Quantum-field-theory calculation of the two-dimensional Ising model correlation function

The equivalence between the two-dimensional Ising model and a free fermion field theory is used to rederive various results for the two-spin correlation function in the critical region.

I. INTRODUCTION nr/4= p,p, (2. 2) The utility of the relation between the two-dimensional Ising model and a relativistic free Fermi field theory' for the computation of the spinspin correlation function, (e"o ",), has been noted recently by several groups. " In Ref. 2 this technique was used to evaluate the scaling part of the correlation function at T = T, as well as leading corrections in (T -T,)~rr'~.
The expressions for the correlation functions at T = T, have been known for some time, 4 as has been the leading term in the (T -T,)~rr'~expansion. ' The entire perturbation expansion, as well as a closed solution for the scaling region correlation function, was presented in a monumental work of Wu, Mccoy, Tracy, and Barouch' and earlier references. "' In this article we obtain a perturbation expansion for the correlation function for T-T"~rr'~-~, and (T -T,)~r -r'~f ixed. This result has been obtained previously. ' ' It may be useful to rederive these results using an approach other than the one of Refs. 6-8. We stay completely within the framework of a relativistic field theory. These techniques may prove to be of use in the study of other lattice field theories, especially in performing mass perturbation where infrared difficulties are severe.
In (1) it was shown that this correlation function is given by [Ref. 2,Eq. (29)] (2.3) In the above c"and c"' are fermion annihilation and creation operators attached to site y. . The sites x and v' are along the same row and the expectation value is in the vacuum state of another set of fermion operators, (, and ('"defined on the reciprocal la,t tie e q = 2 p P/L, p = 0, + 1, + 2, . . . , + L /2a. The ('s satisfy the usual anticommutation relations (2.4) We still need a relationship between the c s of (2.3) and the $'s. In order to achieve this we introduce a set of fields defined on a particular row of the lattice: 1/2~+ q 1/2 |t,(r) = -Q ' e(q)((, e""+ t', e-""), We wish to review some of the results of Refs. 1 and 2. No details will be presented as this section is intended to establish notation. Qur interest is in the two-spin correlation function near the critical point for a square Ising lattice of L/a points on a side; a is the lattice spacing. The critical temperature 1/p, is obtained from the transcendental equation z(q)=q/~q~a nd u is related to m [Eq. (2.2)],   H=g~,(t'tt', --').

By
(2.10) (2.11) g, and |I), may be viewed as the two components of a Euclidean Majorana, field of mass P1l. Introducing the Euclidean time development of the fields by suggests the interaction representation for the coupling of the Majorana field to an external potential A(r, t). Connecting (3.5) to an anticommuting path integral, it becomes apparent that (3.7) S(ni, 0) is the propagator for the free massive field while S(m, A) is the same propagator in the presence of the external potential A. " Normally, when dealing with Dirac fields it is the determinant and not its squa. re root that would appear in (3.7); however, we are dea. ling with a Majorana field with half the degrees of freedom of a Dirac field. As discussed above, we will take the positive square root. We shall study some of these propagators in the next section.

IV. PROPAGATORS
The form of Eq. (2.11}suggests that we introduce the complex variable z and its complex conjugate and that the correlation function is From (2.5) and (2.7) we note that the c's are proportional to un and in the limit x'r large the first term in the exponent of (3.2) may be neglected, while the second one becomes an integral.
(4.4) A = 0. Writing with 6'(z -z') = 6(x -x')6(t -t'). We shall now concentrate on the propagators for the case A =0 and for the massless case in the presence of A. (4.14) The propagator may be expressed in terms of the solutions to Comparing the above with (4.4) we find that 1 1 s -, = 6'(z -z'),  The equation for the propagator in this situation ls As (4. 10) provides us with Green's functions for the differential operators in the complex plane we ob- tion; returning to the discrete case, it is easy to see that the correct replacement in this limit is or lim S(0, 0; z, z') -i la . In order to make the above more precise we would have to treat both x and t as discrete. We do not pursue this further and thus abandon the calculation of the overall magnitude of the correlation function, and satisfy ourselves with its functional dependence. This is analogous to any renormalization calculation in a local field theory, where the magnitude of the field itself is arbitrary and fixed only by placing some conditions on the propagators.
From (3.7) we infer that for /// =0 (4.21) It will prove useful in the next section to generalize the above results to the case where A is multiplied by a real constant X. It follows immediately where for the moment we treat the more general case discussed at the end of the last section. Without any loss of generality we may take x=0 and 'V = P. Though this situation was discussed in Ref. 2, we repeat it using the Majorana formalism. As we shall be discussing singular products, we must at times remember that in reality we are dealing with a cutoff field theory, the cutoff provided by the lattice spacing a. For example, we shall encounter the massless propagator at zero separa-However, the quantities on the right-hand side of (5.5) are too singular for these manipulations and thus we have to isolate these singular parts first. These difficulties occur only in the first few terms in the expansion of (5.2) in power of X. We deal with the identity Parenthetically, we may remark that this result is analogous to the observation that in two-dimensional QED only the one-loop contributions to the propagator are nonvanishing.
Returning to (5.7) and remembering the discussion at the beginning of the last section regarding the regularization of the free propagator, we ob- x' x p -X (6.9) and an analogous expression for the other term. An application of (4.10) to (6.9) yields j.f 2 &r x/g 2 p-x "' x' 28-2S 2a = -6(f)e(x)e(p x-) p 6(t') e(x') e( p -x') .
2')T z~zi 8 p~g x~x x px (6.10) The singularity at x=x' is to be intepreted as the Cauchy principal value in both x and x'. Returning to (6.8) and applying (6.10}we find that G-'Z G = -6(f)e(x)e( p x)6-(f')e(x')e( px') (6.11) The 6 and e functions imply that the evaluation of (6.2) may be restricted to operators acting on the one- Combining all the above we find" that Again, the above operators are restricted to the real interval (0, p). The lower line in (6.12);s the trans pose of the upper one and contributes the same value to the trace removing the factor -, in front of (6.12), ln F(mp) = Trln(1 -o't}, where 0, t are one-dimensional operators The expansion of the logarithm in (6.13) yields the desired perturbation in powers of m and lnm agreeing with that of Ref. 6. We may also note that the calculations for T & T, of the two-point connected correlation functions (i.e., with the subtraction of the square of the magnetization, a quantity vanishing in the scaling region) a.re given by the same formulas as above, with n~treated as a negative quantity.
As an example we shall calculate the first term in this expansion: InF, (mp)= --,P (6.15} The evaluation of the integrals is tedious and gives A = -, ', B = -, '(y -3 ln2) . Separately the two terms on the right-hand side of (7.2) are singular and some regularization scheme has to be introduced to give them individual meaning. If we were to reintroduce a lattice for this purpose the evaluation of (7.2) would parallel the discussion of Ref. 6. Furthermore, it seems reasonable to expect that the methods of this paper can be extended to higher-order correlation functions, yielding a systematic series expansion.