Spectral flow between conformal field theories in 1 + 1 dimensions

NUCLEAR

P H Y S I CS B

Nuclear Physics B 370 (1992) 511—550 North-Holland

Spectral flow between conformal field theories in 1 + 1 dimensions Timothy R. Kiassen

*

Enrico Fermi Institute and Department of Physics, University of Chicago, Chicago, IL 60637, USA

Ezer Meizer

* *

Department of Physics, University of Miami, P.O. Box 248046, Coral Gables, FL 33124, USA Received 16 May 1991 Accepted for publication 22 November 1991

Consider the q5 13-perturbed minimal unitary conformal field theories .,~ç, (with diagonal modular invariant partition function) of central charge c~= 1 —(6/p(p + 1)), p = 3, 4,... It is believed that for one sign of the perturbing parameter they are massive scattering theories of kinks, whereas for the other they are massless but not scale-invariant quantum field theories interpolating between in the UV and ~ in the JR limit. We propose integral equations 4’2k+2 for the on the cylinder exact finite-volume (of circumference energies R).ofThese the first integral and kth equations excitedarestates similar in to çb13-perturbed the thermodynamic Bethe Ansatz equations recently proposed by Al. Zamolodchikov for the ground-state energy in these theories. The behaviour of the conjectured expressions for the finite-volume energies at small and large R, which we investigate analytically and numerically, is in excellent agreement with predictions of conformal perturbation theory around the UV and JR fixed point, respectively, providing strong support for our proposal. In particular, we study in some detail the flow of the spin field (dimension d = ~) of the tricritical Ising model 4’~to the spin field (d = ~) of the critical Ising model .%~. For 4 13-perturbed 4~ we also compare the small-R behaviour of the conjectured first energy-gap with results from the “truncated conformal-space approach”, and discuss the limitations of the latter method when conformal perturbation theory has UV divergencies.

4;

1. Introduction The study of the space of quantum field theories (QFTs) is certainly one of the most important problems of theoretical physics. In higher space-time dimensions not much is known about this space. In 1 + 1 dimensions, however, the situation looks much better. There the fixed points of the renormalization group (RG) are E-mail: [email protected]. Present address: Newman Laboratory, Cornell University, Ithaca, NY 14853, USA. * * E-mail: [email protected]. Present address: ITP, SUNY at Stony Brook, NY 11794, USA. * In 1 + 1 dimensions global scale-invariance implies local scale-invariance, i.e. conformal invariance, *

under weak assumptions [3]. 0550-3213/92/$05.00 © 1992

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Elsevier Science Publishers B.V. All rights reserved

T.R. Klassen, F. Melzer

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conformal field theories (CFTs) wide classes of which are understood in great detail [1,2]. Our knowledge of non-scale-invariant QFTs has also increased significantly in recent years. One approach to study a non-scale-invariant OFT is to consider it as a relevant perturbation of the CFT describing its UV limit [4,5]. The euclidean action of the theory can then be written as ~,

A =A~~+ Afd2~cP(~),

(1.1)

where cP is the spinless, say, perturbing field of scaling dimension d 2 y <2, so that A has mass dimension y (for simplicity we wrote down the case of one perturbing field, extensions are obvious). This in principle uniquely defines the OFT, but extracting its long-distance properties from such a definition is in general only possible using approximate numerical methods (one can use e.g. the “truncated conformal-space approach” (TCSA) of Yurov and Al. Zamolodchikov [61to calculate numerically the finite-size spectrum or at least energy gaps, see subsect. 2.2). However, if the perturbation is integrable, which can be checked perturbatively by constructing non-trivial integrals of motion [7], analytical techniques become available to study the perturbed theory. First of all, using the knowledge of the integrals of motion one can in many cases construct a conjecture for the factorized S-matrix of the perturbed theory by applying the general principles of S-matrix theory and the bootstrap [7—10].These conjectures can then be tested non-perturbatively using the so-called thermodynamic Bethe Ansatz [9,11—14](TBA), which enables one to calculate the exact infinite-volume thermodynamics of a QFT in terms of its S-matrix. This is useful, since the infinite-volume free energy at finite temperature T (and zero chemical potentials) is simply related to the ground state energy E0(R) of the zero-temperature OFT on a cylinder of radius =

—

—

R 1/T. The behaviour of E0(R) for small and large R allows one to read off the central charge and other properties of the CFT describing the short- and long-distance limits, respectively, of the OFT in question. For any QFT with a diagonal S-matrix the derivation of the TBA integral equations for the ground-state energy is quite straightforward (see refs. [9,12,13] for a detailed description). Recently Al. Zamolodchikov [15,16] has applied the TBA to a class of perturbed CFTs with generically non-diagonal S-matrices, namely the 413-perturbed minimal unitary CFTs of central charge c~= 1 (6/p(p + 1)), p 3, 4,... and diagonal modular-invariant partition functions. These perturbed theories will be referred to as 4’A~.They are quite different for opposite signs of the perturbation parameter A, and we will denote them by 4A~~ and ~ respectively, for positive and negative A. The corresponding QFTs can =

—

=

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be obtained [17] by taking scaling-limits of the r =p + 1 RSOS lattice model [18] in regimes IV and III, respectively. is believed [10,19] to describe a massive scattering theory of p 2 types of kink—antikink pairs, which interpolate between the p 1 degenerate vacua of the theory. Explicit conjectures for the (non-diagonal for p 4) S-matrices of these models have been proposed. In these cases the derivation of the ground-state TBA equations is much more involved than in the case of diagonal S-matrices, even though the final equations are quite simple and of the same general form. In fact, a derivation was presented in ref. [151only for the 4’A~~~ case. For p 5 the TBA equations were conjectured as a generalization of the p 4 case, and some consistency checks performed. with p 4, on the other hand, is argued [4,5,20] to be a massless (but of course not scale-invariant) OFT, interpolating between the minimal CFTs 4 in the UV and ~ç-1 in the JR. Except for 4’A~~, where the S-matrix is believed to be diagonal and to describe the scattering of the left- and right-moving modes of an interacting massless particle [16], no conjecture for the S-matrices of these theories has been proposed so far. Nevertheless, TBA equations for the finitevolume ground-state energy in all the 4~’A~J~~ models were conjectured [16] by generalizing the structure seen in the ~ and %‘A~~ cases. From a fundamental point of view the QFTs ~ are of course extremely interesting. To our knowledge 4At~~is the first non-trivial interacting massless OFT for which an exact S-matrix has been proposed. The models £A~, p 5, are even more interesting since they presumably have non-diagonal S-matrices. All results from the TBA approach mentioned so far refer to the integral equations for the finite-volume ground-state energy of factorizable scattering theories. It has long been an open question if similar integral equations (or any other expressions) can be written down for the exact finite-volume energies of excited states. For energies above the ground state only few results are known, mainly analytical large-volume estimates of the finite-size spectrum for massive QFTs (see e.g. refs. [21,22]), in addition, of course, to numerical results for small to moderately large volumes obtained from the simulation of lattice models or using the TCSA [6,22—25]. In this paper we propose integral equations for the exact energies of certain excited states in the models ~ p even, in finite volume with periodic boundary conditions (in the sector of zero total momentum). At the moment we cannot derive these equations from “first principles”, but we present analytical and numerical checks (for small and large volume) of these equations, which, we think, leave little doubt as to their correctness, at least for p 4 and 6. For a similar proposal in another class of perturbed (non-unitary) CFTs, see ref. [27]. The outline of this paper is as follows. In sect. 2 we describe general results useful in the study of the small- and large-R behaviour of (functions that serve as candidates for) energy levels obtained from TBA-like integral equations. In addi—

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tion we discuss some facts about conformal perturbation theory (CPT) that we will need, and also clarify the relation between CPT and the TCSA. This will point out certain limitations but also advantages of the TCSA, concerning the regularization of UV and IR divergencies of CPT. It also leads to a general prescription for calculating (numerically, eventually) CPT expansion coefficients. Sect. 3 summarizes what is known about the models ~ in particular their finite-volume ground-state energies. This serves as an introduction to sect. 4, where we motivate and provide analytical and numerical evidence for our conjecture for the first excitation energy E~±~(R) in ~ The theory ~ is a somewhat special case in the series .,KA~ in particular, the JR behaviour of E~kR) can be studied analytically much more thoroughly than in the other models. The behaviour of E~±~(R) ~ ~(R) for small R will also be compared with results from the TCSA. In sect. 5 we generalize the discussion of the previous sections: We propose equations for the exact finite-volume energies of the first and kth excited states in JA~~2,and present analytical and numerical results supporting them. In this —

—

—

section we also give some numerical evidence for the conjecture of Al. Zamolodchikov for the ground state energies in 4’A~.In sect. 6 we briefly summarize our results and list a number of open questions.

2. General methods Consider a non-scale-invariant relativistic OFT on an infinitely long cylinder of circumference R, which is regarded as the “volume of space”. Let E(R) denote the energy of a generic state in this geometry. It is convenient to use only dimensionless quantities, namely r Rm, where m is a mass scale of the theory, and the scaling functions e(r) defined by =

E(R)

(2ir/R)e(r).

=

(2.1)

(In a generic theory e(r) may depend on other dimensionless couplings in addition to r, which are however absent in the specific models treated in the next sections.) The ground state energy, in particular, will be denoted by E0(R) 2ire0(r)/R. Recall from finite-size scaling [26] that for flows between unitary CFTs =

e0(r)

—‘

—~cuv for —~cIR for 1

r—sO r-~,

(2.2)

where ~ and CIR are the central charges of the CFTs describing the UV and IR limits, cuv > dR [4], of the theory in question (if the theory is purely massive, the IR fixed point is trivial, dIR 0, and e0(r) decays exponentially for large r). Similarly, if E~(R)is the energy of a state interpolating between states created by =

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conformal fields of scaling dimensions ~ and dIR in the UV and JR CFTs, respectively, then ~ e1(0) e0(0) and dIR e1(co) e0(~). =

—

=

—

2.1. TBA-LIKE INTEGRAL EQUATIONS

The equations known or conjectured to give the exact scaling functions for certain states in certain integrable theories are of the form 2)E e(r)

=

—

(r/4~

f

do

Pa(0)

ln(1

+ ta

e~°~),

(2.3)

where the r-dependent functions a(O) are determined by a set of coupled nonlinear integral equations N a(0)

=

r~a(6)

—

~

(Kab

*

ln(1 + tb e~))(O).

(2.4)

b=1

Here

*

denotes the convolution

(f

*

g)(O)

=

(1/2~)f

dO’ f(O

-

O’)g(O’),

(2.5)

the ta ±1, a 1,..., N specify what we will refer to as the “type” t (ta,..., tN) of the system of equations (2.3)—(2.4), the kernel K is a symmetric matrix whose elements are even functions of 0, exponentially decaying as I 0 I and the ~~(O),finally, are of the form =

=

=

—~ ~,

Pa(0) E

{iuia cosh o,

~tha

e~°},

(2.6)

where the tha are some non-negative (dimensionless) mass parameters. The matrix K(O) and the “bare energy terms” Va(0) depend on the S-matrix of the theory in question. For factorizable theories with a diagonal S-matrix there is a direct correspondence between the particles and the va(0): If the theory is purely massive, describing N different kinds of (stable) particles labelled by a 1,.. ., N whose masses are ma, then to each particle there corresponds an energy term lIla cosh 0 where tha ma/rn. In a massless (non-scale-invariant) theory, on the other hand, distinct labels are attached to the left- and right-moving modes of each massless particle and the corresponding energy terms are ~tha e ±6, respectively, where the tha depend on the mass scale. In both cases Kab(O) is, up to a factor of —i, simply the logarithmic derivative of the two-particle S-matrix Sab(0) for the scattering process ab —* ab (9 is the relative rapidity, suitably defined for massless particles, cf. sect. 3). =

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For theories with a non-diagonal S-matrix some ‘~‘a~0) 0 and the relation between Kab(O) and the S-matrix is more complicated; in fact, it is not yet understood in general. Within a given theory, different types t correspond or are conjectured to correspond to different excitations (although, to be sure, neither does an arbitrary type correspond to some excitation, nor can all excitations be obtained by simply varying the type). The ground state, for instance, corresponds to ta 1 for all a. Even though eqs. (2.3)—(2.4) have not been derived using the TBA in all cases of interest, in particular not for any excited state, we will nevertheless refer to them as “TBA equations” for short. Although we do not have general rigorous results concerning the existence and uniqueness of solutions to eq. (2.4), we expect the general features known [9,12,13] from the case where all ta 1 to persist even when some ta 1. Namely, as r 0 the functions —

—

=

=

=

—

—~

(2.7) e~”~6~ approach constants y, Xa~ y~ in the regions 0 ~ —ln(2/r), —ln(2/r) <<0 << ln(2/r), 0 >> ln(2/r), respectively, with two transition regions around ±ln(2/r). The choice of ~a(~) determines whether the functions Ea(0) may exhibit certain symmetries, e.g. the Ea(O) are even if all ~~‘a(0) are, or, if tN+ i—a ta and 1 —a~~) ~‘a( —0) for all a, then the equations have a solution satisfying ~N+1—a(0) Ea(0). Furthermore, Ya± 0 if Va(0) grows exponentially as 0 respectively. Assuming this general behaviour, the constants Xa, )~a±are certain non-negative solutions (respecting the above mentioned symmetries of the Ea(0)) of the equations =

= =

=

+

Xa

—‘

tbxb)~b, y~- (1— =

~)fl

(1

+ tby~),

(2.8)

where (2.9)

Nab= _f(d0/2~T)Kab(0),

and

8 a± ~0 J1 =

~f ~‘a(~) grows exponentially as 0 otherwise.

—‘

±~

(2 10)

It can be shown then, along the lines of the calculations in refs. [9,12], that e(0)

~

=

_(1/4~2)a~i ~

-~(y~)

-~(y~)],

(2.11)

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where .~~(x)=~(~),

~(x)

=

-~(x),

(2.12)

.2’(x) being the Rogers dilogarithm [28] function 1 =

—

~x

I dt

—

2~o

ln

ln(1—t)

t +

(2.13)

.

1—t

t

The next question we have to address is the small-r behaviour of e(r). As argued in ref. [29], it is related to the periodicity properties of the Ya(0). To uncover these properties we now specialize to the case that will be of interest later, namely

K(0)

1 ~

cosh

9)j(N)

(2.14)

j(N)

where I~foris the matrix the simple Liethat algebra i.e. modifications ~ = 8ab+1’ a, bincidence = 1,..., N. We of should remark withAN, slight + these considerations also apply to equations of the form (2.4) with certain other types of kernels K (see e.g. ref. [29]). Taking the Fourier transform of eq. (2.4) (first bringing ri-’a(O) to the l.h.s. in order to ensure existence of the transform for non-zero r’a(O)) using ~(k)

f

do

etkO~(O) =

cosh(~k/2)’

multiplying by 1/~(k),Fourier transforming back, and noticing that —

~i~)

=

(2.15)

~a(~

+

~i7r) +

0 for all ‘~‘a(~) of the form (2.6), one obtains +

~i~)1~(0

—

~i~)

=

fl (1

+ tbYb(0)).

(2.16)

Generalizing an observation in refs. [15,16] for the ground state energy, it follows (by iterating eq. (2.16)) that a=1,...,N,

YJ(9+T(N+3))0a(t)1’N+l_a(O)

(2.17)

where cra(t) = tatN+ 1—a For the type choices we will be interested in later (cf. sects. 4, 5) ~a(~) = 1 for all a. This is true in particular for the ground state. In general, eq. (2.17) of course implies ~‘a(~

+ i~(N+

3))

=

Ya(~)

(2.18)

518

(for N odd

o~N±~)/2(t) =

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1, so the period of

Y~N±l)/

Spectralflow

2(0)is half of that indicated in (2.18)). For the energy terms (2.6) (and weak assumptions on the kernels K) it can be shown that the ~a(~) are entire functions20”~~3~ for all r> 0. It follows thatinthe have around any point theYa(0) complex au-plane, Laurentexcept expansion in terms of u e u = ~, cx• As argued in ref. [29] (cf. also the appendix of ref. [16]),

one therefore expects the scaling functions e(r) to have an expansion in powers of r4~”~3~ for small r up to possibly some irregular terms. In the cases studied so far there has always been a single irregular term, either a “bulk term” proportional to r2 or a term of the form r2 ln r ~. The coefficients of these terms have been explictly calculated (though special considerations are required in different cases). Using the methods discussed in detail in [12,13] one can show that for N even the irregular term is a bulk term Br2, whose coefficient is given by —

N

B

=

(1/16ir2)

~ th~(8~T~ + 6~T~), a=1

(2.19)

where the T±must satisfy N

~

(2.20)

k~T~= 2~Ttha67~,

b—i

being defined by the large 101 expansion Kab(0)= ~kab ~~°1 + .... Note that the Ta~ and B are independent of the type t. If the matrix (kab) is invertible, (2.20) uniquely determines the I~. In the case at hand kab = 2I~’~’, and since for N even the incidence matrix of AN is invertible the T±are calculable. Some scaling function e 1(r), determined by eqs. (2.3)—(2.4) with kernel (2.14), therefore has a small-r expansion of the form 2+ ~ for N even. (2.21) e1(r) =e~(0)+Br kab

—

(For N odd the Br2 term has to be replaced by some other term, cf. subsect. 5.3 for some specific cases.) Directly from the TBA equations one can argue that the sum over n in the above equation starts at n = 2 for the ground state, but at n = 1 for other generic types (cf. the appendix of ref. [161).The sum is expected [12,13]to have a finite non-zero radius of convergence around r = 0. *

More precisely the r2 term is actually an “anti-bulk term”, i.e. it has to be present in the small r expansion so that e(r)/r has a finite limit — in particular no bulk r2 term corresponding to E(R) having a term linear in R — at large r. Similarly one can understand the r2 In r term. Cf. ref. [131for details, and also remarks in subsect. 2.2.

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At present it is not known how to calculate the coefficients a1,~analytically directly from the TBA equations. Estimates for the first few coefficients can however be obtained by solving these equations numerically and fitting the results for the e~(r)at small r to the expansion (2.21). Our iterative procedure to solve equations of the form (2.3)—(2.4) was described in detail in ref. [13] for the case of the ground state. The same method can be used in cases where some or all ta = 1 (though the convergence of the iteration is quite a bit slower then). We should add that we do not have a rigorous proof that the TBA equations have a unique solution for generic ~a(0), Kab(0) and types, but our numerical results indicate that this is true in all cases we considered or at least that our iterative procedure finds the “right” solution should there be several. Up to now we have discussed results pertaining to the small-r expansion of the scaling functions. We conclude this subsection by looking at the large-r behaviour. If the QFT under consideration is purely massive then its JR limit is trivial. From the point of view of the TBA equations this is manifested as follows: In this case all iia(0) are of the form tha cosh 0, so that the Ea(0) with non-vanishing lila grow linearly with r for large r, and all e.(r) decay exponentially as r ~. In particular e0(cx) = 0 implies CIR = 0, and the fact that e1(ce) actually vanishes for any choice of type t in eqs. (2.3)—(2.4) tells us that the e1(r) can at best describe excitations that are degenerate with the ground state in infinite volume (so considering the TBA systems (2.3)—(2.4) with different types may be relevant for a purely massive OFT only if its vacuum is degenerate). On the other hand, if the OFT under consideration is not purely massive, its IR limit can be a non-trivial CFT, and the large-r behaviour of the e1(r) is more interesting. From the TBA equations one sees that it is possible to have e~(cx)>0 if there are some ~a(~) = ~lI’Za e±Owith tha > 0. In fact, in this case the analysis of the IR limit is qualitatively very similar to that of the UV limit. As r ~ the Ya(O) develop plateaus for 0 << ln(r/2), ln(r/2) << 0 <> ln(r/2), the asymptotic values of which we will again denote by y;, Xa, y,~, respectively. A calculation essentially identical to that leading to (2.11) shows that —

—

—~

—~

—

e(~)

where now the

=(1/4~2)a~i[2~(Xa)

~

Xa

—

-~(y~)

-g(Ya~)],

(2.22)

and Ya~~ must satisfy

xa

=

=

(i

—

max6~)~ ±

N b-I

(1 —8~)fl(1

(1

+ tbXb),

+tbyb).

(2.23)

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Note that the second equation here is identical to the second equation in (2.8), the latter being relevant to the UV limit. At present not much is known analytically about the behaviour of the TBA integral equations at large yet finite r in generic cases. We will present (analytical and numerical) results for some specific models in subsects. 4.2 and 5.3. 2.2. CPT AND THE TCSA

If the e1(r) discussed in subsect. 2.1 are supposed to be true scaling functions in some perturbed CFT, the small-r expansion (2.21) has to agree with the predictions of conformal perturbation theory (CPT) on the cylinder (see e.g. ref. [13] for relevant facts of CPT used here). Assume that e~(r)is the scaling function of a state which in the UV limit is created by the conformal field 4~.CPT gives an expansion of e(r) in powers of AR~(recall that we are considering the case of a single spinless perturbing field 1 of scaling dimensions d = 2 y). A is related to the mass scale m of the perturbed theory via —

IAI ~

(2.24)

[A priori one can choose different mass scales for opposite signs of A and correspondingly there are two different K’s. We will see however empirically that in the theories we will consider, K is independent of the sign of A for the most natural definition of the mass scales, implemented in the relevant TBA equations by setting the nonzero lIla equal to 1.] Using this relation the CPT expansion of e,(r) becomes a power series in r~,and we denote the corresponding coefficients for positive and negative A by ~ ~, respectively. Starting from the action (1.1), they can be written as —

1—Y[

=

—‘

~2~‘

\

~

.‘

n.(ilz)

\~1

‘~

f fl

n—i

j—i

d2z

—

n—i

~ (j CD(1, 1)

(2irIz~I) \

fl ~(z

j=1

1, z~)

~) / (2.25)

The correlators here are connected (with respect to the “in- and out-states” created by ~,) critical (n + 2)-point functions on the plane, with (i I ... I I) denoting <41(cc, cc). . . ~ 0)). [To obtain (2.25) one has to transform the correlators from the cylinder to the plane; if 4,, is not primary (note that in a unitary CET 1~is always primary if it is spinless and relevant) the “planar” state I i) appearing *

We might occasionally drop the superscript (±) from the CPT coefficients discussed here and/or attach it to their TBA counterparts of eq. (2.21). This will eliminate the difference in notation between the two sets of coefficients, but should not lead to any confusion, especially since the two sets are equal anyhow if our conjecture for the e~(r)is correct.

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in the above has to be interpreted suitably, taking into account the non-trivial transformation properties of 4,,. For an example where cP is not primary see eq. (4.19) of sect. 4.1 In the framework of CPT it is obvious that a~ = (— 1)”a~,

(2.26)

for any i, n. This implies that up to the possible irregular terms, the scaling functions e,(r) obtained from the “TBA” for opposite directions of the perturbation should be analytic continuations of one another with respect to r~ (corresponding to A —A). From the point of view of the TBA equations, where perturbations with different signs of A are conjectured to correspond to different sets of Va(0) (see sects. 3—5), this is quite a non-trivial prediction which is not yet understood analytically. Comparing the explicit values of the a,~obtained (numerically) from the TBA with those predicted by CPT within a given model is possible only to a limited extent, because it is not easy to obtain more than one or two non-trivial coefficients within the CPT framework, even numerically. Let us restrict our discussion to perturbations of unitary CFTs (generalizations to the non-unitary case are straightforward, see e.g. ref. [13]). Then CPT for the ground state, corresponding to 4,, = 111. in eq. (2.25), immediately gives ~ = 0 and “usually” (see below) allows one to calculate ~ and ~ For excitations one has ~

*-*

*

a~

=

±(2~-)’~KKi I i)’C4.~,

(2.27)

but already ~ is in general very difficult to compute. Comparing with the TBA one uses one of the above analytically calculable coefficients to determine K, the other coefficients then provide a non-trivial consistency check. Actually, if d 1 a finite number of terms in the CPT expansion will diverge. More precisely, if d 2(n 1)/n the term ~ is UV divergent (see e.g. refs. [5,13]) and has to be determined by regularization and renormalization in some scheme. In practice, finite values for ~ and ~ are often obtained simply by analytic continuation in d = 2 y from a region where the corresponding integrals are finite. [Although we are not aware of a rigorous proof, it is presumably a consistent renormalization scheme to define all ~ by analytic continuation in d, as long as d ~ 2(n 1)/n, see below.] This is equivalent to introducing a UV cutoff e into the integrals in eq. (2.25), and after evaluating them taking the limit —

—

—

*

Operator product coefficients C4,,1,.4,5 of arbitrary conformal fields can be calculated using conformal invariance [11given the coefficients for the primary fields, which are known in all minimal CFTs [30].

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TR. Kiassen, E. Meizer

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Spectral flow

0, with terms that diverge like negative powers of e being ignored. Explicitly [5,30], —~

a~=

2(1~K2y2(1

_~(2~)

—

(y/2))y(y —1),

(2.28)

and 2—y 0,3

~~)_

—

~ —48k

~

\3(i~)~)K 3‘y 3

3y—2 4

_______

p

where y(s) = F(s)/F(1 s). However, even the analytically continued integrals do not always exist since they may be singular at certain values of y. [If an UV cutoff is used to regularize the integrals for these special values of y, the result contains terms that diverge logarithmically when the cutoff is removed.] This happens for a~ when d = 1, a~) when d = ~, and in general is expected to happen for a~ when d = 2(n 1)/n for any i, n. It is well known that these divergencies signal the presence of terms non-analytic in r~in e(r). At present no general systematic method is known to explicitly determine these non-analytic terms within a CPT or CPT-inspired framework. We note however the following. Assuming consistency of the analytic-continuation renormalization scheme mentioned earlier, the coefficients a,~of the finite number of terms diverging in CPT, and any possible irregular terms, are in principle uniquely determined by the requirement that the analytic continuation of the small-r CPT expansion to large r approaches const r as r oc (this assumes, as one would expect, that there is a finite number of irregular terms which are all of a “reasonable” form, such as r” ln~r,a, f3 0). In practice, this observation by itself is of course not particularly useful. Within the TBA framework, however, the irregular terms can often be determined analytically in the cases studied so far they are of the form r2 or r2 ln r, as mentioned in subsect. 2.1 and the coefficients a,~of all regular terms can in principle be determined numerically (in practice at least for small n, corresponding to the “difficult”, namely divergent, coefficients in CPT). It is possible to circumvent the need to regularize UV divergencies by writing down the CPT expansion for energy differences, like ê,(r) e,(r) e 0(r). For these objects the coefficients, written as integrals of differences of correlators, are all finite. We will later in this section discuss a general method to determine these coefficients. The coefficients a~) may also be JR divergent, the divergencies coming from regions of integration in (2.25) where fields cP get close to the 4,, located at 0 and cc, corresponding to the infinite “past” and “future” on the cylinder. If present, the divergence is always power-like (logarithmic IR divergencies are absent since the integrated correlators are connected), the power increasing with the dimension d, but also depending on the specifics of the CFT operator-algebra. The nature of —

—

—‘

—

—

—

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these divergencies will be seen more explicitly below, see e.g. eq. (2.33), when we compare CPT from the euclidian field theory viewpoint (as discussed so far) with CPT from a hamiltonian point of view. In the hamiltonian framework IR divergencies obviously never arise (the volume of “space” is finite!), and in fact the comparison will show us that the right way to regularize the possible IR divergencies in eq. (2.25) is by analytic continuation in d, (this recipe is always well-defined since only power-like divergencies occur). There is another method which we can use to study (numerically) the finitevolume spectrum of perturbed CFTs. The “truncated conformal space approach” [6] (TCSA) is a non-perturbative method to calculate the low-lying excitations in an approximation which is good for small to moderately large volume. The idea is simply to truncate the full infinite-dimensional Hilbert space of the (perturbed) CFT to a suitable finite-dimensional subspace, on which one can numerically diagonalize the hamiltonian of the theory

HA(R) ‘Hcp.-ç(R) +AfdxcP(x), (2.30)

H~~(R)=(2~r/R)(Lo+Lo— j1yc). (Here the integral is along a “line of constant time” around the cylinder, and L

0 (L0) are Virasoro modes whose eigenvalues ~l (ii) are left (right) dimensions.) The truncation is usually done by ignoring all states above a certain level 1 in the (left and right) Virasoro representations of the CFT, or above a certain (left and right) dimension. For details see [6,22—25]. As we will now demonstrate, a major drawback of the TCSA (that seems to have been overlooked so far) is that in general it only allows one to estimate scaled energy gaps ê,(r) = e~(r) e0(r), not the individual scaling functions e,(r). This is the case whenever CPT for the e,(r) suffers from UV divergencies, namely for d 1. These divergencies should in principle be handled by renormalization, i.e. the introduction of an UV cutoff and the addition of cutoff-dependent counterterms to the hamiltonian to implement the desired renormalization conditions. This is not easy to do, in general, neither in CPT nor in the TCSA framework. In CPT one can avoid this, in principle, as long as d * 2(n 1)/n, by defining the nth expansion coefficient a1~ by analytic continuation in d (cf. our earlier remarks). For the TCSA, however, there is not such an “easy way out”. What happens, in fact, is that the eigenvalues of the TCSA hamiltonian HA all diverge as the truncation is removed (this was not realized in a previous study [23] of the theory 4A4, for which d = ~ instead, the absence of clear signs for convergence of the energy levels was attributed to truncation effects). In more mathematical language, for A > 0 the hamiltonian HA is not an operator but just a quadratic form on the Hilbert space of the UV CFT a common fate of perturbed operators —

—

—

524

T.R. Klassen, E. Melzer

/ Spectralflow

in infinite dimensions when the perturbing term is not “small” w.r.t. to the unperturbed operator (as signaled in the case at hand by the existence of UV divergencies). Of course the UV divergencies cancel in energy differences like ê,(r), and so when calculated with the TCSA they will converge as the truncation is removed. If one wants to avoid the explicit renormalization of UV divergencies, estimates for the ê1(r) seem to be the best one can obtain from the TCSA, for d 1. We now want to show the divergence of the TCSA for the e,(r) more explicitly, in the process illuminating the connection between the TCSA and CPT. Let e~’kr) denote the scaling functions obtained from the TCSA at truncation level 1. It is clear that for any finite 1, at least, these functions have a small-r expansion of the form e~(r) =

E~

(2.31)

n~0

One can use standard Rayleigh—Schrodinger perturbation theory to calculate the “truncated coefficients” aft We will see that as I cc these coefficients approach the corresponding CPT coefficients without any analytic continuation In 1),will diverge as 1 —sincc~d.This particular, for 1 d < 2, some finite number of the a~ implies that the e~kr)diverge for any r> 0 as 1 cc• [Note that it is not possible —*

—

—*

that the e~(r)are finite but simply not analytic in r-” at 1 = cc, since then either none or infinitely many of the a~),would diverge as 1 cc.] To calculate the ~ choose a basis {4,~)(not necessarily orthonormal) of conformal fields in which the unperturbed hamiltonian HCFT and the (commuting) momentum operator P = (2~/R)(L 0 L0) are diagonal. In other words, the have definite left and right dimensions z~t,and /i~.For our basis to be the A —s 0 limit of an energy-eigenstate basis of the perturbed theory, we choose it to diagonalize the perturbation f dxcP(x) in subspaces of degenerate states. (For our formulas below to hold, we actually only have to diagonalize the perturbation in the subspace degenerate with the unperturbed state I i). If 4,, is primary this subspace is usually 1-dimensional, so that no diagonalization is necessary.) Namely, in terms of the CFT data, we require that C4.~5= 0 for j * k if d3 = dk and = 5~ where s~ = is the conformal spin of ~ Introduce the inner-product matrix g~k= Kj 1k) = C~I~j4k~ It is block-diagonal, in the sense that g1~ = 0 unless = ~k and = ~k’ and furthermore orthonormal in the subspace of primary fields (level 1 = 0) when one uses the conventional [1] normalization for the primary fields. Evaluating matrix elements of the perturbation term in the hamiltonian (2.30) in terms of operator product coefficients of the unperturbed CFT on the plane, we —*

—

—

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525

find for the first five coefficients (cf. ref. [31]) a~’~=d~— ~c, aVl a~=

—

(2ir)~~(±K)(g~1)1C~,

=

g) ~

(2~)2(1—Y)~2(

______

j#i

ii

c..c”.c’j.

C..C~3kC!k.

a~=

3(g

( 2~.)3(1Y)(

a~=

—

(2ir)4°

~~)

11)_i

~K~(

g.) —1

________________

j,k,li

Ii

,j

ki

,,

d~d

—

j,k+i

ji

ki

2 d ii

E’

dii d ki

—

CC’~C’~C’~i

j,kf=i

j,k+i

Ii

‘

ii

—

ki

Ji ~‘

“

ii

‘

i ki

,, ,, ,j

,k

+

>

d3

j+i

(2.32)

ii

where d 1),kCkJ6S.S., and the prime on the 3, =symbols d. d,, indicates C~= C,,,~., C”3 =are ~(g summation that they to be restricted to the truncated Hilbert space of the CFT. Also, restrictions like j * i should be actually understood to exclude summation over states j such that d~= d,, as formally enforced by the vanishing of the corresponding operator product coefficients in the special basis {4,~}we chose. Note that the first two coefficients a~and a~1~ are independent of the truncation level 1; in particular, a~1I coincides with the corresponding CPT coefficient (2.27). To make direct contact between the above formula for ar), and the CPT expansion coefficient a,~,we return to eq. (2.25). For any n, we can restrict the integration region there to I z,,_ 1 I < I z,,_2 I < < I z~I <1, if we multiply by an overall factor n!. This makes the fields in the correlator (radially) time-ordered, and suggests that we insert a decomposition of the identity operator 1 = —

...

~jk I j)(g_i)J~(k I between consecutive ~ fields. One should however take into account the fact that the correlators are connected, which makes things technically a bit more involved for large n (still, using this method it is possible to compute the coefficients of Rayleigh—Schrodinger perturbation theory in a relatively simple way). Since the main points are already seen for n = 2, we will elaborate here only on this case. The projection operator on the conformal state created by 4,, is

TR. K/assets, E. Meizer

526

P1

=

I i)E~(g~) ~kK’c I.

/

Using Kk I ~(z, 2)1 i)

Spectra/flow

=

Ckjz~~ 2—~,2~k—d/2—~, we ob-

tain 2Y2

a

(2’~-) 1

1,2=—

=

-

,.

.~

~

1/ 2YK2 .\

(2~)i_

Kilt, =

=

— (2~)2U~~~2(

—

f J

~

IzI
d2zIzI~(iI~(1, 1)(1-P 1)~(z,

Iz~
g)

(2~)2(1—Y)~2( g)

—

E

~

j#i

k

C11(g

i)jkCkiossfidppdk_di_i 0

C~JC’~1

—

j+i

=

2)Ii)

3’

lima~~.

(2.33)

The next to last equality is established by noticing that for the chosen basis {4,~}the properties of the inner-product matrix and the operator product coefficients such effectively restrict the double summation on the third line to states 4,~, that d 5k = s~.It is important to note that because of the 3 =over dk *p d, and s3 = integral = I z I, this equality is valid in general only after analytic continuation in d, to a region where d 1 is smaller than any scaling dimension dk of fields 4~k contributing to the sum (as dictated by the operator product coefficients and the inner-product matrix). This explains our earlier remark regarding the regularization of JR divergencies of CPT in the euclidean field theory formulation. In certain cases, where the CFT correlators we integrate over are given by an elementary or some familiar special function, one can obtain expressions for the a,~ in terms of more explicit (multiple) infinite sums see refs. [13,32] for examples. These expressions are related to the general formulas (2.32) (with 1 = cc) through non-trivial sum rules for operator product coefficients of descendant fields. The simplest example is that of a 02, where the integrated 4-point function reduces to (0I~(1, 1)1(z, 2)10) = Ii —z I —2d Using this expression one can expand 4ik

—

ao2_(2~r)

2

~

K)

r

2 dzlzl

d—2

Il—zI

—2d

Izi
21 =

_(2~)2

~

_d)

(2.34)

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527

The infinite sum here converges only if d < 1 (and d * 0, —2, —4,...), and then the result is of course equal to the r.h.s. of eq. (2.28). Comparing eq. (2.34) with (2.32) in the case i = 0, where C2~ * 0 only if is in the conformal family of ~, we conclude (cf. ref. [32]) that a~ is given by the sum in eq. (2.34) with cc in the upper limit replaced by 1. So we see explicitly that for d 1(r) 1, will a~ diverge divergesfor when all 1r> —s cc (unlike a~,,with n > 2/(2 d)), implying that e~~ 0 once the truncation is removed. The situation is of course the same for all —

other excitations e~(r),as is clear from eq. (2.25). What we learned about the coefficients a~ is actually useful in practice. Using eq. (2.32), and its straighforward though increasingly tedious extension to higher n, one can obtain estimates for the UV and JR finite expansion coefficients â~,,by calculating az), aW,, for several low truncation levels and extrapolating to I = cc~ We emphasize that this prescription for evaluating the a,,, (or even the as,, if they are UV-finite) is general, in the sense that it does not rely on special “closed form” representations for the corresponding CFT correlators in eq. (2.25). In subsect. 4.1 we will use this method to estimate several expansion coefficients in ~ We finally turn to the IR regime. One may also attempt to explore the neighborhood of r = cc by CPT around the JR CFT. To do so it is necessary to know (or conjecture) along which direction the IR fixed point is approached, i.e. what perturbation of the JR CFT describes at least the large-r behaviour of the theory which was initially specified as a certain perturbation of the UV CFT. IR CPT corresponds to perturbing a CFT by irrelevant operators, and one will therefore have to add more and more higher-dimension counterterms as one goes to higher orders of perturbation theory. It remains to be investigated if JR CPT is really well-defined, and in particular how to determine the coefficients of the higher-dimension counterterms. Nevertheless, knowing just the leading IR perturbing field(s) provides non-trivial predictions for the large-r behaviour of the e,(r), which can then be compared with analytical or numerical results from the TBA equations (cf. subsects. 4.2 and 5.3). —

3. 4’A 4: known results %‘A4 describes the integrable perturbation [7] of the tricritical Ising CFT (TIM) of central charge c = ~ by the relevant spinless primary field 41,3 of scaling dimension ~. The action is 4,i,3(~)’

(3.1)

A,A4=ATIM+Afd2~ where A is a real coupling constant of mass dimension ~. As mentioned in the introduction, 4’A~~~ is believed [4,5,20] to be a massless QFT, interpolating

TK Kiassen, E. Meizer

528

/ Spectral flow

between the TIM in the UV and the critical Ising model (CIM) of c = ~ in the JR. As such, it may be considered [15,16,201 as an irrelevant perturbation of the CIM, with the non-renormalizable action A,A~- = A CIM

+ gfd2~

[Ti’

+

(higher dimension fields)

1

(3.2)

Here (T)T is the (anti)holomorphic component of the stress—energy tensor of the CIM, and accordingly g is of dimension —2. Note that ~ is special in that the IR fixed point is believed to be approached along the direction of TT, whereas in the flows 4 —*4 with p > 4 the IR CFT is presumably approached asymptotically along the direction defined by the operator 4~of (this is known [4] to be true for p>> 1 from perturbative RG calculations). On the other hand, 4’A~~~ was argued [10,19] to be a massive OFT whose spectrum consists of two kink—antikink pairs of degenerate mass mk~flk linking three degenerate (in infinite volume) vacua. The scattering of the (anti)kinks is conjectured to be described by a factorizable RSOS-like S-matrix that was given explicitly in ref. [10]. Recently, Al. Zamolodchikov [15,16] has employed the TBA technique to study the ground state energy E~~(R) of the theories 4’At4~~. In ref. [15] he derived a -

~

system of TBA equations for E~kR), starting from the conjectured S-matrix of ref. [10] and diagonalizing the corresponding “transfer matrix”, which has essentially the structure of the row-to-row transfer matrix of the critical Ising lattice model [33]. The equations for the ground state scaling function e~(r) take the form (2.3)—(2.4) with N = 2, v1(0) = cosh 0, v2(0) = 0, t1 = t2 = 1, and K given by eq. (2.14). In ref. [16] a scattering theory of massless particles for ~ was proposed, where the mass scale M of the theory is determined by the position (on the s-plane) of a resonance pole in the S-matrix of the massless particles. Explicitly, parametrize the on-shell momentum of where right- 0and modes by mass(p°, 1)=(~Me±&,±~M e±S), respectively, canleft-moving be considered as the p less analog of rapidity (note however that zero momentum corresponds to 0 —s F cc, for right/left-moving modes). The only non-trivial matrix element is that for the scattering on left on right modes, given by 0—i7r/2

S(0) =i sinh

2

0+iir/2\

/sinh

2

)‘

(3.3)

where 0 is now the (absolute value of the) relative rapidity of the scattering “particles”. Since the S-matrix is diagonal, one can easily write down the TBA equations for e~(r). They are [161exactly the same as those for e~(r)but now with r’ 6, v 1(0) = ~e 2(0) =

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529

Employing the general machinery of sect. 2 one gets [15,16] e~kO) = 7/120 and ~ = 1/(8ir), ~ = 0. The first non-zero coefficients a~j and a~ 0Vin the expansion (2.21) extracted from the numerical solution [15,16]of the two TBA systems are in perfect agreement with the CPT prediction (2.28)—(2.29) with —

—

4”5

K

=

IA I m

0.148695516112(3).

(3.4)

Here the mass scale m is M, mklflk, respectively, in ~ ~ Furthermore, the coefficients ~ n 8, obtained numerically, satisfy eq. (2.26) to great accuracy. The results [16] of the large-r analysis of the the TBA equations for e~kr)lend further support to their validity. In the particular case of 4’A~~~ one can analytically obtain an asymptotic large-r expansion (see subsect. 4.2 for more details), which is then compared with the IR CPT expansion of e~(r) based on the action (3.2). Contrary to the UV case, here the relation g= —(2/~M)2 between the coupling constant g and the S-matrix mass scale M was found [16] already while constructing the conjectured massless S-matrix. Using this relation the leading three terms in the asymptotic expansion of the TBA result for e~kr) agree with those obtained from JR CPT ~. In particular, the TBA analysis gives the expected central charge for the JR CFT, dIR = 12e~(cc) = Thus we see that the studies of e~~(r), both analytical and numerical, provide strong evidence for many qualitative properties of the theories .%‘A~ 4~ which had not been proven so far, in addition, of course, to giving a detailed quantitative understanding of various aspects of these theories. In sect. 4 we will describe an attempt to extend these non-perturbative studies to the lowest excitation E~±kR). —

4. Lowest excitation in 4’A4 Consider the energy E~±kR) of the first excited state in the theories %‘A~4~, and the corresponding scaling functions e~ ±~(r). We conjecture that integral equations for e~ ±~(r) can be obtained by just changing the type choice in the TBA **

*

The next terms get contributions from the higher-dimension fields in the action (3.2) whose exact

couplings are not known. It might be possible to fix these couplings by comparison with higher terms in the large-r TBA expansion [161. ~ From here on we will refer to a (smooth) function E/R) as the ith excitation energy if it starts out at small R being equal to the ith energy level above the ground state E0(R). We will do so even though possibly at larger volumes E(R) is not the energy of the ith level above the ground state (due to level crossings, that can occur in integrable theories even among states of zero momentum and the same quantum numbers with respect to other “unbidden” symmetries).

T.R. Klassen, E. Melzer

530

equations describing e~kr) to

t

2)

e~U(r)

=

—

=

1

=

t2

2 a~i

=

—

/

Spectral flow

1. Explicitly,

~ ~ do v~~(0) ln(1

—

e_~0)),

(4.1)

(r/4ir

r~~(0)

—

b=i

*

ln(1

—

e~))(0),

(4.2)

where

v~(0)

=

~e°,

~~(0)=cosh0,

t4~(0) = ~e°, v~(0)=O,

and is defined in eq. (2.14). Since at present we have no derivation of the above TBA-like equations for e~—~~(r) (contrary to those for e~—~~(r), see sect. 3), we will just offer the following motivation for writing them down: (1) In the QFT describing the massive scaling limit of the Ising model in the low-temperature phase, corresponding to the theory ~ the complete finitevolume spectrum is known exactly [34] and an analogous recipe for calculating e~~(r) holds ~. Namely, in this theory [13] J(2)~(O)

e~(r) = _(r/47r2)f for i

=

0, 1, where

dO cosh 0 ln(1

+ t(j)

e_~0))

(4.4)

t(

0) = t(i) = 1 and c’-~(0)= r cosh 0. (ii) Klumper and Pearce have recently presented [35] a new method to analytically calculate the eigenvalues of the row-to-row transfer matrix of certain integrable lattice models at criticality in the thermodynamic limit. In particular, one of the models they study in detail is that of tricritical hard squares, which scales to the TIM. Remarkably, although a priori their physical problem looks quite different from ours we are interested in the spectrum of the non-scale-invariant OFT on the cylinder in the end it boils down to solving a very similar mathematical problem. Specifically, in the infinite-volume limit the equations in ref. [35]from which the behaviour of the largest eigenvalue of the transfer matrix is extracted, are essentially the same as the r-independent (!) “kink-version” [9,12] of the TBA equations which is encountered in the calculation of the UV limit e~(0). —

—

*

This is apparently also the motivation in ref. [271,where systems of equations similar to (4.1)—(4.2) for the lowest excitation in a class of 413-perturbed non-unitary minimal CFTs are proposed. The terminology “bosonic TBA equations” used there presumably alludes to the TBA equations for the ground state of a factorizable S-matrix theory involving particles that do not obey an exclusion principle [9,12] (in rapidity space). We think this terminology is misleading, and if taken literally would actually predict certain wrong signs in the TBA equations for the excited state.

TR. Klassen, E. Me/zer

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Spectralflow

531

Both calculations of course lead to the same number the central charge ~ of the TIM. This suggests the following “recipe” to obtain TBA-like equations for the first excited state in 4’A~: Re-introduce the same r-dependence that was lost when passing to the kink-version of the TBA equations for the ground state into the equations of ref. [35] for the next-to-largest eigenvalue of the critical transfer matrix. This indeed gives eqs. (4.1)—(4.2) for the ~ case (written in terms of the functions e~kO)= e~~(0) + i~instead of e~kO)). —

4.1. ULTRAVIOLET BEHAVIOUR

The UV limit r 0 of eqs. (4.1)—(4.2) is immediately obtained from eqs. (2.8)—(2.13): Since ~ko)=~k—0) and the two functions e~(0) are even, we obtain using t1 = t2 = —1 and Nab = 2’,~b~ —~

—

5— 1),

y~=y~=0,

yj =y~ = 1,

(4.5)

y~=y~=l,

(4.6)

x1 =x2= ~(v’ for %‘A~,t~, and xi=x 2=~(v~—1), y~=yj=O, for ~ .2’(l) =

v.2/6

It follows from (2.11), using ~“(0) that for both ~ e~~(0)=

=

=

0,

~

—

1))

=

(d~2 ~c)TIM, —

~,.2/1O

and

(4.7)

where d2,2 = ~ is the scaling dimension of the field 4,2,2 in the TIM. From the general results of subsect. 2.1 we further expect the small-r expansion 2 + ~ a~r~5, e~(r)

=

~

+B~r

(4.8)

n=I

with the same bulk term coefficients Bt~~ = 1/(8ir), ~ = 0 as in the analogous expansions of e~~(r).From the numerical solution of eqs. (4.1)—(4.2) (a sample of our results for e~~(r) is presented in table 1) we estimated the values of the first seven coefficients ~ which we quote in table 2. To test these results, we first of all note that the coefficients corresponding to opposite signs of the perturbation satisfy the relation (2.26), as required by CPT. Another consistency check is performed by combining the CPT prediction for ~ with the result (3.4) for K obtained from the analysis of the UV behaviour of the ground state. Using eq. (2.27) we find —

a~)= ~

=

±0.032765643448(1),

(4.9)

TR. Kiassen, E. Meizer

532

/

Spectral flow

TABLE 1 The scaling functions e~± ~(r) in ~ calculated numerically by solving eqs. (4.1)—(4.2) for the given values of r. The error in the last digit given for the scaling functions is about 2

r 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 0.02 0.04 0.06 0.08 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

e~.~(r) 0.0167975004154638 0.0168949295092867 0.0169829494502597 0.0170654328257782 0.0171440685953867 0.0172197943600199 0.0172932016215320 0.0 173646949543954 0.0174345670526136 0.0175030386155410 0.0181366324404342 0.0192604672472030 0.0202889633497887 0.0212600378331327 0.0221896452057990 0.0264292064187193 0.0302028180541729 0.0336312659342478 0.0367716738047481 0.0396606072131994 0.0423252939706666 0.0447877478320708 0.0470666177890245 0.0491781419697495

eS)(r) 0.0165366540366860 0.0164408544578196 0.0163550186802904 0.0162751753340647 0.0161995 761380189 0.0161272423781548 0.0160575517351934 0.015990075345 6891 0.0159245006888137 0.0158605904552103 0.0152846945736008 0.0143242711648944 0.0 135040879692650 0.0127744255424971 0.0121122975068332 0.0094675887238990 0.0075311289205263 0.0060454016089318 0.0048798522405290 0.0039536301444608 0.0032115100745074 0.0026 135694006936 0.0021298950390135 0.001737530978970

in excellent agreement with the “TBA” result quoted in table 2. As discussed in subsect. 2.2, it is very difficult to calculate the higher coefficients a~j~ in CPT using eq. (2.25) directly with 4,, = 4,2,2 and cl~= ~ Furthermore, ~ actually diverges and we cannot define it by analytic continuation in y, since we do not

TABLE 2 4~5part of the) small-r expansion of e~kr) in The first few coefficients a’j~in the (regular in r 4’A~~’~, obtained from the numerical solution of eqs. (4.1)—(4.2) for r-values between 0.001 and 0.75. The error in the last digit is given in parenthesis

n

a~

1 2 3 4 5 6 7

0.03276564345(2) 0.027159771(2) 0.01121648(3) 0.0016284(4) — 0.000406(3) —0.000116(5) 0.000054(6)

________________ 0.03276564345(2) 0.027159771(3) —0.01121647(6) 0.001628(2) 0.000407(6) —0.00011(1) — 0.00005(1) —

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Spectral flow

533

TABLE 3 1~, ~ — a~’1,,of êç’](r) in the theory ~ for n = 2, 3 and 4. We The expansion coefficients â~ present the values obtained from 1 eq. (2.32) for truncation levels I = 0, 1 5, the results obtained by rational extrapolation to / = ~, and, for comparison, the corresponding TBA results

â~uI

/

4 0 1 2 3 4 5 TBA

0.00600490958383 0.0022892946683 1 0.00061964543465 — 0.00036553844715 — 0.00102914565022 — 0.00151285708543 —0.0058(2) — 0.005786772(2)

0.00552552856535 — 0.003703 69895 695 — 0.00321370576240 —0.00299763018172 —0.00287867366142 —0.00280428778590 —0.00252(2) — 0.00250612(4) —

0.00048758047822 0.00087945728222 0.00111400275873 0.00111278989424 0.00116995945649 0.00116608688950 0.00122(6) 0.0012603(4)

know its meromorphic structure in y, at present. (None of the ~ is JR divergent, so there is no need to worry about analytic continuation in d22.) However, using the formulas of subsect. 2.2 for the truncated coefficients a~), we can estimate a1,, for n = 2, 3 and 4. The ay~for truncation levels 1 = 0, 1,.. . ,5 are extrapolated to l = cc using rational extrapolation [36] w.r.t. the variable (1 + 1)°’, with w chosen to optimize the fit. (Here, as in the TCSA results

TABLE 4 The scaled energy gaps ~ ~(r) in ~ calculated numerically from the 5/4, TCSA with (up K given to level in eq. 5) (3.4). and the Error “TBA” estimates integral for the equations. TCSA results For theare TCSA deduced we used fromr the = Rvariation m = RUt of /k)the êç’1(r) with the truncation level 1 (more precisely, we varied the maximal value of the scaling dimension of the fields up to level 5 allowed in the truncated Hilbert space). The error in the last digit given for the “TBA” results is about 2

r

~~~(r) (TCSA)

ê~~(r) (TBA)

ê~kr)(TCSA)

ê~kr)(TBA)

0.002 0.004 0.006 0.008 0.01 0.02 0.04 0.06 0.08 0.10 0.20 0.40 0.60 0.80 1.0 2.0 3.0

0.0752268(2) 0.075394(1) 0.075545(1) 0.075686(2) 0.075819(3) 0.07642(1) 0.07746(3) 0.07838(5) 0.07923(8) 0.0800(1) 0.0836(3) 0.089(1) 0.093(2) 0.097(3) 0.100(5) 0.11(1) 0.12(3)

0.0752268345108792 0.0753945801472539 0.0755453160397259 0.0756859010467654 0.0758193453289885 0.0764217150729928 0.0774603608837440 0.0783839948041927 0.0792368473659823 0.0800384729420204 0.0835556378326484 0.0891961305246395 0.093/274034212962 0.0974952936573320 0.100677923094200 0.110991634159497 0.116215072382535

0.0747726(2) 0.074604(1) 0.074451(2) 0.074309(2) 0.074173(3) 0.07355(1) 0.07247(3) 0.07149(5) 0.07054(8) 0.0697(1) 0.0656(3) 0.058(1) 0.051(2) 0.045(3) 0.039(5) 0.02(1) 0.01(1)

0.0747726094442890 0.0746037342758520 0.0744514593145803 0.0743089895688543 0.0741733532619985 0.0735561573714244 0.0724726176351653 0.0714879316008110 0.0705605014959597 0.0696723971059167 0.0655775257418039 0.0582619441090885 0.0516210950821149 0.0455165773953252 0.0399201830014164 0.0192440358367457 0.008483787237920

TR. Klassen, E. Melzer

534

/ Spectral flow

presented later, we use the algorithm of ref. [24] to calculate the operator product coefficients of descendant fields up to level 5.) Our results, presented in table 3, are in good agreement with the TBA estimates of the corresponding coefficients, which are also shown in the table. This method can also be applied to the individual a0,, and a1,, when they are finite, namely n 3 in this case. For n = 3 one gets quite good agreement with the TBA estimates, but for n = 4, for instance, going up to level 5 is not enough to allow for reasonable extrapolation to 1 = cc (remarkably however, the differences aç~ a~converge much faster though not monotonically as one sees from table 3). We can also use the TCSA to approximate the gap scaling functions ê~~(r) directly for small to moderately large r; the results are shown in table 4. The excellent agreement of these results with our conjecture for the first gap, as illustrated by table 4, provides further strong support for our proposal. As table 4 shows, the TCSA gives relatively accurate results for ê1(r) up to r 1. In previous studies [6,22,25,27] of theories without UV divergencies, the TCSA (up to the same level 5) has proven to yield good results for the ê,(r), and even e1(r), over a much wider range of r-values. This is perhaps not surprising, since if UV divergencies are present, the ê~’kr)are given as the finite difference of quantities which ultimately diverge, and so they would not be expected to converge rapidly, except for small coupling (= r). —

—

—

4.2. INFRARED BEHAVIOUR

In ~ the lowest excitation should (cf. sect. 1) become degenerate with the ground state in infinite volume, the splitting decaying exponentially with the volume R. Indeed, from eq. (4.2) one can show that for large r and small 0 4, r~), (4.10) = r(1 + cosh 0) + ~ln(81Tr) + 0(02, r0 which implies e_2r e~~(r) = -~-—-~-(1 + O(r_i))

(4.11)

for large r. Note that the ground state energy, for which we find e~(r)

=

-

~ 2K1(r)

+

-

+

+

12~r +

0(r3/2)),

(4.12) falls off much more slowly (the leading term here, involving the Bessel function K1(r), was given in ref. [161).

T.R. K/assen, E. Meizer

/

Spectral flow

535

The large-r behaviour of e~(r)is more interesting. Based on perturbative RG calculations [4,5] for the 4~3-induced flow from .%‘~, to one expects 4,2,2 in to flow to 4,2,2 in ~ Although these calculations are reliable only for p>> 1, the Landau—Ginzburg representation [5,37] of the models 4’A~~,where 4,2,2 is identified with the fundamental field in the lagrangian, leads one to expect that this is also true for small p, and in particular p = 4 corresponding to the flow from the TIM to the CIM. If our conjecture for e~kr) is to agree with this expectation, we should have ~

e~1cc~ 1 ‘~ ) — —

td 2,2 t,

—

—i--c~ 12

— —

1dM

4 13

12 ‘

as d22 = ~ in the CIM. We should furthermore find agreement between the next two terms in the asymptotic expansion of e~kr) obtained from eqs. (4.1)—(4.2) and the JR CPT expansion based on (3.2), as was observed [16] in the analogous analysis of e~(r). To obtain the asymptotic expansion of our e~(r) we rewrite eqs. (4.1)—(4.2), 6, in the using ~)( —0) = e~k0) e(0) and changing variables from 0 to ~= ~r e form e~(r)

=

—

d~ln(1

(1/21r2)f

4~ ~d~’

—

e~),

(4.14)

2

ln(1-e~)

=11+ ~-~e~(r)~

—

~

(~

41

fS(~f)2k

1)k

k—i

d1’

2k±1

r

0

ln(1

—

e~”~).

(4.15)

IT

To get the last expression we expanded the square in the integrand on the 2)2,tbrackets he leading term giving rise to first linethrough of eq. eq. (4.15) in Note powersthat of the (4~~’/r e~~(r) (4.14). sum on the r.h.s. of (4.15) is expected to be only an asymptotic series. Iteratively solving for e~~~(r) by substituting eq. (4.15), with the sum truncated to km~= 0, 2,..., into eq. (4.14) one gets the first 3, 7,... coefficients, respectively, in the asymptotic expansion e~~(t)=

~

b 1

n=0

,,t”,

(4.16)

/

TR. Klassen, E. Meizer

536

Spectral flow

TABLE 5 1, obtained by solving the corresponding Large-r behaviour of the scaling functions e( 0~(r) in 4’A~~ TBA equations numerically. The error in the last digit given for e~4j~(r) is about 2

r

e~,~(r)

1.0 2.0 4.0 6.0 8.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 100.0 110.0 120.0 130.0 140.0 150.0 160.0 170.0 180.0 190.0 200.0

where

t

=

e~’~(r)

—0.05 14997811244503 — 0.0473446948808 103 — 0.0439646731976788 — 0.0428203393378089 — 0.0423400617158524 — 0.0421028450008550 — 0.0417761516333321 — 0.0417152458836187 — 0.0416939704704940 — 0.0416841338833664 —0.04 16787938266075 — 0.041675575 1213585 — 0.0416734865486325 — 0.0416720548577304 — 0.0416710308927720 — 0.0416702733368747 — 0.0416696971892399 — 0.0416692488323849 — 0.0416688930883763 — 0.0416686061009506 — 0.0416683712285591 — 0.0416681765760232 — 0.0416680134580702 — 0.0416678754133261 — 0.041667757554395 1

0.0491781419697495 0.0636469392786868 0.0751419329994049 0.0790966166422270 0.0808032008794417 0.0816678444690990 0.0829017247075258 0.0831403246699290 0.0832245384032651 0.0832636378137238 0.0832849086033682 0.0832977449084385 0.08330608045305 18 0.0833117972215017 0.0833158873622887 0.0833189141240443 0.0833212165205417 0.0833230085032403 0.0833244304967346 0.0833255777594846 0.0833265167571212 0.0833272950075385 0.0833279472113958 0.0833284991874764 0.0833289704672963

2ir/3r2. We find for the first seven coefficients

b 10=~, b11= p1,3

5 =

—

~

+

4

2

~IT

U14

b

4, —

7

i

,

b15 = + For the ground state energy, eS~(t)= ~ [16]) — ~

b12=~,

_1!1,

2

32 —

(4.17) 11 + a similar calculation gives (cf. ref. 2IT4.

16

IT

=

=

—

~

b 00 1. U03

U05

— —

—

7

+

=

—

5

—

—

—

192

441

32017’

2

b01

~,

=

49 2 40017’

2883

245

4

—

~i1,

~, ‘

0,4

j, ‘

‘-‘0,6

— —

— —

b0,2

=

7 192

11 128

—

49

2

1oo~7’ +

539 2 l6O~~

723819 9800

4

In table 5 we present results for e~(r) and e~(r) for large r, obtained from a direct numerical solution of the corresponding TBA equations. One can check that

TR. K/assets, E. Melzer

/ Spectra/flow

537

the first seven terms in the small-i’ expansions of e~(t) and e~kt) lead to excellent agreement with the numerical results in this table for the largest r-values given. For smaller r, best agreement is obtained by taking fewer terms, as expected for an asymptotic series. On the other hand, CPT based on the action (3.2) (with all the higher-dimension fields ignored) predicts 2IT

24

“

n—i

2d2z

filE

b1,~=_—~_(__)

Iz~l

3

X ~4,~(cc,cc)~(1,1)fl~(z1,

2~)4,22(0,0))~,

(4.19)

where 2]1

~fT(z, 2)

=

T(z)T(2) —

2422 dT(z)

—

—

24z2 dT(Z)

+

(241d z

2)2

namely I2ITz/R I 4Y(z, 2) is (up to a factor of —4 the determinant of) the energy—momentum tensor on the cylinder expressed in terms of the coordinates of the plane. Here we used the relations g = —(2/irM)2 = —(6R2/IT3)t. The correlators in eq. (4.19) can be calculated explicitly using conformal Ward identities [1]. In general, for any spinless primary field 4, of scaling dimension d in a CFT of central charge d, we find K4,(cc, cc)~(1,

1)4,(0, 0)) = [~(d

1)9’(z, 2)q5(0, ~

K4,(cc, cc),9’(l,

=

(d—~d)

21

_7+ -tz

-

2

d 2 z(1—z)

+

2

d

2(1—z)

21

4

—

(d—~d) —i 4z

(4.20)

.

It follows that considering the action ACFT + gfTT, the leading three coefficients in the expansion e(t) = Eb,,(d, d)t~(where t = —173g/6R2) of the scaling function corresponding to the state associated with 4, in the CFT are

b

0(d, d) = d

—

*d,

b1(d, d)

=

—

12(d

—

~d),

b2(d, d)

=

288(d

—

id).

(4.21) This formula generalizes the case d

=

0 studied in ref.

[15].

538

TR. K/assen, E. Me/zer

/

Spectra/flow

[In evaluating the UV divergent integral leading to b2 we performed analytic continuation in what can be considered as the scaling dimension of TT. This is done using the “generalized beta function” of the complex number field (cf. ref. [38]):

f d~zz

2(l m/225_m/2(l

—

—

_)t_n/2

z)t+n/

T(s+m/2+1)T(t+n/2+1)F(—s—t+(m+n)/2—1) F(—s+m/2)F(—t+n/2)F(s+t+(m+n)/2+2)

—IT

.

(

.

)

Here m and n are integers, and the r.h.s. is the analytic continuation to all complex s, t of the integral on the l.h.s. which converges in the region where Re s> —1, Re t> —1, and Re(s + t) < —1.] One verifies that in our case, where d = ~ and d = the JR CPT result (4.21) agrees with (4.17) obtained from the “TBA approach”. ~-,

5. Generalizations to As a generalization of the discussion in sects. 3 and 4 consider the theories .%‘A~,with p > 4. Recall that these are the 4~ 3-perturbed CFTs .%, of central charge d~= 1 (6/p(p + 1)), so that y = 2 d13 = 4/(p + 1) *~ For the ground state, Al. Zamolodchikov proposed [15,16] the TBA system (2.3)—(2.4) with N = p 2, all ta = 1, K(0) = J(N)/cosh 0, and —

—

—

v~(0)

=

~e°,

v~’~(0) =cosh 0,

v~*)(0)=0

for

a =2,...,N— 1,

i~’~(0) =0

for

a

t’~~/~(0) =

(5.1)

It is natural now to try and describe excitations in the theories ~ by just changing the type of the TBA system, choosing only the ta differently. However, it turns out that not every choice of type leads to a reasonable conjecture for an excitation energy. As a guide for selecting appropriate choices we used the UV-limit analysis summarized in eqs. (2.8)—(2.13), checking if it yields e(0) = *

In general, the scaling dimension of the (spinless) primary field

4,,,

in .4’~,is given by

2—1 d,,= (r(p+1)— sp) 2p(p+l)

,

i~r~p—1,1~s-~p.

TR. K/assets, E. Melzer / Spectra/flow

539

d ~ with d some scaling dimension in the CFT 4. We discovered two promising families of type choices, on which we elaborate in the following subsections. —

5.1. CONJECTURE FOR

ek(r) 0F4’A

2k±2

Consider the case ta = 1 for all a = 1,..., N = 2k. Eq. (2.4) for %‘A~J~2 is symmetric under the interchange of Ea(0) and EN+1_a(~0). Assuming the solution has this symmetry (numerically we can find such a solution and we believe that it is the unique non-trivial one for all cases we checked) implies that the solutions to eqs. (2.8)—(2.10) (with Nab = ~j~1)) must satisfy Xa ‘~N±i—aand Ya~ YN+1—a’ with y~=y,~= 0 in particular. For 4~A~2,on the other hand, we have the symmetry Ea(0) ,,( —0), resulting in y~’= Ya with y1 = 0. We see therefore that eqs. (2.8)—(2.10) admit solutions Xa =XN+l_a which are identical for both %‘A~J~2, and such that y~=Y~+la of 4’A~J~2 are identical to the Ya of 4’A~~2. The resulting e~~(0), eq. (2.11), will therefore trivially coincide, as required. But even with the above symmetry restrictions, there are several solutions with Xa~ Ya E [0,1]. (This is in contrast to the case of the ground state, where all ta = 1 and there is a unique non-negative solution for the Xa~Y,,, cf. eq. (5.9).) It is easy to see, however, that there is at most one solution with 0
—

~

sin Xa=

and

Xa

X2k+1_a

(a/2)ir

sin

(a/2+1)IT

2k+3 2k+3 2 (k+1)IT sin 2k + 3

for

for a odd. We claim that these ~,

a

Xa

a

even,

(5.2)

together with

JO

for

a

odd

(53

~

for

a

even,

~.

describe the correct UV limit of eq. (2.4). We have checked numerically for the first few k that the solutions YJO) = e~’~°~ of eq. (2.4) for small r indeed develop plateaus of heights given by (5.2) and (5.3) to good accuracy. Using the sum rule for the Rogers dilogarithm

k(k+2)

k

(6/IT2)

~

5f(X 2a~)=

2k

+

(5.4)

/ Spectra/flow

TR. Kiassen, E. Meizer

540

we obtain e~(O) =

2k 2) ~ a-i

(1/2IT

[~‘(x~)

k

5~(Ya)]

=

(

=dk+ik+l

—

~2k+2’

+) (5.5)

where dk+lk±1= k(k + 2)/2(2k + 2)(2k + 3) is the scaling dimension of the field 4,k±i,k+i in the UV CFT “2k±2~ In the UV limit of 4”A~,the field ~mm with 1
—

t’~ ±

—

—

—

2) 2k—i a~i e(O)

=

~~ya)1

[5~(Xa)

=

k+2+ 1) 12(2k

=

dk+ik+i

(1/2IT

—

~d 2k+l,

(5.6) where dk±lk+i = k(k + 2)/2(2k + 1)(2k + 2) is the scaling dimension of the field 4,k+i,k+i ~ A “physical” reason that eq. (5.6) should hold even without having anything to do with the UV limit of .I’A 2k+l will be given in subsect. 5.3. ~

5.2. CONJECTURE FOR e1(r) IN .4’A2k÷2

Now take tk = tk±1 = —1, all other ta with a = 1,..., N = 2k being equal to 1. The TBA equations have the same symmetries as in the cases discussed in subsect. 5.1, and we use the same notation for the r 0 asymptotic values of the functions 1’a(0). Again there are several solutions for the Xa~ Ya~The ones corresponding to the UV limit of eqs. (2.4) are —~

(2k Xa

=

sin

—

2a

2k

+

—

l)IT

.

3 sin sin2k3

(2k

—

2a

2k

+ 3)IT

+~

,

(5.7)

/

TR. Klassen, E. Me/zer

Spectra/flow

541

and {ya}~i

{o,

=

0,1,0,

2k2’

(5.8)

~i’...’~k_2}’

~1’•~~’

where sin

bIT k+1

sin

(b+2)IT k+1 IT

Sifl2k

(5.9)

1

(the last formula gives the unique non-negative solution to the xe-equations in the case N = k 2 with all i’a = 1, cf. ref. [15]). Using the sum rules —

6

k—i

~

x

a~i~(i~a)

k(2k—5) —~(xk)

=

2k+3

(5.10)

and [39] 6

(k—2)(k—1)

k—2

~2’

—

=

(5.11)

,

IT2b 1

l+Xb

k+1

we obtain the UV limit 2)L2k [~*(~a)

e~(O) =

-

~ —

1 5~,(Ya)} = (k + 1)(2k + 3)

-

1 12

(1/2IT

“2,2

1 —

~d2k+2,

where d 22 UV CFT

3/2(2k + 2)(2k + 3) is the scaling dimension of the field 4,2,2 in the Hence we consider the TBA equations (2.3)—(2.4) with N = 2k, tktk+t = —1, i’a = 1 for all other a, and v~~(0), Kab(0) as above to be candidates for describing e~~(r) in 4’A~2~2. =

‘~~2k±2•

5.3. FURTHER CHECKS

Being assured that the equations for the e(r) conjectured in subsects. 5.1 and 5.2 have the right UV limit we will now check various other properties, in particular their behaviour for small and large r. Let us first look at a qualitative property of the e~(r).Namely, since we do not expect a finite-temperature phase transition in the theories .,SfA(,~) in infinite volume, there should be no level crossing involving the ground state scaling function e~kr) in finite volume (we are using here the relation between the

542

T.R. K/assets, E. Melzer

/

Spectra/flow

finite-temperature free energy and the finite-volume ground-state energy, cf. sect. 1). In fact, our conjectured e~ kr) satisfy ±

eS~(r)
(5.13)

in any of the models £A~~2,k 2 (for k 1 e~kr) 0, the only inequality left to verify is e~kr) Oand all a=1,...,N, 0El~l ~‘

=

±

=

=

~)(o) 0. Turning now to the small-r behaviour, we know from the general results of subsect. 2.1 that in £A~~2,N even, the e,(r) have an expansion of the form (2.21) with bulk term coefficients 2/8IT, B~~=0, (5.14) B(~=(_1)Nz~~ for the v~—~k0) given in(5.1). Note that an expansion in powers of r”, y 4/(N + 3) is exactly what is expected from the UV CPT point of view. Using the methods explained earlier we have calculated the coefficients in the UV expansion of the scaling functions in several cases. In fact, we have not only done this for excited states, but also for the e~kr) in ..KAN+ 2 conjectured by Al. Zamolodchikov [15,16], since no checks of the UV expansion were presented for ln 2 r term N> 2. Zamolodchikov argued that for n odd e~kr) contains a B~—~r with coefficient =

B~=

I —

~

—

i\~”°~~ ,

2IT2(N+3)

B~~=

1

—

,

N odd,

(5.15)

2IT2(N+3)

in addition to the standard regular expansion ~ Our results for the coefficients ~ for N 3, 4, 5 are presented in table 6. They are not as accurate as the analogous ones for ~ [15,161,because the CPU times required to get accurate results for e~~(r) increase rapidly with N (the situation is even worse if some ta 1). We see from table 6 that the relation (2.26) is satisfied within numerical accuracy, as required. Furthermore, using the CPT prediction (2.28) =

=

—

T.R. K/assets, E. Melzer / Spectra/flow

543

TABLE 6 t The first few coefficients ~ in the regular part of the small-r expansion of e~—~ ~(r) in (a) 4~A 5~), (b) t 1 .4’A 6~and (c) ..E’A 7~,obtained from the numerical solution of the corresponding TBA equations

n (a) .4’A~5~ ~: 2 3 4 5 6 7 2 3 4 5 6

0.016781684(2) 0.00926790(5) — 0.003626(1) —0.00041(1) 0.00040(2) —0.00011(1)

0.0167816854(3) 0.00926793(2) — 0.0036257(3) 0.000414(4) 0.000412(6) 0.000110)

0.010458395(3) 0.0322795(2) —0.017091(1) 0.00125(2) 0.00038(2)

0.01045840(1) 0.0322794(5) —0.017089(7) —0.00 126(5) 0.00045(8)

0.0070730(3) —0.01771(1) 0.0071(1) 0.0050(5) 0.0010(5)

0.0070727(2) 0.017701(2) 0.00701(3) —0.0055(1)

—

(c) 4’A~7~: 2 3 4 5 6

=

a~

—

1 (b) .4’A 6~:

with y 4/(N estimates * for

a~

+

3), the values of ~

—

0.009(2)

given in the table lead to the following

K

K

0.130234474(2) 0.11334655(2)

,.1A5 .4’A6

0.099267(2)

4’A7,

(5.16)

where we combined the data from 4’A~J~2 and 4’A~J~2. Using these values of K the prediction of CPT for a(~j~ is R0.03227952(2)

..E’A6

~0.017700(1)

4’A7,

(5.17)

in excellent agreement with the values quoted in table 6. (For .4’A5 no CPT prediction is possible, since a03 diverges in this case, even after analytic continua*

2 In r Based on a formal treatment of the diverging CPT coefficient a03, “responsible” for the r term in the small-r expansion of e 0(r) 1~6(2~rY1y(~) in .4’A5, we = 0.13023447336.... arrived at the following Assuming conjecture this isfortrue, the exact CPT value predicts of(using K in eq. this(2.28)) model:~ K == 18 ~(9/~r)4/3y5(~) = 0.016781685345.... It is an intriguing mathematical problem to analytically (dis)prove the last formula directly from the TBA equations.

TR. Kiassen, E. Melzer

544

/

Spectra/flow

TABLE 7 1in the (regular part of the) small-r expansion of e~± kr) and e~1(r), in The first few coefficients a~, 4’A~ 1~, obtained from the numerical solution of the TBA equations

6 n

a~)

a~

1 2 3 4 5

0.0109699(2) 0.005251(6) —0.03 106(7) —0.0164(4) —0.002(3)

—0.0109700(2) 0.005244(7) 0.03096(6) —0.0167(4) —0.001(2)

a~ 0.0321810(4) —

0.00483(1)

0.02920(8) —0.0154(6) —0.001(2) —

—0.032179(2) — 0.00485(3) 0.0293(2) —0.015(1) —0.001(3)

lion in y. In fact, as discussed in subsect. 2.2, for all N ~ 3 does a 03 diverge, and only after analytic continuation in y is it finite for N> 3.) Thus the results summarized in table 6 provide strong support for Al. Zamolodchikov’s conjecture for e~~(r) in ..FA~,for p 5, 6 and 7. For excited states we have done some numerical calculations to check the UV expansion of our conjectures for e~~(r)and e~kr) in ~ The estimated expansion coefficients are shown in table 7. The relation (2.26) is again satisfied within numerical accuracy. Checking the CPT prediction for ~ eq. (2.27), using the value of K given above, we obtain =

a~f~ ±~ =

=

7y2(~)

~

=

(~)~3(~)

±0.010970132(2), =

±0.032181049(6),

(5.18)

±~K(2IT)i~’

in excellent agreement with table 7. For .d’A 2k÷l,k 2, 3, we have also calculated numerically the functions e(r) that passed the prerequisite UV-limit test and so seemed to be promising candidates for the scaling functions of the kth excitation in these models (see end of subsect. 5.1). However, it turns out that the resulting expansion coefficients ~ do not satisfy the required CPT relation (2.26) Another reason one should be suspicious of these e(r) as conjectures for the kth excitation energy even without numerical calculations, is that limit lead4,k,k one into ~2k conclude that the 4,k±i,k-4-i in their JR flows towould the field contrary to conformal field expectations [4,5,371 based on the Landau—Ginzburg picture and large-k RG calculations (e(cc) can be calculated using eqs. (2.22) and (2.23) of subsect. 2.1; see =

*•

~2k±i

*

—

Amusingly, though, within our numerical precision the a~j~ = a~j~ are equal in absolute value to the coefficient (2.27) predicted by CPT when the relevant K values of eq. (5.16) are used (remember that CPT predicts a~j)= — a~,’j~). This feature however does not extend to higher n; in particular ~ I I ~ I for n > 1. So the integral equations for at least one of ‘A1 2~,1are not correct (as we will see next, those for At’A~~~+ are presumably wrong; the case of 4’A~j~~ 1 deserves further investigation).

T.R. K/assen, E. Meizer

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below for some examples which do work). This explains why we restricted ourselves to the case of N even in subsect. 5.1. We now turn to the large-r behaviour of our conjectured e~ kr). Since in infinite volume the vacuum in £A~42 is expected to be (N + 1)-fold degenerate, ±

the finite-volume spectrum should contain N excited levels E~kR) that decay exponentially at large R. It seems reasonable to assume that our conjectured scaling functions e~(r) and e~(r) in ~ correspond to two of these N states, and therefore should decay exponentially as r cc~ Indeed, as already remarked in subsect. 2.1, this general behaviour is obvious from the TBA equations, but we will not pursue the details here. As for the large-r behaviour of e~kr) and e~(r) in ~ consider first the JR limit which is non-trivial in this case (the JR limit of e~~(r) in ~ was discussed in ref. [16] for arbitrary p ~ 4). The limits e~~(cc) and e~7~(cc)are determined using eqs. (2.22)—(2.23). For the kth excitation we immediately note that x1 =X2k 0, whereas the other Xa+i, a 1,... ,2k —2 in eq. (2.23) satisfy the first equation in (2.8) with N 2k 2, relevant for the UV limit of e~~(r)in ..FA2k. This is due to our choice of the ta, and to the fact that the matrix j(N—2) can be obtained from I~ by deleting the latter’s first and last rows and columns. Therefore the ~a’ a 2,... ,2k 1 are equal to the Xa_i of (5.2) with k replaced by k 1. On the other hand, the Ya “Yak Y2k±i-a are the same as in the UV, i.e. given by (5.3) as is. Eq. (2.22) therefore gives (using eq. (5.4)) —~

=

=

=

=

—

—

—

=

k+2 12(2k + 1)

=

(dk+ik+i

—

(5.19)

~d)~2k,.

[More suggestively, the above calculation can be rephrased as follows (see last paragraph of subsect. 5.1): Except for the “decoupled” x1 y1 0, for a 2,. . . ,2k the Xa and Ya are just the Ya—i and Xa_i~ respectively, of the N= 2k—i UV-system (2.8). Since .2’(O) 0, the result (5.19) follows immediately from eqs. (2.22) and (5.6). We emphasize that eq. (5.6) was used in subsect. 5.1 to obtain the UV limit of a function e(r) that eventually fails (see above) to give the scaling function of the kth excitation in ~A2k±i.] This result is in line with the expectation [4,5,37] that in the 4,~3-induced flow 4,k4-1,k+i in the UV CFT flows to the field with from to the field the same Kac labels in the IR CFT’. For the first excitation in ~ a similar calculation, using results of subsect. 5.2, leads to =

=

=

=

~2k-4-2

~2k-4-i

~

‘

1 (k + i)(2k

+

1 1) ———‘d 12 ~ —

2,2

~

520

i2C)~ 2k+,~

consistent with the expectation that

4,2,2

of

~2k4-2

flows to

4,2,2

of

~2k±i~

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T.R. Kiassen, E. Meizer

/ Spectra/flow

Finally consider the behaviour of e~kr)in ~ (where i 0, 1, N/2 for N even but only i 0 for N odd) when r is large but finite. At present we do not have any analytical results based on the TBA equations for N> 2. We can however try to analyze the leading large-r corrections to the IR limit using hints from JR CPT, in the spirit of the treatment of the N 2 case (see subsect. 4.2). In the RG trajectory %, —*4~~, p > 3, the perturbing field 4,1,~is believed [4] to flow to ~ possibly combined with some more irrelevant fields. One may speculate, therefore, that a CPT expansion based on the CFT ~ perturbed by 4~ plus some higher-dimension counterterms (that were argued [15,161 to be spiniess descendants of Ill., and 4,~ in order to ensure the integrability of the perturbation) may reproduce the large-r asymptotics of quantities in .%‘A~.In particular, the above suggests the appearance of certain negative powers ~,,n,,(2 da) of r in large-r expansions of e~(r), where d,, are the scaling dimensions of the perturbing (irrelevant) fields labelled by a, and n,, are non-negative integers. In order to check this idea, we analyzed large-r data of e~(r)in ~ p 5, 6 and e~(r) and e~~kr) in ~ obtained numerically from the corresponding conjectured TBA equations. Although the present precision of our data does not warrant as accurate fittings as in the UV regime (and also because the expansions, whose forms we are not certain about, are expected to be only asymptotic), we can still present the following results with some confidence. (i) For ~ we find =

=

=

=

~

—

=

e~(r)

=

—~

—0.02723(2) ~r2 ln r+0.0173(2) ~r2+

...

for

r>> 1. (5.21)

The appearance of negative integral powers of r, with an r~ term missing, are as expected from a perturbation of .%~ by the field ~ whose RG eigenvalue is y 1. The r2 in r term is also not surprising, since the CPT coefficient a 02 calculated from the action A,4 + g1 J4,~~ diverges even after analytic continuation in y (using eq. (2.28)). CPT based on the pure 4,3 1-perturbation also4’spredicts 4~~S a0,, 0 fortoall~ n odd. This from (2.25), using the fact that in . equivalent which is afollows 7L 2-odd member of the (1, s)-operator-subalgebra of the model, so that all odd n-point functions of 4~ vanish identically. Note that adding a g2TT term to the perturbation will not lead to new negative powers of r in the expansion, because the RG eigenvalue of this operator is y —2 and furthermore it is a ~2-even member of the (1, s)-subalgebra of the model. The above observations make it impossible to perform any JR CPT consistency checks on the numerical values of the expansion coefficients presented in (5.21); we are even unable to extract K1 m I I. =

—

=

=

=

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(ii) In ~ e~~(r)

—

—

0.04796(1) r8”5 + 0.1113(1) r2

e~,~(r)

—

—

0.0196(1) r4”5

=

=

e~(r)

=

+

0.0347(1) r4~ +

—

0.077(2) r~2”5+

+

(5.22)

...,

for r>> 1. Here we can demonstrate rather convincingly that the E,,b~,,r4”5 expansions (whose only first one or two leading terms we see) showing up in e~~(r) originate from a perturbation by g 14,31, of RG eigenvalue y of 4’s; Extracting the corresponding K1 g1 I from b02 we obtain the estimate 0.048755(5), and using it in eqs. (2.29) and (2.27) we arrive at the CPT predictions b03 0.07879(3), b11 0.019498(2), b21 0.034662(3), in good 2 agreeterm, ment with the corresponding terms in (5.22). The appearance of an r which we could identify in the expansion of e~(r), presumably indicates the presence of an additional term g 2TT in the perturbation. However, since till now we do not have sufficiently accurate data could to reliably estimateany theofcoefficients of 2 terms in e~j(r),we not perform the necessary (possibly present) r tests related to this perturbation term. JR CPT consistency =

~,

I

=

K1

—

=

=

=

=

6. Summary and open questions We have provided what we believe is strong analytical and numerical evidence for the correctness of the integral equations we proposed for the exact finite-volume energies of the first and kth excited states in the perturbed minimal models ~A 2k+2, in addition to numerical evidence for the equations conjectured by Al. Zamolodchikov to describe the ground state in ~ In the process we determined rather accurately the relation between the mass scales and the coupling constants of the perturbed CFTs ~ for p 5, 6, 7. The finite-volume energies show explicitly how states flow from the UV CFT 4, to the JR one ~ ~. In particular, our large-volume results provide strong evidence that the JR fixed point is approached asymptotically along the direction defined by the field of at least for p 5, 6 (TT for p 4). Of course, our work should just be considered as a starting point for the investigation of a host of important questions. Besides explicitly proving our conjectures, there is now some hope that similar integral equations exist for arbitrary excitations, at least in the models 4’A~. If so, what is the general pattern? From a mathematical point of view, we think it is quite amazing that the simple-looking integral equations considered in the TBA for the ground state and here for excited states should, for instance, encode information about all (occa=

=

=

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T.R. K/assets, E. Me/zer

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Spectral flow

sionaily regularized!) integrated CFT correlators appearing in the corresponding small-r CPT expansion. There is certainly something to be learned in understanding this from a more conceptual point of view. In addition, of course, one would like to be able to extract the smali-r expansion coefficients analytidally from the integral equations, not just numerically as done so far. It is also desirable to further illuminate the question of renormalization of UV divergencies of CPT (for d ~ 1), by comparison with the manifestly finite smail-r expansions provided by the TBA integral equations. For the large-r expansion of the scaling functions this question is even more accute, since for JR CPT it is at present not even clear how to obtain the higher counterterms to make the large-r CPT expansion of the e,(r) well-defined (as asymptotic series). From the TBA point of view, it remains to be understood analytically how the large-r expansions of the e,( r) in ~ look for p > 4. More detailed numerical studies of the large-r behaviour should also be performed. Looking beyond these somewhat technical issues, there are questions of more general interest for non-perturbative QFT. What are the (non-diagonal) S-matrices of the models £A(J~)(p > 4)? Can one find other factorizable scattering theories involving resonances? Are non-scale-invariant QFTs interpolating between nontrivial CFFs always described by massless scattering theories involving resonances? It would be interesting to study the finite-volume spectrum of such (integrable) theories, corresponding to RG flows between CFTs other than the ones considered in this paper. We thank J.L. Cardy for pointing out ref. [35] to us. This work is supported by the NSF, grants PHY-90-00386 (T.R.K.) and PHY-90-07517 (E.M.).

Note added It has been brought to our attention that CPT was discussed not only in refs. [4,5,32], but also in ref. [40], where the hamiitonian form of CPT is used to study finite-size scaling functions of quantum spin chains.

References [1] A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, NucI. Phys. B241 (1984) 333 [2] J.L. Cardy, P. Ginsparg, in Fields, strings and critical phenomena, ed. E. Brezin and J. Zinn-Justin, Les Houches 1988 (North-Holland, Amsterdam, 1989) [3] J. Polchinksi, NucI. Phys. B303 (1988) 226 [4] A.B. Zamolodchikov, Soy. J. NucI. Phys. 46 (1987) 1090 [5] A.W.W. Ludwig and J.L. Cardy, Nucl. Phys. B285 (1987) 687 16] V.P. Yurov and Al.B. Zamolodchikov, mt. J. Mod. Phys. AS (1990) 3221; Paris preprint ENS-LPS273 (1990)

TR. K/assen, E. Me/zer

[7]A.B. Zamolodchikov, (1989) 1

/

Spectra/flow

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Int. J. Mod. Phys. A3 (1988) 743; Advanced Studies in Pure Mathematics 19

[8] J.L. Cardy and G. Mussardo, Phys. Lett. B225 (1989) 275; P.G.O. Freund, T.R. Klassen and E. Melzer, Phys. Lett. B229 (1989) 243; P. Christe and G. Mussardo, NucI. Phys. B330 (1990) 465; Int. J. Mod. Phys. AS (1990) 4581; G. Sotkov and C.-J. Zhu, Phys. Lett. B229 (1989) 391; V.A. Fateev and A.B. Zamolodchikov, mt. J. Mod. Phys. A5 (1990) 1025; K. Schoutens, NucI. Phys. B344 (1990) 665; P. Fendley, S.D. Mathur, C. Vafa and NP. Warner, NucI. Phys. B348 (1991) 66; A.B. Zamolodchikov, Princeton preprint PUPT-1195 (1990) [9] T.R. Klassen and E. Meizer, NucI. Phys. B338 (1990) 485 [10] A.B. Zamolodchikov, Landau Institute preprint (1989) [11]C.N. Yang and C.P. Yang, J. Math. Phys. 10 (1969) 1115 [12] AI.B. Zamolodchikov, Nucl. Phys. B342 (1990) 695 [13] T.R. Kiassen and E. Melzer, NucI. Phys. B350 (1991) 635 [14]MJ. Martins, Phys. Lett B240 (1990) 404; P. Christie and M.J. Martins, Mod. Phys. Lett. AS (1990) 2189; H. Itoyama and P. Moxhay, Phys. Rev. Lett. 65 (1990) 2102 [15]AI.B. Zamolodchikov, Nuci. Phys. B358 (1991) 497 [16] AI.B. Zamolodchikov, NucI. Phys. B358 (1991) 524 117] D.A. Huse, Phys. Rev. B30 (1984) 3908 [18] G.E. Andrews, R.J. Baxter and P.J. Forrester, J. Stat. Phys. 35 (1984) 193 [19] C. Ahn, D. Bernard and A. LeClair, Nucl. Phys. B346 (1990) 409 and references therein; T. Eguchi and S-K. Yang, Phys. Lett. B235 (1990) 282; N.Yu. Reshetikhin and F.A. Smirnov, Commun. Math. Phys. 131 (1990) 157 [20] D.A. Kastor, E.J. Martinec and S.H. Shenker, Nuci. Phys. B316 (1989) 590 [21] M. LOscher, Commun. Math. Phys. 104 (1986) 177; 105 (1986) 153; in Fields, string and critical phenomena, ed. E. Brezin and J. Zinn-Justin, Les Houches 1988 (North-Holland, Amsterdam, 1989) [22] T.R. Klassen and E. Melzer, NucI. Phys. B362 (1991) 329 [23] M. Lhssig, G. Mussardo and J.L. Cardy, NucI. Phys. B348 (1991) 591 [24] M. Lässig and 0. Mussardo, Hilbert space and structure constants of descendant fields in two-dimensional conformal theories, Santa Barbara preprint UCSBTH-90-40 (1990) [25] M. Lhssig and M.J. Martins, Nuci. Phys. B354 (1991) 666 [26] H.W.J. Bldte, J.L. Cardy and M.P. Nightingale, Phys. Rev. Lett. 56 (1986) 742; I. Affleck, Phys. Rev. Lett. 56 (1986) 746 [27] M.J. Martins, Scaling non-unitary models with degenerate ground state: Thermodynamic Bethe ansatz approach, Santa Barbara preprint UCSBTH-90-73 (1990) [28] L. Lewin, Dilogarithms and associated functions (MacDonald, London, 1958) [29] Al.B. Zamolodchikov, Phys. Lett. B253 (1991) 391 [30] Vl.S. Dotsenko and V.A. Fateev, NucI. Phys. B251 (1985) 691, Phys. Lett. B154 (1985) 291 [31] M. Reed and B. Simon, Methods in modern mathematical physics, Vol. IV, Analysis of operators (Academic Press, New York, 1978) [32] H. Saleur and C. Itzykson, J. Stat. Phys. 48 (1987) 449 [33] R.J. Baxter, Exactly solved models in statistical mechanics (Academic Press, London, 1982) [34] A.E. Ferdinand and M.E. Fisher, Phys. Rev. 185 (1969) 832 [35] A. Klümper and P.A. Pearce, Analytic calculation of scaling dimensions: Tricritical hard squares and critical hard hexagons, Melbourne preprint no. 17-1990 (1990) [36] W.T. Vetterling, S.A. Teukolsky, W.H. Press and B.P. Flannery, Numerical recipes: The art of scientific computing (Cambridge University Press, Cambridge, 1985) [37] A.B. Zamolodchikov, Soy. J. NucI. Phys. 44 (1986) 529 [38] I.M. Gel’fand, MI. Graev and 1.1. Pyatetskii-Shapiro, Representation theory and automorphic functions (Saunders, Philadelphia, 1966); E. Melzer, Int. J. Mod. Phys. A4 (1989) 4877

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/

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[39] A.M. Tsvelick, J. Phys. C18 (1985) 159; AN. Kirillov and N.Yu Reshetikhin, J. Phys. A20 (1987) 1587; AN. Kirillov, J. Soy. Math. 47 (1989) 2450. [40]P. Reinicke, 3. Phys. A20 (1987) 4501; P. Reinicke and T. Vescan, J. Phys. A20 (1987) L653