Boundary transitions of the O(n) model on a dynamical lattice
aa r X i v : . [ h e p - t h ] N ov YITP-09-56IPhT-t09/186
Boundary transitions of the
O(n) model on a dynamical lattice
Jean-Emile Bourgine ∗ , Kazuo Hosomichi ⋆ and Ivan Kostov ∗ ∗ Institut de Physique Th´eorique, CNRS-URA 2306C.E.A.-Saclay,F-91191 Gif-sur-Yvette, France ⋆ Yukawa Institute for Theoretical Physics, Kyoto UniversityKyoto 606-8502, Japan
We study the anisotropic boundary conditions for the dilute O ( n ) loop model with the methods of 2Dquantum gravity. We solve the problem exactly on a dynamical lattice using the correspondence witha large N matrix model. We formulate the disk two-point functions with ordinary and anisotropicboundary conditions as loop correlators in the matrix model. We derive the loop equations for thesecorrelators and find their explicit solution in the scaling limit. Our solution reproduces the boundaryphase diagram and the boundary critical exponents obtained recently by Dubail, Jacobsen and Saleur,except for the cusp at the isotropic special transition point. Moreover, our solution describes the bulkand the boundary deformations away from the anisotropic special transitions. In particular it showshow the anisotropic special boundary conditions are deformed by the bulk thermal flow towards thedense phase. Associate member of the Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 72Tsarigradsko Chauss´ee, 1784 Sofia, Bulgaria
Introduction
The boundary critical phenomena appear in a large spectrum of disciplines of the contemporary the-oretical physics, from solid state physics to string theory. The most interesting situation is whenthe boundary degrees of freedom enjoy a smaller symmetry than those in the bulk. In this case onespeaks of surface anisotropy. The D-branes in string theory are perhaps the most studied exampleof such anisotropic surface behavior. Another example is provided by the ferromagnets with surfaceexchange anisotropy, which can lead to a critical and multi-critical anisotropic surface transitions. Aninteresting but difficult task is to study the interplay of surface and the bulk transitions and the relatedmulti-scaling regimes.For uniaxial ferromagnets as the Ising model, there are four different classes of surface transi-tions: the ordinary, extraordinary, surface, and special transitions [1]. This classification makes sensealso for spin systems with continuous O ( n ) symmetry. It was predicted by Diehl and Eisenriegler[2], using the ε -expansion and renormalization group methods, that the effects of surface anisotropycan be relevant near the special transitions of the d -dimensional O ( n ) model. These effects lead to‘anisotropic special transitions’ with different critical exponents.Recently, an exact solution of the problem for the -dimensional O ( n ) model was presented byDubail, Jacobsen and Saleur [3] using its formulation as a loop model [4, 5]. Using an elaboratemixture of Coulomb gas, algebraic and Thermodynamic Bethe Ansatz techniques, the authors of [3]confirmed for the dilute O ( n ) model the phase diagram suggested in [2] and determined the exactscaling exponents of the boundary operators. A review of the results obtained in [3], which is acces-sible for wider audience, can be found in [6]. Their works extended the techniques developped byJacobsen and Saleur [7, 8] for the dense phase of the O ( n ) loop model.In this paper we examine the bulk and the boundary deformations away from the anisotropicspecial transitions in the two-dimensional O ( n ) model. In particular, we address the question how theanisotropic boundary transitions are influenced by the bulk deformation which relates the dilute andthe dense phases of the O ( n ) model. To make the problem solvable, we put the model on a dynamicallattice. This procedure is sometimes called ‘coupling to 2D gravity’ [9, 10]. The sum over latticeserases the dependence of the correlation functions on the coordinates, so they become ‘correlationnumbers’. Yet the statistical model coupled to gravity contains all the essential information about thecritical behavior of the original model such as the qualitative phase diagram and the conformal weightsof the scaling operators. When the model is coupled to gravity, the bulk and boundary flows, originallydriven by relevant operators, becomes marginal, the Liouville dressing completing the conformalweights to one. This necessitates a different interpretation of the flows. The UV and the IR limitsare explored by taking respectively large and small values of the bulk and boundary cosmologicalconstants. Our method of solution is based on the mapping to the O ( n ) matrix model [11, 12] andon the techniques developed in [13, 14]. Using the Ward identities of the matrix model, we were ableto evaluate the two-point functions of the boundary changing operators for finite bulk and boundarydeformations away from the anisotropic special transitions.The paper is structured as follows. In Sect. 2 we summarize the known results about the boundarytransitions in the O ( n ) loop model. In Sect. 3 we write down the partition function of the boundary O ( n ) model on a dynamical lattice. In particular, we give a microscopic definition of the anisotropicboundary conditions on an arbitrary planar graph. In Sect. 4 we reformulate the problem in terms ofthe O ( n ) matrix model. We construct the matrix model loop observables that correspond to the disktwo-point functions with ordinary and anisotropic special boundary conditions. In Sect. 5 we write aset of Ward identities (loop equations) for these loop observables, leaving the derivation to AppendixA. We are eventually interested in the scaling limit, where the volumes of the bulk and the boundary1f the planar graph diverge. This limit corresponds to tuning the bulk and the boundary cosmologicalconstants to their critical values. In Sect. 6 we write the loop equations in the scaling limit in the formof functional equations. From these functional equations we extract all the information about thebulk and the boundary flows. In particular, we obtain the phase diagram for the boundary transitions,which is qualitatively the same as the one suggested in [3, 6], apart of the fact that we do not observea cusp near the special point. In Sect. 7 we derive the conformal weights of the boundary changingoperators. All our results concerning the critical exponents coincide with those of [3, 6]. In Sect. 8 wefind the explicit solution of the loop equations in the limit of infinitely large planar graph. The solutionrepresents a scaling function of the coupling for the bulk and boundary perturbations. The endpointsof the bulk and the boundary flows can be found by taking different limits of this general solution.The boundary flows relate the anisotropic special transition with the ordinary or with the extraordinarytransition, depending on the sign of the perturbation. The bulk flow relates an anisotropic boundarycondition in the dilute phase with another anisotropic boundary condition in the dense phase. For therational values of the central charge, the boundary conditions associated with the endpoints of the bulkflow match with those predicted in the recent paper [15] using perturbative RG techniques. O ( n ) model on a regular lattice: a summary The O ( n ) model [4] is one of the most studied statistical models. For the definition of the partitionfunction see Sect. 3. The model has an equivalent description in terms of a gas of self- and mutuallyavoiding loops with fugacity n . The partition function of the loop gas depends on the temperaturecoupling T which controls the length of the loops: Z O ( n ) = X loops n loops ] (1 /T ) links occupied by loops ] . (2.1)In the loop gas formulation of the O ( n ) model, the number of flavors n can be given any real value.The model has a continuum transition if the number of flavors is in the interval [ − , . Dependingon the temperature coupling T the model has two critical phases, the dense and the dilute phases. Atlarge T the loops are small and the model has no long range correlations. The dilute phase is achievedat the critical temperature [16, 17] T c = q √ − n (2.2)for which the length of the loops diverges. If we adopt for the number of flavors the standardparametrization n = 2 cos( πθ ) , < θ < , (2.3)then at the critical bulk temperature the O ( n ) model is described by (in general) a non-rational CFTwith central charge c dilute = 1 − θ θ . (2.4)The primary operators Φ r,s in such a non-rational CFT can be classified according to the generalizedKac table for the conformal weights, h rs = ( rg − s ) − ( g − g , g = 1 + θ. (2.5)2nlike the rational CFTs, here the numbers r and s can take non-integer values.When T < T c , the loops condense and fill almost all space. This critical phase is known as thedense phase of the loop gas. The dense phase of the O ( n ) is described by a CFT with lower value ofthe central charge, c dense = 1 − θ − θ . (2.6)The generalized Kac table for the dense phase is h rs = ( r ˜ g − s ) − (˜ g − g , ˜ g = 11 − θ . (2.7)Most of the exact results for the dense and the dilute phases of the O ( n ) model were obtained bymapping to Coulomb gas [5, 16].The boundary O ( n ) model was originally studied for the so called ordinary boundary condition ,where the loops avoid the boundary as they avoid themselves. The ordinary boundary condition isalso referred as Neumann boundary condition because the measure for the boundary spins is free. Theboundary scaling dimensions of the L -leg operators S L , realized as sources of L open lines, wereconjectured for the ordinary boundary condition in [18] and then derived in [19]: ( Ord | S L | Ord ) → Φ B L, ( dilute phase )( Ord | S L | Ord ) → Φ B , L ( dense phase ) . (2.8)Another obvious boundary condition is the fixed , or Dirichlet , boundary condition, which allows,besides the closed loops, open lines that end at the boundary [20]. The dimensions of the L -legboundary operators with Dirichlet and Neumann boundary conditions were computed in [21, 22] bycoupling the model to 2D gravity and then using the KPZ scaling relation [23, 24]: ( Ord | S L | Dir ) → Φ B / L, ( dilute phase )( Ord | S L | Dir ) → Φ B , / L ( dense phase ) . (2.9)From the perspective of the boundary CFT, these operators are obtained as the result of the fusionof the L -leg operator and a boundary-condition-changing (BCC) operator, introduced in [25], whichtransforms the ordinary into Dirichlet boundary condition.Recently it was discovered that the O ( n ) loop model can exhibit unexpectedly rich boundary crit-ical behavior. Jacobsen and Saleur [7, 8] constructed a continuum of conformal boundary conditionsfor the dense phase of the O ( n ) loop model. The Jacobsen-Saleur boundary condition , which wedenote shortly by JS, prescribes that the loops that touch the boundary at least once are taken withfugacity y = n , while the loops that do not touch the boundary are counted with fugacity n . Theboundary parameter y can take any real value. The JS boundary conditions contain as particular casesthe ordinary ( y = n ) and fixed ( y = 1 ) boundary conditions for the O ( n ) spins. Jacobsen and Saleurconjectured the spectrum and the conformal dimensions of the L -leg boundary operators separatingthe ordinary and the JS boundary conditions. These conformal dimensions were subsequently veri-fied on the model coupled to 2D gravity in [13, 14]. If y is parametrized in the ‘physical’ interval ≤ y ≤ n as y = sin π ( r + 1) θ sin πrθ (1 ≤ r ≤ /θ − , (2.10) In the papers [20, 21, 22] the loop gas was considered in the context of the SOS model, for which the Dirichlet andNeumann boundary conditions have the opposite meaning. Φ r,r . Note that r here is not necessarily integer or even rational. The L -leg operatorswith L ≥ fall into two types. The operator S − L creates open lines which avoid the JS boundary. Theoperator S + L creates open lines that can touch the JS boundary without restriction. The two types of L -leg boundary operators are identified as ( Ord | S ± L | J S ) → Φ Br,r ± L ( dense phase ) . (2.11)The general case of two different JS boundary conditions with boundary parameters y and y wasconsidered for regular and dynamical lattices respectively in [26] and [27].The JS boundary condition was subsequently adapted to the dilute phase by Dubail, Jacobsen andSaleur [3]. The authors of [3] considered the loop gas analog of the anisotropic boundary interactionstudied previously by Diehl and Eisenriegler [2], which breaks the symmetry as O ( n ) → O ( n (1) ) × O ( n (2) ) , n (1) + n (2) = n. (2.12)The boundary interaction depends on two coupling constants, λ (1) and λ (2) , associated with the twounbroken subgroups. In terms of the loop gas, the anisotropic boundary interaction is defined byintroducing loops of two colors, (1) and (2) , having fugacities respectively n (1) and n (2) . Each timewhen a loop of color ( α ) touches the JS boundary, it acquires an extra factor λ ( α ) . We will call thisboundary condition dilute Jacobsen-Saleur boundary condition , or shortly DJS , after the authors of[3]. Let us summarize the qualitative picture of the surface critical behavior proposed in [3]. Considerfirst the isotropic direction λ (1) = λ (2) = λ . In the dilute phase one distinguishes three differentkinds of critical surface behavior: ordinary, extraordinary and special. When λ = 0 , the loops inthe bulk almost never touch the boundary. This is the ordinary boundary condition. When w → ∞ ,the most probable loop configurations are those with one loop adsorbed along the boundary, whichprevents the other loops in the bulk to touch the boundary. The adsorbed loop plays the role of aboundary with ordinary boundary condition. This is the extraordinary transition. In terms of the O ( n ) spins, ordinary and the extraordinary boundary conditions describe respectively disordered and or-dered boundary spins. The ordinary and the extraordinary boundary conditions describe the samecontinuous theory except for a reshuffling of the boundary operators. The L -leg operator with or-dinary/extraordinary boundary conditions will look, when L ≥ , as the ( L − -leg operator withordinary/ordinary boundary conditions, because its rightmost leg will be adsorbed by the boundary.The -leg operator with ordinary/extraordinary boundary condition will look like the -leg operatorwith ordinary/ordinary boundary condition because one of the vacuum loops will be partially adsorbedby the extraordinary boundary and the part which is not adsorbed will look as an open line connectingthe endpoints of the extraordinary boundary.The ordinary and the extraordinary boundary conditions are separated by a special transition,which happens at some λ = λ c and describes a conformal boundary condition. For the honeycomblattice the special point is known [28] to be at λ c = (2 − n ) − / T c ( honeycomb lattice ) . (2.13)At the special point the loops touch the boundary without being completely adsorbed by it. The specialtransition exists only in the dilute phase, because in the dense phase the loops already almost surely The spontaneous ordering on the boundary does not contradict the Mermin-Wagner theorem. In the interval < n < the target space, the ( n − -dimensional sphere, has negative curvature and thus resembles a non-compact space. λ is the reshuffling of the spectrum of L -leg operators, which happens in the same way as in the dilute phase.In the anisotropic case, λ (1) = λ (2) , there are again three possible transitions: ordinary, extraor-dinary and special. When λ (1) and λ (2) are small, we have the same ordinary boundary condition asin the isotropic case. In the opposite limit, where λ (1) and λ (2) are both large, the boundary spins be-come ordered in two different ways, depending on which of the two couplings prevails. If λ (1) > λ (2) ,the (1) -type components order, while the (2) -type components remain desordered, and vice versa .The ( λ (1) , λ (2) ) plane is thus divided into three domains, characterized by disordered, (1) -ordered and (2) -disordered, and (2) -ordered and (1) -disordered, which we denote respectively by Ord , Ext (1) and
Ext (2) . The domains
Ext (1) and
Ext (2) are separated by the isotropic line starting at the specialpoint and going to infinity. This is a line of first order transitions because crossing it switches fromone ground state to the other. The remaining two critical lines,
Ord/Ext (1) and
Ord/Ext (2) , are thelines of the two anisotropic special transitions , AS (1) and AS (2) . It was argued in [2, 3, 6], usingscaling arguments, that the lines AS (1) and AS (2) join at the point Sp = ( λ c , λ c ) in a cusp-like shape.The model was solved in [3] for a particular point on AS (1) : λ (1) = 1 + 1 − n (2) + √ − n (1) n (2) √ − n ,λ (2) = 1 + 1 − n (1) − √ − n (1) n (2) √ − n . (2.14)In terms of the loop gas expansion, the anisotropic special transitions are obtained by criticallyenhancing the interaction with the boundary of the loops of color (1) or (2) . The boundary CFT’sdescribing the transitions AS (1) and AS (2) were identified in [3]. A convenient parametrization of n (1) and n (2) on the real axis is n (1) = sin[ π ( r − θ ]sin[ πrθ ] , n (2) = sin[ π ( r + 1) θ ]sin[ πrθ ] (0 < r < /θ ) . (2.15)The loop model has a statistical meaning only if both fugacities are positive, which is the case when < r < /θ − . With the above parametrization, the BCC operators ( AS (1) | Ord ) and ( AS (2) | Ord ) are argued to be respectively Φ Br,r and Φ Br,r +1 . More generally, one can consider the L -leg boundaryoperators, S (1) L and S (2) L , which create L open lines of color respectively (1) and (2) . The Kac labelsof these operators were determined in [3] as follows, ( Ord | S (1) L | AS (1) ) → Φ Br + L,r , ( Ord | S (2) L | AS (1) ) → Φ Br − L,r , ( Ord | S (1) L | AS (2) ) → Φ Br + L,r +1 , ( Ord | S (2) L | AS (2) ) → Φ Br − L,r +1 . (2.16)In the vicinity of the special transitions the theory is argued to be described by a perturbation of theboundary CFT by the boundary operator Φ B , in the isotropic direction and by Φ B , in the anisotropicdirection. The correspondence between our notations and the notations used in [6] is { n (1) , n (2) , r, λ (1) , λ (2) } here = { n , n , r, w , w } there . The boundary O ( n ) model on a dynamical lattice O ( n ) model on a planar graph The two-dimensional O ( n ) loop model, originally defined on the honeycomb lattice [4], can be alsoconsidered on a honeycomb lattice with defects, such as the one shown in Fig. 1a. The lattice repre-sents a trivalent planar graph Γ . We define the boundary ∂ Γ of the graph by adding a set of extra lines(the single lines in the figure) which turn the original planar graph into a two-dimensional cellularcomplex. The local fluctuating variable is an O ( n ) classical spin, that is an n -component vector ~S ( r ) with unit norm, associated with each vertex r ∈ Γ , including the vertices on the boundary ∂ Γ . Thepartition function of the O ( n ) model on the graph Γ depends on the coupling T , called temperature,and is defined as an integral over all classical spins, Z O ( n ) ( T ; Γ) = Z Y r ∈ Γ [ d~S ( r )] Y h rr ′ i (cid:16) T ~S ( r ) · ~S ( r ′ ) (cid:17) , (3.1)where the product runs over the lines h rr ′ i of the graph, excluding the lines along the boundary. The O ( n ) -invariant measure [ d~S ] is normalized so that Z [ d~S ] S a S b = δ a,b . (3.2)The partition function (3.1) corresponds to the ordinary boundary condition , in which there is nointeraction along the boundary.Expanding the integrand as a sum of monomials, the partition function can be written as a sumover all configurations of self-avoiding, mutually-avoiding loops as the one shown in Fig. 1b, eachcounted with a factor of n : Z O ( n ) ( T ; Γ) = X loops on Γ T − length n loops . (3.3)The temperature coupling T controls the length of the loops. The advantage of the loop gas represen-tation (3.3) is that it makes sense also for non-integer n . In terms of loop gas, the ordinary boundarycondition, which we will denote by Ord , means that the loops in the bulk avoid the boundary as theyavoid the other loops and themselves.The
Dirichlet boundary condition , was originally defined for the dense phase of the loop gas[20, 21, 22] and requires that an open line starts at each point on the boundary. The dilute versionof this boundary condition depends on an adjustable parameter, which controls the number of theopen lines. In terms of the O ( n ) spins the Dirichlet boundary condition is obtained by switching ona constant magnetic field ~B acting on the boundary spins. This modifies the integration measure in(3.1) by a factor Y r ∈ ∂ Γ (cid:16) ~B · ~S ( r ) (cid:17) , (3.4)where the product goes over all boundary sites r . The loop expansion with this boundary measurecontains open lines having both ends at the boundary, each weighted with a factor ~B .The dilute anisotropic (DJS) boundary condition is defined as follows. The n components of the O ( n ) spin are split into two sets, (1) and (2) , containing respectively n (1) and n (2) components, with n (1) + n (2) = n . This leads to a decomposition of the O ( n ) spin as ~S = ~S (1) + ~S (2) , ~S (1) · ~S (2) = 0 . (3.5)6 bFigure 1: a) A trivalent planar graph Γ with a boundary b) A loop configuration on Γ for the ordinaryboundary condition. The loops avoid the boundary as they avoid themselves (2)(1) Figure 2:
A loop configuration for the JS boundary condition. The loops in the bulk (in red) have fugacity n , while the loops that touch the boundary (in blue and green) have fugacities n (1) or n (2) depending on theircolor. The DJS boundary condition is introduced by an extra factor associated with the boundary links, Y h rr ′ i∈ ∂ Γ (cid:16) X α =1 , λ ( α ) ~S ( α ) ( r ) · ~S ( α ) ( r ′ ) (cid:17) . (3.6)This boundary interaction is invariant under the subgroup of independent rotations of ~S (1) and ~S (2) .The boundary term changes the loop expansion. The loops are now allowed to pass along the boundarylinks as shown in Fig. 2. We have to introduce loops of two colors, (1) and (2) , having fugacitiesrespectively n (1) and n (2) . A loop of color ( α ) that visits N boundary links acquires an additionalweight factor λ N ( α ) . For the loops that do not touch the boundary, the contributions of the two colorssum up to n (1) + n (2) = n and we obtain the same weight as with the ordinary boundary condition. The disk partition function of the O ( n ) model on a dynamical lattice is defined as the expectation valueof (3.1) in the ensemble of all trivalent planar graphs Γ with the topology of the disk. The measuredepends on two more couplings, ¯ µ and ¯ µ B , called respectively bulk and boundary cosmological con-7tants, associated with the volume | Γ | = cells ) and the boundary length | ∂ Γ | = external lines ) .The partition function of the disk is a function of ¯ µ and ¯ µ B and is defined by U ( T, ¯ µ, ¯ µ B ) = X Γ ∈{ Disk } | ∂ Γ | (cid:18) µ (cid:19) | Γ | (cid:18) µ B (cid:19) | ∂ Γ | Z O ( n ) ( T ; Γ) . (3.7) L -leg boundary operators Our aim is to evaluate the boundary two-point function of two L -leg operators separating ordinaryand anisotropic boundary conditions. The L -leg operator S L is obtained by fusing L spins with flavorindices a , . . . , a n ∈ { , . . . , n } . In terms of the loop gas, the operator S L creates L self and mutuallyavoiding open lines. We would like to exclude configurations where some of the lines contract amongthemselves. This can be achieved by taking the antisymmetrized product S L ∼ det L × L S a i ( r j ) , (3.8)where r , . . . , r L are L consecutive boundary vertices of the planar graph Γ , and we put the label L instead of writing its dependence on a , . . . , a L explicitly. The two-point function of the operator S L is evaluated as the partition function of the loop gas in presence of L open lines connecting the points { r i } and { r ′ i } . The open lines are self- and mutually avoiding, and are not allowed to intersect thevacuum loops.Since the DJS boundary condition breaks the O ( n ) symmetry into O ( n (1) ) × O ( n (2) ) , there aretwo inequivalent correlation functions of the L -leg operators with Ord/DJ S boundary conditions.Indeed, the insertion of S a has different effects depending on whether a belongs to the O ( n (1) ) or the O ( n (2) ) sectors. In the first case the open line created by S a acquires a factor λ (1) each time it visitesa boundary link. In the second case, the factor is λ (2) . Therefore the boundary spin operators (3.8)with Ord/DJ S boundary conditions split into two classes, S (1) L ∼ det[ S a i ( r j )] , a , . . . , a L ∈ (1) ,S (2) L ∼ det[ S a i ( r j )] , a , . . . , a L ∈ (2) . (3.9)We denote the corresponding boundary two-point functions respectively by D (1) L and D (2) L . O ( n ) matrix model The O ( n ) matrix model [11, 12] generates planar graphs covered by loops in the same way as theone-matrix models considered in the classical paper [29] generate empty planar graphs. The modelinvolves the hermitian N × N matrices M and Y a , where the flavor index a takes n values. Thepartition function is given by an O ( n ) -invariant matrix integral Z N ( T ) ∼ Z d M d n Y e − β tr ( M + T ~ Y − M − M ~ Y ) . (4.1)This integral can be considered as the partition function of a zero-dimensional QFT with Feynmanrules given in Fig. 3, where we used the ’t Hooft double-line notations. The graphs made of such We used bars to distinguish from the bulk and boundary cosmological constants in the continuum limit. a i ijjk ki ijjk kij ij i ij j /β /βT β β Figure 3:
Feynman rules for the O ( n ) matrix model i i jj i i j j j ji ik k k k / ¯ µ B λ (1) / ¯ µ B λ (2) / ¯ µ B Figure 4:
The constituents of the DJS boundary generated by M , ~ Y and ~ Y : a non-occupied site, asequence of two sites visited by a loop of color (1) , and a sequence of two sites visited by a loop of color (1) double-lined propagators are known as fat graphs. The ‘vacuum energy’ of the matrix model repre-sents a sum over connected fat graphs, which can be also considered as discretized two-dimensionalsurfaces of all possible genera. As the action is quadratic in the matrices Y a , their propagators arrangein closed loops carrying a flavor a . The sum of all Feynman graphs with given connectivity can beviewed as the sum over all configurations of self and mutually avoiding loops on a given discretizedsurface. The weight of each loop is given by the product of factors /T , one for each link, and thenumber of flavors n . We are interested in the large N limit N → ∞ , β/N = ¯ µ ( fixed ) , (4.2)in which only fat graphs of genus zero survive [30].The basic observable in the matrix model is the resolvent W (¯ µ B ) = 1 β (cid:28) tr µ B − M (cid:29) , (4.3)evaluated in the ensemble (4.1). The resolvent is the one-point function with ordinary boundaryconditions and is related to the disk partition function by W = − ∂ ¯ µ B U .The one-point function with Dirichlet boundary condition is obtained by adding a term ~B · ~ Y whichexpresses the coupling with the magnetic field on the boundary. This leads to a more-complicatedresolvent R (¯ µ B , ~B ) = 1 β (cid:28) tr 1¯ µ B − M − ~B · ~ Y (cid:29) . (4.4)In order to include the anisotropic boundary conditions in this scheme, we decompose the vector ~ Y into a sum of an n (1) -component vector ~ Y (1) and an n (2) -component vector ~ Y (2) as in (3.5): ~ Y = ~ Y (1) + ~ Y (2) , ~ Y (1) · ~ Y (2) = 0 . (4.5)9he one-point function with ordinary and DJS boundary conditions is given by the resolvent H (¯ µ B , λ (1) , λ (2) ) = 1 β * tr µ B − M − µ B P α =1 , λ ( α ) ~ Y α ) + . (4.6)The two extra terms are the operators creating boundary links containing segments of lines of type (1) and (2) , as shown in Fig.4. Each such operator created two boundary sites, hence the factor / ¯ µ B .The matrix integral measure becomes singular at M = T / . We perform a linear change of thevariables M = T (cid:0) + X (cid:1) , (4.7)which sends this singular point to X = 0 . After a suitable rescaling of ~ Y and β , the matrix modelpartition function takes the canonical form Z N ∼ Z d X d n Y e β tr[ − V ( X )+ X ~ Y ] , (4.8)where V ( x ) is a cubic potential V ( x ) = X j =0 g j j x j = − T (cid:16) x + (cid:17) + 12 (cid:16) x + (cid:17) . (4.9)We introduce the spectral parameter x which is related to the lattice boundary cosmological constant ¯ µ B by ¯ µ B = T (cid:0) x + (cid:1) . (4.10)Now the one-point function with ordinary boundary condition is W ( x ) = 1 β h tr W ( x ) i , (4.11)where the matrix W ( x ) def = 1 x − X . (4.12)creates a boundary segment with open ends. In the following we will call x boundary cosmologicalconstant. We also redefine the boundary couplings λ ( α ) in (4.6) as λ ( α ) → λ ( α ) ¯ µ B , α = 1 , . (4.13)Then the operator that creates a boundary segment with DJS boundary condition is H ( y ) def = 1 y − X − λ (1) ~ Y (1) − λ (2) ~ Y (2) . (4.14)The boundary L -leg operators are represented by the antisymmetrized products S L def = Y a Y a · · · Y a L ± permutations . (4.15)The boundary two-point function of the L -leg operator with Ord/Ord boundary conditions is givenby the expectation value D L ( x , x ) def = 1 β (cid:10) tr (cid:2) W ( x ) S L W ( x ) S L (cid:3)(cid:11) . (4.16)10he role of the operators W ( x ) and W ( x ) is to create the two boundary segments with boundarycosmological constants respectively x and x . The two insertions S L generate L open lines at thepoints separating the two segments. It is useful to extend this definition to the case L = 0 , assumingthat S is the boundary identity operator. In this simplest case the expectation value (4.16) is evaluatedinstantly as D ( x , x ) = 1 β (cid:10) tr (cid:2) W ( x ) W ( x ) (cid:3)(cid:11) = W ( x ) − W ( x ) x − x . (4.17)The two-point functions (4.16) for L ≥ were computed in [21, 22].In the case of a DJS boundary condition, the matrix model realization of the two types of bound-ary L -leg operators is given by the antisymmetrized products (4.15), with the restrictions on the com-ponents as in (3.9). The boundary two-point functions with Ord/DJ S boundary conditions areevaluated by the expectation values D ( x, y ) = 1 β D tr (cid:2) W ( x ) H ( y ) (cid:3)E (4.18)and for L ≥ , D (1) L ( x, y ) = 1 β D tr (cid:2) W ( x ) S (1) L H ( y ) S (1) L (cid:3)E , (4.19) D (2) L ( x, y ) = 1 β D tr (cid:2) W ( x ) S (2) L H ( y ) S (2) L (cid:3)E . (4.20)Apart of the DJS boundary parameter n (1) and the boundary couplings λ (1) , λ (2) for the second seg-ment, they depend on the boundary cosmological constants x and y associated with the two segmentsof the boundary. Our goal is to evaluate the two-point functions (4.19) in the continuum limit, when the area andthe boundary length of the disk are very large. They will be obtained as solution of a set of Wardidentities, called loop equations, which follow from the translational invariance of the integrationmeasure in (4.1), and in which the n (1) enters as a parameter. The solutions of the loop equations areanalytic functions of n (1) which can take any real value. We will restrict our analysis to the ‘physical’case ≤ n (1) ≤ n , when the correlation functions have good statistical limit. Here we summarizethe loop equations which will be extensively studied in following sections. The proofs are given inAppendix A. The loop equation for the resolvent is known [31], but we nevertheless recall it here in order to set upa self-contained description of the method. The resolvent W ( x ) splits into a singular part w ( x ) and apolynomial W reg ( x ) : W ( x ) def = W reg ( x ) + w ( x ) . (5.1)11he regular part is given by W reg ( x ) = 2 V ′ ( x ) − nV ′ ( − x )4 − n = − a − a x − a x ,a = − g n = T − n ) ,a = − g − n = T − − n ,a = − g n = T n . (5.2)The function w ( x ) satisfies a quadratic identity w ( x ) + w ( − x ) + nw ( x ) w ( − x ) = A + Bx + Cx . (5.3)The coefficients A, B, C as functions of T , ¯ µ and W = h tr X i can be evaluated by substituting thelarge- x asymptotics w ( x ) = − W reg ( x ) + ¯ µ − x + h tr X i βx + O ( x − ) (5.4)in (5.3). The solution of the loop equation (5.3) with the asymptotics (5.4) is given by a meromorphicfunction with a single cut [ a, b ] on the first sheet, with a < b < . This equation can be solved by anelliptic parametrization and the solution is expressed in terms of Jacobi theta functions [32]. The two-point correlators (4.16) with ordinary boundary conditions are known to satisfy the integralrecurrence equations [21, 22] D L +1 = W ⋆ D L . (5.5)The “ ⋆ -product” is defined for any pair of meromorphic functions, analytic in the right half planeRe ( x ) ≥ and vanishing at infinity, [ f ⋆ g ]( x ) def = − I C − dx ′ πi f ( x ) − f ( x ′ ) x − x ′ g ( − x ′ ) , (5.6)with the contour C − encircling the left half plane Re x < . These equations actually hold for a moregeneral set of two-point correlators, which have ordinary boundary condition on one segment and anarbitrary boundary condition on the other segment [13]. Thus the boundary two-point functions (4.19)and (4.20) for L ≥ satisfy the same recurrence equations D (1) L +1 = W ⋆ D (1) L ,D (2) L +1 = W ⋆ D (2) L . (5.7)Using the recurrence relation, the correlation functions of the L -leg operators can be obtained recur-sively from those of the one-leg operators D (1)1 and D (2)1 .12he correlator D , defined by (4.18), and the correlators D (1)1 and D (2)1 , which we normalize as D ( α )1 ( x, y ) = 1 βn ( α ) X a D tr (cid:2) W ( x ) Y ( α ) a H ( y ) Y ( α ) a (cid:3)E ( α = 1 , , (5.8)can be determined by the following pair of bilinear functional equations, derived in Appendix A. Inorder to shorten the expressions, here and below we use the shorthand notation F ( x ) def = F ( − x ) . (5.9)The two equations then read W − H + D ( x − y ) + X α =1 , n ( α ) λ ( α ) (cid:18) ( λ ( α ) D − D ( α )1 + W D (cid:19) = 0 , (5.10) P + D (cid:0) H − V ′ + W + nW (cid:1) + X α =1 , n ( α ) ( λ ( α ) D − D ( α )1 = 0 . (5.11)The second equation involves an unknown linear function of x : P ( x, y ) def = 1 β (cid:28) tr V ′ ( x ) − V ′ ( X ) x − X H ( y ) (cid:29) = g H ( y ) + g H ( y ) + xg H ( y ) . (5.12)Equations (5.10) and (5.11) can be solved in favor of D (1)1 or D (2)1 . If we define A (1) def = λ (1) D − , B (1) def = ( λ (1) − λ (2) ) n (1) { λ (1) D (1)1 + W } − λ (2) (cid:0) W − V ′ + H (cid:1) − x − y, (5.13) C (1) def = λ (1) λ (2) P + ( λ (1) + λ (2) ) H − x + y − ( λ (1) − λ (2) ) (cid:0) W + n (1) W (cid:1) − λ (2) V ′ , and similarly for A (2) , B (2) , C (2) , with an obvious exchange (1) ↔ (2) , then (5.10) and (5.11) takethe following factorized form: A ( α ) B ( α ) = C ( α ) ( α = 1 , . (5.14)Equations (5.14) are the main instrument of our analysis of the DJS boundary conditions.It is convenient to define the functions D ( α )0 by D ( α )0 def = D − λ ( α ) D ( α = 1 , . (5.15)In Appendix A we show that with this definition the recurrence equations (5.7) hold also for L = 0 .The equation for L = 0 is a consequence of (5.14). On the flat lattice, the DJS boundary condition with n (1) = 1 is equivalent, for a special choice ofthe boundary parameters, to the Dirichlet boundary condition defined by the boundary factor (3.4). Inorder to make the comparison on the dynamical lattice, we will formulate and solve the loop equationfor the two-point function of the BCC operator with ordinary/Dirichlet boundary conditions.13ssume that the magnetic field points at the direction a = 1 . Then the correlator in question isgiven in the matrix model by the expectation value Ω( x, y ) = 1 β h tr W ( x ) R ( y ) i , R ( y ) = 1 y − X − B Y . (5.16)To obtain the loop equation we start with the identity W ( x ) − R ( y ) = ( y − x )Ω( x, y ) − B Ω ( x, y ) , (5.17)where we denoted by R ( y ) the one-point function with Dirichlet boundary condition, R ( x ) = 1 β h tr R ( y ) i , (5.18)and introduced the auxiliary function Ω ( x, y ) = 1 β h tr Y W ( x ) R ( y ) i . (5.19)The function Ω satisfies the identity Ω ( x, y ) = B I C − dx πi Ω( x , y )Ω( − x , y ) x − x , (5.20)which follows from (A.3). After symmetryzing with respect to x we get Ω ( x, y ) + Ω ( − x, y ) = B Ω( x, y )Ω( − x, y ) . (5.21)From here we obtain a quadratic functional equation for the correlator Ω : ( x − y )Ω( x, y ) − ( x + y )Ω( − x, y ) + W ( x ) + W ( − x ) + B Ω( x, y )Ω( − x, y ) = 2 R ( y ) . (5.22)The linear term can be eliminated by a shift G ( x, y ) = B Ω( x, y ) − x + yB . (5.23)The function G satisfies G ( x, y ) G ( − x, y ) = − W ( x ) − W ( − x ) + 2 R ( y ) − x − y B . (5.24)This equation is to be compared with the loop equation (5.14) for Ord/DJS boundary conditions, with n (1) = 1 and λ (2) = 0 : λ (1) A (1) ( x, y ) B (1) ( − x, y ) = − W ( x ) − W ( − x ) + H ( y ) − x − yλ (1) . (5.25)These two equations coincide in the limit B, λ (1) → ∞ . To see this it is sufficient to notice thatin the limit λ (1) → ∞ we have the relation B (1) = y A (1) . The exact relation between A (1) (with n (1) = 1 , λ (2) = 0 ) and G in this limit follows from the definitions of D and Ω : A (1) ( x, y ′ ) = B G ( x, y ) − G ( x, − y )2 y , λ (1) = B , y ′ = y ( B → ∞ ) . (5.26)14 Scaling limit
In this section we will study the continuum limit of the solution, in which the sum over lattices isdominated by those with diverging area and boundary length. The continuum limit is achieved whenthe couplings x , y and ¯ µ are tuned close to their critical values.Once the bulk coupling constants are set to their critical values, we will look for the critical linein the space of the boundary couplings y, λ (1) and λ (2) . After the shift (4.7) the bondary cosmologicalconstant x has its critical value at x = 0 , while the critical value of y in general depends on the valuesof λ (1) and λ (2) . Here we recall the derivation of the continuum limit of the one-point function W ( x ) from the func-tional equation (5.3). Even if the result is well known, we find useful to explain how it is obtained inorder to set up the logic of our approach to the solution of the functional equations (5.14).In the limit x → , the boundary length | ∂ Γ | of the planar graphs in (3.7) becomes critical. Thequadratic functional equation (5.3) becomes singular at x → when the coefficient A on the r.h.s.vanishes. This determines the critical value of the cosmological constant ¯ µ , for which the volume | Γ | of a typical planar graph diverges. The condition that the coefficient B of the linear term vanishesdetermines the critical value of the temperature coupling T = T c for which the length of the loopsdiverges: T c = 1 + q − n n ∈ [1 , . (6.1)Near the critical temperature the coefficient B is proportional to T − T c .We rescale x → ǫx , where ǫ is a small cutoff parameter with dimension of length, and define therenormalized coupling constants as ¯ µ − ¯ µ c ∼ ǫ µ , T − T c ∼ ǫ θ t (6.2)and write (5.1) as W ( x ) def = W reg + ǫ θ w ( x ) . (6.3)The renormalized bulk and boundary cosmological constants are coupled respectively to the renor-malized area A and boundary length ℓ of the graph Γ defined as A = ǫ | Γ | , ℓ = ǫ | ∂ Γ | . (6.4)In the following we define the dimensions of the scaling observables by the way they scale with x . Wewill say that the quantity f has dimension d if the ratio f /x d is invariant with respect to rescalings. Inthis case we write [ f ] = d . The continuous quantities introduced until now have scaling dimensions [ x ] = 1 , [ µ ] = 2 , [ t ] = 2 θ, [ w ] = 1 + θ . (6.5)The scaling resolvent w ( x ) is a function with a cut on the negative axis in the x -plane. It canbe obtained from the general solution found in [32] by taking the limit in which the cut extends tothe semi-infinite interval [ −∞ , − M ] . To determine M as a function of µ and t one has to solve asystem of difficult transcendental equations. A simpler indirect method was given in [33]. We beginby noticing that the term Cx in (5.3) drops out because it vanishes faster than the other terms when15 → , and B = B t , where B depends only on n . Introducing a hyperbolic map which resolvesthe branch point at x = − M , x = M cosh τ , (6.6)we obtain a quadratic functional equation for the entire function w ( τ ) ≡ w [ x ( τ )] : w ( τ + iπ ) + w ( τ ) + n w ( τ + iπ ) w ( τ ) = A + B tM cosh τ . (6.7)This equation does not depend on the cutoff ǫ , which justifies the definition of the renormalizedthermal coupling in (6.2). Then the unique solution of this equation is, up a factor which depends onthe normalization of t , w ( τ ) = M θ cosh(1 + θ ) τ + tM − θ cosh(1 − θ ) τ . (6.8)One finds B = 4 sin ( πθ ) and A = sin ( πθ )( M θ − tM − θ ) for this solution.The function M = M ( µ, t ) can be evaluated using the fact that the derivative ∂ µ W ( x ) dependson µ and t only through M . As a consequence, in the derivative of the solution in µ at fixed x , ∂ µ w = − M ∂ µ M (cid:16) (1 + θ ) M θ − (1 − θ ) tM − θ (cid:17) sinh θτ sinh τ , (6.9)the factor in front of the hyperbolic function must be proportional to M θ − : ∂ µ M (cid:16) (1 + θ ) M θ − (1 − θ ) tM − θ (cid:17) ∼ M θ − . Integrating with respect to µ one finds, for certain normalization of µ , µ = (1 + θ ) M − tM − θ . (6.10)To summarize, the disk bulk and the boundary one-point functions with ordinary boundary condi-tion, − ∂ µ U and − ∂ x U , are given in the continuum limit in the following parametric form: − ∂ x U | µ = M θ cosh((1 + θ ) τ ) + t M − θ cosh(1 − θ ) τ , (6.11) − ∂ µ U | x ∼ M θ cosh θτ,x = M cosh τ, with the function M ( µ, t ) determined from the transcendental equation (6.10). The expression for ∂ µ U was obtained by integrating (6.9).The function M ( t, µ ) plays an important role in the solution. Its physical meaning can be revealedby taking the limit x → ∞ of the bulk one-point function − ∂ µ U ( x ) . Since x is coupled to the lengthof the boundary, in the limit of large x the boundary shrinks and the result is the partition function ofthe O ( n ) field on a sphere with two punctures, the susceptibility u ( µ, t ) . Expanding at x → ∞ wefind − ∂ µ U ∼ x θ − M θ x − θ + lower powers of x (6.12)(the numerical coefficients are omitted). We conclude that the string susceptibility is given, up to anormalization, by u = M θ . (6.13)16he normalization of u can be absorbed in the definition of the string coupling constant g s ∼ /β .Thus the transcendental equation (6.10) for M gives the equation of state of the loop gas on the sphere, (1 + θ ) u θ − t u − θθ = µ . (6.14)The equation of state (6.14) has three singular points at which the three-point function of theidentity operator ∂ µ u diverges. The three points correspond to the critical phases of the loop gas onthe sphere. At the critical point t = 0 the susceptibility scales as u ∼ µ θ . This is the dilute phaseof the loop gas, in which the loops are critical, but occupy an insignificant part of the lattice volume.The dense phase is reached when t/x θ → −∞ . In the dense phase the loops remain critical butoccupy almost all the lattice and the susceptibility has different scaling, u ∼ µ − θθ . The scaling of thesusceptibility in the dilute and in the dense phases match with the values (2.4) and (2.6) of the centralcharge of the corresponding matter CFTs. Considered on the interval −∞ < t < , the equation ofstate (6.14) describes the massless thermal flow [34] relating the dilute and the dense phases.At the third critical point ∂ µ M becomes singular but M itself remains finite. It is given by t c = 1 + θ − θ M θc > , µ c = − θ θ − θ M c < . (6.15)Around this critical point µ − µ c ∼ ( M − M c ) + · · · , hence the scaling of the susceptibility is thatof pure gravity, u ∼ ( µ − µ c ) / . We found the scaling limit of the one-point function (4.11) as a function of the renormalized bulkcouplings, µ and t , and the coupling x characterizing the ordinary boundary. Now, analyzing theloop equation (5.14) for the two-point functions, we will look for the possible scaling limits for thecouplings y , λ (1) and λ (2) , characterizing the DJS boundary.As in the previous subsection, we will write down the conditions that the regular parts of thesource terms C ( α ) vanish. Let us introduce the isotropic coupling λ and the anisotropic coupling ∆ as λ (1) = λ + ∆ , λ (2) = λ − ∆ (6.16)and substitute (5.1) in the r.h.s. of (5.14). We obtain C (1) = c + c x + c x − ∆( w + n (1) w ) , C (2) = c + c x + c x + ∆( w + n (2) w ) , (6.17)where the coefficients c and c are functions of λ and ∆ : c = ( λ − ∆ )( g H + g H ) + 2 λH + y − λg − ∆ g ( n (1) − n (2) )2(2 + n ) (6.18) c = ( λ − ∆ ) g H − − λg + ∆ g ( n (1) − n (2) )2(2 − n ) . (6.19) c = − g (cid:18) λ + n (1) − n (2) n ) ∆ (cid:19) . (6.20)For generic values of the couplings y, λ and ∆ , the coefficient c is non-vanishing. The condition c = 0 determines the critical value y c where the length of the DJS boundary diverges. Once the Indeed, the term c is the dominant term when x → . For c = 0 the solution for A ( α ) and B ( α ) in (5.14) is given bylinear functions of x and w . Such a solution describes the situation when the length of the DJS boundary is small and thetwo-point function degenerates to a one-point function. c = 0 determines thecritical lines in the space of the couplings λ (1) and λ (2) , where the DJS boundary condition becomesconformal. The two equations c ( y, λ, ∆) = 0 , c ( y, λ, ∆) = 0 ( µ = t = 0) (6.21)define a one-dimensional critical submanifold in the space of the boundary couplings { y, m, ∆ } : λ = λ ∗ (∆) . (6.22)Obviously A (1) and A (2) cannot be simultaneously zero. Therefore the curve (6.22) consists of twobranches, which correspond to different conformal DJS boundary conditions, the lines of anisotropicspecial transitions AS (1) and AS (2) : AS (1) : A (1) = 0 , A (2) = 0; AS (2) : A (2) = 0 , A (1) = 0 . (6.23)The branch AS (1) corresponds to ∆ > , while the branch AS (2) corresponds to ∆ < . Considerthe behavior of correlators D ( α )0 on the two branches of the critical line. By the definition (5.15), thetwo correlators D (1)0 and D (2)0 are related by D (1)0 = D (2)0 − ∆ D (2)0 , D (2)0 = D (1)0 D (1)0 . (6.24)On the branch AS (1) the correlator D (1)0 diverges while D (2)0 remines finite, and vice versa . Assumethat λ (1) and λ (2) are positive. Then D ( α )0 are both positive by construction. If ∆ = λ (1) − λ (2) > ,then the coefficients of the geometric series D (1)0 = ∞ X k =0 ∆ k (cid:16) D (2)0 (cid:17) k +1 (6.25)are all positive and D (1)0 diverges, while D (2)0 → / ∆ . Thus the branch AS (1) , where D (1)0 diverges,is associated with ∆ > . On this branch the probability that the loops of color (1) touch the DJSboundary is critically enhanced. In the correlator D (1)0 , the ordinary boundary behaves as a loop oftype (1) and can touch the DJS boundary. The geometrical progression (6.25) reflects the possibilityof any number of such events, each contributing a factor ∆ . Conversely, the ordinary boundary for thecorrelator D (2)0 behaves as a loop of type (2) , since such loops almost never touch the DJS boundary.On the branch AS (2) the situation is reversed.It is not possible to solve explicitly the conditions of criticality (6.21) without extra information,because they contain two unknown functions of the three couplings, H and H . Nevertheless, thequalitative picture can be reconstructed.First let us notice that the form of the critical curve can be evaluated in the particular cases n (1) = n and n (1) = 0 . In the first case n (2) = 0 and the correlation functions do not depend on λ (2) and so thecoefficients c and c depend on λ and ∆ through the combination λ (1) = λ + ∆ / . Similarly oneconsiders the case n (1) = 0 . The phase diagram in these two cases represents an infinite straight lineseparating the ordinary and the extraordinary transitions: λ ∗ (∆) = ( λ c − ∆ / if n (1) = n,λ c + ∆ / if n (1) = 0 . (6.26)18 xt AS AS (2) (2) λλ (1)(1) ∆ Sp λ ExtOrd (1) (2)
Figure 5:
Phase diagram in the rotated ( λ (1) , λ (2) ) plane for n < and n (1) > n (2) . The ordinary and theextraordinary phases are separated by a line of anisotropic special transitions, which consists of two branches, AS (1) and AS (2) . The two extraordinary phases, Ext (1) and
Ext (2) , are separated by the isotropic line ∆ = 0
The critical line crosses the axis ∆ = 0 at the special point λ = λ c . The value of λ c can be evaluatedby solving (6.21) for n (1) = n and λ (2) = 0 . The result is λ c = p (6 + n )(2 − n )1 − n . (6.27)For general n (1) ∈ [0 , n ] we can determine three points of the critical curve: { λ (1) , λ (2) } = { − n − n (1) λ c , } , { , − n − n (2) λ c } , { λ c , λ c } . (6.28)In the two limiting cases considered above the critical line, given by equation (6.26), crosses theanisotropic line without forming a cusp. Is this the case in general? Let us consider the vicinity of thespecial point ( λ, ∆) = ( λ c , . In the vicinity of the special point a new scaling behavior occurs. Inthis regime the term x in (6.17) cannot be neglected. The requirement that all terms in (6.17) havethe same dimension determines the scaling of the anisotropic coupling ∆ : [∆] = 1 − θ. (6.29)In order to determine the form of the critical curve near the special point, we return to the equations(6.21) and consider the behavior near the special point of the unknown functions H ( y ) = β h tr H ( y ) i and H ( y ) = β h tr XH ( y ) i , which depend implicitly on λ and ∆ . For t = µ = 0 , these functions canbe decomposed, just as the one-point function with ordinary boundary condition, W ( x ) , into regularand a singular parts: H ( y ) = H reg ( y ) + h ( y ) , H ( y ) = H reg1 ( y ) + cst · h ( y ) . (6.30)19n the critical curve λ = λ ∗ (∆) the singular part of H vanishes and the coefficient c given by eq.(6.19) can be Taylor expanded in λ − λ c and ∆ : c ( m, ∆) ≡ A ( λ − λ c ) + B ( n (1) − n (2) )∆ + A ( λ − λ c ) + B ∆ + · · · = 0 . (6.31)If A = 0 , the critical curve is given by a regular function of λ and ∆ , the critical curve is a continuousline which crosses the real axis at λ = λ c without forming a cusp. This form of the curve differs fromthe predictions of [2] and [3], where a cusp-like form is predicted by a scaling argument. We will seelater that the fact that the critical curve is analytic at the special point does not contradict the scaling(6.29). ∆ = 0 Consider first the case when the anisotropic coupling ∆ is finite and assume that we are on the branch AS (1) where ∆ > . Then the x term on the r.h.s. can be neglected, because it is subdominantcompared to w ∼ x θ . The scaling limit corresponds to the vicinity of the critical submanifoldwhere the two coefficients c and c scale respectively as x θ and x θ .We are now going to find the scaling limit of the loop equations (5.14). In the scaling limit we canretain only the singular parts of the correlators D (1) , (2) L , which we denote by d (1) , (2) L ( L = 0 , , ... ).We define the functions d (1)0 and d (2)0 by D (1)0 = d (1)0 , D (2)0 = 1∆ − d (2)0 ≈
1∆ + 1∆ d (2)0 . (6.32)Then the relation (6.24) implies d (1)0 d (2)0 = − . (6.33)We define in general d (1) , (2) L as the singular part of D (1) , (2) L , with the normalization chosen so that therecurrence equation (5.7) holds for any L ≥ : d (1) L +1 = w ⋆ d (1) L ,d (2) L +1 = w ⋆ d (2) L . (6.34)On the branch AS (1) the first of the two equations (5.14) becomes singular, since A (1) vanishes while A (2) remains finite. We write this equation in terms of d (2)0 and d (1)1 using that A (1) = λ (1) d (2)0 , B (1) = ∆ λ (1) d (1)1 . (6.35)We get d (2)0 d (1)1 + w + n (1) w = µ B − t B x , (∆ > (6.36)where µ B and t B are defined by c ∆ = µ B , c ∆ = − t B . (6.37)20nce the solution of (6.37) is known, all two-point functions d ( α ) L can be computed by using therecurrence equations (6.34).The scaling limit near the branch AS (2) ( ∆ < ) is obtained by using the symmetry n (1) ↔ n (2) , ∆ ↔ − ∆ . In this case one obtains another equation d (1)0 d (2)1 + w + n (2) w = µ B − t B x . (∆ < (6.38)Note that the relation (6.33) is true on both branches of the critical line.The map { y, λ } → { µ B , t B } defined by (6.18), (6.19) and (6.37) represents a coordinate changein the space of couplings which diagonalizes the scaling transformation. The coupling µ B is therenormalized boundary cosmological constant for the DJS boundary. The coupling t B is the renor-malized boundary matter coupling, which defines the DJS boundary condition. The dimensions ofthese couplings are [ µ B ] = 1 + θ, [ t B ] = θ. (6.39)Once we choose y so that µ B = 0 , the condition t B = 0 gives the critical curve where the anisotropicspecial transitions take place. If the function t B is regular near the critical line λ = λ ∗ (∆) , then it canbe replaced by the linear approximation t B ∼ λ ∗ (∆) − λ. (6.40)The deformations in the directions t B and ∆ , are driven by some Liouville dressed boundaryoperators O Bt B and O B ∆ . Knowing the dimensions of t B and ∆ , we can determine the Kac labels ofthese operators with the help of the KPZ formula. The general rule for evaluating the Kac labels in2D gravity with matter central charge (2.4) is the following. If a coupling constant has dimension α ,then the corresponding operator has Kac labels ( r, s ) determined by α = α r,s = min (cid:18) θ ± (1 + θ ) r − s (cid:19) . (6.41)The details of the identification are given in Appendix B. We find O Bt B = O B , , O B ∆ = O B , . (6.42)Near the special point we have ∆ ∼ t /φB , φ = θ − θ = α , α , < . (6.43)Since t B and λ scale differently, there is no contradiction between the scaling (6.43) and the analyticityof the critical curve near the special point. More precisely, it is a combination of the boundary coupling constant and the disk one-point function with DJS bound-ary. What is important for us is that the condition µ B = 0 fixes the critical value of the bare DJS cosmological constant y .At µ B = 0 , the length of the DJS boundary diverges. .3.2 The scaling limit in the isotropic direction (∆ = 0) Along the isotropic line ∆ = 0 the two functional equations (5.14) degenerate into a single equationfor the correlator D : A ≡ λD − − y − x + λ (2 H + λP − V ′ ) y − x + λ ( W + H − V ′ ) . (6.44)In order to evaluate D = D (1)1 = D (2)1 , we can consider the linear order in ∆ . It is however easier touse the fact that D and D do not depend on the splitting n = n (1) + n (2) . Furthermore, if we choose n (1) = n and λ (1) = λ , the observables do not depend on λ (2) , which can be chosen to be zero. Taking n (1) = n , λ (1) = m and n (2) = λ (2) = 0 , we obtain from (5.14) B (1) (cid:12)(cid:12) n (1) = n ≡ λ (cid:0) λD + nW (cid:1) + x − y = y − x + λ ( H − W − nW ) λD − . (6.45)From these expressions it is clear how the scaling of the singular parts of D and D , which wedenote respectively by d and d , change when we go from λ = 0 to λ = λ c . When λ = 0 wehave H ( y ) = W ( y ) and D is the disk partition function with ordinary boundary conditions and twomarked points on it, eq. (4.17). When λ = λ c and y = y c , d = g x w ∼ x − θ , d = w ( w + n ¯ w ) g x ∼ x θ ( λ = λ c , y = y c ) . (6.46) Now we will focus on the special case n (1) = 1 and compare the scaling behavior with that for theDirichlet boundary conditions. The critical behavior of the two-point correlator in both cases is thesame, but the boundary coupling constants correspond to different boundary operators.Consider the functional equation (5.25) for the correlator with Ord/DJS boundary conditions when n (1) = 1 . The critical value of λ (1) is infinite in this case, see equation (6.28). The scaling limit of(5.25) is d (1)0 ( x, y ) d (2)1 ( − x, y ) = − w ( x ) − w ( − x ) + µ B − xλ (1) . (6.47)The last term remains finite if λ (1) tends to infinity as x − θ . The scaling boundary coupling can beidentified as t B = 1 /λ (1) and equation (6.47) takes the general form (6.36). What is remarkable hereis that the boundary temperature constant need not to be tuned. Equation (6.47) holds for any valueof λ (1) . On the other hand, when t B = 1 /λ (1) is small, the last term describes the perturbation ofthe AS (1) boundary condition by the thermal operator O , with α , = θ . When λ (1) is small, thelast term accounts for the perturbation of the ordinary boundary condition by the two-leg boundaryoperator O , with α , = − θ , whose matter component is an is irrelevant operator.Now let us take the scaling limit of the quadratic functional equation (5.24) for the correlator withOrd/Dir boundary conditions. At x = 0 the equation (5.24) becomes algebraic. The critical value y = y c , where the solution develops a square root singularity, is determined by W (0) − R ( y c ) + y c /B = 0 . We can write equation (5.24) as G ( x, y ) G ( − x, y ) = µ B − w ( x ) − w ( − x ) − x B , (6.48)22here µ B = 2 w (0) + 2[ R ( y ) − R ( y c )] + y − y c B . (6.49)For any finite value of B , the scaling limit of this equation is G ( x, y ) G ( − x, y ) = µ B − w ( x ) − w ( − x ) . (6.50)The x term survives only if B vanishes as x − θ : [ B ] = (1 − θ ) / α , . (6.51)This is the expected answer, because O B , is the one-leg boundary operator which creates an open linestarting at the boundary. We conclude that the Dirichlet and the DJS boundary conditions have thesame scaling limit, but in the first case the boundary coupling λ corresponds to a relevant perturba-tion and it is sufficient give it any finite value, while in the second case the boundary coupling λ (1) corresponds to an irrelevant perturbation and therefore must be infinitely strong. Let us denote by α (1) L and α (2) L the scaling dimensions respectively of d (1) L and d (2) L : α ( α ) L = [ d ( α ) L ] ( L ≥ , a = 1 , . (7.1)The recurrence equations (6.34) tell us that the dimensions grow linearly with L : α ( α ) L = L [ w ] + α ( α )0 . (7.2)These relations make sense in the dilute phase, where [ w ] = 1 + θ , as well as in the dense phase,where [ w ] = 1 − θ . In addition, by (6.33) we have α (1)0 + α (2)0 = 0 . (7.3)Thus all scaling dimensions are expressed in terms of α (1)0 .Let us evaluate α for the branch AS (1) of the critical line. We thus assume that ∆ is finite andpositive sufficiently far from the isotropic special point. Take µ = µ B = t B = 0 and write the shiftequations which follow from (6.36), AS (1) : d (1)0 ( e iπ x ) d (1)0 ( e − iπ x ) = w ( e − iπ x ) + n (1) w ( x ) w ( e iπ x ) + n (1) w ( x ) . (7.4)The one-point function (6.8) behaves as w ∼ x θ in the dilute phase ( t = 0 ) and as w ∼ x − θ in thedense phase ( t → −∞ ). In both cases the r.h.s. is just a phase factor. Since all the couplings except for x have been turned off, d (1)0 ( x ) should be a simple power function of x . Substituting d (1) L ( x ) ∼ x α (1) L in (7.4), we find n (1) = sin π ( α (1)0 ± θ )sin πα (1)0 , (+ for dilute , − for dense ) . (7.5)23his equation determines the exponent α (1)0 up to an integer. In the parametrization (2.15) we have α (1)0 = − θr + j dil in the dilute phase and α (1)0 = θr + j den in the dense phase.The integers j dil and j den can be fixed by additional restrictions on the exponents. Let us assumethat λ (1) and λ (2) are non-negative, n ≥ and the boundary parameter r is in the ‘physical’ interval ≤ r ≤ /θ − , where both n (1) and n (2) are non-negative. These assumptions guarantee thatthe Boltzmann weights are positive and the loop expansion of the observables has good statisticalmeaning. Since all loop configurations that enter in the loop expansion of the one-point function W ( x ) are present in the loop expansions of A ( α ) and B ( α ) , the singularity of these observables when λ → λ ∗ (∆) must not be weaker than that of W . In other words, the scaling dimensions of d (2)0 ∼ A (1) and d (1)1 ∼ B (1) must not be larger than the scaling dimension of the one-point function w : α (2)0 < [ w ] , α (1)1 < [ w ] . (7.6)Since α (2)0 + α (1)1 = [ w ] , this also means that α (2)0 and α (1)1 are non-negative. Taking into account that [ w ] = 1 ± θ in the dilute/dense phase, we get the bound − (1 ± θ ) ≤ α (1)0 ≤ for dilute , − for dense ) . (7.7)This bound determines j dil = 0 and j den = − . As a consequence, on the branch AS (1) of the criticalline the dimensions α (1) L = [ d (1) L ] in the dilute and in the dense phases are given by AS (1) : α (1) L = L (1 + θ ) − θr, α (2) L = L (1 + θ ) + θr (dilute phase) α (1) L = L (1 − θ ) + θr − , α (2) L = L (1 − θ ) − θr + 1 (dense phase) . (7.8)Note that the results for the dense phase are valid not only in the vicinity of the critical line AS (1) , butin the whole half-plane ∆ > .By the symmetry (1) ↔ (2) , the exponents α (1) L on the branch AS (1) and the exponents α (2) L onthe branch AS (2) should be related by n (1) ↔ n (2) , or equivalently r ↔ /θ − r : AS (2) : α (1) L = L (1 + θ ) − θr + 1 , α (2) L = L (1 + θ ) + θr − , (dilute phase) α (1) L = L (1 − θ ) + θr, α (2) L = L (1 − θ ) − θr (dense phase) . (7.9)The scaling exponents of the two-point functions of the O ( n ) model coupled to 2D gravity al-low, through the KPZ formula [23, 24], to determine the conformal weights of the matter boundaryoperators. In the dilute phase, where the Kac parametrization is given by (2.5), the correspondencebetween the scaling dimension α of a boundary two-point correlator and the conformal weight h r,s ofthe corresponding matter boundary field is given by α = (1 + θ ) r − s → h = h r,s (dilute phase). (7.10)In the dense phase, where the Kac labels are defined by (2.7), one obtains, taking into account that theidentity boundary operator for the ordinary boundary condition has ‘wrong’ dressing, α = r − s (1 − θ ) → h = h r,s (dense phase) . (7.11)From (7.10) and (7.11) we determine the scaling dimensions of the L -leg boundary operators (3.9): AS (1) : S (1) L → O Br − L,r , S (2) L → O Br + L,r (dilute phase) S (1) L → O Br − ,r − L , S (2) L → O Br − ,r + L (dense phase) . (7.12)24 S (2) : S (1) L → O Br − L,r +1 , S (2) L → O Br + L,r +1 (dilute phase) S (1) L → O Br,r − L , S (2) L → O Br,r + L (dense phase) . (7.13)These conformal weights are in accord with the results of [7], [13], [14], [3]. We remind that thescaling dimensions are determined up to a symmetry of the Kac parametrization: h r,s = h − r, − s , h r,s − = h r +1 /θ,s +1 /θ (dilute phase) (7.14) h r,s = h − r, − s , h r +1 ,s = h r +1 /θ,s +1 /θ (dense phase) . (7.15)Comparing the scaling dimensions in the dilute and in the dense phase, we see that the bulk thermalflow t O , transforms the boundary operator O r,s in the dilute phase into the boundary operator O s − ,r in the dense phase. For the rational points θ = 1 /p , our results for the endpoints of the bulkthermal flow driven by the operator t O , match the perturbative calculations performed recently in[15].In the O ( n ) model the boundary parameter r is continuous and we can explore the limit r → ,in which the BCC operator O r,r carries the same Kac labels as the identity operator. Let us callthis operator ˜ O B , . The bulk thermal flow transforms the operators ˜ O B , and O B , into two differentboundary operators in the dense phase. Hence there are at least two distinct boundary operators withKac labels (1 , : the identity operator and the limit r → of the operator O Br,r . We are going to study two particular cases where the analytic solution of the functional equations(6.36) and (6.38) is accessible. First we will evaluate the two-point function on the two branches AS (1) and AS (2) of the critical line, where the O ( n ) field is conformal invariant both in the bulk andon the boundary. In this case t = t B = 0 and the boundary two-point function is that of Liouvillegravity. The three couplings are introduced by the world sheet action of Liouville gravity with mattercentral charge (2.4), which we write symbolically as S Liouv = S free + Z bulk µ O , + Z Ord. boundary x O B , + Z DJS boundary µ B O B , . (8.1)Since the perturbing operators in this case are Liouville primary fields, O , ∼ e bφ , O B , ∼ e bφ , the two-point function is given by the product of matter and Liouville two-point functions. Up to anumerical factor, the solution as a function of µ and µ B must be given by the boundary two-pointfunction in Liouville theory [35]. We will see that indeed the functional equation (6.36) is identicalto a functional equation obtained in [35] using the operator product expansion in boundary Liouvilletheory.In the second case we are able to solve, we take µ = µ B = 0 and non-zero matter couplings t and t B . This case is more interesting, because it is not described by the standard Liouville gravity. Thecorresponding world sheet action is symbollically written as S = S free + Z bulk t O , + Z Ord. boundary x O B , + Z DJS boundary t B O B , . (8.2)25he worldsheet theory described by this action is more complicated than Liouville gravity, because itdoes not enjoy the factorization properties of the latter. The boundary two-point correlator does notfactorize, for finite t and/or t B , into a product of matter and Liouville correlators, as is the case for theaction (8.1). This is because the perturbing operators O , and O B , have both matter and Liouvillecomponents: O , ∼ Φ , e b (1 − θ ) φ , O B , ∼ Φ B , e b (1 − θ ) φ . Let us mention that the theory of random surfaces described by the action (8.2) has no obvious directmicroscopic realization. Our solution interpolates between the two-point functions for the dilute ( t =0 ) and the dense ( t → −∞ ) phases of the loop gas, on one hand, and between the anisotropic special ( t B = 0) and ordinary/extraordinary boundary conditions ( t B → ±∞ ), on the other hand. t = 0 , t B = 0 In the dilute phase ( t = 0 ) the solution (6.6) - (6.8) for the boundary one-point function takes the form x = M cosh τ, w ( x ) = M θ cosh(1 + θ ) τ. (8.3)Then the loop equations (6.36) become a shift equation d (2)0 ( τ ) d (1)1 ( τ ± iπ ) + w cosh [(1 + θ ) τ ± iπ (1 − r ) θ ] = µ B − t B M cosh τ , (8.4)where we introduced the constant w = M θ sin πθ sin πrθ . (8.5)At the point t B = 0 , where the DJS boundary condition is conformal, this equation can be solvedexplicitly. After a shift τ → τ ∓ iπ we write it, using (6.33), as d (1)1 ( τ ) = [ w cosh [(1 + θ ) τ ± iπrθ ] − µ B ] d (1)0 ( τ ± iπ ) . (8.6)If we parametrize µ B in terms of a new variable σ as µ B = w cosh(1 + θ ) σ, (8.7)the loop equation turns out to be identical to the functional identity for the boundary Liouville two-point function [35], which we recall in Appendix B. In the Liouville gravity framework, τ and σ parametrize the FZZT branes corresponding to the ordinary and anisotropic special boundary condi-tions.The loop equations for the dense phase ( t → −∞ ), are given by (8.6) with θ sign-flipped. Thisequation describes the only scaling limit in the dense phase. The term with t B is absent in the densephase, because it has dimension θ , while the other terms have dimension − θ . In this case, theloop equation gets identical to the functional identity for the Liouville boundary two-point function ifwe parametrize µ B as µ B = w cosh (cid:0) − θ ) σ. (8.8)26 .2 Solution for µ = 0 , µ B = 0 Here we solve the loop equation (6.36) in the scaling limit with µ = µ B = 0 but keeping x , t and t B finite. Let us first find the expression for the one-point function w ( x ) for µ = 0 . The equation (6.10)has in this case two solutions, M = 0 and M = (1 + θ ) − t θ . One can see [33] that the first solutionis valid for t < , while the second one is valid for t > . Therefore when µ = 0 and t ≤ , thesolution (6.6)-(6.8) takes the following simple form: w ( x ) = x θ + tx − θ ( t ≤ . (8.9)Introduce the following exponential parametrization of x, t, t B in terms of τ, γ, ˜ γ : x = e τ , t = − e γθ , t B = − w e γθ sinh(˜ γθ ) , w = sin( πθ )sin( πrθ ) . (8.10)In terms of the new variables, equation (6.36) with µ B = µ = 0 acquires the form d (1)1 ( τ ) /d (1)0 ( τ ± iπ ) = w (cid:16) − e γθ sinh(˜ γθ ) e τ + e (1+ θ ) τ ± iπrθ − e θγ e (1+ θ ) τ ± iπrθ (cid:17) = 4 w e τ + γθ cosh θ ( τ − γ + ˜ γ ± iπr )2 sinh θ ( τ − γ − ˜ γ ± iπr )2 . (8.11)Taking the logarithm of both sides we obtain a linear difference equation, which can be solved explic-itly. The solution is given by AS (1) : d (1)0 ( τ ) = w e − τ − γ ( rθ − )+ ˜ γ V − r ( τ − γ + ˜ γ ) V θ − r ( τ − γ − ˜ γ ) ,d (1)1 ( τ ) = − e τ + γ ( + θ − rθ )+ ˜ γ V − r ( τ − γ + ˜ γ ) V θ − r ( τ − γ − ˜ γ ) , (8.12)where the function V r ( τ ) is defined by log V r ( τ ) def = − Z dωω (cid:20) e − iωτ sinh( πrω )sinh( πω ) sinh πωθ − rθπω (cid:21) . (8.13)The properties of the function V r ( τ ) are listed in Appendix C.The solution (8.12) reproduces correctly the scaling exponents (7.8) and it is unique, assumingthat the correlators d (1)0 and d (1)1 have no poles as functions of x . Near the branch AS (2) , the functions d (2)0 and d (2)1 are given by the same expressions (8.12), but with r replaced by /θ − r . To explore the scaling regimes of the solution (8.12) we use the expansion (C.2) and return to theoriginal variables, e τ = x, e γθ = ( − t ) θ , e ( γ ± ˜ γ ) θ = ∓ t B r t B − t ! θ . (8.14)Let us define the function ˆ V ( x ) by V r ( τ ) = ˆ V r ( e τ ) . The large x expansion of ˆ V r goes, according to(C.5), as ˆ V r ( x ) = x rθ/ (cid:18) πr sin π/θ x − + sin πrθ sin πθ x − θ + . . . (cid:19) . (8.15)27he expansion at small x follows from the symmetry ˆ V ( x ) = ˆ V (1 /x ) . Written in terms of the originalvariables, the scaling solution near the branch AS (1) takes the form d (1)0 ( x ) = 1 w ( − t ) − r √ x (cid:18) − t B / q t B / − t (cid:19) θ × ˆ V − r " x (cid:18) t B / q t B / − t (cid:19) − θ ˆ V θ − r " x (cid:18) − t B / q t B / − t (cid:19) − θ (8.16) d (1)1 ( x ) = −√ x t − r (cid:18) − t B / q t B / − t (cid:19) θ × ˆ V − r " x (cid:18) t B / q t B / − t (cid:19) − θ ˆ V θ − r " x (cid:18) − t B / q t B / − t (cid:19) − θ . (8.17)The critical regimes of this solution are associated with the limits t → − , −∞ and t B → , ±∞ of the bulk and the boundary temperature couplings. (i) Dilute phase, anisotropic special transitions This critical regime is achieved when both t and t B are small. Using the asymptotics (8.15), wefind that in the limit ( t B → , t → − the expressions (8.16)-(8.17) reproduce the correct scalingexponents (7.8) in the dilute phase: AS (1) : d (1)0 ∼ x − rθ , d (1)1 ∼ x θ − θr ( t B = 0 , t → − . (8.18)The regime AS (2) is obtained by replacing (1) → (2) , r → θ − r . (ii) Dilute phase, ordinary transition At t → − , the leading behavior of d (2)0 = 1 /d (1)0 and d (1)1 for large t B is (we omitted all numericalcoefficients) d (2)0 ∼ t Br (cid:16) t B − θ x + t − B x θ + . . . (cid:17) d (1)1 ∼ t B − r (cid:16) x + t B − θ x + t − B x θ +1 + t − B x θ +1 + . . . (cid:17) ( t → − , t B → + ∞ ) . (8.19)In the expansion for d (2)0 , the first singular term, x θ , is the singular part of D with ordinary/ordinaryboundary conditions. In the expansion for d (1)1 , the first singular term, x θ , is the one-point function w , while the next term, x θ +1 , is the singular part of the boundary two-point function D (1)1 withordinary/ordinary boundary conditions. (iii) Dilute phase, extraordinary transition Now we write the asymptotics of (8.16)-(8.17) in the opposite limit, t → − , and t B → −∞ : d (2)0 ∼ t B − θ + r (cid:16) x + t B − θ x + t − B x θ +1 + t − B x θ +1 + . . . (cid:17) d (1)1 ∼ t B θ +1 − r (cid:16) t B − θ x + t − B x θ + . . . (cid:17) ( t = 0 , t B → −∞ ) . (8.20)This asymptotics reflects the symmetry of the solution (8.12) which maps r → /θ − r, d (2)0 ↔ d (1)1 (8.21)28hich is also a symmetry of the loop equations (5.14). In the limit of large and negative t B , thefunction d (2)0 behaves as the singular part of the correlator D (1)1 with ordinary/ordinary boundary con-ditions, while d (1)1 behaves as the singular part of the correlator D with ordinary/ordinary boundaryconditions.The asymptotics of the solution at t B → ±∞ confirms the qualitative picture proposed in [3] andexplained in the Introduction. When t B is large and positive, the loops avoid the boundary and wehave the ordinary boundary condition. In the opposite limit, t B → −∞ , the DJS boundary tends to becoated by loop(s). Therefore the typical loop configurations for D (1)1 in the limit t B → −∞ will looklike those of D (1)0 in the ordinary phase, because the open line connecting the two boundary-changingpoints will be adsorbed by the DJS boundary. Conversely, the typical loop configurations for D (1)0 will look like those of D (1)1 in the ordinary phase, because free part of the loop that wraps the DJSboundary will behave as an open line connecting the two boundary-changing points.We saw that the solution reproduces the qualitative phase diagram for the dilute phase, shown inFig. 5. Now let us try to reconstruct the phase diagram in the dense phase. (iv) Dense phase, anisotropic special transitions For any finite value of t B , the dense phase is obtained in the limit t → −∞ . The asymptotics of(8.16)-(8.17) in this limit does not depend on t B : d (1)0 ∼ x rθ − ( − t ) θ − r , d (1)1 ∼ x − (1 − r ) θ ( − t ) − r + θ ( t → −∞ ) . (8.22)This means that in the dense phase the DJS boundary condition is automatically conformal for anyvalue of t B . The boundary critical behavior does not change with the isotropic boundary coupling t B , but it can depend on the anisotropic coupling ∆ . The solution (8.16)-(8.17) holds for any positivevalue of ∆ . For negative ∆ we have another solution, which is obtained by replacing (1) → (2) and r → /θ − r . Thus in the dense phase there are two possible critical regimes for the DJS boundary,one for positive ∆ and the other for negative ∆ , which are analogous to the two anisotropic specialtransitions in the dilute phase. The domains of the two regimes are separated by the isotropic line ∆ = 0 .The above is true when t B is finite. If t B tends to ±∞ , we can obtain critical regimes with theproperties of the ordinary and the extraordinary transitions. (v) Dense phase, ordinary and extraordinary transitions If we expand the solution (8.16)-(8.17) for − t ≫ x θ and t B ≫ − tx − θ , the singular parts of thetwo correlators will be the same as the correlators with ordinary boundary condition on both sides.For example, instead of the term t − B x θ in the expression for d (2)0 , we will obtain t − t B x − θ . This isthe singular part of the correlator D with ordinary boundary conditions in the dense phase. Further,the asymptotics of the solution in the limit − t ≫ x θ and t B ≪ tx − θ is determined by the symmetry(8.21). This critical regime has the properties of the extraordinary transition, in complete analogy withthe dilute case. We conclude that the ordinary and the extraordinary transitions exist also in the densephase, but they are pushed to t B → ±∞ .Finally, let us comment on the possible origin of the square-root singularity of the solution (8.16)-(8.17) at t B = 4 t . This singularity appears in the disordered phase, t > , which is outside the domainof validity of the solution. Nevertheless, one can speculate that this singularity is related to the surfacetransition, which separates the phases with ordered and disordered spins near the DJS boundary. Thesingularity in our solution has two branches, t B = ± √ t , while in the true solution for t > thenegative branch should disappear. 29 Conclusions
In this paper we studied the dilute boundary O ( n ) model with a class of anisotropic boundary condi-tions, using the methods of 2D quantum gravity. The loop gas formulation of the anisotropic boundaryconditions, proposed by Dubail, Jacobsen and Saleur (DJS ), involves two kinds of loops having fu-gacities n (1) and n (2) = n − n (1) . Besides the bulk temperature, which controls the length of the loops,the model involves two boundary coupling constants, which define the interaction of the two kinds ofloops with the boundary.The regime where the DJS boundary condition becomes conformal invariant is named in [3]anisotropic special transition. The enhanced symmetry of the model after coupling to gravity sys-tem allowed us to solve the model analytically away from the anisotropic special transition. We usedthe solution to explore the deformations away from criticality which are generated by the bulk andboundary thermal operators.Our main results can be summarized as follows.1) We found the phase diagram for the boundary transitions in the dilute phase of the O ( n ) modelwith anisotropic boundary interaction. The phase diagram is qualitatively the same as the one obtainedin [3] and sketched in the Introduction. The critical line consists of two branches placed above andbelow the isotropic line. Near the special point the critical curve is given by the same equation on bothsides of the isotropic line, which means that the two branches of the critical line meet at the specialpoint without forming a cusp. We also demonstrated that the analytic shape of the critical curve doesnot contradict the scaling of the two boundary coupling constants. This contradicts the picture drawnin [2] on the basis of scaling arguments, which seems to be supported by the numerical analysis of[3]. Of course we do not exclude the possibility that the origin of the discrepancy is in the fluctuationsof the metric.2) From the singular behavior of the boundary two-point functions we obtained the spectrumof conformal dimensions of the L -leg boundary operators between ordinary and anisotropic specialboundary conditions, which is in agreement with [3]. In order to establish the critical exponents weused substantially the assumption that n (1) and n (2) are both non-negative.3) We showed that the two-point functions of these operators coincide with the two-point func-tions in boundary Liouville theory [35]. The functional equation for the boundary two-point functionobtained from the Ward identities in the matrix model is identical to the functional equation derivedby using the OPE in boundary Liouville theory.4) The result which we find the most interesting is the expression for the two-point functionsaway from the critical lines. For any finite value of the anisotropic coupling ∆ , the deviation fromthe critical line is measured by the renormalized bulk and the boundary thermal couplings, t and t B .Our result, given by eqs. (8.16)-(8.17), gives the boundary two-point function in a theory which issimilar to boundary Liouville gravity, except that the bulk and the boundary Liouville interactionsare replaced by the Liouville dressed bulk and boundary thermal operators, t O , and t B O B , . Theboundary flow, generated by the boundary operator O B , , relates the anisotropic special transitionwith the ordinary and the extraordinary ones. The bulk thermal flow, generated by the operator O , ,relates the dilute and the dense phases of the O ( n ) model coupled to gravity. At the critical value ofthe boundary coupling, the bulk flow induces a boundary flow between one DJS boundary conditionin the dilute phase and another DJS boundary condition in the dense phase. For the rational values ofthe central charge, the boundary conditions associated with the endpoints of the bulk flow match withthose predicted by the recent study using perturbative RG techniques [15].Here we considered only the boundary two-point functions with ordinary/DJS boundary condi-tions. It is not difficult to write the loop equations for the boundary ( n + 1) -point functions with one30rdinary and n DJS boundaries. The loop equations for n > will depend not only on the parameterscharacterizing each segment of the boundary, but also on a hierarchy of overlap parameters that definethe fugacities of loops that touch several boundary segments. The loop equations for the case n = 2 were studied for the dense phase in [27]. In this case there is one extra parameter, associated withthe loops that touch both DJS boundaries, which determines the spectrum of the boundary operatorscompatible with the two DJS boundary conditions. In the conformal limit, the loop equations for the (1 + n ) -point functions should turn to boundary ground ring identities, which, compared to thosederived in [36] for gaussian matter field, will contain a number of extra contact terms with coefficientsdetermined by the overlap parameters. It would be interesting to generalize the calculation of [27] tothe dilute case and compare with the existing results [37, 38] for the 3-point functions in Liouvillegravity with non-trivial matter field.The method developed in this paper can be generalized in several directions. Our results wereobtained for the O ( n ) model, but they can be easily extended to other loop models, as the dilute ADEhight models. It is also clear that the method works for more general cases of anisotropic boundaryconditions, with the O ( n ) invariance broken to O ( n ) × · · · × O ( n k ) .Finally, let us mention that there is an open problem in our approach. The loop equations doesnot allow to evaluate the one-point function with DJS boundary conditions, H ( y ) , except in someparticular cases. There are two possible scaling limits for this function, which correspond to the twoLiouville dressings of the identity operator with DJS boundary condition, and it can happen that bothdressings are realized depending on the boundary parameters. This ambiguity does not affect theresults reported in this paper. Acknowledgments
We thank J. Dubail, J. Jacobsen and H. Saleur for useful discussions. This work has been supported inpart by Grant-in-Aid for Creative Scientific Research (project
A Derivation of the loop equations
Here we give the derivation of the loop equations which are extensively discussed in this work.We first summarize our technique to derive loop equations. The translation invariance of thematrix measure implies, for any matrix F made out of X and Y a , the identities β h ∂ X F i + h tr (cid:2) ( − V ′ ( X ) + ~ Y ) F (cid:3) i = 0 , (A.1) β h ∂ Y a F i + h tr (cid:2) Y a ( XF + FX ) (cid:3) i = 0 , (A.2)where the derivatives with respect to matrices are defined by ∂ X F ≡ ∂F ij /∂X ij summed over theindices i and j , and are generally given by sums of double traces. Written for the observable G = − ( XF + FX ) , the second equation states (cid:10) tr (cid:0) Y a G (cid:1)(cid:11) = 1 β I i R dx πi (cid:10) ∂ Y a (cid:2) W ( x ) GW ( − x ) (cid:3)(cid:11) . (A.3) Here it is assumed that the eigenvalues of X are all in the left half plane. This is indeed true in the large N limit of ourmatrix integral. N factorization h tr A · tr B i ≃ h tr A i h tr B i (A.4)to derive various relations among disk correlators. A.1 Loop equation for the resolvent
An equation for the resolvent (4.11) is obtained if we take F = W ( x ) . The identity (A.1) then gives W ( x ) = 1 β (cid:10) tr (cid:2) V ′ ( X ) W ( x ) (cid:3)(cid:11) − n X a =1 β (cid:10) tr (cid:2) Y a W ( x ) (cid:3)(cid:11) . (A.5)Then the identity (A.3) applied to the last term gives (cid:10) tr (cid:2) Y a W ( x ) (cid:3)(cid:11) = 1 β I i R dx ′ πi (cid:10) ∂ Y a (cid:2) W ( x ′ ) W ( x ) Y a W ( − x ′ ) (cid:3)(cid:11) = 1 β I i R dx ′ πi (cid:10) tr (cid:2) W ( x ′ ) W ( x ) (cid:3) tr (cid:2) W ( − x ′ ) (cid:3)(cid:11) = − β Z i R dx ′ πi W ( x ) − W ( x ′ ) x − x ′ W ( − x ′ ) . (A.6)Using the ⋆ -product introduced in (5.6), the last line can be written as β [ W ⋆ W ]( x ) . The equation(A.5) can then be written as W ( x ) − V ′ ( x ) W ( x ) + n [ W ⋆ W ]( x ) = 1 β (cid:28) tr (cid:16) V ′ ( X ) − V ′ ( x ) x − X (cid:17)(cid:29) . (A.7)For a cubic potential the expectation value on the r.h.s. is a polynomial of degree one. Using animportant property of the ⋆ -product [ f ⋆ g ]( x ) + [ g ⋆ f ]( − x ) = f ( x ) g ( − x ) , (A.8)which can be proved by deforming the contour of integration, one obtains a loop equation [31], whichis a quadratic functional equation for W ( x ) . The term linear in W can be eliminated by a shift w ( x ) def = W ( x ) − V ′ ( x ) − nV ′ ( − x )4 − n . (A.9)The loop equation for w ( x ) is given by (5.3). A.2 Loop equations for the boundary two-point functions
The boundary two-point functions of L -leg operators satisfy the recurrence equations D (1) L +1 = W ⋆ D (1) L , D (2) L +1 = W ⋆ D (2) L ( L ≥ which can be derived by applying (A.3) with F = WS L +1 HS L . By applying (A.3) to F = WY a H and WHY a , we find D (1)1 = ( λ (1) D (1)1 + W ) ⋆ D , βn (1) D tr (cid:2) WHY (cid:3)E = D ⋆ ( λ (1) D (1)1 + W ) , (A.10)32nd a similar pair of equations for D and D (2)1 . The first equation of (A.10) can be rewritten into anequation for the discontinuity along the branch cut,Disc D (1)1 ( x ) = D ( − x ) Disc ( λ (1) D (1)1 ( x ) + W ( x )) , (A.11)which implies that the recurrence equation can be extended to L = 0 by defining D (1)0 def = D λ (1) D − , D (2)0 def = D λ (2) D − . Also, by applying (A.1) to F = WH one finds ( W + H ) D = V ′ ( x ) D − P ( x ) − β * tr (cid:2) WH X α =1 , Y α ) (cid:3)+ , (A.12)where P ( x ) is defined in (5.12).By using (A.8) to combine the two equations in (A.10) and noticing thattr (cid:2) WH X α =1 , λ ( α ) Y α ) ] = tr (cid:2) H + ( y − x ) WH − W ] , one finds a quadratic relation ( y − x ) D ( x ) + H − W ( x ) + X α =1 , λ ( α ) n ( α ) D ( α )1 ( − x )= D ( x ) X α =1 , (cid:16) λ ( α ) n ( α ) D ( α )1 ( − x ) + λ ( α ) n ( α ) W ( − x ) (cid:17) . (A.13)By inserting the second of the equation (A.10) into (A.12) one finds another quadratic equation, ( W ( x ) + H ) D ( x ) + P ( x ) − V ′ ( x ) D ( x ) − X α =1 , n ( α ) D ( α )1 ( − x )= − D ( x ) n X α =1 , λ ( α ) n ( α ) D ( α )1 ( − x ) + nW ( − x ) o . (A.14)These two equations are equivalent to (5.10) and (5.11). B 2D Liouville gravity
In 2D Liouville gravity formalism, the c ≤ matter CFTs are coupled to the Liouville theory withcentral charge − c and the reparametrization ghosts. For Liouville theory, we denote the standardcoupling by b and the background charge by Q = b + 1 /b . The central charge is given by − c =1 + 6 Q . In our convention, b is always smaller than one. When the matter CFT is ( p, q ) minimalmodel, we have b = r pq , c = 1 − p − q ) pq . (B.1)In studying the O ( n ) model we used the parametrization n = 2 cos πθ . If θ = 1 /p with p ∈ Z , themodel describes the flow between ( p, p + 1) and ( p − , p ) minimal models corresponding respectivelyto the dilute and dense phase critical points. 33he primary operators in ( p, q ) minimal model fit in the Kac table which has ( p − rows and ( q − columns. The operator Φ r,s has the conformal dimension h r,s = ( rq − sp ) − ( q − p ) pq = ( r/b − sb ) − (1 /b − b ) , (B.2)and are subject to the identification Φ r,s = Φ p − r,q − s . In 2D Liouville gravity coupled to the ( p, q ) minimal CFT, the operator Φ r,s is dressed by the Liouville exponential e α r,s φ or e α r,s φ dependingon whether it is a bulk or boundary operator, so that the total conformal weight becomes one. Thisrequires α r,s ( Q − α r,s ) + h r,s = 1 , α r,s = Q ± (cid:0) r/b − sb (cid:1) . (B.3) B.1 Conformal weight and scaling exponents of couplings
In making the comparison between the matrix model and Liouville gravity, we start from the fact thatthe resolvent w ( x ) and its argument x correspond to the two boundary cosmological constants µ B and ˜ µ B . They couple respectively to the boundary cosmological operators e bφ and e φ/b , and are thereforeproportional to µ and µ b , where µ is the Liouville bulk cosmological constant. Since w ( x ) ∼ x ± θ in the dilute and dense phases, we find(dilute): x = µ B , w ( x ) ∼ x θ = ˜ µ B , b = (1 + θ ) − / , (dense): x = ˜ µ B , w ( x ) ∼ x − θ = µ B , b = (1 − θ ) +1 / . (B.4)Suppose a boundary condition has a deformation parametrized by a coupling which scales like x ρ . Then the corresponding boundary operator should be dressed by the Liouville operator withmomentum α = ρb in the dilute phase and α = ρ/b in the dense phase. Then using (B.3) one candetermine the conformal dimension of the operator responsible for the boundary deformation. Let usapply this idea to the deformations parametrized by µ B , t B and ∆ in Sect. 6. Deformation by µ B . µ B scales like x θ in the dilute phase and x − θ in the dense phase. Sothe corresponding operators are dressed by e (1+ θ ) bφ = e φ/b in the dilute phase and e (1 − θ ) φ/b = e bφ in the dense phase. The matter conformal dimension is zero, i.e., the operator responsible for thedeformation is the identity Φ , . Deformation by t B . t B scale like x θ in the dilute phase and x − θ in the dense phase, so the cor-responding operator gets dressings with Liouville momentum bθ = b − b in the dilute phase and − θ/b = b − b in the dense phase. We identify the operator with Φ , (relevant) in the dilute phaseand Φ , (irrelevant) in the dense phase. Deformation by the anisotropy coupling ∆ . The coupling scales like x − θ in the dilute phase.The corresponding operator is dressed by the Liouville momentum b (1 − θ ) = 2 b − b and thereforeidentified with Φ , . B.2 Conformal weight and scaling exponents of correlators
After turning on the gravity, correlators no longer depend on the positions of the operators insertedbecause one has to integrate over the positions of those operators. The dimensions of the operatorstherefore should then be read off from the dependence of correlators on the cosmological constant µ .34f we restrict to disk worldsheets, the amplitudes with n boundary operators O Bi dressed by Liouvilleoperators e β i φ and λ bulk operators O i dressed by e α j φ scale with µ as (cid:10) O B · · · O Bn O · · · O m (cid:11) ∝ µ b ( Q − P α j − P β i ) . (B.5)Suppose a disk two-point function of a boundary operator O B scales like x ρ . Then O B should bedressed with Liouville momentum β = ( Q − bρ ) in the dilute phase and β = ( Q − ρ/b ) in thedense phase. We thus identify O B with the ( r, s ) operator in the Kac table if ± ρ = r (1 + θ ) − s ( dilute phase ) , ± ρ = r − s (1 − θ ) ( dense phase ) . (B.6) B.3 Boundary two-point function
In Liouville theory with FZZT boundaries, the disk two-point structure constant is d ( β | t, s ) = ( µπγ ( b ) b − b ) ( Q − β ) / b G ( Q − β ) G − (2 β − Q ) S ( β + it + is ) S ( β + it − is ) S ( β − it + is ) S ( β − it − is ) . (B.7)Here the special functions G and S satisfy S ( x + b ) = 2 sin( πbx ) S ( x ) , G ( x + b ) = 1 √ π b − bx Γ( bx ) G ( x ) , (B.8)and similar equations with b replaced by /b . The boundary parameter s is related to the boundarycosmological constant by µ B = (cid:16) µ sin πb (cid:17) cosh(2 πbs ) , ˜ µ B = (cid:16) ˜ µ sin πb − (cid:17) cosh(2 πs/b ) . (B.9)where µ is the bulk cosmological constant and ˜ µ is its dual, ( µπγ ( b ) b − b ) /b = (˜ µπγ ( b − ) b − b − ) b . (B.10)Our loop equation (5.14) can be compared with d ( β | t, s ) d ( Q − β − b | t ± ib , s ) = F ( β ) n cosh 2 πb ( t ± iβ ) − cosh 2 πbs o ,F ( β ) def = r µ sin πb Γ(2 bβ − b − − bβ )Γ( − b ) . (B.11)or the one with b replaced by /b . Similarly, the recurrence equation (5.7) among disk correlators of L -leg operators can be compared with d ( β | t + ib , s ) − d ( β | t − ib , s ) d ( β + b | t, s ) = G ( β ) n cos 2 πib ( t + ib ) − cos 2 πib ( t − ib ) o ,G ( β ) def = − r µ sin πb Γ(1 + b )Γ(1 − bβ )Γ(2 + b − bβ ) . (B.12)35 Properties of the function V r ( τ ) The function V r ( τ ) defined by log V r ( τ ) def = − Z ∞−∞ dωω (cid:20) e − iωτ sinh( πrω )sinh( πω ) sinh πωθ − rθπω (cid:21) (C.1)satisfies the shift relations V r +1 ( τ ) = 2 cosh (cid:16) θ ( τ ± iπr )2 (cid:17) V r ( τ ∓ iπ ) ,V r + θ ( τ ) = 2 cosh (cid:16) τ ± iπr (cid:17) V r ( τ ∓ iπθ ) , (C.2)which follow from the integration formula log(2 cosh t ) = Z ∞−∞ dω ω (cid:20) − e − iωt sinh( πω ) + 1 πω (cid:21) . (C.3)By deforming the contour of integration and applying the Cauchy theorem we can write the inte-gral (C.1) as the following formal series which makes sense for Re[ τ ] > : ln V r ( τ ) = θr τ − ∞ X n =1 ( − ) n n e − nτ sin( nπr )sin( πn/θ ) − ∞ X n =1 ( − ) n n e − nθτ sin( nπθr )sin( nπθ ) . (C.4)The expansion for Re[ τ ] < follows from the symmetry V r ( τ ) = V r ( − τ ) . The expansion of thefunction V r at infinity is V r ( τ ) = e θr | τ | / (cid:18) πr )sin( π/θ ) e −| τ | + sin( πθr )sin( πθ ) e − θ | τ | + . . . (cid:19) , τ → ±∞ . (C.5) References [1] H.W. Diehl, J. Appl. Phys. , 7914 (1982)[2] H. W. Diehl and E. Eisenriegler, “Effects of surface exchange anisotropies on magnetic critical and multicritical behav-ior at surfaces”, Phys. Rev. B (1984) 300[3] J. Dubail, J. L. Jacobsen and H. Saleur, “Conformal boundary conditions in the critical O ( n ) model and dilute loopmodels”, arXiv:0905.1382v1 [math-ph][4] E. Domany, D. Mukamel, B. Nienhuis and A. Schwimmer, “Duality Relations And Equivalences For Models WithO(N) And Cubic Symmetry,” Nucl. Phys. B (1981) 279.[5] B. Nienhuis, “Exact Critical Point And Critical Exponents Of O ( N ) Models In Two-Dimensions,” Phys. Rev. Lett. (1982) 1062.[6] J. Dubail, J. L. Jacobsen and H. Saleur, “Exact solution of the anisotropic special transition in the O ( n ) model in 2D”,Phys. Rev. Lett. (14):145701, 2009 [cond-math,stat-mech 0909.2949][7] J. L. Jacobsen and H. Saleur, “Conformal boundary loop models,” Nucl. Phys. B , 137 (2008) [math-ph/0611078].[8] J. L. Jacobsen and H. Saleur, , “Combinatorial aspects of boundary loop models”, arXiv:0709.0812v2 [math-ph][9] P. Di Francesco, P. Ginsparg, J. Zinn-Justin, “2D Gravity and Random Matrices”, Phys.Rept. 254 (1995) 1-133, hep-th-9306153[10] P. Ginsparg and G. Moore, “Lectures on 2D gravity and 2D string theory (TASI 1992)”, hep-th/9304011.[11] I. K. Kostov, “ O ( n ) vector model on a planar random lattice: spectrum of anomalous dimensions”, Mod. Phys. Lett.A (1989) 217.
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