6 - Criterion of Truncation: Symmetric Systems

Summary

Truncation of basis states is a vital step in the tensor network renormalization. This chapter introduces the concept of reduced density matrices and discusses the criterion of judging which state should be retained and which not in the basis truncation. In a Hermitian system, the reduced density matrix of a quantum state is semi-positive definite, and its eigenvalues measure the probabilities of the corresponding eigenvectors. Therefore, we should do the truncation according to the eigenvalues of the reduced density matrix. This criterion is equivalent to taking a Schmidt decomposition for the wave function of the quantum state and truncating the basis states according to their singular values. It is also equivalent to maximizing the fidelity of the targeted state before and after truncation. We also introduce the edge and bond density matrices and show that they have the same eigen-spectra as the reduced density matrix.