M. N. Chernodub

numerical part is based on collaboration withV. V. Braguta, P. V. Buividovich, A. Yu. Kotov and

M. I. Polikarpov

On electromagnetically superconducting vacuum due to strong magnetic field

Based on: arXiv:1008.1055, arXiv:1101.0117, arXiv:1111.4401, arXiv:1104.3767, arXiv:1104.4404 + ongoing work

+ in preparation + ....

Superconductivity

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

Type I Type II

The claim:

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

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

General features

Some features of the superconducting state of vacuum:

1. spontaneously emerges above the critical magnetic field

Bc ≃ 1016 Tesla = 1020 Gauss 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

2 2 2or

Too strong critical magnetic field?

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)

eBc≃ mρ ≃ 31 mπ ≃ 0.6 GeV2 2 2

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.

LHCLHC

RHIC

eBc

eBc

ultraperipheral

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 field1) 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 fieldIngredients 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 fieldWell-known “magnetic catalysis”:

B0

B0 Bc

attractive channel: spin-0 flavor-diagonal states

attractive channel: spin-1 flavor-offdiagonal states (quantum numbers of ρ mesons)

This talk:

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

enhances chiral symmetry breaking

electrically chargedcondensates: lead to electromagneticsuperconductivity

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 alongthe magnetic field axis nonnegative integer number

projection of spin onthe magnetic field axis

(the external magnetic field is directed along the z-axis)

- Masses of ρ mesons and pions in the external magnetic field

- Masses of ρ mesons and pions:

becomes heavier

becomes lighter

Condensation of ρ mesons (naïve)

masses in the external magnetic field

The ρ± mesons become massless and condense at the critical value of the external magnetic field

Kinematical impossibility of dominant decay modes

stops at certain valueof the magnetic field

- The decay

- A similar statement is true for

The pion becomes heavier while the rho meson becomes lighter

Electrodynamics of ρ mesons = NSSM**NSSM - “naive simplest solvable model”

- Lagrangian (based on vector dominance models):

[D. Djukanovic, M. R. Schindler, J. Gegelia, S. Scherer, PRL (2005)]

- Tensor quantities - Covariant derivative

- Kawarabayashi-Suzuki- Riadzuddin-Fayyazuddin relation

- Gauge invariance

- Tensor quantities

Nonminimal couplingleads to g=2

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

Depend on transverse coordinates only

In terms of quarks, the state implies

(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

- The condensate “locks” rotations around field axis and gauge transformations:

In terms of quarks, the state implies

Abelian gauge symmetryRotations around B-axis

(the same structure of the condensates in the Nambu-Jona-Lasinio model)

(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, solutionsSuperconducting condensate Superfluid condensate (charged rho mesons) (neutral rho mesons)

New objects, topological vortices, made of the rho-condensates

B = 1.01 Bc

similar results in holographic approaches by M. Ammon, Y. Bu, J. Erdmenger, P. Kerner, J. Shock, M. Strydom (2012)

The phases of the rho-meson fields wind around vortex

centers, at which the condensates vanish.

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 NJLM.Ch. (2011)

( )

Numerical simulations in the magnetic field backgroundV.Braguta, P. Buividovich, A. Kotov, M. Polikarpov, M.Ch., arXiv:1104.3767

cond

ensa

te

magnetic field[qualitatively realistic vacuum, quantitative results may receive corrections (20%-50% typically)]

Theory:

Quenched SU(2) +overlap fermions.Spacings: a ~ 0.1 fmLattices: L4 ~ (2 fm)4

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?

- 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).

or:

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”

The color field strength tensor

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:

Ground state: hexagonal lattice arrangement of the vortices along the magnetic field B (naïve)

B

Expected reality: quantum fluctuations move and disturb vortices. They are no more straight parallel and static.

However: other lattice arrangements (square, rhombic, etc) are very close (0.1%) in energy the hexagonal order may be destroyed by fluctuations.

[J. Van Doorsselaere, H. Verschelde, M. Ch, arXiv:1111.4401]

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 valuesof the magnetic field strength)

Real superconductors possess a vortex phase:

Nishizaki, Kobayashi, Supercond. Sci. Technol. 13 (2000) 1

More on correlators:

Superconducting phase, B>Bc, vortex liquid phase (quenched QCD ...)

Insulating phase, B<Bc, “gas” (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

“no condensate”vacuum state

instability due tomagnetic field

rolling towards new vacuum state

rolling back to “no condensate”

vacuum state

Emission of ρ mesons

ρmeson vacuum state between the ions in ultraperipheral collisions

B = 0 B = 0B > Bc B > Bc

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

Lorentz invariants:A scalar function of Lorentz invariants.In this particular model:

(slightly different form of κ function in NJL)

If B is along x3 axis, then we come back to

and