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Published online by Cambridge University Press: 18 January 2024
This chapter discusses the truncation criteria in the RG treatment of a non-Hermitian matrix, starting with a modified definition of the reduced density matrix using the leading left and right eigenvectors. As the reduced density matrix so defined is not Hermitian, there is no theorem to protect or guarantee that its eigenvalues are semi-positive definite. This non-Hermitian problem causes trouble in the determination of an optimized truncation scheme. Three truncation schemes for determining the RG transformation matrices are introduced, relying on the canonical diagonalization of the reduced density matrix, biorthonormalization, and lower-rank approximation of the environment density matrix, respectively. The canonical diagonalization scheme is optimal if the reduced density matrix is semi-positive definite. The scheme of biorthonormalization may not be optimal, but it is mathematically more stable.
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