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Damtp-2014-13

Entanglement Entropy of two dimensional lattice gauge fields

###### Abstract

In this paper, we explore the question of how different gauge choices in a gauge theory affect the tensor product structure of the Hilbert space in configuration space. In particular, we study the Coulomb gauge and observe that the naive gauge potential degrees of freedom cease to be local operators as soon as we impose the Dirac brackets. We construct new local set of operators and compute the entanglement entropy according to this algebra in dimensions. We find that our proposal would lead to an entanglement entropy that behave very similar to a single scalar degree of freedom if we do not include further centers, but approaches that of a gauge field if we include non-trivial centers. We explore also the situation where the gauge field is Higgsed, and construct a local operator algebra that again requires some deformation. This should give us some insight into interpreting the entanglement entropy in generic gauge theories and perhaps also in gravitational theories.

PACS:

## 1 Introduction

The seminal paper [1] initiated the important question of defining the notion of entanglement entropy of a gauge theory. The definition of the entanglement entropy has been heavily based on a tensor product structure in the Hilbert space. To study entanglement in configuration space, it requires a tensor product structure in configuration space. Naively, this is a natural feature in the Hilbert space, since the world we experienced around us is local, and that it gives the impression that the Hilbert space naturally factorizes as a tensor product of spaces defined locally at each point in space. There are clear subtlties when we work with gauge theories, where it is well known that gauge theories are by construction made local by including gauge degrees of freedom. The gauge constraints such as the Gauss law implies that degrees of freedom at different locations are not entirely independent, and thus a naive factorization of the Hilbert space is not possible. This is clearly a significant issue both for gauge theories and gravitational theories [2, 3]. A clear understanding is thus crucial, also in ultimately formulating a theory of quantum gravity.

Until very recently, the replica trick has been the main tool employed to computing the entanglement enropy or the Renyi entropy in field theories, which in turn can be formulated as a path-integral in a conical space. This allows one to momentarily brush off issues of the Hilbert space and obtain some results – until it is realized that the issue in fact re-emerge as some ambiguities with edge modes that are localized at the entangling surface that is only recently understood [4, 5]. See also [6, 7].

Since the introduction of the notion of center in [1] into the discussion of entanglement entropy, it has made sense of entanglement even as the Hilbert space does not admit a factorization. Instead of directly considering the Hilbert space, one formulates the question in terms of the choice of an algebra attached to some region. There is some ambiguity in the selection of the algebra, and they can characterized by different centers. A large amount of work is inspired to understand the physical significance of these difference choices [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], and to demonstrate that these arbitrary choices can approach the same value as the continuous limit is taken in the mutual information for example [1].

The discussion in [19] is mainly phrased directly in terms of gauge invariant degrees of freedom. However, most lattice gauge theories or the field theoretic studies of gauge theories are formulated in terms of the gauge potential, and a Fock space is constructed for the gauge potential. Most of the discussions of entanglement in lattice gauge theories proceed by picking the temporal gauge , and that the gauge potential lives on the links of the lattice, with gauge invariance imposed at the vertex. For example in the original paper [1] and most other references, this is essentially the choice. This suggests a natural question: have we exhausted all the subtleties posed by the non-locality of gauge theories? Do we understand the operator algebra and how they are attached to local regions in an arbitrary gauge? Notwithstanding the introduction of centers, it is still necessary that there is an approximate choice of local operators that can be associated to some region for entanglement in configuration space to be meaningful.

We therefore take the first step in this direction, and explore the quantization of a U(1) gauge theory in Coulomb gauge. It is well known that by imposing the Coulomb gauge, standard Poisson brackets have to be modified by Dirac brackets to take into account the gauge condition which in this case are second class constraints. Perhaps unsurprisingly, the gauge potential and its conjugate momenta cease to be local operators. Their Dirac commutators are non-vanishing even as the operators are separated by very large distances.

To make sense of the entanglement, it becomes necessary to construct local operators. We found a suitable construction making use of the duality relation between a vector and a scalar in 2d. We proposed a construction of local operators in terms of the now non-local gauge potentials, and obtained the entanglement entropy accordingly. It appears that the proposal is robust - it is insensitive to various local prescriptions at the boundary. However the log term associated to corners seem to approach that of a local scalar field in the case of a trivial center, but approaches that of a gauge field when we pick a center mimicking the electric/magnetic center.

Then we consider a U(1) theory in the Higgsed phase, and apply our prescription still picking the Coulomb gauge. We recover a mutual information that decays exponentially according to the ratio of the mass and the lattice scale. A gauge theory coupled to matter in 1+1 d has been considered in [20], although the gauge theory has no dynamical degrees of freedom and the only remnant is Gauss’s law. Here we have an example in which matter interacts with dynamical gauge fields.

Our paper is organized as follows. In section 2 we review [19] briefly the computation of entanglement entropy of a quadratic theory making use of the values of the correlators and commutators. In section 3, we discuss the quantization of the U(1) gauge field in Coulomb gauge and the associated Dirac brackets. We discretize the theory in section 4, and introduce dual scalar variables, which allowed to construct truely local operators. In setion 5, we computed the entanglement entropy based on our prescription. In section 6-8, we generalize our method to the Higgsed gauge theory and computed the entanglement entropy accordingly. We look into mutual information in section 9 and finally conclude in section 10.

## 2 Entropies of Gaussian States in terms of correlation functions

In this section, we briefly review the methods described in [19]. It is demonstrated that expressions for entanglement entropy can be readily expressed in terms of commutators and correlations of some (canonical) variables in a quadratic theory.

We consider the general commutation relations

(2.1) |

and correlation functions

(2.2) |

(2.3) |

(2.4) |

with . In the trivial center case, we have the entanglement entropy of region [19]

(2.5) |

where . and are matrices of correlation functions and are matrix of commutators.

The method can be generalized to the case with center. We consider the algebra generated by , with and with . We assume for ,, such that span the center of the algebra. In the case with center, the entanglement entropy is defined as [21]

(2.6) |

where is an average of quantum contributions and is the classical Shannon entropy. With the expressions of correlation functions and commutators, we have[19]

(2.7) |

(2.8) |

(2.9) |

Here is the commutation matrix (2.1) between with and with . The classical part has the form

(2.10) |

The case of center formed by with can be analyzed in the same way, interchanging .

## 3 Local Operators of Gauge Fields with Coulomb Gauge

We consider the gauge fields with Coulomb gauge in dimensions. The Lagrangian is

(3.1) |

and the Coulomb gauge fixing is

(3.2) |

The temporal component is non dynamical and it needs only to satisfy a constraint following from the Gauss law and the gauge constraint, relating it to the total charge. Because we consider free gauge fields with no charged matter, we can set the temporal component to be 0. We have the canonical momentum

(3.3) |

The Gauss law and the gauge constraint also imply

(3.4) |

To impose the two constraints (3.2) and (3.4), we have to consider the Dirac bracket. A detailed discussion can be found in [22]. The commutators

(3.5) | ||||

and

(3.6) |

Since we consider a 2+1 dimensional theory, there is only one degree of physical freedom in Maxwell fields, also only one polarization. The mode expansions for gauge fields and their canonical momenta are given by

(3.7) |

and

(3.8) |

To satisfy the commutator (3.5), the polarization have to satisfy

(3.9) |

Interestingly, by modifying the brackets by the Dirac method, we have made some rather drastic change to the tensor product structure of the Hilbert space. The gauge potentials and their conjugate momenta, even before applying the Gauss’s constraint, can no longer be considered as a local degree of freedom. i.e. The operators and are not local i.e. From (3.5), one can see that their commutators are not local. They remain non-vanishing even though the fields are separated by large distances. To discuss entanglement entropy, we need to recover a basis of local operators. To our knowledge, we are not aware of a standard method of defining a suitable set of basis in such a situation. We therefore propose the following. Consider the operators

(3.10) |

The two constraints of the new variables and are

(3.11) |

and

(3.12) |

We find that the commutators of new operators and are

(3.13) |

and

(3.14) |

We can see that the operators and are local. We will consider the duality of them in the lattice and calculate the entanglement entropy.

## 4 Gauge Fields with Coulomb Gauge duality in the lattice

In (2+1) dimensional gauge fields, the polarization constraint (3.9) implies a solution

(4.1) |

where in this section. From this solution, we can see that the Maxwell fields are dual to a scalar field in a fixed time slice in the two dimensional spatial slice. The duality is written as

(4.2) |

giving the following identifications

(4.3) |

(4.4) |

From (3.7) and (3.9), we have the mode expansion of

(4.5) |

Therefore, it follows that

(4.6) |

with

(4.7) |

Let us define

(4.8) |

For the operators and , we have the commutators

(4.9) |

and

(4.10) |

For the local operators and , we have similar relations

(4.11) |

and

(4.12) |

Now we discretize the model in a square lattice. We define the operators and associated to horizontal links, and to vertical links, as shown in figure (1). For example, we have associated to a horizontal link and to vertical link, where are coordinates of the initial and final points of the links. For simplicity, we label them with the initial vertex of the vector,

(4.13) |

(4.14) |

The discrete version of (4.11) and (4.12) is also shown in figure (1). The operators and are related to the differences of the scalar field operators and in the orthogonal direction in the dual lattice respectively, such as

(4.15) |

(4.16) |

and

(4.17) |

(4.18) |

Because there are redundant degrees of freedoms in gauge fields, we have two constraints (3.11) and (3.12) with operators and . In the discrete lattice, the two constraints become

(4.19) |

and

(4.20) |

where the sum is over all the links with the common vertex . In the above equations, it is assumed that the field component is the corresponding one to the link direction. The links have orientations, which changes the field attached to it when changing the orientation, such as .

With the above dualities, the non-zero commutators of the discrete version of operators and are

(4.21) |

(4.22) |

(4.23) |

and

(4.24) |

We can see that the discrete version of operators and are almost local. We use these operators to calculate the entanglement entropy in section 5.

From (4.8) and (4.7), the vacuum correlation functions of operators and are found to be

(4.25) |

(4.26) |

The vacuum correlation functions of the discrete version are

(4.27) |

(4.28) |

The vacuum correlation functions of discrete variables and can be expressed with the above correlation functions, such as

(4.29) |

## 5 Entanglement Entropy of two dimensional lattice gauge fields with Coulomb gauge

To calculate the entanglement entropy of some “region” in the lattice, we have to choose an algebra of local operators to define the “region”. In the case of gauge fields, the gauge fields operators are associated to the links. We study four possible choices of algebras, which are shown in figure (2), figure (3) and figure (4). The four choices of algebras are full trivial center , trivial center with one physical degree of freedom removed , center and center . In figure (2), we illustrate graphically the different algebras also in terms of the gauge potential and also their dual scalar variables.

Due to the redundancy of degrees of freedom in the gauge potentials, we have to remove any remaining unphysical degrees of freedom. From the two constraints (4.19) and (4.20), we have to remove one degree of freedom with a vertex. That is, for every vertex, we have to remove one link connected to it. Note that it is not possible to keep all the external links of any regions, since there is an overall constraint. They are not independent.

In the trivial center choices, as shown in figure (3), both operators and are associated to every link in the figure. We keep the same number of and . In the left figure, we keep all the physical degrees of freedom, which is obtained by removing the unphysical degrees of freedom of gauge potential in the left panel of figure (2), while in the right figure, we remove a physical degree of freedom.

To get the center choice, we remove the unphysical degrees of freedom of the gauge field in the right panel of figure (2), in which we remove all the operators associated to the boundary links, and then remove a degree of freedom, as shown in figure (4). In such a choice, all the operators associated to the boundary links commute with the rest of the operators on the algebra. Hence, they form a center.
In the center choice, we do it in the same way, interchanging .

Now let us see the results of four algebra choices. We expect the entropy to take the following form as a function of the square region size ,

(5.1) |

a) Full trivial center algebra

The entanglement entropy with algebra is shown in figure (5). We have the coefficients

(5.2) |

b) Trivial center with one degree of freedom removed algebra

The entanglement entropy with algebra is shown in figure (6). We have the coefficients

(5.3) |

We can see that the entanglement entropy of the two different trivial centers are quite close. The effect of one degree of freedom is very small and can be neglected when the region becomes large. To calculate the entanglement entropy of gauge fields with non-trivial centers, we have to remove one physical degree of freedom after gauge fixing. The results are as follows.

c) center algebra

The entanglement entropy with algebra is shown in figure (7). We have the coefficients

(5.4) |

d) center algebra

The entanglement entropy with algebra is shown in figure (8). We have the coefficients

(5.5) |

We see that, in the two trivial center choices, the logarithmic coefficients are very close, and they are also close to the logarithmic coefficient of entanglement entropy of massless scalar field in [19]. While, in the two non-trivial centers, the logarithmic coefficients of their entanglement entropy are also very close. They are close to the logarithmic coefficient of entanglement entropy of gauge field in [19]. In [19], they calculate the entropy from the gauge invariant electric and magnetic fields, while we from the perspective of and . The final results are very close.

## 6 Local Operators of Gauge Fields coupling with matter

Now we consider gauge fields coupling with matter. We consider the situation of a Higgsed U(1) theory, where the gauge fields gain a mass. The Lagrangian considered is

(6.1) |

We take the gauge fixing

(6.2) |

From the gauge fixing, Gauss’s law reduces to

(6.3) |

With the gauge fixing (6.2), we have the equations of motion

(6.4) |

(6.5) |

and

(6.6) |

From (6.4), we find that there is no dynamic in the temporal component of gauge field. Similarly to the previous case without matter, we can take and keep the degrees of freedom of and . The canonical momenta are

(6.7) |

and

(6.8) |

With , we get two second class constraints

(6.9) |

and

(6.10) |

Because they are the second class constraints, we have to consider the Dirac bracket. The Poisson bracket is defined as

(6.11) |

For the two constraints, we have

(6.12) | ||||

The Dirac bracket is defined as

(6.13) |

where the matric is defined as

(6.14) |

For other Poisson brackets, we have

(6.15) |

(6.16) |

(6.17) |

(6.18) |

(6.19) |

(6.20) |

(6.21) |

(6.22) |

From the above equations and eq(6.13), we have

(6.23) |

(6.24) |

(6.25) |

(6.26) |

and

(6.27) |

To quantize the fields, we canonically quantize, and impose the Dirac brackets to obtain the following commutators

(6.28) |

(6.29) |

(6.30) |

(6.31) |

and

(6.32) |

Here we know, there are 2 degrees of physical freedom in the total fields in 2+1 dimension. For simplicity, we treat as . We do the mode expansion and we get

(6.33) |

and

(6.34) |

Here . From the above Dirac brackets, we have the constraints for the polarizations

(6.35) |

(6.36) |

(6.37) |

(6.38) |

However, from the commutators above, we find that the operators , and their canonical momentums are again not local. As before, we need to construct a set of local operators in this model. We define

(6.39) |

(6.40) |

and consider the operators , and ,. The two constraints of the new variables are

(6.41) |

and

(6.42) |

The commutators of the new operators are

(6.43) |