection of spin on
the magnetic field axis
(the external magnetic field is directed along the z-axis)
- Masses of ρ mesons and pions in the external magnetic field
becomes heavier
becomes lighter
- Masses of ρ mesons and pions:
Condensation of ρ mesons
The ρ± mesons become massless and condense
at the critical value of the external magnetic field
masses in the external magnetic field
Kinematical impossibility
of dominant decay modes
The pion becomes heavier while
the rho meson becomes lighter
- The decay
stops at certain value
of the magnetic field
- A similar statement is true for
Structure of the condensates
In terms of quarks, the state
implies
Depend on transverse coordinates only
(the same structure of the condensates
in the Nambu-Jona-Lasinio model)
- The absolute value
of the condensate:
Second order (quantum) phase
transition, critical exponent = 1/2
Condensates of ρ mesons, solutions
Superconducting condensate
(charged rho mesons)
Superfluid condensate
B = 1.01 Bc
(neutral rho mesons)
New objects, topological vortices, made of the rho-condensates
The phases of the rho-meson
fields wind around vortex
centers, at which the
condensates vanish.
similar results in holographic
approaches by M. Ammon,
Y. Bu, J. Erdmenger, P. Kerner,
J. Shock, M. Strydom (2012)
Anisotropic superconductivity
(via an analogue of the London equations)
- Apply a weak electric field E to an ordinary superconductor
- Then one gets accelerating electric current along the electric field:
[London equation]
- In the QCDxQED vacuum, we get
an accelerating electric current
along the magnetic field B:
(
)
Written for an electric current
averaged over one elementary
(unit) rho-vortex cell
(similar results in NJL)
Numerical simulations in the magnetic field background
V.Braguta, P. Buividovich, A. Kotov, M. Polikarpov, M.Ch., arXiv:1104.3767
condensate
Quenched SU(2) +
overlap fermions.
Spacings: a ~ 0.1 fm
4
4
Lattices: L ~ (2 fm)
Theory:
magnetic field
[qualitatively realistic vacuum, quantitative results may receive corrections (20%-50% typically)]
Ongoing discussion: negative results by Y. Hidaka, A. Yamamoto, arXiv:1209.0007;
counterarguments by M. Ch., arXiv:1209.3587.
The ρ meson condensate is difficult to
observe in lattice simulations due to strong
inhomogeneities of the ground state
1) The phase of the ρ meson condensate
is a lively function of the transverse
coordinates (x,y): its average is
zero (analytically). But the
condensate itself is nonzero.
2) The phase of the condensate
jumps by 2π around the ρ vortices.
3) The ρ vortices move even within
the same configuration (= neither
straight nor static)
Can we visualise the ρ meson condensate?
Inspiration: G.S. Bali, K. Schilling, C. Schlichter,
hep-lat/9409005 (PRD). Observation of chromoelectric flux
(confining) tubes between quark and antiquark on the lattice
Observable:
The color field strength tensor
in the presence of the Wilson
loop (quark source) “W”
Observables:
The ρ meson field
;
in the background of magnetic field B and SU(2) gauge
configuration ASU(2)
transforms as a charged field under the electromagnetic U(1):
Normalized energy:
;
Normalized electric current:
Vortex density (2π singularities in the phase of the ρ meson field):
Expectations:
[J. Van Doorsselaere, H. Verschelde,
B
M. Ch, arXiv:1111.4401]
Ground state: hexagonal lattice
arrangement of the vortices along
the magnetic field B
However: other lattice arrangements
(square, rhombic, etc) are very close
(0.1%) in energy the hexagonal order
may be destroyed by fluctuations.
Expected reality: quantum fluctuations
move and disturb vortices. They
are no more straight parallel and static.
Normalized energy of the ρ meson
condensate in the transverse plane.
Check x-y slice
at fixed time t
and distance z
Instead of a regular lattice structure we see an irregular vortex
pattern (vortex liquid?) The vortices move as we move the slice.
Normalized energy of the ρ meson
condensate along the magnetic field.
Check x-z slice
at fixed time t
and coordinate y
Vortices are not straight and static: they are curvy moving
one-dimensional (in 3d) structures.
Electric currents around vortices:
Theory:
It is indeed a vortex liquid!
Vortex density-density correlators:
(similar pictures for other values
of the magnetic field strength)
Real superconductors
possess a vortex phase:
Nishizaki, Kobayashi,
Supercond. Sci. Technol. 13 (2000) 1
Conclusions
●
In a sufficiently strong magnetic field condensates with
ρ meson quantum numbers are formed spontaneously.
The vacuum (= no matter present, = empty space, = nothing)
● becomes electromagnetically superconducting.
●
●
●
●
●
●
The superconductivity is anisotropic: the vacuum behaves as
a perfect conductor only along the axis of the magnetic field.
New type of topological defects,''ρ vortices'', emerge.
The ground state of ρ vortices is the Abrikosov-type lattice
in transverse (w.r.t. the axis of magnetic field) directions.
The liquid of the vortices is seen in lattice simulations.
(Signatures of) the superconducting phase can be checked at LHC?
Backup
Topological structure of the ρ mesons condensates
Simplest approach: Electrodynamics of ρ mesons
- Lagrangian (based on vector dominance models):
Nonminimal
coupling
leads to g=2
- Tensor quantities
- Covariant derivative
- Kawarabayashi-SuzukiRiadzuddin-Fayyazuddin relation
- Gauge invariance
[D. Djukanovic, M. R. Schindler, J. Gegelia, S. Scherer, PRL (2005)]
Too quick for the condensate to be
developed, but signatures may be seen
due to instability of the vacuum state
ρmeson vacuum state between the ions in ultraperipheral collisions
B=0
“no condensate”
vacuum state
B > Bc
instability due to
magnetic field
B > Bc
rolling towards
new vacuum state
Emission of ρ mesons
B=0
rolling back to
“no condensate”
vacuum state
Anisotropic superconductivity
(Lorentz-covariant form of the London equations)
We are working in the vacuum, thus the transport equations
may be rewritten in a Lorentz-covariant form:
Electric current
averaged over
one elementary
rho-vortex cell
A scalar function of Lorentz invariants.
In this particular model:
Lorentz invariants:
(slightly different form of κ function in NJL)
If B is along x3 axis, then we come back to
and