Ising model on a twisted lattice with holographic renormalization-group ﬂow

In the paper PTEP 2015, 113B02, the partition function of the 2D Ising model is exactly obtained on a lattice with a twisted boundary condition by introducing the so-called shift matrix. Although the strategy of the computation is correct, we found that we need a phase factor in the deﬁnition of the shift matrix in the fundamental representation of Spin ( 2 n : C ) in order that all the elements in the spin representation become real. In addition, there are several typos in the article. We correct them and reﬁne the derivation of the partition function in this erratum.

In the paper PTEP 2015, 113B02, the partition function of the 2D Ising model is exactly obtained on a lattice with a twisted boundary condition by introducing the so-called shift matrix. Although the strategy of the computation is correct, we found that we need a phase factor in the definition of the shift matrix in the fundamental representation of Spin(2n : C) in order that all the elements in the spin representation become real. In addition, there are several typos in the article. We correct them and refine the derivation of the partition function in this erratum.
(1) On page 5, there is a typo in equation (2.19). We replace the paragraph including this equation by the following: In evaluating (2.17), the following fact is useful: Suppose that ∃ A ∈ Spin(2n; C) in the fundamental representation is transformed into the "canonical form" by T ∈ O(2n) as whereĴ μν are the generators of Spin(2n; C) in the fundamental representation and R(θ ) is the twodimensional rotation matrix with the (complex) angle θ given by (A4). Then we can write Tr ± (A) as Thus, we first express H m ± p ± in the fundamental representation and then transform them into the canonical form using appropriate matrices T ± ∈ O(2n).
(2) Corresponding to the change of the definition of T ± mentioned below, det T ± = ±1 written three lines above equation (2.22) on page 6 is replaced by det T ± = 1.

(6) We replace equation (A2) with the following equation and comments:
where the overall factor α ± in the definition ofˆ ± is given by , which are necessary in order to make all the nonzero elements of the shift matrix in the spin representation unity.

(7) We replace the text from equation (A13) to the end of Appendix A with the following:
Since A I , B I , and C I satisfy we can uniquely determine the parameters γ I > 0, θ I ∈ [0, π 2 ], and r = ±1 by where the sign in the definition of C I takes + for 1 ≤ I ≤ n and − for n + 1 ≤ I ≤ 2n. Note that the γ I appearing in (A14) is the same one defined in (2.20). We also note that N 0 = N n = 0 and M 0 and M n are given by where the sign appearing in the expression of M 0 takes + in the disordered phase (ã > b) and − in the ordered phase (ã < b). We can rearrange the matrices (A6) and (A7) by permuting the elements properly. To this end, we introduce the matrices which generate the transportations, . . .