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Condensed Matter, Statistical Mechanics, cond-mat.stat-mech
Abstract:
The complexity of quantum many-body systems is manifested in the vast
diversity of their correlations, making it challenging to distinguish the
generic from the atypical features. This can be addressed by analyzing
correlations through ensembles of random states, chosen so as to faithfully
embody the relevant physical properties. Here we focus on spins with local
interactions, whose correlations are extremely well captured by tensor network
states. Adopting an operational perspective, we define ensembles of random
tensor network states in one and two spatial dimensions that admit a sequential
generation. As such, they directly correspond to outputs of quantum circuits
with a sequential architecture and random gates. In one spatial dimension, the
ensemble explores the entire family of matrix product states, while in two
spatial dimensions, it corresponds to random isometric tensor network states.
We extract the scaling behavior of the average correlations between two
subsystems as a function of their distance. Using elementary concentration
results, we then deduce the typical case for measures of correlation such as
the von Neumann mutual information and a measure arising from the
Hilbert-Schmidt norm. We find for all considered cases that the typical
behavior is an exponential decay (for both one and two spatial dimensions). We
observe the consistent emergence of a correlation length that only depends on
the underlying spatial dimension and not the considered measure. Remarkably,
increasing the bond dimension leads to a higher correlation length in one
spatial dimension but has the opposite effect in two spatial dimensions.
