CORRIGENDUM: Discrete Wigner–Weyl calculus for the finite lattice (2023 J. Phys. A: Math. Theor. 56 395201)

We correct misprints and technical mistakes found in the text of the paper. Our result for the topological invariant N responsible for the quantum Hall effect is valid only when the limit of infinite lattice is taken. The way how this limit is taken is important. Namely, in equation (19) from section ‘Main results’, inside N, in the expression for the Weyl symbol QWM we should first take the limit of infinite N, and only after that differentiate it with respect to momenta. Finite lattice size enters expression for QWM in two ways: through the coordinates that belong to the finite lattice, and directly via dependence on N as on parameter. By the limit N→∞ we mean that the explicit parameter N tends to infinity, while the coordinate arguments are held fixed. Correspondingly, we should calculate GWM as inverse with respect to Moyal product of this truncated QWM. We take the limit of infinite lattice size in the sum over M′ and O′ at the end of the calculation.Equation (7) should read (formula presented) Correspondingly, the same correction to the argument of exponent is to be applied to equations (50), (66), (72), (76), (83), (91) and (95), and to the first line of equation (108).In equations (8) and (9) there should be (2N)D instead of (4N)D in the denominator.Property 3 of Weyl symbol should read (formula presented) All arguments here remain on the lattice M′×O′: the shifts πNek and 12ek are one lattice step of the k–th component of the momentum and of the coordinate, respectively, while N2ek and πek translate these components by half of the corresponding period.In equation (6) we should replace n by −n. With this correction the two representations given by equations (5) and (6) become identical to each other. Correspondingly, in the last line of equation (92) n is to be replaced by −n, while the factor 1+e−2iviπ/N2 entering it should read 1+e2iviπ/N2 (the first line of equation (92) remains unchanged). The same replacements refer to the corresponding intermediate formulas of the derivations of the basic properties of the Weyl symbol. In equations (14) and (126) n is to be replaced by −n as well.In equations (11) and (96) the expression e2iπ/N should be replaced by eiπ/N, i.e. the Weyl symbol of the operator of translation to one lattice spacing is (formula presented) Its limit at N→∞ remains eipj, as stated in equation (12).In the last formula of item 7 of the list of section 2 (as well as in equations (105)–(108)) the factor e2ip−q should read e−2ip−q. Likewise, in the first two lines of equation (85), e2ip−q should read e−2ip−q, while in its third line eiπmq should read e−iπmq.In equations (57) and (101), the factor e2ip1v1+2p2v2 should read e2ip1v1+2ip2v2. Similarly, in equation (104), e2ipv1+2pv2 should read e2ipv1+2ipv2. In the last summation in equation (102), the summation variables should read ∑ni=0,1;v+∈O′. The reason for this is that the derivative with respect to external gauge field of Weyl symbol may be replaced by derivative with respect to momentum only if the artificial terms disappearing in the limit of large lattice are disregarded. Correspondingly, equation (137) is valid only if inside Weyl symbol of Q^ the limit of infinite N is taken before differentiation over momentum. The same refers to the further expressions including the expression for N.Actually it is this corrected expression for the analytical continuation that was used in the derivations of the basic properties of the symbols considered in the paper. Therefore, this correction does not cause any changes in the results of the paper (except for the periodicity property, see item 3 below).The previously announced periodicity properties of the Buot symbol (equation (51)) and of the modified Buot symbols remain valid for the discrete values of arguments ( p∈M′, q∈O′).