Contents

Bi-partite entanglement entropy in massive 1+1-dimensional quantum field theories

Olalla A. Castro-Alvaredo and Benjamin Doyon

Centre for Mathematical Science, City University London,

Northampton Square, London EC1V 0HB, UK

Department of Mathematical Sciences, Durham University

South Road, Durham DH1 3LE, UK

This manuscript is a review of the main results obtained in a series of papers involving the present authors and their collaborator J.L. Cardy over the last two years. In our work we have developed and applied a new approach for the computation of the bi-partite entanglement entropy in massive 1+1-dimensional quantum field theories. In most of our work we have considered these theories to be also integrable. Our approach combines two main ingredients: the “replica trick,” and form factors for integrable models and more generally for massive quantum field theory. Our basic idea for combining fruitfully these two ingredients is that of the branch-point twist field. By the replica trick we obtained an alternative way of expressing the entanglement entropy as a function of the correlation functions of branch-point twist fields. On the other hand, a generalisation of the form factor program has allowed us to study, and in integrable cases to obtain exact expressions for, form factors of such twist fields. By the usual decomposition of correlation functions in an infinite series involving form factors, we obtained exact results for the infrared behaviours of the bi-partite entanglement entropy, and studied both its infrared and ultraviolet behaviours for different kinds of models: with and without boundaries and backscattering, at and out of integrability.

 

1 Introduction

Entanglement is a fundamental property of quantum systems. Its most striking consequence is the fact that performing a local measurement may affect the outcome of local measurements far away; this is one of the main differences between quantum and classical systems. Given that our understanding of the physical world is largely based on everyday experience and that entanglement seems to contradict such experiences, it is not surprising that the existence of quantum entanglement has been a source of controversy and heated scientific debate for some time (see e.g. [1] for the famous EPR-paradox). Today entanglement is a well established and measurable phenomenon, whose reality was experimentally confirmed in the early eighties by the famous experiments of Alain Aspect and collaborators [2, 3], using pairs of maximally entangled photons. In the last decades, many applications of entanglement have developed into successful fields of research such as quantum computation and quantum cryptography. Entanglement lies also at the heart of other interesting phenomena such as quantum teleportation (see [4] for a review).

As a consequence of the prominent role of entanglement in quantum physics, there has been great interest in developing efficient (theoretical) measures of entanglement. For example, a quantity of current interest in quantum models with many local degrees of freedom is the bi-partite entanglement entropy [5], which we will consider in this review. In its most general understanding, it is a measure of the amount of quantum entanglement, in some pure quantum state, between the degrees of freedom associated to two sets of independent observables whose union is complete on the Hilbert space. In the cases considered in this review, the quantum state will mostly be the ground state of some extended 1+1-dimensional (1 space + 1 time dimension) quantum model, and the two sets of observables correspond to the local observables in two connected regions, say and its complement, (we will also briefly discuss the general technique in the case of excited states). Other measures of entanglement exist, see e.g. [5, 7, 8, 9, 10], which occur in the context of quantum computing, for instance. Measures of entanglement are important at a theoretical level, as they give a good description of the quantum nature of a ground state, perhaps more so than correlation functions.

General aspects of the entanglement entropy in extended quantum systems will be discussed in [6]. For our present purposes, prominent examples of extended one-dimensional quantum systems are quantum spin chains, which model physical systems consisting of infinitely long one-dimensional arrays of equidistant atoms characterized by their spin. Their entanglement has been extensively studied in the literature [11, 12, 13, 14, 15, 16, 17, 18, 19].

Figure 1: The entangled region of length of a quantum spin chain. a) the bulk case, b) the boundary case.

In order to provide a formal definition of the entanglement entropy, let us consider the Hilbert space of a quantum model, such as the chain above, as a tensor product of local Hilbert spaces associated to its sites. This can be written as a tensor product of the two Hilbert spaces associated to the regions and :

(1.1)

Then the entanglement entropy is the von Neumann entropy of the reduced density matrix associated to :

(1.2)

We will be interested in analysing the entanglement entropy in the scaling limit of infinite-length quantum chains. The scaling limit gives the universal part of the quantum chain behaviour near a quantum critical point, which is described by a model of 1+1-dimensional quantum field theory (QFT) (which we will assume throughout to possess Poincaré invariance). The scaling limit is obtained by approaching the critical point while letting the length of the region go to infinity in a fixed proportion with the correlation length (these lengths are measured in number of lattice sites). In this limit, the entanglement entropy is in fact divergent. The way the entanglement entropy diverges was understood in [20, 21, 22]. It is controlled by the central charge corresponding to the critical point that is being approached. In general, for every point that separates a connected component of from a connected component of its complement (a boundary point of ), there is a term . We will look at two cases: where the quantum chain is infinite in both directions and is a connected segment, with two boundary points (the bulk case), and where the quantum chain is infinite only in one direction, and is a connected segment starting at the boundary of the chain, with only one true boundary point (the boundary case) – see Fig. 1. In these two cases, the divergent part of the entanglement entropy is respectively:

(1.3)

These divergent terms are not universal (as they depend on the correlation length); the universal terms of the entanglement entropy are hidden in the part. These universal terms, that depend on the proportion to with which is sent to infinity, are in general described by QFT (conformal, massive, etc.). The analysis of the universal terms using CFT techniques in various situations is reviewed in [23]. In the present review, we will explain how to use massive QFT techniques in order to obtain information about the universal terms. We will now overview the main ideas and results of our works on this subject.

1.1 Overview of ideas and results

It is known since some time [20, 21, 22, 24] that the bi-partite entanglement entropy in the scaling limit can be re-written in terms of more geometric quantities, using a method known as the “replica trick”. The essence of the method is to “replace” the original QFT model by a new model consisting of copies (replicas) of the original one, in order to use the formula

(1.4)

The trace in this formula is reproduced by the condition that these copies be connected cyclicly through a finite cut on the region . Then, this trace is the partition function of the original (euclidean) QFT model on a Riemann surface with two branch points, at the points and in , and sheets cyclicly connected (we will provide more explanations about this in the next section). Fig. 2 shows a representation of such a Riemann surface.

Figure 2: A representation of the Riemann surface .

The positions of the branch points correspond to the end-points of the region in the scaling limit. This gives:

(1.5)

Here, is the euclidean distance between and . This formula holds both for our bulk and boundary cases. In the boundary case, one of the branch points (say ) is on the boundary of the model, . The positions and are dimensionful positions in the QFT model. They are naturally at zero imaginary time , (but this is not crucial because the euclidean QFT has rotation invariance), and their -coordinates are related to the ratio between the dimensionless region length and correlation length of the quantum chain by , where is the QFT mass scale associated to (whose only role here is to provide a dimension).

Naturally, this expression implies that we must analytically continue the quantity from , where it is naturally associated to Riemann surfaces, to . The object certainly has a well-defined meaning for any such that . Indeed, is hermitian (and has non-negative eigenvalues summing to 1), so that is the sum of the powers of its eigenvalues (with multiplicities). Note that this is an analytic continuation from positive integers to complex that satisfies the requirements of Carlson’s theorem [25], hence the unique one that does. The scaling limit of this object is what defines the proper analytic continuation of . Finding the correct analytic continuation has been one of the major challenges encountered in our work. It is natural to assume, as it has been done before [20] and discussed in [26] in the present context, that the two branch points just become conical singularities with angle , the rest of the space being flat.

As will be described later in more detail, in [27] we showed that there is a way of associating the branch points at and to local QFT fields: through branch-point twist fields . These twist fields are defined only in the replica model (not in the original model), and are associated to certain elements of the extra permutation symmetry present in the replica model. In terms of these fields we showed that, in the bulk case,

(1.6)

Here denote correlation functions in the -copy model; the state is the vacuum state of the latter. The branch-point twist fields have the CFT normalisation (which we will discuss later). The constant , with , is an -dependent non-universal constant, is a short-distance cut-off which is scaled in such a way that at , and, finally, is the scaling dimension of the counter parts of the fields in the underlying -copy conformal field theory,

(1.7)

which can be obtained by CFT arguments [24, 27] and where is the central charge. The short-distance cut-off is related to the correlation length via for some dimensionless finite non-universal number . This short distance cut-off takes care of the infinite contributions to the partition functions , as compared to the power of , around the points and where a branch point lies. There is one contribution of for each point, corresponding to one contribution of to the entanglement entropy.

In our most recent work [28] we have generalized this understanding to the boundary case, where now the region extends between the origin , where the boundary of the model is located, and the -coordinate . With similar arguments we have showed that

(1.8)

The state is a boundary state, which depends on the particular model and boundary condition under consideration. In the context of integrable models, it was introduced in the seminal work of Ghoshal and Zamolodchikov [29]. Here, we take it with the normalisation . The factors and are defined through similar conditions as in the bulk case.

In terms of the variables , and using the CFT normalisation of the branch-point twist fields as well as large-distance factorisation of correlation functions, we can re-write the logarithmic divergence formula (1.3) in a more precise fashion. The IR (large-) and UV (small-) leading behaviours of the entropy in the bulk are

(1.9)

The term is a model-dependent constant, which we computed exactly for the Ising model

(1.10)

in [27] using QFT methods, reproducing results of [15] obtained on the lattice. In (1.9), the short-distance cut-off is of course non-universal, and it is in general hard to evaluate its exact relation to the correlation length. But the UV leading behaviour provides an unambiguous QFT definition for it, independently of what the correlation length of a particular quantum chain may be. Once this definition is taken, the terms that are added to are universal – these terms constitute the universal part of the entanglement entropy. For instance, the constant is indeed a universal QFT quantity. Note that the IR leading behaviour is interpreted as the saturation of the entanglement entropy at large distances.

In the boundary case, our definitions of the boundary state further imply that the UV and IR leading behaviours of the entanglement entropy are

(1.11)

Again, is non-universal and hard to calculate. But since we know that the constant is universal, the IR behaviour in (1.11) gives an unambiguous definition of . Then, the terms that are added to are the universal part of the entanglement entropy, true QFT quantities. In particular, the UV behaviour in (1.11) provides a universal definition for . Note that the leading asymptotic term at large distance in the boundary case is just a choice. Once this choice is made, the constant is universally fixed. In particular, our short-distance cut-off here is in general different in the bulk and boundary cases – it is related to the correlation length in different ways. Here, is a parameter that represents the boundary condition.

A major focus of our work [27, 26, 28] has been the study of the ratios of partition functions (1.6) and (1.8) at large distances (the infrared (IR) region) for 1+1-dimensional integrable QFTs. Integrability means that in these models there is no particle production in any scattering process and that the scattering () matrix factorizes into products of two-particle -matrices which can be calculated exactly (for reviews see e.g. [30, 31, 32, 33, 34]). This is the factorised-scattering theory of integrable models. Since the scattering matrix and the particle spectrum fully encodes the local definition of QFT, it is also possible to incorporate the presence of boundaries in an integrable model defined in the factorised-scattering way. The study of integrable QFTs with boundaries has attracted a lot of attention in the last two decades (see e.g. [35, 36, 37, 29, 38, 39]). In our work on the boundary case we have made extensive use of the results of Ghoshal and Zamolodchikov [29], particularly the explicit realization of the boundary state which they proposed.

Taking the known -matrix of a model as input it is possible to compute the matrix elements of local operators (also called form factors). This is done by solving a set of consistency equations [40, 41], also known as the form factor bootstrap program for integrable QFTs. In [27], this program was used and generalised in order to compute (1.6) in the case of integrable models with diagonal scattering matrix (that is, without backscattering). In order to do this, the two-point function in (1.6) was expressed as a sum in terms of form factors of the twist fields involved (an expansion using a decomposition in energy-momentum eigenstates). This was then extended to models with backscattering, such as the sine-Gordon model [26], to integrable models with boundaries in [28], and some aspects were generalised to non-integrable models [42].

One of the most interesting results of our works [27, 26, 42] has been the identification of the next-to-leading order correction to the large-distance (large-) behaviour of the entropy of all (unitary) massive two-dimensional theories, that is, the third term in the following large- expansion:

(1.12)

Here, are the masses of the particles in the QFT model, with , and is the model-dependent constant introduced above. The first two terms are the expected saturation of the entanglement entropy, but the interesting feature is the universal third term, where we see that the leading exponential corrections are independent of the scattering matrix, and only depend on the particle spectrum of the model. This is quite striking: for instance, a model of free particles of masses will give the same leading exponential corrections as one with interacting particles of the same masses. The result (1.12) was first obtained using integrable QFT methods [27, 26], then, even more strikingly, it was understood to hold as well outside of integrabilty [42].

Infrared corrections were also studied for integrable models with boundary [28], in which case they are always model-dependent, in particular through the reflection matrices off the boundary.

The results described so far have been extended further for the particular case of the Ising model. For this model the particular form of all infrared corrections to the entropy with and without boundary has been obtained in [28]. In fact, for this model without boundary, like for other free-field QFT models, there is an alternative powerful way of studying the entanglement entropy, see the review [43].

This paper is organized as follows. In section 2 we review the relationship between partition functions on multi-sheeted Riemann surfaces, correlation functions of branch-point twist fields and the entanglement entropy. Employing these relationships, we provide expressions for the bi-partite entanglement entropy of 1+1-dimensional quantum field theories, both in the bulk and boundary cases. In section 3 we introduce the form factor program for branch-point twist fields. We explain how it can be employed to obtain the form factors for these twist fields in integrable models, and how these form factors can be checked for consistency against conformal field theory results. For the two-particle form factors we generalize this program also to 1+1-dimensional non-integrable QFTs. We identify the general structure of the two particle form factors and, in integrable cases, suggest how higher particle form factors may be obtained from lower particle ones. For the Ising model, we give closed formulae for all higher particle form factors. In general, we recall how the form factors can be regarded as building blocks for correlation functions. The correlation functions can then be expressed as series where the leading contributions at large distances arise from the lower particle form factors. In section 4, we used these form factor series in order to analyse the entanglement entropy, both in the bulk and boundary cases. In the bulk case, our most important result is the universal expression (1.12) for the next-to-leading order correction to the entanglement entropy at long distances. Both for the bulk and boundary cases of the Ising model, we evaluate all higher order infrared corrections to the entanglement entropy. In the boundary case, we find the precise relationship between the ultraviolet leading behaviour of the entanglement entropy and the boundary entropy introduced by Affleck and Ludwig [44]. Finally, in section 5 we summarise our conclusions and outlook.

2 Replica trick and entanglement entropy

2.1 Partition functions on multi-sheeted Riemann surfaces

From the considerations in the introduction, it is clear that for the study of the entanglement entropy in the scaling limit, we must study partition functions of (euclidean-signature) quantum field theory on multi-sheeted Riemann surfaces. We will come back to the precise relation in the next section, but for now, let us discuss such partition functions. In particular, we wish to introduce the concept of branch-point twist fields, following [27].

The partition function of a model of 1+1-dimensional QFT, with local lagrangian density , on a Riemann surface is formally obtained by the path integral

(2.1)

Here, is an infinite measure on the set of configurations of some field living on the Riemann surface and on which the lagrangian density depends in a local way, and is a point on the Riemann surface. Consider Riemann surfaces with zero curvature everywhere except at a finite number of points; the points where the curvature is non-zero are branch points. In the case where the initial quantum system we consider is on the line, the Riemann surface forms a multiple covering of ; in the case where the model is on the half-line, it forms a multiple covering of the half-plane, which we will take to be the right half-plane, . For simplicity, we will only consider the case in the following discussion, but the case is entirely similar. Since the lagrangian density does not depend explicitly on the Riemann surface as a consequence of its locality (and the fact that the curvature is zero almost everywhere), it is expected that this partition function can be expressed as an object calculated from a model on , where the structure of the Riemann surface is implemented through appropriate boundary conditions around the points with non-zero curvature. Consider for instance the simple Riemann surface with , composed of sheets sequentially joined to each other on the segment (see Fig. 2). We would expect that the associated partition function involves certain “fields”***Here, the term “field” is taken in its most general QFT sense: it is an object of which correlation functions – multi-linear maps – can be evaluated, and which depends on a position in space – parameters that transform like coordinates under translation symmetries. at the points on given by and . These fields would be implementing the appropriate non-zero curvature at the branch points.

The expression (2.1) for the partition function essentially defines these fields: it gives their correlation functions, up to a normalisation independent of their positions; we only have to put as many branch points as we have such fields in the correlation function. But if we insist in understanding this as the initial model on , this definition makes these special fields non-local. Locality of a field (used here in its most fundamental sense) means that as an observable in the quantum theory, it is quantum mechanically independent of the energy density at space-like distances. In the associated euclidean field theory, this means that correlation functions involving these fields and the energy density are, as functions of the position of the energy density, defined on and continuous except at the positions of the fields. The energy density is simply obtained from the lagrangian density. But clearly, bringing the lagrangian density around a branch point changes the value of the correlation function, since it gets to a different Riemann sheet. Hence, the fields defined by (2.1) seen as fields in a model on with lagrangian density are non-local: the lagrangian density is not well-defined on . Locality is at the basis of most of the results in QFT, so it is important to recover it.

In order to correct the problem, note that if we defined a new lagrangian density on at the point by simply summing the initial lagrangian density over all the points of the Riemann surface that project onto , then with respect to this new lagrangian density, the fields defined above would be local. Hence, the idea is simply to consider a larger model: a model formed by independent copies of the original model, where is the number of Riemann sheets necessary to describe the Riemann surface by coordinates on . Let us take again the simple example of . We re-write (2.1) as

(2.2)

where are continuity conditions on the fields restricting the path integral:

(2.3)

where we identify . What appears in the action of that path integral can now be seen as the lagrangian density of the multi-copy model,

The energy density of that model is the sum of the energy densities of the individual copies. The expression (2.2) with (2.3) does indeed define the insertion of local fields at and in the multi-copy model, since the energy density is the same on both sides of the segment according to the conditions .

The local fields defined by (2.3) are examples of “twist fields”. Twist fields exist in a QFT model whenever there is a global internal symmetry (a symmetry that acts the same way everywhere in space, and that does not change the positions of fields): for some lagrangian density . Their correlation functions can be formally defined through the path integral:

(2.4)

where represent insertions of other local fields at different positions. The path integral continuity conditions produce a cut on a half-line starting at the point :

(2.5)

The proportionality constant is an infinite constant that is independent of the position and of those of the other local fields inserted, present in order to render the path integral finite (so that it may represent a correlation function). For insertion of many twist fields, we just add more continuity conditions on different half-lines, starting at different points. The fact that is a symmetry ensures that is local, since it ensures that the energy density is continuous through the cut produced by the continuity condition. Also, it ensures that the result is in fact invariant under continuous changes of the shape of this cut, up to symmetry transformations of the local fields that are being swept. This is because inside a loop, we can always apply a symmetry transformation without changing the result of the path integral, up to transformations of local fields present inside the loop. In this way, we can modify the continuity conditions through the loop. By drawing a line, where the usual continuity holds, starting and ending on a twist-field cut (or possibly on its end-points), we form a loop with the twist-field cut. Applying a symmetry transformation inside this loop, we erase part (or all) of the twist-field cut and make the line a new twist-field cut. In this way, it is possible to move the twist-field cut to any shape. Hence our choice of a half-line extending to the right in the definition of the twist field is just for convenience.

A consequence of the formal definition (2.4) is that correlation functions with some local fields are defined, as functions of (continuous except at positions of other local fields), on a multi-sheeted covering of with a branch point at , whenever . More precisely, twist fields have the property that a clockwise continuous displacement of a local field around back to its initial projected point on is equivalent to the replacement in any correlation function. If , then is said to be “semi-local” with respect to . This twist-field property is satisfied by a large family of fields, not only the one obtained through the formal definition (2.4). For instance, we could have inserted a field in the path integral, leading to the same twist property (this is a descendent twist field). However, the twist property, along with the condition that has the lowest scaling dimension and be invariant under all symmetries of the model that commute with (that is, that it be a primary field in the language of conformal field theory), uniquely fixes the field up to a normalisation. These conditions lead to a definition that is in agreement with the path-integral definition (2.4). In fact, these conditions constitute a more fundamental way of defining the primary twist field than the path integral, as they do not require the existence of a lagrangian density. In particular, they lead to unambiguous definitions in any quantisation scheme (we will discuss what the twist condition is in the quantisation on the line when we discuss form factors). We will take this general point of view in the following, but we will continue to think of a model of QFT through its lagrangian density for clarity.

Twist fields associated to internal symmetries have been largely studied in the context of CFT: they correspond to twisted modules for vertex operator algebras [46, 47, 48], at the basis of so-called orbifold models. In the context of massive integrable QFT, only the simplest (standard) cases are well known. The Ising order and disorder fields are twist fields in the free massive Majorana fermion theory, and in the equivalence between the massive Thirring model and the sine-Gordon model, the bosonic exponential fields of the latter are twist fields of the former. But in fact, for fields with more general twist propertes, the form factor equations of integrable QFT were written in [49]. The first extensive study of form factors of non-standard twist fields (see below) was done in [27, 26], and the first time twist fields form factors were considered beyond integrability was in [42]; we will describe these works in the next section.

In the -copy model with lagrangian , there is a symmetry under exchange of the copies. The twist fields defined by (2.2), which we call branch-point twist fields, are twist fields associated to the two opposite cyclic permutation symmetries and (). We will denote them simply by and , respectively:

(see Fig. 3 for the case ).

Figure 3: The effect of on other local fields.

More precisely, we have

(2.6)

This can be seen to be correct by observing that for , consecutive copies are connected through due to the presence of the cut produced by , whereas for , the additional cut produced by cancels this, and copies are connected to themselves through . Hence, indeed only a finite cut between and remains. The precise proportionality constant was discussed around equation (1.6).

More generally, the identification holds for correlation functions in the model on , this time with an equality sign:

(2.7)

where is the field in the model coming from the copy of , and is the projection of onto .

Note that it is easy to transform the twist field into , and vice versa: we only have to apply the “flip” symmetry transformation by which the order of the copies is inverted (this is another element of the permutation symmetry group). Note also that this construction can also be generalised to Riemann surfaces with more branch points is straightforward, but this will not be needed here.

The conformal dimension of branch-point twist fields is an important characteristic of these fields. It was essentially calculated in [22], although branch-point twist fields were not introduced; only the non-local fields discussed above were considered. Here, we reproduce the derivation of [27], which makes explicit reference to the branch-point twist fields, but otherwise follows closely [22].

Consider the model to be a conformal field theory (CFT). Then also is a CFT. There are fields in that correspond to the holomorphic stress-energy tensors of the copies of , and in particular the sum is the holomorphic stress-energy tensor of . The central charge of is , if is that of .

Consider the holomorphic stress-energy tensor in . We can evaluate the one-point function by making a conformal transformation from in to in (here and are complex coordinates, and ) given by

We have

where the Schwarzian derivative is

Using , we obtain

Since, by (2.7), this is equal to for all , we can evaluate the correlation function involving the stress-energy tensor of by multiplying by :

From the usual CFT formula for insertion of a stress-energy tensor

we identify the scaling dimension of the primary fields and (they have the same scaling dimension) using :

(2.8)

This dimension is the lowest possible dimension for fields with the branch-point twist property, as it is the dimension of a primary field in this family.

Hence, the branch-point twist fields and are unambiguously defined by specifying their associated symmetry, and , namely that they are invariant under other global symmetry transformations that commute with , and by specifying their scaling dimension to be (2.8). This is a definition up to normalisation; we will come back to the normalisation in the next section. There are of course many other twist fields associated to other elements of the permutation symmetry group, and these may have smaller scaling dimension than , but these are not the branch-point twist fields that are used in order to represent the partition function on multi-sheeted Riemann surfaces.

2.2 Bulk entanglement entropy

Partition functions on Riemann surfaces with branch points can be used in order to evaluate the entanglement entropy. We start by considering the bulk case, where the quantum model is on the line, and the corresponding (euclidean) QFT model is on . We are ultimately interested in the scaling limit of the ground-state entanglement entropy, (1.2). However, we will provide here general arguments valid not only for the ground state, by also for excited states. Hence, in place of in (1.2), we will take some arbitrary, possibly excited state . In the scaling limit, the ground state maps to the QFT vacuum state , but excited states map to QFT asymptotic states characterised by the particle content and their momenta (in general, we will keep the notation for the corresponding state in QFT). We will later specialise to the ground state.

The main idea in order to evaluate the entanglement entropy in the scaling limit of a quantum model is to use the replica trick [20, 21, 22, 24]; we will follow more precisely the method of [22]. That is, we use the identity (1.4), where we start by considering the trace for positive integer , evaluate it in QFT (in the scaling limit), then take the appropriate analytic continuation in in order to evaluate the limit of the derivative. Considering positive integer in QFT is useful, because there, the trace is directly related to a partition function on a multi-sheeted Riemann surface, studied in the previous subsection. The way this works is as follows.

First, in order to construct in the scaling limit, consider the QFT Hilbert space as a space of field configurations on . A state can be written as a linear combination of field configurations on where the coefficients are obtained by path integrals on the lower half of the plane, . The boundary condition on is determined by the coefficient we are looking at, and the asymptotic condition is determined by . That is,

(2.9)

where is the action of the model on , and the condition is

Here, represents the asymptotic condition corresponding to the state ; for instance, for the vacuum state, otherwise it reproduces wave packets corresponding to asymptotic particles. The number is the “partition function” of the QFT model in the state :

where

It is a true partition function if is the ground state; otherwise, it corresponds to an excited-state normalisation. The factor is inserted here in order that we maintain the proper normalisation (the states are taken with the natural delta-functional normalisation). Note that the coefficients in the expansion of the dual vector are simply obtained from (2.9) by complex conjugation, along with the inversion of the imaginary time – this naturally gives a path integral on the other half of .

Next, the density matrix is constructed by tracing over degrees of freedom on . We sum the diagonal matrix elements of , but only in the space , formed by field configurations on . This has the effect of “connecting”, on the part of their boundaries, the two surfaces on which the path integrals used to represent matrix elements of and of are defined. Hence, we get a path integral on the whole , with continuity on , but with an open slit on . We are left with a matrix element of , determined by field configurations on both sides of the slit on . That is, we find

(2.10)

where

The power of ,

can be obtained by taking copies of with their own independent path integrals over , and by connecting in a sequential way one side of the slit on of one copy to the other side of the slit on of the next copy (that is, having continuity of the fields through the slit in this sequential way). Finally, taking the trace connects the last copy to the first, so that we obtain the partition function on a multi-sheeted Riemann surface. Taking to consist of only one interval for simplicity, with end-points at positions and in , we then find, in agreement with (1.5),

(2.11)

where the partition function is (2.2). In this derivation, we referred to the definition of the QFT model given by its lagrangian density , but as mentioned in the previous subsection, this is just for clarity of exposition; the existence or not of a lagrangian density does not affect any of the results.

As we saw in the previous subsection, the partition function above can be computed as a two-point correlation function of local fields in the model given by the lagrangian density using (2.6). In the case of the ground state, using (1.6), we have

(2.12)

In the case where is an excited state, the relation (2.6) still holds, because the derivation of the previous subsection did not make any reference to the asymptotic conditions on the field as . However, in (2.6), we have to understand the correlation function as an excited-state diagonal matrix element of the product of twist fields, in order to implement the appropriate asymptotic conditions in the path integral. Hence, what replaces (1.6) is

(2.13)

so that we obtain

(2.14)

In this formula, the normalisation of is not the standard one in the case of an excited state, because of the normalisation factor in (2.13). Indeed, this factor normalises away the infinite-volume divergencies coming from colliding rapidities of asymptotic states in the disconnected terms of the two-point function. We will not discuss further here the case of excited states, and concentrate solely on the ground state entanglement entropy.

2.3 From bulk to boundary entanglement entropy

We are now interested in the situation where the system is on the half-line and the connected region has an end-point on the boundary of the system . However, it will be most easy and instructive to start with a slightly different problem, where the region lies entirely in the bulk, with two boundary points on the half line . Since the system is on the half-line, one needs to provide an additional boundary condition at in order to fully define the model. There are various ways of implementing such a boundary condition. From the viewpoint of the path integral, the boundary condition is implemented by a restriction on the allowed values of the fields and its derivatives at the boundary , along with, possibly, an extra term in the action that is supported on , . From the viewpoint of the quantised theory on the half-line , with the imaginary time, the boundary condition determines the whole Hilbert space, the vacuum and all excited states. Finally, crossing symmetry gives, from the latter, the viewpoint of the quantised theory on the full line with imaginary time . There, the boundary condition corresponds to a boundary state , a state in the usual Hilbert space on the full line.

The derivations of the previous two subsections, connecting the entanglement entropy to partition functions on multi-sheeted Riemann surfaces, and then connecting the latter to correlation functions of branch-point twist fields, can be directly generalised to the boundary situation. First, retracing the steps of the last subsection in the path-integral formulation of the model on the boundary, we obtain again (2.11), where now the partition functions are path integrals over configurations of the field with , with a boundary condition at and possibly an extra boundary term in the action. Second, the derivation of subsection 2.1 can also be directly reproduced, and we obtain, instead of (1.6),

(2.15)

Here is the ground state in the -copy QFT model on the half line, and and are the twist fields with the same fundamental definition as before (the twist property, invariance under other symmetry transformations, and lowest scaling dimension), but as operators on the half-line Hilbert space. Hence we obtain

(2.16)

From this, there are two ways to obtain the entanglement entropy for a region starting from