Mutual information area laws for thermal free fermions

Abstract

We provide a rigorous and asymptotically exact expression of the mutual information of translationally invariant free fermionic lattice systems in a Gibbs state. In order to arrive at this result, we introduce a novel frameworkfor computing determinants of Toeplitz operators with smooth symbols, and for treating Toeplitz matrices with system size dependent entries. The asymptotically exact mutual information for a partition of the one-dimensional lattice satisfies an area law, with a prefactor which we compute explicitly. As examples, we discuss the fermionic XX model in one dimension and free fermionic models on the torus in higher dimensions in detail. Special emphasis is put onto the discussion of the temperature dependence of the mutual information, scaling like the logarithm of the inverse temperature, hence confirming an expression suggested by conformal field theory. We also comment on the applicability of the formalism to treat open systems driven by quantum noise. In the appendix, we derive useful bounds to the mutual information in terms of purities. Finally, we provide a detailed error analysis for finite system sizes. This analysis is valuable in its own right for the abstract theory of Toeplitz determinants.

42 pages, 4 figures, replaced by final version

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• Quantum many-body systems in thermal equilibrium
• Generalized mutual informations of quantum critical chains
• Computable Rényi mutual information: Area laws and correlations
• Typical Correlation Length of Sequentially Generated Tensor Network States
• Large-scale behaviour of local and entanglement entropy of the free Fermi gas at any temperature
• Diagnosing Topological Edge States via Entanglement Monogamy
• Mutual information for fermionic systems
• Universality in the tripartite information after global quenches: (generalised) quantum XY models
• Thermal Area Law in Long-Range Interacting Systems
• Many-Body Localization: Concepts and Simple Models
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• Clustering of conditional mutual information and quantum Markov structure at arbitrary temperatures
• Non-Conformal Behavior of Holographic Entanglement Measures
• Entanglement of Free Fermions and Bosons at Finite Temperature
• Distilling momentum-space entanglement in Luttinger liquids at finite temperature
• Universality in the tripartite information after global quenches: spin flip and semilocal charges
• Asymptotic behaviour of determinants through the expansion of the Moyal star product