Aharonov-Bohm phases in a quantum LC circuit

# Aharonov-Bohm phases in a quantum LC circuit

ChunJun Cao Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA 91125, USA    Yuan Yao Department of Physics and Astronomy, University of British Columbia, Vancouver, B.C. V6T 1Z1, Canada    Ariel R. Zhitnitsky Department of Physics and Astronomy, University of British Columbia, Vancouver, B.C. V6T 1Z1, Canada
###### Abstract

We study novel types of contributions to the partition function of the Maxwell system defined on a small compact manifold. These contributions, often not addressed in the perturbative treatment with physical photons, emerge as a result of tunneling transitions between topologically distinct but physically identical vacuum winding states. These new terms give an extra contribution to the Casimir pressure, yet to be measured. We argue that this effect is highly sensitive to a small external electric field, which should be contrasted with the conventional Casimir effect where the vacuum photons are essentially unaffected by any external field. Furthermore, photons will be emitted from the vacuum in response to a time-dependent electric field, similar to the dynamical Casimir effect in which real particles are radiated from the vacuum due to the time-dependent boundary conditions. We also propose an experimental setup using a quantum LC circuit to detect this novel effect. We expect physical electric charges to appear on the capacitor plates when the system dimension is such that coherent Aharonov-Bohm phases can be maintained over macroscopically large distances.

###### pacs:
11.15.-q, 11.15.Kc, 11.15.Tk

## I Introduction. Motivation.

It has been recently argued [1, 2, 3, 4] that some novel terms in the partition function emerge when a pure Maxwell theory is defined on a small compact manifold. These terms are not related to the propagating photons with two transverse physical polarizations, which are responsible for the conventional Casimir effect (CE)[5]. Rather, they occur as a result of tunneling events between topologically different but physically identical topological sectors. While such contributions are trivial in Minkowski space-time , they become important when the system is defined on certain small compact manifolds. Without loss of generality, consider a manifold which has at least one non-trivial direct factor of the fundamental group, e.g., . The presence of the topological sectors , which play a key role in our discussions, arise precisely from the presence of such nontrivial mappings for the Maxwell gauge theory. The corresponding physically observable phenomenon has been termed the topological Casimir effect (TCE).

In particular, it has been explicitly shown in [1] that these novel terms in the topological portion of the partition function lead to a fundamentally new contribution to the Casimir vacuum pressure that appears as a result of tunneling events between topological sectors . Furthermore, displays many features of topologically ordered systems, which were initially introduced in the context of condensed matter systems (see recent reviews [6, 7, 8, 9, 10]): demonstrates the degeneracy of the system which can only be described in terms of non-local operators [2]; the infrared physics of the system can be studied in terms of non-propagating auxiliary topological fields [3], analogous to how a topologically ordered system can be analyzed in terms of the Berry’s connection (also an emergent rather than fundamental field), and the corresponding expectation value of the auxiliary topological field determines the phase of the system. Finally, one can show [4] that if the same system is placed in a time-dependent magnetic field , real photons will be emitted from the vacuum, similar to the so-called dynamical Casimir effect (DCE) [11, 12, 13]). The difference from the conventional DCE is that the dynamics of the vacuum in our system defined on a small compact manifold is not related to the fluctuations of the conventional degrees of freedom, the virtual photons. Rather, the radiation here arises from tunneling events between topologically different but physically identical sectors in a time-dependent background.

As we review in section II, the relevant vacuum fluctuations which saturate the topological portion of the partition function are formulated in terms of topologically nontrivial boundary conditions. Classical instantons formulated in Euclidean space-time satisfy the periodic boundary conditions up to a large gauge transformation and provide a topological magnetic instanton-flux in the -direction. These integer magnetic fluxes describe the tunneling transitions between physically identical but topologically distinct sectors. Precisely these field configurations generate an extra Casimir vacuum pressure in the system. What happens to this complicated vacuum structure when the system is placed in the background of a constant external magnetic field ? The answer is known [1]: the corresponding partition function as well as all observables, including the topological contribution to the Casimir pressure, are highly sensitive to small magnetic fields and demonstrate periodicity with respect to magnetic flux represented by the parameter where is the area of the system . This sensitivity to external magnetic field is a result of the quantum interference of the external field with the topological quantum fluctuations. Alternatively, one can see this as resulting from a small but non-trivial overlap between the conventional Fock states, constructed by perturbative expansions around each sector, and the true energy eigenstates of the theory, which are only attainable in a non-perturbative computation that takes the tunneling into account. This strong “quantum” sensitivity of the TCE should be contrasted with conventional Casimir forces which are practically unaltered by any external field due to the strong suppression (see [1] for the details).

The main goal of the present work is to study the dynamics of electric instanton-fluxes in contrast with the magnetic ones considered previously in [1, 2, 3, 4]. Placing metallic plates at the opposing ends in the -direction endows our system with the geometry of a small quantum capacitor. Periodic boundary conditions up to a large gauge transformation can be enforced by connecting the two plates of the capacitor with an external wire. Thus, the system forms a quantum LC circuit where the wire provides the inductance. Formal computation of the partition function with electric-type instantons is very similar to previous studies. Like their magnetic counterparts, these electric instantons also describe tunneling transitions between physically identical but topologically distinct sectors in Euclidean space-time. Both types of instanton-fluxes give rise to an extra repulsive pressure, the opposite of the conventional CE, and both exhibit oscillatory behavior in pressure, induced field, and susceptibility in response to an external field represented by an effective theta parameter, .

However, the behaviour of the system as well as the physical interpretation drastically changes when one considers the system placed in an external field: while the magnetic field in space-time with Euclidean or Minkowski signatures remains unaltered, the electric fluxes pick up an imaginary in the transition from Minkowski to Euclidean space-time. Most notably, whereas the magnetic oscillations have universal -periodicity for all system sizes, the periodicity in the electric system varies with the system size. Compared to the magnetic system, an advantage of the electric system is that the system size can be much more easily optimized to produce topological effects of order one.

The topological instanton configurations which describe the tuneling transitions are formulated in terms of periodic boundary conditions on the gauge field up to a large gauge transformation. These boundary conditions correspond to persistent fluctuating charges that reside on the capacitor plates. These charges can be interpreted as a consequence of the Aharonov-Bohm phases of the topological vacuum. Positive and negative charges on each plate cancel exactly in the absence of an external electric field, but with a non-vanishing external field, each plate acquires a nonzero net charge, giving rise to an induced dipole moment.

We also study the emission of real photons from the topological vacuum when the external electric field varies slowly (in the adiabatic limit). The external field interferes with the topological vacuum configurations and generates real photons whose energy ultimately comes from the time-varying external field. One can also explain this emission in terms of the dipole moment generated by the fluctuating electric charges. As the external field takes on time-dependence, so does the induced dipole moment. And a time-varying dipole moment naturally implies electromagnetic radiation.

The structure of our presentation is as follows. In section II, we review the construction of magnetic-type instantons and the partition function in [1, 2, 3, 4]. We also explain how an external magnetic field enters . In section III, we construct the electric-type instantons that are the focus of the present studies, and their extra vacuum energy contribution. In section IV, we consider the system placed in an external electric field. In particular, we explain in detail how to correctly treat an external field applied in Minkowski space-time in the Euclidean description. We also compute the various physical parameters of the system such as the induced electric field and corresponding susceptibility which measures the response of the system with respect to the applied field. In section V, we provide an alternative interpretation of the topological vacuum configurations in terms of fluctuating boundary charges, and comment on the connection with a previously discussed phenomenon of persistent currents. In section VI we propose an experimental setup using quantum LC circuits to detect these topological effects, and numerically estimate the magnitude of the boundary charges and the associated dipole moment in systems with experimentally accessible dimensions. In section VII we discuss the emission of real photons as the static external field takes on time-dependence. And in section VIII, we conclude the present studies and speculate on the possible relevance of the TCE for cosmology. In particular, the de Sitter behavior in the inflationary epoch could be an inherent property of the topological sectors in QCD in the expanding universe, rather than a result of some ad hoc dynamical field such as inflaton. The emission of real physical degrees of freedom from the inflationary vacuum in a time-dependent background (the so-called reheating epoch) in all respects resembles the effect considered in the present work where real photons can be emitted from the vacuum in a time-varying external electric field.

## Ii Topological partition function. Magnetic type instantons

Our goal here is to review the Maxwell system defined on a Euclidean 4-torus with sizes in the respective directions. This 4-torus provides the infrared regularization of the system, which plays a key role in the proper treatment of the topological terms related to tunneling events between topologically distinct but physically identical sectors.

We follow [1] in our construction of the partition function where it was employed to compute the corrections to the Casimir effect due to these topological fluctuations. The crucial point is that we impose periodic boundary conditions on the gauge field up to a large gauge transformation. In what follows we simplify our analysis by considering a clear case with winding topological sectors in the -direction only. The classical instanton configuration in Euclidean space which describes the corresponding tunneling transitions can be represented as follows:

 Aμtop=(0, −πkeL1L2x2, πkeL1L2x1, 0), (1)

where is the winding number that labels the topological sector, and , are the dimensions of the plates in the - and -directions respectively, assumed to be much larger than the plate separation in the -direction, . The terminology “instanton” is adapted from similar studies in 2d QED [1] where the corresponding configuration in the gauge describes the interpolation between pure gauge vacuum winding states . We use the same terminology and interpretation for the 4d case because (2) is the classical configuration saturating the partition function , in close analogy with the 2d case (details in [1]). This classical instanton configuration satisfies the periodic boundary conditions up to a large gauge transformation, and provides a topological magnetic instanton-flux in the -direction:

 →Btop = →∇×→Atop=(0, 0, 2πkeL1L2), (2) Φ = e∫dx1dx2Bztop=2πk.

The Euclidean action of the system is quadratic and has the form

 12∫d4x{→E2+(→B+→Btop)2}, (3)

where and are the dynamical quantum fluctuations of the gauge field. We call the configuration given by Eq. (1) the instanton-fluxes describing the tunneling events between topological sectors . These configurations saturate the partition function (see (6) below) and should be interpreted as “large” quantum fluctuations which change the winding states , in contrast with “small” quantum fluctuations which are topologically trivial and are expressed in terms of conventional virtual photons saturating the quantum portion of the partition function .

The key point is that the topological portion decouples from the quantum fluctuations, , such that the quantum fluctuations do not depend on topological sectors and can be computed in the trivial topological sector, . Indeed, the cross term vanishes,

 ∫d4x →B⋅→Btop=2πkeL1L2∫d4x Bz=0, (4)

because the magnetic portion of the quantum fluctuations in the -direction, represented by , is a periodic function as is periodic over the domain of integration. This technical remark in fact greatly simplifies our analysis as the contribution of the physical propagating photons is not sensitive to the topological sectors. This is, of course, a specific feature of quadratic action (3), in contrast with non-abelian and non-linear gauge field theories where quantum fluctuations do depend on the topological sectors.

The classical action for configuration (2) then takes the form

 12∫d4x→B2top=2π2k2βL3e2L1L2. (5)

To further simplify our analysis in computing , we consider a geometry where , similar to the construction of the conventional CE. In this case our system is closely related to 2d Maxwell theory by dimensional reduction: taking a slice of the 4d system in the -plane will yield precisely the topological features of the 2d torus considered in great detail in [1]. Furthermore, with this geometry our simplification (2) where we consider exclusively the magnetic instanton-fluxes in the -direction is justified as the corresponding classical action (5) assumes a minimum possible value. With this assumption we can consider very low temperatures, but still we cannot take the formal limit in the final expressions because of the technical constraints.

With these additional simplifications the topological partition function becomes

 Ztop=√2πβL3e2L1L2∑k∈Ze−2π2k2βL3e2L1L2=√πτ∑k∈Ze−π2τk2, (6)

where we have introduced the dimensionless system size parameter

 τ≡2βL3/e2L1L2. (7)

Eq. (6) is essentially the dimensionally reduced expression of the topological partition function for the 2d Maxwell theory analyzed in [1]. One should also note that the normalization factor which appears in (6) does not depend on the topological sectors , and essentially represents our normalization convention in the limit , which corresponds to a convenient setup for Casimir-type experiments. The simplest way to demonstrate that in the limit is to use the dual representation (10), see below.

Next, we introduce an external magnetic field to the Euclidean Maxwell system. Normally, in the conventional quantization of electromagnetic fields in infinite Minkowski space, there is no direct coupling between the fluctuating vacuum photons and an external magnetic field due to the linearity of the Maxwell equations. Coupling with fermions generates a negligible effect as the non-linear Euler-Heisenberg effective Lagrangian suggests (see [1] for the details and numerical estimates). In contrast, the external magnetic field does couple with the topological fluctuations (2) and can lead to effects of order unity.

The corresponding partition function can be easily constructed for the external magnetic field pointing along the -direction, as the crucial decoupling of the background field from the quantum fluctuations assumes the same form (4). In other words, the physical propagating photons with non-vanishing momenta are not sensitive to the topological sectors , nor to the external magnetic field, similar to the discussions after Eq. (4). Additionally, since a real-valued external magnetic field applied in Minkowski space-time remains the same after analytic continuation to Euclidean space-time, this can be used to represent both the Minkowski external field and the Euclidean one.

The classical action in the presence of this uniform static magnetic field therefore takes the form

 12∫d4x(→Bext+→Btop)2=π2τ(k+θeff2π)2 (8)

where the effective theta parameter is defined in terms of the external magnetic field . And the partition function can be easily reconstructed from (6):

 Ztop(τ,θeff)=√πτ∑k∈Zexp[−π2τ(k+θeff2π)2]. (9)

The dual representation for the partition function is obtained by applying the Poisson summation formula such that (9) becomes

 Ztop(τ,θeff)=∑n∈Zexp[−n2τ+in⋅θeff]. (10)

Eq. (10) justifies our notation for the effective theta parameter as it enters the partition function in combination with integer . One should emphasize that the in the dual representation (10) is not the integer magnetic flux defined in Eq. (2) which enters the original partition function (6). Furthermore, the parameter which enters (9, 10) is not the fundamental parameter normally introduced into the Lagrangian in front of the operator. Rather, should be understood as an effective parameter representing the construction of the state for each slice with non-trivial in the 4d system. In fact, there are three such parameters representing different slices of the 4-torus and their corresponding external magnetic fluxes. There are similarly three parameters representing the external electric fluxes (in Euclidean space-time) as discussed in [2] and as we will discuss in section IV, such that the total number of parameters classifying the system is six, in agreement with the total number of hyperplanes in four dimensions111Since it is not possible to have a 3D spatial torus without embedding it in 4D spatial space, the corresponding construction where all six possible types of fluxes are generated represents a pure academic interest..

We shall not elaborate on this classification in the present work. Instead, we limit ourselves to a single describing the partition function in the presence of an (Euclidean) external electric field pointing in the -direction. So essentially in what follows, we consider the manifold , where is an interval in the - and -directions with length and respectively, while a single spatial circle with size points in the -direction. Later in the paper, we interpret the obtained results by switching back to the physical Minkowski space-time. As we shall see, there are significant differences in the behavior of the system in comparison with the previously considered magnetic case [1].

## Iii Electric type instantons

We formulate the electric system on a Euclidean 4-manifold with size . Two parallel conducting plates form the boundary in the -direction, endowing the system with the geometry of a small quantum capacitor that has plate area and separation at an ambient temperature of . These two plates are connected by an external wire to enforce the periodic boundary conditions (up to large gauge transformations) in the -direction, and so the system can be viewed as a quantum LC circuit where the external wire forms an inductor L. The quantum vacuum between the plates (where the tunneling transitions occur) represents the object of our studies.

### iii.1 Construction of topological partition function

The classical instanton configuration in Euclidean space-time which describes tunneling transitions between the topological sectors can be represented as follows:

 Aμtop(t) = (0, 0, 0, 2πkeL3βt) (11) A3top(β) = A3top(0)+2πkeL3,

where is the winding number that labels the topological sector and is the Euclidean time. This classical instanton configuration satisfies the periodic boundary conditions up to a large gauge transformation, and produces a topological electric instanton-flux in the -direction:

 →Etop=˙→Atop=(0, 0, 2πkeL3β). (12)

This construction of these electric-type instantons is in fact much closer (in comparison with the magnetic instantons discussed in [1]) to the Schwinger model on a circle where the relevant instanton configurations were originally constructed [14]. The Euclidean action of the system takes the form

 12∫d4x{(→E+→Etop)2+→B2}, (13)

where, as in the magnetic case, and are the dynamical quantum fluctuations of the gauge field. Because periodic boundary conditions have been imposed on the system, the topological and quantum portions of the partition function again decouple: . One can explicitly check that the cross term vanishes:

 ∫d4x →E⋅→Etop=2πkeβL3∫d4x Ez=0, (14)

since and is periodic over the domain of integration. Hence, the classical action for configuration (11) becomes

 12∫d4x→E2top=2π2k2L1L2e2L3β=π2k2η (15)

where is the key parameter characterizing the size of this electric system, defined as

 η≡2L1L2e2βL3. (16)

This dimensionless parameter is related to the parameter in the magnetic case (7) by . With topological action (15), we next follow the same procedure as in the magnetic case to construct the topological partition function,

 Ztop(η)=∑k∈Ze−π2ηk2, (17)

with normalization , such that no topological effect survives in the limit . As a result, Eq.(17) differs from the partition fuction in the magnetic case (17) by a -independent prefactor. Since the total partition function is represented by the direct product, , any -independent factor in the normalization of can be moved to .

The Poisson summation formula can be invoked to obtain the dual expression for the partition function:

 Ztop(η)=1√πη∑n∈Ze−n2η. (18)

### iii.2 Topological Casimir pressure

The topological pressure between the capacitor plates can be obtained by differentiating the free energy of the system with respect to plate separation. First, we consider the pressure in the absence of an external electric field:

 Ptop(η)=1βL1L2∂∂L3lnZtop(η) = e22L21L22π2η2Ztop(η)∑k∈Zk2e−π2ηk2,

plotted in Fig. 1 as a function of the system size parameter . The pressure peaks around and has a narrower window compared to the magnetic case [1]. And like in the magnetic case, this pressure is repulsive, the opposite of the attractive conventional Casimir pressure. In the large limit one can keep only two terms in expression (III.2) with , and the pressure reduces to

 Ptop≈e2π2η2L21L22e−π2η, (20)

with the familiar exponential suppression representing the typical behavior of tunneling processes.

To get a sense of the magnitude of this topological pressure, we compare it to the well-known expression for the conventional Casimir pressure between two parallel conducting plates with separation :

 P=−π2240L43. (21)

From Fig. 1, the maximum topological pressure (corresponding to ) is about , so the maximum ratio between the two pressures is

 Rmax=|Pmaxtop||P|≈12e2L43π2L21L22=48απL43L21L22. (22)

This ratio, even at its maximum, is very small in a typical Casimir experiment setup where ; besides, the numerical prefactor further suppresses it by an order of magnitude.

To sum up this section, we have found an additional contribution to the Casimir pressure that cannot be attributed to any physical propagating degrees of freedom but instead results from the topological excitation of the gauge field. Specifically, this contribution occurs when the system is defined on a compact manifold with nontrivial boundary conditions. As these topological tunneling transitions are described in terms of integer electric fluxes in Euclidean space-time (12), they exhibit exponential suppression in the conventional geometry, (20). And even at its maximum, this topological pressure is orders of magnitude smaller than the conventional Casimir pressure (22). However, we show in the next section that a unique feature of the topological effect is that an external electric field effectively couples with the electric instanton-fluxes (12), making the topological effect highly sensitive to an applied electric field. Similar sensitivity to external fields is absent from the conventional CE, where the linearity of the Maxwell equations forbids vacuum photon fluctuations from coupling to external fields.

## Iv θ vacua and external electric fields

As shown in the previous section, the pressure produced by the topological CE is many orders of magnitude smaller than the conventional Casimir pressure, making its measurement very difficult. We know that in the magnetic case, as reviewed in section II, a weak external magnetic field interferes with the integer magnetic fluxes describing the tunneling events (2) and enters the partition function (9, 10) as an effective theta parameter, eventually producing oscillatory behavior in the physical observables [1]. In this section, we consider the effect of similarly placing the quantum capacitor in a uniform external electric field in the -direction. However, whereas a magnetic field stays unchanged under analytic continuation between Minkowski and Euclidean formulations, an electric field acquires an additional factor of as it involves the zeroth component of four-vectors, i.e. .

First we consider an external electric field in Euclidean space-time, which simply adds an term to the topological action (15). Note that the quantum fluctuations still decouple from the topological and external fields due to the periodicity of the former over the domain of integration:

 (Ezext+2πkeβL3)∫d4xEz=0. (23)

The partition function then becomes

 Ztop(η,θEeff)=∑k∈Zexp⎡⎣−π2η(k+θEeff2π)2 ⎤⎦, (24)

where the external Euclidean electric field enters the partition function through the combination

 θEeff=eL3βEext. (25)

In what follows we also need a normalization at non-vanishing external field. Since the portion of the partition function proportional to is also -independent, we move it to . To avoid confusion with notation we use for the partition function with this term removed. It is likewise normalized to one in the large limit in the background of a non-vanishing external source, i.e.

 Ztop(η,θEeff)≡exp[−η(θEeff)24]×¯Ztop(η,θEeff) ¯Ztop(η,θEeff)≡∑k∈Zexp[−π2η(k2+k θEeffπ) ] ¯Ztop(η→∞,θEeff)=1. (26)

One can interpret as the partition function with the external source contribution removed from the free energy of the system (see Appendix. A for further justification in the Hamiltonian formulation). Our normalization corresponds to the geometry when tunneling events are strongly suppressed, i.e., physical phenomena discussed in the present work are trivial for systems in such limit.

The dual representation for the partition function is obtained by applying the Poisson summation formula such that (24), (IV) become

 Ztop(η,θEeff) = 1√πη∑n∈Zexp[−n2η+in⋅θEeff] ¯Ztop(η,θEeff) = exp[η(θEeff)24]×Ztop(η,θEeff). (27)

Unfortunately, we cannot directly calculate physically meaningful thermodynamic properties of the system from this partition function, since does not represent a physical electric field living in Minkowski space-time. Rather, we need to first switch to a Minkowski field by the formal replacement . Explicitly, the partition function in the presence of a real Minkowski electric field is given by

 (28)

where the physical Minkowski electric field enters the partition function through the combination

 θMeff=eL3βEMinkext=−iθEeff. (29)

Our interpretation in this case remains the same: in the presence of a physical external electric field represented by the complex source , the path integral (24) is saturated by the Euclidean configurations (12) describing physical tunneling events between the topological sectors .

With an external Minkowski electric field, the -direction topological pressure computed from the partition function (28) exhibits oscillatory behaviour with respect to . A notable feature of the pressure is that the oscillation period depends on the dimensionless parameter characterizing the system (16), in marked contrast with the magnetic case [1] where there is a universal -periodicity for all values of .

To illustrate the -dependence of the observables, first consider a simple case with , where the magnitude of all effects are exponentially small. In this limiting case for sufficiently small external field one can keep only the lowest branch with in Eq. (28) such that the expression for the pressure assumes the form

 Ptop(0≤ηπθMeff≤π)≈e2π2η2L21L22exp(−π2η) (30) ×[cos(πηθMeff)+(πηθMeff)π2ηsin(πηθMeff)].

This formula is valid only for the first branch when the electric field is small, . When the external field becomes stronger one should be more careful with formal differentiation of the partition function (28) with respect to external parameters. This is because the partition function is a periodic function of , and this periodicity must be respected by all physical observables222To be more precise, the physics is periodic under the shift . Similar periodicity with respect to an external parameter is quite generic in many gauge theories, including QCD, where all physical observables are periodic functions of the fundamental parameter. This periodicity can be formally enforced in our case in the path integral by replacing in Eq. (28) and summing over . This is in fact a conventional procedure in similar systems, see for e.g. Eq. (52) in [15] where theoretically controllable computations have been performed in a weakly coupled non-abelian gauge theory. A generic consequence of this periodicity is that the system shows cusp singularities at specific points, normally . The origin for such behavior is the two-fold degeneracy emerging in the system at these points. As a result of this degeneracy, level crossing occurs precisely at these points when the branch with the minimum free energy (corresponding to the ground state) changes , see [15] for a large number of references on the original literature.. Using the periodicity, one can always separate out an integer multiple of from stronger fields, , such that the pressure in the second branch, , assumes the form

 Ptop(π≤ηπθMeff≤2π)≈e2π2η2L21L22exp(−π2η) (31) ×[cos(πηθMeff)+(2π−πηθMeff)π2ηsin(2π−πηθMeff)].

Thus, the second term in the square brackets in (30) and (31) explicitly exhibits the cusp singularity at the point . Such behavior is, in fact, a quite generic feature of gauge systems as a result of two-fold degeneracy and level-crossing phenomena, see footnote 2 for more comments and references.

Indeed, the degeneracy can be observed from the partition function (28) which explicitly exhibits the double degeneracy at when the level crossing occurs. One can approach the point from two sides using a mirror-like symmetry: . It can be easily checked that this does not modify the partition function itself as the corresponding sign-flip in the topological term is equivalent to relabeling in the sum in Eq. (28). An analogous phenomenon is known to occur in the 2d Schwinger model where the system shows two-fold degeneracy at . It also occurs in the 4d Maxwell system saturated by the magnetic instantons as discussed in details in [2].

Numerically, for sufficiently small where the effect is expected to be at its maximum, the pressure as a function of is shown in Fig. 2 and cusp singularities emerge at . One can explicitly see that the system is symmetric under , and it also exhibits the periodicity , as expected.

One can also compute the induced electric field in response to the external source in Minkowski space by differentiating the partition function with respect to :

 ⟨EMinkind⟩=−1βV∂ln¯Ztop∂EMinkext=−eL1L2∂ln¯Ztop∂θMeff = 1¯Ztop∑k∈Z2πkeL3βe−ηπ2k2sin[πkηθMeff],

plotted in Fig. 3.

It is quite obvious that if . The difference in comparison with the analogous expression from [1] is that our present definition of the induced field does not include the external piece , in contrast with the definition used in [1] where the induced field of the system was defined as the total field (i.e., the truly induced field plus the external piece). One can observe that the induced field changes the sign under the mirror-like symmetry in contrast with expression for the pressure presented in Fig. 2. This is because the induced field flips the sign under this symmetry, in contrast with the pressure .

A few comments are in order. First, formula (IV) represents the physical induced field defined in Minkowski space-time. In addition, the oscillations now occur with periodicity , which can be identically rewritten as

 πηθMeff=2πm    ⇒    L1L2EMinkext=em. (33)

Note that this value for the physical electric field is indeed consistent with the results from the canonical Hamiltonian approach in Appendix A. It further supports our formal manipulations in the transition from the Euclidean to Minkowski description.

We should emphasize that the quantization of the physical electric field (33) is drastically different from the quantization of the Euclidean instanton-fluxes (12) saturating the partition function. Indeed, the instanton-fluxes have a different normalization factor along with the geometric factor . It should be contrasted with the magnetic case [1] where the external field and the instanton-fluxes have identically the same periodic properties. This is precisely the reason why formulae from [1] exhibit the universal periodic properties, in contrast with (IV).

In the large limit one can again keep only two terms in Eq. (IV) with such that it assumes the following simple form

 ⟨EMinkind⟩=4πe−ηπ2eL3βsin(πηθMeff). (34)

As expected, the induced term has non-analytical behaviour and cannot be seen in perturbation theories as it originates from the tunneling events. It is exponentially suppressed when as expected, and obviously vanishes in the limit.

Similarly, we can compute the (topological) susceptibility of the system, which measures the response of free energy to the introduction of a source term, represented by in our case:

 χE=−2η∂2∂θMeff2ln¯Ztop(η,θMeff), (35)

shown in Fig. 4.

In the large limit, one can explicitly check using the analytical expression (28) that is strongly suppressed, consistent with our expectation that no electric correlations exist in the conventional Casimir effect setup. The is symmetric under the mirror-like symmetry as the second derivative does not flip the sign at the degeneracy point .

## V Induced dipole moment and surface charges

Although the electric instanton-fluxes describing the tunneling events between topological sectors in our system are formulated in Euclidean space-time (12), we have shown in the previous section that a real-valued electric field will be induced in Minkowski space-time in response to a non-vanishing . In this section, we represent this induced electric field in the bulk of the system in terms of the surface effects. In other words, we want to reformulate the fluctuations of the topological electric fluxes using fluctuating surface charges on the plates.

Similar reformulation of the problem was carried out for the magnetic case in [4] where it was explicitly shown that the tunneling instanton effects can be understood in terms of fluctuating topological non-dissipating currents which unavoidably will be generated on the boundaries. In [4], the system was a cylinder with radius and height with the topological current flowing in the -direction on its infinitely thin boundary. It has also been noted that these topological currents are very similar in nature, but not the same as the well-known persistent currents normally observed on metallic (not superconducting) rings, see references on the original experimental and theoretical studies on persistent currents in [4]. In all cases the effects are due to the coherent Aharonov-Bohm phases correlated on macroscopically large distances, although the nature of the long-range coherence is different for our topological non-dissipating currents and for the well-known persistent currents.

The duality between magnetic and electric fields in the Maxwell system strongly suggests that similar topological effects must be present in the electric systems as well. Essentially we attempt to study the effects which are EM-dual to the persistent currents observed on metallic rings. The relevant formal construction indeed can be easily carried out as we have shown in sections III and IV.

We start with an explicit demonstration that the existence of an induced electric field (generated by the electric instantons) in Minkowski space (IV) suggests that one can effectively recast the mathematics into an equivalent form where physical electric charges are induced on each plate of the capacitor333We briefly remark that the charges indeed physically reside on the actual plates as long as the periodic boundary conditions are enforced using the quantum LC circuit configuration.. These charges have pure quantum origin and are, in all respects, very similar to the persistent non-dissipating currents induced by the magnetic instantons discussed in [4].

An important technical comment here is that the induced electric field (IV) can be thought of as the polarization of the system per unit volume, i.e. , since the definition for is identical to (IV) up to a minus sign because it enters the Hamiltonian as . Therefore, we arrive at the following expression for the induced electric dipole moment of the system in the presence of an external electric field ,

 ⟨pMinkind⟩=−⟨EMinkind⟩L1L2L3 = −1¯Ztop∑k∈Z2πkL1L2eβe−ηπ2k2sin(πkηθMeff).

Based on this interpretation, one can view Fig. 3 as a plot for the induced electric dipole moment in units of which represents the correct dimensionality for the electric dipole moment.

One can understand the same formula (V) using the original expression for the coupling between the external field and the topological instantons

 ∫d4x(→Eext⋅→Etop)=∫d4x Vext(→∇⋅→Etop), (37)

where we have neglected a total divergence term. The cross term written in the form (37) strongly suggests that can be interpreted as surface charges generated on the plates. Indeed, if we use Eq. (12) for the topological instantons describing the tunneling transition to the sector, we arrive at the following formula for the surface charge density

 σMinkind(k)=2πkeβL3[δ(x3)−δ(x3−L3)]. (38)

This formula implies that an electric dipole moment will be generated in each topological sector , given by

 pMinkind(k)=−2πL1L2eβk, (39)

which reproduces the relevant term in (V), derived in a quite different way without any mention of surface charges.

Furthermore, one can explicitly check that the cross term (37) expressed in terms of the surface charge exactly reproduces the corresponding term in the action of the partition function (28) computed in terms of the bulk instantons. Indeed, the cross term in the action is

 ∫d4x(→Eext⋅→Etop)=∫d4x Vext(→∇⋅→Etop) (40) =∫d4x(Eextx3)(2πkeβL3)δ(x3−L3)=πηkθEeff,

where we have substituted and expressed the external field in terms of . Eq. (40) precisely coincides with the cross term in the action for the partition function (IV). In the same way the classical instanton action in (IV) can be also understood in terms of the surface charges.

The basic point of our discussions in this section is that the expression for the induced electric dipole moment (V) can be understood in terms of an induced field according to (IV). The same effect can be also interpreted in terms of the surface charges of the system as (38) and (39) suggest. However, the origin of the phenomena is not these charges but the presence of the topological sectors in Maxwell electrodynamics formulated on a compact manifold with nontrivial mappings . Such sectors exist and transitions between them always occur even if charged particles are not present in the bulk of the system. The secondary role played by the charged particles is in particular illustrated by the fact that the extra contribution to the Casimir vacuum pressure generated by survives a vanishing external field, but the fluctuating positive and negative charges cancel each other exactly. The fundamental explanation is still the tunneling transitions between vacuum winding states which occur regardless of the value of the external field.

To conclude this section, we would like to remark that it is quite typical in condensed matter physics that topologically ordered systems allow such a complementary formulation in terms of the physics on the boundary. Therefore, it is not a surprise that we can reformulate the original instanton fluctuations saturating in terms of the boundary surface charges which always accompany these instanton transitions. See more discussions on the relation between descriptions in the bulk and on the surface for the magnetic system in [4].

## Vi Numerical estimates for a quantum LC circuit

Our goal now is to discuss a possible design where such electric effects can be (at least in principle) studied. In what follows we consider a two-plate capacitor with area separated by distance at temperature . The two plates are connected by an external wire such that charges can freely move from one plate to the other. The system can be viewed as a quantum LC circuit when conventional quantum transitions (due to ordinary degrees of freedom) are replaced by tunneling transitions as described in previous sections. The vacuum between the plates (where these tunneling transitions occur) represents the object of our studies.

We would like to make a few numerical estimates for illustration purposes only. The first set of parameters is motivated by the accurate measurement of the CE using parallel plates [16] (see also [17] where historically the first accurate measurement was performed but for a different geometry). The second set of parameters is motivated by the experiments on persistent currents [18] where the correlation of the Aharonov-Bohm phases is known to be maintained. While the persistent current is a magnetic phenomenon, the EM duality strongly suggests that a similar electric effect should also occur when the coherent Aharonov-Bohm phases are correlated on macroscopically large distances. Therefore, for the second set of parameters we adopt the typical sizes of the magnetic system (where persistent currents have been observed) to estimate the topological effects for our electric capacitor.

First, we adopt the system sizes from the first accurate measurement of the Casimir effect using parallel plates [16]. Unlike the magnetic system studied in [1], the dimensions of this electric system as represented by can be fairly easily optimized to maximize the topological effect. In this experiment a small capacitor was formed using chromium-coated surfaces with area and separation in the range. This geometry corresponds to and for reasons explained previously, we expect the topological effect to be vanishingly small. However, if plate separation could be increased to (such that our approximation is still marginally satisfied) and the ambient temperature set to (corresponding to ), then

 η(I)=2L1L2e2βL3=1.2×$1.2mm2$2πα($180mm$)($0.4mm$)≈0.4, (41)

where the intensity of the topological tunneling transitions assumes the maximum values, see Fig. 1. The effect is still much smaller than conventional Casimir effect as we already discussed in section III.2. The main point, however, is that the effect is highly sensitive to the external field, in contrast with the conventional CE, as argued in section IV. Precisely this sensitivity might be the key element for observing this fundamentally novel phenomenon when the vacuum energy in the bulk is not associated with any physical degrees of freedom propagating in the bulk.

Now we want to consider a second set of parameters motivated by observing the persistent currents [18]. It has been mentioned previously [4] that the topological non-dissipating currents, while similar in nature to the observed persistent currents [18], are nevertheless not identically the same and represent an independent additional contribution to the non-dissipating currents444In the conventional case, the Aharonov-Bohm coherence is determined by the dynamics of the electrons residing on the ring, while in our case it is determined by the dynamics of the vacuum, i.e. tunneling transitions between the winding sectors. This difference, in particular, manifests itself in the properties of the induced magnetic moment: it is quantized in our case, while it can assume any value for conventional persistent currents.. Based on EM duality, one could argue that if Aharonov-Bohm coherence has been established for the magnetic system, it is likely to hold in electric systems with similar geometric sizes as well. Therefore, we expect the topological tunneling transitions to be present in the system when the ring area in the magnetic system [18] is replaced by the electric capacitor plate area , and the ring width replaced by the plate separation . With this correspondence, we estimate the key dimensionless parameter to be

 η(II)=2L1L2e2βL3=2π(1.2μm)24πα(0.6cm)(0.1μm)≈0.16, (42)

where we have used and cm corresponding to the temperature mK below which the electron phase coherence length is sufficiently large and temperature-independent555One should remark here that there are related effects when the entire system can maintain coherent Aharonov-Bohm phases at very high temperatures K [19].. This parameter falls into the region where the intensity of the topological tunneling transitions assumes its maximum values, see Fig. 1. Therefore, one should anticipate a nonzero value for the induced electric dipole moment when an external field is applied.

Assuming appropriate boundary conditions (i.e., periodic up to a large gauge transformation) are established, the induced electric dipole moment depends on the applied external field. The corresponding dependence of on external field can be easily established from Fig. 3 where the plot should be understood as the induced dipole moment in units of . Numerically, these units for our two sets of parameters can be estimated as

 ⟨pMinkind⟩(I) ≈ eL32η(I)∼0.1(e⋅mm) ⟨pMinkind⟩(II) ≈ eL32η(II)∼0.01(e⋅μm). (43)

The numerical estimates for the induced dipole moment due to the coherent tunneling effects (VI) do not look very promising if measurements are performed with a static external electric field. In the next section we consider an option when the external field varies with time. In this case one should anticipate the emission of real propagating photons which leave the system, and which hopefully (in principle) can be detected.

## Vii Electrodynamics and E&M radiation

Up till this point, we have implicitly assumed that the external electric field is static. However, Eq. (V) still holds in the case of a dynamical external field as long as its time dependence is adiabatically slow compared to all relevant time scales of the system. Then this time-dependence simply enters the partition function through . In has been argued in [4] that as the induced magnetic dipole moment varies in response to an adiabatically oscillating external source, real photons will be emitted through magnetic dipole radiation. Following the same reasoning, we can also compute the electric dipole radiation produced by our capacitor.

Therefore, one can use the well-known expressions for the intensity and total radiated power for the electric dipole radiation when the dipole moment (V) varies with time:

 →S=I(t)sin2θ4πr2→n,         I(t)=23c2⟨¨pMinkind(t)⟩2 (44)

In the case where the external electric field plotted in Fig. 3 is almost linear, the approximate induced dipole moment, , varies linearly as one can see from Fig.3. In particular, if then . In this case one can easily compute the average intensity over large number of completed cycles with the result

 ⟨I⟩∼ω43c2⟨pMinkind⟩2, (45)

where is given by (V).

This electromagnetic radiation has a simple explanation in terms of the fluctuating charges that reside on the capacitor plates (38). For static external field , the charge density on each plate also stays constant, giving rise to a time-independent electric dipole moment. But when starts to fluctuate, the induced dipole moment also acquires time-dependence, naturally leading to the emission of real photons.

The detection of this radiation could be made easier by taking advantage of the system’s two-fold degeneracy at corresponding to half integer fluxes, where the system’s ground state reconstructs itself [2]. As the external field slowly sweeps through these points, quickly changes polarity as Fig. 3 shows, producing high amounts of radiation. This will hopefully set the topological emission apart from the uninteresting radiation due to the fluctuating external source. In addition, since the power radiated electrically usually substantially exceeds that radiated magnetically for systems with comparable dimensions, we expect the radiation from this present electric configuration to be much more readily detectable than the one in [4].

## Viii Conclusion and Speculations

In this work we have discussed a number of very unusual features exhibited by the Maxwell theory formulated on a compact manifold with nontrivial topological mappings , which was termed the topological vacuum (). All these features originate from the topological portion of the partition function which cannot be described by a construction of conventional photons expanded near a classical vacuum . In other words, all effects discussed in this paper have a non-dispersive nature.

The computations of the present work along with previous calculations in [1, 2, 3, 4] imply that the extra energy, not associated with any physical propagating degrees of freedom, may emerge in a gauge system if some conditions are met. This fundamentally new type of energy emerges as a result of the dynamics of pure gauge configurations.

The new idea advocated in this work is that this new type of energy might be related to electric-type instantons, in contrast with magnetic-type instantons studied in the previous papers. While the modification may look minor, it leads to a number of novel effects. Most notably, whereas the magnetic oscillations have universal -periodicity for all system sizes, the periodicity in the electric system varies with the system size. It remains to be seen if the effects discussed in the present work can be experimentally observed.

In addition, the electric system studied in the present work could be considered as a simple toy model where the topological tunneling transitions can be theoretically studied. These novel effects could have some profound consequences for astrophysics and cosmology if they persist in the Standard Model (SM). Indeed, the concept of gauge symmetry is central to the SM. As a result, the existence of such non-trivial homotopy mappings implies that the ground state of the system should be represented by a superposition of the topological winding sectors. The corresponding path integral which determines the system therefore must also include the sum over all the topological sectors, accounting for the physics describing the tunneling transitions between them.

All the effects studied in sections III and IV were precisely attributed to the physics of tunneling transitions which generate an additional vacuum energy. The effects are obviously non-local in nature. Moreover, the corresponding contributions to the correlation functions are represented by non-dispersive terms which fundamentally cannot be expressed in terms of any local propagating degrees of freedom. Furthermore, the radiation of real physical photons described in section VII represents a fundamentally novel mechanism of particle production (à la the dynamical Casimir effect) when the emission occurs from the topological vacuum due to the tunneling transitions, rather than from the decay of some local propagating degrees of freedom.

We note that many conventional approaches to describe the dark vacuum energy (or vacuum inflation) are based on effective local field theories. The production of real particles (during the so-called reheating epoch) is also normally considered as a result of conventional local interaction of real particles.

The unique features of the system studied in the present work when an extra energy is not related to any physical propagating degrees of freedom was the main motivation for a proposal [20, 21] that the vacuum energy of the Universe may have, in fact, precisely such non-dispersive and non-local nature666This new type of vacuum energy which cannot be expressed in terms of propagating degrees of freedom is well known to the QCD community and has been well studied in QCD lattice simulations, see [20, 21] for references and details. studied in the present work in a toy model. This proposal where an extra energy cannot be associated with any propagating particles should be contrasted with the conventional description where an extra vacuum energy in the Universe is always associated with some ad hoc physical propagating degrees of freedom, such as inflaton777There are two instances in the evolution of the universe when the vacuum energy plays a crucial role. The first instance is identified with the inflationary epoch when the Hubble constant was almost constant, which corresponds to the de Sitter-type behavior with exponential growth of the size of the Universe. The second instance when the vacuum energy plays a dominant role corresponds to the present epoch when the vacuum energy is identified with the so-called dark energy which constitutes almost of the critical density. In the proposal [20, 21] the vacuum energy density can be estimated as , which is amazingly close to the observed value. .

## Acknowledgements

This research was supported in part by the Natural Sciences and Engineering Research Council of Canada. C.C. was in part supported by the Walter Burke Institute for Theoretical Physics at Caltech, the DOE grant DE-SC0011632, and the Gordon and Betty Moore Foundation through Grant 776 to the Caltech Moore Center for Theoretical Cosmology and Physics. Y.Y. acknowledges a research award from the UBC Work Learn Program.

## Appendix A Hamiltonian Approach

We check the topological partition function with external electric field using canonical quantization so there is no ambiguity regarding the physical meaning of the electric field. Recall the classical lagrangian density for the EM field

 L(x)=12(E2−B2), (46)

where and . We then proceed with the usual Hamiltonian formalism where we realize that is non-dynamical and imposes the Gauss’ law constraint as a Lagrange multiplier. Hence the Hamiltonian density is given by

 H(x)=12(E2+B2)−A0(x)∇⋅E(x). (47)

We can fix the gauge and define the Hamiltonian .

Then, we follow the usual quantization procedure for the EM field by imposing the equal time commutation relations

 [^Aj(x),^Πk(y)]=iδ(x−%y)δjk. (48)

The conjugate momentum and potential are now promoted to operators, and

 ^Ej=iδδAj(x). (49)

We also note the potentials related by a local time-independent gauge transformation are physically equivalent. Namely,