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Available online at www.sciencedirect.com

ScienceDirect

Nuclear Physics B 913 (2016) 879–911

www.elsevier.com/locate/nuclphysb

Branch point twist ﬁeld correlators in the massive free

Boson theory

Davide Bianchini, Olalla A. Castro-Alvaredo

∗

Department

of Mathematics, City University London, Northampton Square, EC1V 0HB, UK

Received 29

August 2016; received in revised form 16 October 2016; accepted 21 October 2016

Available

online 27 October 2016

Editor: Hubert

Saleur

Abstract

Well-known

measures of entanglement in one-dimensional many body quantum systems, such as the en-

tanglement

entropy and the logarithmic negativity, may be expressed in terms of the correlation functions of

local ﬁelds known as branch point twist ﬁelds in a replica quantum ﬁeld theory. In this “replica” approach

the computation of measures of entanglement generally involves a mathematically non-trivial analytic con-

tinuation

in the number of replicas. In this paper we consider two-point functions of twist ﬁelds and their

analytic continuation in the 1 + 1 dimensional massive (non-compactiﬁed) free Boson theory. This is one

of the few theories for which all matrix elements of twist ﬁelds are known so that we may hope to compute

correlation functions very precisely. We study two particular two-point functions which are related to the

logarithmic negativity of semi-inﬁnite disjoint intervals and to the entanglement entropy of one interval.

We show that our prescription for the analytic continuation yields results which are in full agreement with

conformal ﬁeld theory predictions in the short-distance limit. We provide numerical estimates of universal

quantities and their ratios, both in the massless (twist ﬁeld structure constants) and the massive (expectation

values of twist ﬁelds) theory. We ﬁnd that particular ratios are given by divergent form factor expansions.

We propose such divergences stem from the presence of logarithmic factors in addition to the expected

power-law behaviour of two-point functions at short-distances. Surprisingly, at criticality these corrections

give rise to a log(log ) correction to the entanglement entropy of one interval of length . This hitherto

overlooked result is in agreement with results by Calabrese, Cardy and Tonni and has been independently

derived by Blondeau-Fournier and Doyon [25].

Corresponding author.

E-mail

addresses: davide.bianchini@city.ac.uk (D. Bianchini), o.castro-alvaredo@city.ac.uk

(O.A. Castro-Alvaredo).

http://dx.doi.org/10.1016/j.nuclphysb.2016.10.016

The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

880 D. Bianchini, O.A. Castro-Alvaredo / Nuclear Physics B 913 (2016) 879–911

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

1. Introduction

The problem of quantifying the amount of entanglement which may be “stored” in the ground

state of a many body quantum system has attracted the interest of the quantum information and

theoretical physics communities for a long time. Measuring entanglement is of interest both if

we are to employ entanglement as a quantum computing resource and if we w

ant to learn more

about the fundamental features of quantum states of highly complex quantum systems. Among

such systems, 1 +1-dimensional many body quantum systems have received considerable atten-

tion over the past decade. Much work in this area has been inspired by the results of Calabrese

and Cardy [1] which used principles of Conformal Field Theory (CFT) to study a particular mea-

sure of entanglement, the entanglement entrop

y (EE) [2]. In this seminal work, they generalised

previous results [3] and provided theoretical support for numerical observations in critical quan-

tum spin chains [4]. Before we proceed an

y further a few deﬁnitions are in order: let | be a

pure state describing the ground state of quantum spin chain at zero temperature. Consider a bi-

partition of the chain such as in Fig. 1(a) (suppose there are periodic boundary conditions). Then

the entanglement entropy associated to re

gion A may be expressed as S() =−Tr(ρ

log ρ

)

where ρ

= Tr

(||) is the reduced density matrix associated to subsystem A and  is the

subsystem’s length.

One of the main results of [3,4,1] describes the entanglement entropy of 1 +1 dimensional

many body quantum systems (such as spin chains) in the continuous limit at criticality. Such

systems are described by CFT and their EE displays universal features expressed by the now

famous formula: S() =

log





. That is, the EE of a subsystem of length  of an inﬁnite critical

system diverges logarithmically with the size of the subsystem, with a universal coefﬁcient which

is proportional to the central charge of the CFT, c. There are non-universal constant corrections

to this leading behaviour which may be encoded by a short-distance cut-of

f . This behaviour

has been numerically and analytically studied for a plethora of spin chain models in works such

as [4–13].

Another popular measure of entanglement is the logarithmic negativity (LN) [14–18]. Con-

sider again a quantum spin chain in a pure state | and a partition such as depicted in Fig. 1(b).

Then, the LN is a measure of the amount of entanglement between the two non-complementary

sub-systems A and B. Its formal deﬁnition depends on the reduced density matrix ρ

A∪B

E(

, 

) = log(Tr|ρ

A∪B

|) where T

represents partial transposition with respect to subsys-

tem B and |ρ| is the trace norm of ρ, that is the sum of the absolute values of its eigenvalues.

The LN of 1 +1 dimensional critical systems has been studied numerically in [21–23] and more

recently, both numerically and analytically exploiting fundamental conformal ﬁeld theory prin-

ciples, in [19,20]. Since then man

y particular models have been analysed (see e.g. [26–29]).

However, for general conﬁgurations such as in Fig. 1(b) there is no known analytic formula for

generic CFTs. There are however particular limits which are easier to treat such as the limit of

Fig. 1. Typical conﬁgurations for the entanglement entropy of one interval and the logarithmic negativity.

D. Bianchini, O.A. Castro-Alvaredo / Nuclear Physics B 913 (2016) 879–911 881

adjoint intervals (

→ 0) and the limit of semi-inﬁnite disjoint intervals (

, 

→∞keeping



ﬁnite). The former has been studied in [19,20] for generic CFT yielding the simple expres-

sion E (

, 0, 

) =

log



(

+

)

whereas the latter is harder to treat in critical systems but is

of interest in the study of quantum systems near criticality. Such systems are described by 1 +1

dimensional massive quantum ﬁeld theories which, unlike CFT, allow for the existence of a ﬁnite

correlation length. The negativity of such systems wa

s ﬁrst studied in [30] where new results for

both of the limits above in near-critical systems were obtained.

In this paper we will be interested in a particular prescription for the calculation of both the EE

of a single interval and the LN of semi-inﬁnite disjoint regions. It turns out that both quantities

may be expressed in terms of two-point functions of a particular class of ﬁelds known as branch

point twist ﬁelds [1,32]. This relationship comes about through a technique commonly kno

wn as

the “replica trick”. The replica trick may be applied to both the computation of the EE and of the

LN. It involves a rewriting of the deﬁnitions above as follows

S() =− lim

n→1

Tr(ρ

) and E(

,

) = lim

→1

log(Tr(ρ

A∪B

)

), (1)

where the symbol n

in the second formula means n even, that is the limit n to 1 must be carried

out by analytically continuing the function from even, positive values of n to n = 1. The repre-

sentations above were used ﬁrst in [3] for the EE and in [19,20] for the LN. The advantage of

such representations is that both Tr(ρ

) and Tr(ρ

A∪B

)

admit a natural physical interpretation

as partition functions in “replica” theories. The replica theory is a new model consisting of n

non-interacting copies of the original theory. In this context it is natural for n to take positive in-

teger values. However, the deﬁnitions (1) require that such traces be analytically continued from

n inte

ger (and in the LN case, also even) to n real and positive. Hence, formulae (1) are advanta-

geous in that partition functions in replica theories may be computed systematically by various

approaches, but also disadvantageous because the analytic continuations involved are often very

difﬁcult to perform and there is no generic proof of e

xistence and uniqueness.

It wa

s ﬁrst noted in [1] that the function Tr(ρ

) may be expressed as a two-point function of

ﬁelds with conformal dimension given by





n −



. (2)

In fact such ﬁelds had been previously discussed in the context of the study of orbifold CFT

where they emerge naturally as symmetry ﬁelds associated to the permutation symmetry of the

theory [33,34]. In [32] such ﬁelds were named branch point twist ﬁelds and studied in the context

of 1 +1 dimensional massi

ve QFT. Their connection to the cyclic permutation symmetry of the

replica theory was made explicit by formulating their exchange relations with the fundamental

ﬁelds of a generic replica QFT. For integrable QFT this allowed for the formulation of twist ﬁeld

form factor equations whose solutions are matrix elements of twist ﬁelds. Let T be a twist ﬁeld

associated to the c

yclic permutation symmetry j → j + 1 and

T its conjugate, associated with

the permutation symmetry j → j −1 with j =1, ..., n. Then, we may write:

Tr(ρ

) =

4

T (0)

T ()

and

Tr(ρ

A∪B

)

= 

8

T (−

)

T (0)

T (

)T (

+

)

. (3)

At criticality, these formulae may be used directly to derive the expressions for S() and

E(

, 0, 

) given above. The same formulae may be used to study QFT beyond criticality as

882 D. Bianchini, O.A. Castro-Alvaredo / Nuclear Physics B 913 (2016) 879–911

done in [32,30]. In this paper we will analyse the short-distance (e.g.   1) behaviour of the

correlators T (0)

T ()

and T (0)T ()

=

T (0)

T ()

in a massive free Boson theory. At

short-distances we expect the massive QFT to be described by its corresponding ultraviolet limit

(that is, the massless (non-compactiﬁed) free Boson CFT). Thus, we expect these two-point

functions to exhibit power-law behaviours with powers related to the dimension of twist ﬁelds.

Extracting these short-distance beha

viours from a form factor expansion (which is eminently a

large-distance expansion) is generally highly non-trivial and can seldom be done with precision

for any ﬁelds. However, as we will see, this can be done with great precision for the massive free

Boson, on account of the theory’s simplicity and the special properties of the twist ﬁeld form

actors. For the massive free Boson all form factors of twist ﬁelds, that is objects such as

T |j

...j

(θ

, ···,θ

;n) :=0|T (0)|θ

, ···,θ



...j

/T 

, (4)

are known explicitly. Here 0| represents the vacuum state and |θ

, ···, θ



...j

represents an

in-state of k particles with rapidities θ

, ..., θ

and quantum numbers j

...j

. In the free Boson

case, these quantum numbers are just the copy number of the Boson in the replica theory. Here

we have chosen to normalise all form factors by a constant (the vacuum expectation value of the

twist ﬁeld T 

). This will be convenient for later computations.

By reconstructing the short-distance (po

wer-law) behaviour of the correlators T (0)

T ()

and T (0)T ()

for n ≥ 1, integer or not, we will provide strong evidence for our approach to

performing the analytic continuation of the correlators in n. This will provide support for our

methodology and will allow us to examine twist ﬁeld two-point functions further and extract

values of universal quantities such as e

xpectation values and structure constants of twist ﬁelds.

The paper is or

ganized as follows: In sections 2 and 3 we review basic CFT and QFT re-

sults, regarding the short distance behaviour of two-point functions of twist ﬁelds and how these

two-point functions may be expressed in terms of the form factors (4). In section 4 we show

how the power-law decay of tw

o-point functions of twist ﬁelds may be obtained exactly from

form factors in the massive free Boson theory for n ≥ 1 real. In section 5 we provide form factor

expansions for the constant (universal) coefﬁcients that multiply the leading power-law in the

two-point functions of twist ﬁelds. We employ these e

xpansions to obtain numerical predictions

for the ratio of the structure constant C

and the expectation value T 

, analytically continued

from n odd and for the structure constant C

analytically continued from n even. We compare

our values of C

for n even to analytical values obtained in [20] and ﬁnd good agreement. We

numerically examine the limit lim

→1

and compare to an analytical prediction given in

[20]. In section 6 we present an interpretation of the emergence of divergent sums in the rep-

resentation of particular ratios of expectation values and structure constants of the massive free

Boson theory. We propose that such divergences must be related to the presence of log

arithmic

corrections to the two-point functions at criticality. We conclude that such corrections will give

rise to an additional log(log ) term in the EE and the Rényi entropy of one interval in the mass-

less (non-compactiﬁed) free Boson theory. This is in full agreement with previous results for the

LN [20] and the EE [24] of the compactiﬁed massless free Boson in the limit of inﬁnite compact-

iﬁcation ratio. Fo

r the EE the presence of such corrections has also been established analytically

by a different method in [25] but had been overlooked in [31]. In section 7 we compare our nu-

merical estimates of the va

lue of lim

→1

as well as the analytical value given in [20] to

a value that can be read off from numerical results in [35] for the LN of a harmonic chain out

of equilibrium and their CFT interpretation [36]. We present our conclusions in section 8. Ap-

pendix A collects some useful summation formulae which feature in the form f

actor expansions

D. Bianchini, O.A. Castro-Alvaredo / Nuclear Physics B 913 (2016) 879–911 883

of sections 4 and 5. Appendix B provides a discussion and assessment of the error of some of

our numerical procedures.

2. Conformal ﬁeld theory recap

As described in the introduction, we wish to study the two-point functions T (0)

T ()

and

T (0)T ()

and examine their short-distance behaviour. This behaviour is entirely predicted by

CFT and may be expressed as

log



T (0)

T ()

T 



m1

=−4

log  −2logT 

. (5)

Similarly

log



T (0)T ()

T 



m1

⎧

⎪

⎨

⎪

⎩

−2

log  +log

T 

for n odd

−4(

−

) log  +log

T 

for n even

(6)

Note that by examining the next-to-leading order (-independent) corrections above we may

extract values for universal QFT quantities such as the twist ﬁeld expectation value T 

and

the structure constants C

and their ratios. These are difﬁcult to compute by other methods,

demonstrating once more that the form factor approach in particularly powerful in this context.

The dif

ference between the n odd and n even cases was ﬁrst discussed in [19,20] and follows

from the leading term in the conformal OPE of the ﬁeld T with itself, which takes the form

T (0)T () ∼ C



−4

+2

(0) +··· (7)

This leading term is characterized by a new twist ﬁeld T

of conformal dimension 

which is

associated with the permutation symmetry j → j +2for j = 1, ..., n. As discussed in [19,20]

the nature of this ﬁeld is very different depending on whether n is odd or even. Whereas for n

odd, the ﬁeld T

is equivalent to the ﬁeld T (the permutation j → j +2 still allows for visiting

all copies, albeit in a different order), for n even the permutation j → j + 2 divides even- and

odd-labelled copies so that T

is equivalent to two copies of T acting on a

-replica theory.

Consequently the conformal dimension of T

is 

= 

for n odd and 

= 2

for n

even. For the same reasons T



=T 

for n odd and T



=T 

for n even. This simple

interpretation also shows how the analytic continuations (1) from n even and n odd should be

different. Note that, T 

= 1 both for massive and massless theories as the twist ﬁeld becomes

the identity ﬁeld at n =1.

In massive theories, the correlator T (0)T ()

encodes the -dependent part of the negativity

E(∞, , ∞) of semi-inﬁnite disjoint regions. This follows simply from the deﬁnition (3) and the

factorisation of correlation functions at large distances in massive QFT.

In this paper we will use a form f

actor expansion of these correlators to extract the leading term

(the log  term). We will turn our attending to the next-to-leading order corrections in section 5.

3. Form factor expansion of two-point functions

In a massive integrable QFT such as the massive free Boson, the functions (5)–(6) admit a

natural large m expansion in terms of form factors. In general we have that the (normalised)

logarithm of the two-point function of local ﬁelds O

, O

admits and expansion of the form

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