Entanglement Entropy from Corner Transfer Matrix in Forrester Baxter non-unitary RSOS models
EEntanglement Entropy from Corner Transfer Matrix inForrester Baxter non-unitary RSOS models
Davide Bianchini a and Francesco Ravanini b,c September 2015 a Dept. of Mathematics, City University London, Northampton Square, EC1V 0HB,London, UK b Dept. of Physics and Astronomy, University of Bologna, Via Irnerio 46, 40126Bologna, Italy c I.N.F.N. - Sezione di Bologna, Via Irnerio 46, 40126 Bologna, Italy
Abstract
Using a Corner Transfer Matrix approach, we compute the bipartite entan-glement R´enyi entropy in the off-critical perturbations of non-unitary conformalminimal models realised by lattice spin chains Hamiltonians related to the ForresterBaxter RSOS models [30] in regime III. This allows to show on a set of explicitexamples that the R´enyi entropies for non-unitary theories rescale near criticalityas the logarithm of the correlation length with a coefficient proportional to theeffective central charge. This complements a similar result, recently establishedfor the size rescaling at the critical point [23], showing the expected agreementof the two behaviours. We also compute the first subleading unusual correction to the scaling behaviour, showing that it is expressible in terms of expansionsof various fractional powers of the correlation length, related to the differences∆ − ∆ min between the conformal dimensions of fields in the theory and theminimal conformal dimension. Finally, a few observations on the limit leading tothe off-critical logarithmic minimal models of Pearce and Seaton [49] are put forward.PACS numbers: 03.65.Ud, 05.70.Jk, 05.50.+q, 02.30.Ik Email: [email protected], [email protected] a r X i v : . [ h e p - t h ] S e p ntroduction Entanglement is a very specific and intriguing feature of quantum systems [1]. It de-scribes truly quantum correlations between parts of a system. Besides the questions ofprincipal nature that it arises in the interpretation and behaviour of quantum mechanics[2, 3], it finds very interesting and promising applications in quantum information theoryand quantum computing [4], in condensed matter physics [5, 6], as well as in the physicsof black holes [7] and has even risen some interest in biological systems [8].Many proposals have been formulated to quantify it (see for example [9] and refer-ences therein). In this paper we focus on bipartite systems, composed of a subsystem A and its complement ¯ A . In such case, the most convenient measure of entanglement isthe Von Neumann Entropy restricted to A , the so-called Entanglement Entropy [4]. Itis used as a measure of entanglement for pure quantum states. The behaviour of thisquantity has been deeply studied in a wide range of systems from analytical, numeri-cal and experimental points of view. For example, interesting experimental protocolsin many body systems have been recently proposed [10, 11]. It has been evaluated indifferent dimensions and in different regimes, but it is in 1 + 1 dimensions that it showsits most amazing mathematical properties. In particular, in the case of Integrable Mod-els, it reflects many mathematical features that otherwise would be difficult to probe.For example, the study of Entanglement Entropy in critical one-dimensional quantumsystems is one of the most powerful techniques for the identification of the central charge c of the conformal field theory describing the low energy excitations of the model. Thisevaluation can be performed both analytically and numerically (e.g. DMRG [12]) andusually requires less information than the study of the scaling of the ground state energyfor the identification of the universality class.In the last decade there has been a particular focus on the evaluation of EntanglementEntropy in Integrable models, in particular in Conformal Field Theory (CFT) [13, 14],in their massive perturbations [15, 16] and in lattice spin chains [17, 18, 19, 20, 21, 22].Recently also non unitary models have been taken into account, both in the critical [23]and in the massive [24] regimes.At first sight, non unitary theories might be considered just as non physical math-ematical curiosities, but there are many very interesting examples where they actuallydo play a physical role. For example, it is known that a strongly interacting 2D electrongas in a magnetic field produces edge modes described by CFT minimal models, in whatis known as Fractional Quantum Hall Effect (FQHE). In some cases it has been shownthat these edge modes are described by non unitary CFT [25]. In [26], for example, theminimal model M , has been considered. The non-unitarity arises from the fact thatin this particular case the bulk is gapless (while in the ordinary case it is gapped) andthen the edge can dissipate into the bulk.These excitations can be represented using generalised spin chains: the critical Fi-bonacci Chain (the “golden chain” [27], minimal model M , ), which represents theFQHE with filling ν = and its non-unitary generalisation to the Lee-Yang Model M , [28] are among the most known chain representations. These critical chains all2elong to the wide class of U q ( sl (2)) invariant spin chains [29] in the universality class ofconformal minimal models M m,m (cid:48) , unitary or not. They can be further generalised inorder to include their massive Φ perturbations. The off-critical Hamiltonians of suchchains have been recently obtained [30] and they are related to the RSOS models [31, 32],through the usual connection between the time evolution operators of 1D quantum spinchains and the row-to-row transfer matrix of 2D classical lattice models.The Hamiltonians of these chains are generically of the pseudo-hermitian type, i.e.they are not hermitian but have real eignevalues. In recent years they attracted a lot ofinterest in the description of many physical phenomena, ranging from optical effects tonon-equilibrium systems, etc... Similarly, in Quantum Field Theory it has been widelydiscussed [33] how the scattering matrices of apparently non-hermitian theories can bephysically well defined. In particular, using the definitions of [33], non-unitary theoriescan be classified as real or non real ; in both cases the inner product is indefinite, but inreal theories eigenenergies are real and the eigenvalues of the S matrix are pure phases,while in non real models the energies are not real and the S matrix has eigenvalues whichare not pure phases. Notice that if the S matrix is a pure phase, there is a conserva-tion of probability through the scattering process; this conservation is a fundamentalrequirement for having a real, well-defined quantum physical theory.Our goal in this paper is to compute the bipartite Entanglement Entropy of theground state for spin chain hamiltonians of the type introduced in [30] on an infiniteone dimesional lattice, taking as A the negative semi-axis and as ¯ A the positive one.For non-Hermitian hamiltonians this may sound ambiguous, as non Hermitian operatorsshare the same right and left eigenvalues but have different right and left eigenvectors.However, one can argue [23] using PT symmetry and chiral factorization of CFT that, atleast at criticality, the left and right ground states do coincide, thus leading to a correctpositive definition of the Entanglement Entropy for this state.This bipartite Entanglement Entropy calculation can be achieved by consideringthe Corner Transfer Matrix (CTM) introduced by Rodney Baxter [34] in integrablestatistical systems on a 2D square lattice. This tool, originally used for the evaluationof partition functions and one point functions of the 2D models, has been extended tothe evaluation of the Entanglement Entropy of the corresponding 1D chains, followingan approach developed in [17].It is known that these chains, as well as their classical 2D counterparts, the class ofRSOS r,s models, show different regimes with different physical behaviour. In the unitarycase s = 1, Andrews, Baxter and Forrester (ABF) [31] classified four regimes. There aretwo second order phase transitions: one between regimes I and II and the other betweenregimes III and IV. The identification of universality classes at the transition pointshas been studied by Huse [35]. Approaching the III-IV transition the RSOS r, models,for different r , are described by the universality class of unitary CFT minimal models M r − ,r . In the I-II phase transition, instead, the approach to criticality is governed bythe universality class of Z L parafermionic CFTs [36] with L = r − s (cid:54) = 1 explored in [32], the classification of regimes is morecomplicated. However, still there are regimes III and IV (or X if r odd and s = r − )3ith a phase transition between them identified [37, 38] with the universality class ofnon-unitary minimal models M r − s,r . The identification of the universality class of theregime I-II phase transition is still an open problem in the s (cid:54) = 1 case. We shall notaddress this problem here and in the following we restrict our calculations to regime III.In the case of unitary ABF RSOS models, Franchini and De Luca [39] have recentlyused the CTM technique to compute the Entanglement Entropy, checking that in thephase transition III-IV it correctly gives the results expected in the conformal unitaryminimal models M r − ,r while in the regime I-II phase transition, it gives those expectedfor the Z r − parafermionic CFTs.In this paper we extend this Entanglement Entropy calculation to the regime III of s (cid:54) = 1 RSOS models. The main goal of this investigation is to test on a concrete exampleof non unitary integrable theories where the calculations can be carried out exactly,the conjecture that the Entanglement Entropy scales logarithmically in the correlationlength in a manner that parallels the scaling on a finite interval at criticality (for detailsof what we mean here see subsect. 2).The paper is organised as follows: • In section 1 we recall a few notions on minimal CFT models (unitary or not) andtheir characters, emphasising their relation with modular forms. • Then in section 2 we introduce the basic concept and formulae of R´enyi and VonNeumann Entanglement Entropies. We discuss in particular their behaviour forfinite size of the subsystem at criticality and for infinte size but off-criticality. Weconsider in both cases the dominating logarithmic term, but also the form of theexpected corrections. • In section 3 we describe the Forrester-Baxter (FB) RSOS models whose continuumscaling limit is described by the perturbed minimal model M m,m (cid:48) + t Φ , (where t > , the least relevant operator). • We briefly summarise in section 4 the Corner Transfer Matrix construction and re-view its use as a tool for the evaluation of partition functions and R´enyi Entropy. Inparticular, specialising to the RSOS models, we emphasise the connection betweenthe blocks in the partition functions on multi-sheeted surfaces with the conformalcharacters of the corresponding CFT at criticality. • Applying the CTM tool to the FB RSOS model, in section 5 we evaluate theR´enyi Entropy for these theories. Our main results are the confirmation of thepresence of the effective central charge in the leading scaling of the Entropy [23].The difference from the unitary case where the usual central charge appears isdue to the fact that the physical ground state and the conformal vacuum do notcoincide in non unitary models. • In section 6 we evaluate explicitly the most relevant corrections – nowadays tradi-tionally called unusual corrections following [40] – to the dominating logarithmic4ehaviour. They exhibit power law decays with exponents given by a sort of effec-tive dimensions , i.e. the difference between the conformal dimensions of some ofthe relevant fields of the CFT universality class and the lowest (negative) conformaldimension that always appears in non unitary CFTs. • In section 7 we briefly comment on the evaluation of R´enyi Entropy in off-criticalLogarithmic Minimal Models, which can be seen as particular limit realisations ofthe FB RSOS model. The observed absence of the double log behaviour is relatedto the destruction of the Jordan block structures responsible of the logarithmicbehaviour as soon as the log-CFT are perturbed off-criticality. • Finally, in section 8 we trace our conclusions and perspectives for future work.
We recall here some basic facts about conformal minimal models that we need in thefollowing. The minimal models M m,m (cid:48) are conformal theories whose Hilbert space isbuilt up of two chiral parts each one composed of a finite number of irreducible highest-weight representations (HWR) of Virasoro algebra at a given value of its central charge c . They are labelled by two coprime integers m, m (cid:48) such that 2 ≤ m < m (cid:48) and havecentral charge c = 1 − m − m (cid:48) ) mm (cid:48) (1.1)The (left) conformal families [Φ a,a (cid:48) ] are labelled by two integers a, a (cid:48) running on thedomain J = { ( a, a (cid:48) ) : 1 ≤ a ≤ m − , ≤ a (cid:48) ≤ m (cid:48) − } . The conformal dimensions ofthe primary states | a, a (cid:48) (cid:105) of such families are given by∆ a,a (cid:48) = ( am (cid:48) − a (cid:48) m ) − ( m − m (cid:48) ) mm (cid:48) (1.2)The Z symmetry ∆ a,a (cid:48) = ∆ m − a,m (cid:48) − a (cid:48) is present in the whole series of models.The models are unitary for m (cid:48) = m + 1, non unitary otherwise. In the non unitarycase the state of lowest conformal dimension is not the conformal vacuum | (cid:105) ≡ | Φ , (cid:105) but a different state | min (cid:105) with negative conformal dimension (that can be proven alwaysto exist and be unique) ∆ min = 1 − ( m − m (cid:48) ) mm (cid:48) (1.3)Correspondingly, an effective central charge can be defined c eff = c − min = 1 − mm (cid:48) (1.4)Notice that while the central charge c can be negative in non unitary models, the effectiveone c eff is always positive. 5he characters of the Virasoro HWRs can be written as χ a,a (cid:48) ( q ) = q ∆ a,a (cid:48) − c ( q ) ∞ ∞ (cid:88) k = −∞ (cid:104) q k ( kmm (cid:48) + am (cid:48) − a (cid:48) m ) − q ( km + a )( km (cid:48) + a (cid:48) ) (cid:105) (1.5)where q = e πiτ and ( q ) ∞ = (cid:81) ∞ j =1 (1 − q j ). They form a unitary representation of themodular group [41] P SL (2 , Z ) whose generators areˆ T : τ → τ + 1ˆ S : τ → − τ ˆ T χ a,a (cid:48) ( q ) = χ a,a (cid:48) ( qe πi ) = e πi ( ∆ a,a (cid:48) − c ) χ a,a (cid:48) ( q )ˆ S χ a,a (cid:48) ( q ) = χ a,a (cid:48) (˜ q ) = (cid:88) ( b,b (cid:48) ) ∈J S b,b (cid:48) a,a (cid:48) χ b,b (cid:48) ( q ) (1.6)where ˜ q = e − πiτ . S is the modular matrix S b,b (cid:48) a,a (cid:48) = 2 (cid:114) mm (cid:48) ( − ab (cid:48) + a (cid:48) b sin (cid:16) π mm (cid:48) ab (cid:17) sin (cid:18) π m (cid:48) m a (cid:48) b (cid:48) (cid:19) (1.7)Introducing the elliptic function E ( z, q ) = (cid:88) k ∈ Z ( − k q k ( k − z k = ∞ (cid:89) n =1 (1 − q n − z )(1 − q n z − )(1 − q n ) (1.8)they can equivalently be written as χ a,a (cid:48) ( q ) = q ∆ a,a (cid:48) − c ( q ) ∞ (cid:110) E (cid:16) − q m (cid:48) a − a (cid:48) m + mm (cid:48) , q mm (cid:48) (cid:17) − q aa (cid:48) E (cid:16) − q m (cid:48) a + a (cid:48) m + mm (cid:48) , q mm (cid:48) (cid:17)(cid:111) (1.9)The function E ( z, q ) is related to the first Jacobi theta function ϑ ( u ; p ) = (cid:88) n ∈ Z ( − n − p ( n + ) e (2 n +1) iu = 2 p sin u ∞ (cid:89) n =1 (1 − p n cos 2 u + p n )(1 − p n )(1.10)by the relation E (cid:0) e iw , q (cid:1) = iq − e iw ϑ ( w ; q ) (1.11)The first Jacobi theta function, under a modular ˆ S tranformation, behaves as ϑ ( z, e − iπτ ) = − i √ iτ e iτz π ϑ ( τ z ; e iπτ ) (1.12)These formulae will be useful later in sect. 5 for the computation of the R´enyi Entropy.The minimal models can be perturbed off-criticality by picking up combinationsof their relevant fields, resulting in super-renormalizable theories. In particular, theperturbation by the least relevant field Φ , preserves the integrability of the model off-criticality. It is this integrable perturbation that represents the scaling limit of the RSOSmodels in the off-critical vicinity of the phase transition between regimes III and IV.6 R´enyi Entanglement Entropies
Consider a bipartite quantum system whose Hilbert space is H = H A ⊗ H ¯ A . If thesystem is in a pure state | Ψ (cid:105) (i.e. one with density matrix ρ = | Ψ (cid:105)(cid:104) Ψ | ), the entanglementbetween the two parts A and its complement ¯ A can be described by the family of R´enyiEntanglement Entropies S n ( n ∈ R > ) S ( n ) A = 11 − n log tr A ρ nA (2.1)where ρ A = tr ¯ A ρ is the reduced density matrix of the subsystem A . The Von NeumannEntanglement Entropy S = lim n → S ( n ) A = − tr A ( ρ A log ρ A ) is a special case of the R´enyiEntropies.In [13, 14] it has been shown that the R´enyi Entropy for a bipartite one-dimensionalquantum system scales as S ( n ) A ∼ c n + 1 n log l(cid:15) (2.2)where l is the size of the subsystem A , and (cid:15) is some ultraviolet cut-off. This scaling holdswhen the size of the whole system is much larger than the size l of subsystem A and whenthe correlation length ξ is much larger than l . This condition on the correlation length isequivalent to assume that the system is approaching criticality with a universality classidentified by a CFT with central charge c . In [40] the subleading contributions to thisbehaviour were estimated and look like S ( n ) A ∼ c n + 1 n log l(cid:15) + a ( n ) + b ( n ) l − x/n + b (cid:48) ( n ) l − x + ... (2.3)The constants a ( n ) , b ( n ) and b (cid:48) ( n ) are non-universal, but the exponent x is, and hasto be identified with the scale dimension x = ∆ + ¯∆ of some operator Φ of the CFTuniversality class having the lowest conformal dimension among those concurring to thecorrections. Higher order corrections are expected and they are due to other primaryfields and to descendents. For n > l − x term is subleading with respect to the l − x/n one. Viceversa for n <
1. Taking into account both terms is necessary to ensurea smooth limit for n → ξ is finite and the size l → ∞ we have[42] a similar expansion S ( n ) A ∼ c n + 1 n log ξ + A ( n ) + B ( n ) ξ − h/n + B (cid:48) ( n ) ξ − h + ... (2.4)which represents the entanglement between two infinite parts of an infinite system. Inthis case the size l of the subsystem A has been replaced by the correlation length. Thisis due to the fact that the leading term of the Entropy is given by the smallest physicallength which plays a role in the system. The factor of 2 difference in the leading termsand in the corrections is simply explained by the fact that the number of boundarypoints dividing A from ¯ A is 2 in the first case and 1 in the second. The non-universal7oefficients of the corrections A ( n ) , B ( n ) and B (cid:48) ( n ) in general can be different from the a ( n ) , b ( n ) , b (cid:48) ( n ) of eq.(2.3). Also, the universal exponent h can differ from the x above[21, 39].Recently [23] this evaluation has been extended in order to take into account nonunitary systems. The Hamiltonian operator ˆ H describing a CFT system of Hilbert space H put on a cylinder of circumference R is given byˆ H = 2 πR (cid:16) L + ¯ L − c (cid:17) (2.5)where L n , ¯ L n are Virasoro algebra generators and c is their central charge. In unitaryCFT the conformal vacuum | (cid:105) (defined as the unique state such that L n | (cid:105) = 0, ∀ n ≥−
1) is the physical ground state, i.e. ∀| ψ (cid:105) ∈ H : (cid:104) | ˆ H | (cid:105) ≤ (cid:104) ψ | ˆ H | ψ (cid:105) ). Thus the groundstate energy will scale as E = − πc R [43, 44].In non unitary CFT instead, there is at least one primary state | min (cid:105) with negativeconformal weight ∆ min ( (cid:104) min | L | min (cid:105) = ∆ min < | (cid:105) has the lowest possible energy on the cylinder and is thetrue ground state of the theory. For this reason the ground state energy scales as E min = − πc eff R , with the so called effective central charge c eff = c − min ≥ c [45]. The difference between the conformal vacuum and thephysical ground state plays an important role in the definition and calculation of theEntanglement Entropy. For a detailed discussion, see [23]. As a result, in non-unitaryCFT the R´enyi Entropy scales as S ( n ) A ∼ c eff n + 1 n log l(cid:15) (2.6)Although this result seems natural, its proof in a generic non unitary CFT is far fromtrivial and can be acheived by a careful analysis of how to correctly define twist operatorsin such case through an orbifold approach [23].In the case of infinite subsystem, one is tempted to replace l(cid:15) by ξ , the correlationlength, as in the unitary cases. In this paper our aim is to verify that this conjecture iscorrect in a concrete lattice realization of an off-critical non-unitary model. We use theRSOS non-unitary models of Forrester Baxter in regime III as lattice regularizations ofthe perturbed CFT M m,m (cid:48) + t Φ , (where t > S ( n ) A ∼ c eff n + 1 n log ξ + A ( n ) + B ( n ) ξ − hn + B (cid:48) ( n ) ξ − h (2.7)calculating it from the exact expression for the R´enyi Entanglement Entropy obtainableby a Corner Transfer Matrix approach combined with the evaluation of the correlationlength ξ on the lattice. We expect that the exponent h is affected by the non unitarityin a way similar to the change from c to c eff in the leading logarithmic term.8 jkl Figure 1: Square tile
The Restricted-Solid-On-Solid RSOS r,s model [31, 32] is defined on a square lattice, foreach pair of coprime positive integers r, s such that r > ≤ s ≤ r −
1. On each site i ∈ Z there is a variable, called local height , (cid:96) i that takes values (cid:96) i = 1 , , . . . , r −
1. Localheights of two neighbouring sites i and j are restricted to differ by one | (cid:96) i − (cid:96) j | = 1. Localheights can be thought as encoded on a A r − Dynkin diagram. Each node representsa possible value that (cid:96) i can take and it is linked to the possible values at neighbouringsites. This allows to generalise the model, following Pasquier [46], to other simply laced A, D, E
Dynkin diagrams. For the scope of the present paper, however, we focus on the A case only. The RSOS models belong to the wide class of 2D classical lattice modelof IRF (Interaction Round a Face) type. In IRF models, the interaction is on nearestneighbour, so that one can define local Boltzmann weights that depend on the four sites i, j, k, l around a tile (see fig. 1) W (cid:18) (cid:96) l (cid:96) k (cid:96) i (cid:96) j (cid:19) = e − βε lkij (3.1)where β is the inverse temperature, ε lkij is the energy of the configuration of the fourvertices.For the RSOS models, the Boltzmann weights have been calculated from Yang-Baxter equation, that must be satisfied by any integrable lattice model, in [31, 32].They depend on a spectral parameter u , measuring the anisotropy of the lattice, on the crossing parameter λ = sr π (3.2)ruling their behaviour when the lattice is rotated by π W (cid:18) (cid:96) l (cid:96) k (cid:96) i (cid:96) j (cid:12)(cid:12)(cid:12)(cid:12) u (cid:19) = (cid:115) s ( (cid:96) i λ ) s ( (cid:96) k λ ) s ( (cid:96) j λ ) s ( (cid:96) l λ ) W (cid:18) (cid:96) k (cid:96) j (cid:96) l (cid:96) i (cid:12)(cid:12)(cid:12)(cid:12) λ − u (cid:19) (3.3)and on a temperature-like parameter p , ( − < p < t = p is measuring thedeparture from criticality. The system is critical for p = 0. The Boltzmann weights turnout to be expressible in terms of elliptic theta functions for which the parameter p playsthe rˆole of the nome. Here and below s ( u ) = ϑ ( u ; p ), the first Jacobi theta function91.10). The non-zero Boltzmann weights can be put in the following form (in the socalled symmetric gauge) W (cid:18) (cid:96) ± (cid:96)(cid:96) (cid:96) ∓ (cid:12)(cid:12)(cid:12)(cid:12) u (cid:19) = s ( λ − u ) s ( λ ) W (cid:18) (cid:96) (cid:96) ± (cid:96) ∓ (cid:96) (cid:12)(cid:12)(cid:12)(cid:12) u (cid:19) = (cid:112) s (( (cid:96) ∓ λ ) s (( (cid:96) ± λ ) s ( (cid:96)λ ) s ( u ) s ( λ ) (3.4) W (cid:18) (cid:96) (cid:96) ± (cid:96) ± (cid:96) (cid:12)(cid:12)(cid:12)(cid:12) u (cid:19) = s ( (cid:96)λ ± u ) s ( (cid:96)λ )The periodicity and modular properties of the Boltzmann weights determine the funda-mental ranges of the parameters. The classification of possible regimes of the models isquite complicated [32] but in this paper we concentrate on regime III (0 < p < < u < λ ) only, leaving the others for future investigation.For s = 1 the model, often denoted simply RSOS r , has been investigated by Andrews,Baxter and Forrester in [31]. The R´enyi Entanglement Entropy for the coorespondingspin chains has been computed in [39], where it is shown that in the regime III approach-ing the critical point p = 0 it scales like (2.4) with the central charge of unitary minimalmodels M r − ,r . In the regime II transition, instead, it scales as the parafermions Z r − .Here we are interested in the generalizations of these results for other values of s .The spin chains Hamiltonians corresponding to the RSOS r,s models have been recentlywritten in [30]. As the notation for them is quite complicated and not relevant for thefollowing, we do not write them here and invite the interested reader to refer to thepaper [30] for an exhaustive presentation. For spin chains on an infinite 1D lattice, we consider the Renyi Entanglement Entropybetween two semi-infinite halves, the negative one conventionally called A while thepositive is ¯ A . For chains related to a 2D classical lattice model of IRF (InteractionRound a Face) type, an efficient method to compute the reduced density matrix ρ A , asproposed in [17], is to make use of the Corner Transfer Matrix approach [34]. The FBRSOS models are of this kind, therefore we adopt this approach here.Consider the 2D “diamond shaped” lattice of fig.2 and divide it in four quadrants.The central site is denoted by 0. In the lower-left quadrant A introduce an operator (seefig. 2) A ( N ) (cid:96) , (cid:96) (cid:48) ( u ) = δ (cid:96) ,(cid:96) (cid:48) (cid:88) (cid:96) i | i = •∈ A (cid:89) (cid:3) ∈ A W (cid:18) (cid:96) l (cid:96) k (cid:96) i (cid:96) j (cid:12)(cid:12)(cid:12)(cid:12) u (cid:19) (4.1)where (cid:96) = ( (cid:96) , (cid:96) , (cid:96) , ..., (cid:96) N ) and (cid:96) (cid:48) = ( (cid:96) (cid:48) , (cid:96) (cid:48) , (cid:96) (cid:48) , ..., (cid:96) (cid:48) N ) are vectors collecting all thevariables along the two inner boundaries, i.e. on the sites denoted by ◦ in fig. 2. Thesum is performed over all possible values of (cid:96) i on internal sites (signed by a black dot • infig. 2). The variables at the outer boundary sites are assigned fixed values determining10 Figure 2: The 4 corner transfer matrix operatorsa boundary condition uniquely. The product is over all tiles (cid:3) of the quadrant. Noticethe (cid:96) = (cid:96) (cid:48) obvious constraint on the central site 0.Analogously, define in the other quadrants the operators B ( N ) (cid:96) (cid:48) , (cid:96) (cid:48)(cid:48) , C ( N ) (cid:96) (cid:48)(cid:48) , (cid:96) (cid:48)(cid:48)(cid:48) and D ( N ) (cid:96) (cid:48)(cid:48)(cid:48) , (cid:96) .Organizing the product of W ’s to be performed diagonally (thus the diamond shape ofthe lattice), a corner transfer matrix like A ( N ) (cid:96) , (cid:96) (cid:48) may be expressed in terms of smallercorner transfer matrices A ( N − (cid:96) , (cid:96) (cid:48) and A ( N − (cid:96) , (cid:96) (cid:48) (defined for reduced quadrants). Thisrecursion relation allows, in principle, the iterative calculation of the corner transfermatrix for any lattice quadrant of finite size.The partition function can be expressed in terms of the four CTM operators as Z ( N ) = tr( A ( N ) B ( N ) C ( N ) D ( N ) ) (4.2)There are other interesting quantities that can be computed form the CTM method, likee.g. the height probabilities. Here we are interested in the calculation of the reduceddensity matrix ρ A relative to a dominion A coinciding with the negative axis of thehoriziontal direction of the infinite lattice that results taking the thermodynamic limit N → ∞ of the present construction. Following [17], the unnormalised reduced densitymatrix on the subsystem A can be written as (cid:37) A, (cid:96)(cid:96) (cid:48) = ( ABCD ) (cid:96)(cid:96) (cid:48) (4.3) Please notice the difference of the symbols (cid:37) (unnormalised density matrix) and ρ (normalised densitymatrix) A = lim N →∞ A ( N ) and analogously for B , C , D . Obviously tr A (cid:37) A = Z is thepartition function in the thermodynamic limit Z = lim N →∞ Z ( N ) (4.4)To normalise, one has to divide by the partition function ρ A, (cid:96)(cid:96) (cid:48) = ( ABCD ) (cid:96)(cid:96) (cid:48) Z (4.5)For the R´enyi Entanglement Entropy we need to compute tr A ρ nA . Defining the highergenus partition functions tr A (cid:37) nA = Z n one sees thattr A ρ nA = Z n Z n (4.6)and the expression for the R´enyi Entropy is S ( n ) A = 11 − n log Z n Z n (4.7)As shown by Baxter [34], the Yang-Baxter equation that has to be satisfied bythe Boltzamnn weights of any integrable model, implies that the four CTM operators,for N → ∞ , commute each other for different values of their parameters and theireigenvalues, up to a common factor, accommodate into a diagonal matrix of the generalform (cid:37) diag A, (cid:96)(cid:96) (cid:48) = R ( (cid:96) ) T ( (cid:96) ) δ (cid:96) , (cid:96) (cid:48) (4.8)The diagonalisation can be performed along the lines illustrated in [34]. In particular,for FB RSOS models it has been performed in the original paper [32]. It is simplified ifone introduces new variables x , y that in regime III are defined as y = e π p , x = e π p sr = y sr (4.9)Define Φ( (cid:96) ) = N (cid:88) k =1 kγ ( (cid:96) k , (cid:96) k +1 , (cid:96) k +2 ) (4.10)where γ ( (cid:96), (cid:96) ± , (cid:96) ) = ∓ (cid:22) (cid:96)sr (cid:23) , γ ( (cid:96) ± , (cid:96), (cid:96) ∓
1) = 12 (4.11)In regime III we haveΦ( (cid:96) ) = N (cid:88) k =1 (cid:26) k | (cid:96) k − (cid:96) k +2 | (cid:96) k − (cid:96) k +1 ) (cid:22) (cid:96) k sr (cid:23)(cid:27) (4.12) (cid:98) x (cid:99) denotes the integer part of x . s = 1 case simply because (cid:98) (cid:96) k /r (cid:99) isalways 0. Consider now a system with size 2 N + 3. The unormalised reduced densitymatrix in regime III, as computed in [32], is given by (cid:37) ( N ) A, (cid:96)(cid:96) (cid:48) = ( ABCD ) ( N ) (cid:96)(cid:96) (cid:48) = E (cid:16) x (cid:96) , y (cid:17) x (cid:96) ) δ (cid:96)(cid:96) (cid:48) (4.13)where (cid:96) is the central height and (cid:96) = ( (cid:96) , . . . , (cid:96) N +1 ) is a vector of all local heights fromthe central to the boundary (cid:96) N +1 . Here E ( · , · ) is the modular form defined in (1.8). As disccussed in section 2, to compute the R´enyi Entropy we need to consider not only (cid:37) A but also its n -th powers (cid:37) nA .Let us set the boundary condition to (cid:96) N = b and (cid:96) N +1 = c = b ±
1. The n -th powerof the reduced density matrix is given by( (cid:37) ( N ) A ) n (cid:96)(cid:96) (cid:48) = E (cid:16) x (cid:96) , y (cid:17) n x n Φ( (cid:96) ) δ (cid:96)(cid:96) (cid:48) (5.1)The trace can be performed summing over all allowed configurations (cid:96) Z ( N ) n = (cid:88) (cid:96) E (cid:16) x (cid:96) , y (cid:17) n x n Φ( (cid:96) ) ≡ r − (cid:88) ∗ a =1 E ( x a , y ) n D N (cid:0) a, b, c ; x n (cid:1) (5.2)where D N ( a, b, c ; q ) = (cid:80) (cid:96) ··· (cid:96) N − q Φ( (cid:96) ) with (cid:96) = a , (cid:96) N = b and (cid:96) N +1 = c . The (cid:80) ∗ symbolmeans that the sum is restricted to even or odd only values of a accordingly to theparity of the boundary conditions. In particular notice the the central height a and theboundary height b have to be compatible with the requirement | (cid:96) i − (cid:96) i +1 | = 1.The limit N → ∞ can be performed, keeping track of the boundary condition (cid:96) N ≶ (cid:96) N +1 : lim N →∞ q − kN D N ( a, b, b + 1; q ) = 1( q ) ∞ q b ( b − − ( k − b F ( a, b − k ; q ) (5.3)with k = (cid:98) s ( b +1) r (cid:99) = (cid:98) s(cid:96) N +1 r (cid:99) if (cid:96) N < (cid:96) N +1 orlim N →∞ q kN D N ( a, b + 1 , b ; q ) = 1( q ) ∞ q b ( b +1)4 − k ( b +1)2 F ( a, b − k ; q ) (5.4)with k = (cid:98) sbr (cid:99) = (cid:98) s(cid:96) N +1 r (cid:99) if (cid:96) N > (cid:96) N +1 .The function F is defined as F ( a, d ; q ) = q a ( a − − ad (cid:104) E (cid:16) − q rd +( r − a )( r − s ) , q r ( r − s ) (cid:17) − q ad E (cid:16) − q rd +( r − a )( r + s ) , q r ( r − s ) (cid:17)(cid:105) = q a ( a − − ad ( q ) ∞ q − c +∆ da χ da ( q ) (5.5)13here the Virasoro characters χ da ( q ) (see eq.(1.9)) are taken here for m = r − s and m (cid:48) = r . Thus lim N →∞ q ∓ kN D N ( a ; q ) b,d = b − k = q f ∓ ( b,d ) q a ( a − − ad + c − ∆ da χ da ( q ) (5.6)where f + ( b, d ) = b ( b − − b ( k − f − ( b, d ) = b ( b + 1)4 − k ( b + 1)2 (5.7)do not depend on a .Since the physical (normalised) density operator is given by ρ A ≡ (cid:37) A Tr A (cid:37) A = ABCD
Tr(
ABCD ) (5.8)the quantity in which we are interested istr A ρ An = Z n Z n = ˆ Z n ˆ Z n (5.9)where ˆ Z n = r − (cid:88) ∗ a =1 E ( x a , y ) n x n (cid:16) a ( a − − ad − ∆ da (cid:17) χ da (cid:0) x n (cid:1) (5.10)since the factor (cid:0) f ∓ + c (cid:1) n appears in both the numerator and the denominator andcancels out.Notice that the critical point is reached for p →
0, or x →
1. In order to catch thecritical behaviour we transform (5.10) into a new expression which is more suitable forexpansion near p = 0. First of all we can use the relations (1.11) and (1.12) for thefunction E W n = r − (cid:88) ∗ a =1 ϑ (cid:16) πasr , √ p (cid:17) n χ da (cid:0) x n (cid:1) (5.11)where, again, W n is proportional to ˆ Z n and a common factor with ˆ Z n has been droppedso that tr A ρ An = Z n Z n = ˆ Z n ˆ Z n = W n W n (5.12)Furthermore we can perform a modular ˆ S transformation (1.6) on the conformalcharacter χ a,a (cid:48) (˜ q ) = (cid:88) ( b,b (cid:48) ) ∈J S b,b (cid:48) a,a (cid:48) χ b,b (cid:48) ( q ) (5.13)14ince x = e π p sr = e πi (cid:16) − i π log p sr (cid:17) , its modular ˆ S -transform is ω ≡ (cid:103) ( x ) = e log p rs = p r s (recall that, in regime III, p > (cid:93) ( x n ) = ( p r s ) n = ω /n . We have W n = r − (cid:88) ∗ a =1 (cid:88) d (cid:48) a (cid:48) ϑ (cid:16) πasr , √ p (cid:17) n S d (cid:48) a (cid:48) da χ d (cid:48) a (cid:48) (cid:16) ω n (cid:17) (5.14)which is suitable for a p → h = ( d (cid:48) , a (cid:48) ) ∈ J as an index which spans the Kac table we have W n = (cid:88) h ∈J χ h (cid:16) ω n (cid:17) f h ( n, p )= χ min (cid:16) ω n (cid:17) f min ( n, p ) (cid:88) h (cid:54) =min χ h (cid:16) ω n (cid:17) χ min (cid:16) ω n (cid:17) f h ( n, p ) f min ( n, p ) (5.15)where f h ( n, p ) = r − (cid:88) ∗ a =1 ϑ (cid:16) πasr , √ p (cid:17) n S d (cid:48) a (cid:48) da (5.16)and h = min refers to the primary field with the lowest conformal dimension ∆ min .Taking the logarithm, we havelog W n W n = log χ min (cid:16) ω n (cid:17) χ min ( ω ) n + log f min ( n, p ) f min (1 , p ) n + log (cid:88) h (cid:54) =min χ h (cid:16) ω n (cid:17) χ min (cid:16) ω n (cid:17) f h ( n, p ) f min ( n, p ) − n log (cid:88) h (cid:54) =min χ h ( ω ) χ min ( ω ) f h (1 , p ) f min (1 , p ) (5.17)Expanding χ min and f min of the first two terms of the equation above near p, ω = 0 weobtain the leading scaling and the constant coefficient for the R´enyi Entropy S ( n ) A = − c eff n + 1 n log ω + ˜ A ( n ) + 11 − n log (cid:88) h (cid:54) =min χ h (cid:16) ω n (cid:17) χ min (cid:16) ω n (cid:17) f h ( n, p ) f min ( n, p ) (5.18) − n − n log (cid:88) h (cid:54) =min χ h ( ω ) χ min ( ω ) f h (1 , p ) f min (1 , p ) (5.19)15he constant ˜ A ( n ) is given by˜ A ( n ) = 11 − n log r − (cid:88) ∗ a =1 S min da sin n πasr (cid:32) r − (cid:88) ∗ a =1 S min da sin πasr (cid:33) n (5.20)which is well defined in the n → The next step is the evaluation of power law corrections to the logarithmic scaling of theR´enyi Entropy. First, notice that χ h (cid:28) χ min for ω →
0. In this regime the argument ofthe logarithms in the second and third line of (5.18) is close to 1 and then it can be Taylorexpanded. Taking into account only the most relevant contribution and assuming, to fixideas, that n >
1, we have S ( n ) A = − c eff n + 1 n log ω + ˜ A ( n ) + (cid:88) h (cid:54) =min ˜ B ( n ) h ω ∆ h − ∆min n + · · · (6.1)with ˜ B ( n ) h = 11 − n (cid:80) a sin n πasr S h da (cid:80) a sin n πasr S min da (6.2)and the most relevant contribution is given by the second smallest conformal dimensionamong those appearing in the expansion of 5.18, denoted ∆ in the following. Takinginto account only this correction we have S ( n ) A = − c eff n + 1 n log ω + ˜ A ( n ) + ˜ B ( n )1 ω ∆1 − ∆min n + · · · (6.3)In the case n < S ( n ) A = − c eff n + 1 n log ω + ˜ A ( n ) + ˜ B (cid:48) ( n )1 ω ∆ − ∆ min + · · · (6.4)with ˜ B (cid:48) ( n ) h = n − n (cid:80) a sin πasr S h da (cid:80) a sin πasr S min da = n ˜ B (1) h (6.5)In order to interpret these results from a physical point of view, i.e. to have an expressionfor the entropy in terms of the correlation length (or in term of the mass), we need arelation between the parameter ω = p r s and the correlation length ξ . In other words,we need to know the critical exponent ν in regime III: m = ξ − ∼ p ν (6.6)16sing perturbative CFT it is possible to evaluate the critical exponent ν . In thecontinuum limit, the model is described by the perturbed CFT: S = S CFT + t ˆ d x Φ , ( x ) (6.7)where | t | = p measures the departure from criticality. A simple dimensional analysisof this action tells us that ν = − ∆ , ) = r s for the minimal model M r − s,r at thetransition between regime III and IV.This value for the critical exponent ν can also be extracted extending previous results[47] to the Forrester Baxter models. Taking into account the necessary modifications,the calculation can be carried out along the same lines of [47] to get e − ξ = k (cid:48) ( p ν ) (6.8)where k (cid:48) ( q ) is the conjugate elliptic modulus for the elliptic nome qk (cid:48) ( q ) = ∞ (cid:89) (cid:96) =1 (cid:18) − q (cid:96) − q (cid:96) − (cid:19) (6.9)Expanding (6.8) around p = 0 we get ξ − = 8 p ν + 323 p ν + 485 p ν + 647 p ν + O ( p ν ) (6.10)in perfect agreement with the perturbative CFT prediction.Using (4.9) and (6.10) we have ω = (8 ξ ) − + · · · which gives S ( n ) A = c eff n + 1 n log ξ + A ( n ) + B ( n )1 ξ − n (∆ − ∆ min ) + nB (1)1 ξ − − ∆ min ) + ... (6.11)where constants A ( n ) and B ( n )1 are just trivial rescaling of ˜ A ( n ) and ˜ B ( n )1 : A ( n ) = ˜ A ( n ) + c eff n + 1 n log 2 (6.12) B ( n )1 = 8 − n (∆ − ∆ min ) ˜ B ( n )1 (6.13)From the equation (6.12), it is immediately possible to read the correction to the loga-rithmic scaling. For unitary theories the correction is expected to scale as ξ − n where ∆is the conformal dimension of the field related to the correction [39]. In the non unitarycase this term is affected by the fact that the ground state is no more the conformal vac-uum | (cid:105) but another state | min (cid:105) = Φ min (0 , | (cid:105) . The formula ∆ − ∆ min shows that thepresence of a non trivial ground state modifies the scaling of the correction. It is known[23] that this feature affects also the logarithmic scaling of the entropy in non-unitarymodels, where the central charge c is replaced by the effective one c eff . Similarly, it isnot surprising that the conformal dimensions are replaced by some sort of “effective”dimensions ∆ − ∆ min . Notice that the unitary case can be immediately recovered in(6.12) by setting ∆ min = 0. 17 A comment on off-critical Logarithmic Minimal Models
An interesting feature of Forrester Baxter RSOS models is that, for a particular choiceof parameters r and s , they provide a lattice realisation of Logarithmic Minimal models[48] and their off-critical thermal perturbations [49]. In particular it has been shownthat in the limit r, s → ∞ , keeping the ratio r/s fixed to some rational number R/S ,the underlying model becomes the so called logarithmic minimal model LM R − S,R andits off-critical thermal perturbation LM R − S,R + t Φ , .Taking such a limit for the entropy, the result is not affected from the functionalpoint of view: it maintains the same structure and the effective central charge c eff = lim r,s →∞ (cid:18) − r ( r − s ) (cid:19) (7.1)is identically 1 for any choice of the ratio R/SS ( n ) A = 112 n + 1 n log ξ + ¯ A ( n ) + ¯ B ( n )1 ξ − n (∆ − ∆ min ) + n ¯ B (1)1 ξ − − ∆ min ) + · · · (7.2)Here ¯ B ( n )1 = lim r,s →∞ B ( n )1 . When taking the limit r, s → ∞ the S modular matrix becomesill defined because the prefactor (cid:113) r ( r − s ) = r (cid:113) − sr tends to zero in this limit. Whilethis feature does not affect the limit of the coefficent B ( n )1 (6.2) – as S appears withthe same power both at the numerator and at the denominator – it implies some extraattention for the limit of the constant A ( n ) (5.20). For this reason, we need to modifythe definition of the R´enyi Entropy for Logarithmic models multiplying the partitionfunction by r in a sort of renormalisation procedure¯ A ( n ) = lim r,s →∞ (cid:18) A ( n ) − log 1 r (cid:19) (7.3)This multiplication can be seen in the same spirit of defyning generalised order param-eters in [49], while a better understanding of this renormalistion of the Entropy is stillmissing and goes far beyond the aim of this work. In any case, it only affects the nu-merical value of the non-universal constant ¯ A ( n ) and not the functional shape of the ξ dependence of the entropy.This ξ dependence result (7.2) disagrees with the general prediction for the R´enyiEntropy l dependence in logarithmic CFT [23], where a double logarithmic (log log) termis expected. The difference is due to the fact that the logarithmic feature of the systemis related to the presence of non diagonalisable Jordan blocks in the Hamiltonian. It hasbeen noticed in [49] that such non diagonalisability feature disappears as soon as thesystem is perturbed thermally out of the critical point. Thus the so called off-criticallogarithmic minimal models are not really logarithmic and thus we should not expectdouble log term in the computation of the entropy. In the case of logarithmic minimalmodels, then, we expect that, while the dependence in the subsystem size l at criticalityrescales with a double logarithmic term, this is absent in the rescaling off-criticality in18 while approaching the logarithmic critical point. The two behaviours in this case aredifferent, while in the usual minimal models (unitary or not) they show a similar (singlelog rescaling) pattern. We have computed the Entanglement R´enyi Entropy for Forrester-Baxter RSOS modelsin the regime III, a set of lattice systems whose continuum limit is described by non-unitary Minimal Conformal Models perturbed by their Φ , operator. Our evaluationfocused on the scaling with the correlation length ξ of the Entropy for a infinite, bipartite,quantum system. The computation of Entanglement Entropy in non unitary models hasbeen recently addressed in [23] and [24] where a logarithmic scaling with the size l of thesubsystem has been found. In particular it has been shown in [23] that the coefficientof the logarithm is given by the effective central charge c eff . For unitary theories thecomputation of the Entropy for a finite critical system ( l < ∞ , m − = ξ = ∞ ) canbe easily translated [14] in the infinite subsystem with a small but non zero mass gap( l = ∞ , m − = ξ < ∞ ) using arguments similar to the Zamolodchikov c-theorem.In the non unitary case such a translation cannot be performed a priori , since someassumptions about the positivity of certain correlation functions are no more valid inthe non unitary case. For this reason, the scaling we have found, where the proper length l is replaced by the correlation length ξ gives a strong hint about the renormalisationflux properties in the non unitary case. A proof of a sort of c eff -theorem in the nonunitary case goes far beyond the aim of this paper. Nevertheless, our result supportsevidence, in a non trivial set of non unitary models, that the ξ scaling follows the samelogarithmic law of the l scaling, exactly like in the unitary models.We have also studied the power law corrections to the logarithmic scale of the En-tropy. The interesting part is not in the coefficients of the expansion, that are nonuniversal, but in the exponent of the power of ξ . In the unitary case the expansion or-ganises into many series expansions in terms of powers of ξ ∆ h or ξ ∆ h /n , with h labellingthe various conformal primary fields. In the non unitary case we find that the seriesexpansions are in terms of ξ ∆ h − ∆ min or ξ (∆ h − ∆ min ) /n . In other words, also the conformaldimensions here take an “effective” value shifted by ∆ min , exactly like the central chargeis replaced by c eff .We have also briefly considered the limiting case of the off-critical Logarithmic Min-imal Models. Even if a double logarithmic term has been expected from the literature[23], where the l rescaling has been considered, we did not find here such a term: thescaling in ξ behaves exactly as in the non-logarithmic case. This discrepancy is dueto the fact that the perturbation outside the critical point destroys the Jordan blocksresponsible for the creation of the logarithmic features, as already pointed out in [49].It their seminal work on the FB RSOS model [32], Forrester and Baxter classifiedmany regimes, while we have restricted our analysis to regime III only. While it isknown [37, 38] that the regime III is a lattice realisation of the perturbed MinimalModels M r − s,r + Φ , , the physical interpretation of the other regimes has still to be19xplored. In the unitary ABF RSOS model [31], the number of regimes is smaller (justfour, compared to ten in the FB case) and two kind of universality classes have beenidentified. The critical line between the Regime III and IV can be described by theUnitary Minimal Models M r − ,r , while a unitary parafermionic CFT Z r − underlies thecritical line between regimes I and II. It would be very interesting to try to understand ifsome critical line of the FB RSOS model can be classified using, maybe, the non unitaryparafermionc CFT introduced in [50] or if they obey instead some other classificationscheme. The evaluation of Entanglement Entropy has been demonstrated to be a pow-erful tool for the identification of universality classes and it could be a valid help also inthe study of the unknown regimes of the FB RSOS model.Of course, further generalizations of these results to the lattice realisations of highercoset conformal filed theories perturbed by their Φ , , opertors, or even for models basedon cosets of higher algebras, like W n -algebra based series, including Pasquier like gener-alizations and their dilute versions, etc... can all be the subject of future investigation.As non unitary coset models as critical points of fusions of FB RSOS models are underconstruction right in these days [51], the exploration of R´enyi Entanglement Entropiesfor this set of models is viable.The bipartite Entanglement Entropy can give a lot of useful information, but a morecomplete understanding of entanglement needs also the knowledge of the dependenceon the size of the system, as well as its behaviour in finite intervals. We know howto deal with these problems in CFT [14] and, with a form factor expansion, also inoff-critical integrable quantum field theories [15]. However, in integrable spin chains, auseable procedure to compute entanglement entropies on a finite interval is still lackingin general. This would be a serious progress in our understanding of entanglement inone-dimensional systems.Also, the interest in entanglement measurements with more than two subsystemsintroduces new quantites to be evaluated, like negativity [52, 53]. To find a way tocompute these quantites in a consistent lattice integrable approach is a challenge forfuture research. Acknowledgments
We are endebted to O. A. Castro-Alvaredo and B. Doyon for very useful discussions. Inparticular, we thank the fruitful and deep exchange of ideas with P. A. Pearce, as well ashis continuous and encouraging interest in this work. We acknowledge E. Ercolessi forthe advice and the deep discussions during the elaboration of the M.Sc. thesis of one ofus (D.B.), where many of the results presented here were in nuce obtained. F.R. thanksINFN, and in particular its grant GAST, for partial financial support. D.B. thanks CityUniversity London for his Doctoral Scholarship and the University of Bologna for thewarm hospitality when part of this work has been done.20 eferences [1] E. Schr¨odinger,
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