Universal behavior of the Shannon and Rényi mutual information of quantum critical chains

Abstract

We study the Shannon and Rényi mutual information (MI) in the ground state (GS) of different critical quantum spin chains. Despite the apparent basis dependence of these quantities we show the existence of some particular basis (we will call them conformal basis) whose finite-size scaling function is related to the central charge $c$ of the underlying conformal field theory of the model. In particular, we verified that for large index $n$, the MI of a subsystem of size $\ell$ in a periodic chain with $L$ sites behaves as $\frac{c}{4}\frac{n}{n-1}\ln\Big{(}\frac{L}π\sin(\frac{π\ell}{L})\Big{)}$, when the ground-state wavefunction is expressed in these special conformal basis. This is in agreement with recent predictions. For generic local basis we will show that, although in some cases $b_n\ln\Big{(}\frac{L}π\sin(\frac{π\ell}{L})\Big{)}$ is a good fit to our numerical data, in general there is no direct relation between $b_n$ and the central charge of the system. We will support our findings with detailed numerical calculations for the transverse field Ising model, $Q=3,4$ quantum Potts chain, quantum Ashkin-Teller chain and the XXZ quantum chain. We will also present some additional results of the Shannon mutual information ($n=1$), for the parafermionic $Z_Q$ quantum chains with $Q=5,6,7$ and $8$.

11 Pages, 12 figures, v2: a new figure added and some other minor additions

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• Entanglement entropy after selective measurements in quantum chains
• Magic transition in measurement-only circuits
• Stabilizer Rényi Entropy and Conformal Field Theory
• Mutual information for fermionic systems
• Finite size corrections to scaling of the formation probabilities and the Casimir effect in the conformal field theories
• Universal logarithmic correction to Rényi (Shannon) entropy in generic systems of critical quadratic fermions
• 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
• Explicit Pfaffian Formula for Amplitudes of Fermionic Gaussian Pure States in Arbitrary Pauli Bases
• Shannon entropy in quasiparticle states of quantum chains
• Efficient Representation of Gaussian Fermionic Pure States in Non-Computational Bases
• Equivalence of Stabilizer and Shannon Rényi Entropies: Exact Results for Quantum Critical Chains