8,206 research outputs found
Subsystem eigenstate thermalization hypothesis for entanglement entropy in CFT
We investigate a weak version of subsystem eigenstate thermalization
hypothesis (ETH) for a two-dimensional large central charge conformal field
theory by comparing the local equivalence of high energy state and thermal
state of canonical ensemble. We evaluate the single-interval R\'enyi entropy
and entanglement entropy for a heavy primary state in short interval expansion.
We verify the results of R\'enyi entropy by two different replica methods. We
find nontrivial results at the eighth order of short interval expansion, which
include an infinite number of higher order terms in the large central charge
expansion. We then evaluate the relative entropy of the reduced density
matrices to measure the difference between the heavy primary state and thermal
state of canonical ensemble, and find that the aforementioned nontrivial eighth
order results make the relative entropy unsuppressed in the large central
charge limit. By using Pinsker's and Fannes-Audenaert inequalities, we can
exploit the results of relative entropy to yield the lower and upper bounds on
trace distance of the excited-state and thermal-state reduced density matrices.
Our results are consistent with subsystem weak ETH, which requires the above
trace distance is of power-law suppression by the large central charge.
However, we are unable to pin down the exponent of power-law suppression. As a
byproduct we also calculate the relative entropy to measure the difference
between the reduced density matrices of two different heavy primary states.Comment: 28 pages, 4 figures;v2 change author list;v3 related subtleties about
weak ETH clarified; v4 minor correction to match JHEP versio
Dissimilarities of reduced density matrices and eigenstate thermalization hypothesis
We calculate various quantities that characterize the dissimilarity of
reduced density matrices for a short interval of length in a
two-dimensional (2D) large central charge conformal field theory (CFT). These
quantities include the R\'enyi entropy, entanglement entropy, relative entropy,
Jensen-Shannon divergence, as well as the Schatten 2-norm and 4-norm. We adopt
the method of operator product expansion of twist operators, and calculate the
short interval expansion of these quantities up to order of for the
contributions from the vacuum conformal family. The formal forms of these
dissimilarity measures and the derived Fisher information metric from
contributions of general operators are also given. As an application of the
results, we use these dissimilarity measures to compare the excited and thermal
states, and examine the eigenstate thermalization hypothesis (ETH) by showing
how they behave in high temperature limit. This would help to understand how
ETH in 2D CFT can be defined more precisely. We discuss the possibility that
all the dissimilarity measures considered here vanish when comparing the
reduced density matrices of an excited state and a generalized Gibbs ensemble
thermal state. We also discuss ETH for a microcanonical ensemble thermal state
in a 2D large central charge CFT, and find that it is approximately satisfied
for a small subsystem and violated for a large subsystem.Comment: V1, 34 pages, 5 figures, see collection of complete results in the
attached Mathematica notebook; V2, 38 pages, 5 figures, published versio
Tracking nitrogen losses in a greenhouse crop rotation experiment in North China using the EU-Rotate_N simulation model
Vegetable production in China is associated with high inputs of nitrogen, posing a risk of losses to the
environment. Organic matter mineralisation is a considerable source of nitrogen (N) which is hard to
quantify. In a two-year greenhouse cucumber experiment with different N treatments in North China,
non-observed pathways of the N cycle were estimated using the EU-Rotate_N simulation model.
EU-Rotate_N was calibrated against crop dry matter and soil moisture data to predict crop N uptake, soil
mineral N contents, N mineralisation and N loss. Crop N uptake (Modelling Efficiencies (ME) between
0.80 and 0.92) and soil mineral N contents in different soil layers (ME between 0.24 and 0.74) were
satisfactorily simulated by the model for all N treatments except for the traditional N management. The
model predicted high N mineralisation rates and N leaching losses, suggesting that previously published
estimates of N leaching for these production systems strongly underestimated the mineralisation of N
from organic matter
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