Information theoretic aspects of the two-dimensional Ising model

Abstract

We present numerical results for various information theoretic properties of the square lattice Ising model. First, using a bond propagation algorithm, we find the difference $2H_L(w)-H_{2L}(w)$ between entropies on cylinders of finite lengths $L$ and 2L with open end cap boundaries, in the limit $L\to \infty$. This essentially quantifies how the finite length correction for the entropy scales with the cylinder circumference $w$. Secondly, using the transfer matrix, we obtain precise estimates for the information needed to specify the spin state on a ring encircling an infinite long cylinder. Combining both results we obtain the mutual information between the two halves of a cylinder (the "excess entropy" for the cylinder), where we confirm with higher precision but for smaller systems results recently obtained by Wilms et al. -- and we show that the mutual information between the two halves of the ring diverges at the critical point logarithmically with $w$. Finally we use the second result together with Monte Carlo simulations to show that also the excess entropy of a straight line of $n$ spins in an infinite lattice diverges at criticality logarithmically with $n$. We conjecture that such logarithmic divergence happens generically for any one-dimensional subset of sites at any 2-dimensional second order phase transition. Comparing straight lines on square and triangular lattices with square loops and with lines of thickness 2, we discuss questions of universality.

12 pages, including 17 figures

References in corpus (5)

• Area laws in quantum systems: mutual information and correlations
• Rényi entropy of a line in two-dimensional Ising models
• Finite size behaviors of critical Ising model on a rectangle with free boundaries
• "Exact" Algorithm for Random-Bond Ising Models in 2D
• Bond-Propagation Algorithm for Thermodynamic Functions in General 2D Ising Models

Cited by in corpus (10)

• Participation spectroscopy and entanglement Hamiltonian of quantum spin models
• Universal behavior of the Shannon and Rényi mutual information of quantum critical chains
• Generalized mutual informations of quantum critical chains
• Analysis of the phase transition in the $2D$ Ising ferromagnet using a Lempel-Ziv string parsing scheme and black-box data-compression utilities
• Statistical properties of the Green function in finite size for Anderson Localization models with multifractal eigenvectors
• Mutual information, quantum phase transition, and phase coherence in Kondo systems
• Shared information in classical mean-field models
• Pure and Random Quantum Ising Chain : Shannon and Renyi entropies of the ground state via real space renormalization
• Star junctions and watermelons of pure or random quantum Ising chains : finite-size properties of the energy gap at criticality
• Excess entropy and central charge of the two-dimensional random-bond Potts model in the large-Q limit