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hep-th/9708135 ITP-97-080, Minimal Models with Integrable Local Defects Andre LeClair1;a and Andreas W.W. Ludwiga;b a Institute for Theoretical Physics University of California Santa Barbara, CA 93106-4030 b Department of Physics University of California Santa Barbara, CA 93106-4030 We describe a general way of constructing integrable defect theories as perturbations of conformal eld theory by local defect operators. The method relies on folding the system onto a boundary eld theory of twice the central charge. The classi cation of integrable defect theories obtained in this way parallels that of integrable bulk theories which are a perturbation of the tensor product of two conformal eld theories. These include local defect perturbations of all c < 1 minimal models, as well as of the coset theories based on SO(2n), obtained in this way. We discuss in detail the former case of all the Virasoro minimal models. In the Ising case our construction corresponds to having a spin eld as a defect operator; in the folded formulation this is mapped onto an orbifolding of the boundary sine-Gordon theory at 2 =8 = 1=8, or a version of the anisotropic Kondo model. 8/97 1 On leave from Cornell University, Newman Laboratory, Ithaca, NY 14853. 1. Introduction Two dimensional quantum eld theories with impurities or defects have received a great deal of attention in the past few years. In the (1 + 1) quantum context, such theories have many important applications, including `weak links' in in nite s = 1=2 Heisenberg Quantum Spin chains 1], local impurity potentials in interacting 1D electron systems (Quantum Wires) 2] 3], and tunneling point contacts in Fractional Quantum Hall Devices 4]. Notably, for the latter system, the methods of Exact Integrability as applied to impurities and defects have recently proven to be a powerful tool for providing nonperturbative answers to important strongly interacting quantum systems, observed in the Solid State laboratory 5]. In the 2D statistical mechanics context, the simplest example is the 2D Ising model in the full plane, with a defect interaction on the real axis 6] 7] 8]. Generally, a quantum eld theory with defect can be formulated in terms of an action: Sdefect = Z 1<x<1 dxdt Lbulk + Z dt D(0; t); (1:1) where D(0; t) is a eld operator located at the defect at x = 0. The integrability of such theories poses some special problems in comparison with boundary theories on the halfline x > 0 9]. If D is a local operator, then the action (1.1) breaks translation invariance; one thus expects the theory to have non-trivial transmission and re ection at the defect. Unfortunately, the algebraic Yang-Baxter like constraints involving both transmission and re ection have very limited solutions, and generally require the bulk theory to be a free eld theory 10] 8]. In particular, if D de nes an integrable perturbation of the bulk CFT, this does not at all imply that the corresponding defect is integrable as well. This nogo theorem was circumvented in the works 3] 4] by exploiting the following two special features of the bulk theory2 . Namely, if the bulk theory is a massless conformal eld theory (CFT), the defect theory can be folded onto a boundary eld theory with twice the central charge (the tensor product of two identical copies of the original CFT), on the half-line. Secondly, when the bulk theory consists of a free scalar eld, then the folded boundary theory consists of an even and odd combination of the original scalar eld and the odd combination decouples from the boundary. ( Thus, in this case, only half the central charge of the tensor product couples to the defect after folding, giving a boundary sine-Gordon model.) A larger class of theories that can be treated this way was 2 consisting there of free massless scalar elds 1 studied in 11], and can be thought of as corresponding to defect operators that are purely chiral (left or right-moving); in this situation the theory is purely transmitting in the defect formulation, and the transmission S-matrices can be mapped onto the re ection S-matrices of the boundary formulation. (In some cases, this map requires the introduction of defect degrees of freedom.) In this paper we consider a general situation where the bulk theory is a CFT, and the defect operator D is local (having both, left and right moving factors). By folding the system, we show that the class of integrable defect theories of this type is in one-toone correspondence with integrable bulk perturbations of two copies of the CFT. A large class of such integrable bulk perturbations was identi ed in 12]. The resulting integrable theories include defects in minimal models and in coset theories3 based on SO(2n). As opposed to a defect in a theory of free scalar elds (as in 3], 4]), the full central charge of the tensor product of the original CFT couples to the defect after folding in this general situation. In this paper we focus in more detail on the case when the bulk is a c < 1 minimal unitary CFT and the defect operator is the primary eld ` 1;2' or ` 2;1'. In the Ising case this corresponds to taking the spin eld or the energy operator, respectively, as the defect perturbation D. For the spin perturbation, this corresponds to a line of magnetic eld in the bulk of the sample. The case of the energy perturbation corresponds to a free eld theory4 . In contrast, as described below, the spin perturbation cannot be solved in the free fermion basis. Rather, it is related to the sine-Gordon theory at 2 =8 = 1=8, and a version of the anisotropic Kondo model, where these two cases correspond, as explained below in more detail, to di erent choices of boundary conditions in the ultra-violet (`continuous Neumann' and `continuous Dirichlet' 13]). We end this introduction by discussing a general conceptual aspect of integrable massless renormalization group (RG) ows of defect theories and of their corresponding boundary theories, obtained after folding. In general, a massless ow between two conformally invariant boundary conditions on a given CFT in the bulk is characterized by the following data: (i) the bulk CFT, (ii) the particular conformally invariant boundary condition chosen on this CFT before perturbation ( i.e. the ultra-violet limit of the ow), and (iii) a particular relevant boundary operator chosen to perturb this boundary condition. Since a given authors (see e.g. 6], 7], 13]), and the case of a massive bulk was considered in 8]. 3 These range from a c = 1 orbifold, to the level-one current algebra with c = n. 4 For a massless bulk, this perturbation is exactly marginal and has been studied by many 2 bulk CFT may in general have a large number of conformally invariant boundary conditions, several ows may be possible5 (of course, always consistent with the `g-theorem' 14]). If two or more of these ows are integrable, then, there must exist di erent re ection matrices, satisfying the bulk-boundary Yang-Baxter equations with the same bulk S-matrix, corresponding to all the possible integrable boundary ows. A complete classi cation of all these solutions is an open problem. However, the case of a spin- eld defect in an Ising model, analyzed in section 4, is precisely an example of this non-trivial situation (perhaps the rst to be understood completely). In this case we identify two integrable ows, connecting boundary conditions with di erent ratios gUV =gIR of `ground state degeneracies' in the ultraviolet and the infrared . We nd two di erent re ection matrices, one related to that of the boundary sine-Gordon model, the other related to the anisotropic Kondo model. The paper is organized as follows: in Section 2 we brie y review basic ideas of integrability in the bulk and at the boundary, and discuss the folding procedure as applied to general defect theories in CFT's. Then we establish that the class of integrable defect theories is in one-to-one correspondence with integrable bulk perturbations of the tensor product of two copies a CFT. In section 3 we apply the general results of section 2 to the special case of defect perturbations of (Virasoro) minimal models. In section 4, we work out in detail the Ising case (the lowest minimal model). In particular, we obtain the bulk S-matrices, as well as the boundary re ection matrices for two di erent integrable massless boundary ows, and verify that those satisfy the bulk-boundary bootstrap, and give the correct values of the boundary entropies. 2. General Aspects of Integrability In this section we outline a general strategy for constructing integrable local defect theories. We rst review some features about bulk and boundary integrability that we will need. possible conformally inv. boundary conditions on a free scalar eld, von Neumann and Dirichlet. 5 In the boundary sine-Gordon model there is only a single boundary ow, connecting the only 3 2.1. Bulk and Boundary Integrability One can de ne integrable bulk theories as suitable perturbations of conformal eld theory (CFT) 15]: Z CFT + Sbulk = Sbulk dxdt O(x; t): (2:1) CFT Here, Sbulk denotes a formal action for a speci c CFT in the bulk, and O is a suitably chosen local perturbation making (2.1) integrable. For many in nite classes of CFT, the integrable perturbations O are known. For example, for the c < 1 minimal models, O can be the primary eld 1;3; 1;2 or 2;1. As usual, in Euclidean space let z = t + ix, z = t ix. The integrability of (2.1) implies there are an in nite number of conserved currents JL (z); JR (z), which are chiral in the conformal limit and in the perturbed theory satisfy @z JL = @z H; @z JR = @z H; (2:2) for some H; H. The local operator O(x; t) can be factorized into left and right moving components in the CFT: (2:3) O(z; z) = OL (z)OR (z): The conservation laws (2.2) are a consequence of the fact that for each JL (z) the residue of the operator product expansion of JL(z) with OL (w) is a total derivative, and similarly for JR and OR 15]. We now turn to boundary theories on the half line x 0, with a boundary interaction at x = 0. In the conformal limit, the CFT must come equipped with a conformally invariant boundary condition satisfying TL (z) = TR (z) at x = 0, where TL; TR are the left and right moving energy momentum tensors. This implies that at x = 0, the left and right moving operators are identi ed. We make the following claim: If the bulk theory (2.1) is integrable, then the boundary theory de ned by: Z CFT + Sboundary = Sbound dt O(L) (0; t) (2:4) CFT is also integrable.6 (Here, Sbound denotes the formal action of the bulk CFT including the conformally inv. boundary condition.) The reason is that the properties of the operator 6 We distinguish between OL on the in nite plane and its boundary counterpart O(L) . 4 product expansion of JL (z) with OL described above, which ensure integrability of the bulk theory, also ensure that in the boundary theory one has J (L) (0; t) J (R) (0; t) = i@t ; Z1 0 (2:5) for some , and this implies that a conserved charge Q can be constructed from J (L) ; J (R): Q= dx J (L) + J (R) + : (2:6) Using @z J (L) = @z J (R) = 0, one easily sees that @t Q = 0. One can prove (2.5) using conformal perturbation theory techniques outlined in 9]. 2.2. Defect Theories We now describe how to use the above facts to construct integrable local defect theories. Consider a defect theory on the full line 1 x 1 with a defect at x = 0. This can be formulated as a perturbation of a CFT: CFT Sdefect = Sdefect + Z dt D(0; t) (2:7) CFT Here Sdefect denotes a bulk CFT equipped with a conformally inv. b.c. at the location of the defect, and D(0; t) is an allowed operator at this defect. In the simplest case, discussed CFT CFT below, Sdefect = Sbulk is a bulk CFT without any defect at all7. In this case the perturbing eld D(0; t) is an operator of the bulk theory, placed at the location of the defect. We now fold the defect theory onto a boundary theory on the half line x 0. In general, any bulk eld can be decomposed into its components ( ) on either side of the defect: (x; t) = (+) (x; t) (x) + ( ) (x; t) ( x): (2:8) Let L (z) and R (z) denote any left and right-moving elds, respectively, in the defect ( ( CFT, and L ) (z), R ) (z) their components for each side of the defect. From these we de ne four boundary elds in the region x 0: (L) 1 (x; t) = (L) 2 (x; t) = (+) L (x; t); R ( )( x; t); 5 (R) () 1 (x; t) = L ( x; t) (R) (+) 2 (x; t) = R (x; t): (2:9) 7 named \periodic b.c. " in 3] L L R (1) fold R R L (2) defect boundary Figure 1. Graphical Representation of Folding (L) (R) The boundary elds 1;2 are functions of z = t+ix, whereas 1;2 are functions of z. This folding is represented graphically in gure 1. For the energy momentum tensor, the defect conformal boundary condition is ( (+) TL ) (0; t) = TL (0; t); ( (+) TR ) (0; t) = TR (0; t): (2:10) These imply the conformal boundary conditions T1(L) (z) = T1(R) (z); T2(L) (z) = T2(R) (z); (x = 0); (2:11) individually, on the two copies. In the defect CFT we can factorize the perturbing eld D = DL DR : (2:12) ( ( To fold the theory, we let D = (D(+) + D( ) )=2, with D( ) = DL ) DR ) , and use the map (2.9). One obtains: CFT1 Sboundary = Sbound CFT2 + Z 2 (L) (R) (R) (L) dt D1 D2 + D1 D2 ; (2:13) CFT1 CFT2 where Sbound, Sbound denote the two copies of the original CFT, with boundary conditions (2.11). We further assume that the boundary condition (2.11),(2.9),(2.8) allows us to 6 (L) (R) (L) (R) identify D1 = D1 ; D2 = D2 on the boundary; this is expected to be true up to some possible co-cycles. One then obtains CFT1 Sboundary = Sbound CFT2 + Z (L) (L) dt D1 D2 ; (2:14) The boundary thus couples the two copies of the CFT. More generally, one may consider the unperturbed theory ( = 0) in (2.14) to be equipped with any conformally inv. b.c. on the tensor product of the two copies of the CFT. Tracing back the steps, this de nes the more general case8 of a conformally inv. CFT defect theory Sdefect in (2.7). Based on the discussion in section 2.1, we can make the following statement. Let C denote the (bulk) CFT of the defect theory in (2.7). Then the defect theory (2.7) is integrable if the following bulk perturbation of two copies of C is integrable: CC Sbulk = Sbulk + Z dxdt D1 D2 ; (2:15) where D1;2 are the local elds D from copies 1; 2 of C. In the situation when D is purely chiral: D = DL , or DR = 1, copy 2 of C decouples from the boundary (2.14). Thus, in this situation if DL DR de nes an integrable perturbation of one copy of C, the defect theory is integrable. This is the situation studied in 16], and implicit in the works 3] 4]. In the more interesting situation where D is local as in (2.12), the above requirements are extremely restrictive since they require known integrable perturbations of two copies of a CFT. Note in particular that if D de nes an integrable bulk perturbation of one copy of C, then this does not at all ensure that the defect version is also integrable. Nevertheless, there are large classes of such integrable perturbations. A rst example was provided by Vaysburd who showed that two coupled minimal models are integrable 17]. A more general, and systematic scheme for coupling two (or more) copies of a conformal eld theory in a way that leads to an integrable theory was described in 12], and is based on cutting and pasting of Dynkin diagrams for the associated a ne Toda theories. This procedure leads to a large number of new and highly non-trivial massless integrable ows in defect theories. We nish this section by describing two examples which are limiting cases of the integrable defect perturbations of all the minimal models considered in the next sections. below, is an exampe of this situation. 8 The perturbation of the `continuous Neumann' boundary condition in the Ising case discussed 7 Ising model in a defect magnetic eld. This model is de ned by the action Z Ising + Sdefect = Sdefect dt (0; t); (2:16) where is the local spin eld of dimension 1=8. Two copies of Ising is a c = 1 orbifold at the radius R = 1. An integrable perturbation of a scalar eld is the sine-Gordon theory. Thus we expect the boundary version of this theory to be related to the boundary sineGordon theory at 2 =8 = 1=8 since it is at this coupling that the boundary perturbation has dimension 1=8. We will consider this theory in detail below. SU(2) Current Algebra at level 1. This model is de ned by Z X (L) (R) Sdefect = Sk=1 + dt m m; m= 1=2 (2:17) (L) where Sk=1 is the SU(2) WZW model at level 1 and m is the primary eld in the spinor representation with scaling dimension 1=4. Here c = 1. This current algebra can be bosonized, thus the integrability follows from the usual folding of free bosonic elds. The folded c = 2 theory is the SO(4) level-one current algebra, and can thus be formulated as 4 real free fermions. Since the dimension of the perturbation is 1=2, the boundary version of this model is related to the boundary sine-Gordon theory at the free fermion point. 3. Defect Perturbations of Minimal Models 3.1. The Models We now apply the ideas of the last section to defect perturbations of the c < 1 minimal series of unitary CFT. We let Ck denote the k-th minimal model with 6 (3:1) ck = 1 (k + 2)(k + 3) ; k = 1; 2; ::: In Ck there exists the local primary elds 1;2 and e 2;1 , with the scaling dimensions 1 3 dim ( ) = 2 = 2 4 1 k + 3 (3:2) 1 1+ 3 : dim (e) = 2 e = 2 4 k+2 (Here, dim refers to the sum of the left and right conformal dimensions.) We de ne two defect theories, denoted Dk and De which are defect perturbations of the minimal models k by the above operators: Z Ck + Sdefect = Sdefect dt (0; t); (3:3) and similarly with ! e. 8 3.2. Integrability Upon folding, the defect theories Dk become boundary theories, which we will denote Bk , with the action Z (L) (L) Ck Ck + Sboundary = Sbound dt 1 2 : (3:4) All of the statements of this section apply with ! e, and are implied. The arguments of the previous section indicate that Dk are integrable if the bulk theories which are de ned as bulk perturbations of Ck Ck by O = 1 2 are integrable. We will refer to these bulk theories as Mk . Remarkably, the bulk Mk theories are in fact integrable 17] 12]. One way to explain this is as follows. The Ck minimal model can be formulated as an SU(2) coset: Ck = SU(2)k SU(2)1 ; SU(2) k+1 (3:5) where SU(2)k is the WZW model at level k. Now we use the fact that SU(2)k SU(2)k = SO(4)k : This implies (3:6) (3:7) As explained by Vaysburd, there is an unconventional way in which to a nize SO(4), extending the Dynkin diagram by the highest weight of the vector representation rather than adjoint, leading to the twisted a ne algebra d(2) .9 The spectrum and S-matrices of 3 the bulk theories Mk can be obtained as RSOS restrictions of the dual c(1) a ne Toda 2 (2) , as was done in 12]. theory which has quantum a ne symmetry q d3 The limiting cases are the Ising model in a defect magnetic eld (2.16), which occurs at k = 1, and the current algebra with defect (2.17), occurring at k = 1. It was shown in 17] 12] that the bulk theory Mk=1 is equivalent to 4 real massive free fermions, and this is consistent with our previous remarks for the model (2.17). To solve the folded defect theories Bk , one must start with the bulk spectrum of particles that diagonalizes the boundary interaction; this spectrum is dictated by the bulk theory Mk . Given this spectrum and the bulk massless S-matrices, one then nds boundary re ection S-matrices that are consistent with the algebraic constraints described 9 One has the identi cation a(2) = d(2) . 3 3 Ck Ck = SO(4)k SO(4)1 : SO(4) k+1 9 in the next section. Alternatively one can think of solving the bulk massive theory Mk with the boundary interaction of Bk , and then taking the bulk massless limit ! 0, as was done for sine-Gordon in 18]; this is more complicated however since the boundary Yang-Baxter equation is more complicated in the massive versus massless case. The bulk S-matrices for the models Mk are mostly known 17] 12], and in general have an RSOS form. In the next section, we work out the Ising case where the bulk S-matrix is diagonal. 4. Ising Case In this section we work out the Ising case at k = 1. The defect problem is described by the action (2.16). By the arguments of the last section, we must rst consider the folded boundary theory with the action Ising1 Sbound = Sbound Ising2 + Z x0 dxdt 1 2+ Z 2 dt (L) (L) 1 2; (4:1) where the subscripts refer to copies 1; 2 of the Ising CFT. As explained above, the original defect theory corresponds to massless bulk term = 0, and the presence of only serves to determine the bulk spectrum which diagonalizes the boundary interaction. It is important to realize that a theory is not completely de ned by the action (4.1); the theory is only completely speci ed once the boundary CFT in the ultra-violet is equipped with a conformal boundary condition. The Ising Ising CFT is equivalent to a c = 1 orbifold at radius 1. The possible conformal boundary conditions for the orbifold are richer than for the non-orbifolded theory and were classi ed in 13]. There it was shown that the possible boundary conditions are `continuous Neumann' and `continuous Dirichlet' depending on the continuous parameters e0 and 0 respectively, as well as tensor products of the known free and xed Ising boundary conditions. For a generic non-orbifolded scalar eld , on the other hand, the only possible conformal boundary conditions are Neumann and Dirichlet and the dependence on zero modes 0 ; e0 can be removed by a shift in ; for the orbifold case this is not possible because of the identi cation . The Ising case has additional complexity due to the existence of exactly marginal bulk and boundary directions which do not exist for higher k. Namely, consider adding to the action (4.1) the terms S= 0 Z x0 dxdt "1 "2 + 0 10 Z dt "(L) "(L) ; 12 (4:2) where " is the energy operator with scaling dimension 1. (In terms of Majorana fermions, " = L R .) Both 0 and 0 have scaling dimension zero, and are actually completely marginal. The parameter 0 corresponds to moving along the Ashkin-Teller line of bulk xed points, and corresponds to a modi cation of the sine-Gordon coupling below. The parameter 0 can be shown 13] to correspond to the parameters 0 and e0 of the `continuous Dirichlet' and the `continuous Neumann' boundary conditions. The dimension of the defect operator (coupling , in (4.1)) varies continuously with these latter parameters. Henceforth we assume that 0 and 0 are zero and consequently the dimension of the defect operator is 1=8. Equivalently, the parameters 0 and e0 are taken to be xed to the values corresponding to 0 = 0. A particular scattering theory describes a ow between two conformal boundary conditions. One expects that in the infrared when ! 1 the two copies of Ising each have a xed boundary condition, i.e. in the infra-red the boundary condition is xed- xed. Below we will propose two scattering theories which possess either the `continuous Neumann' or the `continuous Dirichlet' boundary condition in the ultra-violet. We rst describe the bulk theory and the constraints on boundary massless scattering. 4.1. Bulk Theory The bulk theory is a special case of the coupled minimal models studied in 17] 12]. The bulk massive S-matrices are the same as for the sine-Gordon theory at 2 =8 = 1=8,10 up to some minus signs in the soliton sector 12]. These signs can be traced to the fact that Ising Ising is an orbifold CFT of a scalar eld 19]. The sine-Gordon theory at this coupling is described by the bulk action 1 Z dxdt 1 (@ )2 + cos ( =2) : (4:3) S=4 2 The spectrum consists of two solitonic particles s1; s2, and 6 breathers with mass ratios a = 1; 2; ::; 6: (4:4) ma = 2ms sin a ; 14 In contrast to the sine-Gordon case where the solitons s1; s2 carry U(1) charge 1, here the solitons are not charge conjugates of each other, but rather are their own anti-particle. The bulk S-matrices are Ss1s1 = Ss2 s2 = Ss1 s2 = F1 ( )F2 ( )F3 ( ); 7 7 7 2 (4:5) 10 The coupling =4 . is normalized in the conventional way where the free fermion point occurs at 11 where tanh 1 ( + i ) 2 ; tanh 1 ( i ) 2 and is the rapidity E = m cosh ; P = m sinh . For the breathers one has F ( )= 2 (4:6) Sab ( ) = ja 14 bj 4 k=1 Sas1 ( ) = Sas2 ( ) = ( 1) min(a;b) 1 Y ja bj + 2k 14 32 5 a+b 14 a1 aY 7 k=0 a + 2k ; 14 (4:7) where ( ) F ( ). The structure of bound states also di ers from the sine-Gordon theory; in 12] this structure was obtained from the restricted q d(2) symmetry. For our problem only the even 3 breathers (2; 4; 6) are s1 s1 or s2 s2 bound states. This is to be compared with the sine-Gordon case where all breathers are s1 s2 bound states. 4.2. Algebraic constraints on massless boundary scattering We will need the the algebraic constraints on the massless boundary scattering matrices. These can be obtained by an appropriate massless limit of the equations in 9], as we now describe. b In the massive case the boundary re ection S-matrices Ra satisfy the crossing and unitarity constraints: c b b Ra( )Rc( ) = a (4:8) ab K ab( ) = Sa0 b0(2 )K b0a0 ( ); b where K ab ( ) = Ra (i =2 ). The massless limit can be taken by replacing ! + , and letting ! 1 and m ! 0 keeping me =2 = held xed. This gives the dispersion relation for right movers Ea = a e , Pa = a e , where the a have the same ratios as the bulk masses(4.4). To obtain left-movers, one lets ! , and takes the same limits b leading to Ea = a e , Pa = a e . Thus, given some re ection S-matrices Ra in a massive theory, we de ne the massless scattering matrices as follows: b ea Rb ( ) = !1;m!0 Ra ( + ); lim b Rb ( ) = !1;m!0 Ra ( lim ); a for right movers for left movers: (4:9) 12 The right hand sides of (4.9) depend on B for right-movers, and + B for left-movers, where e B is de ned as a physical boundary energy scale, as described in 18]. The bulk S-matrix in the massless limit becomes an S-matrix SLL (SRR ) for left (right) movers, both of which are the same as (4.7). We will also need the `braiding' matrix: cd Bab = ! 1 Sab ( ): lim cd (4:10) In general B is a solution of the braiding relations. The two equations (4.8) become b Rc ( )Rb ( ) = a a ec ab c eb Ra (i =2 ) = Bcd Rd (i =2 + ) : (4:11) These can be combined into a single equation for R: eb Bcd Re ( a b i =2) Rc ( + i =2) = a : d ab ab Bcd = Bab c d ; b i =2) Rc ( + i =2) = a : b (4:12) (4:13) (4:14) For diagonal bulk scattering, one has X Another constraint comes from the boundary bootstrap. If the particle of type c is a bound state of particles a; b then the bulk S-matrix has a pole at iuc : ab a0 Sabb0 ( ) c fab fca0 b0 : i iuc ab c Bcb Rc ( a (4:15) The fusing angles satisfy uc + ua + ub = 2 . Taking the massless limit of the boundary ac ab bc bootstrap equation in 9] one obtains b fcabRc ( ) = fd 1 a1 Ra2 ( + iub )Rb2 ( ac b d a1 ba iua )Bb12a2 ; bc (4:16) where u = u. For diagonal bulk scattering, b fcab Rc ( ) = fd 0 a0 Ra0 ( + iub )Rb0 ( a d ac b 13 iua )Bb0a : bc (4:17) A subset of the fusing angles we used to check our solutions below are the following: 7; 4.3. Boundary re ection S-matrices u21s1 = 57 ; s u2 = 11 u41s1 = 37 ; u61 s1 = 7 s s u4 = 27 ; u3 = 3 : 22 12 14 (4:18) In our problem the bulk theory has the properties: Bs1s1 = Bs2s2 = Bs1 s2 = 1; s1 = s1; s2 = s2 ; (4:19) whereas for sine-Gordon Bsisj = 1 and s1 = s2 . We now describe two boundary scattering theories that are both consistent with the above constraints. Boundary Sine-Gordon-like Solution In 18] re ection S-matrices for the boundary sine-Gordon (BSG) theory were obtained as the massless limit of the results in 9] 20]. We nd that by modifying some phases in these re ection S-matrices we can continue to satisfy (4.14) with the new conditions (4.19), and also the bootstrap equation(4.17). The result is e 7 =2 Rs2 = Rs1 = 2 cosh(7 =2 i =4) ei 0 Y ( ) s1 s2 Rs1 = Rs2 = 2 cosh(7e =2 i =4) ei Y ( ) s1 s2 1 Y F2l 7 =2 R2k ( ) = ( 1)k R2k 1 ( ) = i( where for the breathers Ra k l=1 14 8 () (4:20) 1)k 1 f 1 2 () k1 Y l=1 F2l14 7 ( ); Ra , and a 3 14 1 ( )f 2 ( ); Y( )=F f() ei = e i 0 = ei =4 (4:21) sinh 1 ( 2 1 sinh 2 ( +i ) : i) (For the boundary sine-Gordon theory one has instead ei 0 = i; ei = 1.) The constraints of crossing-unitarity(4.14) and the bootstrap(4.17), are easily checked using F = f f1 = F1 , f = f +2 , f f = 1, and f ( + i )f ( i ) = f f + . 14 Kondo-like solution sector: Another solution starts from the minimal one in the soliton Closing the boundary bootstrap on the breathers using (4.17) gives 1 Rs1 = Rs2 = if 2 ( ) = tanh 1 ( s1 s2 2 i =2); a=14 ( ): Rs2 = Rs1 = 0: s1 s2 (4:22) (4:23) Ra ( ) = F This is essentially the same as for the anisotropic Kondo model 21] 22], except that in the s1 s2 latter Rs1 = Rs2 = 0:11. 4.4. Boundary Entropy In order to determine the ultraviolet (UV) and infra-red (IR) xed points of the ( massless) ows described by the above scattering theories, we examine the so-called `ground state degeneracies' g 14]. Consider the partition function Z 0 on a cylinder of circumference L and length R, with boundary conditions and 0 at the ends of the cylinder. If one formulates this in a picture where the hamiltonian evolves the system in the direction along the length of the cylinder, then Z 0 = hB je HR jB 0 i; (4:24) where jB ; 0 i are boundary states. In the limit of large ratio R=L, Z 0 = g g 0 ! hB j0ih0jB 0 i; (4:25) where g is the `ground state degeneracy' for the boundary condition . For the Ising Ising theory, it is known 13]that the `continuous Neumann', `continuous Dirichlet', and ` xed- xed' conformal boundary conditions have `ground state degeneracies' p g = 2, g = 1, and g = 1=2 respectively. For the scattering theory the ratio of ultra-violet to infra-red boundary entropies can be computed using the boundary version of the thermodynamic Bethe ansatz (TBA) 18] 23]. If the scattering theory is diagonal, the result is gUV = X 1 Z d @ (log R ) log 1 + e "a ( ) ; log g a IR a2i 15 (4:26) 11 We thank H. Saleur for suggesting this possibility. where "a are the bulk TBA pseudo-energies for particle of type a, satisfying the integral equations in 24]. For our problem, the analysis proceeds much as in 18], where the ow in the boundary entropy was computed for the BSG theory at the re ectionless points. One nds 6 gUV = X I (a) log(1 + 1=x ) + (I (+) + I ( ) ) log(1 + 1=x ); log g (4:27) a IR a=1 where xn are related to the constant ultra-violet values of "n , xn exp("n (1)), and R I (n) = d @ Rn ( )=2 i. The I ( ) and x come from the soliton sector; in the BSG case the boundary scattering is diagonal in the basis ( ) = s1 s2. The xn are bulk properties and are known to be xa = (a + 1)2 1 and x = 7. It is not di cult to show that the e ect of the phase di erences in (4.20) in comparison to the BSG case is merely to distribute the contributions to g from the solitons di erently, but does not modify the sum of the contributions from s1 and s2. Namely, I (+) +I ( ) = 7=2 and I (a) = a=2. Thus for the BSG-like solution(4.20) one has Here, the scattering theory is conjectured to describe the ow between a continuous Dirichlet and xed- xed boundary condition. Further arguments in favor of these conjectures are as follows. Though our bulk theory is an orbifolding of the sine-Gordon theory which modes out the U(1) symmetry by Z2, this does not modify signi cantly the bulk S-matrices, and this suggests that the boundary re ection S-matrices also have U(1) properties that are similar to the BSG case. It is known that for the non-orbifolded BSG theory the Neumann boundary condition breaks the U(1) symmetry and this allows Rs2 6= 0, as in (4.20). On the other hand, the s1 Dirichlet boundary condition preserves the U(1) symmetry, as does (4.22). We end with a discussion of the e ect of the continuous parameter characterizing the `continuous Dirichlet' xed line. As one moves along this continuous line of ultraviolet xed 16 gUV = 8 1=2 = 2p2: (4:28) 2 gIR We thus conjecture that this describes the ow between the continuous Neumann and xed- xed boundary condition. In the Kondo-like solution(4.22)(4.23), one nds instead I (a) = 1, I (s1) = I (s2) = 1=2. This leads to gUV = 2: (4:29) g IR points of the ow, parametrized by '0 , the scaling dimension b ('0 ) of the perturbing boundary operator varies continuously with '0 . This is reminiscent of the situation in the anisotropic Kondo model. Indeed, we propose that the boundary re ection S matrices are those of the anisotropic Kondo model at the appropriate value of , namely12 2 =8 = b ('0 ). Note that the anisotropic Kondo boundary S matrix leads to a ratio of (ground state degeneracy) g-values gUV =gIR = 2, independent of the value of , precisely as required for the ow from the `continuous Dirichlet' xed line to the xed- xed Ising boundary condition. 5. Conclusions We have described a framework for constructing integrable massless quantum eld theories with local defects which generalizes the folding technique for obtaining integrable defect theories, that was previously understood only for free scalar elds. The classi cation of such theories parallels the classi cation of integrable bulk perturbations of two copies of a conformal eld theory, as pursued in 12], and corresponds to a new class of integrable theories, obtained by using an analysis of the (extended) Dynkin diagram of a ne Lie algebras. This approach has allowed us to propose a solution to the problem of an Ising model in a defect magnetic eld. The Ising case extends to all the minimal models of unitary conformal eld theory with magnetic defects. The more general class includes also defects in all coset minimal models based on SO(2n). We would like to thank I. A eck, F. Lesage, G. Mussardo, and H. Saleur for discussions. A.Lec. and A.W.W.L. would like to thank the Institute for Theoretical Physics in Santa Barbara for the opportunity to carry out this work during the program Quantum Field Theory in Low Dimensions: From Condensed Matter to Particle Physics. A.W.W.L. also thanks the Aspen Center for Physics for hospitality. This research is supported in part by the National Science Foundation under grant no. PHY94-07194. A. Lec. is also supported in part by the National Young Investigator Program and A. W. W. L. by the A. P. Sloan Foundation. (4.22), on the other hand, remains unchanged. 12 Equation (4.23)is to be replaced with Ra ( ) = F ag=(2 2g) Acknowledgments ( ), where g = 2 =8 . Equation 17 References 1] 2] 3] 4] 5] 6] 7] 8] 9] 10] 11] 12] 13] 14] 15] 16] 17] 18] 19] 20] 21] 22] 23] 24] I A eck and S Eggert, Phys. Rev. Lett. 75 (1995) 934. C.L. Kane and M.P.A. Fisher, Phys. Rev. B 46 (1992) 15233 E. Wong and I. A eck, Nucl. Phys. B417 (1994) 403. P. Fendley, A. W. W. Ludwig, and H. Saleur, Phys. Rev. Lett. 74 (1995) 3005; Phys. Rev. B52 (1995) 8934; Statphys. 19, p.137 (World Scienti c, 1996). F.P. Milliken, C.P. Umbach and R.A. Webb, Solid State Commun. 97 (1996) 309. R. Bariev, Sov. Phys. JETP 50 (1979) 613. B. McCoy and J. H. H. Perk, Phys. Rev. Lett. 44 (1980) 840. G. Del no, G. Mussardo and P. Simonetti, Nucl. Phys. B432 (1994) 518. S. Ghoshal and A. Zamolodchikov, Int. J. Mod. Phys. A9 (1994) 3841. P. 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