Topological order and Berry connection for the Maxwell Vacuum on a four-torus
aa r X i v : . [ h e p - t h ] N ov Topological order and Berry connection for the Maxwell Vacuum on a four-torus
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 ona small compact manifold such as torus. These new terms can not be described in terms of thephysical propagating photons with two transverse polarizations. Rather, these novel contributionsemerge as a result of tunnelling events when transitions occur between topologically different butphysically identical vacuum winding states. These new terms give an extra contribution to theCasimir pressure. The infrared physics in the system can be described in terms of the topologicalauxiliary non-propagating fields a i ( k ) governed by Chern-Simons -like action. The system can bestudied in terms of these auxiliary fields precisely in the same way as a topological insulator can beanalyzed in terms of Berry’s connection A i ( k ). We also argue that the Maxwell vacuum defined ona small 4-torus behaves very much in the same way as a topological insulator with θ = 0. PACS numbers: 11.15.-q, 11.15.Kc, 11.15.Tk
I. INTRODUCTION. MOTIVATION.
The main motivation for present studies is as fol-lows. It has been recently argued [1, 2] that if the freeMaxwell theory (without any interactions with chargedparticles) is defined on a small compact manifold thansome novel terms in the partition function will emerge.These terms are not related to the propagating photonswith two transverse physical polarizations, which are re-sponsible for the conventional Casimir effect. Rather,these novel terms occur as a result of tunnelling eventsbetween topologically different but physically identicalstates. These states play no role when the system isdefined in Minkowski space-time R , . But these statesbecome important when the system is defined on a finitecompact manifold such as torus T .In particular, it has been explicitly shown in [1, 2] thatthese novel terms lead to a fundamentally new contri-butions to the Casimir vacuum pressure, which can notbe expressed in terms of conventional propagating phys-ical degrees of freedom. Instead, the new vacuum contri-butions appear as a result of tunnelling events betweendifferent topological sectors | k i . Mathematically, thesesectors emerge as a result of non-triviality of the funda-mental group π [ U (1)] ∼ = Z when the system is definedon a torus.The crucial for the present studies observation is as fol-lows. While the Maxwell Electrodynamics is the theoryof massless particles (photons), the topological portionof the system decouples from dynamics of these masslesspropagating photons. Indeed, as we discuss below, the to-tal partition function Z can be represented as a product Z = Z × Z top . The conventional partition function Z describing physical photons is not sensitive to the topo-logical sectors | k i of the system which itself is describedby Z top . The topological portion of the partition func-tion Z top behaves very much as topological quantum fieldtheory (TQFT) as argued in [2]. Furthermore, it demon-strates many features of topologically ordered systems,which were initially introduced in context of condensedmatter (CM) systems, see original papers [3–7] and re- cent reviews [8–12].In particular, Z top demonstrates the degeneracy of thesystem which can not be described in terms of any localoperators. Instead, such a degeneracy can be formulatedin terms of some non-local operators [2]. Furthermore,our system exhibits some universal sub-leading correc-tions to the thermodynamical entropy which can not beexpressed in terms of propagating photons with two phys-ical polarizations. Instead, the corresponding universalcontribution to the entropy is expressed in terms of the“instantons” describing the tunnelling events betweentopologically different but physically identical topolog-ical sectors | k i .As a result of these similarities, the key question ad-dressed in the present work is as follows. It has beenknown for sometime [4–12] that some key features oftopologically ordered systems can be formulated in termsof the so-called Berry’s connection in momentum space.Does a similar description exist for the Maxwell vacuumdefined on a compact manifold?To address this question we formulate the topologi-cal features of the system in terms of an auxiliary fields.Such a formulation exhibits a close mathematical simi-larity between the auxiliary topological field describingthe Maxwell vacuum state and the Berry’s connection(which is emergent, not a fundamental field) in topolog-ically ordered CM systems. Such a similarity looks veryinstructive and suggestive, and further supports our ar-guments [2] that the ground state of the Maxwell theorydefined on a small compact manifold behaves as a TQFT.The structure of our presentation is as follows. In thenext section II, we review the relevant parts of the two di-mensional Maxwell “empty” theory which does not haveany physical propagating degrees of freedom. Still, itdemonstrates a number of very nontrivial topological fea-tures present in the system. In section III we generalizeour description for 4d Maxwell theory defined on fourtorus. In our main section IV we introduce the auxil-iary fields which effectively account for the topologicalsectors of the system. We study the behaviour of theseauxiliary fields in the far infrared (IR) at small k → II. MAXWELL THEORY IN TWODIMENSIONS AS TOPOLOGICAL QFT
The 2d Maxwell model has been solved numerous num-ber of times using very different techniques, see e.g.[13–15]. It is known that this is an “empty” theory in asense that it does not support any propagating degreesof freedom in the bulk of space-time. It is also knownthat this model can be treated as a conventional topo-logical quantum field theory (TQFT). In particular, thismodel can be formulated in terms of the so-called “BF”action involving no metric. Furthermore, this model ex-hibits many other features such as fractional edge ob-servables which are typical for TQFT, see e.g.[14]. Weemphasize on these properties of the 2d Maxwell theorybecause the topological portion of the partition function Z top in our description of 4d Maxwell system, given insection III, identically the same as the partition func-tion of 2d Maxwell system. Such a relation between thetwo different systems is a result of decoupling of physicalpropagating photons from the topological sectors in 4dsystem.Our goal here is to review this “empty” 2d Maxwelltheory with nontrivial dynamics of the topological sectorswhen conventional propagating degrees of freedom arenot supported by this system. A. Partition function and θ vacua in 2d Maxwelltheory We consider 2d Maxwell theory defined on the Eu-clidean torus S × S with lengths L and β respectively.In the Hamiltonian framework we choose a A = 0 gaugealong with ∂ A = 0. This implies that A ( t ) is the onlydynamical variable of the system with E = ˙ A . Thespectrum for θ vacua is well known [13] and it is given by E n ( θ ) = (cid:0) n + θ π (cid:1) e L , such that the correspondingpartition function takes the form Z ( V, θ ) = X n ∈ Z e − e V ( n + θ π ) , (1)where V = βL is the two-volume of the system. We want to reproduce (1) using a different approachbased on Euclidean path integral computations becauseit can be easily generalized to similar computations 4dMaxwell theory defined on 4 torus. Our goal here is tounderstand the physical meaning of (1) in terms of thepath integral computations.To proceed with path integral computations one con-siders the “instanton” configurations on two dimensionalEuclidean torus with total area V = Lβ described asfollows [15]: Z d x Q ( x ) = k, eE ( k ) = 2 πkV , (2)where Q = e π E is the topological charge density and k is the integer-valued topological charge in the 2d U (1)gauge theory, E ( x ) = ∂ A − ∂ A is the field strength.The action of this classical configuration is12 Z d xE = 2 π k e V . (3)This configuration corresponds to the topological charge k as defined by (2). The next step is to compute thepartition function defined as follows Z ( θ ) = X k ∈ Z Z D A ( k ) e − R d xE + R d xL θ , (4)where θ is standard theta parameter which defines the | θ i ground state and which enters the action with topologicaldensity operator L θ = iθ Z d x Q ( x ) = iθ e π Z d x E ( x ) . (5)All integrals in this partition function are gaussian andcan be easily evaluated using the technique developed in[15]. The result is Z ( V, θ ) = r πe V X k ∈ Z e − π k e V + ikθ , (6)where the expression in the exponent represents the clas-sical instanton configurations with action (3) and topo-logical charge (2), while the factor in front is due to thefluctuations, see [1, 2] with some technical details andrelevant references. While expressions (1) and (6) lookdifferently, they are actually identically the same, as thePoisson summation formula states: Z ( θ ) = X n ∈ Z e − e V ( n + θ π ) = r πe V X k ∈ Z e − π k e V + ikθ . (7)Therefore, we reproduce the original expression (1) usingthe path integral approach.The crucial observation for our present study is thatthis naively “empty” theory which has no physical prop-agating degrees of freedom, nevertheless shows some verynontrivial features of the ground state related to thetopological properties of the theory. These new proper-ties are formulated in terms of different topological vac-uum sectors of the system | k i which have identical phys-ical properties as they connected to each other by largegauge transformation operator T commuting with theHamiltonian [ T , H ] = 0. As explained in details in [1, 2]the corresponding dynamics of this “empty” theory rep-resented by partition function (7) should be interpretedas a result of tunnelling events between these “degener-ate” winding | k i states which correspond to one and thesame physical state.It is known that this model can be treated as TQFT,e.g. supports edge observables which may assume thefractional values, and shows many other features whichare typical for a TQFT, see [14] and references therein.The presence of the topological features of the modelcan be easily understood from observation that entiredynamics of the system is due to the transitions betweenthe topological sectors which themselves are determinedby the behaviour of surface integrals at infinity H A µ dx µ .These sectors are classified by integer numbers and theyare not sensitive to specific details of the system suchas geometrical shape of the system. Therefore, it is notreally a surprise that the system is not sensitive to spe-cific geometrical details, and can be treated as TQFT.The simplest way to analyze the corresponding topologi-cal features of the system is to introduce the topologicalsusceptibility χ and study its property, see next subsec-tion. B. Topological susceptibility
The topological susceptibility χ is defined as follows, χ ≡ lim k → Z d x e ikx h T Q ( x ) Q (0) i , (8)where Q is topological charge density operator normal-ized according to eq.(2). The χ measures response ofthe free energy to the introduction of a source term de-fined by eq. (5). The computations of χ in this simple“empty” model can be easily carried out as the partitionfunction Z ( θ ) defined by (4) is known exactly (7). Tocompute χ we should simply differentiate the partitionfunction twice with respect to θ . It leads to the followingwell known expression for χ which is finite in the infinitevolume limit [2, 15, 16] χ ( V → ∞ ) = − V · ∂ ln Z ( θ ) ∂θ | θ =0 = e π . (9)A typical value of the topological charge k which satu-rates the topological susceptibility χ in the large volumelimit is very large, k ∼ √ e V ≫ h Q ( x ) Q (0) i = e π δ ( x ) . (10) It represents the non-dispersive contact term which cannot be related to any propagating degrees of freedom. Inthis simplest case of the 2d Maxwell system this commentis quite obvious as 2d Maxwell theory does not supportany propagating degrees of freedom. The δ ( x ) functionin (10) should be understood as total divergence relatedto the infrared (IR) physics, rather than to ultraviolet(UV) behaviour. Indeed, χ = e π Z δ ( x ) d x = e π Z d x ∂ µ (cid:18) x µ πx (cid:19) = e π I S →∞ d l µ ǫ µν (cid:16) x ν πx (cid:17) = e π . (11)In other words, the non-dispersive contact term (10) isdetermined by IR physics at arbitrary large distancesrather than UV physics which can be erroneously as-sumed to be a source of δ ( x ) behaviour in (10). Thecomputations of this contact term in terms of the delo-calized instantons (2) explicitly show that all observablesin this system are originated from the IR physics.One should also remark that the same contact term (9)and its local expression (10) can be also computed usingthe auxiliary ghost field, the so-called Kogut-Susskind(KS) ghost, as it has been originally done in ref. [17],see also [2, 16] for relevant discussions in the present con-text. This description in terms of the KS ghost implicitlytakes into account the presence of topological sectors inthe system. The same property is explicitly reflected bysummation over topological sectors k ∈ Z in direct com-putations (4,6) without introducing any auxiliary fields. III. TOPOLOGICAL PARTITION FUNCTIONIN 4D
Our goal here is to analyze the Maxwell system on aEuclidean 4-torus with sizes L × L × L × β in therespective directions. It provides the infrared (IR) reg-ularization of the system. This IR regularization playsa key role in proper treatment of the topological termswhich are related to tunnelling events between topolog-ically distinct but physically identical states. First, wewant to review the previously known results on the vac-uum structure of this system. As the second step, wewant to reproduce these known results on Maxwell vac-uum state using a different technique based on the aux-iliary fields to be developed in next section IV. As weargue in section IV C precisely these auxiliary topologi-cal fields have exactly the same mathematical propertiesas emergent Berry’s connection in topologically orderedCM systems. A. Construction
We follow [1, 2] in our construction of the partitionfunction Z top where it was employed for computation ofthe corrections to the Casimir effect due to these noveltype of topological fluctuations. The crucial point is thatwe impose the periodic boundary conditions on gauge A µ field up to a large gauge transformation. In what followswe simplify our analysis by considering a clear case withwinding topological sectors | k i in the z-direction only.The classical configuration in Euclidean space which de-scribes the corresponding tunnelling transitions can berepresented as follows: ~B top = ~ ∇ × ~A top = (cid:18) , , πkeL L (cid:19) , (12)Φ = e Z dx dx B z top = 2 πk in close analogy with the 2d case (2).The Euclidean action of the system is quadratic andhas the following form12 Z d x (cid:26) ~E + (cid:16) ~B + ~B top (cid:17) (cid:27) , (13)where ~E and ~B are the dynamical quantum fluctuationsof the gauge field. The key point is that the classicaltopological portion of the action decouples from quantumfluctuations, such that the quantum fluctuations do notdepend on topological sector k and can be computed intopologically trivial sector k = 0. Indeed, the cross term Z d x ~B · ~B top = 2 πkeL L Z d x B z = 0 (14)vanishes because the magnetic portion of quantum fluc-tuations in the z -direction, represented by B z = ∂ x A y − ∂ y A x , is a periodic function as ~A is periodic over the do-main of integration. This technical remark in fact greatlysimplifies our analysis as the contribution of the physi-cal propagating photons is not sensitive to the topolog-ical sectors k . This is, of course, a specific feature ofquadratic action (13), in contrast with non-abelian andnon-linear gauge field theories where quantum fluctua-tions of course depend on topological k sectors. The au-thors of ref. [18] arrived to the same conclusion (on de-coupling of the topological terms from conventional fluc-tuating photons with non-zero momentum), though in adifferent context of topological insulators in the presenceof the θ = π term.The classical action for configuration (12) takes theform 12 Z d x ~B = 2 π k βL e L L (15)To simplify our analysis further in computing Z top weconsider a geometry where L , L ≫ L , β similar to con-struction relevant for the Casimir effect [1, 2]. In thiscase our system is closely related to 2d Maxwell theoryby dimensional reduction: taking a slice of the 4d sys-tem in the xy -plane will yield precisely the topologicalfeatures of the 2d torus considered in section II. Further-more, with this geometry our simplification (12) when we consider exclusively the magnetic fluxes in z directionis justified as the corresponding classical action (15) as-sumes a minimal possible values. With this assumptionwe can consider very small temperature, but still we cannot take a formal limit β → ∞ in our final expressionsas a result of our technical constraints in the system.With these additional simplifications the topologicalpartition function becomes [1, 2]: Z top = r πβL e L L X k ∈ Z e − π k βL e L L = √ πτ X k ∈ Z e − π τk , (16)where we introduced the dimensionless parameter τ ≡ βL /e L L . (17)Formula (16) is essentially the dimensionally reduced ex-pression for the topological partition function (6) for 2dMaxwell theory analyzed in section II. One should notethat the normalization factor √ πτ which appears in eq.(16) does not depend on topological sector k , and essen-tially it represents our convention of the normalization Z top → L L → ∞ which corresponds toa convenient set up for the Casimir -type experiments asdiscussed in [1, 2]. B. External magnetic field
In this section we want to generalize our results for theEuclidean Maxwell system in the presence of the exter-nal magnetic field. Normally, in the conventional quan-tization of electromagnetic fields in infinite Minkowskispace, there is no direct coupling between fluctuating vac-uum photons and an external magnetic field as a conse-quence of linearity of the Maxwell system. The couplingwith fermions generates a negligible effect ∼ α B ext /m e as the non-linear Euler-Heisenberg Effective Lagrangiansuggests, see [1] for the details and numerical estimates.The interaction of the external magnetic field with topo-logical fluctuations (12), in contrast with coupling withconventional photons, will lead to the effects of order ofunity as a result of interference of the external magneticfield with topological fluxes k .The corresponding partition function can be easily con-structed for external magnetic field B ext z pointing along z direction, as the crucial technical element on decou-pling of the background fields from quantum fluctuationsassumes the same form (14). In other words, the physi-cal propagating photons with non-vanishing momenta arenot sensitive to the topological k sectors, nor to the ex-ternal uniform magnetic field, similar to our discussionsafter (14).The classical action for configuration in the presence ofthe uniform external magnetic field B ext z therefore takesthe form12 Z d x (cid:16) ~B ext + ~B top (cid:17) = π τ (cid:18) k + θ eff π (cid:19) (18)where τ is defined by (17) and the effective theta pa-rameter θ eff ≡ eL L B z ext is expressed in terms of theoriginal external magnetic field B z ext . Therefore, the par-tition function in the presence of the uniform magneticfield can be easily reconstructed from (16), and it is givenby [1, 2] Z top ( τ, θ eff ) = √ πτ X k ∈ Z exp " − π τ (cid:18) k + θ eff π (cid:19) . (19)This system in what follows will be referred as the topo-logical vacuum ( T V ) because the propagating degrees offreedom, the photons with two transverse polarizations,completely decouple from Z top ( τ, θ eff ).The dual representation for the partition function isobtained by applying the Poisson summation formula (7)such that (19) becomes Z top ( τ, θ eff ) = X n ∈ Z exp (cid:20) − n τ + in · θ eff (cid:21) . (20)Formula (20) justifies our notation for the effective thetaparameter θ eff as it enters the partition function in com-bination with integer number n . One should emphasizethat integer number n in the dual representation (20)is not the integer magnetic flux k defined by eq. (12)which enters the original partition function (16). Fur-thermore, the θ eff parameter which enters (19, 20) is nota fundamental θ parameter which is normally introducedinto the Lagrangian in front of ~E · ~B operator. Rather,this parameter θ eff should be understood as an effectiveparameter representing the construction of the θ eff statefor each slice in four dimensional system. In fact, thereare three such θ eff parameters representing different slicesand corresponding external magnetic fluxes. There aresimilar three θ i parameters representing the external elec-tric fluxes as discussed in [2], such that total number of θ parameters classifying the system equals six, in agree-ment with total number of hyperplanes in four dimen-sions. IV. BERRY CONNECTION
The main goal of this section is to argue that our
T V -configuration represents a simplest version of a topolog-ically ordered phase very similar to CM systems [4–12].We want to reformulate the topological features of thesystem (analyzed in section III) in terms of the Berry’sconnection and Berry curvature normally computed inmomentum space in CM literature. Such a deep relationbetween the two very different descriptions will demon-strate once again that the ground state for the Maxwelltheory defined on a compact manifold exhibits all thefeatures which are normally attributed to a topologicallyordered system. We make this relation much more pre-cise by introducing the auxiliary topological fields whichcan be identified with Berry’s connection. With such an interpretation the complex phase in the dual representa-tion (20) can be thought as the Berry’s phase which isknown to emerge in many quantum systems.We start our study in section IV A by reviewing thewell-known CM results on the Berry’s connection. In sec-tion IV B we describe the ground state of the two dimen-sional Maxwell theory by using the auxiliary topologicalfields. We observe a deep mathematical similarity be-tween the Berry’s connection computed for CM systems(including the monopole-type behaviour in momentum k space) and the corresponding formulae computed for theground state in the Maxwell theory in terms of the aux-iliary topological fields. We generalize the correspondingconstruction to four dimensional Maxwell system definedon a four -torus in section IV C. A. Berry phase in CM systems
In this subsection we review the computations of theBerry connection in some CM systems. In context ofthe topological insulators and quantum Hall systems thecorresponding studies have been carried out in two, threeand four dimensions [4–12], see also [2, 19] with relateddiscussions of the ground state in 2d Maxwell theory .In the simplest D=1 case the expression for the Berry’sphase (which is the accumulated geometric phase of theband electrons under the process when the winding of thegauge field is increased by one unit) can be computed asfollows[18]. In the physical A = 0 gauge it correspondsto a slow variation of gauge filed eA from πnL to π ( n +1) L where L is the size of a torus along x direction. Therelevant formula is given by [18] φ Berry = i Z dA h Ψ θ | ∂∂A | Ψ θ i , (21)where | Ψ θ i is the full wave function of the system whichcan be expressed in terms of single particle wave func-tions. One can explicitly demonstrate [18] that φ Berry = − πP with P being the polarization of the system suchthat θ is shifted as follows θ → ( θ − πP ). The key ob-servation in this computation is that the integration overslow varying gauge fields in eq. (21) is reduced to in-tegration over allowed momentum k covering the wholeBrillouin zone (BZ), i.e. φ Berry = i Z BZ dk h Ψ θ | ∂∂k | Ψ θ i ≡ Z BZ dk A ( k ) , (22) Not to be confused with conventional CM notations, where it isa customary to count the spatial number of dimensions, ratherthan total number of dimensions. For our 2d system this conven-tion corresponds to (D + 1) Maxwell theory with D = 1. Similarstudies have been carried out for topological insulators for D=1and D=3, see e.g. [18] with many references on the original re-sults. For D=2 the corresponding computations of the Berry’sconnection for the quantum Hall systems have been reviewed in[11]. where A ( k ) is the so-called Berry’s connection in the mo-mentum space. A simple technical explanation of this keytechnical step (related to the change of variables) is thatthe large gauge transformation formulated in terms of A can be expressed in terms of a shift of the momentum k when the system returns to the physically identical (buttopologically different) state.Similar computations can be also carried out for inte-ger quantum Hall system for D=2, in which case thecorresponding formula for the Berry’s connection andBerry’s curvature takes the form, see e.g. [11]: A j ( k ) = τ ǫ ij k i k , B ( k ) = τ δ ( k ) , (23)where τ = ± ǫ ( k ) ∼ | k | . One can identifythe behaviour (23) with magnetic monopoles in momen-tum space with half-integer magnetic charges. As weshall see below in section IV B a very similar structurealso emerges in description of the ground state of the2d Maxwell system, when the auxiliary topological fieldsplay the role of the Berry’s connection (23).One should emphasize that in CM literature the corre-sponding A j ( k ) fields are the emergent gauge fields. Thereal source for these emergent gauge configurations is thestrongly coupled coherent superposition of the physicalelectrons. In contrast, in our case a formula to be derivedbelow (and which mathematically identical to eq. (23))will arise from the topologically non-trivial gauge config-urations of the underlying fundamental gauge theory. Inother words, in our case the formula similar to (23) willemerge as a result of the topologically non-trivial vacuumgauge configurations which are present in the system ir-respectively to existence of the fermions.In the following subsection IV B we reformulate theknown results about the ground θ state in 2d Maxwellsystem using the topological auxiliary (non-propagating)fields. The corresponding technique, as we shall see belowin section IV C, can be easily generalized to four dimen-sional Maxwell system, which is the main subject of thepresent work. B. Auxiliary topological fields in 2d Maxwelltheory
We wish to derive the topological action for theMaxwell system in 2d by using a standard conventionaltechnique exploited e.g. in [7] for the Higgs model inCM context or in [20] for the so-called weakly coupled“deformed QCD”. We shall reproduce below the well-known results for this “empty” 2d system including anon-vanishing expression for the topological susceptibil-ity (9,10) using the corresponding auxiliary fields in mo-mentum space. It turns out that the corresponding con-nection and curvature computed using these auxiliaryfields play the same role as the Berry’s connection and Berry’s curvature play in CM systems. To be more pre-cise, the unique topological features of the auxiliary fieldis precisely the key element which allows to representthe accumulated geometric phase in terms of the aux-iliary field sensitive to the boundary conditions. Anexplicit demonstration of such a relation between theBerry’s phase and auxiliary topological fields is preciselythe main subject of this section.Our starting point is to insert the delta function intothe path integral with the field b ( x ) acting as a Lagrangemultiplier δ h Q ( x ) − e π ǫ jk ∂ j a k ( x ) i ∼ (24) Z D [ b ] e i R d x b ( x ) · [ Q ( x ) − e π ǫ jk ∂ j a k ( x ) ]where Q ( x ) = e π E ( x ) in this formula is the topologicalcharge density operator. It will be treated as the origi-nal expression for the field operator entering the action(4) with topological term (5). At the same time a k ( x ) istreated as a slow-varying external source effectively de-scribing the large distance physics for a given instantonconfiguration. The insertion (24) of the delta functionassumes that the path integral computations must in-clude all the classical k-instanton configurations (2),(3)along with quantum fluctuations surrounding them. Inother words, we treat Q ( x ) as a fast degree of freedom,while a k ( x ) are considered as slow degrees of freedomrepresenting an external background field.One should remark here that the corresponding formalmanipulation is not a mathematically rigorous procedureas a k ( x ) must be singular somewhere to support non-vanishing topological charges in the system . The pres-ence of such singularity is very similar to emergent singu-larities in description of the Berry’s connection, Dirac’sstring, or the Aharonov Bohm potential. It is not a goalof the present work to search for a more rigorous mathe-matical tools for corresponding problems. The most im-portant argument for us that our procedure representedby eq.(24) is correct is the fact that the topological sus-ceptibility (29), (30) as well as the expectation value ofthe electric field (31), (32) are precisely reproduced whencomputations are performed with our formal approachutilizing the auxiliary topological fields.Another point worth to be mentioned is as follows. Aswe stated above, the auxiliary field a k ( x ) is treated as aslow field, while Q ( x ) is treated as a fast degree of free-dom. At the same time, formally, these fields are propor-tional to each other Q ( x ) ∼ ǫ jk ∂ j a k ( x ) according to (24),and therefore, it is not obvious how these fields could betreated so differently. The answer lies in the observationthat our auxiliary fields a k ( x ) , b z ( x ) are non-dynamicalfields, have no kinetic terms, and do not propagate, incontrast with conventional gauge fields. Formally, these I am thankful to anonymous referee for pointing this out. fields do not have their conjugate momenta, as they areauxiliary non-dynamical fields of the system.The simplest way to understand this construction isthrough analogy with well known and well understoodmodel in particles physics, the so-called Nambu-Jona-Lasino model. In this case an auxiliary σ field withoutkinetic term is introduced into the system, analogous to(24). The σ ∼ < ¯ ψψ > field is treated as a slow field andin mean field approximation represents the chiral conden-sate of the fermi-fields. Our auxiliary fields a k ( x ) , b z ( x )should be understood exactly in the same way as σ fieldis is understood in Nambu-Jona-Lasino model.Now we are coming back to our proposed formula (24).Our task now is to integrate out the original fast “instan-tons” and describe the large distance physics in termsof slow varying fields b ( x ) , a k ( x ) in form of the effectiveaction S top [ b, a k ] formulated in terms of slow auxiliaryfields b ( x ) , a k ( x ). We use conventional well establishedprocedure of summation over k-instantons reviewed insection II with final result (6). The only new elementin comparison with the previous computations is thatthe fast degrees of freedom must be integrated out inthe presence of the new slow varying background fields b ( x ) , a k ( x ) which appear in eq. (24). Fortunately, thecomputations can be easily performed for such externalsources. Indeed, one should notice that the backgroundfield b ( x ) enters eq. (24) exactly in the same manneras external parameter θ enters (5). Therefore, assumingthat b ( x ) , a k ( x ) are slow varying background fields wearrive to the following expression for the partition func-tion: Z top = Z D [ b ] D [ a ] e − e π · R d x [ θ + b ( x )] − S top , (25)where b ( x ) represents the slow varying background aux-iliary field which is assumed to lie in the lowest n = 0branch, | b ( x ) | < π . Correspondingly, in formula (25) wekept only the asymptotically leading term in expansion(1) with n = 0 in large volume limit, ( e V ) ≫
1. Thetopological term S top [ b, a k ] in eq. (25) reads S top [ b, a k ] = i e π Z d x (cid:2) b ( x ) ǫ jk ∂ j a k ( x ) (cid:3) . (26)Our goal now is to consider a simplest application ofthe effective low energy topological action (25), (26) wejust derived. We want to reproduce the known expressionfor the topological susceptibility (9),(10) by integratingout the b and a k fields using low energy effective de-scription (25), (26), rather than an explicit summationover the instantons, which was employed in the originalderivation (9),(10). The agreement between of two dras-tically different approaches will give us a confidence thatour formal manipulations with the auxiliary fields is acorrect and self-consistent procedure. With this confi-dence, as a next step, we will study the behaviour of theauxiliary topological fields in the IR, which correspondsto k → k → a k ( x ) governedby the action (25), (26) with emergent Berry’s connection A j ( k ) given by eq.(23).To proceed with this task we compute the topologicalsusceptibility at θ = 0 as follows, h Q ( x ) Q ( ) i = 1 Z Z D [ b ] D [ a ] e − S tot [ b,a k ] e π · h ǫ jk ∂ j a k ( x ) , ǫ j ′ k ′ ∂ j ′ a k ′ ( ) i , (27)where S tot [ b, a k ] determines the dynamics of auxiliary b and a k fields, and it is given by S tot [ b, a k ] = Z d x (cid:20) e π b ( x ) + i e π b ( x ) ǫ jk ∂ j a k ( x ) (cid:21) . (28)The obtained Gaussian integral (27) over R D [ b ] can beexplicitly executed, and we are left with the followingintegral over R D [ a ] h Q ( x ) Q ( ) i = 1 Z Z D [ a ] e − R d x [ ǫ jk ∂ j a k ( x ) ] · e π h ǫ jk ∂ j a k ( x ) , ǫ j ′ k ′ ∂ j ′ a k ′ ( ) i . (29)The integral (29) is also gaussian and can be explicitlyevaluated with the following final result h Q ( x ) Q ( ) i = e π δ ( x ) , (30) Z d x h Q ( x ) Q ( ) i = e π . Few comments are in order. First, formula (30) pre-cisely reproduces our previous expression (9),(10) de-rived by explicit summation over fluxes-instantons, andwithout even mentioning any auxiliary topological fields b ( x ) , a k ( x ). It obviously demonstrates a self-consistencyof our formal manipulations with auxiliary topologicalfields. As we shall see below, the reformulation of thesystem in terms of the auxiliary topological fields is ex-tremely useful for studying some other (very non-trivial)topological features of the gauge system.Secondly, the expression (30) for the topological sus-ceptibility represents the contact non-dispersive termwhich can not be associated with any physical propagat-ing degrees of freedom as we discussed in section II B. Thenature of this contact term can be understood in terms ofthe tunnelling transitions between topologically differentbut physically identical | k i states. As we already men-tioned in section II B the same contact term can be alsounderstood in terms of the propagating Kogut-Susskindghost[17], which effectively describes the tunnelling tran-sitions in terms of an auxiliary Kogut-Susskind ghostwhich however, does not belong to the physical Hilbertspace, see [16] for the details in given context.To proceed with our task on establishing the relationbetween the topological auxiliary fields and the Berry’sconnection we want compute the expectation value forthe topological charge density operator h Q i ≡ h e π E i atnon-vanishing θ = 0. The corresponding computationscan be easily performed using the same technique de-scribed above. The only new element which occurs isnecessity to compute the path integral at non-vanishing θ as the entire final result will be proportional to θ , seeeq. (31) below. However, the presence of θ in the effec-tive action does not produce any technical difficulties asthe emergent path integral remains to be the Gaussianintegral determined by the quadratic action (25) evenfor non-vanishing θ . The corresponding computation at θ = 0 can be easily executed by a conventional shift ofvariables b ( x ) , a k ( x ). The result is:lim k → Z d x e i kx h Q ( x ) i (31)= lim k → (cid:16) e π (cid:17) Z d x e i kx h ǫ ij ∂ i a j ( x ) i = ie θ π V, where V is the total volume of the system playing the roleof the IR regulator in all computations in 2d Maxwell sys-tem as reviewed in section II. The obtained formula (31)reproduces the well-known result that a non-vanishing θ corresponds to non-vanishing background electric field E ≡ ǫ ij ∂ i a j in the system [21], h E i Eucl . = ieθ π , h E i Mink . = eθ π , (32)see also [2] with some comments in the given context.The non-vanishing expectation value of the gauge in-variant operator (31) is highly non-trivial phenomenon asthe operator Q ( x ) itself is a total divergence. Naively, allcorrelation functions with operator Q ( x ), including theexpectation value of h Q ( x ) i itself must vanish in k → | k i sectors .Such a strong IR sensitivity implies that the Fouriertransform of the auxiliary topological field a j saturating Furthermore, one can argue that the topological auxiliary field a i ( k ) introduced above can be expressed in terms of the Kogut-Susskind ghost. Apparently, such a relation is very generic fea-ture of many gauge theories. In fact, an analogous relation canbe explicitly worked out in four dimensional gauge theory, in theso-called weakly coupled “deformed QCD” where the auxiliarytopological fields, similar a i ( x ) , b ( x ) fields from (28) are relatedto the Veneziano ghost [20]. The Veneziano ghost was postulatedin QCD long ago [22] with the sole purpose to saturate the non- the expectation value (31) has the singular behaviour atsmall momentum k → a j ( k → ≡ V Z d x e i kx a j ( x ) → (cid:18) eθ π (cid:19) ǫ ij k i k , (33)in spite of the fact that the system does not supportany physical massless propagating degrees of freedom,which erroneously can be associated with the pole (33).The source of this pole is obviously related to the sametopological instanton-like long ranged configurations (2)saturating the contact term in the topological suscepti-bility (11). The singular behaviour (33) which simplyrepresents a non-vanishing expectation value (31), (32),obviously implies that the integral in momentum spacearound k ∼ k → e I | k |→ a j ( k ) dk j = 1 e Z d k (cid:2) ǫ ij ∂ k i a j ( k ) (cid:3) = θ Z d k ∂ k i (cid:18) k i π k (cid:19) = θ Z d k δ ( k ) = θ, (34)which essentially represents the same well-known state-ment about non-vanishing gauge invariant expectationvalue (31), (32), but written in the different terms involv-ing the auxiliary topological fields in momentum space.From (34) one can easily recognize that the auxil-iary field e a j ( k ) in momentum space strongly resemblesthe Berry’s connection (23), while e ǫ ij ∂ k i a j ( k ) can bethought as the Berry’s curvature discussed previously inCM physics, see e.g. [11] for review. The fundamentaldifference between analysis of our system and the com-putations of the Berry phase in CM literature is that theBerry connection (23) in CM systems is a collective phe-nomenon with accumulation of the geometric phase ofthe band electrons. It is represented, as a matter of con-venience rather than necessity, in terms of the emergentgauge field A i ( k ). In contrast, in our case, the topologi-cal fields a i ( k ) represent some fundamental (though aux-iliary, non-propagating) fields describing the ground stateof the underlying gauge theory. These fields are presentin the system even without any matter fields. The topo-logical features of the auxiliary fields in our case emergeas a result of the summation of the topological sectors inpath integral formulation rather than a result of a com-plex interaction of the band electrons in CM systems.Nevertheless, as we observed above, there is verystrong mathematical similarity, between these two, phys-ically very different, entities. These similarities, in par-ticular, include the following features: while a i ( k ) and A i ( k ) are gauge-dependent objects, the correspondingintegrals (22) and (34) are gauge invariant (modulo 2 π ) dispersive (contact) term in topological susceptibility, similar instructure to eq. (30). As it is known this contact term plays thekey role in the resolution of the so-called U (1) A problem in QCD[22, 23]. observables describing the same property related to thepolarization. The 2 π periodicity for all observables inboth systems also has very simple physical explanation.For our system the 2 π periodicity follows from the parti-tion function (1), (7), while in CM context [11, 18] the 2 π periodicity corresponds to the adiabatic process when themany body wave function returns to its physically identi-cal (but topologically different) state. Furthermore, themain features of the systems are formulated in terms ofglobal rather than local behaviour, as formulae (22) and(34) suggest. One should comment here that an explicitcomputations of the Berry’s connection for a specific CMsystem very often requires some tedious microscopicallocal computations, though the final result is in fact de-scribes the global behaviour of the system, not sensitiveto any local characteristics.We conclude this section with the following generalcomment. We have not produced any new physical re-sults in this section as the relevant questions in 2d QEDsuch as the expectation value of the electric field at non-zero θ (represented by eqs. (31), (32)), or non-dispersive(contact) contribution to the topological susceptibility(30) have been computed long time ago . Our contribu-tion in this section is much more modest. We reproducedthese known results by using a different technique: weexpressed the relevant correlation functions in terms ofthe auxiliary topological fields a i ( k ). We established thephysical meaning of these fields, and argued that theseauxiliary objects play the same role as Berry’s connection A i ( k ) in CM systems.As we shall discuss below, the technical tools developedand tested in this subsection (by reproducing the knownresults) will be very useful in our study of a similar phe-nomena in physically relevant four dimensional Maxwelltheory formulated on the torus. This mathematical sim-ilarity occurs as a result of dimensional reduction (to beused below) which essentially translates the correspond-ing 4d problems into 2d analysis developed in presentsection. C. Auxiliary topological fields in 4d Maxwellsystem
We wish to derive the topological action for the 4dMaxwell system by using the same technique exploitedin previous subsection IV B. Our starting point is to in-sert the delta function, similar to eq. (24), into the pathintegral with the field b z ( x ) acting as a Lagrange multi-plier δ (cid:2) B z ( x ) − ǫ zjk ∂ j a k ( x ) (cid:3) ∼ (35) Z D [ b z ] e iL β R d x b z ( x ) · [ B z ( x ) − ǫ zjk ∂ j a k ( x ) ] In particular, formula (30) can be derived using the Kogut-Susskind ghost formalism [17]. where B z ( x ) in this formula is treated as the original ex-pression for the field operator entering the action (13),including all classical k-instanton configurations (12,15)and quantum fluctuations surrounding these classicalconfigurations. In other words, we treat B z ( x ) as fastdegrees of freedom. At the same time a k ( x ) is treated asa slow-varying external source effectively describing thelarge distance physics for a given instanton configuration.Our task now is to integrate out the original fast “fluxes”(12,15) and describe the large distance physics in termsof slow varying fields b z ( x ) , a k ( x ) in form of the effectiveaction similar to (28) derived for 2d system. The phys-ical meaning of these formal manipulations is explainedin the previous section IV B after eq. (24), and we shallnot repeat it here.To proceed with computations, we use the same pro-cedure by summation over k-instantons as described insection III. The only new element in comparison with theprevious computations is that the fast degrees of freedommust be integrated out in the presence of the new slowvarying background fields b z ( x ) , a k ( x ) which appear ineq. (35). Fortunately, the computations can be easilyperformed if one notices that the background field b z ( x )enters eq. (35) exactly in the same manner as exter-nal magnetic field enters (19). Therefore, assuming that b z ( x ) , a k ( x ) are slow varying background fields we arriveto the following expression for the partition function forour T V system: Z top ( τ, θ eff ) = √ πτ X k ∈ Z Z D [ b z ] D [ a ] e − S − S top (36)where quadratic action S [ b z , a k ] is defined as S [ b z , a k ] = π τ Z T d x L L (cid:18) k + φ ( x ) + θ eff π (cid:19) , (37)while the topological term S top [ b z , a k ] in eq. (36) reads S top [ b z , a k ] = iL β Z T d x (cid:2) b z ( x ) ǫ zjk ∂ j a k ( x ) (cid:3) . (38)In formula (37) we rescale the slow varying backgroundauxiliary b z field such that φ ( x ) ≡ eL L b z ( x ). Param-eter θ eff ≡ eL L B ext z represents the external magneticfield while T represents the two torus defined on (1 , k → z direc-tion only. It is naturally to assume that a more gen-eral construction would include fluxes in all three di-rections which would lead to a generalization of action(38). Therefore, it is quite natural to expect that theaction in this case would assume a Chern-Simons likeform iβ R T d x (cid:2) ǫ ijk b i ( x ) ∂ j a k ( x ) (cid:3) which replaces (38). Asimilar structure in CM systems is known to describea topologically ordered phase. Therefore, it is not re-ally a surprise that we observed in [2] some signaturesof the topological order in the Maxwell system definedon a compact manifold. The emergence of the topologi-cal Chern-Simons action (38) further supports this basicclaim that the Maxwell system on a compact manifoldbelongs to a topologically ordered phase as the auxiliarytopological fields entering (38) play the same role as theBerry’s connection in topologically ordered CM systems.Now we can follow the same procedure which we testedfor 2d system in section IV B to compute the expectationvalue of the magnetic field at non-vanishing θ eff . Thecorresponding result is known [1]: it has been derived byusing conventional computation of the path integral bysummation over all “instanton-fluxes”. Our goal now isto reproduce this result by using the auxiliary topologicalfields governed by the action (38). We follow the sameprocedure as before and define the induced magnetic fieldin the system in the conventional way h B z ind ( τ, θ eff ) i = − βV ∂ ln Z top ( τ, θ eff ) ∂B ext (39)= 2 πeL L (cid:28) k + θ eff + φ ( x )2 π (cid:29) , where the last expectation value must be evaluated usingthe partition function (36). The corresponding Gaussianintegral over auxiliary φ ( x ) field can be easily executedwith the result h B z ind ( τ, θ eff ) i = lim k → Z d x L L e i kx (cid:10) − iǫ zjk ∂ j a k ( x ) (cid:11) , (40)where the corresponding expectation value h ... i should becomputed using the following partition function deter-mined by the action S tot [ a k ] (which includes both: thequadratic and topological terms), S tot [ a k ] = L β Z T d x (cid:0) ǫ zjk ∂ j a k ( x ) (cid:1) (41) − i (2 πk + θ eff ) L βeL L Z T d x (cid:0) ǫ zjk ∂ j a k ( x ) (cid:1) . The path integral integral (40) is gaussian, and can beexecuted by a conventional shift of variables in the action S tot [ a k ] defined by (41) (cid:0) ǫ zjk ∂ j a ′ k ( x ) (cid:1) = (cid:0) ǫ zjk ∂ j a k ( x ) (cid:1) − i (2 πk + θ eff ) eL L . (42)Exact evaluation of the gaussian path integral (40) withaction (41) leads to the following final result for h B z ind ih B z ind i = 2 πeL L √ πτ Z top X k ∈ Z (cid:18) k + θ eff π (cid:19) e − π τ (cid:16) k + θ eff2 π (cid:17) , (43) where the partition function Z top for our T V system inthis formula is determined by eq. (19). As expected,the expression (43) exactly reproduces the correspond-ing formula derived in ref[1] by explicit summation overfluxes-instantons. We reproduced the results of ref[1] us-ing drastically different technique as our computations(43) in this section are based on calculation of the pathintegral defined by the partition function (36) formulatedin terms of the auxiliary topological fields b z ( x ) , a i ( x ).Agreement between the two approaches obviously sup-ports the consistency of our formal manipulations withthe path integral and auxiliary fields.Important new point (which could not be seen withincomputational technique of ref[1]) is the expression (40)for the induced field h B z ind i in terms of the auxiliary ob-ject a i ( x ). As we shall see in a moment precisely thisconnection allows us to identify the auxiliary topologi-cal non-propagating field a i ( k ) in momentum space withthe Berry’s connection A i ( k ) from section IV A as bothentities have very similar properties.Before we proceed to establish such a connection, wewould like to make few comments. First, as one cansee from (43) the expression for h B z ind i accounts for thetotal field in the system, including the external field aswell as the induced field due to the interference of theexternal field with the topological fluxes (12). However,in the absence of the external field θ eff ≡ eL L B z ext =0 the contributions to the expectation value (43) fromthe fluxes with positive and negative signs cancel eachother, and h B z ind i vanishes. For θ eff = 0 the cancellationdoes not hold, and the field h B z ind i 6 = 0 will be obviouslyinduced.The effect must vanish when the tunnelling transitionsdue to the fluxes are suppressed at e → τ ≫
1. It is very instructive to see howit happens. The corresponding expression which is validfor τ ≫ θ eff ≪ h B z ind i = θ eff eL L " − πe − π τ sinh( πτ θ eff ) θ eff . (44)One can explicitly see from eq. (44) that the tunnellingeffects are suppressed in large τ limit, and magnetic fieldin this case in the system is entirely determined by ex-ternal source h B z ind i → B z ext at τ ≫
1, as expected.The key point for the present analysis is the expres-sion (40) for h B z ind i in terms of the auxiliary fields a k ( x ).This formula in all respects is very similar to expression(31) previously analyzed in 2d system. One can follow We note that k -independent numerical factor √ πτ enters bothequations: (19) and (43). This numerical factor simply repre-sents our normalization’s convention, and does not affect thecomputations of any expectation values, such as (43). Our nor-malization corresponds to the following behaviour of the topo-logical partition function: Z top → L L → ∞ .Such a convention corresponds to the geometry of the originalCasimir setup experiment, see [1] for the details. a k ( x ) can be thought as theBerry’s connection (similar to (33) from 2d analysis) withthe following singular behaviour at small k → a i ( k → ≡ L L Z d x π e i kx a i ( x ) (45)= h B z ind i ǫ zij k j π k ⇒ (cid:18) θ eff eL L (cid:19) ǫ zij k j π k , where in the last line we use the asymptotical behaviour(44) which is valid for large τ ≫ ǫ zij ∂ k i a j ( k ) ∼ δ ( k ) plays the same role as the Berry’s curvature inCM physics, see (23) and [11] for review. One shouldemphasize that these similarities in the IR behaviour intwo very different systems should not be considered asa pure mathematical curiosity. In fact, there is a verydeep physical reason why these two, naively unrelated,entities, must behave very similarly in the IR. Indeed, asit is known the Berry’s phase in CM systems effectivelydescribes the variation of the θ parameter θ → θ − πP as a result of coherent influence of strongly interactingfermions which polarize the system, i.e. P = ± /
2, seee.g. [18]. The auxiliary topological field a i ( x ) in our T V system with similar IR behaviour essentially describesthe same physics. To be more precise, the interferencebetween the external magnetic field and fluxes lead tothe magnetic polarization formulated in terms of a i ( x )fields, similar to the generation of polarization P = ± / A i ( x ). Our key observation is that the polarizationfeatures of the T V system in our case are represented byeqs. (40), (45). These equations play the same role asequations (22), (23) in CM systems.This close analogy (mathematical and physical), infact, may have some profound observational and experi-mental consequences as an electrically charged probe in-serted into our system characterized by θ eff would behavevery much in the same way as a probe inserted into a CMsystem characterized by non-vanishing Berry’s phase. Inother words, our T V system must demonstrate a num-ber of unusual features which are typical for topologicallyordered phases in CM systems. One of such properties,the degeneracy of the system, which can not be describedin terms of any local operator (but rather is character-ized by a non-local operator) has been already established[2]. It must be other interesting experimentally observ-able effects in 4d Maxwell theory, similar to a numberof profound effects which are known to occur in topologi-cally ordered phases in CM systems [4–12]. We leave thissubject for future studies.
V. CONCLUSION AND FUTURE DIRECTIONS
In this work we discussed a number of very unusualfeatures exhibited by the Maxwell theory formulated on a 4 torus, which was coined the topological vacuum (
T V ).All these features are originated from the topological por-tion of the partition function Z top ( τ, θ eff ) and can not beformulated in terms of conventional E & M propagatingphotons with two physical transverse polarizations. Indifferent words, all effects discussed in this paper have anon-dispersive nature.The computations of the present work along with pre-vious calculations of refs.[1, 2] imply that the extra en-ergy (and entropy), not associated with any physicalpropagating degrees of freedom, may emerge in the gaugesystems if some conditions are met. This fundamentallynew type of energy emerges as a result of dynamics ofpure gauge configurations at arbitrary large distances.This unique feature of the system when an extra energyis not related to any physical propagating degrees of free-dom was the main motivation for a proposal [16, 24, 25]that the vacuum energy of the Universe may have, infact, precisely such non-dispersive nature . This pro-posal when an extra energy can not be associated withany propagating particles should be contrasted with aconventional description when an extra vacuum energyis always associated with some new physical propagatingdegree of freedom, such as inflaton.The main motivation for the present studies is to testthese ideas (about fundamentally new type of vacuumenergy) using a simple quantum field theory (QFT) set-ting which nevertheless preserves the crucial element, thedegeneracy of the topological sectors, responsible for thisnovel type of energy. This simplest possible setting canbe realized in the Maxwell theory formulated on a 4 torus.Most importantly, the effect with this simplest settingcan be, in principle, tested in a tabletop experiment ifthe corresponding boundary conditions can be somehowimposed in a real physical systems. Otherwise, our con-struction should be considered as the simplest possible 4dQFT model when an extra vacuum energy is generated.The crucial point is that this extra vacuum energy cannot be associated with any physical propagating degreesof freedom, as argued in the present work.Essentially, the proposal [16, 24, 25] identifies the ob-served vacuum energy with the Casimir type energy,which however is originated not from dynamics of thephysical massless propagating degrees of freedom, butrather, from the dynamics of the topological sectors There are two instances in evolution of the universe when thevacuum energy plays a crucial role. First instance is identifiedwith the inflationary epoch when the Hubble constant H was al-most constant which corresponds to the de Sitter type behaviour a ( t ) ∼ exp( Ht ) with exponential growth of the size a ( t ) of theUniverse. The second instance when the vacuum energy playsa dominating role corresponds to the present epoch when thevacuum energy is identified with the so-called dark energy ρ DE which constitutes almost 70% of the critical density. In the pro-posal [16, 24, 25] the vacuum energy density can be estimated as ρ DE ∼ H Λ QCD ∼ (10 − eV) , which is amazingly close to theobserved value. b z ( x ) , a k ( x ) fields from (28), (38) and which effectivelydescribe the dynamics of the topological sectors in theexpanding background [25]. As we discussed in lengthin section IV C these auxiliary topological fields play thesame role as the Berry’s connection in CM systems. The b z ( x ) , a k ( x ) fields do not propagate, but they do con-tribute to the vacuum energy. It would be very excitingif this new type of the vacuum energy not associated withpropagating particles could be experimentally measuredin a laboratory, as we advocate in this work.Aside from testing the ideas on vacuum energy of theUniverse, the Maxwell system studied in the present workis interesting system on its own. Indeed, being a “free”Maxwell theory, it nevertheless shows a number of veryunusual features which are normally attributed to a CMsystem in a topologically ordered phase. In particular, itshows the degeneracy of the system which can not be de-tected by any local operators, but characterized by a non-local operator [2]. Furthermore, in the present work weargued that the auxiliary topological fields b z ( x ) , a k ( x )fields in 4d Maxwell system behave very much in thesame way as the Berry’s connection in CM systems. Morethan that, a charged probe particle inserted into our sys-tem would feel the topological features of the b z ( x ) , a k ( x )fields in the same way as a probe inserted into a CMsystem characterized by a nontrivial Berry’s connection A i ( x ).Therefore, it would be very exciting if one could find asystem where a charged probe inserted into our 4-torus(filled by vacuum) would behave similarly to a probe in-serted into a much more complicated CM system, wherethe corresponding nontrivial Berry’s connection is emer-gent as a result of a coherent many body physics. A simplest possible setup we can imagine is as follows.Normally, in condensed matter literature one considers ajunction between a conventional insulator ( I ) and topo-logical insulator ( T I ). One can also consider the
T I which is sandwiched between two conventional insula-tors, i.e. one can consider a system like
I − T I − I . Ourclaim essentially is that the
T I in this system can be re-placed by the
T V configuration considered in this work.In other words, one considers a system like
I − T V − I .Our claim is that this system would behave very much asthe
I − T I − I , because
T V behaves very much in thesame way as a
T I as advocated in this work . Thesesimilarities include such nontrivial features as the degen-eracy (characterized by a non-local operator), the Berry’sconnection and the presence of the effective θ eff state,among many others things. Therefore, it is naturally toexpect that while I − T I − I and
I − T V − I systemsare very different in composition, the behaviour of thesesystems will be very much the same in the IR. We leavethis exciting subject on possible applications of our
T V system, which we believe belongs to a topologically or-dered phase, for future investigations.
ACKNOWLEDGEMENTS
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