Table Of ContentVortex liquid in magnetic-field-induced
superconducting vacuum of quenched lattice QCD∗
V. V. Braguta
IHEP,Protvino,Moscowregion,142284Russia
ITEP,B.Cheremushkinskayastr. 25,Moscow,117218Russia
P. V. Buividovich
InstituteofTheoreticalPhysics,UniversityofRegensburg,Universitätsstrasse31,D-93053
Regensburg,Germany
3
1 M. N. Chernodub†‡
0
2 CNRS,LaboratoiredeMathématiquesetPhysiqueThéorique,UniversitéFrançois-Rabelais
Tours,ParcdeGrandmont,37200Tours,France
n
a DepartmentofPhysicsandAstronomy,UniversityofGent,Krijgslaan281,S9,B-9000Gent,
J Belgium
8
2 A. Yu. Kotov and M. I. Polikarpov
ITEP,B.Cheremushkinskayastr. 25,Moscow,117218Russia
]
t MIPT,Institutskiiper. 9,Dolgoprudny,MoscowRegion,141700Russia
a
l
- In the background of the strong magnetic field the vacuum is suggested to possess an electro-
p
e magnetically superconducting phase characterised by the emergence of inhomogeneous quark-
h
antiquark vector condensates which carry quantum numbers of the charged ρ mesons. The ρ-
[
meson condensates are inhomogeneous due to the presence of the stringlike defects (the ρ vor-
1
tices) which are parallel to the magnetic field (the superconducting vacuum phase is similar to
v
0 themixedAbrikosovphaseofatype-IIsuperconductor). Inagreementwiththeseexpectations,
9
we have observed the presence of the ρ vortices in numerical simulations of the vacuum of the
5
6 quenchedtwo-colorlatticeQCDinstrongmagneticfieldbackground. Wehavefoundthatinthe
.
1 quenchedQCDtheρ vorticesformaliquid. Thetransitionbetweentheusual(insulator)phaseat
0
lowBandthesuperconductingvortexliquidphaseathighBturnsouttobeverysmooth,atleast
3
1 inthequenchedQCD.
:
v XthQuarkConfinementandtheHadronSpectrum
i
X 8-12October2012
r TUMCampusGarching,Munich,Germany
a
∗TheworkofP.V.B.wassupportedbytheS.KowalewskajaawardfromtheAlexandervonHumboldtFoundation
(Germany);theworkofM.N.C.waspartiallysupportedbygrantNo.ANR-10-JCJC-0408HYPERMAGofAgenceNa-
tionaledelaRecherche(France).TheworkoftheMoscowgroupwassupportedbyGrant"LeadingScientificSchools"
No. NSh-6260.2010.2, RFBR-11-02-01227-a, FederalSpecial-PurposeProgram"Cadres"oftheRussianMinistryof
ScienceandEducationandbyagrantfromtheFAIR-RussiaResearchCenter. Numericalcalculationswereperformed
attheITEPcomputersystems“Graphyn”and“Stakan”(authorsaremuchobligedtoA.V.Barylov,A.A.Golubev,V.
A.Kolosov,I.E.KorolkoandM.M.Sokolovforthevaluablehelp).
†Speaker.
‡OnleavefromITEP,Moscow,Russia.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
VortexliquidinsuperconductingvacuumofquenchedlatticeQCD M.N.Chernodub
1. Introduction
Recentlyitwassuggestedthatinthemagneticfieldbackgroundthevacuumbecomeselectro-
magneticallysuperconductingifthestrengthofthemagneticfieldexceedsthecriticalvalue[1,2]:
B =m2/e≈1016Tesla, (1.1)
c ρ
wherem =775.5MeVisthemassoftheρ meson. Themagneticfieldswhicharetwo-threetimes
ρ
strongerthanthecriticalvalue(1.1)areexpectedtobeexperimentallyreachableattheLHC[3].
If the strength B ≡ |(cid:126)B| of the uniform static magnetic field (cid:126)B = (0,0,B) is higher then the
critical value (1.1), then the usual vacuum ground state should experience a tachyonic instability
towardstheemergenceofanewgroundstatecharacterisedbythepresenceofthefollowingquark-
antiquarkcondensates:
(cid:104)u¯γ d(cid:105)=ρ(x ), (cid:104)u¯γ d(cid:105)=−iρ(x ). (1.2)
1 ⊥ 2 ⊥
The complex scalar field ρ is a function of the (transverse) spatial coordinates x =(x ,x ). The
⊥ 1 2
quark-antiquarkcondensates(1.2)carrythequantumnumbersoftheelectricallychargedρ mesons
sothatthisphenomenonmayalsobeinterpretedastheρ-mesoncondensation.
Since the condensed pairs (1.2) are electrically charged states, their condensation implies,
almost automatically, the emergence of the electromagnetic superconductivity in the new vacuum
state at B>B . The emerging superconducting vacuum state has quite unusual properties (for a
c
detailedreview,seeRef.[5]):
(i) Anisotropy: the ground state is a perfect conductor for the electric currents directed strictly
alongthemagneticfieldaxis. Inthetransversedirectionsthesuperconductivityisabsent.
(ii) Inhomogeneity: thetransportcoefficientsdependonthetransversecoordinatesx .
⊥
(iii) AbsenceoftheMeissnereffect: theinducedsuperconductivitycannotscreenthebackground
magneticfieldduetothementionedspatialanisotropy.
(iv) In-tandemsuperfluidity: onecanarguethatthesuperconductingstate(1.2)shouldalwaysbe
accompaniedbyasuperfluidityoftheneutralρ(0) mesons[1,4].
(iv) Optically,thevacuumsuperconductorismetamaterialwithaperfectlensproperties[6].
Inthemagnetic-field-inducedvacuumsuperconductivity,thequark-antiquarkcomposites(1.2)
playthesameroleastheCooperpairsplayinaconventionalsuperconductor. Thisanalogymaygo
evendeeper: inaverystrongmagneticfieldaconventionaltype-IIsuperconductorwassuggestedto
enteraquantumlimitof“reentrantsuperconductivity”characterisedbya p-wavespin-tripletpair-
ing, absence of the Meissner effect, and a superconducting flow along the magnetic field axis [7].
Itisencouragingthattheseareexactlythefeatureswhichweexpecttoberealisedinthesupercon-
ductingvacuumstateatB>B .
c
Thereareindicationsfromholographic[8,9]andnumerical[10]approachesthattheρ-mesons
shouldbecondensedinthestrongmagneticfield;seealsotheongoingdiscussioninRefs.[11,12].
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VortexliquidinsuperconductingvacuumofquenchedlatticeQCD M.N.Chernodub
2. Vortexlatticegroundstateinthemean-fieldapproximation
Theρ-mesoncondensationcanbecharacterisedbyasinglescalarfunction,
1
ρ(x)= u¯(x)γ d(x), γ =γ +iγ , (2.1)
+ + 1 2
2
where the combination γ of the Dirac matrices corresponds to the s =+1 projection of the ρ-
+ z
mesonspinontothemagneticfieldaxis. Inthesmall-condensatelimit,|ρ|(cid:28)m ,thegroundstate
ρ
oftheρ-mesoncondensatescanbedescribedbythefollowinggeneralform[1,2,4,9]:
(cid:114)
ρ(z)= ∑ C exp(cid:110)− π (cid:0)|z|2+z¯2(cid:1)−πν2n2+2πνn z¯ (cid:111), L = 2π , (2.2)
n 2L2 L B eB
n∈ZZ B B
where L is the magnetic length and ν is a real parameter. The solution (2.2) is similar to – and
B
inspiredby–theAbrikosovvortexlatticeconfigurationwhichappearsinamixedstateofatype-II
superconductorinthemagneticfieldbackground[18].
In Eq. (2.2) the complex coefficientsC are fixed by the energy minimisation condition. It is
n
usuallyassumedthatthecoefficientsC obeytheN–foldsymmetry,whereN isaninteger[18]:
n
C =C , N =1,2,... (2.3)
n+N n
sothatthecondensate(2.2)hasaperiodicstructureinthetransverse(x ,x )plane. Acondensateof
1 2
theform(2.2)possessesaninfinitesetofzeros,whichmarkthecentresofthetopologicalstringde-
fectscalledtheρ vortices[1]. SimilarlytotheAbrikosovvorticesinconventionalsuperconductors,
thecondensateρ acquiresthephaseshift2π aswecircumventthesezeros.
Figure1: (left)Theequilateraltriangularvortexlatticeinthemean-fieldapproximationtothegroundstate
ofQCDand(right)itssuggestedmeltingduetothepresenceofquantumand/orthermalfluctuations.
The simplest configuration with N =1 (allC ’s are equal) corresponds to the square vortex
n
lattice. However,thetruemean-fieldgroundstateisgivenbyanequilateraltriangularvortexlattice
(whichissometimesalsocalled“hexagonallattice”)withthefollowingsetofparameters[4,9,18]:
√ √
N =2, C =±iC , ν = 4 3/ 2≈0.9306. (2.4)
1 0
Thus,inthemeanfieldapproximation,theρ-mesonvortexstateissimilartothetruegroundstate
oftheAbrikosovlatticeinatype-IIsuperconductor[18];seeFig.1(left)foranillustration.
3
VortexliquidinsuperconductingvacuumofquenchedlatticeQCD M.N.Chernodub
The analytic results of Ref. [4] suggest that the ρ vortices are weakly interacting with each
other in the vortex lattice state. This fact means that the presence of thermal and/or quantum
fluctuations may either melt the vortex lattice state to a liquid vortex state or even evaporate it by
formingavortexgas(suchphenomenaareknowntohappenwiththeAbrikosovvortexlatticesin
conventionalsuperconductors[14]). Thevortexliquidstateisasuperconductingstatecharacterised
by the brokenU(1) electromagnetic gauge symmetry while the vortex gas state is, generally,
e.m.
a normal (non-superconducting) state with the unbroken U(1) group. In the context of the
e.m.
solidstatephysics,thevortexlattice-liquid-gasphasediagramformagnetic-field-inducedreentrant
superconductivitywasdiscussedinRef.[13].
3. Numericalsimulations
InournumericalsetupwebasicallyfollowRef.[10]. WeuselatticeMonte-Carlosimulations
of SU(2) Yang-Mills lattice gauge theory. The quark fields are introduced by the overlap lattice
Dirac operator D with exact chiral symmetry [16], and – due to the presence of the magnetic
field – with twisted spatial boundary conditions [17]. The quarks are treated in the quenched
approximation so that the vacuum quark loops are absent in our approach. We have studied 184
and194 latticesinawiderangeofthemagneticfieldstrengths,eB=(0...2.14)GeV2 withlattice
spacingsa≈0.11fm. Wehaveused20gaugeconfigurationspereachvalueofthemagneticfield.
Anexplicitmanifestationofthesuperconductingphasewouldbethepresenceoftheconden-
sate(1.2). Unfortunately,theobservable(1.2)cannotbecomputeddirectlyinourapproach,while
itcanbeaccessedviathefollowingsimplestρ-mesoncorrelator:
(cid:18) (cid:19)
φ(x)≡φ(x;A,B)=(cid:10)ρ†(0)ρ(x)(cid:11) ≡Tr 1 γ 1 γ , (3.1)
A,B D (A,B)+m µ D (A,B)+m ν
u d
where the ρ meson field is defined by Eq. (2.1). The subscripts in Eq. (3.1) indicate that the
correlationfunctionφ(x)iscomputedinthefixedbackgroundofboththenon-Abeliangaugefield
AandtheAbelianmagneticfieldB. Equation(3.1)representsthisvectorcorrelatorintermsofthe
(overlap)DiracpropagatorsinthebackgroundofbothAbelianandnon-Abeliangaugefields.
WenoticethatundertheAbeliantransformationfromtheelectromagneticgaugegroupU(1)
e.m.
thefield(3.1)transformsasachargedscalarfield1: φ(x)→eieω(x)φ(x). However,theeffectivefield
φ(x)isstillatwo-pointcorrelationfunctionwhichfallsoffexponentiallyasthedistancexincreases.
This property is not a desired behaviour for a genuine local scalar field so that the quantity (3.1)
cannot,strictlyspeaking,beassociatedwiththeρ-mesonfielditself.
Fortunately, we may get an insight from Ref. [15] where a qualitatively similar issue was
encountered. InthatworkthechromoelectricfluxtubewasstudiedusingarectangularWilsonloop
W asasourceandthelocalenergydensityoperatorO asaprobe. Althoughtheexpectationvalue
oftheWilsonloopfallsoffexponentiallyastheareaoftheWilsonloopgrows,theenergydensity
in the presence of the Wilson loop, given by the normalised energy ratio (cid:104)O(cid:105) =(cid:104)OW (cid:105)/(cid:104)W (cid:105),
W
has,generally,anon-vanishingprofileastheareaoftheWilsonloopgrows.
1Thegaugetransformationattheorigin,ϕ(x)→eieω(0)ϕ(x),actsasaglobalphasewhichisnotessentialforour
interpretationoftheeffectivefieldφ(x).
4
VortexliquidinsuperconductingvacuumofquenchedlatticeQCD M.N.Chernodub
By analogy with Ref. [15], we consider the normalised scalar energy of the ρ-meson field
E(x), the normalised electric (super)current j (x) generated by the ρ-meson field, and the local
µ
vortex density υ(x) in transversal (x,y) plane, respectively (we use the continuum notations in
ordertosimplifytheexpressions):
|D φ(x)|2
µ
E(x) = , D =∂ −ieA , (3.2)
|φ(x)|2 µ µ µ
→− ←−
φ∗(x)D φ(x)−φ∗(x)D φ(x)
µ µ
j (x) = , (3.3)
µ 2i|φ(x)|2
εab ∂ ∂
υ(x) = singargφ(x)≡ argφ(x), a,b=1,2. (3.4)
2π ∂x ∂x
a b
(x,y)plane (x,z)plane
Figure2: Twotypesofthe“probe”cross-sectionsofthe(expected)meltedρ-vortexlattice,Fig.1(right).
Insearchofsignaturesofthe(perhaps,melted)ρ-vortexlatticewehavestudied(configuration-
by-configuration)thebehaviourofthenormalisedenergydensity(3.2)inthe(x,y)and(x,z)planes
(we remind that the magnetic field is directed along the z axis); see Fig. 2. In the center of a
physical ρ vortex the energy density is higher than the energy density outside the vortex. Thus,
if the physical ρ vortices are formed in the (sufficiently strong) magnetic field background, than
we may expect the formation of the pointlike lumps of the energy density in the (x,y) plane [see
Fig.2(left)]andtheformationofthelinelikestructuresinthe(x,z)plane[seeFig.2(right)].
Typicalexamplesofthebehaviouroftheenergydensityinthe(x,y)and(x,z)planesareshown
for weak (eB = 0.356GeV2), moderate (eB = 1.07GeV2) and high (eB = 2.14GeV2) magnetic
fields in Fig. 3. In accordance with our qualitative expectations, at low magnetic field the vortex
lattice is not formed. At the moderate magnetic field the formation of a coherent vortex structure
is seen while the vortices are not strictly ordered in the transversal plane and they are not quite
parallel to the magnetic field. At higher magnetic field the physical picture is visually consistent
withthepresenceofameltedlattice(liquid)oftheρ vortices;seeFig.1(right).
Thepeaksintheenergydensity(3.2)arecorrelatedwiththeρ vortexpositions(3.4)andthat
theρ vorticesareencircledbythesupercurrents(3.3). ThelatterfeatureisshowninFig.4. Thus,
thenumericallyobservedvorticesdoindeedbeartheessentialfeaturesofthephysicalvortices.
The nature of the ρ-vortex state may be characterised by the normalised vortex-vortex cor-
relation function (cid:104)υ(0)υ(R)(cid:105)/(cid:104)υ(0)(cid:105)2, where the ρ vortex density is given in Eq. (3.4). At
low magneticfields thisfunction isa monotonicallyrising functionof theinter-vortex distanceR,
Fig.5(left),implyingthattheρ vorticesconstitutea(nonsuperconducting)gas.
5
VortexliquidinsuperconductingvacuumofquenchedlatticeQCD M.N.Chernodub
Figure 3: Typical behaviour of the energy density in the (x,y) planes (the left column) and in the (x,z)
planes (the right column) for eB=0.356GeV2 (the upper panel), eB=1.07GeV2 (the middle panel) and
eB=2.14GeV2(thelowerpanel). Thenumberofelementaryfluxesisn=eBL L /(2π). Intheleftcolumn
x y
themagneticfieldisperpendiculartothepageandintherightcolumnthemagneticfieldisdirectedvertically.
6
VortexliquidinsuperconductingvacuumofquenchedlatticeQCD M.N.Chernodub
Figure4: Examplesofthesuperconductingcurrents(thebluelines),Eq.(1.6),aroundthevortices(thered
squares), Eq.(2.7), inthe(x,y)planesateB=1.07GeV2 (left)andeB=2.14GeV2 (right). Accordingto
the analytical expectations [1, 4] the currents should encircle the ρ vortices in the clockwise direction (an
exampleofthemean-fieldsolution,Eqs.(1.4),(1.6),isshownintheinsetoftherightfigure,fromRef.[4]).
At higher magnetic fields a non-monotonic behaviour manifests itself via the formation of a
widemaximumatintermediatedistances; seeFig.5(right). Theappearanceofthepeakindicates
thepresenceoftheanticipatedsuperconductingρ-vortexliquid.
Figure5: Thenormalisedvortex-vortexcorrelationfunction(cid:104)υ(0)υ(R)(cid:105)/(cid:104)υ(0)(cid:105)2 inthe(x,y)plane. The
monotonic(nonmonotonic)behaviourofthecorrelatorsignalsthepresenceofthegas(liquid)vortexstate
atlow(high)valuesofthemagneticfieldBasshownintheleft(right)plot.
The melting of the vortex lattice in quenched QCD may make it difficult to observe the sug-
gested ρ vortex condensation using the standard numerical tools. Indeed, in the vortex lattice
state[describedbyEqs.(2.2)andEq.(2.4)]thephaseoftheρ-mesonfieldchangesby2π around
each vortex so that the ρ-field is an oscillatory function of the transverse coordinates x and y.
Thus,inthesuperconductingstateatB>B ,thespace-averaged(bulk)condensateisalwayszero,
c
(cid:104)ρ(x)(cid:105) ≡0,despitethefactthatthelocalρ-mesoncondensateislargeandthegroundstateisa
bulk
superconductor(thesameistruefortheAbrikosovmixedstateinatype-IIsuperconductor[18]).
However, if the vortices were strictly straight, then the ρ-meson condensation could in prin-
ciple still be determined by studying a long-distance limit (taken along the straight vortex world-
sheets) of the correlation function (3.1) averaged over gluon fields. In this straight-vortex case
theoscillatingphasewouldnotcontributetothecorrelationfunctionsothatlong-distancecorrela-
tor should generally be nonzero, lim (cid:10)ρ(0)ρ(x(cid:107))(cid:11) ∼ |(cid:104)ρ(0)(cid:105)|2 with x(cid:107) = (0,0,x,t). However,
x(cid:107)→∞
7
VortexliquidinsuperconductingvacuumofquenchedlatticeQCD M.N.Chernodub
in the liquid vortex phase the vortex worldsheets are not flat surfaces so that the vortex wob-
bling may, in general, add large phase fluctuations to the correlator of the ρ-meson field, hence
lim (cid:10)ρ(0)ρ(x(cid:107))(cid:11)≡0inthissuperconductingstate. Thus,onemayencounteratechnicaldifficulty
x(cid:107)→∞
indeterminationoftheexactρ-mesoncondensateintheliquidstatebyusingtheρ-fieldcorrelators.
4. Conclusions
Wehavenumericallyobservedtheformationoftheρ-vortexliquidinthevacuumofquenched
two-colorQCDinstrongmagneticfieldbackground. Thevortexliquidphaseisanelectromagnet-
ically superconducting phase characterised by the inhomogeneous order parameter (the ρ meson
condensate), similarly to the mixed (Abrikosov) phase of an ordinary type-II superconductor. We
arguethatinthisphasethecalculationofthe(highlyinhomogeneous)ρ-mesoncondensatebyus-
ing the standard methods should be taken with care. The transition between the usual (insulator)
phaseatlowBandthesuperconductingvortexliquidphaseathighBturnsouttobeverysmooth.
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