Sine-Gordon field theory for the Kosterlitz-Thouless transitions on fluctuating membranes.

In the preceding paper, we derived Coulomb-gas and sine-Gordon Hamiltonians to describe the Kosterlitz-Thouless transition on a fluctuating surface. These Hamiltonians contain couplings to Gaussian curvature not found in a rigid flat surface. In this paper, we derive renormalization-group recursion relations for the sine-Gordon model using field-theoretic techniques developed to study flat space problems. PACS numbers: 05.70.Jk, 68.10.-m, 87.22.Bt Typeset using REVTEX 1 I. THE SINE-GORDON HAMILTONIAN The Hamiltonian and associated partition function describing p-atic order on a fluctuating surface were derived and analyzed in the Coulomb gas model in the previous paper [1]. The renormalization-group (RG) recursion relations for K, κ, and y were also derived from the Coulomb gas model. In addition, we described there how the two-dimensional Coulomb gas model can be transformed into a sine-Gordon field theory. Here we describe to what extent these results can be verified by a well-controlled renormalization procedure based on the sine-Gordon field theory. The two-dimensional Coulomb gas model can be converted into the sine-Gordon Hamiltonian via a Hubbard-Stratonovich transformation. The advantage of this transformation is that it makes available to us standard field theory diagrammatics and renormalization procedures [2,3]. This opens up a systematic way of obtaining results for the RG recursion relations. The p-atic membrane partition function in the sine-Gordon field theory is written as Z = ∫ DφDRe−βHκ−βHφ−i(p/2π) ∫ du √ , (1.1) where β is the inverse temperature, βHκ = 1 2 βκ ∫ du √ gH, (1.2) and βHφ = 1 2βK ( p 2π )2 ∫ du √ gg∂αφ∂βφ− 2y a2 ∫ du √ g cosφ (1.3) with 1 2 H = 1 2 K α the mean curvature and S = detK α β the Gaussian curvature. In the Monge gauge, the metric tensor gαβ is written as gαβ = ∂αR · ∂βR = 