PoS(FacesQCD)021 K 12 http://pos.sissa.it/ mesons should also induce an ± ). The physical mechanism of the ρ 2 1GeV ∼ eB mesons. The vacuum superconductivity should 0 ρ Tesla ( 16 meson. Such quark-antiquark pairs condense to form an anisotropic ± ρ ∗† [email protected] This work was supported by GrantOn No. leave ANR-10-JCJC-0408 from HYPERMAG. Institute for Theoretical and Experimental Physics, Moscow, Russia. inhomogeneous superfluidity of the neutral exotic vacuum superconductivity is as follows:quarks and in antiquarks strong is magnetic effectively field one-dimensionaltend the because these to dynamics electrically of move charged virtual along particles theattraction lines between of a the quark magnetic and field.a an colorless antiquark In spin-triplet of bound one different state spatial flavors (a inevitably dimensionof vector leads an analogue a electrically of to gluon-mediated charged the formation Cooper of pair)inhomogeneous with quantum superconducting numbers state similar to theconductor. Abrikosov The vortex onset lattice of in the a superconductivity type-II of super- the charged survive at very high temperatures of typical Quantum Chromodynamics (QCD) scale of 10 A superconductor is a materialtivity that and conducts magnetism electric are known current to withternal be no magnetic antagonistic resistance. field phenomena: (the superconductors Superconduc- Meissner expel effect)destroys weak while superconductivity. ex- a In sufficiently a strong seemingly magneticmagnetic contradictory field, field statement, in can we general, show turn that anquired a empty for very space this strong effect into should a be superconductor. about 10 The external magnetic field re- (100 MeV). We propose the phase diagram of QCD in the plane "magnetic field - temperature". ∗ † Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. c

The many faces of QCD November 2-5, 2010 Gent Belgium CNRS, Laboratoire de Mathématiques et PhysiqueFédération Théorique, Denis Université Poisson François-Rabelais, - CNRS, Parc deDepartment Grandmont, of Université Physics de and Tours, Astronomy, 37200 UniversityBelgium France of Gent, Krijgslaan 281, S9,E-mail: B-9000 Gent, M. N. Chernodub Can nothing be a superconductor and a superfluid? PoS(FacesQCD)021 - - Cooper pair Cooper + + + + + + ]. The effect emerges 2 , + + + + 1 + + - + + + + + + + + + + + + + + + + + + - + Very schematically: the distortion of the ion lat- + + + + + superconducting state [ 2 1 + + + + + + Figure 1: tice around one electron attractsformation another of electron, the leading Cooper to pair. ” in order to distinguish the proposed superconducting state of the vacuum . The formation of the Cooper pairs is promoted by the fact that the 1 electromagnetic Below we describe the physical mechanism of the magnetic-field-induced superconductivity We stress the word “ There are three important ingredients of the standard mechanism of the Cooper pair formation: In a conventional superconductor Recently, we suggested that the vacuum in a sufficiently strong magnetic field may undergo a 1 B) the reduction of physics from (3+1) toC) (1+1) dimensions; the attractive interaction between the like-charged particles. A) the presence of carriers of electric charge (of electric current); of the vacuum and we demonstrateanism. that all these ingredients, A, B and C, are present in our mech- from the “color superconductivity” (which mayductivity” exist associated in with a confining sufficiently features dense of quark the matter) gluonic and fields from in the the “dual pure supercon- Yang-Mills vacuum. dynamics of the electrons neardimension the the Fermi bound surface states is are formed basically for one-dimensional, arbitrarily and weak in attraction one (the Cooper spatial theorem). the superconductivity appears as asult re- of condensation ofThe Cooper Cooper pair pairs. is a bound state ofelectrons two which is formed due toattractive small force between the electrons. The attraction is mediated by a phonon exchange: one electron deforms atice lat- of the positivelya charged metal, ions creating of a localpositive excess charge. of the Thisanother excess electron, attracts and thecess whole of pro- thebe attractive viewed as interaction an exchange can of collective excitations of the ion lattice (phonons) between the electrons, Figure spontaneous transition to an electromagnetically Can nothing be a superconductor and a superfluid? 1. The physical mechanism of electric superconductivity of vacuum as a result ofBelow an we interplay discuss a between standard strongattention mechanism fundamental to of the the forces suggested conventional and superconductivity superconductivity, of and electromagnetic empty then interactions. space. turn our 1.1 Conventional superconductivity PoS(FacesQCD)021

d d

u

d

d u gluons ]. Below we show that a u The vacuum of QCD: a boil- 3 virtual quarks virtual and antiquarks u ), and some of these particles may 2 Figure 2: ing soup of virtual quarks and gluons. 3 exchange. Ob- -like interactions be- photon phonon (1+1) dimensional reduction. ]. Despite of the net electric neutrality, the vacuum may (super)conduct since 2 → , 1 Firstly, let us consider the purely electromagnetic sec- Secondly, we notice that the strongly interacting sec- become real in certain cases. The simplestit relevant example is is energetically the Schwinger favorable effect: for itindependent the turns background out vacuum that of to a produce sufficiently electron-positronsimilar strong effect pairs external exists in in electric a a field background uniform [ ofcertain time- a electrically strong charged magnetic field condensates which of forcescharged condensate quark-antiquark the automatically vacuum pairs. implies to the develop The presence ofmagnitude. presence a of negatively As charged the condensate a of positively result, equal thevacuum is energy zero of [ the vacuum is lowered, while the net electric charge of the C. Attractive interaction between the like-chargedAll particles. possible types of virtualsector particles of "boil" the in vacuum the contains quantum virtualinteracting vacuum. electron-positron sector pairs consists The and purely of virtual electromagnetic virtual photons,particles and quark-antiquark and the pairs weaker strongly interactions). and virtualnamics The leaves gluons quarks a (we trace are ignore in electrically the heavier charged electromagnetic particles sector as so well. that theirtor dy- described by Quantumelectrons Electrodynamics repel each (QED). other due The to the tor of the vacuum doesforce. contain It an provides analogue an ofthe attractive the pairs interaction phonon. called between mesons. the It The quarks is quarks and a and anti-quarks antiquarks gluon, and of a binds the carrier them same in of electric the charge can strong also be bound a weak external electric field pushesdirections, the thus positively creating and a negatively net charged electric condensates current in along opposite theB. electric Dimensional field. reduction. It is the presence of thetron Fermi dynamics and surface facilitates that the leads formation to ofthe the the Cooper vacuum effective pairs dimensional all in reduction chemical a of conventional potentials superconductor. the area In elec- zero sufficiently and, strong obviously, Fermi external surfaceslow-energy magnetic do particles not field may exist. should move However, reduce onlythe along the required the motion (3+1) lines of of electrical the charges: magnetic field. the This effect guarantees viously, there are no attractive tween two electrons in the vacuumCooper and, pairs consequently, cannot the form (the samepositrons). statement is Electron-positron valid bound for statesare (positronium) not interesting forcally us neutral. because This is such a states simplest reasonperconductivity are why cannot the electri- emerge vacuum su- in the pure QED vacuum. Can nothing be a superconductor and a superfluid? 1.2 Electromagnetic superconductivity of the vacuum A. Presence of electric charge carriers. In fact, there are no chargeboiling carriers soup in of virtual the particles vacuum (“quantum in fluctuations”, normal Figure conditions. However, the vacuum is a PoS(FacesQCD)021 ¯ d meson. ± mesons via ρ 3 and the ± / ρ e 2 mesons = + !

u q mesons is a real state. moments moments of the of spins quarks and antiquarks direction of direction the magnetic meson, q=-e meson, q=+e ! ρ ! positively charged positively negatively charged charged negatively u d ). e ). This bound state should be: 3 3 are bound by the gluon-mediated interaction, / u d e = + 4 ρ q gluons = + quark with the electric charge d u q ≡ − ¯ ); d u d 2 q u d

meson with the electric charge

d l e i f c i t e n g a ρ m Very schematically: formation of quark-antiquark pairs with quantum numbers of state would be infinite due to nonperturbative QCD effects); would be electrically neutral If the state is composed of the quark and antiquark of the same flavor then the electric charge of this state is zero. The condensate of the quark-antiquark pairs is an energetically favorable state. Similarly to 2 1) a colorless state due to the quark confinement property (otherwise the2) energy a of the state, bound composed of the quark and the antiquark3) of a different vector (spin-triplet) flavors state (otherwise (in the order to state occupy a lowest energy state). the Schwinger effect, the charged particlesvacuum. (the quark-antiquark In pairs contrast in to our the case) Schwingervirtual emerge effect, nature from these of the pairs the form quantum a vacuum condensate. fluctuations, And, the in condensate contrast of to the the the gluon exchange in abounds states background (mesons) of may a move only very along strong the magnetic axis field. of the The magnetic field. quarks and, consequently, their Figure 3: Can nothing be a superconductor and a superfluid? by the gluon exchange (forantiquark example, with the the electric charge The physically interesting bound state has quantum numbers of an electrically charged forming the A+B+C. Stable charged bound states dueThe to physical gluon interaction mechanism and of dimensionalnetic reduction. the field exotic the vacuum dynamics superconductivity ofthese is electrically virtual as charged quarks particles follows: and tend antiquarks totial in move is dimension strong along a effectively the mag- one-dimensional gluon-mediated lines because attraction offormation of the between a magnetic a quark-antiquark field. quark bound In state and one (Figure an spa- antiquark inevitably leads to PoS(FacesQCD)021 ) ρ ρ ]. ' ]. r 7 2 1.3 , 1) is , the , (1.1) (1.2) (1.3) s 6 . The 1 s 5fm = -meson mesons . s 0 ρ = ρ ,..., z s ' s ρ − r -meson mass). = ρ z )] which supports 0 and s = 1.1 ]. Both the z mesons (with 5 n ρ -meson condensation. 0, ] suggest that a sufficiently ρ 1 = z 5MeV is the , quarks at the critical field ( . p 2 d m 775 + ) and then we discuss the on the field’s axis . | and = 2 e s , u ρ eB meson cannot be treated as a pointlike | . m ) 1 ρ bosons in the electroweak model have the ), the ground state energy of the charged eB Tesla + meson is a composite particle which, in the − z 16 W 1.3 meson should decrease with the increase of the s ], and to the magnetic-field-induced Ambjørn– ρ 2 ρ mesons is similar to the Nielsen-Olesen instabil- 2 4 10 ρ mesons is supported by various arguments [ m 5 ρ − ≈ ρ n e 2 ) = / meson stable against known decay modes. Below we 2 [explicitly implemented in Eq. ( B 2 ρ ( + ( ρ ± m = 2 z 2 ρ particle moving in a background of the external magnetic g p = bosons, the m 2 for the s c , and the electric charge B z = W ) = mesons becomes zero ( p z g p , the projection of the spin 45fm, respectively, which are comparable with the the radius ± ( . m z -bosons in the standard electroweak model [ ρ s 0 , 2 n ) that the ground state corresponds to of the particle are: W ε ' ). We show that both approaches give qualitatively the same results. ε 3 c 1.1 , d r 0 labels the energy levels, and other quantities characterize the properties > n meson has so short lifetime, so that it is often called a “resonance”, not even a . When the magnetic field reaches the critical strength, ] (Section B 8 ρ . The radii of the lowest Landau levels of the 1 32fm and . − fields the strongly interacting (QCD) sector of the vacuum spontaneously develops the 0 ) meson itself. Thus, generally speaking, the mesons using a bosonic effective model (Section c ' B ρ ρ . The energy levels c , > B u The suggested condensation of the charged In contrast to the gluons and In the absence of the external magnetic field the quark-antiquarks pairs are unstable. For Consider a free relativistic spin- It is clear from Eq. ( Thus, the ground state energy of the charged As the field increases above the critical value ( r B -meson condensates. Since the condensed mesons are electrically charged, their condensation 400 MeV condensation beyond the pointlike approximation using(NJL) a model more [ fundamental Nambu–Jona-Lasinio in QCD, the gluonsanomalously in large Yang-Mills gyromagnetic theory, ratio, and the the mentioned effects. The value are of the Olesen condensation of the absence of the external field, has a finite radius of the order of a typical QCD scale, particle at the required strong magneticof fields. the Nevertheless, we work with a pointlike description mesons becomes purely imaginary thus signalingAt a tachyonic instability of theρ QCD ground state. implies, almost automatically, an electromagnetic superconductivity of the condensed state [ ( particle. However, even very simplified kinematical arguments of Ref. [ field ity of the gluonic vacuum in Yang-Mills theory [ example, the Can nothing be a superconductor and a superfluid? 1.3 Energetic arguments strong magnetic field makes thegive charged simple energetic arguments in favor of the magnetic-field-induced “minimal mass”, corresponding to the ground state energy of the charged where the integer of the pointlike particle: mass momentum along the field’s axis, magnetic field the ground state energy of the PoS(FacesQCD)021 , ) ) ], , ± 0 2 ρ ]: ) ( x ρ 9 0 10 m , ( ρ µν , 1 (2.1) ρ 2 and 6 x . ( 4 √ µν / . coupling, v F

) ) e ) R B s 2

, ( , t µ g e r 0 a ρ 2 , induces both mesons to the p ,

i ρππ l c 2 in the 0 a 0) do emerge [ e . / + ρ B ], Figure − R ) ) 6= gauge invariance: ) 1 µ 2 = ( x ) 1 > 3 ρ ) ( ( 0 , µ ] can be found in a i ), perpendicular to the B ( ~ 0 1 | 1 ρ B 2) of the charged ω ρ ( ρ + µ ) eB 88 is the 0 U 1 = | ∂ . ( µ = ( ρ / 5 g ρ µ π ) + 2 ρ ≈ 2 ρ vortices” [ = ( x 2 ( -meson condensates have a spatially- m p v v ρ

µ ρ m ρππ + I =

A , t g r B a µν p Typical structure of the superconduct- L )

→ ≡ y 0 r ( a ) s ≡ n i x g ρ g l , the fields ( ) a ] 0 µ m ν ( I µν

A

A ρ | v | , e u l a v e t u l o s b , A µ 1 4 [ plane ( Figure 4: ing condensate background magnetic field ∂ − , µ = ) 6 ]. On the contrary, our ρ x † µ , ( µν 11 of ) ρ µ F are 3 ± . ρ 2 ρ (2.2) 2 ρ ρ com- ) ρ ) x † 3 m 2 ( m 3 . How- possess ρ ρ ω 3 ρ i i + 2 e ] mesons is accompanied by superfluidity of the neutral + ρ ν ∓ 0 ± 1 mesons based on the vector meson dominance property [ ρ ρ → and , ρ ρ † 0 µ ) ρ ( [ and . The eigenval- 0 ρ is the covariant derivative, x ) becomes unstable against the condensation if the magnetic field 2 ρ ( ( 1 D . The model possesses the electromagnetic 1 µ 2 ρ ] † µ ρ M √ ). The emerging condensate has a typical structure of the Abrikosov ) ν 1.2 ρ ] m ρ ν , ieA , = 0), and they appear in completely empty space. 1.3 + ρ µ † , [ j − 6= ± ! , implying the anomalous gyromagnetic ratio ( µ ρ [ ) ρ 2 ρ µ s , 2 ρ 0 : 1 ( i j D µ A . ig ρ ( ieB ρ , m s are, respectively, as follows: 2 1 . M − e † i ) ig 2 ρ ) − ] eB 0 ρ ieB m 1 ( M ν m 1 + ( µν − ± ρ = , 2 µ j U ∑ 2 ρ F , µ ∂ i [ are, respectively, the fields of the charged and neutral vector mesons with the mass m ∂ µν = ]. Thus, our calculations suggest that the external magnetic field, ) ], composed of the new topological objects in QCD, the “ = F 3 = 1 = ( µ µ state with the mass ( ) = 12 4 1 ) and the corresponding eigenvectors 2 ρ ± D µ 0 − ( M µν µ ± ρ − Notice that in a dense isospin–asymmetric matter the longitudinal condensates (with The condensation of the electrically charged particles leads to the electric superconductivity, A simple realization of the magnetic-field-induced superconductivity [ The quadratic part of the energy density, ρ ≡ ρ ( µ 3 is the photon field with the field strength ) ) = 2 0 ( ( µ 0 µ ε A The last term in the above Lagrangian describes a nonminimal coupling of the superconductivity and superfluidity of the empty space. and electromagnetic field mesons. The last term plays a crucial role in the mechanism of the vacuum superconductivity. leading to an electromagnetically superconducting state [ transverse structure (with The exceeds the critical value ( lattice [ which, in the case of the charged where ρ mesons [ Can nothing be a superconductor and a superfluid? 2. Electromagnetic superconductivity of vacuum in vector meson dominance model quantum electrodynamics for the L shows that the massponents terms are for diagonal and their prefactors unaltered by the external magnetic field ever, the Lorentz components the non–diagonal mass matrix ues the mass matrix PoS(FacesQCD)021 is ]. ) 2 ]: 3 (3.2) (3.3) (3.1) (3.4) is the / e , 13 ) ¯ z , − 1. Apart B n ]: 8 , ( π 2 and , q 2 3 [ ≈ the induced and electric c / i + m 3) [ c q d 0 2 2 e B n J
µ B 2 C π = γ < , ψ ¯ − u ≡ c i < are the Pauli ma- (+ 0 e B τ N ) n B Q 5 = 3 c = γ τ µ ∞ µ y diag , 0 if ρ ∞ − 2 ¯ ∑ + τ = J 1 (which corresponds to -meson electrodynamics = ψγ n , ) =  . ρ 0 1 ) 2 d = = c 2 ) we apply a weak external ) plane, and ρ τ ¯ z
+ q x 2 B n , + 2 x c 2 u 3.2 , |
J > = ( τψ z q ~ 1 | 5 ( ( ψ one gets x τ i B ~ γ 2 π c i τ 51 is a constant, and − . ¯ B -meson field µ . ψ e 0 diag ρ B ¯ > ψγ ) and in the = , ≈ , + B 2 z ˆ 2 h φ ) = / Q
E 3.4 z 1 0 C (
=  3 i c K ¯ c is the magnetic length and the complex field ψψ B B B . One can easily find that at ∑ |  -meson condensate: should be fine-tuned by the minimization of the − B V S − eB 2 ρ 7 n B | 2 G ) can be rewritten in a Lorentz-covariant form [ , 1 c G  / 3  −  is zero, naturally indicating that the vacuum stays in π vortices. These vortices are composed of the quark- + e E 3.4 2 ) 2 i 3 f B ρ ) q ix V ψ ( B p π C ψ q
G L + 2 2 0 µ = m 1 ( ˆ γ x M φ f B =  ¯ C L ψ − are, respectively, the electric charge density 0 h ) K θ ) i f z ) / z e , A q is the doublet of the light quarks, t B ˆ t ( d ( 2) Nambu-Jona-Lasinio model with three colors ( , averaged over the perpendicular ( Q J , z to the vacuum, T 0 ∂ ) , u ) ) = z y ). However, typical lattice conformations in type-II superconductors ρ ) + = = J d z J f B ) is a good approximation to the true energy minimum. is the bare quark mass matrix, f , / , ∂ ( E ∂ = J 4 y i ) ) has a typical structure of the Abrikosov lattice in a type-II superconduc- 3.2 u 0 ∑ , , N J i 0 d y x ρ , + ¯ d are, respectively, the scalar and vector couplings of four-quark interactions. x E ψ m 3.2 2 . An empty space becomes an anisotropic inhomogeneous superconductor. = ( J ) , , J ) = γ c V z x 0 u ( ¯ x , u B ψ E G ( t ) is a London equation for an anisotropic superconductivity. Thus, we have just z h m ) = = ( ( i ≡ µ ( ¯ ∂ > ψ ~ J J determines the magnitude of the − ~ Q 3.4 = ( , and B ) J ∂ ψ = 2 E diag S ( x i , G = d ) are identical. The transport law ( 1 are complex parameters, and L 1 2 0 is an arbitrary coordinate-independent phase, x γ n ( ˆ ¯ 0 c u 0 Q M ]. The ground state of the superconducting vacuum has an inhomogeneous periodic structure h θ ρ The condensate ( In order to probe the conducting properties of the condensate ( Equation ( The signatures of the vacuum superconductivity in strong magnetic field can also be found in The strong magnetic field induces the condensate of the 12 = 0 where the insulating phase at weak magnetic fields. However, at found that the strong magnetic fieldthe induces the vacuum if new electromagnetically superconducting phase of quark mass which is a smooth function of the magnetic field antiquark vector condensates. The parameters electric current made of new type of topological defects, cost a few percents of total condensation energy, so that the choice tor [ Here energy with the ansatz ( the square lattice, Figure electric field (Section ρ an extended two-flavor ( Can nothing be a superconductor and a superfluid? 3. Electromagnetic superconductivity of vacuum in Nambu–Jona-Lasinio model where the charge matrix, where current density trices, and from prefactors, the transport laws in the NJL model ( PoS(FacesQCD)021 ]. 17 ], supported by ]. The behavior

1 r 14 ]

2

o

t

n

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i c

t

i

s

u

n

a d

r [GeV

t

n

n

g

n ]. There is also evidence that

o

n

o i o c

i eB

i

t

t

i i t

i

c

c s 15

1.5

s

u

n p

n

d a

a r

n t o r

t

o n

r

r Anisotropic

t

c o

o

t

i

r t n

t

i

c e e

s

o

u p

m n

d

u

s a e

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o At low temperatures and high magnetic

i

superconductor

f

l

c According to analytical estimates and nu-

-

n n a

r

r

i o

o

t c

h A

a e

l C 1.0

u D

s

n I 8 c At strong values of the magnetic field and, simulta- eB 0.5 , should not encounter principal obstacles in numerical lattice . The phase diagram has the following specific features: 5 ]. 5 16 Quark-Gluon Quark-Gluon plasma (conductor) Cold Cold hadronic vacuum (insulator) ] and found numerically in recent simulations of quenched QCD [ 2 0 50

250 150

200 100 ] V e M [ T Qualitative phase diagram of finite-temperature QCD in strong magnetic field. Figure 5: neously, at high temperatures aphase new, asymmetrically the conducting vacuum phase behaves may asinsulator appear. in a the In conductor directions this perpendicular along tolattice the the simulations axis. axis of The of Ref. presence [ the of this magnetic phase field, was found and in as an Anisotropically conducting phase. of the chiral transition agrees wellthe with strong the enhancement magnetic of field, the known chiral asthese symmetry transitions the breaking may magnetic by split catalysis at [ finite magnetic field, although the splitting may be very small. arguments of Ref. [ Asymmetric superconductor and superfluid. merical simulations of latticetransitions QCD, are the increasing critical functions temperatures of the of strength deconfinement of and the magnetic chiral field [ fields the vacuum behavesThe as vacuum a behaves superconductor as andinsulator a in a superconductor the superfluid, perpendicular along directions. the as This axis we phase was of discussed suggested the in above. Ref. magnetic [ field, and as an Chiral and deconfinement phase transitions. Finally, we would like to mention that QCD in Euclidean space at finite magnetic field has no We think that the phase diagram of finite-temperature QCD in strong magnetic field should 2. 3. 1. simulations. sign problem of the fermionic determinantthe contrary predicted to phase the diagram, finite-density Figure QCD. Thus, verification of have the structure plotted in Figure Can nothing be a superconductor and a superfluid? 4. Phase diagram of finite-temperature QCD in strong magnetic field PoS(FacesQCD)021 ]. In ). 18 1.3 ]. The critical magnetic field was found to be mesons was also found in numerical simulations 17 9 ρ ]. The spontaneous generation of quark condensates ]. 20 19 ). In fact, the conventional Meissner effect is caused by 3.4 Tesla which is quite close to the theoretical expectation in QCD ( -meson condensation in QCD were found in nonperturbative holographical 16 ρ 10 · ] of the vacuum superconductivity has exactly the same qualitative features). ) 2 , 13 1 . 0 ]. Such fields may, presumably, have also arisen in the course of evolution of early ], and they may have imprints in the large-scale structure of the magnetic fields in the ± 21 22 56 . 1 Signatures of the The hadron-scale-strong magnetic fields may emerge in the heavy-ion collisions at the Rela- Experimentally, the reentrant superconductivity is observed in certain materials. A recent ex- Somewhat similar effect, which is known as the magnetic-field-induced, or “reentrant”, super- All known superconductors expel weak external magnetic field. This property is known as the = ( c of SU(2) lattice gauge theory (quenched QCD) [ 5.3 Nonperturbative studies of QCD: holography and lattice gauge theory approaches based on gauge/gravity duality [ with quantum numbers of electrically charged Universe [ present-day Universe. B tivistic Heavy-Ion Collider at Brookhavenat National CERN Laboratory [ and at the Large Hadron Collider most superconductors an external magneticPauli field pair suppress breaking superconductivity effects, via so diamagneticever, that and in in a a very strong strong magneticenter magnetic field a the field quantum superconductivity the limit is AbrikosovMeissner of lost. flux effect, the lattice How- a low of spin-triplet Landau a pairing,(our level proposal type-II and [ dominance, superconductor a may characterized superconducting by flow along the the absence magnetic of fieldample the of axis such material is a uraniumtriplet) superconductor superconductivity URhGe, at which may magnetic exhibit fields (presumably,state spin- up in the to region 2 between Tesla, 8 and and 13 comes Tesla back [ to a superconducting 5.4 Nature: Early Universe and heavy-ion collisions conductivity, is proposed to be realized in type-II superconductors in a quantum regime [ large superconducting currents which aremagnetic induced in field. the superconductor These by currentsaxis) the plane, circulate external generating in magnetic a the back-reacting magnetic perpendicularthe field (with which absence screens respect of the the to external magnetic transverse the field.the superconductivity magnetic longitudinally In the superconducting field external state magnetic of field the cannot vacuum. be screened by 5.2 A similar effect in condensed matter: Reentrant superconductivity Meissner effect. So why the very strongwithout magnetic being field suppressed is by able to the penetrate superconducting theis state QCD rather of superconductor the simple: vacuum? the The absence answer ofof to the this the true question Meissner vacuum effect superconductivity is a ( natural consequence of the anisotropy Can nothing be a superconductor and a superfluid? 5. Discussion 5.1 Provocative question: Where is Meissner effect? PoS(FacesQCD)021 2425 (1989); , 061 (2003) , 054027 63 82 , 0312 , 4525 (1990); Phys. 5 ibid. , 012001 (2005). 95 , 1442 (1957)]. , 033001 (2004) 32 70 , 516 (2009) [arXiv:0810.2316 , 5925 (2009); V. Voronyuk et al, 680 , 249 (1996) [hep-ph/9509320]; 004 (2011), these Proceedings. 24 , 094007 (2004) [hep-ph/0405235]; 462 046 (2011), arXiv:1102.3103 [hep-ph], 70 ibid. , 1343 (2005); D. Aoki, T. D. Matsuda, V. ´ c, M. Rasolt, L. Xing , 114031 (2010); A. J. Mizher, 309 , 045006 (2002) [hep-ph/0205348]. 81 66 , 163 (2001) [arXiv:astro-ph/0009061]. 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