Abstract
We present an analytic calculation of the layer (parallel) susceptibility at the extraordinary transition in a semi-infinite system with a flat boundary. Using the method of integral transforms put forward by McAvity and Osborn [Nucl. Phys. B455 (1995) 522] in the boundary CFT, we derive the coordinate-space representation of the mean-field propagator at the transition point. The simple algebraic structure of this function provides a practical possibility of higher-order calculations. Thus we calculate the explicit expression for the layer susceptibility at the extraordinary transition in the one-loop approximation. Our result is correct up to order O(ε) of the ε = 4 − d expansion and holds for arbitrary width of the layer and its position in the half-space. We discuss the general structure of our result and consider the limiting cases related to the boundary operator expansion and (bulk) operator product expansion. We compare our findings with previously known results and less complicated formulas in the case of the ordinary transition. We believe that analytic results for layer susceptibilities could be a good starting point for efficient calculations of two-point correlation functions. This possibility would be of great importance given the recent breakthrough in bulk and boundary conformal field theories in general dimensions.
Highlights
Approach and OPE have been used in a comprehensive study of the operator algebra and multipoint correlation functions in a series of papers by Lang and Rühl, including [33]
We present an analytic calculation of the layer susceptibility at the extraordinary transition in a semi-infinite system with a flat boundary
We calculate the explicit expression for the layer susceptibility at the extraordinary transition in the one-loop approximation
Summary
We set the “mass” τ0 to zero, as we are interested only in the theory right at the transition point T = Tc. In (2.3), m0(z) is the critical order-parameter profile in the Landau (mean-field) approximation. The propagator G0(p; z, z ) is that of Gaussian fluctuations of the order parameter around the mean-field solution m0(z) It is the connected two-point function in the (unperturbed) zero-loop, or tree approximation of the Landau-Ginzburg theory. There is the IR-stable fixed point u∗ = 0 in this theory, but one has to keep in mind that the running coupling constant u(l), which replaces u0 in renormalizationgroup flow equations, behaves as u(l) ∼ | ln l|−1 when the flow parameter l → 0 This generally leads to logarithmic corrections to the mean-field behavior in d = 4 [115, section VIII.B], [116, section 9-6], [117, section 5.6]. We reproduce the needed information in a few words below
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