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On the S-matrix of the Sub-leading Magnetic Deformation of

arXiv:hep-th/9108024v1 27 Aug 1991ISAS/94/91/EPNORDITA 91/47On the S-matrix of the Sub-leading Magnetic Deformation ofthe Tricritical Ising Model in Two DimensionsF. Colomo1∗, A. Koubek2, G. Mussardo1†1NORDITA, Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark2International School for Advanced Studies, Strada Costiera 11, 34014 Trieste, ItalyAbstractWe compute the S-matrix of the Tricritical Ising Model perturbed by the sub-leading magnetic operator using Smirnov’s RSOS reduction of the Izergin-Korepinmodel.

The massive model contains kink excitations which interpolate betweentwo degenerate asymmetric vacua. As a consequence of the different structure ofthe two vacua, the crossing symmetry is implemented in a non-trivial way.

We usefinite-size techniques to compare our results with the numerical data obtained bythe Truncated Conformal Space Approach and find good agreement.June 1991∗Angelo Della Riccia Foundation’s fellow.†On leave of absence from: International School for Advanced Studies, Strada Costiera 11, 34014Trieste, Italy.

1IntroductionIt has been pointed out by Zamolodchikov that certain deformations of minimal modelsof Conformal Field Theories (CFT) produce integrable massive field theories, which arecharacterized on mass-shell by a factorizable S-matrix [1].A particularly interestingsituation occurs for the “Tricritical Ising Model” (TIM) at the fixed point perturbedby the operator Φ2,1.This field has anomalous dimensions (∆, ∆) = ( 716, 716) and isidentified with the sub-leading magnetic operator1, odd with respect to the Z2 spin-reversal transformation. Hence, this deformation explicitly breaks the Z2 symmetry ofthe tricritical point and the corresponding massive theory can exhibit “Φ3-property”,i.e.

the absence of a conserved current of spin 3 and the possibility to form a boundstate through the process AA →A →AA.The counting argument supports thispicture, giving for the spin of the conserved currents the values s=(1,5,7,11,13) [1, 2]. Theinteresting features of this massive field theory have been first outlined in ref.

[2] wherethe model was studied by the “Truncated Conformal Space Approach” (TCSA), proposedin ref. [3].

This approach consists in the diagonalization of the perturbed HamiltonianH = HCF T + λZΦ2,1(x) dx(1.1)on a strip2 of width R, truncated at a certain level of the Hilbert space defined by theconformal field theory at the fixed point. The lowest energy levels are given in fig.

1. Thetheory presents two degenerate ground states (which correspond to the minima of theasymmetric double-well Landau-Ginzburg potential in fig.

2) and a single excitation B ofmass m below the threshold at 2m. The double degeneracy of the vacuum permits twofundamental kink configurations | K+⟩and | K−⟩and, possibly, bound states thereof.

Ifthe two vacua were related by a symmetry transformation, one would expect a doubledegeneracy of the breather-like bound state | B⟩in the infrared regime R →∞. However,the absence of a Z2 symmetry makes it possible that in this case only one of the twoasymptotic states | K+K−⟩or | K−K+⟩is coupled to the bound state | B⟩.1In the Landau-Ginzburg theory, the TIM represents the universality class of the ϕ6 model.

Theoperator Φ2,1 is represented by the third-order field ϕ3.2We consider in the following only the case of periodic boundary condition.

A scattering theory for such model with asymptotic states | K+⟩, | K−⟩and | B⟩was first conjectured in [4]. On the other hand, a general framework for the Φ1,2 andΦ2,1 deformations of CFT has been recently proposed in [5].

It is based on the RSOSreduction of the Izergin-Korepin model [6]. In section 3 we explicitly work out the RSOS-like S-matrix in the case of the TIM perturbed by the subleading magnetic operator.

Thescattering theory we obtain is different from the one of ref. [4], although both of themhave the following features:1. the existence of two fundamental kink configurations with mass m;2. the appearance of only one bound state of the above, with the same mass m.Hence, both scattering theories give rise to a picture that qualitatively agrees with theresults of ref.

[2]. Therefore a more detailed analysis is required in order to decide whichof the two theories is appropriate for the description of the scaling region of the TIMin presence of a third-order magnetic perturbation.As suggested in ref.

[4], possibleinsight comes from the study of finite-size effects for the theory defined on the cylinder[8]. These finite-size effects, which can be directly related to the scattering data, controlthe exponential decay of the one-particle energy level to its asymptotic value.The paper is organized as follows: in sect.

2 we outline Smirnov’s RSOS reduction ofthe Izergin-Korepin model. In sect.

3 we present our results on the S-matrix of the Φ2,1perturbation of TIM. Sect.

4 contains the analysis of the phase shifts and the asymptoticbehaviour of the RSOS S-matrix. Sect.

5 consists of a short summary of Zamolodchikov’sproposal for the S-matrix [4]. This proposal, together with ours, will be checked againstnumerical data in sect.

6, where we perform an accurate analysis of the energy levelsobtained from the TCSA using the finite-size theory. Our conclusions are in sect.

7.2Smirnov’s RSOS reduction of the Izergin-KorepinmodelIn a recent paper, Smirnov has related the Φ1,2 and Φ2,1 deformations of a CFT to a RSOSprojection of the R-matrix of the Izergin-Korepin (IK) model [5, 6]. His results can be

summarized as follows. The R-matrix of the IK model cannot be directly interpretedas the S-matrix of the massive excitations of the perturbed CFT because, introducing aHilbert space structure, it does not satisfy the unitarity requirement.

But it is possibleto use the quantum SL(2)q symmetry of the IK model to study the RSOS restrictionof the Hilbert space. This happens when q is a pth root of unity.

The RSOS reductionpreserves the locality of an invariant set of operators and yields S-matrices which have asensible physical interpretation. The RSOS states appearing in the reduced model| β1, j1, k1, | a1 | β2, j2, k2, .

. .

| an−1 | βn, jn, kn⟩(2.1)are characterized by their rapidity βi, by their type k (which distinguish the kinks fromthe breathers), by their SL(2)q spin j and by a string of integer numbers ai, satisfyingai ≤p −22,| ak −1 |≤ak+1 ≤min(ak + 1, p −3 −ak) . (2.2)In the case of Φ2,1 deformation, using the following parameterization of the central chargeof the original CFTc = 1 −6 πγ + γπ −2!,(2.3)the q-parameter is given byq = exp(2iγ) .

(2.4)For the unitary minimal CFT, γ = πrr+1 (r = 3, 4, . .

. ).

It is also convenient to definethe quantityξ = 23 π22γ −π!. (2.5)The S-matrix of the RSOS states is given by [5]

Saka′kak−1ak+1 (βk −βk+1) =i4 S0(βk −βk+1)1ak−1ak1ak+1a′kq× exp 2πξ (βk+1 −βk)!−1!q(cak+1+cak−1−cak−ca′k+3)/2(2.6)− exp −2πξ (βk+1 −βk)!−1!q−(cak+1+cak−1−cak−ca′k+3)/2!+ q−5/2(q3 + 1)(q2 −1)δak,a′k.Herein, ca are the Casimir of the representation a, ca = a(a + 1) and the expression ofthe 6j-symbols is that in ref. [7].

S0(β) has the following integral representationS0(β)= sinh πξ (β −iπ) sinh πξβ −2πi3!−1× exp−2iZ ∞0dxxsin βx sinh πx3 coshπ6 −ξ2xcosh πx2 sinh ξx2. (2.7)3RSOS S-matrix of Φ2,1 perturbed TIMIn the TIM perturbed by the subleading magnetization operator, r = 4 and ξ = 10π9 .From eq.

(2.2), the only possible values of ai are 0 and 1 and the one-particle states arethe vectors: | K01⟩, | K10⟩and | K11⟩. All of them have the same mass m. Notice thatthe state | K00⟩is not allowed.

A basis for the two-particle asymptotic states is| K01K10⟩, | K01K11⟩, | K11K11⟩, | K11K10⟩, | K10K01⟩. (3.1)The scattering processes are| K01(β1)K10(β2)⟩=S1100(β1 −β2) | K01(β2)K10(β1)⟩| K01(β1)K11(β2)⟩=S1101(β1 −β2) | K01(β2)K11(β1)⟩| K11(β1)K10(β2)⟩=S1110(β1 −β2) | K11(β2)K10(β1)⟩(3.2)| K11(β1)K11(β2)⟩=S1111(β1 −β2) | K11(β2)K11(β1)⟩+ S1011(β1 −β2) | K10(β2)K01(β1)⟩| K10(β1)K01(β2)⟩=S0011(β1 −β2) | K10(β2)K01(β1)⟩+ S1011(β1 −β2) | K11(β2)K11(β1)⟩

Explicitly, the above amplitudes are given by❅❅❅0011= S1100(β) = i2 S0(β) sinh95β −iπ5(3.3.a)❅❅❅0111= S1101(β) = −i2 S0(β) sinh95β + iπ5(3.3.b)❅❅❅1111= S1111(β) = i2 S0(β)sinπ5sin2π5 sinh95β −i2π5(3.3.c)❅❅❅1101= S0111(β) = −i2 S0(β)sinπ5sin2π512sinh95β(3.3.d)❅❅❅1100= S0011(β) = −i2 S0(β)sinπ5sin2π5 sinh95β + i2π5(3.3.e)The function S0(β) can actually be computed. It is given byS0(β)=−sinh 910(β −iπ) sinh 910β −2πi3−1× wβ, −15wβ, + 110wβ, 310(3.4)× tβ, 29tβ, −89tβ, 79tβ, −19,wherew(β, x) =sinh910β + iπxsinh910β −iπx ;t(β, x) = sinh 12(β + iπx)sinh 12(β −iπx) .It is easy to check the unitarity equations:S0011(β) S0011(−β) + S0111(β) S1011(−β) = 1 ;

S1011(β) S0111(−β) + S1111(β) S1111(−β) = 1 ;S1011(β) S0011(−β) + S1111(β) S1011(−β) = 0 ;(3.5)S1110(β) S1110(−β) = 1 ;S1100(β) S1100(−β) = 1 .An interesting property of this S matrices is that the crossing symmetry occurs in anon-trivial way, i.e.S1111(iπ −β)=S1111(β) ;S0011(iπ −β)=a2 S1100(β) ;(3.6)S0111(iπ −β)=a S1101(β) ;wherea = −s15s2512,(3.7)and s(x) ≡sin(πx).The above crossing-symmetry relations may be seen as due to a non-trivial chargeconjugation operator (see also [9]). In most cases, the charge conjugation is implementedtrivially, i.e.

with a = ±1 in eq. (3.6).

Here, the asymmetric Landau-Ginzburg potentialdistinguishes between the two vacua and gives rise to the value (3.7).The amplitudes (3.3) are periodic along the imaginary axis of β with period 10 πi. Thewhole structure of poles and zeros is quite rich.

On the physical sheet, 0 ≤Im β ≤iπ,the poles of the S-matrix are located at β =2πi3and β =iπ3 (fig. 3).

The first polecorresponds to a bound state in the direct channel while the second one is the singularitydue to the particle exchanged in the crossed process. The residues at β = 2πi3 are givenbyr1=Resβ= 2πi3S1100(β) = 0 ;r2=Resβ= 2πi3S1101(β) = is25s152ω ;r3=Resβ= 2πi3S1111(β) = i ω ;(3.8)

r4=Resβ= 2πi3S0111(β) = is25s1512ω ;r5=Resβ= 2πi3S0011(β) = is25s15 ω ;whereω = 59s15s110s49s19s2 518s310s118s718s229. (3.9)Their numerical values are collected in Table 1.In the amplitude S1100 there is no bound state in the direct channel but only thesingularity coming from to the state | K11⟩exchanged in the t-channel.

This is easilyseen from Fig. 4 where we stretch the original amplitudes along the vertical direction (s-channel) and along the horizontal one (t-channel).

Since the state | K00⟩is not physical,the residue in the direct channel is zero. In the amplitude S1101 we have the bound state| K01⟩in the direct channel and the singularity due to | K11⟩in the crossed channel.

InS1111, the state | K11⟩appears as a bound state in both channels. In S0111 the situation isreversed with respect to that of S1101, as it should be from the crossing symmetry property(3.6): the state | K11⟩appears in the t-channel and | K01⟩in the direct channel.

Finally,in S0011 there is the bound state | K11⟩in the direct channel but the residue on the t-channel pole is zero, again because | K00⟩is unphysical. This situation is, of course, thatobtained by applying crossing to S1100.4Energy levels, phase shifts and generalized statis-ticsThe one-particle line a of fig.

(1.a) corresponds to the state | K11⟩. This energy levelis not doubly degenerate because the state | K00⟩is forbidden by the RSOS selectionrules, eq.

(2.2). With periodic boundary conditions, the kink states | K01⟩and | K10⟩areprojected out and | K11⟩is the only one-particle state that can appear in the spectrum.Consider the threshold line.

On a strip with periodic boundary conditions we ex-pect this energy level to be doubly degenerate. This because the states | K01K10⟩and

| K10K01⟩are identified under such boundary conditions whereas the state | K11K11⟩re-mains distinct. The conformal operators creating these states in the u.v.

limit are Φ 716, 716and Φ 610 , 610, respectively (see lines marked b and c in fig. 1.a).

We have checked that forlarge values of R these two lines approach each other faster than 1/r (as would be thecase if the lowest line b was the only threshold line and c a line of momentum). Howeverthe truncation effects already present in this region prevents us from showing that theyreally approach each other exponentially3.

In the following we assume that ideally thesetwo levels are exponentially degenerate in the infrared limit.The identification of the threshold lines described above holds only for the staticconfiguration of two kinks with zero relative momentum. The situation is indeed differentfor the lines of momentum which approach the threshold.

In fact, the S-matrix actingon | K01K10⟩can only produce | K01K10⟩as final state whereas | K10K01⟩and | K11K11⟩can mix through the processes of eq. (3.2).

In order to determine the pattern of theenergy levels obtained from TCSA and to relate the scattering processes to the data ofthe original unperturbed CFT (along the line suggested in [12]), we would need a higher-level Bethe ansatz technique. This is because our actual situation deals with kink-likeexcitations in contrast to that of ref.

[12] which considers only diagonal, breather-like Smatrices. The Bethe-ansatz technique gets quite complicated in the case of a S-matrixwith kink excitations.

For the moment, it has been applied in few cases [13, 14]. Inthe light of these difficulties, we prefer here not to pursue such a program and insteadconcentrate on some properties of the phase shifts and generalized statistics which arisesfor kink excitations.For real values of β, the amplitudes S1100(β) and S1101(β) are numbers of modulus 1.

It3Consider, for instance, figure 10 of ref. [2] (the case of low-temperature phase of thermal perturbationof the TIM).

One observes that the onset of the exponential approach of degenerate excited levels usuallyoccurs quite far from the value of R at which the two ground state energies coincide. As we discuss insect 6, the situation of the subleading magnetic deformation of the TIM is even worse from a numericalpoint of view, because the anomalous dimension of the subleading magnetic operator is approximately1/2.

In this case, it is therefore possible that the onset of the exponential approach of the higher levelstakes place in a region of R strongly dominated by truncation effects.

is therefore convenient to define the following phase shiftsS1100(β)≡e2iδ0(β) ;(4.1)S1101(β)≡e2iδ1(β) .The non-diagonal sector of the scattering processes is characterized by the 2×2 symmetricS-matrixS1111(β)S0111(β)S0111(β)S0011(β). (4.2)We can define the corresponding phase shifts by diagonalizing the matrix (4.2).

Theeigenvalues turn out to be the same functions in (4.1),e2iδ0(β)00e2iδ1(β). (4.3)The phase shifts, for positive values of β, are shown in fig.

5. Asymptotically, they havethe following limitslimβ→±∞e2iδ0(β)=e± 6πi5 ;(4.4)limβ→±∞e2iδ1(β)=e± 3πi5 .There is a striking difference between the two phase shifts: while δ1(β) is a monotonicdecreasing function, starting from its value at zero energy δ1(0) =π2, δ0(β) shows amaximum for β ∼π3 and then decreases to its asymptotic value 3π5 .

Its values are alwayslarger that δ0(0) =π2.Such different behaviour of the phase shifts is related to thepresence of a zero very close to the real axis in the amplitude e2iδ0(β), i.e. at β = iπ9.This zero competes with the pole at β = iπ3 in creating a maximum in the phase shift.Similar behaviour also occurs in non-relativistic cases [10] and in the case of breather-likeS matrices which contains zeros [11].

The presence of such a zero is deeply related to theabsence of the pole in the s-channel of the amplitude e2iδ0(β). For the amplitude e2iδ1(β),the zero is located at β = 4πi9 (between the two poles) and therefore its contribution tothe phase shift is damped with respect to that one coming from the poles.

The net resultis a monotonic decreasing phase shift.

Coming back to the 2 × 2 S-matrix of eq. (4.2), a basis of eigenvectors is given by| φ1(β1)φ1(β2)⟩=A(β12) (| K11(β1)K11(β2)⟩+ χ1(β12) | K10(β1)K01(β2)⟩) ; (4.5)| φ2(β1)φ2(β2)⟩=A(β12) (| K11(β1)K11(β2)⟩+ χ2(β12) | K10(β1)K01(β2)⟩) .where A(β12) is a normalization factor.

In the asymptotic regime β →∞χ1=−e−2πi5a2 + e6πi5a(4.6)χ2=−e−2πi5a2 + e3πi5a,and the probability P1001 to find a state | K10K01⟩in the vector | φ2φ2⟩w.r.t.theprobability P1111 to find a state | K11K11⟩is given by the golden ratioP1001P1111= 1a2 = 2 cosπ5. (4.7)For the state | φ1φ1⟩, we haveP1001P1111= a2 =12 cosπ5 .

(4.8)The “kinks” φ1 and φ2 have the generalized bilinear commutation relation [15, 16, 17]φi(t, x)φj(t, y) = φj(t, y)φi(t, x) e2πisijǫ(x−y) . (4.9)The generalized “spin” sij is a parameter related to the asymptotic behaviour of theS-matrix.

A consistent assignment is given bys11=35 = δ0(∞)π;s12=0 ;(4.10)s22=310 = δ1(∞)π.The implications of these generalized statistics will be discussed elsewhere. Here we onlynotice that, interesting enough, the previous monodromy properties are those of the chiralfield Ψ = Φ 610,0 of the original CFT of the TIM.

The operator product expansion of Ψwith itself readsΨ(z)Ψ(0) = 1z65 1 + CΨ,Ψ,Ψz35Ψ(0) + . .

. (4.11)

where CΨ,Ψ,Ψ is the structure constant of the OPE algebra. Moving z around the origin,z →e2πiz, the phase acquired from the first term on the right hand side of (4.11) comesfrom the conformal dimension of the operator Ψ itself.

In contrast, the phase obtainedfrom the second term is due to the insertion of an additional operator Ψ. A similarstructure appears in the scattering processes of the “kinks” φi: in the amplitude of thekink φ1 there is no bound state in the s-channel (corresponding to the “identity term” in(4.11)) whereas in the amplitude of φ2 a kink can be created as a bound state for β = 2πi3(corresponding to the “Ψ term” in (4.11)).

In the ultraviolet limit, the fields φi shouldgive rise to the operator Ψ(z), similarly to the case analyzed in [17]. The actual proofrequires the analysis of the form factors and will be given elsewhere.5Zamolodchikov’s S-matrix for Φ2,1 perturbed TIMThe problem of finding a theoretical explanation for the energy levels of the Φ2,1 perturbedTIM was first discussed in a remarkable paper by Zamolodchikov [4].

In his notation, theone particle states are given by| K+⟩, | K−⟩, | B⟩,(5.1)which we can identify with our | K10⟩, | K01⟩and | K11⟩, respectively. The two-particleamplitudes of the scattering processes were defined in [4] to be| B(β1)B(β2)⟩=a(β1 −β2) | B(β2)B(β1)⟩+ b(β1 −β2) | K+(β2)K−(β1)⟩| K−(β1)B(β2)⟩=c(β1 −β2) | K−(β2)B(β1)⟩| B(β1)K+(β2)⟩=c(β1 −β2) | B(β2)K+(β1)⟩(5.2)| K−(β1)K+(β2)⟩=d(β1 −β2) | K−(β2)K+(β1)⟩| K+(β1)K−(β2)⟩=e(β1 −β2) | K+(β2)K−(β1)⟩+ b(β1 −β2) | B(β2)B(β1)⟩They are in correspondence with those of eq.

(3.2) if we make the following assign-mentsa(β)→S1111(β) ;

b(β)→S1011(β) ;c(β)→S1101(β) ;(5.3)d(β)→S1100(β) ;e(β)→S0011(β) . .In order to solve the Yang-Baxter equations which ensure the factorization of the scat-tering processes, Zamolodchikov noticed that the above amplitudes coincide with thedefinitions of the Boltzmann weights of the “Hard Square Lattice Gas”.

Therefore, heborrowed Baxter’s solution [18] in the case where it reduces to trigonometric forma(β)=sin2π5 + λβsin2π5R(β) ;b(β)=eδβsin (λβ)hsin2π5sinπ5i 12 R(β) ;c(β)=e−δβ sinπ5 −λβsinπ5R(β) ;(5.4)d(β)=e−2δβ sinπ5 + λβsinπ5R(β) ;e(β)=e2δβ sin2π5 −λβsin2π5R(β) .Here δ and λ are arbitrary parameters and R(β) is an arbitrary function. In order to fixcompletely the amplitudes, Zamolodchikov imposed the following requirements:1. the unitarity conditions, eqs.(3.5);2.

the absence of a pole in the direct channel of the amplitude d(β);3. crossing symmetry, implemented in the following forma(β)=a(iπ −β) ;b(β)=c(iπ −β) ;(5.5)d(β)=e(iπ −β) .

The final form of the S matrices is given bya(β)=e−2iπδ sin 2π−6iβ5sin3π+6iβ5sinπ−6iβ5sin2π+6iβ5 ;b(β)=e−δ(iπ−β)sin6iβ5sin 3π+6iβ5sinπ−6iβ5sin 2π+6iβ5 ;c(β)=e−δβ sinπ+6iβ5sin3π+6iβ5sin π−6iβ5sin 2π+6iβ5 ;(5.6)d(β)=e−2δβ sin 3π+6iβ5sin2π+6iβ5 ;e(β)=e−2δ(iπ−β) sin 3π+6iβ5sinπ−6iβ5 .Herein δ is an imaginary number satisfyinge−2πiδ =s15s25 . (5.7)All amplitudes but d(β) have a simple pole at β = 2πi3 .

Their residues are given byτ1=Resβ= 2πi3a(β) = i 56s153s252 ;τ2=Resβ= 2πi3b(β) = −i 56s15 136s25 76;τ3=Resβ= 2πi3c(β) = i 56s15 43s25 13 ;(5.8)τ4=Resβ= 2πi3d(β) = 0 ;τ5=Resβ= 2πi3e(β) = i 56s15 43s25 13 .Their numerical values are collected in Table 2.In the asymptotic limit β →∞, all amplitudes but a(β) have an oscillating behavioura(β)∼e−2πiδ ;b(β)∼e−δiπ eδβ e3πi5;

c(β)∼e−δβ e−3πi5;(5.9)d(β)∼e−2δβ e−iπ5 ;e(β)∼e−2δiπ e2δβ e−4πi5.Such oscillating behaviour was also found by the same author for a scattering model withZ4 symmetry [19]. There, it was suggested that in the ultraviolet limit, the Z4 theoryhas a limit cycle.

Whether or not this is also the case for the S-matrix proposed byZamolodchikov for the subleading magnetic perturbation of TIM, it is not clear to usthat it is possible to match such oscillating behaviour to either a definite CFT in theultraviolet limit, or a choice of generalized statistics for the kinks.6Finite-size effectsThe study of the scaling region around a fixed point is simplified along those directions(in the space of coupling constants) which define an integrable massive field theory. Insuch cases, the knowledge of the theory on mass-shell (the S-matrix) makes it possibleto characterize completely the dynamics even off-shell.

In particular, using the Thermo-dynamical Bethe Ansatz method [20] one could compute the ground-state energy E0(R)for the theory on a cylinder of width R. If this computation could be performed forthe subleading magnetic perturbation of TIM, it would become easy to decide which ofthe two proposed scattering theories is the correct one. Unfortunately, the Bethe-ansatztechnique has not been extented to the case of an S-matrix with kink excitations.

Onlyfew examples have been worked out [13, 14]. However, we can get around this difficultyusing the “Truncated Conformal Space Approach” (TCSA) [2, 3, 22] and the predictionsof finite-size theory [4, 8].The TCSA allows us to study the crossover from massless to massive behaviour ina theory with the space coordinate compactified on a circle of radius R. The methodconsists in truncating the infinite-dimensional Hilbert space of the CFT up to a level Λin the Verma modules.

Then, the off-critical HamiltonianH(λ, R) = H0(R) + V (R) . (6.1)

is numerically diagonalized. Here, H0(R) is the Hamiltonian of the fixed point [21]H0(R) = 2πRL0 + L0 −c12,(6.2)and V (R) is the interaction term given by the perturbationV (R) = λZ R0 Φr,s(x) dx .

(6.3)The matrix elements of V (R) are computed in terms of the three-point functions of thescaling fields of the fixed point. An efficient algorithm has been developed for performingsuch computation [22].In our case, the truncation is fixed at level 5 in the Vermamodules.

The parameter λ in (6.3) is a dimensionful coupling constant, related to themass scale of the perturbed theory[λ] = m2−2∆r,s . (6.4)In the following we fix the mass scale by λ = 1.The energy levels Ei(R) of the Hamiltonian (6.1) have the scaling formEi(R) = 1RFi(mR) ,(6.5)with the asymptotic behaviourEi(R) ≃2πR2∆i −c12, mR ≪1 ;ǫ0m2R + Mi, mR ≫1 .

(6.6)Here, ∆i is the anomalous dimension of some scaling field in the ultraviolet regime andMi is the (multi)particle mass term in the infrared limit. However, the above infraredasymptotic behaviour holds only in the ideal situation when the truncation parameterΛ goes to infinity.

In practice, for finite Λ, the linear behaviour of eq. (6.6) is realizedonly within a finite region of the R axis.

The large R behaviour is dictated by truncationeffects. In order to find the physical regions, we make use of the following parameter(introduced in ref.

[2])ρi(R) = d log Ei(R)d log R.(6.7)The parameter ρi is between the values ρi = −1 (in the ultraviolet region) and ρi = 1 (inthe infrared one). In the limit of large R (the truncation-dominated regime), ρi = 1−2∆.

The “window” in R where the linear infrared behaviour holds depends upon theperturbing field and, for the case of operators with anomalous dimension ∆≥12, it canbe completely shrunk away. This phenomenon is related to the divergences which appearin a perturbative expansion of the Hamiltonian (6.1), which must be renormalized.

Underthese circumstances, it is more convenient to consider the differences of energies, whichare not renormalized.In the case of the subleading magnetic perturbation of TIM, the anomalous dimensionof Φ2,1 is ∆=716, which is near 12. Looking at fig.

(1.a), we see that the onset of theinfrared region of the two lowest levels is around R ∼2 and persists only for few unitsin R. In this region one can check that they approach each other exponentially [2]E1 −E0 ∼e−mR ,(6.8)and extract in this way the numerical mass of the kinksm = 0.98 ± 0.02 . (6.9)From fig.

6, we see that for the third level, that of one-particle state, the ultravioletbehaviour extends till R ∼0.5. The crossover region is in the interval 0.5 ≤R ≤2.Beyond this interval, the infrared regime begins but the “window” of infrared behaviouris quite narrow and is in the vicinity of R ∼3.

In such small region, it becomes hardto extract any sensible result. To overcome this difficulty, it is better to consider thedifferences of energies with respect to those of the degenerate ground states.

From fig. (1.b) one can easily read offthe mass-gap and see that is consistent with the valueextracted from the exponential approach of the two lowest levels.

In fig. (1.b), the thirdline defines the threshold, with a mass-gap 2m.In the ideal situation Λ →∞, the crossover between the intermediate region (mR ∼1) and the infrared one (mR ≫1) is controlled by off-mass shell effects and has anexponential behaviour.

The computation of these finite-size corrections has been putforward by L¨uscher [8].In the case of one-particle state, there are two leading off-mass-shell contributions coming from the processes shown in fig. 7.

The first correctioninvolves the on-mass-shell three-particle vertex Γ, which is extracted from the residue at

β = 2πi3 of the amplitudes S1111(β) (in the case of RSOS S-matrix) and a(β) (in the caseof Zamolodchikov’s S-matrix). The second correction comes from an integral over themomentum of the intermediate virtual particle, interacting via the S-matrix (S1111(β) forthe RSOS S-matrix and a(β) for that of Zamolodchikov).

Therefore we have∆E(R)≡E2(R) −E0(R) = m + i√3m2Γ2 exp −√3mR2! (6.10)−mZ ∞−∞dβ2π e−mR cosh β cosh βSβ + iπ2−1.We have done the following.

First we have computed numerically the integral on theintermediate particles in both cases of RSOS and Zamolodchikov’s S-matrix and we havesubtracted it from the numerical data obtained from the TCSA. After this subtraction,we have made a fit of the data with a function of the formG(R) = A + Be−√32 mR + Ce−mR .

(6.11)The first term should correspond to the mass term. The coefficient of second one is thequantity we need in order to extract the residue of the S-matrix at β = 2πi32√3m B = i Resβ= 2πi3S1111(β)a(β).

(6.12)The third term is a subleading correction related to the asymptotic approach of the lowestlevels of our TCSA data to the theoretical vacuum energy E0(R).In the case of RSOS S-matrix, the best fit gives the following valuesA=0.97 ± 0.02 ;B=−0.29 ± 0.02 ;(6.13)C=−0.36 ± 0.02 .The corresponding curve is drawn in fig. 8, together with the data obtained from TCSA.The mass term agrees with our previous calculation (eq.

(6.9)). The second term givesfor the residue at β = 2πi3 the value 0.34 ±0.02.

This is consistent with that of the RSOSS-matrix. In our fit procedure, the value of the residue we extracted through (6.12) isstable with respect to small variation of the mass value.

Increasing (decreasing) m, B

increases (decreases) as well, in such a way that the residue takes the same value (intothe numerical errors). This a pleasant situation because it permits an iterative procedurefor finding the best fit of the data: one can start with a trial value for m (let’s say m = 1)and plug it into (6.11).

From the A-term which comes out from the fit, one gets a newdetermination of the mass m that can be again inserted into (6.11) and so on. Continuediteration does not affect significantly the value we extract for the residue, but convergesto an accurate measurement of the mass.

The values in (6.13) were obtained in this way.With Zamolodchikov’s S-matrix the best fit of the data (with the same iterativeprocedure as before) gives the resultA=0.96 ± 0.02 ;B=−1.10 ± 0.02 ;(6.14)C=1.14 ± 0.02 .The residue extracted from these data (1.29±0.01) is not consistent with that one of theamplitude a(β). The situation does not improve even if we fix the coefficient of e−√3m2tobe that one predicted by Table 2, namely B = −0.158 and leave as free parameters fora best fit A and C. In this case, our best determination of A and C were A = 0.965 andC = −0.046.

The curve is plotted in fig. 8 together with the data obtained from TCSA.7ConclusionsThe S-matrix proposed by Zamolodchikov for the subleading magnetic perturbation ofTIM is a particular solution of the “Hard Square Lattice Gas” [18], fitted in such a waythat it matches the physical picture coming from the TCSA data, i.e.

the presence oftwo fundamental kinks and only one bound state thereof. But also the RSOS S-matrixcomputed in sect.

3 is a particular set of Boltzmann weights for the “Hard Square LatticeGas” and it reproduces the same features. The key difference is how the crossing sym-metry comes about: for the RSOS S-matrix, one finds a non-trivial charge conjugationmatrix, eq.

(3.6), whereas for Zamolodchikov’s proposal the crossing symmetry occurs ina standard way, eq. (5.5).

Their asymptotic behaviour for large real values of β is also

quite different: the RSOS S-matrix goes to a definite limit while the S-matrix proposedby Zamolodchikov is oscillating. For the RSOS case, the finite limit of the S-matrixallows us to introduce generalized statistics of the kink excitations [15, 16, 17].The real discrimination between the two scattering theories proposed for the sublead-ing magnetic deformation of TIM is seen by comparing them with a numerical “experi-ment”, i.e.

from the study of the finite-size corrections of the energy levels obtained bythe Truncated Conformal Space Approach. We have investigated this problem in sect.6.

The result suggests that the RSOS S-matrix gives a more appropriate description forthe scattering processes of the massive excitations of the model. Though, the interest-ing question as to what kind of system the S-matrix of Zamolodchikov corresponds to,remains open.AcknowledgmentsWe thank T. Miwa, A. Schwimmer and Al.B.

Zamolodchikov for useful discussions. Weare grateful to P. Orland for a critical reading of the paper.

One of us (F.C.) thanks P.Di Vecchia for warm hospitality at NORDITA.

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