Current correlation functions from a bosonized theory in 3/2+1 dimensions

Within the context of a bosonized theory, we evaluate the current-current correlation functions corresponding to a massive Dirac field in 2+1 dimensions, which is constrained to a spatial half-plane. We apply the result to the evaluation of induced vacuum currents in the presence of an external field. We comment on the relation with the purely fermionic version of the model, in the large-mass limit.

Bosonization is a useful tool which, in 1 + 1 space-time dimensions, allows for the solution of some non-trivial Quantum Field Theory models (see [1] for a comprehensive review and useful references).
For a massive Dirac field in 2 + 1 dimensions, the situation we are concerned with here, the path integral bosonization framework may be used to derive the exact bosonization rule for the current. The (dual) bosonic action, is gauge-invariant and, in the massive case, local, what determines the form of the possible terms in a mass expansion 1 . Thus, to the leading order, it is a Chern-Simons term, while the next-to-leading one corresponds, in the Abelian or non Abelian cases, to a (local) Maxwell [2]- [3] or Yang-Mills term [4], respectively. We note that the need for the CS term has been shown explicitly, even in a massless theory, as a consequence of an η function regularization, required to have a consistent gauge invariant theory [5].
In a previous work [6], we have applied the functional bosonization approach to a system consisting of a massive Dirac field constrained to a 2 + 1 dimensional spacetime manifold U, with non-trivial conditions on its boundary M ≡ ∂U. Those conditions, when imposed on the dual (bosonized) version of the theory, amounted to the vanishing, at each point of M, of the normal component of the (bosonized) current. The bosonization rules, formulated in terms of an Abelian gauge field A µ , were shown to be the same as in the no-boundary case, while the existence of the boundary manifested itself through the fact that the gauge field satisfied perfect-conductor conditions on M.
The exact bosonization of a 1 + 1 dimensional model with a boundary, i.e., on a half-line, has been implemented in [7]. In this article, following [6], we apply the bosonization approach above to the calculation of current correlation functions, in a concrete geometry: a massive Dirac field confined to a spatial half-plane (so that, following [7], we dub the associated space-time as '3/2 + 1 dimensions').
We do not dwell here with a massless theory, where there seems to be, in principle, no natural mass to use in the expansion, and the low energy terms can be non-local. In spite of this, the program could be implemented also in this case (see [8] for a discussion), by using the renormalization mass scale µ as the expansion parameter. Our study of a bosonized Dirac field in 3/2 + 1 dimensions, which takes into account the leading and sub-leading terms in the mass expansion, encompasses the evaluation of the currentcurrent correlation function, in the context of functional bosonization. We also apply it to the determination of the induced current in the presence of an external gauge field, presenting the general form of the result, as well as more explicit expressions for some particular cases.
We consider a massive Dirac field in 2 + 1 dimensions which, in its fermionic incarnation, is described by an Euclidean action S f (ψ, ψ), given by: on a spacetime manifold U which, in terms of the coordinates x = (x 0 , x 1 , x 2 ), corresponds to the region determined by x 2 > 0. Dirac's γ-matrices are defined according to the conventions: Here, and in what follows, letters from the middle of the Greek alphabet are assumed to run over the values 0, 1, 2. The Euclidean metric has been assumed to be the identity matrix δ µν , and ǫ µνλ denotes the Levi-Civita symbol, with ǫ 012 = +1. . The functional bosonization approach singles out the conserved Noether current J µ ≡ψγ µ ψ corresponding to (1), namely, J µ =ψγ µ ψ, while the existence of the boundary is reflected in the vanishing of J n ≡n µ J µ M , its normal component. Here, M ≡ (x , 0), with x = (x 0 , x 1 ), and the (outer) unit normaln µ isn µ = −δ µ2 .
Then, one defines the fermionic generating functional, by adding two ingredients: first, we add to S f a term S J : including a source s µ , to be able to generate current correlation functions. Another term, S M , depending on an auxiliary field ξ(x ), is added in order to impose the condition on the normal current: which can be also written as a term which couples the fermionic current to a vector field c µ (ξ, x), which is completely determined by the auxiliary field and the boundary; indeed: Although we have explicitly indicated here the dependence of c µ on ξ, for the sake of notational convenience, writing that dependence explicitly will be omitted henceforward.
Note that the functional integral over ξ yields a (functional) δ of the normal current: The advantage of interpreting S M as a coupling between the current and a field c µ stems from the fact that the fermionic generating functional Z(s) may be written as follows: with We then follow the procedure devised in [6], of which we provide a lightning review here (with minor changes in conventions). That procedure allows one to disentangle s µ + c µ out of the fermionic action in (7). To that end, we first perform the change of variables: and integrate over α, to obtain: Then, the integration over α is substituted by one over a vector field b µ since the manifold is simply connected). This condition is implemented by means of another auxiliary field, A µ : Thus, Finally, we make the shift b → b − c − s, to obtain: where W (b) denotes the effective action: This leads to a bosonized representation for the generating functional, which may be rendered as follows: where the bosonized action S B (A) is determined by the expression: This leads to the bosonization rule: with a bosonized action S B (A) yet to be determined. Since that depends on the knowledge of W (b), an exact expression of which is unknown, we use a possible approximation to it. The usual approach is to use a largemass expansion, retaining just the leading contribution, a Chern-Simons (CS) term. This term is m-independent. Since we are interested here in dealing with a situation where there is another scale present, namely, the distance to the boundary, and to allow for a possible interplay, we will also include the next-to-leading term, which has the form of a Maxwell action: Inserting this into the expression for the bosonized action S B (A), (17), and working consistently up to the same order in the mass expansion, leads to: Recalling then (16), the generating functional Z(s) requires the evaluation of an integral including the perfect-conductor constraint, what is implemented by the auxiliary field. That integral may be exactly calculated, for example by integrating out A µ firstly, and then over ξ (a Gaussian). The result may be presented as follows: where with Π (1) µµ ′ and Π µµ ′ denoting qualitatively different contributions: Π (1) µµ ′ is identical to the contribution one would obtain for a Dirac field in the absence of boundaries. Π (2) µµ ′ , on the other hand, depends on the existence of the boundary. Therefore, it cannot be translation invariant along the x 2 coordinate. We have found it convenient to represent both Π (1) and Π (2) in terms of their Fourier transforms with respect to the x coordinates (for which there is translation invariance). Assuming that indices from the beginning of the Greek alphabet (α, α ′ , . . . ) run over the values 0 and 1, the explicit form of those terms may be shown to be: with: and where we assumed that x 2 > 0 and x ′ 2 > 0 (which corresponds to the region of interest).
We have explicitly checked that each term, Π µµ ′ and Π (2) µµ ′ , satisfies a Ward identity separately. Namely, In a mass expansion, and keeping just the leading and sub-leading terms, one sees that those two objects adopt the form: and where, in the above two expansions, we have introduced which approximates Dirac's δ-function in the large-m limit. We have kept a number of terms which is consistent with the Ward identities (note that, to verify this, one must use the property that −6mδ m (x 2 ) is an approximates of δ ′ .) Let us apply the above result to the determination of the induced vacuum currents in the presence of a border and of an external electromagnetic field.
We begin by pointing out that Π µµ ′ satisfies: This is consistent with the boundary conditions imposed on the normal component of the current. Indeed, the vacuum expectation value of the current in the presence of an external gauge field a µ , is given by: or, Thus, (30) guarantees that the expectation value above vanishes on M. An important point we would like to stress is that, in the presence of borders, the large mass expansion can be problematic, in the sense that the boundary conditions involve a limit, and the current correlation functions contain singular functions. Thus, we argue that in the presence of boundaries it is safer to take the large-mass limit only after calculating observables (for example, an induced current). Let us apply the general result to the evaluation of the 0-component of the current, i.e., the charge density, in the presence of a point-like static magnetic vortex, located at (x 1 , x 2 ) = (h 1 , h 2 ), which is minimally coupled to the current. Namely, an external field a µ such that: where φ denotes the magnetic flux piercing the plane at the vortex location. We chose the gauge: a 0 = 0, a 1 = 0, and a 2 = φ θ(x 1 − h 1 ) δ(x 2 − h 2 ) (θ ≡ Heaviside's step function) to find that J 1 = J 2 = 0, and Using the explicit form of Π (1,2) , we see that: From this, we wee that the interplay between boundary conditions and parity breaking implies that the induced charge density vanishes at the boundary x 2 = 0, since it is the sum of two contributions, one of them being the reflected opposite of the other.
We see that the infinite-mass limit does produces the expected result, namely, Let us also consider the induced vacuum current in the presence of a electric field of magnitude E in the direction of the x 2 coordinate. Using the gauge field choice a 0 (x 2 ) = −E x 2 , it is straightforward to show that the only non-vanishing component of the current is along the x 1 direction: a paritybreaking effect. Since the gauge field is static and translation-invariant along x 1 , one sees that: Inserting the form of Π 10 (0; x 2 , x ′ 2 ), we see that: i.e., a Hall current exponentially concentrated on the border.

Discussion
A first issue that we comment here is the form of the current-current correlation function, from the point of view of the fermionic theory. The contribution of a massive fermion may be written in terms of the fermion propagator S F (k ; x 2 , x ′ 2 ), which satisfies bag-like boundary conditions. For the case at hand, that condition adopts the form: Therefore, (40) Following the massive version of the procedure followed in [9] for the massless case, it is rather straightforward to show that the fermion propagator is given by: where where ω(p ) = p 2 + m 2 , and U(p ) = (−i p + m)/ω(p ). To the contribution of a massive fermion, one should add the parity-anomaly one. The form of the anomalous contribution, on the other hand, is again a local Chern-Simons term. Indeed, it may only proceed from the UV-divergent part of the calculation. And that corresponds to a fermion loop involving just the S F 0 term, in the large-mass limit. Indeed, UV divergences appear in the coincidence (x 2 → x ′ 2 ) limit, and S F 1 has large-momentum (exponential) suppression for any x 2 , x ′ 2 > 0. At the border, it may indeed contribute with a localized contribution, which is the form of the terms we have found in the bosonized form of the problem: indeed, Π (2) is non vanishing only when x 2 = x ′ 2 = 0 (in the large-mass limit). We have found an expansion for the current-current correlation function which involves continuous approximations to the δ-function. This behaviour exhibits the role of the next to leading term included in the expansion, which here regulates the behaviour of the kernels in the effective action.
Finally, we have shown that the current-current correlation function may be expanded, for a large mass, in a way that preserves the Ward identity.
In recent years, dualities have been applied to analyze different condensed matter systems, like topological insulators, superconductors, and fractional quantum Hall effect systems [5], [10]- [11]. In these studies, bosonization in 2 + 1 dimensions in the presence of a boundary like the one considered here may be relevant to the applications [12], [13], [14], [15].