Entanglement entropy for a Maxwell field: Numerical calculation on a two dimensional lattice
We study entanglement entropy (EE) for a Maxwell field in dimensions. We do numerical calculations in two dimensional lattices. This gives a concrete example of the general results of our recent work  on entropy for lattice gauge fields using an algebraic approach. To evaluate the entropies we extend the standard calculation methods for the entropy of Gaussian states in canonical commutation algebras to the more general case of algebras with center and arbitrary numerical commutators. We find that while the entropy depends on the details of the algebra choice, mutual information has a well defined continuum limit as predicted in . We study several universal terms for the entropy of the Maxwell field and compare with the case of a massless scalar field. We find some interesting new phenomena: An “evanescent” logarithmically divergent term in the entropy with topological coefficient which does not have any correspondence with ultraviolet entanglement in the universal quantities, and a non standard way in which strong subadditivity is realized. Based on the results of our calculations we propose a generalization of strong subadditivity for the entropy on some algebras that are not in tensor product.
In a recent paper  we have analyzed the problem of defining a local entropy for gauge fields. The inconveniences caused by the constraint equations of the physical degrees of freedom pointed to a natural setting within an algebraic approach for states and local algebras.
The entropy on a region of the space is usually understood as the von Neumann entropy of the density matrix reduced to the degrees of freedom on that region. From the algebraic point of view, this is the entropy which results from the density matrix on a local algebra associated to the region. In general, this algebra may have a center , a set of operators that commutes both with the operators in the algebra and its commutant . Typically, the center is produced by the constraint equations. Only the case with trivial center admits a bipartition of the Hilbert space as tensor product of subspaces of inner and outer degrees of freedom, and in this case the local entropy is an entanglement entropy for a global pure state.
Of course, there is not a unique way to assign a local algebra to a region and different assignations give rise to ambiguities in the entropy. Even if these ambiguities are present in all theories, when the elementary excitations are not point like or more precisely, the operators are attached not to vertices but to links in the lattice, the standard prescription of identifying the region with the subset of operators attached to vertices within the region has to be revised. This is the case in lattice calculations for gauge fields, where the local gauge invariant algebra is generated by the electric link operators and Wilson loops along closed paths. The constraint equations give extra relations among the variables. A particular choice of local algebra with electric center has been discussed previously in the literature in a way unrelated to the algebraic formulation .
In this paper we show how this general scheme applies in a specific example. We consider a Maxwell field theory in dimensions. In order to evaluate the entropies for general algebraic prescriptions for the local algebras we generalize the formulas for Gaussian states in canonical commutation algebras to the case where the the commutators are arbitrary matrices and the algebras have center. The techniques can be used for free (uncompactified) gauge fields (and more generaly massive or massless tensor fields) in any dimensions. We avoid using Wilson loop variables for this free model.
We are able to show in detail the main prediction of : The large ambiguities in the entropy introduced by the uncertainties of the algebra choice and the universality of the mutual information in the continuum limit. This universality is a consequence of the fact that mutual information is ordered by inclusion of algebras. More technically, in the cases with center, we also show the classical Shannon term is not relevant for the continuum limit of the mutual information, and the calculation can be reduced to a unique arbitrary sector in the central decomposition.
Hence, the continuum limit eliminates the ambiguities in the relation between algebras and regions. In a certain sense, a geometric region has only meaning in terms of the content of the model once the continuum limit has been achieved.
We also study some universal terms that can be obtained from the entropy itself, as the usual logarithmically divergent term due to the corners on the region boundary in dimensions. We find the logarithmic coefficient has also a curious additional contribution proportional to the number of connected components of the region. This should be regarded as related to the peculiarities of the gauge field in three dimensions. Surprisingly, this term is at the same time ultraviolet divergent and non local. However, we argue the ultraviolet nature of this term is not captured by any universal quantity in the model, i.e., the short distance behavior of mutual information. A related logarithmic term for the compactified Maxwell field in the limit of decompactification has been discussed in the literature in relation with the -theorem [3, 4].
This three dimensional model is dual to a “truncated” scalar field. The algebra generated by the electric and magnetic physical operators coincides with the one of time and space like derivatives of the scalar field, where the field operator itself has been removed. We compare several universal terms of the gauge field (and truncated scalar) model with the model of a full scalar field.
Even if with some specific choices of generating operators the local algebras have trivial center, the constraints reappear in other interesting phenomena. For example, the usual geometric expression of strong subadditivity (SSA) property has to be reinterpreted in algebraic terms and generalized with respect to its usual form.
We conclude with some discussion. In particular we revisit the issue of defining a topological entanglement entropy in a lattice using Levin and Wen  prescription in an algebraic way. This is connected with our discussion of strong subadditivity.
2 Lattice Maxwell field in dimensions
The physical operators and of the dimensional Maxwell theory are written as
in terms of the tensor field . The canonical commutation relations (in Lorentz gauge)
give the gauge invariant commutation relations
The Hamiltonian in terms of and is
Now, we discretize the model in a square lattice. The standard procedure for a gauge field is to assign the electric field variables to the links of the lattice and elementary Wilson loop operators to the plaquettes. For the non-compact Maxwell field we can consider directly the magnetic field operator (corresponding to the magnetic flux on the plaquette) and associate it to the dual lattice vertices in the middle of the plaquettes as shown in figure 1. Using directly the electric and magnetic variables allow us to profit from the Gaussianity of the model.
More precisely, we define the electric operators associated to horizontal and vertical links respectively, as and , where are the coordinates of the initial and final points of the link. This notation is useful but redundant since we can define the electric variables named by the initial vertex of the vector,
The magnetic operator is denoted by the left down corner of the plaquette (see figure(1)).
Hence, there are twice as many electric variables than magnetic variables. However, half the electric variables are redundant because of the constraint equations of electric flux (Gauss law) in two dimensions. This gives the relations
where the sum is over all links with common vertex . In this equation, it is assumed that the electric field component is the corresponding one to the link direction and also that links have orientation which changes the sign of the electric field attached to it .
In the lattice theory, the commutation relations become
Finally, the Hamiltonian writes
where the sum is over the vertices for the magnetic variables and over the links for the electric ones. In contrast to the lattice Hamiltonian for a scalar field, this Hamiltonian is trivial as a bilinear form in the variables. All the dynamics is hidden in the constraint equations and non trivial commutation relations.
2.1 Maxwell-scalar field duality in the lattice
In dimensions, the Maxwell theory is dual to the theory of the derivatives of a massless scalar . The duality is written
giving the following identifications
This gives a complete one to one map between the theories, including the commutation relations and Hamiltonians. Note however, that the electromagnetic fields do not capture the full scalar theory but only the derivatives of the field. Hence the algebra of operators is strictly smaller than the one of the full scalar which includes .
The discrete version of the above relations is expressed defining the scalar field variables on the sites of the dual lattice as shown in figure (2). The electric link operators are related to the differences of the scalar field operators in the orthogonal direction in the dual lattice
and the magnetic operators are given by the corresponding momentum operators
2.2 Truncated scalar theory
The duality relations (15) and (16) show it is equivalent to consider the gauge fields or the gradients of the scalar. We find convenient this later expression of the model for the entropy calculations. Summarizing, this model consists of the subalgebra of the scalar field algebra (truncated scalar algebra) generated by
The commutation relations
where the sum is over the links on a plaquette , with the same orientation along a curve encircling the plaquette.
The lattice Hamiltonian for the scalar field is
From this Hamiltonian, the vacuum correlation functions for the field and momentum operators are found to be 
The correlators for the new variables and can be easily written in terms of the ones of the variables, for example
These correlators are equivalent to correlators for lattice electric and magnetic fields and play an important role in the EE calculation we discuss later in Section 4.
3 Local algebras and regions
A “region” in the lattice has to be defined by the physical content of the model. That is, we must choose an algebra of local operators that defines it. In this sense, the assignation of algebras to regions is subject to ambiguities and several choices are possible. In the case the lattice operators are attached to vertices (such as a scalar field), a natural election seems to be to choose the local algebra associated to the region as the one generated by the operators attached to vertices within the region. Of course, this is just one possible choice. As we will discuss later, even in this case, we can think in different possibilities for the local algebra.
In the case of gauge fields, we study three possible choices defined in  which are shown in figure (3): The algebras with electric, trivial, and magnetic center. Figure (3) also shows the equivalent description of these algebras in terms of the dual truncated scalar algebra.
In the electric center choice, we keep all the operators, inside and along the boundary. The constraints applied to the sites on the boundary show the electric fields normal to the boundary are automatically included in the algebra. As they commute with the rest of the operators on the algebra, they form the center. The electric center choice coincides with some constructions developed in the literature to define entropies in gauge theories .
The trivial center case, consists in choosing all the electric and magnetic operators within the region and only one electric link operator along the boundary. This corresponds to remove the link operators along a boundary maximal tree . This election provides a good partition as tensor product between inside and outside degrees of freedom, giving place to a purely quantum entanglement entropy. However, this is not unique, we still have the possibility to vary the position of the electric field chosen at the boundary.
The magnetic center choice corresponds to the case where all the links along the boundary are removed. The center then, consists in the boundary Wilson loop, or equivalently the sum of the magnetic operators living in the interior plaquettes.
For the truncated scalar algebra, we have analogue cases. The electric center, where the center is given by all the boundary links, the trivial center case, where the number of links and momentum operators are the same, and the magnetic case, with a one variable center given by the sum of all momentum operators on the region.
4 Entropies of Gaussian states in terms of correlation functions
In a general algebra, the center produces superselection sectors which cannot be changed by the local operators. The global state is then reduced into these sectors to give a block diagonal density matrix
The entropy associated to the algebra has a precise definition given by 
where the first term corresponds to the classical Shannon entropy
of the probability distribution of the variables which simultaneously diagonalize the operators in the center. The second term is an average of the corresponding purely quantum contributions
In the following, we are going to compute explicitly these entropies for the case of Gaussian states in algebras of coordinate and momentum operators. Hence, we consider cases with center containing operators with continuum spectrum (for example ) and the above formulas are generalized by converting the sum over discrete sectors into integrals.
4.1 Algebra of canonical conjugated variables
with . In this case, the entropy can be calculated in terms of the correlators
where , and , are the correlators matrices (restricted to the algebra).
Algebra of canonical conjugated variables with non trivial center
The entropy of algebras with center formed by operators with continuous spectrum suffers from ambiguities due to the lack of a mechanism to fix the field normalization. However, mutual information between two algebras in tensor product (corresponding to two separated regions in a lattice model for example)  is free from these ambiguities. Here, we deduce the general expressions for a set of coupled harmonic oscillators which we will use later in Sections 5 and 6 to calculate the mutual information between two sets for a scalar and a Maxwell field.
Consider a set of harmonic oscillators with variables , . We choose the algebra as the one generated by all the operators but only a subset of the momentum operators with . Hence this algebra has a center formed by the field with .
We want to compute the entropy on this algebra for a state in that we assume is a Gaussian state. Then, it is convenient to write the density matrix in in a basis which simultaneously diagonalizes all elements in the center. In this case, we choose the coordinate basis. We have for a Gaussian state
where is a normalization constant, and due to hermiticity, and are real symmetric. The relation of these matrices with correlation functions on follows from
All higher point functions are obtained by Wick’s theorem for a Gaussian state. Inversely, we have
Now, the probability density of a particular value for the variables on the center is again fixed by the correlation of the field in this region ,
The reduced density matrix in corresponding to this value of the variables on the center follows from (35) by fixing these values for the fields on , and a change on normalization,
we get the density matrix
Evidently, this change of variables does not change the entropy. Very conveniently, the density matrix (46) is independent of the values of the field at the center. Hence, the average of the quantum entropy in over the values of the field on the center is trivial, and we get for the quantum part of the entropy
The whole entropy contains also a classical Shannon term due to the center probability distribution (42)
This classical Shannon term can only have unambiguous meaning in relative entropy quantities, for example the relative entropy of two states or a mutual information between two regions for the same state. This is because the normalization of the fields in the center are not fixed by the commutation relations. For example, choosing the field instead of we get
The mutual information between two regions and , with centers formed by the fields in , , is given by
This gives the desired expression for the mutual information of two algebras with center purely in terms of the correlation function matrices.
4.2 Generalization for arbitrary commutators and constraints
In the case the variables satisfy canonical commutation relations and the algebra has a trivial center, the entanglement entropy associated to a region , can be calculated in terms of the correlators restricted to according to eq. (34). In the case the algebra has a non trivial center the entropy is given by the sum of the quantum (47) and classical (50) parts. In this section, we show how these formulas can be extended to the case of conjugated variables having general numeric commutators. This is the case of the physical variables in Maxwell theory, or the variable , in the truncated scalar model.
Consider an operator algebra with non canonical commutation relations
Suppose we are interested in a subalgebra without center with . We can define new canonical variables , , using
The correlation functions restricted to the region will be
The entropy is then calculated in terms of as (34).
It is important to notice that the entropy is a function of the algebra and the global state. Hence, if instead of the variables , , we take arbitrary linear combinations of these, we end up with the same entropy, as long as we consistently change the correlation matrices and commutators. The same can be said for the case where there are constraints. For example, we can have more variables than because some combinations of the variables are zero, as happens for the electromagnetic field and the truncated scalar.
The case with center
The general case in gauge theories involves algebras with center and non trivial commutators. The previous discussion about the entropy can be generalized to this case. Consider the algebra generated by , with and with . We assume for , , in such a way that , span the center of the algebra. Using again the transformation (57) we arrive at the case studied in (4.1.1) and we find the quantum contribution to the entropy is given by
Here is the commutation matrix (53) between with and with . The classical contribution has the same form as before
The case for a center formed by with is analyzed in the same way, interchanging .
4.3 Correlators for the vacuum state
We are interested in vacuum entropies. Here, we show how to compute the correlators for simple quadratic Hamiltonians relevant for the Maxwell field.
The vacuum correlators for Gaussian states can be directly calculated from the kernel of the quadratic Hamiltonian  for the free scalar field. It is easy to show that this result can be generalized for the case of variables with non canonical commutation relations. Consider a theory with Hamiltonian
for the canonical conjugated variables , p, with
and defined in (53). Changing variables, the Hamiltonian takes the form
The two point correlation functions for the fundamental state are given in terms of and as ,
This gives for the original variables
In the case is the identity matrix, which is relevant for the Maxwell field, we have
In section (2.2) we showed how correlation functions for the Maxwell field in dimensions are obtained from scalar correlation functions. Of course, this coincides with the above formulas when applied directly to the gauge field. These formulas can be used to obtain the lattice correlators of electric and magnetic fields in other dimensions. Notice that formulas (75) and (76) for the correlators do not have singularities for non invertible correlator matrix. This means we can use them for the gauge fields directly without necessity of solving for the constraints. Indeed we have for the Maxwell Hamiltonian expressed in the variables and where the constraints have not been used. Then, the commutator matrix is in general rectangular, but this does not affect the validity of (75) and (76).
5 Some examples with a massless scalar field
Before considering the gauge field and the truncated scalar, we exemplify the methods discussed above with a massless scalar fiels and several different algebra choices.
We consider a simple case of two square regions and of size lattice points, separated by the same number of lattice sites, and compute the mutual information for four different algebras for each square as shown in figure (5):
a) The full algebra of the squares of size with trivial center, already studied in .
b) The algebra which results by removing all the operators from the boundary with a center formed by the remaining along the boundary.
c) The opposite case where the removed fields are the and we have an algebra with momentum center.
d) The algebra resulting from the elimination of all operators of the boundary (that is, we consider squares of side in the center of the original squares).
We have for these algebras
The mutual information is monotonously increasing with the algebra. Hence, we expect to have
where and are the two squares.
In figure (6) we show the numerical calculation of the mutual information between two squares of the same size and separated by a distance , being the number of vertices, for the different algebra choices. Here, we use the correlators (25), (26) and formula (52) to calculate the mutual information. The figure shows the ordering relations (79) are obeyed. We expect a convergence of the mutual information for large to the continuum limit. In fact, the limit values of the mutual information obtained by a fit of the form are , , and for the algebras , , and , respectively, showing a remarkably fast convergence to a common constant value already for sets of size . This mutual information is a very small number, approximately bit for infinitely many degrees of freedom in the continuum limit. This reflects the locality of the theory.
Figure (6) illustrates our general argument  on why mutual information must have a unique continuum limit disregarding the details of the algebra choice for the region. This is because some prescription for algebra choice (in this case and ) can be bounded above and below by another prescription with slightly different size for the regions (in this case and ). As the continuum limit is reached, necessarily, all these prescriptions lead to the same values.
On the other hand, regarding the entropy, different choices of local algebra result in dramatic changes. In dimensions, we expect the entropy for massless theories and polygonal sets to have the following form as a function of the overall size ,
For a square region and a massless scalar field, we have
a) Trivial center - full algebra
where we have allowed for an additional term in the fit and taken squares of size up to . The presence of a center and in consequence, a classical contribution to the total entropy, results in relevant changes on the constant and area terms while the only preserved term is the logarithmic one. This can be seen in the coefficients we find for two different center choices,
b) Center of
c) Center of
where is the normalization constant in the formula for the Shannon entropy of the center (51).
Clearly, the area and constant terms suffer large changes with the change of algebra prescription. Of course, these non universal terms suffer other ambiguities in the continuum limit, for example, they are not rotational invariant. It is remarkable the area term (and hence the full entropy) can easily turn to be negative due to the choice of the center prescription. Negative entropies were found for non minimally coupled scalars and gauge fields in early calculations using the replica trick . The classical center hints to a natural explanation for these puzzling results.
6 Entropy and mutual information for the gauge model
We consider now the gauge model and calculate the entropies and mutual information for different geometries. Calculations are equivalently done in the dual model of a truncated scalar described in Section 2 using the tools developed in Section 4.
First, we calculate the mutual information between two squares of equal sizes separated by a distance equal to the squares size for the three different algebra choices of figure (3). The result is shown in figure (7). As expected, the electric center has larger mutual information than the trivial center, and this is in turn larger than the magnetic center, in agreement with the monotonicity property of mutual information and the fact that the algebras in figure (3) are ordered by inclusion.
for the electric, trivial, and magnetic centers respectively.
The contribution of the classical Shannon term to these universal numbers is shown in figure (8). We have that goes to zero both for the electric and magnetic centers. It falls much faster for the magnetic center because it contains only one degree of freedom in contrast to the electric center which contains an area increasing number of degree of freedom. This also confirms general expectations that has zero continuum limit  because it is bounded above by the mutual information of regions on the boundary with lattice spacing (cutoff) width and fixed distance in the continuum interpretation. Hence, the Shannon term does not seem to have physical meaning in the continuum limit. This, together with the fact that all the superselection sectors for a given center give place to the same entropy (as shown in section 4) leads us to the conclusion that mutual information can be computed in the continuum limit from the reduced density matrix for just only one arbitrary sector, for example a fixed arbitrary normal electric field at the boundary in the electric center choice.
Note the numerical value for the gauge field mutual information is around times smaller than the one corresponding one to the same two squares for the scalar field, . In fact, the gauge model is (locally) identical to a subalgebra of the scalar field, the one generated by the gradient field. Hence, we expect the mutual informations for any two regions always satisfy
We can learn more about the similitudes and differences between the scalar and gauge models by studying two limits on the mutual information.
This is an area term for the mutual information in the coincident limit. Now, we can argue that the constant coefficient must be the same for the scalar and the gauge fields, as follows. The calculation of (89) for a scalar starts by realizing that in the small limit the term (89) is extensive in the direction of the coordinate parallel to the two nearby sides of the squares. Therefore, we can replace the two squares by two half-spaces for the sake of this computation. Then, we can compactify the space in a circle with large radius in the direction without changing the extensive part of the entropies (see  for details). Decomposing the scalar into Fourier modes
in the large direction, the mutual information turns into a sum over the mutual informations of massive dimensional scalar fields, where the mass is produced by the momentum in the transverse direction, . We have
where is the entropic -function of a massive scalar in , and is the entanglement entropy for an interval of size and a field of mass . Now, for the truncated scalar we can do the same calculation. For any mode with in (90) the model produced by the truncated scalar is the same as the one of the full scalar. This is because for non-zero momentum the operators in (90) belong to the truncated algebra as well, since the integral of these modes on the direction is exactly zero. Hence, the coefficient for the truncated scalar and the gauge field are given by the same integral (91), differing only in a measure zero set at , and we have
The second limit we want to look at is when the two regions and have large separations. For two squares this is the limit of , with the separation distance. In this case we have that mutual information falls as the square of the correlation function of the lowest dimension operator  (see also  for the scalar case). This is for the scalar model and for the gauge field. Thus, we expect
We can confirm these expectations for two squares in figure (9), where we have plotted mutual information against . The short and long distance behavior nicely approach (89) and (93), (94). For two squares we can get the coefficients in (93), (94) approximately as , .
Summarizing, the mutual information for the scalar is always larger than the one for the gauge model. At short distances they have the same leading ultraviolet divergent term (area law) because they have the same ultraviolet modes, while at larger distances the scalar field has much larger mutual information. This is because the gauge field does not contain the scalar “center of mass” mode, , which controls the largest share of mutual information for large distances.
6.1 Logarithmic term in the entropy
Let us compute the logarithmic terms on the entropy for different choices of center. These are generally universal terms, and we have seen they do not depend on the center for the scalar field. Again we calculate the entropy for squares as in figure (3) and fit with a function of the form with squares of size up to points. We get for the trivial center case (see figure 10)
where is the logarithmic coefficient for a scalar. Surprisingly, for the gauge model we get plus the logarithmic coefficient for the scalar on the square. For the electric and magnetic centers we get similarly
Thus, the logarithmic term is independent of the center choice. We have also checked it is rotational invariant in the lattice (that is, it does not change for rotated squares).
However, as we have pointed out, the entropies for algebras with continuum center are not well defined, in the sense that its classical contribution (64) can vary with field normalizations. To understand the validity of these results for the logarithmic term we notice that a change in normalization by a factor changes the entropy of the square by , where is the number of degrees of freedom in the center. This grows with the area in the electric center and is just one for the magnetic center. Hence, no changes in the logarithmic term are expected if we change the normalization by a factor independent of the number of points in the square. Changing the normalization by a factor depending on does not seem to be fair, in the sense that it would be a prescription for doing computations which includes information a priori on the object on which one wants to compute the entropy.
In this regard, it is interesting to note that the classical entropy of the center for the electric choice does indeed give a non zero contribution to the logarithmic term (this is in fact a large fraction of the logarithmic term). Hence, even if the mutual information does not depend on the classical terms in the continuum limit, the logarithmic term in the entropy is sensible to the classical contributions.
To discern how is this contribution related to the presence of angles in the square, we have computed the logarithmic term for several other shapes illustrated on figure 11. Here, we list the results for for the different regions and compare with the coefficient for the same regions in the full massless scalar theory (which is always a sum over the contributions of the different angles):
a) Entropy of two equal squares separated by a distance of the size of the square:
b) Square of size , with a centered square hole of size :
c) Square of size with a removed corner square of size :
In figure (10) we also show the logarithmic term in the entropies of circles in the square lattice. In this case, no angle term is expected and we get for circles of radius up to . Of course, we have considerable noise for circles, but still the logarithmic term is clearly seen and can be extracted from the data with good precision.
From the above results and many other checks, including triangular regions with angles different from , we conclude the general form of the logarithmic coefficient is
where is the number of connected components in the region, and is exactly the same function giving the logarithmic coefficient due to the angles in the full scalar theory. The presence of the same ultraviolet logarithmic term depending on the angles as in the scalar case can be explained by the same argument as for the coefficient in the mutual information (previous subsection). This is because the coefficient is computed by dimensional reduction in spherical symmetry  and all relevant dimensionally reduced modes for the scalar coincide with the ones for the truncated scalar.
On the other hand, the new contribution is topological and “counts” the number of components of the region independently of the shape. This contribution to the entropy is rather puzzling in two respects. The first one is that this is a term proportional to and hence apparently has an ultraviolet origin, but at the same time it does not look local on the boundary on geometrical grounds.
where is the region perimeter. For the truncated scalar we should have the same formula, with the same coefficients, plus the term . However, this last term is clearly impossible, since the mutual information of the truncated scalar is bounded above by the one of the scalar, and this term would violate this inequality for small enough . A direct check of this on the lattice is difficult because we have to go to the small limit but having first reached the continuum limit. We will say a bit more about this topic and its possible relation with the -theorem in [3, 4] in the discussion section.
The second puzzle is the positive sign of the logarithmic term