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I'll try to explain why there could be a critical line and not just a critical point, and hopefully that will answer your question. If you think about the Ising model, we have the standard Hamiltonian: $$-\beta H = J_1\sum_{<i,j>}s_i s_j + h\sum_{i}s_i$$ where $\sum_{<i,j>}$ is a sum over nearest neighbors. This model ...
In CFT, we are interested in the continuum limit, where we can classify classes of models at their critical points. By means of the Jordan-Wigner transformation one can construct a fermion operator out of the spin operators of the usual 2D Ising model. Then, the continuum critical Ising model is described by a massless real fermion: $$S=\frac 12\int d^2 z\... 2 I agree that the language is very confusing - I'm a native English speaker, and it also took me a while to understand what they were saying. When they talk about the dimension of "the statistical system itself," they mean the spacetime dimension. So if a system has two spatial dimensions, then it has three dimensions total (including time), and the ... 1 Recall that the (global) conformal group is given by$${\rm Conf}(p,q)~\cong~O(p\!+\!1,q\!+\!1)/\{\pm {\bf 1} \},\tag{1} $$cf. e.g. this Phys.SE post. Using the embedding \imath: \mathbb{R}^{p,q}\hookrightarrow \overline{\mathbb{R}^{p,q}} into the conformal compactifification \overline{\mathbb{R}^{p,q}}, one may show after a short calculation that the ... 2 Comments to the post (v2): Ref. 1 is considering the d-dimensional real Euclidean space (\mathbb{R}^d,|\cdot|^2) with the standard norm$$|x|^2~:=~\sum_{\mu=1}^d (x^{\mu})^2~=~\sum_{\mu,\nu=1}^d x^{\mu}\eta_{\mu\nu}x^{\nu}, \qquad \eta_{\mu\nu} ~=~{\rm diag}(1,\ldots, 1),\tag{A}$$and inner product$$\langle x ,y\rangle~:=~\sum_{\mu,\nu=1}^d x^{\mu}\...