Entanglement in quantum impurity problems is non perturbative

# Entanglement in quantum impurity problems is non perturbative

H. Saleur Institut de Physique Théorique, CEA, IPhT and CNRS, URA2306, Gif Sur Yvette, F-91191 Department of Physics, University of Southern California, Los Angeles, CA 90089-0484    P. Schmitteckert Institute for Nanotechnology, Karlsruhe Institute of Technology, 76344 Eggenstein-Leopoldshafen, Germany    R. Vasseur Institut de Physique Théorique, CEA Saclay, 91191 Gif Sur Yvette, France LPTENS, 24 rue Lhomond, 75231 Paris, France
July 6, 2019
###### Abstract

We study the entanglement entropy of a region of length with the remainder of an infinite one dimensional gapless quantum system in the case where the region is centered on a quantum impurity. The coupling to this impurity is not scale invariant, and the physics involves a crossover between weak and strong coupling regimes. While the impurity contribution to the entanglement has been computed numerically in the past, little is known analytically about it, since in particular the methods of conformal invariance cannot be applied because of the presence of a crossover length.

We show in this paper that the small coupling expansion of the entanglement entropy in this problem is quite generally plagued by strong infrared divergences, implying a non-perturbative dependence on the coupling. The large coupling expansion turns out to be better behaved, thanks to powerful results from the boundary CFT formulation and, in some cases, the underlying integrability of the problem. However, it is clear that this expansion does not capture well the crossover physics.

In the integrable case – which includes problems such as an XXZ chain with a modified link, the interacting resonant level model or the anisotropic Kondo model – a non perturbative approach is in principle possible using form-factors. We adapt in this paper the ideas of Doyon1 (); Doyon () to the gapless case and show that, in the rather simple case of the resonant level model, and after some additional renormalizations, the form factors approach yields remarkably accurate results for the entanglement all the way from short to large distances. This is confirmed by detailed comparison with numerical simulations. Both our form factor and numerical results are compatible with a non-perturbative form at short distance.

###### pacs:
05.70.Ln, 72.15.Qm, 85.35.Be

## I Introduction

Quantum entanglement has given rise to much work in the condensed matter community as a new way to explore interesting aspects of physical systems. The Kondo problem for instance has been revisited along these lines, with studies addressing the interplay between the impurity screening and the information shared between the impurity and the bath AffleckLaflorencie1 (); AffleckLaflorencie2 (). It is certainly reasonable to expect that entanglement – together with other quantities inspired by quantum information theory, such as the Loschmidt echo or the work distribution – might shed new light on, and offer new experimental/numerical probes of, the key physical features of the Kondo and other problems ChoKenzie (); AffleckLaflorencie1 (). A particularly interesting question in this direction is whether the Kondo screening cloud – which has had so elusive an appearance in standard thermodynamic quantities Affleck () – might play a bigger role in quantum information aspects. Other aspects of interest in the context of 2 level systems interacting with gapless excitations – generalizing the Kondo problem – apply to the decoherence of qubits interacting with the environment LeHur1 (); LeHur2 (); LeHur3 ().

A large part of the work combining entanglement and quantum impurities has been numerical so far. Indeed, apart from the scale invariant situations, where conformal invariance techniques have led to spectacular progress CardyCalabrese1 (); CardyCalabrese2 (), the general situations involving crossover are very difficult to tackle. This is mostly because the entanglement is a different kind of quantity, not amenable to simple Bethe ansatz calculations, for instance. There is, however, another reason for the relative lack of analytical results in this area: entanglement, being a zero temperature quantity, is naturally plagued by IR divergences, which make it non perturbative in the impurity strength. In that respect, it does behave somehow like some properties of the Kondo screening cloud studied in AffleckBarzykin (); Affleck ().

In order to clarify the main features of entanglement in the presence of impurities – in particular its scaling properties, and flow from small to strong coupling – we focus in this paper on a couple of representative situations, which we handle by a mix of analytical and numerical techniques. The lessons learned will be put to use in forthcoming papers, with applications of more direct physical interest.

The paper is organized as follows. In section II we discuss the basic models we want to study, and define precisely the entanglement entropy. In section III we put together the perturbative calculation of the entanglement at small coupling, and show that it is plagued by strong IR divergences. In section IV we discuss this difficulty in a more general context. In section V we show how the non perturbative nature of the entanglement entropy can be obtained using general conformal field theoretic arguments. In section VI we recall the principles of the large coupling expansion proposed in AffleckLaflorencie2 () and carried out to high order in FretonIR (). When the dimension of the perturbation is , we develop in section VII the form-factor approach using the results of Doyon (); Doyon1 (), and obtain non perturbative approximations for the entanglement extrapolating all the way from the UV to the IR limit. Finally, in section VIII we compare our results with those of exact numerical calculations on large spin chains. The conclusion contains some last comments and prospect for future work. Finally, appendix A contains a discussion of the equivalence between our impurity models when the dimension of the perturbation is to the boundary Ising model with a boundary magnetic field at special values of the coupling.

## Ii Models and questions

The main problem we study in this paper – though it has various, mathematically equivalent formulations, see below – is the calculation of the entanglement of a region of length centered on an ‘impurity’ in an otherwise one dimensional, gapless quantum system. We characterize this entanglement by the von Neumann entropy , where is the reduced density matrix that has been formed by tracing over the degrees of freedom outside of the segment of length .

An example of this setup is obtained by taking two semi infinite XXZ chains coupled by a weak link:

 H=−1∑−∞J[SxiSxi+1+SyiSyi+1+ΔSziSzi+1]+∞∑1J[SxiSxi+1+SyiSyi+1+ΔSziSzi+1]+J′[Sx0Sx1+Sy0Sy1+ΔSz0Sz1]. (1)

The bulk chains are in a gapless Luttinger liquid phase for . We shall consider the case of anisotropy , where the tunneling between the two half infinite chains is a relevant perturbation, and one observes healing at large scales. The case is exactly marginal. We parametrize , . We focus on the physics at energies much smaller than the band-width, where field theoretic results can be applied.

Consider then the entanglement of a region of length centered on the modified link. We can easily surmise what this entanglement will look like in the high and low energy limits from the existing literature. Indeed, at high energy, the system is effectively cut in half. Using the well known formula for the entropy of a region on the edge of a conformal invariant system we have

 SUV=2×[16ln2Lϵ+s12+lng]=13ln2Lϵ+s1+lngUV, (2)

where we used that the central charge is unity, is a UV cutoff of the order of the lattice spacing , and where is the boundary entropy AffleckLudwig () associated with the conformal boundary condition corresponding to an open XXZ spin chain. The remaining constant term is non-universal as it obviously depends on the definition of the cutoff .

On the other hand, at low energy, healing has taken place, the system behaves just as one ordinary quantum spin chain, and the entropy obeys the general form for a region of length in the bulk of a conformal invariant system:

 SIR=13ln2Lϵ+s1+lngIR, (3)

Here, we have allowed for a term , which can be thought of as a residual contribution of the weak link at low energy. In general, comparing entanglements for bulk and boundary theories is indeed difficult, since the dependency of the cutoff on the physical cutoff (the lattice spacing in the spin chain , which is the same in both geometries) is not universal, and not necessarily the same in the bulk and boundary cases. This important aspects is discussed in detail in Doyon (), see in particular section 6.2.1 in that reference. The point for us is that the quantity is well defined, and its value can be easily obtained from the folded version of the system (see eq. (6) below).

More generally, since the bulk behavior of the entanglement entropy is not modified, it is natural to expect the existence of a scaling relation

 S(L)−SIR=Simp(LTB), (4)

where the crossover scale, , is expected to be related to the coupling . should be a monotonic function extrapolating between at small values of the argument and at large values.

To proceed, and conveniently describe the field theory limit AffleckEggert (), we first observe that the problem, at low energy, can be turned into a purely chiral one. Indeed, in the low energy limit, each half chain is equivalent to a combination of L and R moving excitations, and we formally map via a canonical transformation the L moving sector into a R moving one so as to have two chiral ‘wires’ representing the two half chains. The additional tunneling between the two chains becomes, in this language, a hopping term between two chiral wires. Bosonizing, forming odd and even combinations of the bosons for each wire, one finds that the odd combination decouples, while for the even one obtains the simple hamiltonian,

 H=∫∞−∞dx (∂xϕR)2+λcosβϕR(0). (5)

where is the conformal weight of the perturbation, and we have set the Fermi velocity . The dimension of the perturbation being , we see that . One can also fold back this problem into the boundary sine-Gordon model (BSG) with Hamiltonian

 HBSG=∫0−∞dx12[(∂xΦ)2+Π2]+λcosβ2Φ(0). (6)

This shows equivalence to a large variety of other problems, including the one of tunneling between edge states in the Fractional Quantum Hall Effect (FQHE) WenFQHE (); ConductanceFQHE (). In this case, is the filling fraction. The RG flows from Neumann () to Dirichlet () boundary conditions (BC), and the boundary entropy associated with these conformally invariant BC satisfy , as claimed earlier.

An interesting variant involves modifying two successive links on the chain:

 H = −2∑−∞J[SxiSxi+1+SyiSyi+1+ΔSziSzi+1]+∞∑1J[SxiSxi+1+SyiSyi+1+ΔSziSzi+1]+ (7) + J′[Sx−1Sx0+Sy−1Sy0+ΔSz−1Sz0+Sx0Sx1+Sy0Sy1+ΔSz0Sz1].

This is equivalent to tunneling through a resonant level at the origin. This time, the dimension of the tunneling operator is half what it is in the previous situation, is always relevant for , and the system is always healed at low energy. The same series of manipulations – ‘unfolding the two half chains’, forming odd and even combinations, decoupling the odd one and bosonizing – lead to the hamiltonian formulation

 H=∫∞−∞dx(∂xϕR)2+λ[eiβ√2ϕR(0)S−+e−iβ√2ϕR(0)S+], (8)

where we recall that (Post-publication footnote added 111Minor erratum: unfortunately, the field theory (8) does not correspond to the XXZ spin chain with two successive weak links as claimed here. The bosonization of the XXZ chain with two weak-links leads instead to a field theory with two chiral bosons (one for each half-chain) tunneling through a spin- impurity. The dimension of the perturbation is also and the predictions of this paper can be applied to that case as well. The anisotropic Kondo Hamiltonian (8) can be realized on the lattice as an interacting resonant level model (IRLM), with two half-infinite chains of non-interacting fermions tunneling through a resonant level model with Coulomb interactions on the dot only. In the XXZ language, this corresponds to an XX spin chain with two weak links, with anisotropy on the two weak links. All the conclusions of this paper remain unchanged. ) . In this case, the dimension of the perturbation is (note the factor compared with the first case). The problem can also be folded back into the anisotropic Kondo problem

 HAK=∫0−∞dx12[(∂xΦ)2+Π2]+λ[eiβ2√2Φ(0)S−+e−iβ2√2Φ(0)S+]. (9)

Of particular interest is the case , which corresponds to free fermions. While the chain with one weak link is marginal, the chain with two weak links describes an interesting flow, and is in fact equivalent to a widely studied problem – that of the resonant level model (RLM). Indeed, fermionization in this case leads to

 H=−J(−2∑−∞c†m+1cm+h.c.)−J(∞∑1c†m+1cm+h.c.)−J′(c†−1c0+c†0c−1+c†0c1+c†1c0), (10)

where we have redefined the couplings and . When going to the continuum limit, the site behaves like a two level impurity, and the hamiltonian reads

 H=∫0−∞i[ψ†1L∂xψ1L−ψ†1R∂xψ1R]dx+∫∞0i[ψ†2L∂xψ2L−ψ†2R∂xψ2R]dx+λ[(ψ†1(0)+ψ†2(0))d+h.c.], (11)

with , same for the second species, 222Having the Fermi velocity , corresponds here to in eq (10).. In contrast with the case of the XX chain with a single defect, this non-interacting problem not scale invariant. The coupling flows, and the system again exhibits healing: at low energy, the impurity level is completely hybridized with the two half chains.

Let us go back to the general case . Proceeding like before, we can write the limiting behaviors of the entanglement entropy. At low energy, the impurity is hybridized, the system behaves just as one non chiral wire and a hybridized impurity, and the entropy obeys the general form for a region of length in the bulk of a conformal invariant system decoupled from the two baths, so

 SIR=13ln2Lϵ+s1+lngIR, (12)

where once again, we included a term that accounts for the remaining boundary condition at of the hybridized impurity. Meanwhile, at high energy, the impurity is completely decoupled from the wires, and one gets

 SUV=2×[16ln2Lϵ+s12+lng]=13ln2Lϵ+s1+lngUV. (13)

Using the folded (boundary) version of the system (9), one can easily argue that , as a decoupled impurity has two degrees of freedom. One thus expects a behavior entirely similar to (4), where the crossover scale, , is expected to be proportional to a power of the coupling square, , and should be a monotonic function extrapolating between at small values of the argument and at large values.

Finally, we note that in the boundary versions (6,9), the entanglement impurity we have discussed is now the entanglement of a region of length on the edge of the system with the rest. If one were to start from an (anisotropic) Kondo version, this would be the most natural point of view AffleckLaflorencie1 ().

There are of course other variants of the problem, for instance involving a slightly modified link in the antiferromagnetic XXZ chain with , interactions in the RLM model, etc. In all these cases, we should stress that the geometry we are considering is probably not the most interesting: considering the entanglement of the two halves connected by a weak link or a quantum dot is probably more physical. This latter problem is however significantly more difficult technically. We will discuss it in our next paper, relying on the present work as a stepping stone.

## Iii UV perturbation

The most natural to explore the behavior of between the fixed points is to use perturbation theory. The required calculation is a modification of the one proposed in Holzhey (); CardyCalabrese1 (); CardyCalabrese2 (). Using the well-known replica trick, one first observes that the entanglement entropy can be obtained from the Renyi entropies by considering .The Renyi entropies in turn can be obtained as , where is the partition function on a -sheeted Riemann surface with the sheets joined at a cut corresponding to the segment of length . The difference between the problem at hand and the conformal case is that now there is a perturbation inserted at the origin in the Hamiltonian formulation, which corresponds to the insertion of a perturbation along an imaginary time line for each of the sheets. The modified partition functions can in principle be expanded in powers of the coupling constant, and perturbative corrections to the Renyi entropies and the entanglement entropies finally obtained.

To see what happens in more detail, we consider first hamiltonian (5). We start with sheets, and write formally the Renyi entropy as a functional integral for a pair of chiral bosons as

 R2=Z2(Z1)2=∫twist[Dϕ1][Dϕ2]exp{−A[ϕ1]−A[ϕ2]−λ∫∞−∞[V1(x=0,y)+V2(x=0,y)]dy}(∫[Dϕ]exp{−A[ϕ]−λ∫∞−∞V(x=0,y)dy})2, (14)

where is the perturbation, and is the free action. Note we are working in the chiral version, but have suppressed the ‘R’ label in the fields for simplicity of notation. Finally, the label twist means the functional integral is evaluated with conditions around the cut

 ϕ1(−L≤x≤L,y=0+)=ϕ2(−L≤x≤L,y=0−),ϕ2(−L≤x≤L,y=0+)=ϕ1(−L≤x≤L,y=0−). (15)

If , the ratio (14) is nothing but the correlation function of an (order two) twist operator corresponding to (15), which we will write then Holzhey (); CardyCalabrese1 ()

 R2(λ=0)=z2z21=⟨τ2(L,0)~τ2(−L,0)⟩L(2),C∝L−1/8, (16)

where means that the correlator is to be evaluated in the plane (worldsheet) with the Lagrangian . We also recall that in general, the scaling dimension of the twist operator reads . We now consider the perturbation expansion in powers of . For the denominator, we have immediately

 D=d2;   d=z1[1+λ22∫dydy′12|y−y′|2μ+…] (17)

where the factor in the integral comes from the ’s in the cosines, and we recall that is the conformal weight of the perturbation.

For the numerator, things are a little more complicated since we have two types of fields on the plane, with and contractions, in the presence of the glueing conditions along the cut. To proceed, we uniformize. We start with the complex coordinates , and introduce

 z=(w−uw−v)1/2, (18)

where and are the complex coordinates of the cut’s extremities. This maps the whole 2-sheeted Riemann surface to the -complex plane . We then write (14) as

 R2R2(λ=0)=1+λ22∫dwdw′⟨cosβϕ(w)cosβϕ(w′)⟩R2,1(1+λ22∫dydy′12|y−y′|2μ)2+… (19)

where the spatial integrals in the numerator are now over (worldsheet), and we have a unique boson instead of and . Here the integrals in the numerator correspond to insertions along two lines, corresponding to the two copies of the theory, so overall there are four possible terms (contractions). The perturbation is a primary operator, so we can calculate the correlations on by using the conformal mapping (18). We have

 2×22μ⟨cosβϕ(w)cosβϕ(w′)⟩R2,1=(u−v)2μN2μ12(w−u)μ/2(w−v)μ/2(w′−u)μ/2(w′−v)μ/2, (20)

Here,

 N12=(w−u)1/2(w′−v)1/2−(w′−u)1/2(w−v)1/2. (21)

In (19) we have to integrate both over the imaginary axis , but also over the second sheet, which is obtained by sending and same for . This means we end up with two integrals where are on the same sheet, and two where they are on different sheets.

Replacing everything by the particular choice of coordinates, and expanding the denominator in (19) we get

 R2R2(λ=0)=1+λ24×2∫∞−∞dydy′[Gsame(y,y′)+Gdiff(y,y′)]+…, (22)

where the factor comes since there are two sheets, and insertions can be on same or different sheets, so

 Gsame(y,y′)=122μ|y−y′|2μ[(iy−L)1/2(iy′+L)1/2+(iy′−L)1/2(iy+L)1/2]2μ(y2+L2)μ/2((y′)2+L2)μ/2−1|y−y′|2μ, (23)

while will be the same expression with a minus in the numerator’s bracket, and no subtraction (the two point function of the fields in different copies in the denominator of course vanish identically).

It is convenient to introduce new variables via , so the integral becomes

 L2−2μ∫π/2−π/2dθdθ′(cos2θ)μ−1(cos2θ′)μ−1(cos2θ−θ′2)μ−1(sin2(θ−θ′))μ (24)

 L2−2μ∫π/2−π/2dθdθ′(cos2θ)μ−1(cos2θ′)μ−11(cos2θ−θ′2)μ (25)

Both integrals are UV convergent for a relevant perturbation . They are however both IR divergent (here the IR region being ). This means that, although formally the perturbation at small coupling looks like it should be an expansion in powers of , this might actually not be the case and, as we shall see later, is not. In fact, we will see that the entanglement is simply non perturbative in , and cannot be obtained via this perturbation theory.

This result could appear as a surprise. On the one hand, the entanglement is a quantity, and such quantities are often plagued by IR divergences. On the other hand, we are looking for an dependent quantity, and it would be natural to expect that would act as an effective IR cutoff, rendering the perturbation expansion finite. This is however definitely not what happens. The situation is reminiscent of similar divergences encountered in the Kondo screening cloud problem Affleck ().

We stress finally that the argument applies almost without modification to the Hamiltonian (8). All that changes is that the perturbation is of the form instead of , so exponentials of opposite signs have to alternate in the imaginary time insertions, modifying some of the numerical coefficients, but not the integrals or their divergences.

## Iv IR divergences in quantum impurity problems

To gain a better understanding of the situation, it is useful to start by discussing another observable 333This section somewhat lies outside of the main flow of this paper, as it does not deal directly with the computation of the entanglement entropy. It does contain however some very important points for our purpose, but the reader interested only in entanglement may wish to skip this section. than the entanglement entropy for (5). We turn briefly to the boundary formulation (6), and consider the one point function which appears, for instance, in the determination of Friedel oscillations for impurities in Luttinger liquids. Simple scaling arguments suggest the general form

 ⟨:cosβΦ2:⟩=(2|x|)μ/2F(λ|x|1−μ), (26)

where is a universal function obeying , so the field sees Dirichlet boundary conditions, and the bulk normalization has been chosen appropriately.

Determining the function is also a difficult problem. The most natural is once again to attempt perturbation theory in . This would share many features of the calculation of one and two point functions in the bulk sine-Gordon theory KL (). There, it is well known that (in fact, the result essentially goes back to Coleman), provided (that is, the perturbing operator is not irrelevant), there are no UV divergences in the calculation. All divergences coming from bringing together two insertions of the perturbing term are exactly canceled by similar divergences coming from the expansion of the denominator (the partition function and associated bubble diagrams). In general, the divergences are indeed controlled by the Operator Product Expansion (OPE),

 eiβΦ(y)/2e−iβΦ(y′)/2=|y−y′|−2μ(1+…−πμ(y−y′)2(∂Φ(y))2+…),

with all fields at , the coordinate along the boundary. The leading order comes from the contribution of the identity operator and leads to a disconnected piece subtracted off by a similar term in the denominator. The stand for higher orders, or lower orders that vanish after integration. The overall singularity at order thus behaves as , it comes with dimension and for the integrals are UV finite. Other singularities (when several points are brought together at once etc KL ()) behave similarly.

However, there will always appear IR divergences at a certain order, depending on the exact value of the conformal weight . Of course, we expect in the end the scaling form to hold, and thus to depend only on , . What will happen in general is that the divergences in the perturbative expansion have to be resummed before the proper scaling form can be obtained. The latter, in general, will thus behave non perturbatively in the coupling .

This is nicely illustrated in the case , where the the exact form of the one point function is known, thanks to a mapping to the boundary Ising model (see below), together with a very clever argument by Chatterjee and Zamolodchikov CZ (). One finds LLS ()

 ⟨:cosβΦ2:⟩(x)=4λ√π(x2)1/4Ψ(1/2,1;8πλ2x) (27)

where is the degenerate hypergeometric function. The asymptotics follows from , where is the usual modified Bessel function, so that we find

 ⟨:cosβΦ2:⟩(x) ∼ (2x)1/4, x≫1, ⟨:cosβΦ2:⟩(x) ∼ x−1/427/4λx1/2×−ln(λ2x), x≪1. (28)

We thus see that this function exhibits a non perturbative dependence at small coupling . The non analyticity in arises from the IR divergence of the first perturbative integral.

There is a general way to understand the non analyticity of course. Whenever a bulk operator (of conformal weights ) is sent to the boundary where it becomes a boundary field of weight , one has

 O(x)≈xhB−2hOB+… (29)

In our case, the cosine of the bulk field simply goes over to the cosine of the boundary field. We have thus and , while . We thus expect that . The dependence of the one point function of the boundary field on is non analytic in , and non perturbative - of course, because again of IR divergences. This problem is the cousin of a similar problem in bulk massive theories, and has been studied in Zamo (); FLZZ (). We deduce from this that, to leading order,

 ⟨O(x)⟩∝xhB−2hλhB/1−μ (30)

More generally, we can write

 O(x)≈∑ixhiB−2hOiB (31)

so

 ⟨O(x)⟩≈∑ixhiB−2hciBThiBB=x−2h∑ixhiBciBThiBB≡x−2hG(xTB) (32)

with

 F(y)∝yhB∝λhB/1−μxhB (33)

Going back to the case of Friedel oscillations, we have therefore . This leading dependence in replaces the expected perturbative one, which would be linear in .

The foregoing argument applies in the generic case. Whenever there are “resonances” and the parameter takes special rational values , extra logarithmic terms appear in the one point functions of the operators right on the boundary, which translates in logarithms in the one point functions of operators at as well. This is the case precisely when .

It is important to stress also that the IR divergences naturally disappear at finite temperature, providing a natural cutoff. Once again this is illustrated in the case, where one finds LLS (); SM (), for Friedel oscillations at finite temperature

 ⟨:cosβΦ2:⟩(x)=f(2λ2/T)(4πTsinh(2πTx))1/4F(12,12;1+2λ2T,1−coth2πxT2) (34)

Here is the usual hypergeometric function, is a function whose existence and value were determined in SM (). The right hand side admits a perturbative expansion in powers of , whose leading term, at fixed , goes as when . The coefficient of thus diverges in the zero temperature limit, in agreement with the fact that the true expansion is then in .

The general IR behavior can easily be investigated. One finds that at order , there is no IR divergence provided . Only when – that is, the boundary perturbation is exactly marginal, and the bulk is a Fermi liquid – are all orders finite. In this case, the Friedel oscillations admit a perturbative expansion in powers of  Egger ().

While the nature of the divergences is quite generic, the quantities for which they occur depend on the problem at hand. For instance, for the screening cloud in the (anisotropic) Kondo model, divergences occur even when the boundary perturbation has dimension one - in that case, it is marginally relevant Affleck ().

## V The small coupling behavior of the entanglement entropy

We now go back to the calculation of the entanglement entropy for hamiltonian (5). We see that, to obtain the non perturbative UV behavior, we must discuss twist fields and their OPEs. We follow the paper CCT () but focus more directly on the question at hand. Imagine we have a single interval for which we want to calculate the entanglement with the rest of the system, and introduce accordingly the -sheeted Riemann surface ( replicas) . In the limit where the interval of length shrinks, we expect the presence of the two sewing points to decompose like an operator product expansion of the form

 I=∑{kj}C{kj}n∏j=1Ψkj(zj) (35)

where we allowed for fields inserted at points , the point on the sheet, and the set denotes a complete set of local fields for one copy of the CFT. Recall that the cut in the Riemann surface corresponds to the insertion of twist fields in the complex plane, so that and (35) should be considered as the OPE of these twist fields. What (35) means more precisely is that, if we have other operators inserted elsewhere, we can expect to have

 Zn(L)Zn1⟨n∏j=1Oj⟩Rn,1=⟨In∏j=1Oj⟩Cn=∑{kj}C{kj}n∏j=1⟨Ψkj(zj)Oj⟩Cj (36)

where designates operators inserted on the sheet, and is the copy of the complex plane. Note indeed that the expectation on the right is taken in a fully factorized theory.

Restricting now to the that make an orthonormal basis (so in particular they are all quasiprimary), and choosing shows that the structure constant will not vanish only if the average of on the Riemann surface does not vanish. It is useful to make things concrete now, so for instance we see that there is no term with a single primary operator on the right hand side of (35) since the corresponding one point function on vanishes. There is, however, at least one term with a single operator, the stress energy tensor, since we know that . Apart from this, the most important terms will be those involving the same primary operator on two different sheets , whose average on will be non zero in general. If the field has conformal weights , we will thus have that

 C∝L−4hnL2×(h+¯h) (37)

where is the conformal weight of the twist field. The crucial point is that involves twice the scaling dimension of primary fields, in contrast with ordinary OPEs where only the scaling dimension would appear.

The discussion carries over to the boundary case. One can, for instance, think of it after unfolding the system so as to keep only chiral fields as in (5). Everything then formally goes through after setting . The question is then, what kind of fields (the chiral part of ) can appear in the OPE of two twist fields. The one copy bulk theory is a compact boson which allows for the fields on the boundary. This means that the radius is , and thus the bulk conformal weights are given by

 Δwk=2π(βk8π−wβ)2,  ¯Δwk=2π(βk8π+wβ)2 (38)

Restricting to scalar operators we get or . For instance, the first values of correspond to fields , or, for the chiral part, .

We now go back to the entropy calculation in the folded, non-chiral theory (6). Upon folding, the chiral vertex operators become , as . Recall also that the non-chiral twist field in the folded version can be thought of as the chiral part of in the unfolded theory. Hence, going through the discussion of short distance expansions we find, for the non chiral twist field

 (39)

where we used the fact CCT () that the two fields in the twist OPEs must belong to different copies. We are only interested in terms whose one point function acquires a non zero value in the presence of the perturbation. This means the first term with cannot contribute, and thus we need , . Taking derivative with respect to gives then the leading term for the entanglement correction, which should go as

 Simp−ln2∝(LTB)2μ∝L2μλ2μ/(1−μ) (40)

For in particular, this can be corrected by a resonance, and it is tempting to speculate then that one has

 Simp−ln2∝LTB[cst+cstln(LTB)] (41)

Finally, we note once again that the RLM or the various (anisotropic) Kondo versions will behave identically, the presence of the operators not modifying in any essential way the OPE argument – but one will have to be careful with the dimensions of the operators involved, and their relationship with . In the end, we find that for the RLM (41) is expected to hold as well.

## Vi Large coupling expansion

While the small coupling expansion is plagued with IR divergences, a large coupling expansion is possible. It is now finite in the IR, and exhibits UV divergences which are easily taken care of using integrability and analyticity. Let us recall how the calculation goes at leading order in the anisotropic Kondo case ImpEnt () (see also e.g. Eriksson ()). The leading IR perturbation is nothing but the stress energy tensor The correction to the Renyi entropy can therefore be expressed as

 −δZn=nπTB∫+∞−∞dτ⟨T(w=iτ)⟩Rn,1=n−n−124πTB∫+∞−∞(2L)2(iτ−L)2(iτ+L)2dτ=112LTB(n−1n). (42)

The first correction to the entanglement entropy thus reads

 Simp=16LTB+… (43)

It is quite remarkable that this result does not depend on the anisotropy parameter (recall that in the XXZ language). It turns out that this IR expansion can be generalized to higher orders FretonIR (). The results for the Kondo case are as follows

 Simp = 16ln(1+1LTB)−1835(πg4)2(2LTB)6(4α4−8α2+9)+O((LTB)−7) (44)

where the coefficient has the following dependence on the dimension of the tunneling operator

 g4=μ12π2(Γ(μ/2(2−μ))Γ(1/(2−μ)))3Γ(3/(2−μ))Γ(3μ/2(2−μ)),  α=(2−μ)√2μ. (45)

Note that in (44), the first term in the right hand side has to be truncated at order 6.

While in principle higher orders in the IR expansion could be determined, the complexity of the calculations increases considerably. Moreover, the convergence properties of this expansion are not clear. Finally, we observe that, in this point of view, the pure BSG case turns out to be quite different, because different operators appear in the IR effective description. The corresponding result has not even been worked out yet.

Making analytical progress therefore requires developing non perturbative approaches. The problems we are interested in are indeed integrable, at least in their boundary versions. While it is natural to expect that this can be used in some way, integrability has been mostly used to calculate local properties such as magnetization, energy or impurity entropies. Von Neumann entanglement is non local, and therefore much harder to obtain in general.

## Vii Form factor approach to the entanglement entropy

We will in what follows restrict to the case where the dimension of the perturbation is : this corresponds to () for the problem of tunneling between XXZ chains, and to – the RLM () – for the tunneling through an impurity. These cases are closely related to the boundary Ising model with a boundary magnetic field (see Appendix A). While the problem of calculating the entanglement non perturbatively remains extremely difficult – entanglement still involving non local observables in the fermionic language – it can be tackled using the idea of form-factors.

It has been known for many years that correlation functions of local observables in massive integrable theories can be calculated using the form-factors approach, where the integrable quasiparticles provide a basis of the Hilbert space, and the form-factors (FF) – that is, the matrix elements of the operators in that basis – can be obtained using an axiomatic approach based on the knowledge of the S matrix and the bootstrap. It is a natural idea to extend this approach to the case of entanglement entropy. Indeed, the Von Neumann entanglement is obtained form the Renyi entropy by taking an derivative at , and the Renyi entropies can be considered formally as correlation functions of twist operators that live in copies of the theory of interest. The integrability of a single theory carries over to integrability of the copies, and a calculation similar to the one of ordinary correlators can be set up, after some additional work to determine the form factors of the twist operators  Doyon (); Doyon1 ().

We are interested here in a variant where the bulk is massless. The form-factors technique in this case is more delicate to use, since particles can have arbitrarily low energies, and the convergence of the approach is not guaranteed. Various regularization tricks have to be used in the calculation of local quantities (eg the charge density for Friedel oscillations) Skorik (); LS (), and we will see below that the situation for the entanglement is not better. Nevertheless, can be calculated for , by using the Ising model formulation, and relying heavily on the work Doyon (); Doyon1 ().

To fix ideas, and explore the feasibility of form-factors calculations in our problem, we first discuss briefly the bulk case and the massless limit. One can find in Doyon (); Doyon1 () the first order contribution to the two point function of the bulk Ising model twist field in the bulk

 ⟨τ(r)~τ(0)⟩=⟨τ⟩2+12n∑i,j=1∫dθ12πdθ22π|Fτ|ij2(θ12,n)|2e−mr(coshθ1+coshθ2)+… (46)

where is the number of copies, is the rapidity of the particle with energy and momentum , and is the two-particle form factor of the twist field

 Fτ|ij2(θ12,n)=⟨0∣∣τ(0)Z†i(θ1)Z†j(θ2)∣∣0⟩. (47)

In this last expression, we have used the notation for the usual Faddeev-Zamolodchikov creation operators (here, the fermions) living in the copy. Since the theory is integrable, the form factors can be computed exactly and are conveniently expressed using the function

 K(θ)=Fτ|112⟨τ⟩=−icosπ2nsinhθ2nnsinhiπ+θ2nsinhiπ−θ2n (48)

which vanishes when . The other form factors can then be obtained from by shifting appropriately by a factor of . Going to variables one can perform one integration, and be left with

 ⟨τ(r)~τ(0)⟩=⟨τ⟩2(1+n4π∫∞−∞dθf(θ,n)K0(2mrcosh(θ/2)))+… (49)

where

 ⟨τ⟩2f(θ)≡|Fτ|112(θ)|2+n−1∑j=1|Fτ|112(θ+2iπj)|2 (50)

Doyon et al. then argue the crucial result that

 ddnnf(θ,n)∣∣∣n=1=π22δ(θ). (51)

Taking the derivative of the two point twist correlation function meanwhile should give, at short distances, the entanglement entropy of the CFT. Since (46) is only a first order approximation where contributions with a larger number of particles have not been included, we get an approximation to the entanglement entropy of a segment of length in the bulk with the rest of the system Doyon1 ()

 SA=…−K0(2mr)8+…≈…+lnr8+… (52)

and thus the expected factor is approximated by at this order.

Since in this paper we are interested in bulk CFTs, we need to take an limit. This corresponds formally to describing the CFT using massless particles and massless scattering. We thus set and send . Only two types of excitations remain at finite energies: those for which with finite. In the first case, one obtains right moving particles with and in the second case left moving particles with . Conformal fields factorizing into left and right components are not expected to mix the L and R sectors. Indeed,

 limθ→∞K(θ)=0, (53)

so only the LL and RR sectors will contribute in the massless limit of (46). Therefore, setting (say for the R sector)

 θ1,2=θ0+β1,2, (54)

and introducing we obtain

 ⟨τ(r)~τ(0)⟩=⟨τ⟩2+12n∑i,j=1∫dβ+2πdβ−2π|Fτ|ij2(β−,n)|2e−2Mreβ+/2cosh(β−/2)+… (55)

where the coming from the Jacobian was canceled by the fact that there are two integrals, the L and the R one. Using (51) we get the correction to the entanglement entropy as

 SA=…−116∫∞−∞dβ+e−2Mreβ+/2=…−18∫∞0dxxe−2Mrx (56)

This integral is divergent at small energy, a feature which is quite general in the use of massless form-factors. We regularize by considering the integral

 ∫∞0dxxα−1e−2Mrx=1(2Mr)αΓ(α)=(1α+…)(1−αln(2Mr)+…) (57)

so the finite part of the integral is and thus we recover

 SA=…+18lnr+… (58)

Let us now consider the Ising model with a boundary magnetic field as in (85), and to start assume that the bulk is massive. The form factors approach can be applied to this case as well. The first non trivial contribution reads then Doyon ()

 ⟨0|τ(r)|B⟩=⟨τ⟩+12n∑i=1∫dθ2πR(iπ2−θ)e−2mrcoshθF