Strong Coupling Expansion of the Entanglement Entropy of Yang-Mills Gauge Theories

We calculate the entanglement entropy of the $SU(N)$ Yang-Mills gauge theories on the lattice under the strong coupling expansion in powers of $\beta=2N/g^{2}$, where $g$ is the coupling constant. Using the replica method, our Lagrangian formalism maintains gauge invariance on the lattice. At $O(\beta^{2})$ and $O(\beta^{3})$, the entanglement entropy is solely contributed by the central plaquettes enclosing the conical singularity of the $n$-sheeted Riemann surface. The area law emerges naturally to the highest order $O(\beta^{3})$ of our calculation. The leading $O(\beta)$ term is negative, which could in principle be canceled by taking into account the"cosmological constant"living in interface of the two entangled subregions. This unknown cosmological constant resembles the ambiguity of edge modes in the Hamiltonian formalism. We further speculate this unknown cosmological constant can show up in the entanglement entropy of scalar and spinor field theories as well. Furthermore, it could play the role of a counterterm to absorb the ultraviolet divergence of entanglement entropy and make entanglement entropy a finite physical quantity.


I. INTRODUCTION
Entanglement entropy is a measure of the level of entanglement between the degrees of freedom in two subregions of a physical system. In some systems, entanglement entropy plays the role of an order parameter to characterize quantum phase transitions [1][2][3] , while in others it demonstrates the scaling behavior [4,5]. In field theory, a widely used method to calculate the entanglement entropy is the replica method [6]. This method calculates the trace of the reduced density matrix to the n-th power in the path integral formalism, which amounts to computing the free energy of the system on a n-sheeted Riemann surface, or equivalently on a cone with a conical angle of 2nπ. The entanglement entropy is then obtained as a response of the free energy to the change of the conical angle at n = 1.
In previous studies, while the computation of entanglement entropy for the scalar and spinor fields are considered straight forward, the gauge field case is more subtle. In the Hamiltonian approach [7][8][9][10], one might need to impose constraints such as the Gauss law or gauge fixing in order to get rid of the unphysical degrees of freedom. Then if one uses the gauge link language to discretize the theory, there is an ambiguity about which side the gauge links on the boundary belong to. Also, it was argued that the inability to decompose all gauge invariant states into direct products of gauge invariant states living in the two subregions might be a problem. It was proposed that these ambiguities might be compensated by edge modes which live in the interface of the two subregions [7].
This situation reminds us the difficulty of the canonical quantization of gauge fields. It is desirable to explore other approaches which might shed light on the problem from a different angle. Therefore, in this paper, we study the entanglement entropy on a lattice using the Lagrangian formalism [11,12].
Our starting point is the replica method, which is assumed to be valid as long as the quantum field theory is local. Then we use the Wilson gauge action which has the advantage of being gauge invariant on a discrete lattice. This action is the sum of plaquettes which are Wilson loops living on the n-sheeted Riemann surface. There are two types of plaquettes on our squared lattice. If a plaquette encircles the tip of the cone, then it is formed by 4n links and called a central plaquette. Otherwise it is formed by 4 links, and is called a regular plaquette. When the conical angle or n changes, only those plaquettes with 4n links response to this change. Therefore entanglement entropy necessarily involves those central plaquettes. The fact that those central plaquettes all live in the interface between the two subregions naturally hints to the area law of entanglement entropy which states that the leading contribution to entanglement entropy scales as the area of the interface.
The connection to the area law can be further demonstrated order by order diagrammatically under the strong coupling expansion. Interestingly, we find the leading term in the strong coupling expansion to be negative. However, symmetries of the action allow a two-dimensional cosmological constant living in the interface [13] which could provide a positive contribution at an even lower order. We speculate this unknown two-dimensional cosmological constant is corresponding to the ambiguities encountered in the Hamiltonian approach.
We further speculate that the two-dimensional cosmological constant can show up in the entanglement entropy of scalar and spinor field theories as well. They can play the role of a counterterm to absorb the ultraviolet divergence of entanglement entropy and make entanglement entropy a finite physical quantity. This paper is organized as follows. In Sec. II, we briefly review the notion of entanglement entropy and the replica method. We carry out the calculation of the entanglement entropy of Yang-Mills fields on the lattice under the strong coupling expansion in Sec. III. In Sec.
IV, we discuss the cancellation of the negative term by taking into account the cosmological constant, and the ambiguity arising from it and conclude.
where ρ A = Tr B [ρ] is the reduced density matrix by tracing out the degrees of freedom in region B. This expression is called the replica method because it involves n copies of ρ A .
An elegant path integral formulation to compute the entanglement entropy using the replica method was first introduced in [6] (see also [14]). In this set up, one recalls that ρ ij = In this paper, we will just concentrate on the simplest case with the sizes of space and (Euclidean) time to be both infinite (i.e. T = 0) and the interface between A and B to be a flat infinite plane. In this limit, the n-sheeted Riemann surface has a conical structure as shown in Fig. 1(B) with the time and longitudinal spacial direction (the direction that is perpendicular to the interface) lying on the cone while the space on the interface transverse to the cone.

As a result, Tr[ρ n
A ] becomes a partition function Z n on the n-sheeted Riemann surface, or, in our case, a cone with 2nπ conical angle, normalized by n-copies of the partition function on the ordinary Euclidean space Z n 1 : which ensures that as n = 1, Tr[ρ n A ]=1. The entanglement entropy is then given by Note that n is taken as an integer in the integral of Z n . After one obtains the analytic expression for T r[ρ n ], then n can be analytically extended to non-integers to carry out the differentiation at n = 1.

III. CALCULATION ON AN N-SHEETED LATTICE MANIFOLD
We now discrete the spacetime on the n-sheeted manifold by a squared lattice. Firstly, the spacetime is decomposed into the direct product of 1 + 1 dimensional n-sheeted lattice Recall that the partition function of the lattice gauge theory on a one-sheet manifold is given by where β = 2N/g 2 and p is denoted as the index of plaquette. The plaquette U p is the Wilson loop at location p composed of the ordered product of four gauge links, U p = l∈∂p U l . The action recovers the Yang-Mills action in the continuum limit by setting the lattice spacing a → 0.
To construct the lattice system in the general D + 1 dimensions whose 1 + 1 dimensions is an n-sheeted manifold, we will rewrite the partition function in (4)  p =0 (x ⊥ ) and the central plaquettes U(x ⊥ ) = l∈∂(p =0) U l which enclose the conical singularity (as shown in Fig. (1A)). We also introduce an extra 1/n factor to the central plaquette terms. This is because a central plaquette is composed of 4n gauge links. It encircles the F 01 flux over an area na 2 and it will contribute a factor ∝ n 2 a 4 F 2 01 to the action which is inconsistent with the contribution from the transverse plaquettes which scales as na 4 . Therefore the central plaquette contribution is multiplied by a factor 1/n to compensate this effect. The partition function on the n-sheeted Riemann surface now reads where we have used the boundary condition i.e. the field strength on each sheet should be equal. This condition is natural if we impose 2π rotation symmetry on the n-sheeted surface.
Expanding Eq.(6) to the second order of β, we have Here N is the number of plaquettes on a single sheet for plaquettes not encircling the conical singularity, including both the parallel and transverse plaquettes. N ⊥ is the number of plaquettes encircling the conical singularity. For the SU(N) (N > 2) gauge theory, In the last line we have usedˆd As a result,ˆD Similarly,ˆD Note that this result is independent of n. This means no matter how we discretize the lattice, as long as the conical singularity is encircled by the same number of plaquettes in the action, the result will be the same. But if one chooses to put the conical singularity on a cite with no plaquette encircling the conical singularity, then those effects will all vanish in that calculation.
Taking the combination then the Renyi entropy is Using the identitiesˆd we can find the non-vanishing contribution coming from the N-th order of β. It is obtained viaˆD and alsoˆD This type of contribution appears at the N-th order of β, where U (k) p =0 includes the transverse plaquettes U (k) p ⊥ and the parallel plaquettes U (k) p =0 (x ⊥ ). For SU(2) theory, we have an additional O(β 2 ) contribution, For SU(3) theory, So Renyi entropy for SU(N) theory to the order of β 3 is As a result, entanglement entropy of SU(N) gauge theory in the strong coupling expansion where A ⊥ is the area of interface, a ⊥ is lattice spacing in the transverse space and where λ = g 2 N is the t'Hooft coupling. As we argued after Eq. (13), this result is independent of how the lattice is discretize, as long as the conical singularity is encircled by the same number of plaquettes in the action. The additional term δS EE will be explained in Sec. (IV).
Likewise, the U(1) result can also be obtained using the same method:

IV. DISCUSSION AND CONCLUSION
In the result Eqs. (26-28) and (29), one finds that the first order term in S EE has a negative contribution while all the rest orders contribute to entanglement entropy positively.
It turns out with the conical structure of space time, we are allowed to introduce more local operators in the continuum action The c 4 term is a four dimensional cosmological constant which does not contributes to the entanglement entropy. However, the c 2 term, which breaks the translational symmetry while moving on the cone, can contribute to the entanglement entropy. Assuming c 2 is a smooth function of n, then where we have used the fact that c 2 should vanish at n = 1 where translational symmetry is recovered. Therefore there is an extra unknown contribution to the entanglement entropy which also obey the area law: where we have shown the N dependence explicitly since different theories would have different c 2 counterterms.
The negative term in our result could in principle be compensated by the two dimensional cosmological constant. This uncertainty resembles the ambiguity of edge modes in the Hamiltonian method of the lattice gauge theory's entanglement entropy.
We further speculate that the two-dimensional cosmological constant can show up in the entanglement entropy of scalar and spinor field theories as well. Also, they could play the role of a counterterm to absorb the ultraviolet divergence of entanglement entropy and make entanglement entropy a finite physical quantity.