Differential equation for local magnetization in the boundary Ising model

We show that the local magnetization in the massive boundary Ising model on the half-plane with boundary magnetic field satisfies second order linear differential equation whose coefficients are expressed through Painleve function of the III kind.

describes the continuum limit in the vicinity of the critical point of the lattice Ising model with zero magnetic field in the bulk and with boundary magnetic field being suitably rescaled. Let us briefly list known results about the local magnetisation in this theory defined on the half-plane.
The form factor expansion for local magnetizationσ (t) was written down in [7] using exact expression for boundary state obtained in [6]: where m ∼ T − T c is the mass of a particle, h -scaling boundary magnetic field, ydistance from the boundary, σ 0 -magnetization on the infinite plane, A = 1.28243 . . . is Glaisher's constant. Here and later on we always consider the low temperature phase T < T c , unless it is specially pointed out. It is also implied conformal normalization of spin field: With this normalizationσ (t, λ) → σ 0 as t → ∞. The expansion (1)-(3) was first obtained in [9] from lattice model calculations. It was also shown in [8], [9] that in the cases of "free" (h = 0) and "fixed" (h → ±∞) b. c. local magnetization can be expressed through Painleve function of the III kind: where ϕ (r) is the solution of radial sinh-Gordon equation: satisfying asymptotic conditions: where γ is the Euler's constant, K 0 (x) is the modified Bessel function of zeroth order. As is known, ϕ (x) is related to Painleve function of the III kind η (x) as η (x) = e −ϕ(2x) . More about this function see [10], [11].
In the case when bulk is critical (m = 0) it was shown in [12] that: is a solution of degenerate hypergeometric equation. The qualitative behavior ofσ (t, λ) is well understood [7]. On the whole interval (0, ∞) σ f ree (t) monotonically increases andσ f ixed (t) monotonically decreases, both approaching σ 0 as t → ∞. For small t: 1σ For 0 < λ < 2σ (t, λ) remains monotonically increasing. Its values near the boundary are somewhat enhanced by the presence of boundary magnetic field, the leading term of its short distance asymptotic become dressed by logarithm: For λ > 2 it possesses a maximum in some point. As λ → ∞ this maximum turns into a very sharp peak located in the region t ∼ λ −1 near the boundary, its shape being described by (10), (11). For t ≪ λ −1σ (t, λ) behaves as (14), while for t ≫ λ −1 its behavior coincides with one under "fixed" b. c. (6). This dependence reflects the renormalization group cross-over between "free" and "fixed" b. c. The main result of this paper is that for arbitrary λ: whereσ f ree (t) is given by (5), and u (t, λ) is the solution of differential equation: satisfying asymptotic condition: (Here ϕ (t) is the same function as in (5), (6) and the strokes stand for detivatives with respect to t.) Let us make some remarks on the equation (16). One can see that when λ = 0, the only solution of (16) satisfying (17) is u (t) = 1. When λ → ∞ (16) turns into a first order differential equation which upon integrating and fixing integration constant with the help of (17) yields (6). In the massless limit (t → 0, λ → ∞, tλ kept fixed) (16) turns into a degenerate hypergeometric equation. Its solution can be fixed by "sewing" its asymptotic as tλ → ∞ with asymptotic of (6) as t → 0, and this yields (10), (11). The fact that in massless limit we reproduce the result of [12] is not very surprising because the approach we used to derive (15), (16) is a generalization of one used in [12]. Concerning the relation between the form factor expansion (1), (2), (3) and our result (15), (16) we just note that it seems to be very difficult to show directly that (1), (2) satisfy (15), (16). In any case, it is beyond the analytic abilities of the author. 1 Note that comparing the coefficient in the short distance asymptotic of σ f ixed (t), that follows from (6), with the result σ f ixed (y) = 2 Being second order linear differential equation, (16) possesses two linearly independent solutions. Their asymptotics as t → ∞ are u 1 (t) ∼ 1 and u 2 (t) ∼ e (λ−1)t . Hence, for λ > 1 the condition that u (t) → 1 as t → ∞ is sufficient to fix the solution uniquely. For λ ≤ 1 more strict condition (17) is required, which follows from form factor expansion (1), (2). Another linearly independent solution in this case also has physical meaning. As explained in [6], for λ < 1 there exists metastable state characterized by asymptotic behaviorσ (t) → −σ 0 as t → ∞ and corresponding to the boundary bound state in the hamiltonian picture with "space" being half-line and "time" axis being parallel to the boundary. The local magnetizationσ 1 (t, λ) in this state can be obtained fromσ (t, λ) by analytic continuation h → −h. Clearly, it is also a solution of (16). As it was shown in [14] its asymptotic as t → ∞ is: The presence of exponential term ∼ e −(1−λ)t in (18) agrees with (16).
In the rest of the paper we present the details of our derivation of (15), (16).

Ising field theory in the bulk
In this section we briefly recall some well known facts [1] about the structure of the Ising field theory in the bulk needed for further computations. As is known the Ising field theory in zero magnetic field is equivalent to the free Majorana fermion theory with euclidean action: Here we have assumed that the theory is defined on an infinite plane R 2 , whose points x are labelled by cartesian coordinates (x, y) = (x (x) , y (x)), and d 2 x ≡ dx dy. Complex coordinates are defined as z (x) = x + iy,z (x) = x + iy, and the derivatives ∂,∂ in (19) stand for ∂ z = 1 2 (∂ x − i∂ y ) and ∂z = 1 2 (∂ x + i∂ y ) respectively. The ciral components ψ,ψ of fermi field satisfy Dirac's equations: Their normalization in the action (19) corresponds to the following short-distance limit of the operator products The order σ (x) and disorder µ (x) fields are semi-local with respect to the fermi fields; the products acquire a minus sign when the point x is taken around zero point. The fields ψ (x) and ψ (x) in the products (22) can be expanded in the complete set of solutions of Dirac's equations (20) having this monodromy property: (here r, θ are polar coordinates, i. e. z = re iθ ,z = re −iθ and I ν are modified Bessel functions). The coefficients a n ,ā n in (23) are understood as operators acting on the space of fields. It can be easily shown that for any two solutions of Dirac's equations which change sign after the point x is taken around zero the integral over a contour C 0 encircling zero (in counter-clockwise direction) does not change under continuous deformation of C 0 and therefore defines a bilinear form on the space of such solutions. The solutions U n = (u n ,ū n ) and V n = (v n ,v n ) satisfy the following orthogonality properties with respect to this bilinear form: Let us also write down the following differentiation formulas which are useful in computations with U n and V n : (here we denote ∂U n ≡ (∂u n (x) , ∂ū n (x)), etc.). Using relations (27) one can express operators a n ,ā n in terms of contour integrals: This representation can be used to show that they satisfy canonical commutation relations: The fields σ and µ are "primary" with respect to the algebra (31), i. e. they satisfy relations: a n σ = 0 ,ā n σ = 0 , a n µ = 0 ,ā n µ = 0 (32) for n > 0, as well as where ω = e iπ/4 andω = e −iπ/4 . These equations define the fields σ and µ up to normalization. In what follows we will assume conformal normalization of fields σ and µ: As it is shown in [1], first and second order descendants of σ and µ with respect to the algebra a n ,ā n are expressed in terms of coordinate derivatives of σ and µ: This observation is very important for the method of [1] to work. The Majorana theory (19) corresponds to both high and low temperature phases of the Ising model in the vicinity of its critical point T c depending of the choice of the sign of the mass parameter m in (19). Our definition in (33) corresponds to the identification of the case m > 0 with the ordered phase T < T c , while the case m < 0 is identified with the disordered phase T > T c . From now on we will consider the ordered phase m > 0.

"Free" and "fixed" boundary conditions
In this section we consider the Ising field theory defined on the half-plane y > 0. As a warm-up exercise let us first rederive formulas (5), (6) for local magnetization in the cases of "free" and "fixed" b. c. Explicit expressions for boundary states for "free" and "fixed" b. c. was obtained in [4]. It follows from this expressions that the fields ψ andψ satisfy the following b. c.: ψ +ψ y=0 = 0 (for "fixed" b. c.) Suppose that (χ (x) ,χ (x)) is a double-valued solution of Dirac's equations defined on the half-plane y > 0 (with punctured point x 0 ) such that it changes sign when x is taken around x 0 , decays sufficiently fast as |x| → ∞, and satisfies b. c.: (χ −χ)| y=0 = 0 (for "free" b. c.) Then the following identity holds: where C x 0 is a contour encircling the point x 0 . This is because the integral on the lefthand side of (41) does not changes under continuous deformations of the contour C x 0 and therefore one can deform it in such a way that it constitutes of two parts: C ∞ which tends to infinity and C b which passes along the boundary. Then the integral along C ∞ is zero because χ (x) andχ (x) decay at infinity and the integral along C b is zero due to (39) or (40). On the other hand one can shrink C x 0 to a small circle around the point x 0 and express the left-hand side of (41) in terms of descendents of µ using the operator product expansions of ψ (x),ψ (x) with µ (x 0 ). If the singularity of (χ (x) ,χ (x)) at the point x 0 is not too strong only descendents of not higher than the second order appear in this expression. Since they are expressed in terms of coordinate derivatives of σ this will lead to a differential equation for local magnetization σ (x) . The only problem is to find a solution of Dirac's equations (χ (x) ,χ (x)) satisfying the above conditions. For this purpose we will use the following trick. We will search for this solution in the form of the linear combination: where . . . 0 denotes a correlation function in the Ising field theory defined on the plane and P denotes the reflection in the line y = 0 (i. e. P (x, y) = (x, − y)). Obviously each term in (42) is non-zero and as a function of x is a solution of Dirac's equations, change sign when x is taken around x 0 and decay at infinity. The coefficients in this linear combination can be determined from the requirement that it satisfies (39) or (40). Note that we do not need to know the functions χ (x) andχ (x) in explicit form. What we really need is several terms of their short-distance asymptotics as x → x 0 , but the latter can be expressed in terms of the two-point functions σ (x 0 ) σ (P x 0 ) 0 ≡ G (2my 0 ) and µ (x 0 ) µ (P x 0 ) 0 ≡G (2my 0 ). As is well known [10] (see also [1]) this functions can be expressed in terms of Painleve function of the III kind as follows: where ϕ (t) is the same function as in (5), (6). Under parity transformation P fermi fields transform as: We have therefore the following identities which follows from the invariance of correlation functions under parity transformation: It follows from this identities and Dirac's equations that from all correlation functions that present in the expression (42) only four are linearly independent functions of x on the line for "free" b. c., and for "fixed" b. c. It is now straightforward but somewhat tedious exercise to substitute (50) and (51) in (41) and evaluate the left-hand side. One has to expand ψ andψ using (23), than to evaluate contour integrals using (27), (28) and than to act by the operators a n ,ā n on σ and µ using (32), (33) and (35). Due to (32) all terms with descendants of higher than the first order vanish. Finally, taking into account that σ (x 0 ) ≡σ (2my 0 ) depends only on y 0 due to translation invariance, one obtains the following differential equations: (the stroke denotes derivative with respect to t = 2my 0 ). Integrating this equations, substituting (43), (44) and fixing integration constants with the help of asymptotic condition σ (t) → σ 0 as t → ∞ one obtains (5) and (6).
Let us now consider the high temperature phase T > T c . The differential equations in this case can be obtained from (52), (53) by substitution m → −m, G ⇄G: The only solution of (54) that does not grow exponentially as t → ∞ isσ f ree = 0, while from (55) we obtain:σ f ixed, T >Tc = e − 1 2 tσ f ixed, T <Tc (56) in agreement with [9]. This confirms our identification of the case m > 0 with the low temperature phase. Had we chosen the other choice, we would obtain the exponentially growing solution forσ f ixed in the high temperature phase.

Boundary magnetic field
Let us now consider the general case of "free" b. c. perturbed by boundary spin operator σ B . The latter is identified with degenerate primary boundary field with dimension ∆ = 1/2 [4]. It can be written in terms of fermion fields as follows [6]: where a (x) is additional fermionic degree of freedom with two-point function The action of the theory has therefore the following form: It leads to the following b. c. for fermion fields [6]: where λ = 4πh 2 /m. We can now proceed in the same way as in the previous section but now instead of (39) or (40) we should require the functions χ andχ to satisfy the condition: in order to write down the Ward identity (41). It turns out that in this case it is necessary to include also terms with second order derivatives in the linear combination (42) in order to satisfy (61). As a result one obtains the following linear combination: Substituting it in (41) and evaluating the left-hand side we obtain the following differential equation for local magnetizationσ (t): It can be brought to a simpler form (16) by means of substitution (15).

Acknowledgements
I am especially grateful to Y. P. Pugai for his interest to my work and encouragement. I am also grateful to my scientific advisers A. I. Bugrij and V. N. Shadura for giving me creative freedom and to N. Z. Iorgov and V. N. Shadura for pointing out the paper [14].