On the renormalisation group for the boundary Truncated Conformal Space Approach
aa r X i v : . [ h e p - t h ] A p r KCL-MTH-11-04
On the renormalisation group for the boundary TruncatedConformal Space Approach
G´erard M. T. Watts ∗ Department of Mathematics, King’s College London,Strand, London WC2R 2LS – UK
Abstract
In this paper we continue the study of the truncated conformal space approachto perturbed boundary conformal field theories. This approach to perturbationtheory suffers from a renormalisation of the coupling constant and a multiplica-tive renormalisation of the Hamiltonian. We show how these two effects can bepredicted by both physical and mathematical arguments and prove that they arecorrect to leading order for all states in the TCSA system. We check these resultsusing the TCSA applied to the tri-critical Ising model and the Yang-Lee model.We also study the TCSA of an irrelevant (non-renormalisable) perturbation andfind that, while the convergence of the coupling constant and energy scales areproblematic, the renormalised and rescaled spectrum remain a very good fit to theexact result, and we find a numerical relationship between the IR and UV cou-plings describing a particular flow. Finally we study the large coupling behaviourof TCSA and show that it accurately encompasses several different fixed points. ∗ Email: [email protected]
Introduction
The Truncated Conformal Space approach (TCSA) of Yurov and Zamolodchikov [1] is awidely-used method to study the finite-size dependence of perturbed two-dimensional con-formal field theories. It is based on truncating the infinite dimensional Hilbert space toa finite-dimensional system on which the Hamiltonian is studied numerically. It has beenknown for a long time that the method has various convergence problems which can reduceits effectiveness [4]. The principal problems are a renormalisation of the coupling constant,a renormalisation of the energy scale and differences between ground state contributions indifferent sectors, all of which depend on the size of the truncated system (we shall always cutoff the size of system by taking all states whose unperturbed energy above the ground state isless than or equal to a given number, which we shall call the truncation level). In a previouspaper we showed how the renormalisation of the coupling constant in perturbed boundaryconformal field theories could be studied using a variant of standard perturbed-conformal fieldtheory methods [12]. In this paper we extend this study to the second effect, and show howthe leading energy scale renormalisation is an overall multiplicative renormalisation whichcan be considered equivalent to a renormalisation of the size of the system. We find ‘physical’arguments based on the operator product expansions and also more rigorous arguments basedon an analysis of the eigenvalues of the perturbed Hamiltonian and show that these give iden-tical results. We test our results using two integrable conformal field theories, the tri-criticalIsing model and the Yang-Lee model, as in both cases the finite-size spectrum has been foundusing TBA methods and this provides an accurate quantitative check of the proposed results.In the next section we briefly review the case of the tri-critical model to illustrate therenormalisation issues to be solved and to find numerical estimates for the coupling renormal-isation and energy rescaling. We then show that these have scaling forms and find numericalestimates for the associated exponents.In the third section we show how these can be derived, to one loop order, from considera-tions of the renormalisation of the perturbed action and in the fourth section show how thesecan be proven, to leading order, for all energy levels, from analysis of the eigenvalues of theperturbed Hamiltonian. In the subsequent two sections we check these predictions againstthe numerical data in the tri-critical Ising model and the Yang-Lee modelIn section 8 we consider the case of an irrelevant (non-renormalisable) perturbation andin section 9 we consider the flows beyond the fixed points and speculate on the exponentsthat have been found numerically. Finally in section 10 we present our conclusions.
We start with a CFT defined on a strip 0 ≤ y ≤ L of width L in the upper half plane withcoordinate z = x + iy . We take the strip to have conformally invariant boundary conditionsso that the system is conformally invariant, the Hilbert space carries an action of the Virasoroalgebra and decomposes into a direct sum of representations of the Virasoro algebra. Therepresentations occurring depend on the boundary conditions on the strip. In this paper we study the original form due to Yurov and Zamolodchikov, not the revised version of [2, 3]. H = Z L T xx d y π . (2.1)We will map the strip to the upper half plane with coordinate w = exp( πz/L ) in terms ofwhich the CFT Hamiltonian is H = πL (cid:16) L − c (cid:17) , (2.2)where L is the zero mode of the Virasoro algebra.We are interested in perturbations by one or more boundary fields φ i ( x ) living on thebottom edge of the strip, y = 0. We take these to be quasi-primary fields of conformaldimension h i . If the coupling to these fields are µ i then the perturbation is given by anaddition to the action δS = Z X i µ i φ i ( x ) d x . (2.3)When mapped to the upper half plane this gives the perturbed Hamiltonian as H = πL " (cid:16) L − c (cid:17) + X i µ i (cid:18) Lπ (cid:19) y i φ i (1) , (2.4)where y i = 1 − h i . Note that (2.4) is only correct if the fields φ i are primary; there are correc-tions if they are quasi-primary but not primary. We will normally consider the dimensionlessoperator H = (cid:18) Lπ (cid:19) H = (cid:16) L − c (cid:17) + X i λ i π y i φ i (1) , (2.5)where λ i = µ i L y are dimensionless coupling constants.The TCSA approach is to restrict this Hamiltonian to a finite dimensional space of ex-citation level n or lower. If we take the projector onto this space to be P n then the TCSAHamiltonians are H n = (cid:16) L − c (cid:17) + X i λ i π y i P n φ i (1) P n . (2.6)These Hamiltonians can be diagonalised numerically and their eigenvalues and eigenstatesform the TCSA approximations to the perturbed system. Any quantity that can be consideredin the perturbed system can also be considered in the TCSA system; the only question is howgood the approximation is, and whether the dependence on the truncation level is either smallor can be estimated efficiently. In the next section we present the two leading truncation effects— the renormalisation of the coupling constant and the rescaling of the energy levels — inthe case of the tri-critical Ising model. We take as our example model example the boundary tri-critical Ising model on a strip.We shall review the boundary conditions and their flows in section 5; for the moment it issufficient to know that, amongst others, there are conformal boundary conditions labelled(11), (21) and (12), and the (12) boundary condition can be perturbed by a field of weight3 / ←− (12) −→ (12) . (2.7)3he spectrum of the Hamiltonian on a strip with an unperturbed (11) and a perturbed(12) boundary condition has also been analysed from the TBA approach by Feverati andcollaborators in [7]. Using the results in [21] and [22, 23], we can relate the TBA and conformalperturbation theory parameters and so we can compare the TCSA spectrum to the exactperturbed conformal field theory spectrum for this system.The most immediate difference is that there is an overall shift in the energy levels. Forfinite perturbations for which there are no divergences in the perturbation expansion thiscorresponds to a measurable free energy per unit length. In this case the perturbation theoryis divergent and the overall shift in the energies is not meaningful: in TCSA it depends onthe truncation level and does not converge to a finite value. As a consequence, in figure 1 wejust show the difference between the ground state and the excited states - the energy gaps:in figure 1(a) we give the gaps as calculated using the TBA method, and in figure 1(b) theTCSA gaps. - - (a) TBA data - - (b) TCSA data from truncation level 14 Figure 1: The energy gaps for the strip with boundary conditions (11) and (12) + λφ (13) asgiven by the TBA and TCSA methods plotted against λ .On inspection of figure 1, one can also discern the two other effects. In the TCSA plot,there is an apparent convergence of many levels at or near λ = 2. This corresponds to a fixedpoint of the flow where the (12) boundary condition has reached its (21) fixed point. Thisis only reached at λ = ∞ in the TBA flow. The two couplings are related by a non-trivialrenormalisation. Less obvious but still measurable is the overall rescaling of the energy levels.At λ = − | λ | .We can find numerical estimates for the renormalisation (of the coupling constant) andrescaling (of the energy scale) by comparing the TCSA and TBA data. Firstly we consideronly the energy gaps to remove the unphysical ground state energy contributions in the TCSA4cheme. Next, since both the overall scale and the value of the coupling are different, weconsider the ratio of two gaps to find a quantity which is independent of the energy rescalingand use this to determine the effective coupling constant. We can then use this in turn tofind the energy rescaling.If we denote the coupling constant in TCSA by λ n and that in the TBA by λ ∞ , and the n th energy gaps by ∆ nn and ∆ n ∞ respectively, then we can calculate f ∞ ( λ ∞ ) = ∆ ∞ ∆ ∞ , f n ( λ n ) = ∆ n ∆ n , (2.8)We can then find the effective (TBA) coupling corresponding to a given TCSA coupling fromthe function g n defined by λ ∞ = f − ∞ ( f n ( λ n )) = λ n g n ( λ n ) , (2.9)and the energy rescaling r n ( λ n ) can then be found from r n ( λ n ) = ∆ n ( λ n )∆ ∞ ( λ ∞ ) . (2.10)In figure 2 we plot these functions for the flows starting from the (12) boundary condition forthree different truncation levels. It is the aim of this paper to show how these two effects canpredicted from physical and mathematical arguments. - - - - (a) The renormalised coupling constant λ n g n ( λ n )plotted against λ n - - (b) The energy rescaling function r n ( λ n ) plottedagainst λ n Figure 2: The numerical coupling constant renormalisation and energy rescaling found forthe tri-critical Ising model at truncation levels 6 (red, solid), 14 (green, dotted) and 22 (blue,dashed). 5
Physical Arguments
In [12], we presented an argument deriving the leading term in the coupling constant renormal-isation based on standard considerations of the partition function and the operator productexpansion of the perturbing field. We repeat this argument here in a little more generality,both as a derivation of the coupling constant renormalisation which we shall check in thenext section, but also to show that this does not in fact give the leading term to the energyrescaling, as might have been thought was the case.We consider a strip geometry of width L with complex coordinate z = x + iy with 0 ≤ y ≤ L . We take a general boundary perturbation of the form X i µ i φ i ( x ) , (3.1)where µ i are the couplings to the boundary fields φ i which have conformal weights h i . Wedefine y i = 1 − h i .We consider a truncation to level n which we can treat using a projection operator P n onto states of excitation level n or below. We can use the projection operator to restrictthe perturbation to the truncated Hilbert space so that the TCSA approximation to theperturbation 3.1 is then given by X i µ i ( P n φ i P n ) . (3.2)The effect of the perturbation is to introduce the following operator P exp − Z ∞ x = −∞ X i µ i ( P n φ i ( x ) P n ) d x ! , (3.3)into all expectation values, where P is path ordering. Mapping the strip to the upper halfplane by w = exp( πz/L ), this operator becomes P exp − Z ∞ x =0 X i λ i ( P n φ i ( x ) P n ) d x ( xπ ) y i ! , (3.4)where λ i = µ i L y i is the dimensionless coupling constant.Expanding this exponential out to second order we get1 − X i λ i Z ∞ x =0 P n φ i ( x ) P n d x ( xπ ) y i + X jk λ j λ k Z ∞ x =0 Z xx ′ =0 P n φ j ( x ) P n φ k ( x ′ ) P n d x ( xπ ) y j d x ′ ( x ′ π ) y k + O ( λ ) . (3.5)We define the TCSA renormalisation group by the requirement that the partition functionis unchanged when the truncation level is increased. This introduces a level dependence intothe coupling constants, which we now denote by λ i ( n ).6ue to translation invariance, we do not need to compare the whole expression in equation3.5, instead we can just compare the integrand with respect to x , at x = 1, which is X i λ i ( n ) π y i P n φ i (1) P n − X jk λ j ( n ) π y j λ k ( n ) π y k Z x ′ =0 P n φ j (1) P n φ k ( x ′ ) P n d x ′ ( x ′ ) y k + O ( λ ) . (3.6)We can further simplify matters by assuming that the Hilbert space contains a vacuum state | i and a state | φ i i corresponding to the field which satisfy h φ i | φ j (1) | i = δ ij , h φ i | φ j (1) φ k ( x ) | i = (1 − x ) x i − x j − x k C ijk . (3.7)Sandwiching equation 3.6 between the states h φ i | and | i , we get λ i ( n ) − X jk λ j ( n ) λ k ( n ) Z x =0 h φ i | φ j (1) P n φ k ( x ) | i ( πx ) y i − y j − y l d x + O ( λ ) . (3.8)This must be invariant under changes in n so that two second order in λ . Using the fact that h φ k | φ i (1)( P n +1 − P n ) φ j ( x ) | i is the coefficient of x n +1 in the expansion of (1 − x ) h i − h j − h k andis equal to Γ( h j + h k − h i + n + 1)Γ( h j + h k − h i )Γ( n + 2) = n y i − y j − y k Γ( h j + h k − h i ) (1 + O (1 /n )) , (3.9)we find the TCSA renormalisation group equations n d λ i d n ≃ n ( λ i ( n + 1) − λ i ( n )) = X jk λ j λ k ( nπ ) y i − y j − y k C ijk Γ( h j + h k − h i ) + O ( λ ) . (3.10) In the case of the perturbation by a single field with self-coupling C , the TCSA renormalisationgroup equation (3.10) becomes n d λ d n = C Γ( h )( nπ ) y λ + O ( λ ) , (3.11)with solution λ ( ∞ ) = λ ( n )1 − Cy Γ( h )( nπ ) y λ ( n ) . (3.12)There are two predictions from this calculation - firstly the general prediction that the couplingrenormalisation has a particular scaling form, namely that the function g n in equation 2.9has the form g n ( x ) = g ( xn − y ) , (3.13)and secondly a particular prediction for the 1-loop behaviour, g ( x ) = (cid:18) − Cxy Γ( h ) π − y (cid:19) − + O ( x , n − y ) (3.14)We test these predictions for the tri-critical Ising model in section 5.1. We will find thatthe functions log( g n ( − xn y )) are indeed almost identical for various values of n and in goodagreement with the prediction, confirming both the prediction of the scaling form and theapproximate numerical expression for this scaling function.7 .3 The energy rescaling We can try to apply the same arguments to derive the energy rescaling, assuming that thistoo comes from the operator product of the perturbing fields giving a contribution to thebare action (corresponding to the kinetic term in a standard Lagrangian theory). The BareHamiltonian is ( π/L )( L − c/ L which is a mode of theenergy-momentum tensor T ( z ). This field appears in the operator product of the boundaryperturbing field as φ ( z ) φ ( w ) ∼ z − w ) h + C ( z − w ) h φ ( w ) + h/c ( z − w ) h − T ( w ) + . . . , (3.15) T ( z ) φ ( w ) ∼ h ( z − w ) φ ( w ) + 1 z − w φ ′ ( w ) + . . . , (3.16)where we have also included the OPE of T with the perturbing field.If we denote the coupling to the perturbing field by λ and to the energy momentum tensoron the boundary by λ T we then have h T = 2, y T = − n d λ d n = C ( nπ ) y Γ( h ) λ + (4 hnπ ) λλ T + . . . , (3.17) n d λ T d n = 2 h/c ( nπ ) y Γ( − y ) λ + . . . . (3.18)To find the leading dependence on the induced coupling λ T we can take λ to be constant andusing λ T ( ∞ ) = 0, find λ T ( n ) = − Z ∞ n d λ T d n d n = − h/c Γ( − y )(1 + 2 y ) λ ( nπ ) y +1 + O ( λ ) . (3.19)The addition of the term Z µ T T ( x )d x , (3.20)to the action will give the following term to the Hamiltonian (after mapping from the stripto the upper half plane) δH = µ T T (1) = πL (cid:18) − h/c Γ( − y )(1 + 2 y ) λ ( nπ ) y n (cid:16) L − c/ (cid:17) + other modes (cid:19) , (3.21)so that the Hamiltonian becomes H + δH = πL (cid:18) − h/c Γ( − y )(1 + 2 y ) λ ( nπ ) y n (cid:19) (cid:16) L − c/ (cid:17) + . . . , (3.22)This gives the rescaling function as r n ( λ ) = 1 − h/c Γ( − y )(1 + 2 y ) λ ( nπ ) y n + . . . . (3.23)8nfortunately, this has the wrong n -dependence - it is one order of n too small. If we plotlog( r n ( λ ) −
1) against log( n ) for fixed λ for the functions shown in figure 1(b), in the tri-criticalIsing model, we find that the prediction for the leading exponentlog( r n ( λ ) −
1) = α log( n ) + . . . , (3.24)is approximately α = − .
85, which is close to − y = − . − − y = − .
8. Asa consequence, we deduce that the coupling of the perturbing fields to the energy-momentumtensor is not the leading source of the correction to the energy rescaling function.Instead we look to the constant ground-state energy contribution coming from the couplingto the identity operator. As we show in the next section, this is actually not a constant becauseof the presence of the projector P n and this can also give a correction proportional to thebare Hamiltonian. In perturbation theory where the perturbing field has a weight greater than 1/2 one normallyignores the ground state energy as it is a divergent unphysical quantity. In this case theTCSA cutoff makes the contribution finite, and further more, the presence of the projector P n means it is not constant.We are interested in the correction to the coupling to the identity operator which ariseswhen the expression 3.6 acts on a state of excitation level E . If we denote this state by | E i and the coupling to the identity by λ ( E, n ), then we are interested in the expression λ ( E, n ) π − λ ( n ) π y Z x ′ =0 h E | φ (1) P n φ ( x ′ ) | E i d x ′ ( x ′ ) y (3.25)Requiring that this be invariant as we change the truncation level n , we getd λ ( E, n )d n ≃ ( λ ( E, n + 1) − λ ( E, n ))= λ ( n ) π y Z x =0 h E | φ (1)( P n +1 − P n ) φ ( x ) | E i d x ( x ) y , (3.26)where we only take the contribution to the OPE on the right hand side that comes from theidentity channel. The projector ( P n +1 − P n ) picks out the state at level n + 1 in the actionof the field φ ( x ) on the state | E i . The action of the field φ ( x ) on the state | E i has a modeexpansion φ ( x ) | E i = X φ m x m − h | E i , (3.27)so that the term at excitation level n is just the coefficient of x n +1 − E . From the operatorproduct expansion (3.15), the coefficient of x n +1 − E in h E | φ (1) φ ( x ) | E i isΓ( n + 1 − E + 2 h )Γ( n + 2 − E )Γ(2 h ) , (3.28)This means that the leading contribution to the RG equation (3.26 for the coupling λ isd λ ( E, n )d n ≃ λ ( n ) π y Z x =0 Γ( n + 1 − E + 2 h )Γ( n + 2 − E )Γ(2 h ) x n +1 − E d x ( x ) y λ π y ( n − E ) h − Γ(2 h ) + . . . ≃ λ ( nπ ) y h ) (cid:18) y En + . . . (cid:19) (3.29)The first term is the constant contribution which we shall now ignore; the second is theexcitation-level dependant term which we are interested in. We can integrate equation (3.29)using the boundary condition λ ( E, ∞ ) = 0, to find λ ( E, n ) | E = − Z ∞ n λ π y h ) 2 yEn − y − d n = λ ( nπ ) y h ) E . (3.30)We can now replace the excitation level E by the operator ( L − c/ E is approximatelythe eigenvalue of ( L − c/ r n ( λ )Firstly, we can see that this coupling will lead directly to a change in the Hamiltonian;the new Hamiltonian is πL (cid:20) (1 + λ ( nπ ) y h ) )( L − c/
24) + λπ y φ (1) (cid:21) , (3.31)giving the energy rescaling function as r n ( λ ) = 1 + λ ( nπ ) y h ) + . . . , r ( x ) = 1 + x π y h ) + . . . . (3.32)This has the correct n dependence (in agreement with the numerical data) and is also a verygood fit to the actual rescaling function as we see in section 5.2.An alternative viewpoint of the energy rescaling function is that it represent an effectivechange in the geometry of the system. Coordinate transformations are implemented in CFTby changes to the action (see e.g. [13]). The coordinate change x µ → α µ + α µ corresponds tothe change in the action δS = − π Z T µν ∂ µ α ν d x . (3.33)The energy-dependant correction to the identity operator we have just calculated can alsobe put in this form. On the upper-half-plane, with complex coordinate w = r exp( iθ ), thecorrection is δS = − π Z λ ( E, n ) d rr = − λ ( nπ ) y Γ(2 h ) Z L d rr . (3.34)Firstly, we note that on the upper half plane L = πL Z πθ =0 ( w T ( w ) + ¯ w ¯ T ( ¯ w )) d θ π . (3.35)Combining equations (3.34) and (3.35) and transforming to the strip with coordinate z = x + iy , the correction to the action becomes δS = − λ ( nπ ) y Γ(2 h ) Z T xx d x d y π , (3.36)10hich in turn corresponds to the change in coordinates (cid:18) xy (cid:19) → (cid:18) x (cid:16) λ ( nπ ) y Γ(2 h ) (cid:17) y (cid:19) = (cid:18) r n ( λ ) xy (cid:19) . (3.37)If we consider the theory on a strip of length R then this change in coordinates is an effectiveincrease in the length by a factor r n ( λ ). To the order in λ to which we are working, this isequivalent to a reduction in the strip width by the same factor and consequently a rescalingof the eigenvalues of the Hamiltonian by the same factor, r n ( λ ), exactly in accordance withequation (3.32). It is also possible to derive the coupling constant renormalisation and energy rescaling byexamining the lowest three eigenvalues of the perturbed, truncated Hamiltonian directly.This gives exact expressions in terms of integrated four-point functions of the conformalfield theory. We can then find the large- n behaviour of these eigenvalues by a saddle-pointapproximation. It turns out that these are given in terms of the crossed four-point functionsand the leading large- n behaviour is given by the leading terms in the crossed four-pointfunctions, which are in turn given by the operator product expansion coefficients. In this waywe see that the “physical” arguments of the previous section are indeed correct. Furthermore,we can show that the corrections we have found are the leading order corrections for all theeigenvalues of the perturbed Hamiltonian, not just the lowest three.We start from the expression (2.6) for the dimensionless TCSA Hamiltonian: H n = ( L − c
24 ) + ˜ λP n φ (1) P n . (4.1)We have introduced ˜ λ = λπ − y for notational convenience.For simplicity we consider a case where the Hilbert space is a single highest weight rep-resentation of the Virasoro algebra with conformal weight H , and that the two lowest levelstates are | ψ i , L − | ψ i . (4.2)We will assume that the the energy-rescaling and coupling constant renormalisation take theforms (3.13) and g ( x ) = 1 + bπ − y x + O ( x , n − y ) , r ( x ) = 1 + aπ − y x + O ( x , n − y ) . (4.3)We also assume that the first energy gap takes the exact form∆ ( λ ) = E ( λ ) − E ( λ ) = 1 + α ˜ λ + β ˜ λ + O ( λ ) , (4.4)so that the TCSA approximation at truncation level n is given by∆ n ( λ ) = r ( λn − y )∆ ( λg ( λn − y ))= 1 + α ˜ λ + ˜ λ ( β + an − y + bαn − y ) + O ( λ , n − y ) . (4.5)11his means that we can find the coefficients a and b appearing in the rescaling and renor-malisation functions (4.3) from the n dependence in the second order term in the first energygap. From the physical arguments of the previous section, we expect a = 1Γ(2 h ) , b = Cy Γ( h ) . (4.6)By standard perturbation theory, we find the following expressions for the ground stateand first excited state energies, to second order: E = H + ˜ λC ′ − (˜ λC ′ ) Z ( F − z h ) d zz y ,E = ( H + 1) + ˜ λC ′ (cid:18) h − h H (cid:19) +(˜ λC ′ ) (cid:20) h H − Z d zz y (cid:18) F − h / (2 H ) z − y − (2 H + h ( h − (2 H ) z − y (cid:19)(cid:21) . (4.7)Here C ′ is the boundary coupling constant in the operator product expansion φ ( x ) | ψ i = C ′ x h | ψ i + . . . , (4.8)and the function F is the four point chiral block F ( z ) = ψ φ ψ φz ψ = z − h (1 + . . . ) , (4.9)that appears in the boundary two-point function on the strip. Since we have chosen boundaryconditions for which the Hilbert space of the model on the strip has only a single representationwith highest weight | ψ i , the boundary two point function is given as the product of thestructure constants and the single chiral block with the representation ψ in the intermediatechannel: h φ (1) φ ( z ) i strip = h ψ | φ (1) φ ( z ) | ψ i = ( C ′ ) F ( z ) . (4.10) F is the function that appears in the two point function of the field φ ( z ) in the first excitedstate, h ψ | L φ (1) φ ( z ) L − | ψ ih ψ | L L − | ψ i = ( C ′ ) F ( z ) ,F ( z ) = 12 H (cid:26)(cid:20) z ( z −
1) dd z + 2 zh − (cid:21) (cid:20) ( z −
1) dd z + 2 h (cid:21) + 2 H (cid:27) F . (4.11)If h , the conformal dimension of the perturbing field, is greater than or equal to 1/2 then theintegrals in the expressions (4.7) diverge, but their difference, the energy gap ∆ , is finite.We can read off the coefficients α and β that appear in (4.4) as α = h ( h − H C ′ , (4.12) β = ( C ′ ) (cid:26) h H − Z dzz y (cid:20)(cid:18) F − h / (2 H ) z − y − (2 H + h ( h − (2 H ) z − y (cid:19) − (cid:18) F − z h (cid:19)(cid:21)(cid:27) (4.13)12hese are the exact coefficients in the perturbative expansions of the energy gap; we areinterested in the n -dependence in the TCSA approximation (4.5). The n -dependence comesfrom replacing the functions F and F in (4.13) by the TCSA truncations. We see that α is independent of n but β does depend on n ; we shall denote the TCSA truncation by β ( n )and we expect from (4.5) that β ( n ) = β + an − y + bαn − y + O ( n − y ) . (4.14)As before, in section 3.4, the TCSA truncations of the functions F and F come fromrestricting their Taylor expansions to include only modes up to power z n − h . It is easierto analyse the change in the functions as the TCSA level is increased than to find the n -dependence directly, so we consider the change in β given by the taking just the coefficient of z n + y in the integrand of (4.13)d β d n ≃ β ( n + 1) − β ( n ) = ( C ′ ) n I d z πiz n +1 h z h ( F − F ) i . (4.15)The contour in (4.15) is a small circle around the origin. The function z h ( F − F ) is singlevalued around the origin but has a cut from z = 1 to z = ∞ , and can be expanded inincreasing powers of (1 − z ) − . We use the result that I d z πiz n +1 (1 − z ) − α = Γ( n + α )Γ( α )Γ( n + 1) = n α − Γ( α ) (1 + O (1 /n ) ) , (4.16)to see that the dependence of the function (4.15) on n is determined by the expansion of theintegrand in powers of (1 − z ). The expansion of F in (1 − z ) is determined by taking thealternative conformal block expansion of the boundary two point function:( C ′ ) F ( z ) = ψφ φψ id + CC ′ ψφ φψφ + . . . = (1 − z ) − h (cid:8) O (1 − z ) (cid:9) + CC ′ (1 − z ) − h { O (1 − z ) } + . . . (4.17)The further terms in (4.17) correspond to further fields in the operator product expansion of φ ( x ) with itself. We assume that these are less singular than the two shown, as is the case forperturbations by the unitary minimal model field φ . Calculating the integral (4.15) we getd β d n = 2( h − h ) n − y − + CC ′ h ( h − H Γ( h ) n − y − + . . . , (4.18)where it is worth noting that the leading term which one would expect to have dependence n h − = n − y has zero coefficient and the term in n − y − is the sub-leading term. We cannow integrate (4.18) to find β ( n ) = β ( ∞ ) − Z ∞ n d β d n d n = β ( ∞ ) + 1Γ(2 h ) n − y + CC ′ h Hy Γ( h ) n − y + . . . . (4.19)13his exactly agrees with the expected form (4.5), so that the rigorous mathematical derivationof the leading n -dependence of the TCSA approximation to the first energy gap agrees withthe heuristic derivation of the coupling constant renormalisation and energy rescaling fromphysical arguments.It is important to note that these effects are independent of the representation ψ chosen,and so, in particular, they should apply to all of the flows considered by Feverati et al.It is in fact possible, by considering the effect of replacing the state L − | ψ i by suitablestates at arbitrary levels, to extend this calculation to cover all energy gaps, not just the firstenergy gap. This means that if ∆ ni ( λ ) is the TCSA approximation to the i -th energy gap,then one can prove that ∆ ni ( λ ) = r ( λn − y )∆ i ( λg ( λn − y )) + O ( λ ) , (4.20)where the functions r and g are those given in (4.3) with coefficients (4.6). As mentioned before, we shall perform most of our checks in the tri-critical Ising modelbecause the excited state spectra of the strip with a perturbed boundary condition has beenstudied in detail using the TBA approach by Feverati et al. [7, 8, 9] allowing us to comparethe TCSA results with the ‘exact’ TBA spectrum. The tri-critical Ising model is a unitaryconformal field theory with central charge 7 /
10 and the Virasoro algebra has six unitaryhighest weight representations listed in table 1. The model has six fundamental, or “Cardy”,conformally invariant boundary conditions [10, 11] listed in table 1. These can be labelledeither by representation of the Virasoro algebra or by the allowed values of the boundaryspins in its realisation as a spin-1 Ising model. The boundary flows were first given by Affleckin [5] and are shown in figure 3.Virasoro label (11) (21) (31) (12) (13) (22)Conformal weight 0
716 32 110 35 380
Boundary spins ( − ) (0) (+) ( −
0) (0+) ( − d )Boundary fields (11) (11) , (31) (11) (11) , (13) (11) , (13) (11) , (13) , (12) , (31)Table 1: The representations of the Virasoro algebra in the tricritical Ising model and prop-erties of the corresponding conformal boundary conditionsFrom these boundary conditions and their flows one can easily construct superpositions ofboundary conditions and further flows using the action of topological defects [6]. In the casein hand, the topological defects and the boundary conditions are labelled by representationsof the Virasoro algebra, and both the action of the defects on the boundary conditions andthe representation content of the strip Hilbert space are given in terms of the fusion ofrepresentations. If we denote the representations of the Virasoro algebra by a , b etc, the14 − − +& − ∗∗ ∗ (1 , , , ,
2) + (1 , (cid:27) Integrable
Figure 3: The space of boundary flows in the tricritical Ising modeldefects by D a and the boundary conditions by B a then the defects act as D a · B b = B a ∗ b = X c N abc B c , (5.1)where N abc are the Verlinde fusion numbers. Likewise, the representation content of a stripwith conformal boundary conditions a and b , which we denote H ( a,b ) is H ( a,b ) = ⊕ c N abc L c , (5.2)where L a is the highest weight representation with label a . The commutativity of the fusionalgebra means that the strip with boundary conditions ( a, b ∗ c ) has the same Hilbert space asthe strip with boundary conditions ( b ∗ a, c ), which we can interpret as the statement that theaction of a defect of type b on either of the two boundary conditions a or c leaves the Hilbertspace unchanged. This means that the six separate flows considered by Feverati et al. cannow be group into three pairs which can be found as the spectrum of the strip with the basicset of flows (2.7) on one edge and one of the three ‘fixed’ conformal boundary conditions ( r D ( r on the (11) boundary condition, they have alternative interpretations as the spectra of stripswith perturbed boundary conditions of type (12), (13) and (22) coupled with an undeformed(11) boundary condition on the second edge as we show in table 2. In the tri-critical Ising model we first consider the renormalisation of the coupling λ of theperturbing field φ / which generates the flows(11) − λφ ←− (12) + λφ −→ (21) . (5.3)We recall that there are two predictions from this calculation - firstly the general predictionthat the coupling renormalisation has a particular scaling form, namely that the function g n in equation 2.9 has the form g n ( x ) = g ( xn − y ) (5.4)15pectral flow Strip configuration 1 Strip configuration 2(11) ← (12) → (21) (11; 11) ← (12; 11) → (21; 11) (11; 11) ← (12; 11) → (21; 11)(31) ← (13) → (21) (31; 11) ← (13; 11) → (21; 11) (11; 31) ← (12; 31) → (21; 31)(21) ← (22) → (11)+(31) (21; 11) ← (22; 11) → (11+31; 11) (11; 21) ← (12; 21) → (21; 21)Table 2: The possible interpretations of the flows considered by Feverati et al. Configurationone has the boundary condition (11) on one side of the strip; configuration two has a fixedboundary condition of type ( r
1) on one side and the basic flow (11) ← (12) → (21) on theother.and secondly a particular prediction for the 1-loop behaviour, g ( x ) = (cid:18) − Cxy Γ( h ) π − y (cid:19) − + O ( x , n − y ) (5.5)The values for the tri-critical Ising model are C = − Γ( − )Γ( ) / Γ( )Γ( − ) / = 0 . ... , h = 3 / , y = 2 / . (5.6)For the TCSA approach, we need to specify two boundary conditions. On one we of coursetake the perturbed boundary conditions (5.3); on the other we can take any of the “fixed”boundary conditions ( r
1) and we will obtain one of the cases investigated by Feverati et al.For simplicity, we start with the boundary condition (11) on the other. For the flow withnegative λ , that is for (11) ← (12), we plot the functions log( g n ( − xn y )) against log( − x )for several values of n together with the 1-loop prediction for log( g ( − x )) in equation (5.5).These are shown in figure 4(a) We see that the functions log( g n ( − xn y )) are indeed almostidentical for various values of n and in good agreement with the prediction, confirming boththe prediction of the scaling form and the approximate numerical expression for this scalingfunction. For positive values of λ , we plot log( g n ( xn y )) against log( x ) for the same values of n , again together with the 1-loop prediction for log( g ( x )) in equation (5.5) in figure 4(b) withthe same results for small values of λ .The scaling form does clearly break down for larger values of λ positive, where the func-tions g n ( xn y ) have n -dependent maxima at n -dependent values of x . This corresponds phys-ically to the boundary condition approaching close to the (21) fixed point but then movingaway in the direction of the (13) fixed point. Presumably the flow in this region is governedby the critical exponents around the (21) fixed point and different arguments are required toanalyse this behaviour, but we can make some suggestions about the energy-level-dependenceof the position of the fixed point, which we do later in this section.Finally we can check whether the coupling constant renormalisation and energy re-scalingsdepend on the second boundary condition on the strip. In figure 5(a) we present the couplingconstant renormalisation as calculated from the three different choices of second boundarycondition, (11), (21) and (31). We see that, modulo numerical inaccuracies, the three differentTCSA strip configurations give the same coupling constant renormalisation, as would beexpected on physical grounds, and that this is in agreement wit the 1-loop calculation.16 - - - - - - - - - - - - (a) The scaling functions log g n ( − xn y ) and the pre-diction for log( g ( − x )) plotted vs. log( − x ) i.e. fornegative coupling constant - - - - (b) The scaling functions log g n ( xn y ) and the predic-tion for log( g ( x )) plotted vs. log( x ) for positive cou-pling constant Figure 4: The numerical coupling constant renormalisation found for the tri-critical Isingmodel with (11) boundary condition on the other edge, for truncation levels 8 ( ◦ ) 15 ( • ) and22 ( (cid:3) ) together with the 1-loop predictions (3.14) shown as a solid line. We can also check the prediction 3.32 for the form of the energy rescaling in the tri-criticalIsing model. This is shown in figure 5(b) where log( r ( − x )) is plotted together with thenumerical estimates for log( r n ( − xn y )) for various different truncation levels and for differentchoices of the non-perturbed boundary condition. This confirms both the scaling form of r n and also the numerical coefficient calculated in this section. The Lee-Yang model two conformal boundary conditions which are connected by an integrableboundary flow which has been studied in great detail [14]. The model is nonunitary and theboundary field generating the boundary flow has conformal weight − /
5, which means thatthe perturbation is UV finite but IR divergent. Since h = − / y = 1 − h = 6 / n − y ,decay very rapidly and the TCSA quickly becomes very accurate. The truncation does stillhave an effect, of course, and the accuracy of the numerical method allows us to check thepredictions for the coefficients of λ in the first and second energy gaps. The results are shownin 6, where we show the coefficients extracted from the TCSA method and also the predictionbased on the two leading terms in n − y . We see very good agreement so that the leading termsdo indeed give the correct behaviour. We have also checked this in the tri-critical Ising modelwhere we also get good agreement, but the corrections from the terms in n − y are larger.17 - - - - - - - - - (a) The TCSA coupling constant renormalisationlog( g n ( − xn y )) plotted against log( − x ) for the tri-critical Ising model and choices of second boundarycondition (11) = ( − ), ( ◦ , level 22), (21) = (0) ( • ,level 14) and (31) = (+) ( (cid:3) , level 18) together withthe 1-loop prediction (3.14) shown as a solid line. - - - - (b) The energy rescaling function log r n ( − xn y ) andthe prediction for log( r ( − x )) plotted vs. log( − x ) fornegative coupling for the flow (11; 11) ← (12; 11) fortruncation levels 8 ( ◦ ) 15 ( • ) and 22 ( (cid:3) ). Figure 5: The coupling constant and energy rescaling functionsWe have used the following expression for the second energy level in the Lee-Yang model,which is valid for any system where the representation ψ of conformal weight H is degenerateat level 2 and has only a single state at that level which we have chosen to be L − | ψ i : E = ( H + 2) + ˜ λC ′ (cid:18) h ( h − N (cid:19) + (˜ λC ′ ) (cid:20) h N + h (2 h − H N − Z d zz y (cid:18) F − h N z − y − h (2 h − (2 H N ) z − y − (4 h ( h − N ) ( N ) z − y (cid:19)(cid:21) ,F = 1 N (cid:18)(cid:20) z ( z −
1) dd z + (3 z + 1) h − (cid:21) (cid:20) ( z − /z ) dd z + h (3 + 1 /z ) (cid:21) + N (cid:19) F ,F = ψ φ ψ φz ψ , N = c H , (6.1)where for the lee-Yang model h = H = − / F = z / (1 − z ) / F ( , ; ; z ) , C ′ = (cid:18) Γ(1 / / − / / (cid:19) / . (6.2)18 - - - - - - - - - - - - - Figure 6: The coefficients of λ in the first and second gap for the Yang-Lee model. The dotsare from TCSA data, the solid curve the prediction using the two leading terms in 1 /n . Thestraight line is the exact value. One of the principal effects of the coupling constant renormalisation in the TCSA approach isthe possibility to move the IR fixed point of the finite-size scaling flow from infinite couplingconstant to finite coupling constant. From equations (2.9) and (3.14), we see that for C positive, the IR fixed point at λ ∞ = ∞ is brought to the finite value λ ∗ n = y Γ( h ) C ( πn ) y , (7.1)However, just as the coupling to the identity operator depends on the level of the state, sothe coupling constant renormalisation depends on the level of the state, so that the one-loopeffective coupling constant renormalisation of a state at level E is not given by (3.13), butrather by λ ∗ n ( E ) = A ( n )( n − E ) y , (7.2)where the one-loop prediction for A ( n ) is A ( n ) = y Γ( h ) π y /C . The exact position of the fixedpoint is not determined very accurately by the 1–loop calculation, (indeed we shall see insection 8.2 that the position of the fixed point in the tri-critical Ising model is approximately λ ∗ n = B √ n ) but we might conjecture is that the dependence on energy level still has thisform. In figure 7(a) we show the full TCSA spectrum for the tri-critical Ising strip withboundary conditions (12) and (11). The level crossings seen for positive λ occur at the IRfixed point, and their position depends on the excitation level. We have also shown a fit tothese positions of the form λ FP ( n, E ) = A ( n )( n − E ) y . This at least appears to capture thequalitative behaviour quite well.A second prediction is that the position of this fixed point is independent of the unper-turbed boundary condition. In figure 7(b) we show the TCSA spectra of the three stripswith one boundary being the perturbation (12) → (21) and the other being one of the three19xed boundary conditions ( r (a) The full TCSA spectrum showing the variationof the fixed point with energy level; also shown isthe best fit to these positions in the scaling form λ crit = A ( n − E ) y , in the case n = 14. - - - - (b) A simultaneous plot of the TCSA spectrum ofthe basic flow (12) → (21) on one edge and the fixedboundary condition (11) (red), (21) (blue) and (31)green on the other edge for truncation level 14. Figure 7: Energy-dependence and boundary-condition independence of the position of thefixed point in the flow (12) → (21). When we consider a flow with a UV and an IR fixed point, we can attempt to describe it interms of perturbations of either fixed point. A perturbation of an IR fixed point will be by anirrelevant field which is a non-renormalisable perturbation. This means that the perturbationtheory will have divergences that require regulation and that a possibly infinite number ofcounterterms will be required to allow one to remove the regulator. For this reason, theyhave only infrequently been studied in the literature [18, 19, 20]. In TCSA, however, there isno need to introduce counterterms, one can simply compute the spectrum of the truncatedHamiltonian.We have investigated this in the case of the tri-critical Ising model. There are three IRfixed points in the basic sequence (9.1). In the case of the (11) and (31) boundary conditions,the fields on the boundary comprise the vacuum representation of the Virasoro algebra soit is expected that the flow will be generated by the field T ( x ), the boundary stress-energytensor, as in [20]; for the (21) boundary condition, the boundary fields form the (11) and (31)representations and it is to be expected that the flow is generated by the field φ (31) of weight20 /
2, mirroring the case in the bulk flows considered in [19]. We shall present some results forthis last case, and for the flow (21) → (13), in particular. Firstly, we find that the TCSA spectrum is in very good agreement with the “exact” TBAspectrum; we show the results for the spectrum in figure 11. Secondly, we can compare theTCSA couplings λ (21) n at truncation level n with the effective coupling λ (13) ∞ for the descriptionof the system as a perturbation of the (13) by the field φ (13) . On dimensional grounds, weexpect a relation of the form λ (13) ∞ = ( λ (21) n ) − / g ( λ (21) n n / ) , (8.1)and this indeed what we find. In figure 8(a) we plot the effective couplings against theTCSA coupling for various truncation levels, and in figure 8(b) we plot the scaling function λ (13) ∞ ( λ (21) n ) / . - - - - - - - - - (a) A plot of the effective coupling log( λ (13) ∞ ) againstlog( λ (21) n ). - - - - - - - - - - - - (b) A plot of the scaling form λ (13) ∞ ( λ (21) n ) / againstlog( λ (21) n n / ) together with the limiting value of − . Figure 8: Truncation level dependence and scaling functions for the irrelevant coupling λ (21) at truncation levels 8 (red, dotted), 13 (green, dashed) and 18 (blue, solid).The scaling form (8.1) does, however, have the major consequence that the TCSA couplingconstant does not converge to a fixed value as the truncation level is increased. For anirrelevant perturbation, h > y <
0. If the function g ( x ) in (3.13) is approximatelyequal to one for a range | x | < A , then the TCSA scaling function g n ( λ ) is approximately equalto one for | λ | < An − y . For y positive, this range grows with increasing n ; for an irrelevantperturbation it shrinks. In this case, we see that λ (13) ∞ ( λ (21) n ) / ≃ .
06 for λ (21) n < e − √ n . (8.2) While (8.1) and (8.2) relate the TCSA parameter λ (21) n to the “exact” parameter λ (13) ∞ , it is alsointeresting to compare the two TCSA parameters. In this case we find the problem that the21anges in which the TCSA parameters are good approximations to the “exact” parameters donot overlap. The IR parameter is good for λ (21) < . n − / , which equates to λ (13) > . n / ,while we can see from figure 4(a) that the UV parameter is good for λ (13) < . n / (where wehave used the Z symmetry to relate λ (13) to − λ (12) .) As a consequence, there are no valuesof the TCSA coupling constants for which (8.2) holds. Instead we find the interesting plot9(a); as n increases, the relation between the two TCSA couplings changes. For either of thecouplings small, there is an approximate linear relationship. We can elucidate this by notingthat the TCSA coupling relation itself takes a scaling form as we show in figure 9(b), wherewe see that, to a good approximation, λ (13) n √ n ≃ F ( λ (21) n / √ n ) , (8.3)where F ( x ) ≃ . − .x for x small, F ( x ) ≃ . − . x for F ( x ) small . (8.4)The linear relation is to be expected, as the perturbing fields at one end (UV or IR) will flowinto the perturbing field at the other (IR or UV, respectively) and so the coupling at one endwill end up being linearly related to the coupling at the other. Quite why this relation takesthe form (8.3) is unclear at the moment. (a) A plot of λ (13) n against λ (21) n (b) A plot of λ (13) n / √ n against λ (21) n √ n ) Figure 9: The relation between λ (13) n and λ (21) n for truncation levels 8 (blue, short dashes), 10(dotted), 12 (dot-dashed), 14 (long dashes) and 22 (solid). As has been remarked on before [12, 20], the TCSA spectra have the remarkable propertythat they extend beyond the IR fixed points and can encompass several different perturbativeflows. These appear to follow the sequence of flows found first by Lesage et al in [15] and22 - - Figure 10: The scaled gaps for the perturbation of the (21) boundary condition on a stripwith boundary conditions (21; 11) by λ (21) φ (31) , plotted against λ (21) at truncation level 14.The four dashed lines are the approximate positions of the fixed points (in order) (11), (12),(13) and (31).which also appear in Fredenhagen et al [16] and Dorey et al [17] and which in the simplestform applicable to the tri-critical Ising model is(12) (11) ← (12) → (21) ← (13) → (31) L99 (13) (9.1)where the dashed arrows reflect what is seen in TCSA and are a natural extension of thesequence in the papers cited. The same sequence is seen no matter which boundary conditionwe take as the starting point from which we perturb. This applies also to the case of irrelevant,non-renormalisable perturbations starting from the ‘fixed’ ( r
1) boundary conditions. Thisenables us to put coordinates on the sequence, in the sense that every pair of flows is covered asthe standard perturbation of a boundary condition, and we can relate the coupling constantson successive overlapping pairs:(12) + µ (12) φ (13) + µ (13) φ z }| { z }| { (12) (11) ← (12) → (21) ← (13) → (31) L99 (13) | {z } | {z } | {z } (11) + µ (11) T + .. (21) + µ (21) φ + .. (31) + µ (31) T + .. (9.2)where the ellipsis in the perturbations of the fixed boundary conditions shows that furthercounterterms may be needed as these perturbations are non-renormalisable. As an exampleof the way the TCSA flows can encompass several fixed points, in figure 9 we show the scaledgaps for the perturbation of the (21) boundary conditions by the irrelevant field φ (31) ofconformal weight 3 /
2. As can be seen, all five fixed points of (9.1) appear.To give support to this picture, we present the TCSA spectra for the flow (13) → (21)obtained from the three separate routes: the relevant perturbation of the (13) boundary23 - - Figure 11: The scaled gaps for the flow from the (13) boundary to the (21) boundary condition,as found from the relevant perturbation of the (13) boundary condition ( • ), from the irrelevantperturbation of the (21) boundary condition ( ◦ ), and from the extension of the flow generatedby the relevant perturbation of the (12) boundary condition ( (cid:3) ) at truncation level 22. Theyare plotted against ξ , the boundary parameter in Pearce et al.condition, the irrelevant perturbation of the (21) boundary condition, and the continuationof the flow (12) → (21) given by the relevant perturbation of the (12) boundary condition.These are all shown in figure 11.The TBA spectra are given as functions of the boundary reflection parameter θ B in [21] or ξ of Feverati et al. [8] (note ξ = θ B ). Using the results in [22, 23], we can find the exactexpression for θ B in terms of the perturbative coupling constant λ (13) of the relevant pertur-bation of the (13) boundary condition. We can also find the approximate numerical result forthe boundary parameter in terms of the TCSA coupling λ (21) , which is only valid for smallcoupling. These are summarised belowBoundary θ B (13) (cid:0) log a − log C (cid:1) + log λ (13) = 1 . ... + log λ (13) (21) − . − . log λ (21) + . . . (9.3)The constants in the first relation are C = π (cid:2) Γ( ) (cid:3) − , a = π sin( π )Γ( ) Γ( ) sin( π ) sin( π ) . (9.4) The equations in [15, 21] for the g -function are known not to be correct, but they do describe the changesunder a purely boundary perturbation. See [17] for the correct equations when there is a simultaneous bulkperturbation. λ (21) and λ (12) for this region, which is λ (12) n √ n − . √ nλ (21) n ≃ . , (9.5)which is in agreement with (8.4) to the accuracy that is obtainable for the flow from the (12)boundary condition beyond the (21) fixed point.
10 Conclusions
We have made further progress in understanding the errors in the truncated conformal spaceapproach to perturbed boundary conformal field theory on a strip. The principal quantitycalculated with TCSA is the spectrum as a function of the coupling constant. The errors areof two principal sorts - a renormalisation of the coupling constant and a change in the energyscale. We have found perturbative expressions for these from physical arguments and showedthey are correct using an analysis of the perturbative spectrum.One important aspect of these predictions is that they are independent of the secondboundary condition on the strip and we have verified this by considering different choices forthe unperturbed boundary condition and finding the same results for each choice.Furthermore we have investigated the behaviour of the TCSA spectrum for large couplingconstant. We have reported before that this appears to show a sequence of RG flows in thesame pattern as found by Lesage et al [15], and we know show that this is quantitativelycorrect as well, in the case of the tri-critical Ising model.It would be good to find some way to describe the full RG flows such as in figure 9 interms of some beta-function which has the sequence of fixed points as its zeroes, but at themoment we are unsure how to do that. In the lattice model, the perturbation parameter is aboundary magnetic field and the sequence of fixed points (9.1) (excluding the points joinedby the dashed lines) can be found simply by varying this field [24]; of course in the quantummodel this is not simply the case - the natural sequence of flows splits up into overlappingpairs of flows in both the TBA and the TCSA descriptions. It may be helpful to use thecoordinates on the full moduli space given by the values of the g -function and the excited g -function which corresponds to the overlap with the bulk spin field. This is something wehope to return to shortly,Finally the ideas on the perturbative treatment of the corrections to the TCSA presentedhere can be easily adapted to the case of bulk flows which we plan to address in [25].
11 Acknowledgements
I am very grateful to G. Feverati, F. Ravanini and P.A. Pearce for providing the numericaldata for the TBA flows from their papers [7, 8], without which this paper would not havebeen possible.I would like to thank P. Giokas, G. Tak´acs and G.Zs. T´oth for helpful discussions, B. Doyonand G. Tak´acs for comments on the manuscript and STFC grant ST/G000395/1 for support.All numerical calculations were performed using Mathematica [26].25 eferences [1] V.P. Yurov and A.B. Zamolodchikov,
Truncated conformal space approach to the scal-ing Lee-Yang model , Int. J. Mod. Phys A5 (1990) 3221–3245.[2] R.M. Konik and Y. Adamov, A Numerical Renormalization Group for ContinuumOne-Dimensional Systems , cond-mat/0701605[3] G.P. Brandino, R.M. Konik and G. Mussardo,
Energy level distribution of perturbedconformal field theories , J. Stat. Mech. (2010)
P07013 , arXiv:1004.4844[4] M. L¨assig, G. Mussardo and J.L. Cardy,
The scaling region of the tricritical Isingmodel in two dimensions , Nucl. Phys. B (1991) 591-618.[5] I. Affleck,
Edge critical behaviour of the two-dimensional tri-critical Ising model , J.Phys.
A33 (2000) 6473-6479 [cond-mat/0005286].[6] K. Graham and G.M.T. Watts,
Defect lines and boundary flows , JHEP (2004)019 [hep-th/0306167].[7] G. Feverati, P.A. Pearce and F. Ravanini,
Excited Boundary TBA in the TricriticalIsing Model , Int. J. Mod. Phys.
A19S2 (2004) 155-167 [hep-th/0306196][8] G. Feverati, P.A. Pearce and F. Ravanini,
Exact φ , boundary flows in the tricriticalIsing model , Nucl. Phys. B (2003) 469-515 [hep-th/0308075][9] G. Feverati, Exact ( d ) → (+) & ( − ) boundary flow in the tricritical Ising model ,J. Stat. Mech.0403: P001,2004 [hep-th/0312201][10] J.L. Cardy, Boundary conditions, fusion rules and the Verlinde formula , Nucl. Phys.B (1989) 581–596.[11] L. Chim,
Boundary S-matrix for the Tricritical Ising Model , Int. J. Mod. Phys.
A11 (1996) 4491, hep-th/9510008v1[12] G. Feverati1, K. Graham, P.A. Pearce, G.Zs. T´oth and G.M.T. Watts,
A renormalisa-tion group for the truncated conformal space approach , J. Stat. Mech. (2008) P03011[hep-th/0612203][13] J.L. Cardy,
Conformal Invariance and Statistical Mechanics , in
Exact Methods inLow-Dimensional Statistical Physics and Quantum Computing , Les Houches lectures2008, [arXiv:0807.3472][14] P. Dorey, A. Pocklington, R. Tateo and G. Watts,
TBA and TCSA with boundariesand excited states , Nucl. Phys. B (1998) 641 [hep-th/9712197].[15] F Lesage, H. Saleur and P. Simonetti,
Boundary flows in minimal models , Phys. Lett.
B427 (1998) 85–92, hep-th/9802061.[16] S. Fredenhagen, M.R. Gaberdiel and C. Schmidt-Colinet,
Bulk flows in Virasoro min-imal models with boundaries , J. Phys.
A42 :495403, 2009, arXiv:0907.2560[17] P. Dorey, R. Tateo and R. Wilbourne,
Exact g-function flows from the staircase model ,Nucl. Phys. B (2011) 724-752, arXiv:1008.1190[18] Al.B. Zamolodchikov,
From tricritical Ising to critical Ising by thermodynamic Betheansatz , Nucl. Phys. B (1991) 524–5462619] G. Feverati, E. Quattrini and F. Ravanini,
Infrared Behaviour of Massless IntegrableFlows entering the Minimal Models from ϕ , Phys. Lett. B374 (1996) 64-70, hep-th/9512104v2[20] G.Zs T´oth,
A study of truncation effects in boundary flows of the Ising model on astrip , J. Stat. Mech. (2007) P04005, hep-th/0612256[21] R.I. Nepomechie and C. Ahn,
TBA boundary flows in the tricritical Ising field theory ,Nucl. Phys. B (2002) 433-470, hep-th/0207012[22] V.V. Bazhanov, S.L. Lukyanov and Al.B. Zamolodchikov,
Integrable structure of con-formal field theory, quantum KdV theory and thermodynamic Bethe ansatz,
Commun.Math. Phys. (1996) 381-398, hep-th/9412229[23] V.V. Bazhanov, S.L. Lukyanov and Al.B. Zamolodchikov,
Integrable quantum fieldtheories in finite volume: Excited state energies , Nucl. Phys. B (1997) 487-531,hep-th/9607099[24] P. Giokas,
Mean field theory for boundary Ising and tri-critical Ising models, preprintkcl-mth-11-05.[25] P. Giokas and G.M.T. Watts, in preparation.[26] Wolfram Research Inc,