Entanglement in a Maxwell Theory coupled to a non-relativistic particle

We consider electromagnetism in a cylindrical manifold coupled to a nonrelativistic charged point-particle. Through the relation between this theory and the Landau model on a torus, we study the entanglement between the particle and the electromagnetic field. In particular, we compute the entanglement entropy in the ground state, which is degenerate, obtaining how it varies in the degeneracy subspace.

the Hilbert space that are not spatial ones, as, for example, between right and left moving excitations [21][22][23] or between winding modes [24].
When the field theory presents gauge symmetry, spatial partitions are subtle: it is not possible to make them and still preserve gauge invariance. This difficulty is due to the fact that gauge theories contain non local degrees of freedom such as Wilson loops. Hence, when a spatial partition is made, these loops are necessarily broken. So we are left with an arbitrary choice (and therefore an ambiguity) of deciding to which of the subregions the broken degrees of freedom belong. Many aspects of this problem have been addressed in the literature since the work [25]. It has been discussed in the context of lattice gauge theory giving rise to different prescriptions for computing EE [26][27][28][29][30][31]. In the continuum, one possibility is to calculate EE using the replica trick after extending the Hilbert space in a particular way [32][33][34]. Without resorting to the replica trick, some alternative approaches have also been considered. For example, in [35], EE in 2+1 dimensions was studied employing gaugeinvariant variables. In [36], the zoo of prescriptions for computing EE was unified using an algebraic approach, defining it in terms of a subalgebra of gauge-invariant operators associated to each subregion. Another algebraic framework based on the Gel'fand-Naimark-Segal construction was proposed formerly in [37,38] in order to treat systems of identical particles. This method was also applied to analyse the ambiguities of EE in systems with gauge symmetries [39,40].
The works mentioned in the previous paragraph concern pure gauge theories, without coupling to matter. By including matter, one may not only study spatial entanglement [41] but also the entanglement between the gauge field and the matter sectors. The present paper is dedicated to the latter situation. This problem has also been investigated recently in [42] where the entanglement between a quantum harmonic oscillator and a quantized electromagnetic field was analysed.
Here we consider a non-relativistic particle coupled to an Abelian Yang-Mills (YM) theory in 1+1 dimensions with compactified spatial coordinate, i.e., space-time is a cylinder R×S 1 .
In order to compute the EE between the particle and the field, we map the theory to a quantum mechanical system consisting of a charged particle moving on a torus with a uniform transverse magnetic flux. This is the Landau problem [43] on a torus [44]. In fact, as shown in [45], the field dynamics of a pure YM theory defined on a cylinder can be reduced to that of a free particle moving along the gauge group manifold. This means that we can reduce the quantum field theory problem to a quantum mechanical one. For simplicity, here we restrict to the Abelian case, in which the field theory is mapped to a particle moving on a circle [46]. From the field theory point of view, the only gauge-invariant observable is the Wilson loop along S 1 . If we were to partition this circle to compute some kind of spatial entanglement, then we would break the gauge invariance and we would need to apply the techniques cited above. The fact that there is only one gauge invariant observable in the field theory implies that there is a single degree of freedom associated to the gauge field in the quantum mechanical theory. Thus if we do not consider matter, we have only one degree of freedom (a particle moving on a circle) and, therefore, it is impossible to make any partitions in this setup. Including a non-relativistic particle adds another degree of freedom (the particle moves now on a torus) and the possibility of making a partition. Hence the goal of this work is to understand the entanglement between the degree of freedom associated to the gauge field and that corresponding to the non-relativistic particle.
The paper is organised as follows: in the next section, we show the equivalence between electromagnetism on a space-time cylinder coupled to a non-relativistic charged point particle and the Landau problem on a torus. In section II B, we obtain the solution of the Schrödinger equation of the latter, finding that the ground state is degenerate. In section III, we study the entanglement entropy in this degeneracy subspace. This is equivalent to measuring the entanglement between the particle and the electromagnetic field in the ground state of the field theory. In particular, we perform both an analytical and a numerical analysis of this quantity. We find that the reduced density matrix in the degeneracy subspace can be approximated by that of a two-level system. This observation allows us to obtain an analytical expression for the entanglement entropy in this subspace whose accuracy is checked numerically. We also study the entanglement entropy in the state that is invariant under the symmetry transformation associated to the degeneracy subspace. Finally, in section IV, we present the conclusions and outlooks.

LANDAU MODEL ON A TORUS
In this section, we introduce the model to be discussed later and fix the notation. The model is electromagnetism in a space-time cylinder coupled to a non-relativistic charged point particle. We review its relation with the Landau model on a torus, that is, a charged particle moving on a torus subject to a transverse magnetic field.

A. Electromagnetism Coupled to a Charged Point Particle
We consider a space-time cylinder with coordinates s = (s 0 , s 1 ) ≡ (t, s), where t ∈ R and s ∈ [0, 2πR). If τ is the proper time of the point particle, with electric charge q and mass m, then its classical trajectory can be parametrized as r(τ ) = (r 0 (τ ), r 1 (τ )) ≡ (t(τ ), r(τ )).
The electromagnetic field is described by If we align the proper time τ with the time coordinate of the particle, so that t = τ , then we have J 0 = q δ s − r(t) ≡ ρ and J 1 = q δ s − r(t) ṙ ≡ j. Therefore, in local coordinates, the field equations read For simplicity, we consider a non-relativistic charged point particle. Its equation of motion reads 2 mr = qE. From now on, we set m = 1.
The above equations of motion are obtained from the Lagrangian In the Coulomb gauge 3 , ∂ s A 1 ≈ 0, the Gauss law (2) becomes ∂ 2 s A 0 = ρ. Therefore, A 0 is not dynamical and its only role is enforcing the Gauss law. This constraint can be readily solved as where G(s, s ) is the Green's function of the operator ∂ 2 s . Generically, finding this Green's function depends on the boundary conditions. Here we will not need to find G(s, s ) explicitly. What we will need is only the fact that, for any choice of boundary conditions, G(0, 0) 5 is a constant. For example, for G(0, s ) = G(2πR, s ) = 0, if follows that and G(0, 0) = 0.
In the Coulomb gauge, the field A 1 (t, s) does not depend on the spatial coordinate s, i.e., Moreover, gauge invariance implies that a(t) is valued on a circle of length 1 eR . In fact, consider a gauge transformation g = e ieΛ(t,s) , where e denotes the elementary electrical charge, that winds around the spatial dimension. In order to be a single-valued transformation, Λ(t, s) must satisfy A possible solution is Λ(t, s) = ns eR . In this case, the gauge field transforms as The equivalence of configurations of the field related by gauge transformations implies that we can restrict a(t) to 0 ≤ a(t) < 1 eR .
Now going back to the Lagrangian, the Coulomb gauge allows us to rewrite L EM as where we have used that ∂ 2 s A 0 = ρ. Applying (5), we then obtain Therefore the Lagrangian can be written as where G ≡ G(0, 0) and the boundary terms are constants which can be adjusted to zero.
After completing the square, we finally arrive at From now on, we define the elementary electric charge as e = 2π and write the charge of the particle as q = −eθ = −2πθ. We also choose the specific value R = 1 2π , such that the spatial direction has unit length. Observe that, under these considerations the gauge field a(t) is also valued on a circle of length one. Moreover, a may be replaced by x and the position of the particle r by y. Therefore, x ∈ [0, 1), y ∈ [0, 1) and the Hamiltonian corresponding to the Lagrangian (12) reads where Since we are interested in studying the entanglement between the electromagnetic and the matter sector, in the following sections we will consider the quantum version of the Hamiltonian (13). It can be straightforwardly found through canonical quantization, i.e., by promoting x, y and p x , p y to operators acting on L 2 ([0, 1) × [0, 1)) that satisfy the canonical

B. Landau Problem on a Torus
The Hamiltonian (13) is also obtainable from the problem of a particle of charge 1 moving on a torus with local coordinates 0 ≤ x, y < 1 in presence of a constant magnetic field B z = 2πθ in the transverse direction. This is the famous Landau problem on a torus.
In Appendix B of [44] this model is studied in detail for the case θ = 1 (see also [46]).
Note that the Hamiltonian (13) is written in the gauge A y = 2πθx and A x = 0, so that (we denote the gauge field as A to avoid confusion with section II A). Moreover, we can define the momenta There is a second set of translation operators that commute with the above momenta and therefore with the Hamiltonian (13). These translations are generated by v x = p x + 2πθy and v y = p y .
Let us now obtain the eigenfunctions of the Hamiltonian (13) in the coordinate basis.
First, we set up the boundary conditions. This Hamiltonian is self-adjoint if the boundary conditions are ψ(0, y; t) = e 2πiθy ψ(1, y; t) , Similar conditions apply to the first derivative of the wave function.
Recall that the length of the torus in each direction is 1. Hence a full rotation around each direction of the torus is performed by Now, if one starts with a wave function at point (x, y) ≡ (0, 0) and transports it to (1, 1), there are two possible paths, Equivalently, Notice that the charge 2πθ plays the role of a central charge for the translations on the torus.
Moreover, since the final result at (1, 1) should be independent of the path, we obtain θ ∈ Z.
This is electrical charge quantization in the field theory and magnetic flux quantization in the Landau model.
The above quantization has also implications for the degeneracy of the Hamiltonian. The reason is that there is a further discrete translation on the torus obtained by the θ-th root of the translation operator V x . We define the operator V θ such that The operator V θ commutes with the Hamiltonian (13). Then we may consider the quantum The solution to the stationary Schrödinger equation Hψ k (x, y) = Eψ k (x, y) may be written as where with f satisfying a harmonic oscillator equation with angular frequency 2πθ. The energy levels are given by E λ = 2πθ(λ + 1/2) with λ ∈ Z * . Note that the wave functions are not defined for θ = 0, which would correspond to zero transverse magnetic field.
In the following section, we will be interested in the ground state solutions, λ = 0, where ϕ nk (x) ∼ e −πθ(x+n+ k θ ) 2 . Therefore, the ground state wave functions are given by where N is a normalization constant. The wave function (27) can be rewritten using the Jacobi ϑ function,

III. ENTANGLEMENT ENTROPY
In this section, we study the entanglement between the charged particle and the electromagnetic field in the ground state of the theory described by the Lagrangian (4). According to the analysis performed in section II, this is equivalent to measuring the entanglement between the two degrees of freedom, x and y, of the Landau model on a torus defined by the Hamiltonian (13). We shall compute the entanglement entropy in the ground state of the latter system. As we have seen in the previous section, the ground state is degenerate. We shall then analyse the entanglement entropy in the degeneracy subspace {|ψ k } θ−1 k=0 where the vectors |ψ k are such that ψ k (x, y) ≡ x, y|ψ k with {|x, y } being the coordinate basis and ψ k (x, y) being the ground state wave function given in (27).
In order to define the entanglement entropy, we need the associated density matrix ρ k = |ψ k ψ k |, whose entries in the coordinate basis are ρ k (x, y; x , y ) = ψ k (x, y)ψ k (x , y ) .
Now we have to trace out one of the degrees of freedom, say y. This is equivalent to tracing out the degrees of freedom of the particle in the gauge field theory. Then we obtain the reduced density matrix k associated with the gauge field A µ , Finally, the (von Neumann) entanglement entropy is defined as If we had chosen to trace out the degree of freedom x that corresponds to the gauge field, the reduced density matrix would be associated with the charged particle. Nevertheless, the resulting entanglement entropy does not depend on which reduced density matrix we consider.
Inserting the explicit expression of the wave function ψ k (x, y), see (25) and (27), we find that the reduced density matrix (29) is of the form or, in terms of the ϑ function, Now the direct way to obtain the entanglement entropy would be to compute the eigenvalues of k and then plug them in (30). However, this is in principle a difficult task that we shall bypass approximating k in two different ways.
First, observe that k (x, x ) is made of peaks localized along the line x = x, as Fig. 1 (a) illustrates for θ = 3 and the three possible values for k. In Fig. 1 (b) we represent separately k (x, x ) for each value of k delimiting the interval [0, 1), which is the domain where the variables x, x are defined. Observe that as k grows the peaks of k (x, x ) move down along the line x = x. Each peak of k (x, x ) comes from one of the modes in the sum (31). Therefore, only the modes that correspond to a peak inside the square [0, 1) × [0, 1) contribute to k (x, x ). For example, in Fig. 2 we can see that for θ = 3 the only modes that contribute are n = −2, −1, 0.
In fact, for any value of k and θ, one can see that only two n modes are significant in the interval x ∈ [0, 1) so that we can neglect the rest of them in the calculation of the reduced density matrix. Thus, we can treat the latter as that of a two-level system.   For k/θ < 1/2, only the peaks corresponding to the modes n = 0 and n = −1 are relevant in the interval [0, 1). Therefore, the reduced density matrix can be approximated as If k/θ > 1/2, then the non-neglectable peaks in the interval [0, 1) correspond to the modes n = −1 and n = −2 and In the case k/θ = 1/2, only the peak with n = −1, that it is at x = 1/2, gives a significant contribution. Thus These approximations may be written in terms of normalised functions u nk (x). To do so, we define p nk , where erf(z) denotes the error function. In terms of p nk we can write the approximations as: • For k/θ < 1/2, and u nk (x) = 1 λ nk ϕ nk (x) .
• For k/θ = 1/2, Note that it follows from the definitions of λ kn and λ kn that λ −1k = 1 − λ 0k and λ −2k = 1 − λ −1k . Therefore, the entanglement entropy can be expressed as The latter case only happens when θ is an even number.
Using the identity between the error function and the confluent hypergeometric function of the first kind M (a, b, z) (see, e.g., Eq. 13.6.7 in [47]), we have where we have introduced the notation M(z) ≡ M (1/2, 3/2, z) and χ k = k/θ. Hence we find , .
From these expressions it is clear that the entanglement entropies for k/θ < 1/2 and for k/θ > 1/2 are related by the transformation k/θ → 1 − k/θ.
Let us check numerically the accuracy of the above results. This is done by expanding the functions ϕ nk (x) in Fourier modes, In the basis of Fourier modes the entries of the reduced density matrix (29) are given bỹ In order to compute numerically the entanglement entropy we truncate the matrix (˜ k (p, p )) restricting the indices p, p ∈ Z to the interval −N ≤ p, p ≤ N . Then we calculate the eigenvalues of this sub-matrix and we we plug them in the expression of the entanglement entropy (30).
The value obtained numerically for the entanglement entropy should converge to that predicted by the expression (32) as we increase the cut-off N . In Fig. 3 we compare the results for a given θ and k varying from 0 to θ − 1. As we can see the results agree for N large enough. Notice also that the entanglement entropy varies with k. This means that there is an ambiguity associated to the entanglement entropy of the ground state.
It is also interesting to analyse how the entanglement entropy behaves as a function of θ for fixed k. Recall that θ is proportional to the electrical charge of the particle (we have set q = −2πθ). In Fig. 4, we plot S k in terms of θ for several fixed values of k using the analytical approximation (32). The initial point of the curve for each k corresponds to θ = k + 1. Observe that, due to the symmetry k/θ → 1 − k/θ, the initial points of all the curves with k > 1 also belong to the curve for k = 1. As θ increases, S k decreases until θ = 2k, where it vanishes. From this point, S k increases tending to log 2 when θ → ∞ (infinite transverse magnetic flux in the associated Landau model). Note that as k is larger, the entropy saturates more slowly to the asymptotic value log 2. We can conclude that there is an upper bound for the entanglement entropy in the degeneracy subspace of the ground state, which is exactly that of a maximally entangled two-level quantum system.
Another interesting case is θ = 0. It corresponds to a particle with zero electrical charge (zero transverse magnetic flux in the Landau model). For θ = 0, the analytical approximation (32) is not well defined. Nevertheless, since the particle and the gauge field are decoupled, the degrees of freedom of the Landau model, x and y, are separable; that is, the wave function of the ground state, that now is not degenerate, can be factorized in the form ψ(x, y) = X(x)Y (y). This implies that the entanglement entropy is zero for θ = 0.

A. Entanglement Entropy for a V θ -invariant Mixed State
We have seen that the entanglement entropy changes inside the degeneracy space of the ground state, that is, it varies under the translation defined by V θ . However, there is a state The reduced density matrix after the partial trace in y is Recall that the entanglement entropy is not necessarily additive. So one must be a bit careful. Indeed, it is easy to see that states for different k's are not mutually orthogonal.
Nevertheless, observe that the Fourier coefficients of the n-modes ϕ nk (x), satisfy the identitiesφ and hence Using this property we can rewrite the reduced density matrix (31) as We then obtain that the entropy of for the reduced density matrix (35) is Observe that the entropy of the mixed state (34) (without the partial trace) is log θ. This entropy comes from the sum over the degenerate states |ψ k considered to be equiprobable.
Hence, in the entropy of the reduced density matrix, the term log θ is due to the "classical" or statistical average over the states |ψ k . Therefore, we can conclude that in S θ the term associated to entanglement is log 2.
It is a striking result that the entanglement entropy is precisely related to the number of quantum degrees of freedom Ω Q of the system, that is, log Ω Q ≡ log 2. Moreover, the other term of the entropy S θ is related to the central charge θ. Recall that the central charge appears due to the statistical average associated with the invariance under translations generated by V θ .

IV. CONCLUSIONS
In this work, we studied the ground state entanglement entropy between an electromagnetic field and a charged non-relativistic particle on a space-time cylinder. In order to compute this entropy, we resorted to the fact that a Yang-Mills field theory defined on a space-time cylinder can be mapped to the problem of a free quantum particle moving on the gauge group manifold. In our case, we considered an electromagnetic field, for which the gauge group is U (1), and therefore the corresponding manifold is the unit circle. Since here the gauge field is coupled to a non-relativistic particle, the associated quantum mechanical problem is a particle moving on a torus with a transverse magnetic field: the Landau model on a torus. The two degrees of freedom of the particle on the torus correspond respectively to the gauge field and to the non-relativistic particle in the field theory problem.
Therefore, the computation of the entanglement entropy between the electromagnetic field and the non-relativistic particle was reduced to taking the partial trace of one of the degrees of freedom in the wave function of the particle on the torus and computing the entropy from the corresponding reduced density matrix. Since the ground state of the Landau model is degenerate, we analysed the entanglement entropy in the degeneracy subspace. We performed this analysis treating the reduced density matrix of the states that generate this subspace as that of a two-level system. We obtained an approximate analytical expression for the entanglement entropy which was checked numerically. In particular, we found that, when the electromagnetic field and the particle are decoupled, the entanglement entropy is zero while, when the particle's charge goes to infinity, the entanglement entropy tends to log 2.
We also considered the state that is invariant under the group of translations V θ behind the existence of the ground state degeneracy. The only way to construct a V θ -invariant state is by considering the equiprobable ensemble of the pure states that generate the subspace.
Since this is a mixed state, the entropy of the reduced density matrix is not a proper measure of the entanglement. Nevertheless, we were able to distinguish between the contributions due to the statistical average and due to entanglement. The statistical contribution only depends on θ (the electrical charge in the field theory/the transverse magnetic flux in the Landau model) that is precisely the central charge of the translations V θ on the torus. Hence, we concluded that the contribution from entanglement is log 2, which could be interpreted as the logarithm of the number of degrees of freedom in the Landau model.
The natural continuation of this work is to take a non-Abelian Yang-Mills theory instead of an Abelian one and study how the results obtained here generalise to the SU (N ) gauge group. In particular, the YM theory would be mapped to a particle moving along a different gauge group manifold. For example, for SU (2) we would have a particle moving along S 3 .
Solving its dynamics would then mean working with a non-trivial set of Wong's equations [48]. Another interesting aspect to analyse is the evolution of the entanglement between the matter and the gauge sectors after a quantum quench. This could be done by preparing the system in the ground state in which the gauge field and the particle are decoupled and then suddenly turning on the interaction term, for example. Indeed, the non-equilibrium dynamics of a 1+1 dimensional U (1) gauge theory coupled either to fermions [49] or to bosons [50] has recently been investigated and it was observed that the system may not thermalize. We plan to tackle these problems in the future.