Critical exponents of block-block mutual information in one-dimensional infinite lattice systems

Abstract

We study the mutual information between two lattice-blocks in terms of von Neumann entropies for one-dimensional infinite lattice systems. Quantum q-state Potts model and transverse field spin-1/2 XY model are considered numerically by using the infinite matrix product state (iMPS) approach. As a system parameter varies, block-block mutual informations exhibit a singular behavior that enables to identify critical points for quantum phase transition. As happens with the von Neumann entanglement entropy of a single block, at the critical points, the block-block mutual information between the two lattice-blocks of ℓ contiguous sites equally partitioned in a latticeblock of 2ℓ contiguous sites shows a logarithmic leading behavior, which yields the central charge c of the underlying conformal field theory. 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. 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. For a given lattice-block size ℓ, the critical exponents for the same universality classes seem to have very close values each other. Whereas the critical exponents have different values to a degree of distinction for different universality classes. As the lattice-block size becomes bigger, the critical exponent becomes smaller.

Link representation of the entanglement entropies for all bipartitions

“…Notice that the even values are correctly represented, but not the odd ones, which are zero in the free-fermionic approximation. link strengths, J i, j ≈ exp(−|i − j|/ξ), for some correlation length ξ [34]. Indeed, this is our main result, but the technical details are also relevant.…”

“…From the comparison we note that J cont r almost coincides with J opt r for all r and h z values. We may conclude that the link strengths for MPS, when they are approximated using either equations (34) or (43) decay exponentially with a certain correlation length associated to the second maximal eigenvalue of the transfer matrix. In fact, this property may be considered the hallmark of the area law.…”

Quantum coherence and spin nematic to nematic quantum phase transitions in biquadratic spin-1 and -2 XY chains with rhombic single-ion anisotropy

“…where S A/A∪B = −Tr̺ A/A∪B log 2 ̺ A/A∪B are the von Neumann entropies with the reduced density matrix ̺ A/A∪B for one site A and two sites A ∪ B, respectively. This quantum mutual information can be used to detect and characterize quantum phase transitions [41][42][43][44][45][46]. From our iMPS groundstates for the Hamiltonian of Eq.…”

“…Though a few correlations are known involving for quantum phase transitions in quantum many-body systems such as in the transverse-field Ising model, general quantum manybody systems have various correlations of very different nature that result from interactions and thus finding proper correlation related to phase transition is to characterize quantum phase transition. In this sense, from the information theory perspective, for instance, quantum mutual information measuring the sum of quantum and classical correlations can be used as a general tool to identify quantum phase transitions because it is not required to know a priori what the right correlation function is in a given many-body system [41][42][43][44][45][46].…”