We discuss a relation between the exponents of block-block mutual information and correlation with the Shatten one-norm of block-block correlation. As the lattice-block size becomes bigger, the critical exponent becomes smaller. Whereas the critical exponents have different values to a degree of distinction for the different universality classes. For a given lattice-block size ℓ, the critical exponents for the same universality classes seem to have very close values each other. The critical exponent of block-block mutual information in the thermodynamic limit is estimated by extrapolating the exponents of power-law decaying regions for finite truncation dimensions. As the separation between the two lattice blocks increases, the mutual information reveals a consistent power-law decaying behavior for various truncation dimensions and lattice-block sizes. It seems that two disjoint blocks show a similar logarithmic growth of the mutual information as a characteristic property of critical systems but the proportional coefficients of the logarithmic term are very different from the central charges. As happens with von Neumann entanglement entropy of single block, at critical points, block-block mutual information for two adjacent blocks show a logarithmic leading behavior with increasing the size of the blocks, which yields the central charge c of the underlying conformal field theory, as it should be.
As a system parameter varies, block-block mutual information exhibit singular behaviors that enable us to identify the critical points for the quantum phase transitions. Quantum q-state Potts model and transverse-field spin- 1 / 2 X Y model are considered numerically by using the infinite matrix product state approach. We study the mutual information between two lattice blocks in terms of von Neumann entropies for one-dimensional infinite lattice systems.