Download On magnetic-field-induced electromagnetic superconductivity of

On electromagnetically superconducting
vacuum due to strong magnetic field
M. N. Chernodub
numerical part is based on collaboration with
V. V. Braguta, P. V. Buividovich, 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
+ in preparation + ....
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≃
2
2
2
mρ ≃ 31 mπ ≃ 0.6 GeV
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.
Accounts in the literature
Similar to:
1. Nielsen-Olesen instability of the gluon fields in the Savvidy
vacuum in Yang-Mills theory since the gluons have an
anomalously large chromo-magnetic moment), in 1980.
2. Ambjorn-Olesen W-condensation in the Electroweak theory
in strong magnetic fields (W-bosons also have a large
magnetic moment), in 1988.
3. A possibility of rho-meson condensation in QCD was
noticed by S. Schramm, B. Müller and A. Schramm in 1992.
(many thanks to Berndt for the reference!)
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]
[Bu, Erdmenger, Shock, Strydom(Munich, Germany), arXiv:1210.6669]
5. Numerical simulation of vacuum
ITEP Lattice Group, Moscow, arXiv:1104.3767 + (this talk)
Conventional BCS superconductivity
1) The Cooper pair is
the relevant degree
of freedom!
2) The electrons are
bounded into the
Cooper pairs by
the (attractive)
phonon exchange.
Three basic ingredients:
Real vacuum, no magnetic field
1) Boiling soup of everything.
Virtual particles and antiparticles
(electrons, positrons, photons,
gluons, quarks, antiquarks …)
are created and annihilated
every moment.
2) Net electric charge is zero.
An insulator, obviously.
3) We are interested in “strongly
interacting” sector of the theory:
a) quarks and antiquarks,
i) u quark has electric charge qu=+2 e/3
ii) d quark has electric charge qd=- e/3
b) gluons (an analogue of photons, no electric charge) “glue”
quarks into bounds states, “hadrons” (neutrons, protons, etc).
The vacuum in strong magnetic field
Ingredients needed for possible superconductivity:
A. Presence of electric charges?
Yes, we have them: there are virtual particles
which may potentially become “real” (= pop up from the vacuum)
and make the vacuum (super)conducting.
B. Reduction to 1+1 dimensions?
Yes, we have this phenomenon: in a very strong magnetic field
the dynamics of electrically charged particles (quarks, in our case)
becomes effectively one-dimensional, because the particles tend
to move along the magnetic field only.
C. Attractive interaction between the like-charged particles?
Yes, we have it: the gluons provide attractive interaction between
the quarks and antiquarks (qu=+2 e/3 and qd=+e/3)
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) - effective models
V. P. Gusynin, V. A. Miransky and I. A. Shovkovy ('94, '95, '96,...) - real QCDxQED
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: lead
to electromagnetic
superconductivity
0
B
Bc
Key players: ρ mesons and vacuum
- ρ mesons (= condensates with their quantum numbers):
• 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 (naïve)
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
Electrodynamics of ρ mesons = NSSM*
*NSSM - “naive simplest solvable model”
- 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)]
Homogeneous approximation (“NSSM”)
- Energy density:
- Disregard kinetic terms (for a moment) and apply Bext:
mass matrix
- Eigenvalues and eigenvectors of the mass matrix:
±
At the critical value of the magnetic field: imaginary mass (=condensation)!
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 = ½ (?)
Too naïve “mean field”: next talk by Y. Hidaka
Symmetries
In terms of quarks, the state
implies
(the same structure of the condensates
in the Nambu-Jona-Lasinio model)
Abelian gauge symmetry
Rotations around B-axis
- The condensate “locks” rotations around field axis and gauge transformations:
(similar to “color-flavor locking” in color superconductors at high quark density)
No light goldstone boson: it is “eaten” by the electromagnetic gauge field!
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)
Topological structure of the ρ mesons condensates
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
M.Ch., (2010)
similar results in NJL
M.Ch. (2011)
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: 1. results by Y. Hidaka, A. Yamamoto, arXiv:1209.0007 – WV theorem;
2. (counter)arguments by M. Ch., arXiv:1209.3587 – U(1)em was forgotten in “1”
3. next talk by Y. Hidaka (thanks for discussions!)
4. quenched QCD (overlap fermions vs. Wilson fermions)
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)
“ … strong inhomogeneities of the ground
state” - What does this mean, really?
or:
- Exactly the same situation happens in the mixed state of a type-II
superconductor: on average, the superconducting condensate is
zero but … the theory is still superconducting in this state.
- Why? Because the condensate is inhomogeneous, the phase of
the condensate is a sign-changing function of coordinates. On
average the condensate is zero, but … look, the ground state still
superconducts!
- Sounds too unrealistic? Hesitating?
(Answer: Abrikosov, Nobel prize 2003).
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 (quenched QCD, sorry ...):
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 (naïve)
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
More on correlators:
Insulating phase, B<Bc, “gas” (quenched QCD ...)
Superconducting phase, B>Bc, vortex liquid phase (quenched QCD ...)
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
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