Author Correction: Topological quantum criticality in non-Hermitian extended Kitaev chain (Scientific Reports, (2022), 12, 1, (6993), 10.1038/s41598-022-11126-7)

Document Type

Article

Publication Title

Scientific Reports

Abstract

Correction to: Scientific Reports, published online 28 April 2022 The original version of this Article contained a repeated error, where the sign “CP” was incorrectly given as “Cq”. As a result of this error, in the Results, “The Hermitian model consists of three critical lines (solid black line) “AB”, “BD” and “Cq” distinguishing topological phases W = 0, W = 1 and W = 2.” now reads: “The Hermitian model consists of three critical lines (solid black line) “AB”, “BD” and “CP” distinguishing topological phases W = 0, W = 1 and W = 2.” And, in the Results section, under the subheading ‘Zero mode analysis for topological characterization’, “From the Fig. 1, it can be clearly observed that the critical line “Cq” is not present for the non-Hermitian case (critical lines presented in red color).” now reads: “From the Fig. 1, it can be clearly observed that the critical line “CP” is not present for the non-Hermitian case (critical lines presented in red color).” Additionally, the original version of this Article contained errors in the sign in the zero mode solutions in the Method section, where Zero mode solutions. The model Hamiltonian can be written as, (Formula presented.) where (Formula presented.) and (Formula presented.) Substituting the exponential forms of (Formula presented.) and (Formula presented.) , Eq. (9) becomes, (Formula presented.) We replace (Formula presented.) , Eq. (10) becomes, (Formula presented.) To find the zero mode solutions, we make (Formula presented.). By solving the Eq. (11), we get, (Formula presented.) Equation 12 shows that there will be more than one solution. Considering the Eq. (11) and squaring both sides with (Formula presented.) , we get, (Formula presented.) Substituting back the exponential forms to the respective terms, we get, (Formula presented.) Simplifying the Eq. (14), we end up with a quadratic equation, (Formula presented.) Simplifying the Eq. (15) to a quadratic form and substituting (Formula presented.) , (Formula presented.) The roots of this quadratic Equation is given by, (Formula presented.) The roots Eq. (17) are the solutions of zero modes. now reads: Zero mode solutions. The model Hamiltonian can be written as, (Formula presented.) where (Formula presented.) and (Formula presented.) Substituting the exponential forms of (Formula presented.) and (Formula presented.) , Eq. 1 becomes, (Formula presented.) We replace (Formula presented.) , Eq. 2 becomes, (Formula presented.) We make (Formula presented.) 31, to obtain the zero solutions for certain q where (Formula presented.) and (Formula presented.) square to 1 or become 0 due to anticommutation. (Formula presented.) Simplifying the Eq. 4, we end up with a quadratic equation, (Formula presented.) Simplifying the Eq. 5 to a quadratic form and substituting (Formula presented.) , (Formula presented.) The roots of this quadratic Equation is given by, (Formula presented.) Finally, the original version of this Article contained an error in the legend of Fig. 2. “Zero mode solutions plotted with respect to the parameter λ1 shows both W = 0 to W = 1(λ2 = 0.5) and W = 1 to W = 2(λ2 = 2.0) topological phase transitions. The red dot (p1 and p2) represents the transition points. The ZMS, X+ (red) and X− (blue) are plotted in y-axis.” now reads: “Zero mode solutions plotted with respect to the parameter λ1 shows both W = 0 to W = 1(λ2 = 0.5) and W = 1 to W = 2(λ2 = 2.0) topological phase transitions. Blue dots (p1 and p2) represent the transition points. The ZMS, X+ (red) and X− (blue) are plotted in y-axis.” The original Article has been corrected.

DOI

10.1038/s41598-023-49938-w

Publication Date

12-1-2024

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