Dynamical Casimir Effect in a small compact manifold for the Maxwell vacuum. Ariel R. Zhitnitsky Department of Physics & Astronomy, University of British Columbia, Vancouver, B.C. V6T 1Z1, Canada We study novel type of contributions to the partition function of the Maxwell system defined on a small compact manifold M such as torus. These new terms can not be described in terms of the physical propagating photons with two transverse polarizations. Rather, these novel contributions emerge as a result of tunnelling events when transitions occur between topologically different but physically identical vacuum winding states. These new terms give an extra contribution to the Casimir pressure, yet to be measured. We argue that if the same system is considered in the background of a small external time- dependent magnetic field, than there will be emission of photons from the vacuum, similar to the DynamicalCasimirEffect(DCE)whenrealparticlesareradiatedfromthevacuumduetothetime- dependentboundaryconditions. ThedifferencewithconventionalDCEisthatthedynamicsofthe vacuum in our system is not related to the fluctuations of the conventional degrees of freedom, the 5 virtual photons. Rather, the radiation in our case occurs as a result of tunnelling events between 1 topologically different but physically identical |k(cid:105) sectors in a time -dependent background. We 0 2 comment on relation of this novel effect with the well-known, experimentally observed, and theo- reticallyunderstoodphenomenaofthepersistentcurrentsinnormalmetalrings. Wealsocomment n on possible cosmological applications of this effect. a J PACSnumbers: 11.15.-q,11.15.Kc,11.15.Tk 9 2 I. INTRODUCTION. MOTIVATION. Instead, such a degeneracy can be formulated in terms ] of some non-local operators, see few comments on this h t classification in Appendix A. Furthermore, the infrared - The main motivation for present studies is as follows. p physics of the system can be studied in terms of aux- It has been recently argued [1–3] that if free Maxwell e iliary topological non-propagating fields [3] precisely in theory (without any interactions with charged particles) h the same way as a topologically ordered system can be [ is defined on a small compact manifold than some novel analyzed in terms of the Berry’s connection (which is 1 terms in the partition function will emerge. These terms alsoemergentratherthanafundamentalfield). Further- are not related to the propagating photons with two v more, the corresponding expectation value of the auxil- transverse physical polarizations, which are responsible 3 iarytopologicalfileddeterminesthephaseofthesystem. fortheconventionalCasimireffect(CE)[4]. Rather,these 0 6 noveltermsoccurasaresultoftunnellingeventsbetween As we review in section II, the relevant vacuum fluc- 7 topologically different but physically identical |k(cid:105) topo- tuations which saturate the topological portion of the 0 logical sectors. These states play no role when the sys- partition function Z are formulated in terms of topo- 1. tem is defined in infinitely large Minkowski space-time logicallynon-trivialbtoopundaryconditions. Theseconfigu- 0 R1,3. But these states become important when the sys- rationssatisfytheperiodicboundaryconditionsongauge 5 tem is defined on a small compact manifold. Without field up to a large gauge transformation such that the 1 loosing any generality we shall call this manifold M, it tunnelling transitions occur between physically identi- v: couldbethefour-torusT4, oritcouldbeanyothercom- cal but topologically distinct |k(cid:105) sectors. Precisely these Xi pact manifold with a non-trivial mapping π1[U(1)] = Z. field configurations generate an extra Casimir vacuum Precisely this non-trivial mapping for the Maxwell U(1) pressure in the system. What happens to this compli- r gauge theory implies the presence of the topological sec- a cated vacuum structure when the system is placed into tors |k(cid:105) which play the key role in our discussions. The thebackgroundofexternalconstantmagneticfieldBz ? ext correspondingphenomenonwascoinedasthetopological The answer on this question is known: the correspond- Casimir Effect (TCE). ing partition function Z as well as all observables, in- top In particular, it has been explicitly shown in [1] that cluding the topological part of the Casimir pressure, are these novel terms in the topological portion of the par- highly sensitive to small magnetic field and demonstrate tition function Z lead to a fundamentally new contri- the 2π periodicity with respect to magnetic flux repre- top butions to the Casimir vacuum pressure, which can not sented by parameter θ ≡ eSBz where S is the xy eff ext be expressed in terms of conventional propagating phys- area of the system T4. This sensitivity to external mag- ical degrees of freedom. Furthermore, the Z shows netic field is a result of the quantum interference of the top many features of topologically ordered systems, which externalfiledBz withtopologicalquantumfluctuations ext were initially introduced in context of condensed matter describing the tunnelling transitions between |k(cid:105) sectors. (CM) systems, see recent reviews [5–9]. In particular, This strong “quantum” sensitivity of the TCE should be Z demonstrates the degeneracy of the system which contrasted with conventional Casimir forces which are top can not be described in terms of any local operators [2]. practically unaltered by the external field due to very 2 strong suppression ∼B2 /m4, see [1] for the details. observed in small rings S1, see reviews [14]. Our com- ext e The main goal of the present work is to the study the menthereisthatsimilarcurrentsflowingalongtherings dynamicsofthesevacuumfluctuationsinthepresenceof of S1 which represents the boundary of M can be inter- a time-dependent magnetic field Bz (t). We would like preted as a result of topological vacuum configurations ext toargueinthepresentworkthattherewillbearadiation intimately related to Aharonov Bohm phases when the of real photons emitted from the vacuum (described by system is defined on a topologically nontrivial manifold. the partition function Z [Bz (t)]) as a result of this We elaborate on this connection (between the persistent top ext time dependent external source Bz (t). currents and our description in terms of the topological ext A simple intuitive picture of this emission can be ex- vacuum configurations) further in the text. plained as follows. Imagine that we study conventional The only comment we would like to make here is as CE with static metallic plates. When these plates move follows. The persistent clockwise and anti-clockwise cur- or fluctuate, there will be emission from the vacuum, rentscanceleachotherincaseofvanishingexternalmag- whichiswellknownandwellstudiedphenomenonknown netic field. This cancellation does not hold in the pres- as the Dynamical Casimir Effect (DCE), see original pa- ence of a time-independent external magnetic field Bz ext pers [10] and reviews [11]. The DCE has been observed perpendiculartotheringS1,inwhichcasethepersistent inrecentexperimentswithsuperconductingcircuits[12]. current I will be generated. One can view this system 0 Basically, the virtual photons which are responsible for as generation of a static magnetic moment mz = I S. ind 0 conventional Casimir pressure may become the real pho- It is quite obvious now that if the external magnetic tons when the plate is moving or fluctuating. fieldBz (t)becomesatime-dependentfunction,thecor- ext The novel effect which is the subject of this work is responding induced magnetic moment mz (t) also be- ind that the topological configurations describing the tun- comes a time-dependent function. The corresponding nelling transitions between |k(cid:105) sectors will be also modi- time-dependence in mz (t) obviously implies that the ind fiedwhenthereisatimedependentexternalinfluenceon system starts to radiate physical photons with typical the system. This time dependent impact on the system angular distribution given by the magnetic dipole radi- canberealizedbymovingtheplatesoftheoriginalman- ation. This radiation is ultimately related to the topo- ifoldM,incloseanalogywithDCE.Thetime-dependent logical vacuum configurations describing the tunnelling impact may also enter the system through the quantum transitions between |k(cid:105) sectors. These vacuum configu- interference of the external field with topological con- rations get modified in the presence of a time-dependent figurations saturating the partition function Z . Such field Bz (t), which is precisely the source for the radia- top ext quantum interference, as we mentioned above, is prac- tion of physical photons. In all respects the idea is very tically absent in conventional CE but is order of one in similartoDCEwiththe“only”differenceisthatthecon- the TCE. Therefore, this quantum interference gives us ventionalvirtualphotons(responsiblefortheCE)donot a unique chance to manipulate with the Maxwell vac- interfere with external magnetic field Bz (t), while the ext uum defined on M using a time-dependent external elec- topological vacuum instanton-like configurations (satu- tromagnetic source. This effect leads to the production rating the TCE) do. We coin the corresponding phe- of the real photons with transverse polarizations emit- nomenon of the emission of real photons from vacuum tedfromthetopologicalquantumvacuumconfigurations configurationssaturatingZ [Bz (t)]inthepresenceof top ext saturating the partition function Z in the presence of time dependent source [Bz (t)] the non-static (or dy- top ext time-dependent magnetic field Bz (t). namical) topological Casimir effect (TCE) to discrimi- ext As this effect is very novel and quite counter-intuitive, nate it from the conventional DCE. we would like to present one more additional explana- Thestructureofourpresentationisasfollows. Innext tion supporting our claim that there will be emission of section II we review our previous results [1–3] on con- realphotonsfromtheMaxwellvacuumwhenthesystems struction of the partition function Z describing the top (described by the partition function Ztop) is placed into tunnelling transitions between |k(cid:105) sectors. We also ex- the background of time-dependent source Bz (t). plain how this partition function is modified in the pres- ext Our second explanation goes as follows. The topo- ence of external static magnetic field Bz . After that ext logicalconfigurationswhichdescribethetunnellingtran- in section III we generalize the construction to include sitions are formulated in terms of the boundary condi- the slow time-dependent fields, which allow us to com- tions on gauge field up to a large gauge transformations. putetheinducedmagneticdipolemomentofthesystem. These boundary conditions correspond to some persis- This time dependent induced magnetic moment radiates tentfluctuatingcurrentswhichcanflowalongthemetal- real physical photons from vacuum. We also elaborate lic boundaries corresponding to the edges of M. In fact, on relation of our construction with persistent currents the possibility that such persistent current may occur in section IIIB. Finally, in sections IIIC, IIID we make in small rings with topology S1 have been theoretically few simple numerical, order of magnitude estimates in predicted long ago [13], though with very different mo- ordertogetsomeinsightsonpotentialprospectsofmea- tivation from the one advocated in present work. Fur- suring the effect, which crucially depends on property of thermore, the corresponding persistent non-dissipating degeneracy of the system. As this feature of degeneracy currents in different materials have been experimentally iscrucialforpotentialexperimentalstudiesofthiseffect, 3 we make few comments on this property in Appendix A. gauge vacuum winding states |k(cid:105). We use the same ter- The corresponding description of the system (when it is minology and interpretation for 4d case because (2) is characterized by a global, rather than local observables) the classical configuration saturating the partition func- is quite different from conventional classification scheme tion Z in close analogy with 2d case as discussed in top when a system is characterized by an expectation value details in [1].This classical instanton-flux configuration of a local operator. satisfies the periodic boundary conditions up to a large Our conclusion is section IV where we speculate on gauge transformation, and provides a topological mag- possiblerelevanceofthisnoveleffectforcosmologywhen netic instanton-flux in the z-direction: appropriatetopologyisπ3[SU(3)]=Zreplacingthenon- (cid:18) 2πk (cid:19) trivial mapping π1[U(1)]=Z considered in present work B(cid:126)top =∇(cid:126) ×A(cid:126)top = 0, 0, eL L , (2) for studying the Maxwell theory on a compact manifold. 1 2 (cid:90) To be more concrete, we speculate that the de Sitter Φ=e dx dx Bz =2πk. behaviour in inflationary epoch could be just inherent 1 2 top property of the topological sectors in QCD in expand- The Euclidean action of the system is quadratic and has ing Universe, rather than a result of dynamics of some the following form ad hoc dynamical field such as inflaton. The emission of real physical degrees of freedom from the inflationary 1(cid:90) (cid:26) (cid:16) (cid:17)2(cid:27) vacuum in time dependent background (the so-called re- 2 d4x E(cid:126)2+ B(cid:126) +B(cid:126)top , (3) heatingepoch) in allrespectsis verysimilartothe effect considered in the present work when the real photons where E(cid:126) and B(cid:126) are the dynamical quantum fluctuations canbeemittedfromvacuuminthebackgroundofatime of the gauge field. We call the configuration given by dependent magnetic field Bz (t). ext eq. (1) the instanton-fluxes describing the tunnelling events between topological sectors |k(cid:105). These configura- tionssaturatethepartitionfunction(6)andshouldbein- II. TOPOLOGICAL PARTITION FUNCTION terpreted as “large” quantum fluctuations which change thewindingstates|k(cid:105),incontrastwith“small”quantum Our goal here is to review the Maxwell system defined fluctuationswhicharetopologicallytrivialandexpressed onaEuclidean4-toruswithsizesL ×L ×L ×β inthe 1 2 3 in terms of conventional virtual photons saturating the respective directions. It provides the infrared (IR) reg- quantum portion of the partition function Z . quant ularization of the system. This IR regularization plays The key point is that the topological portion Z de- top a key role in proper treatment of the topological terms couples from quantum fluctuations, Z = Z ×Z quant top which are related to tunnelling events between topologi- such that the quantum fluctuations do not depend on cally distinct but physically identical |k(cid:105) sectors. topological sector k and can be computed in topologi- cally trivial sector k =0. Indeed, the cross term A. Construction (cid:90) 2πk (cid:90) d4x B(cid:126) ·B(cid:126) = d4x B =0 (4) top eL L z 1 2 We follow [1–3] in our construction of the partition vanishes because the magnetic portion of quantum fluc- function Z where it was employed for computation of top tuations in the z-direction, represented by B =∂ A − the corrections to the Casimir effect due to these novel z x y typeoftopologicalfluctuations. Thecrucialpointisthat ∂yAx, is a periodic function as A(cid:126) is periodic over the do- weimposetheperiodicboundaryconditionsongaugeAµ mainofintegration. Thistechnicalremarkinfactgreatly field up to a large gauge transformation. In what follows simplifies our analysis as the contribution of the physi- we simplify our analysis by considering a clear case with cal propagating photons is not sensitive to the topolog- winding topological sectors |k(cid:105) in the z-direction only. ical sectors k. This is, of course, a specific feature of The classical instanton configuration in Euclidean space quadratic action (3), in contrast with non-abelian and which describes the corresponding tunnelling transitions non-linear gauge field theories where quantum fluctua- can be represented as follows: tions of course depend on topological k sectors. Theclassicalactionforconfiguration(2)takestheform (cid:18) (cid:19) πk πk Aµtop = 0, − eL1L2x2, eL1L2x1, 0 , (1) 1(cid:90) d4xB(cid:126)2 = 2π2k2βL3 (5) 2 top e2L L where k is the winding number that labels the topologi- 1 2 calsector, andL , L arethedimensionsoftheplatesin To simplify our analysis further in computing Z we 1 2 top thexandy-directionsrespectively,whichareassumedto considerageometrywhereL ,L (cid:29)L ,β similartocon- 1 2 3 be much larger than the distance between the plates L . struction relevant for the Casimir effect. In this case our 3 This terminology (“instanton”) is adapted from similar systemiscloselyrelatedto2dMaxwelltheorybydimen- studiesin2dQED[1]wherecorrespondingconfiguration sional reduction: taking a slice of the 4d system in the in A =0 gauge describe the interpolation between pure xy-plane will yield precisely the topological features of 0 4 the 2d torus considered in great details in [1]. Further- Theclassicalactionforconfigurationinthepresenceof more, with this geometry our simplification (2) when we the uniform static external magnetic field Bext therefore z consider exclusively the magnetic instanton- fluxes in z takes the form directionisjustifiedasthecorrespondingclassicalaction 1(cid:90) (cid:16) (cid:17)2 (cid:18) θ (cid:19)2 (5) assumes a minimal possible values. With this as- d4x B(cid:126) +B(cid:126) =π2τ k+ eff (8) sumption we can consider very small temperature, but 2 ext top 2π still we can not take a formal limit β → ∞ in our final whereτ isdefinedby(7)andtheeffectivethetaparameter expressions as a result of our technical constraints in the θ ≡ eL L Bz is expressed in terms of the original system. eff 1 2 ext external magnetic field Bz . Therefore, the partition With these additional simplifications the topological ext function in the presence of the uniform magnetic field partition function becomes [1–3] : can be easily reconstructed from (6), and it is given by (cid:114) [1–3] Ztop = e22πLβ1LL32 k(cid:88)∈Ze−2πe22Lk21βLL23 =√πτk(cid:88)∈Ze−π2τk2, (6) Z (τ,θ )=√πτ(cid:88)exp(cid:34)−π2τ(cid:18)k+ θeff(cid:19)2(cid:35). (9) top eff 2π where we introduced the dimensionless parameter k∈Z This system in what follows will be referred as the topo- τ ≡2βL /e2L L . (7) 3 1 2 logicalvacuum(TV)becausethepropagatingdegreesof freedom, the photons with two transverse polarizations, Formula (6) is essentially the dimensionally reduced ex- completely decouple from Z (τ,θ ). pression for the topological partition function for 2d top eff The dual representation for the partition function is Maxwell theory analyzed in [1]. One should note that √ obtained by applying the Poisson summation formula the normalization factor πτ which appears in eq. (6) such that (9) becomes doesnotdependontopologicalsectork,andessentiallyit represents our convention of the normalization Ztop →1 (cid:88) (cid:20) n2 (cid:21) Z (τ,θ )= exp − +in·θ . (10) inthelimitL1L2 →∞whichcorrespondstoaconvenient top eff τ eff set up for the Casimir -type experiments as discussed in n∈Z [1–3]. The simplest way to demonstrate that Z → 1 top Formula(10)justifiesournotationfortheeffectivetheta in the limit τ →0 is to use the dual representation (10), parameter θ as it enters the partition function in com- eff see below. bination with integer number n. One should emphasize that integer number n in the dual representation (10) is not the integer magnetic flux k defined by eq. (2) which B. External static magnetic field enters the original partition function (6). Furthermore, theθ parameterwhichenters(9,10)isnotafundamen- eff Inthissectionwewanttogeneralizeourresultsforthe tal θ parameter which is normally introduced into the Euclidean Maxwell system in the presence of the exter- Lagrangian in front of E(cid:126) ·B(cid:126) operator. Rather, this pa- nal magnetic field. Normally, in the conventional quan- rameter θeff should be understood as an effective param- tization of electromagnetic fields in infinite Minkowski eter representing the construction of the |θeff(cid:105) state for space,thereisnodirect couplingbetweenfluctuatingvac- each slice with non-trivial π1[U(1)] in four dimensional uum photons and an external magnetic field as a conse- system. In fact, there are three such θMi parameters eff quence of linearity of the Maxwell system. The coupling representing different slices and corresponding external with fermions generates a negligible effect ∼α2B2 /m4 magnetic fluxes. There are similar three θEi parameters ext e eff as the non-linear Euler-Heisenberg Effective Lagrangian representing the external electric fluxes as discussed in suggests, see [1] for the details and numerical estimates. [2], such that total number of θ parameters classifying In contrast with conventional photons, the the external thesystemequalssix,inagreementwithtotalnumberof magnetic field does couple with topological fluctuations hyperplanes in four dimensions. We shall not elaborate (2). It leads to the effects of order of unity as a result of onthisclassificationinthepresentwork. Inthisworkwe interference of the external magnetic field with topologi- limit ourselves with a single θ parameter entering (9), eff cal fluxes k. (10), and corresponding to the magnetic external field Thecorrespondingpartitionfunctioncanbeeasilycon- Bz pointing in z direction. ext structed for external magnetic field Bext pointing along z z direction, as the crucial technical element on decou- plingofthebackgroundfieldsfromquantumfluctuations III. INDUCED MAGNETIC DIPOLE MOMENT assumes the same form (4). In other words, the physi- AND E&M RADIATION calpropagatingphotonswithnon-vanishingmomentaare not sensitive to the topological k sectors, nor to the ex- The main goal of this section is to estimate the in- ternal uniform magnetic field, similar to our discussions duced magnetic dipole moment of the system in a time after eq.(4). dependentbackground. First,insectionIIIAwederivea 5 formulaforinducedmagneticmomentinthebackground The corresponding formulae for Z (τ,θ ) replacing top eff of a uniform static external magnetic field. We shall dis- (9), (10) with new geometry assume the form: cuss a different geometry in this section (in comparison with4-torusdiscussedinpreviousreviewsectionII).This Z (τ,θ )=√πτ(cid:88)exp(cid:34)−π2τ(cid:18)k+ θeff(cid:19)2(cid:35) is because our generalization for a time dependent back- top eff 2π ground cannot be consistently introduced on 4 torus. As k∈Z the corresponding formula for the induced magnetic mo- (cid:88) (cid:20) n2 (cid:21) = exp − +in·θ , τ ≡2βL /e2πR2,(13) mentplaysanimportantroleinthepresentwork,weoffer τ eff 3 n∈Z acomplementaryinterpretationofthesameexpressionin section IIIB in terms of the dynamics on the boundaries where L is the length of the cylinder, β is the in- 3 ofthesystem,ratherthanintermsofthebulk-instantons verse temperature, and effective theta parameter θ ≡ eff (1), (2). In section IIIC we provide some numerical esti- eπR2Bz is expressed in terms of the original external ext mates, and make few comments on relation with exper- magnetic field Bz similar to (8). ext imentally observed persistent currents. Finally, we gen- Fewcommentsareinorder. First,thetopologicalpor- eralize the expression for induced magnetic moment for tion of the partition function for the cylinder given by a slowly varying field. A time dependent magnitude for (13) plays the same role as (9), (10) plays for the 4- theobtainedmagneticmomentautomaticallyimpliesthe torus. There are few differences, though. In case for radiation of the real physical photons from this system the 4-torus we could, in principle, introduce 6 differ- as we show in sections IIID. ent θ parameters corresponding to 6 different hyper- i planes and 6 different nontrivial mappings π [U(1)]=Z 1 for each slice in 4d Euclidean space. In contrast, with new geometry there is just one type of magnetic fluxes A. Magnetization of the system. The basics. (12). Another difference is that our treatment of 4-torus in previous section neglects all other types of instan- tons. This was achieved by imposing geometrical con- Our goal here is to construct the topological portion dition L ,L (cid:29) β,L to guarantee that action (5) as- Z for the partition function similar to (9), (10), but 1 2 3 top sumes a minimal possible value (maximal contribution for a different geometry. To be more specific, we want to the partition function), while other types of instan- to consider a solenoid (cylinder) with opened ends to tons would produce parametrically smaller contribution have an option to place our system under influence of to the partition function Z (τ,θ ). In present case we time variable magnetic field (which is not possible for top eff donothaveanyothertypesofinstantonswhichmaycon- 4-torus). This geometry also gives us an opportunity tribute to Z (τ,θ ). However, we must assume that to discuss the relation between the topological currents top eff L (cid:29) R to justify our approximate for (finite L ) solu- derived from Z in next section IIIB and persistent 3 3 top tion (11), (12) when a contribution from outside region currents considered long ago [13]. To proceed with this canbeneglectedintheaction(13). Finally,aswealready goal we consider a cylinder with cross section of area √ πR2 which replaces the area L L from original compu- mentioned, the normalization factor πτ in Ztop(τ,θeff) 1 2 does not depend on topological sectors k, and in fact, is tations [1] on 4 torus. Furthermore, for this geometry a matter of convention. It is more appropriate for the it is convenient to consider the instanton solution de- √ given geometry to define Z(cid:48) = Z / πτ such that scribing the transitions between topological |k(cid:105) sectors top top Z(cid:48) (τ → ∞,θ = 0) → 1. However, we opted to inCylindricalcoordinatesratherthaninCartesiancoor- top eff preserve our original normalization because our results dinates (1). The corresponding “instanton”-like config- which follow do not depend on this convention. uration describes the the same physics as we discussed Our next step is to analyze the magnetic response of before in section II, and it is given by thesystemunderinfluenceoftheexternalmagneticfield. Theideabehindthesestudiesistheobservationthatthe kr k A(cid:126) = φˆ, r <R; A(cid:126) = φˆ, r ≥R(11) externalmagneticfieldactsasaneffectiveθ parameter top eR2 top er eff as eq. (13) suggests. Therefore, one can differentiate with respect to this parameter to compute the induced whereφˆisunitvectorinφˆdirectioninCylindricalcoordi- magnetic field nates. Byconstruction,thevectorpotentialA(cid:126) = k∇(cid:126)φ top e 1 ∂lnZ e ∂lnZ is a pure gauge field on the boundary with nontrivial (cid:104)B (cid:105)=− top =− top (14) winding number k. The topological magnetic flux in the √ind βV ∂Bezxt βL3 ∂θeff z direction is defined similar to (2), and it is given by = τπ (cid:88)(cid:18)B + 2k (cid:19)exp(cid:20)−τπ2(k+ θeff)2(cid:21). Z ext R2e 2π top k∈Z 2k B(cid:126) = zˆ, r <R; B(cid:126) =0, r >R, (12) top eR2 top As one can see from (14), our definition of the in- (cid:90) (cid:73) duced field accounts for the total field which includes Φ=e dx dx Bz =e A(cid:126) ·d(cid:126)l=2πk. 1 2 top top both terms: the external part as well as the topological r=R 6 portion of the field. In the absence of the external field in the presence of external magnetic field Bz , ext (Bext = 0), the series is antisymmetric under k → −k 1∂lnZ and (cid:104)B (cid:105) vanishes. It is similar to the vanishing expec- (cid:104)mz (cid:105)= top =−(cid:104)B (cid:105)L πR2 (15) ind ind β ∂Bext ind 3 tation value of the topological density in gauge theories √ z (cid:18) (cid:19) (cid:20) (cid:21) when θ = 0. One could anticipate this result from sym- =−L32π τπ (cid:88) θeff +k exp −τπ2(k+ θeff)2 . metry arguments as the theory must respect P and CP e Z 2π 2π top k∈Z invariance at θ =0. One can view Fig. 1 as a plot for the induced magnetic Theexpectationvalueoftheinducedmagneticfieldex- moment in units eL3c which represents correct dimen- 4πα hibits the 2π periodicity from the partition function and sionality e·cm2. it reduces to triviality whenever the amount of skewing s results in an antisymmetric summation, i.e. (cid:104)B (cid:105) = 0 ind for θ ∈ {2nπ : n ∈ Z}. The point θ = π deserves eff eff B. Interpretation special attention as this point corresponds to the degen- eracy, see [2] with detail discussions. This degeneracy As formula (15) plays a key role in our discussions, we can not be detected by an expectation value of any local would like to interpret the same expression for (cid:104)mz (cid:105) operator, but rather is classified by a nonlocal operator, ind butindifferentterms. Tobemoreprecise: equation(15) similar to studies of the topological insulators at θ =π. describes the magnetization properties of the system in terms of the vacuum configurations describing the tun- nelling transitions between topological sectors |k(cid:105). The corresponding partition function Z (τ,θ ) which gov- top eff erns these vacuum processes is given by (13). A non- trivial behaviour of the magnetization of the system in terms of the induced magnetic dipole moment (15) is di- rect consequence of the basic properties of the partition function Z (τ,θ ). We would like to understand the top eff same properties in more intuitive way in terms of the fluctuatingcurrentswhichunavoidablywillbegenerated on the boundaries, as we discuss below. Indeed,thecrosstermintheeffectiveaction(8)which describes coupling of the external field with topological instanton-likeconfigurationcanberepresentedasfollows (cid:90) (cid:16) (cid:17) (cid:90) (cid:16) (cid:17) d4x B(cid:126) ·B(cid:126) = d4x A(cid:126) · ∇(cid:126) ×B(cid:126) ,(16) ext top ext top where we neglected a total divergence term. The cross FIG. 1. A numerical plot of the induced magnetic field in term written in the form (16) strongly suggests that units πRc2e asafunctionofexternalfluxθeff. Thesameplot (cid:16)∇(cid:126) ×B(cid:126) (cid:17) can be interpreted as a steady current flow represents the induced magnetic moment −(cid:104)mz (cid:105) in units top ind eL3c, see text with details. Unit magnetic flux corresponds along the boundary. Indeed, 4πα to θ =2π. The plot is adapted from [1]. eff 2k (cid:126)j (k)=−∇(cid:126) ×B(cid:126) =−δ(r−R) φˆ, (17) top top eR2 where we used expression (13) for B(cid:126) describing the We now return to analysis of eq. (14). The topologi- top tunnelling transition to |k(cid:105) sector. Formula (17) is very cal effects, as expected, are exponentially suppressed at suggestive and implies that the vacuum transitions for- τ (cid:28) 1 and τ (cid:29) 1 according to eq.(13). The effect is mulated in terms of the fluxes-instantons (11), (12) can much more pronounced in the range where τ (cid:39) 1, see be also interpreted in terms of accompanied fluctuating Figure 1, where we plot the induced magnetic field in topological currents (17). The total current (in topologi- units (πR2e)−1 as a function of external flux θ (point eff cal k sector) which flows along the infinitely thin bound- θ =πshouldbeconsideredseparatelyaswementioned eff ary of a cylinder radius R and length L in our ideal above). 3 system is given by Importantcommenthereisthattheinducedmagnetic (cid:90) L3 (cid:90) 2kL field defined as (14) can be thought of as the magnetiza- Jφ(k)= dz drjφ =− 3. (18) top eR2 tion of the system per unit volume, i.e. (cid:104)M(cid:105)=−(cid:104)B (cid:105), 0 ind as the definition for (cid:104)M(cid:105) is identical to (14) up to a This current in topological k sector produces the follow- minus sign because it enters the Hamiltonian as H = ing contribution to the magnetic dipole moment: p−rm(cid:126)esisnidon·B(cid:126)foerxtt.heThinedreufcoerem,awgeneatrircivmeotmoetnhteoffoltlhoewisnygsteexm- mzind(k)=πR2Jφ(k)=−2πeL3k. (19) 7 This formula precisely reproduces the term proportional charged particles in the presence of external magnetic to k in the parentheses in eq. (15) which was originally field leads to such pronounced effect as persistent cur- derived quite differently, see previous subsection IIIA. rents [13, 14]. The secondary, rather than fundamental Fewcommentsoneq. (18). Theexpectationvalue(cid:104)Jφ(cid:105) role of the charged matter particles in this phenomenon of the topological current obviously vanishes at zero ex- manifests, in particular, in a fact that Z generates top ternal magnetic field when one sums over all topological an extra contribution to the Casimir vacuum pressure k-sectors,inagreementwithourpreviousexpression(15) even at zero external magnetic field. At the same time withθ =0. Itisquiteobviousthatthetopologicalcur- the persistent current can not be generated at B = 0 eff ext rentshavepurequantumnatureastheyeffectivelyrepre- as clockwise and counterclockwise currents cancel each sent the instantons describing the tunnelling transitions other. in the path integral computations. The currents could Furthermore, the persistent currents in the original have clockwise or anticlockwise direction, depending on works [13, 14] were introduced as a response of the elec- signofintegernumberk,similartofluctuatinginstanton trons (residing on the ring) on external magnetic flux solutions (11), saturating the topological portion of the with nontrivial Aharonov -Bohm phase. In contrast, the partition function (13). non-trivialgaugeconfigurations(andaccompaniedtopo- Furthermore, one can explicitly check that the cross logical currents (17) in k sectors) in our system are gen- term (16) computed in terms of the boundary current eratedevenwhennoexternalfieldnorcorrespondingex- (cid:126)j (k)exactlyreproducesthecorrespondingterminthe ternal Aharonov Bohm vector potential are present in top action for the partition function (15) computed in terms the system. In addition, the correlation length in con- of the bulk instantons (11), (13). Indeed, ventional persistent currents [13, 14] is determined by dynamics of the electrons residing on the ring, while in (cid:90) d4xA(cid:126) ·(cid:16)∇(cid:126) ×B(cid:126) (cid:17)= 2k(2πβL3)(cid:90) rdrδ(r−R)Aφ our case it is determined by the dynamics of the vac- ext top eR2 ext uum described by the partition function Z . The cor- top 2k(2πβL ) (cid:18)Bz R2(cid:19) (cid:18)θ (cid:19) responding topological fluctuations (described in terms = 3 · ext =2τπ2k eff , (20) of the instantons (13)) also generate the persistent topo- eR2 2 2π logical currents on the boundary (17). However, this ad- where the vector potential Aφ (r) = rBz corresponds ditional contribution should be treated separately from ext 2 ext totheexternaluniformmagneticfieldincylindricalcoor- conventional persistent currents [13, 14], as it is abso- dinates. Ourfinalresultineq. (20)isexpressedinterms lutely independent contribution which is generated due of the external flux θ ≡ eπR2Bz and dimensionless to the unavoidable coupling of topological gauge config- eff ext parameter τ ≡2βL /e2πR2. One can explicitly see that urations with charged particles on the boundary of the 3 the cross term in action in eq. (15) is reproduced by eq. system1. (20) derived in terms of the boundary currents, rather Finally, the induced magnetic moment (19) due to the then in terms of the bulk instantons. topological currents flowing on the boundary is quan- The classical instanton action is also reproduced in tized. Indeed, mzind/L3 assumes only integer numbers in terms of the boundary currents. Indeed, by substitut- units of 2π. This is because the corresponding induced e ing the expression for the current (17) to the classical currents always accompany the quantized instanton -like action one arrives to fluctuations (12). In contrast, a similar induced mag- neticmomentduetotheconventionalpersistentcurrents 1(cid:90) (cid:16) (cid:17) 1(cid:90) (cid:16) (cid:17) d4x B(cid:126) ·B(cid:126) = d4x A(cid:126) · ∇(cid:126) ×B(cid:126) , [13, 14] is not quantized, and can assume any value. 2 top top 2 top top To conclude this subsection we would like to comment = k(2πβL3)(cid:90) rdrδ(r−R)Aφ (r)=τπ2k2, (21) that it is quite typical in condensed matter physics that eR2 top the topologically ordered systems exhibit such a com- which is precisely the expression for classical instanton action entering the topological portion of the partition function (15). 1 In many respects this situation is very similar to QCD when The basic point of our discussions in this section is thepresenceofthetopologicalsectorsisabsolutelyfundamental that the expression for the induced magnetic moment basicpropertyofthegaugesystem. Atthesametime,verypro- (15) can be understood in terms of the topological cur- nouncedconsequencesofthisfundamentalfeatureareexpressed intermsofthematterfermifields,ratherthanintermsoforig- rents flowing along the boundaries of the system. How- inal gauge configurations. These well noticeable properties of ever, the origin of the phenomena is not these currents the system are basically the consequence of the Index Theorem butthepresenceofthetopological|k(cid:105)sectorsinMaxwell which states that the fermions in the background of nontrivial U(1)electrodynamicswhenitisformulatedonacompact gaugeconfigurationshavechiralzeromodes. Preciselythesezero manifold with nontrivial mapping π [U(1)] = Z. Such modesplayextremelyimportantroleinexplanationofmanyef- 1 fects such as generation of the chiral condensate in QCD, the |k(cid:105)sectorsexistandtransitionsbetweenthemalwaysoc- resolutionoftheso-calledtheU(1)A problem,etc. However,the cur even if charged particles are not present in the sys- root,theoriginofthesepropertiesofthesystemisthepresenceof tem. Thecouplingofthenon-trivialgaugeconfigurations topologically non-trivial gauge configurations, while the matter describing the transitions between the |k(cid:105) sectors with fieldsplaythesecondaryrole. 8 plementary formulation in terms of the physics on the current (30 nA) can be estimated as follows boundary. Our system (TV) can be also thought as a topologically ordered system as argued in [2, 3] because m (cid:39)πI R2 ∼π(30 nA)·(cid:18)2.4 µm(cid:19)2 it demonstrates a number of specific features which are persistent 0 2 inherent properties of topologically ordered systems. In (cid:18)e cm2(cid:19) particular, Z (τ,θ ) demonstrates the degeneracy of ∼0.7·104 . (22) top eff s the system which can not be described in terms of any local operators. Furthermore, the infrared physics of Itisinstructivetocomparethismomentwithfundamen- (cid:16) (cid:17) the system can be studied in terms of auxiliary topo- tal Bohr magneton µ =e(cid:126)/2m (cid:39)0.6· e cm2 , which B e s logical non-propagating fields precisely in the same way providesacrudeestimateofanumberofeffectivedegrees as a topologically ordered system in condensed matter of freedom ∼ 104 which generate the persistent current physics can be analyzed in terms of the Berry’s con- I forthisspecificsample. Thisestimateshouldbetaken nection. Therefore, it is not a surprise that we can re- 0 withsomeprecautionbecausetheeffectofpersistentcur- formulate the original instanton fluctuations saturating rents is entirely determined by the properties of the ma- Z (τ,θ ) in terms of the boundary persistent currents top eff terial (such as the electron phase coherence length l ) which always accompany these instanton transitions. φ which is beyond of the scope of the present work. Before we estimate (cid:104)mz (cid:105) from eq. (15) to compare it ind with (22) we would like to get some insights about the numerical magnitude of the dimensionless parameter τ for the ring with parameters used in the estimate (22), 2βL 2(0.1µm)(0.6cm)(cid:18) 2 (cid:19)2 C. Numerical estimates τ ≡ 3 ∼ (cid:29)1, (23) e2πR2 4π2α 2.4 µm where we use for L ∼0.1µm and β ∼0.6 cm which cor- 3 We want to make some simple numerical estimates by respondstothetemperatureT (cid:39)300mKbelowwhichl φ comparingthemagneticinducedmoment(cid:104)mz (cid:105)fromeq. issufficientlylargeandtemperatureindependent2. Large ind (15) with corresponding expression m (cid:39) πI R2 magnitudeofτ impliesthatforthechosenparametersfor persistent 0 withmeasuredpersistentcurrentI . Oneshouldempha- the system the vacuum transitions between the topolog- 0 size that the conventional persistent currents are highly ical sectors are strongly suppressed as the expression for sensitive to the properties of the material. More than the partition function(9), (10)states. Inthis regime the that, the properties of the condensed matter samples es- effect (22) is entirely determined by conventional mech- sentially determine the magnitude of the measured cur- anism [13], [14]. rents. At the same time, in all our discussions above we If somehow we could manage to satisfy our ideal assume that the “ideal” boundary conditions can be ar- boundary conditions and could adjust parameters of the ranged,suchthattheMaxwellvacuumdefinedonacom- system such that τ ∼ 1 than the magnitude of (cid:104)mz (cid:105) ind pact manifold is well described by the partition function from eq. (15) is determined by parameter 2πL /e such 3 (9), (10). Moreover, as we discussed in previous section that IIIBthetopologicalboundarycurrents(17)inourframe- 2πL L ce (cid:18)e cm2(cid:19) work should be considered as an independent additional (cid:104)mz (cid:105)∼ 3 ∼ 3 ∼1.5·107 , (24) ind e 2α s contribution to conventional persistent currents. There- fore, the corresponding numerical estimates taken from which is at least 3 orders of magnitude larger than the earlywork[15]arepresentedherefordemonstrationpur- value(22)ofthemagneticmomentgeneratedduetocon- poses only. ventional persistent current when the correlation length is determined by the physics of the ring. The crucial Themeasurementofthetypicalpersistentcurrentwas element in our estimate (24) is that the key parameter reported in [15]. The measurements were performed on τ should be order of one, τ ∼ 1. This would guaran- single gold rings with diameter 2R =2.4 and 4 µm at tee that the vacuum transitions would not be strongly a base temperature of 4.5 mK. Reported values for the suppressed. We really do not know if it could be real- currents are I = 3 and 30 nA for these two rings. It 0 ized in practice. The answer hopefully could be positive should be contrasted with expected current ∼ 0.1 nA. as Aharonov Bohm phase coherence can be maintained We are not in position to comment on this discrepancy, at sufficient high temperature, which can drastically de- as the effect is basically determined by the properties of crease parameter τ from (23), see footnote 2. the material, which is not subject of the present work. Our goal is in fact quite different. We want to compare themagneticmomentwhichisinducedduetothispersis- tent current with induced magnetic moment due to the 2 One should remark here that there are related effects when the topological vacuum configurations (cid:104)mz (cid:105) from eq. (15). entiresystemcanmaintaintheAharonovBohmphasecoherence ind Numerically,amagneticmomentforthelargestobserved atveryhightemperatureT (cid:39)79K[16]. 9 Weemphasizeonceagainthateq. (24)describesanew the system even when persistent currents are not gener- contribution to the magnetic moment originated from ated in the system (for example in absence of external tunnelling transitions between topological k sectors. It field). shouldbecontrastedwithconventionalpersistentcurrent Our final comment in this subsection is as follows. which also contributes to magnetic moment (22). These As we discussed above the energy for E&M radiation two contributions originated form very different physics: eventuallycomesfromtime-dependentexternalmagnetic in case eq. (22) the correlation in the system is achieved field. One could suspect that it would be very difficult by the dynamics of the electrons on the boundary, while to discriminate a (non-interesting) direct emission origi- in our case it is achieved by the tunnelling transitions nated from B (t) (cid:54)= 0 and the (very interesting) emis- ext between gauge k sectors and described by the vacuum sion resulted from the Maxwell vacuum which itself is instantons (13) saturating Z (τ,θ ). excited due to the quantum interference of the vacuum top eff configurations describing the topological |k(cid:105) sectors with external magnetic field. First (non-interesting) term is D. E&M Radiation represented by θ = eB (t)πR2 in the parentheses in eff ext eq. (15), while the second (very interesting) term is rep- Important comment we would like to make is as fol- resented by term ∼k in eq. (15). lows. Formula (15) has been derived assuming that the Fortunately, one can easily discriminate between (the external field is static. However, formula (15) still holds very interesting) emission from the vacuum and (abso- even in the case when the time dependence is adiabat- lutelynon-interesting)backgroundradiation. Thepoint ically slow, i.e. (dBditnd)/Bind = ω is much smaller than is that the induced magnetic dipole moment (cid:104)mzind(t)(cid:105) anyrelevantscalesoftheproblem,tobediscussedbelow. is the periodic function of B (t). Exactly at the point ext Therefore, one can use the well -known expressions for θ =πtheinducedmagneticdipolemoment(cid:104)mz (cid:105)sud- eff ind the intensity S(cid:126) and total radiated power I for the mag- denlychangesthesignasonecanseefromFigure1. This netic dipole radiation when dipole moment (15) varies isaresultofcompletereconstructionofthegroundstate with time: in the vicinity of θeff =π when the level crossing occurs, which eventually results in the double degeneracy of the sin2θ 2 S(cid:126) =I(t) (cid:126)n, I(t)= (cid:104)m¨z (cid:105)2 (25) system at this point. As this is the key element of the 4πr2 3c2 ind construction which leads to the important observational consequences related to the topological features of the In case when the external magnetic field in the vicin- system, we elaborate on this issue with more details in ity of θ ∼ 2πn the behaviour of the induced magnetic eff moment almost linearly follows Bz (t) as one can see Appendix A. ext from Figure 1. In particular, if Bz (t) ∼ cosωt than Therefore, one could slowly change the external field ext (cid:104)m¨z (cid:105) ∼ ω2cosωt. In this case one can easily compute Bext(t) in vicinity of θeff = π which corresponds to the ind the average intensity over large number of complete cy- halfintegerflux,toobservethevariationinintensityand cles with the result polarizationofradiation. Thecorrespondingbackground radiation must vary smoothly, while the emission from (cid:104)I(cid:105)∼ ω4 (cid:104)mz (cid:105)2, (26) vacuum should change drastically. One could hope that 3c2 ind these drastic changes may serve as a smoking gun for discoveryofafundamentallynoveltypeofradiationfrom where (cid:104)mz (cid:105) is given by (15). ind topological Maxwell vacuum, similar to DCE. Few comments are in order. First of all, the magnetic dipoleradiationcanbeeasilyunderstoodintermsofper- sistentcurrents[13,14]flowingalongthering. Forstatic external magnetic field the corresponding persistent cur- IV. CONCLUSION AND FUTURE DIRECTIONS rent I is also time independent. The magnetic dipole 0 moment generated by this current can be estimated as m (cid:39) πI R2. When the external magnetic field In this work we discussed a number of very unusual persistent 0 starts to fluctuate, the corresponding current I (t) as features exhibited by the Maxwell theory formulated on 0 well as magnetic dipole moment m (t) also be- a compact manifold M with nontrivial topological map- persistent come time-dependent functions. It obviously leads to ping π [U(1)], which was coined the topological vacuum 1 the radiation of real photons which is consistent with (TV). All these features are originated from the topo- our analysis. However, we should emphasize that the in- logical portion of the partition function Z (τ,θ ) and top eff terpretation of this phenomenon (which we coin as non- can not be formulated in terms of conventional E&M stationary TCE) in terms of topological persistent cur- propagating photons with two physical transverse polar- rents (17) is the consequential, rather than fundamental izations. In different words, all effects discussed in this explanation. The fundamental explanation, as empha- paper have a non-dispersive nature. sized in the previous section IIIB is based on topologi- The computations of the present work along with pre- cal instanton-like configurations interpolating between k vious calculations of refs. [1–3] imply that the extra topologicalsectors. Thesetunnellingtransitionsoccurin energy (and entropy), not associated with any physical 10 propagatingdegreesoffreedom,mayemergeinthegauge configurations describing the tunnelling transitions be- systems if some conditions are met. This fundamentally tween |k(cid:105) sectors, rather than from physical propagat- new type of energy emerges as a result of dynamics of ing degrees of freedom, which would correspond to con- pure gauge configurations at arbitrary large distances. ventional DCE when the virtual photons become real The new idea advocated in this work is that this new photons in a time dependent background. This is pre- type of energy can be, in principle, studied if one place cisely the difference between DCE and non-stationary the system in time-dependent background. In this case TCE considered in the present work. weexpectthatthevacuumtopologicalconfigurationscan radiate conventional photons which can be detected and analyzed. ACKNOWLEDGEMENTS Thisuniquefeatureofthesystemwhenanextraenergy isnotrelatedtoanyphysicalpropagatingdegreesoffree- I am thankful to Maxuim Chernodub for discussions domwasthemainmotivationforaproposal[17,18]that and useful comments. I am also thankful to Alexei Ki- thevacuumenergyoftheUniversemayhave,infact,pre- taevforlongandinterestingdiscussionsonrelationofthe cisely such non-dispersive nature3. This proposal when persistent currents [13, 14] and the topological configu- an extra energy can not be associated with any propa- rations saturating Z . This research was supported in top gatingparticlesshouldbecontrastedwithaconventional part by the Natural Sciences and Engineering Research descriptionwhenanextravacuumenergyintheUniverse Council of Canada. isalwaysassociatedwithsomeadhocphysicalpropagat- ing degree of freedom, such as inflaton4. Essentially,theproposal[17,18]identifiestheobserved Appendix A: Classification of the vacuum states, vacuumenergywiththeCasimirtypeenergy,whichhow- Degeneracy, and the Topological order. everisoriginatednotfromdynamicsofthephysicalprop- agating degrees of freedom, but rather, from the dynam- The main goal of this Appendix is to review and elab- ics of the topological sectors which are always present in orateonimportantpropertyofdegeneracyofthesystem gauge systems, and which are highly sensitive to arbi- under study. As suggested in section IIID the feature of trary large distances. Furthermore, the radiation from degeneracymayplayanimportantroleindiscrimination the vacuum in a time-dependent background (which is ofnovelandinterestingeffectofemissionofrealphotons the main subject of this work) is very similar in all re- from vacuum (as a result of the non-static Topological spects to the radiation which might be responsible for Casimir Effect) from the background radiation. the end of inflation in that proposal, see [18] for the de- The starting point is to analyze the symmetry proper- tails. The present study, in fact, is motivated by the ties of partition function Z (τ,θ ) defined by eq.(13). top eff cosmological ideas [18] which hopefully can be tested in One can easily observe that the θ = π is very special eff a tabletop experiment when the vacuum energy in time- point as even and odd “n”- terms in the dual represen- dependent background can be transferred to real propa- tationforZ (τ,θ )contributeequallytothepartition top eff gatingdegreesoffreedomassuggestedinsectionIIID.In function. Itobviouslyleadstothedegeneracyofthesys- cosmology the corresponding period plays a crucial role tem at θ =π. The conventional, non-topological, part eff and calls the reheating epoch which follows the inflation of the partition function Z (τ,θ ) isnot sensitive to quant eff when the vacuum energy is the dominating component thetopologicalsectorsatall,asdiscussedinsectionIIA. of the Universe. Therefore,thispropertyofdegeneracyisanexactfeature To conclude, the main point of the present studies is of the system. This double degeneracy implies that Z 2 that the radiation may be generated from the vacuum symmetry is spontaneously broken. Whatisthesymmetrywhichisspontaneouslybroken? What is order parameter which classifies two physically distinct states? One can not formulate the correspond- 3 This new type of vacuum energy which can not be expressed ing symmetry breaking effect in terms of any local oper- intermsofpropagatingdegreesoffreedomhasbeeninfactwell ators and their vacuum expectation values as argued in studied in QCD lattice simulations, see [17] with large number [2]. Rather, a proper classification of the ground state ofreferencesontheoriginallatticeresults. 4 There are two instances in evolution of the universe when the is formulated in terms of non-local operators. Indeed, a vacuum energy plays a crucial role. First instance is identified corresponding order parameter which characterizes the withtheinflationaryepochwhentheHubbleconstantH wasal- system is mostconstantwhichcorrespondstothedeSittertypebehaviour a(t) ∼ exp(Ht) with exponential growth of the size a(t) of the e (cid:73) 1 (cid:104) A dx (cid:105) =+ (A1) Universe. The second instance when the vacuum energy plays 2π i i θeff=π−(cid:15) 2 a dominating role corresponds to the present epoch when the vacuumenergyisidentifiedwiththeso-calleddarkenergyρDE (cid:104) e (cid:73) A dx (cid:105) =−1 whichconstitutesalmost70%ofthecriticaldensity. Inthepro- 2π i i θeff=π+(cid:15) 2 posal [17, 18] the vacuum energy density can be estimated as ρDE ∼ HΛ3QCD ∼ (10−4eV)4, which is amazingly close to the where computations should be carefully carried out by observedvalue. approaching θ = π in partition function Z (τ,θ ) eff top eff