A Useful formulas

OU-HET-931, YITP-17-27

Entanglement Entropy

[3pt] for 2D Gauge Theories with Matters

Sinya Aoki, Norihiro Iizuka, Kotaro Tamaoka and Tsuyoshi Yokoya

Center for Gravitational Physics,

Yukawa Institute for Theoretical Physics, Kyoto 606-8502, JAPAN,

Department of Physics, Osaka University

Toyonaka, Osaka 560-0043, JAPAN

saoki at yukawa.kyoto-u.ac.jp, iizuka at phys.sci.osaka-u.ac.jp,

k-tamaoka, yokoya at het.phys.sci.osaka-u.ac.jp

We investigate the entanglement entropy in 1+1-dimensional gauge theories with various matter fields using the lattice regularization. Here we use extended Hilbert space definition for entanglement entropy, which contains three contributions; (1) classical Shannon entropy associated with superselection sector distribution, where sectors are labelled by irreducible representations of boundary penetrating fluxes, (2) logarithm of the dimensions of their representations, which is associated with “color entanglement”, and (3) EPR Bell pairs, which give “genuine” entanglement. We explicitly show that entanglement entropies (1) and (2) above indeed appear for various multiple “meson” states in gauge theories with matter fields. Furthermore, we employ transfer matrix formalism for gauge theory with fundamental matter field and analyze its ground state using hopping parameter expansion (HPE), where the hopping parameter is roughly the inverse square of the mass for the matter. We evaluate the entanglement entropy for the ground state and show that all (1), (2), (3) above appear in the HPE, though the Bell pair part (3) appears in higher order than (1) and (2) do. With these results, we discuss how the ground state entanglement entropy in the continuum limit can be understood from the lattice ground state obtained in the HPE.

Contents:

1 Introduction

Entanglement is an key feature, which distinguishes quantum worlds from classical worlds. Simply saying, entanglement allows us to know detailed information about subsystem A once we measure the other subsystem B, even though we know nothing about each subsystem A & B separately before we make a measurement. Recently these entanglement were caught attention since it becomes more and more clear that the notion of entanglement is one of the key feature to understand the gauge/gravity duality [1] and emerging smooth space-time (see for example, [2]). Needless to say, all of the forces except for gravity in Nature are described by gauge theories, and furthermore due to the gauge/gravity duality, quantum gravity in asymptotic anti-de Sitter space is also equivalent to certain gauge theory non-perturbatively. In order to understand how the space-time emerges through the idea of entanglement and gauge/gravity duality, deepening our understanding of entanglement in gauge theory must be crucial.

Entanglement in spin system is well-defined and there is no ambiguity for its definition. Decomposing the Hilbert space into “inside” and “outside”, and by tracing out the “outside” Hilbert space, we obtain the density matrix of the “inside” states. Its von Neumann entropy is the entanglement entropy between “inside” and “outside”. However the situation is a bit more subtle in gauge theories. In gauge theories, Hilbert space cannot be decomposed into two gauge invariant subsystem properly, due to the local gauge invariance condition, which gives non-local constraints for the allowed states. As a result, there exists non-local operators such as Wilson loops which spread both “inside” and “outside”, and thus restrict Hilbert spaces of “inside” and “outside” through Gauss’s law constraints. The absence of the gauge invariant decomposition brought some confusions for how to define the entanglement entropy in gauge theories.

The main problem of how to define the entanglement entropy associated with the non-product nature of the Hilbert space in gauge theories is now solved through recent works [3, 4, 5, 6]. For Abelian gauge theory, Casini et al. in [3] pointed out that the presence of a non-trivial center, which commute with all the operators on the “inside” (Hilbert space), characterizes the ambiguity of the entanglement entropy in gauge theories. Clearly this center corresponds physically to gauge invariant Wilson loop operators penetrating the boundary. They connect “inside” and “outside” Hilbert spaces, and also split the “inside” Hilbert space into several different superselection sectors labeled by fluxes of the penetrating loop . In each superselection sector, the Hilbert space can now be written as a tensor product of “inside” and “outside” Hilbert spaces , . They allow us to define reduced density matrix such that , where is the label for different superselection sectors, specifying the penetrating gauge flux ‘representations’ at all boundaries. Then the definition of the entanglement entropy is given as [3]

(1.1)

where the second term is the weighted average of the “genuine” entanglement on each sector with the probability , which we mean EPR Bell pairs obtained in entanglement distillation,

(1.2)

while the first term is the classical Shannon entropy for the probability distribution of the variables on the center1. Note that this classical entropy is different from the “genuine” entanglement entropy.

The “extended Hilbert space” definition of the entanglement entropy is given in [4, 5, 6]. In these, we literary extend the Hilbert space in such a way that the Hilbert space is no more restricted to gauge invariant state only

(1.3)

As a result of this extension, the Hilbert space can now be decomposed as a tensor products of two (gauge non-invariant) subsystems without ambiguity. In the lattice formulation of gauge theories, the extended Hilbert space can be identified to the Hilbert space of a spin system, so that one can define the entanglement entropy unambiguously. For example in case, the explicit calculation becomes possible [7, 8], and it has been shown in [4] that this definition agrees with (1.1).

In non-Abelian gauge theory, however, the extended Hilbert space entanglement entropy definition needs an extension of (1.1), which consists of three terms as [8]

(1.4)

The first and third term are essentially the same as (1.1), while the peculiarity of the non-Abelian gauge theory appears in the second term, which contains the sum over boundary vertices index , where runs all boundary vertices and is the irreducible representation of the penetrating gauge loop at that boundary with being the dimension of the representation .2 Thus the second term vanishes for the abelian case since all representations are 1-dimensional. Here the superselection sector is labelled by . Since the representations in non-Abelian gauge theory are no more one dimensional, the requirement of wave function being gauge invariant (singlet) at the boundaries generates a new type of “entanglement” between “inside” and “outside” states. In other words, the non-Abelian gauge theory has a new term in (1.4), which is the entanglement entropy associated with “color” at each boundary.

Although the appropriate definition is given, definitely more detailed aspects of the entanglement entropy, especially for non-Abelian gauge theories, need to be better understood both qualitatively and quantitatively. A purpose of this paper is twofold: one is to deepen our understanding of the formula (1.4) in non-Abelian gauge theories with various matter fields, by explicitly evaluating the contributions to each of the three terms in (1.4). This is because the non-Bell pair contributions, i.e., the first and second terms of (1.4) are less familiar. The other is to study the vacuum entanglement entropy of non-abelian gauge theories through the lattice formulation. Gauge theories are well-defined on the lattice, and moreover, once we employ the extended Hilbert space definition, the gauge theory on the lattice effectively reduces to the one essentially equivalent to the usual spin system.

The entanglement entropy for the ground state in non-Abelian gauge theories is especially interesting and it is well studied by the strong coupling expansion in the lattice formulation[9, 10, 11, 6]. In the formulation by Kogut-Susskind [12], the Hamiltonian for pure gauge theories (without matter fields) in lattice regularization is given by [13]

(1.5)

where is the lattice spacing, is the bare gauge coupling on the lattice, and is the generator of the gauge transformation at the vertex for the link , which satisfies . In the strong coupling limit that , the ground state, which we call the strong coupling ground state , is given by the tensor product of the ground state of each link as

(1.6)

where satisfies . Therefore there is no entanglement for the strong coupling ground state. Note that plaquette terms disappear in 2 dimension, so that one can always obtain this as a ground state in 2-dimensional pure gauge theories at an arbitrary value of the coupling constant. In other words, not only the ground state obtained in higher dimensional () pure gauge theories at strong coupling limit but also that of 2-dimensional gauge theories at an arbitrary coupling ground state are given by on the lattice.

On the other hand, the vacuum in continuum gauge theories, which we call the continuum ground state, is manifestly entangled: tracing out the subsystem makes the rest subsystem into mixed states like the Bogoliubov transformation. This is not a contradiction, however, since the lattice gauge theories at the strong coupling limit is far from the continuum limit. Due to the asymptotic freedom of gauge theories, the continuum gauge theory with non-zero renormalized coupling (the IR theory) is obtained from the lattice gauge theory in the limit of zero bare gauge coupling (the UV theory).

Therefore, it is important to understand how the strong coupling ground state approaches the entangled continuum ground state in the process of the continuum limit. In generic dimensions, however, solving the gauge theory on the lattice analytically is very hard exercise, unless we take the strong coupling limit or the expansion around it. That is why people use numerical simulations in lattice gauge theories, which are shown to be very successful. This situation is a little different in 2-dimensions, since a 2-dimensional pure gauge theory is in some sense “trivial” due to the absence of local physical degrees of freedom. As a result, we can calculate entanglement entropy for any states at an arbitrary coupling constant [14], so that we can take the continuum limit analytically. Unfortunately, “genuine” entanglement, i.e., the third term in (1.4), vanishes in 2-dimensional pure gauge theories even in the continuum limit [14] as is expected.

Once we add matter fields to pure gauge theories in 2-dimensions, “genuine” entangled states emerge due to the existence of local degrees of freedom. We thus take these gauge plus matter theories as toy models of pure gauge theories in higher dimensions, since gauge plus adjoint matters in 2-dimensions, for example, are expected to have analogous behaviors as higher dimensional pure Yang-Mills theories with compactified extra () dimensions. While pure gauge theories plus matters can not be solved analytically even in 2-dimensions,3 we can include effects of matter fields order by order in the hopping parameter expansion (HPE) for the small hopping parameter , where is the bare mass of matter field and must be large for the HPE to work.4

In this paper, using the HPE but at an arbitrary gauge coupling, we demonstrate how the “genuine” entanglement entropy emerges for the ground state of gauge plus matter fields in 2-dimensions. We mainly consider matter fields in the fundamental representation, but an essential idea works similarly for adjoint matters and other representations. Adding adjoint matters is an interesting set-up, since it resembles the large D1-brane gauge theory, which is dual to the string theory in the curved space-time [16].

The organization of this paper is as follows. In §2, we review the lattice study in [6] for the pure gauge theory in 2-dimensions, which has no local physical degrees of freedom. Therefore, there is no “genuine” entanglement in 2-dimensional pure gauge theory. Then in §3 and 4, we add matter fields, and study entanglement of various meson excited states. §5 gives a short summary of the first part. Then in §6, we show at the leading order of HPE that these mesons states appear in the ground state of this theory, which is the eigenstates of the “transfer matrix” with the largest eigenvalues. The transfer matrix is the time translation operator on the lattice with one time unit and is related to the Hamiltonian as . Then later in §7, we consider the higher order corrections of HPE and show that the strong coupling ground state and lattice meson states mix to form the true ground state, and at the order, the ground state of the transfer matrix shows nonzero “genuine” entanglement, and we end with discussion in §8 on our picture of how the strong coupling ground state, which has no entanglement, is connected to the continuum entangled ground state.

Throughout this paper5, we consider gauge theory with .

2 Entanglement entropy for pure gauge theory in lattice formulation

In this section, we briefy illustrate how the second terms of the entanglement entropy in eq. (1.4) appear in the 2-dimensional pure gauge theory on the lattice formulation [14], using explicit examples.

We will consider the 7 vertex spatial lattice given in Fig. 1 as a simple example, which is good enough to see the essential points, and one can easily generalize the results in this section to more general cases.

Figure 1: Toy seven vertex lattice setup. Black vertices and solid lines belongs to “inside” and white vertices and dotted lines to “outside”.

Consider following wave function

(2.1)

where is the spatial gauge link variable between the vertices and , which satisfies , and is the character for the ‘fundamental representation’ .6

Straightforward calculation shows that the reduced density matrix becomes

(2.2)

where we used (6.19) and integrated out “outside”-link variables . Therefore the square of the reduced density matrix is

(2.3)

where again we used (6.19). This implies

(2.4)

As a result, we obtain an entanglement entropy as

(2.5)

This is consistent with the “area-law” of the entanglement entropy [17], where the boundary is consists of two sites, i.e., site 3 and 7, so the “boundary site number” . To see this further, as an example of , we consider a different separation of in and out regions in such a way that link 2-3 and 5-6 are outside and others are inside. Then using (6.19) and (6.20), it is straightforward to check the reduced density matrix and its square become

(2.6)
(2.7)

so that we obtain

(2.8)

for . It is easy to see in general that

(2.9)

where is the dimension of the irreducible representation R. This is the essential results of [5, 14]. Before we end this section, we have several comments.

Since there is no physical degrees of freedoms in the 2-dimensional pure gauge theory, the result (2.9) cannot represent the “genuine” entanglement in the sprint of the information theory, which is equivalent to the number of Bell pairs obtained in the entanglement distillation. See §4 of [6], for example.

All calculations in the above are done in the extended Hilbert space definition [4, 5, 6]. The Hilbert space in the gauge theory cannot be written as a tensor product of “inside” Hilbert space and “outside” Hilbert space. In above calculations, however, we trace over all of the out states without worrying about the gauge constraint. This is possible only in the extended Hilbert space.

In the extended Hilbert space, we can define the entanglement entropy, which consists of three contributions as is given (1.4). Different superselection sectors are distinguished by the electric flux for the Abelian gauge theory and by the quadratic Casimir for the non-Abelian gauge theory at each boundary, and the different Casimir corresponds to the different “spin”, or representation. Due to the Gauss’s law in 1+1 dimension, we have only one sector, , in our wave function (2.1), restricted in the fundamental representation. Therefore (2.9) gives only the second term in (1.4), as the first and the third term in (1.4) vanish.

Clearly this entanglement entropy (2.9) is associated with the fact that in and out link variables connected with each other at the boundary vertex cannot take values freely due to the gauge invariance constraint, and this gauge invariance correlates the two link variables. As a result, this correlation produces the entanglement obtained in (2.9), which is the “color entanglement”.

3 Entanglement entropy for single meson states

3.1 2d gauge theory with the fundamental scalar field

Now we consider the 2-dimensional gauge theory with the fundamental scalar field. Again we consider the Fig. 1 lattice setup. For each vertex , there is a scalar field , in addition to the link variable on each link .

Let us consider the following wave function,

(3.1)
(3.2)

where is the normalization constant. This is a single “meson” state composed by a scalar “quark” (at site ) and “anti-quark” (at site ) pair. For the wave function of the scalar field to be normalizable, we have introduced the Gaussian suppression factor with the Gaussian parameter . The normalization constant is obtained from the condition

where we use (A.1) and (A.10). Similarly, using (A.1), (A.2) and (3.2), the reduced density matrix becomes

(3.3)

and a square of the reduced density matrix thus is given by

(3.4)

where we have performed the integral using the formula (A.10) in the third equality, and then the integral using (A.2) in the fourth equality. From eq. (3.4), the entanglement entropy is obtained as

(3.5)

Here simply represents the color charge entanglement between scalar quark and anti-quark in the fundamental representation.

A few comments are in order.

  • This term corresponds to the 2nd term of (1.4). First of all, since a color is neither physical nor observable, this term cannot be the “genuine” entanglement related to the Bell pair, i.e., the 3rd term in (1.4). A reason why eq. (3.5) does not satisfy the area-law of the entanglement is simply because the flux takes the fundamental representation at the “boundary vertex” 3 only but the trivial representation at the “boundary vertex” 7. Furthermore, since we have already fixed the representation in this setup, the 1st term of (1.4) can not appear in eq. (3.5).

  • We can easily show the followings. The entanglement entropy is given again by (3.5) for the wave functions

    (3.6)

    instead of (3.1), while it vanishes if all fields (quark, anti-quark and all link variables) belong to either “inside” or “outside” such that

    (3.7)

    as expected.

  • The situation is very similar to the pure gauge theory in §2. Regarding that the link variable made up of two scalar fields and as , the result in eq. (2.5) can be understood as follows. The argument of for the entanglement entropy is the dimensions of the representation, i.e., the entanglement associated with color numbers. The coefficient in front of counts a number of boundary vertices in which the gauge flux penetrates. As we will see in the next subsection, the adjoint matter field gives the instead of contribution to the entanglement entropy.

3.2 2d gauge theory with the adjoint scalar field

For completeness, we show the result with the adjoint matter field . We take

(3.8)

for the wave function with the adjoint scalar field at the vertex 1 and 5, where is the Gaussian suppression factor. The lattice setup is same as Fig. 1.

Applying (A.7) and (LABEL:eq:U4d) to the condition

(3.9)

the normalization constant is determined as

(3.10)

Then, the reduced density matrix is given by

(3.11)

and its square becomes

(3.12)

Therefore, the entanglement entropy is obtained as

(3.13)

which confirms that the argument of counts a dimension of the representation for the flux at the boundary vertex.

3.3 Entanglement entropy for a single meson with the multiple splitting

Let us consider the situation where vertices and links belong to “inside” and the rest belong to “outside”. See Fig. 2.

Figure 2: Black vertices and solid lines belong to “inside” and white vertices and dotted lines to “outside” as before. Scalar quark/anti-quark are at vertices 2 and 7 and gluon is penetrating at the boundary vertices 3, 4, and 6 but not 1. The color indices for the reduced density matrix can be seen pictorially in the right figure.

Let us consider the following wave function

(3.15)

It is straightforward to show

(3.16)

This reduced density matrix can be shown pictorially in Fig. 2. We thus obtain

(3.17)
(3.18)

which is again consistent with the second term in (1.4), since a number of boundaries on which the penetrating flux of the fundamental representation exists is (at vertices 3, 4, and 6). The boundary 1 does not contribute since there is no penetrating flux there.

So far, we obtain the entanglement entropy

(3.19)

where is the dimension of the representation R, and is the number of boundaries on which there is nontrivial flux in the representation R of the gauge group.

4 Entanglement for multiple meson states

We next consider multiple meson states and evaluate their entanglement entropy. In §4.1, we first consider a two meson state where two meson excitations do not overlap each other. Next in §4.2, we consider a various types of overlapped two meson states whose excited fluxes go through the same boundary. We classify these states in Fig. 3, and consider the entanglement entropy for all of these possibilities. In §4.3, we finally consider a four meson state where all excited fluxes penetrate the same boundary.

One of the main differences between these multiple meson excitations and single meson excitations in the previous section is that we need to decompose the product of the same link variables of multiple meson excitations at the same boundary into a sum of irreducible representations. As a results of this decomposition, we have several different superselection sectors, labeled by the irreducible representation R of the penetrating flux. This results in nonzero contribution to the 1st term of the entanglement entropy in (1.4), which is the Shannon entropy associated with the superselection sector distribution.

In this section, we again use the lattice setup in Fig. 1.

4.1 Two mesons without overlapping

We first consider a two meson state without overlap. Explicitly, let us consider the following wave function,

(4.1)
(4.2)

A straightforward calculation shows that the reduced density matrix and its square are given by

(4.3)
(4.4)

Thus the entanglement entropy is

(4.5)

which is simply the twice of the single meson result (3.5), and can be understood from (3.19).

4.2 Two mesons sharing the same boundary

We next consider several types of two overlapping meson states whose excited fluxes penetrate the same boundary, as shown in Fig. 3.

Case (a): Opposite meson direction with 4 (anti)quarks at different positions

Figure 3: Two meson configurations which we consider in this subsection.

Let us consider the following state corresponding to Fig. 3 (a),

(4.6)
(4.7)

Overlapping links need to be decomposed into a sum of irreducible representations. Explicitly, let us consider the link variable between 3 and 4 vertices. Since there are one fundamental () and one anti-fundamental () links, this state split into a sum of “singlet” and “adjoint” states as follows. Let us first rewrite our state as

(4.8)

where ’s are defined by

(4.9)
(4.10)
(4.11)

We then decompose this state as

(4.12)

where

(4.13)

As mentioned, the first and the second terms in the r.h.s. of (4.12) represent the adjoint and the singlet states, respectively.

The reduced density matrix for this state becomes

(4.14)

where