Strong decay of correlations for Gibbs states in any dimension
- URL: http://arxiv.org/abs/2401.10147v1
- Date: Thu, 18 Jan 2024 17:20:29 GMT
- Title: Strong decay of correlations for Gibbs states in any dimension
- Authors: Andreas Bluhm, \'Angela Capel, Antonio P\'erez-Hern\'andez
- Abstract summary: We show that systems with short-range interactions that are above a critical temperature satisfy a mixing condition.
This condition is stronger than other commonly studied measures of correlation.
We show that many notions of decay of correlations in quantum many-body systems are equivalent under the assumption that there exists a local effective Hamiltonian.
- Score: 0.27309692684728604
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: Quantum systems in thermal equilibrium are described using Gibbs states. The
correlations in such states determine how difficult it is to describe or
simulate them. In this article, we show that systems with short-range
interactions that are above a critical temperature satisfy a mixing condition,
that is that for any regions $A$, $C$ the distance of the reduced state
$\rho_{AC}$ on these regions to the product of its marginals, $$\| \rho_{AC}
\rho_A^{-1} \otimes \rho_C^{-1} - \mathbf{1}_{AC}\| \, ,$$ decays exponentially
with the distance between regions $A$ and $C$. This mixing condition is
stronger than other commonly studied measures of correlation. In particular, it
implies the exponential decay of the mutual information between distant
regions. The mixing condition has been used, for example, to prove positive
log-Sobolev constants. On the way, we investigate the relations to other
notions of decay of correlations in quantum many-body systems and show that
many of them are equivalent under the assumption that there exists a local
effective Hamiltonian. The proof employs a variety of tools such as Araki's
expansionals and quantum belief propagation.
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