Insight into hydrochemistry: a multi-catchment comparison using Horizontal Visibility Graphs

Long time series of environmental variables are reflecting the dynamics and functioning of ecosystems. Here, we investigate data from a long-term monitoring site in Germany, the Bramke valley in the Harz mountains, where time series of ion concentrations in stream water are obtained since the 1970ies at four measurement locations from three small adjacent forested catchments. Since for (only) one of the catchments daily runoff rates are also available, we invent a method to generate time series of nutrient output from the catchments. Both concentrations and outputs show a number of remarkable long-term changes, including ones not obviously related to changes in atmospheric deposition, management or properties of the forest stands. For the analysis of the Bramke data, we investigate Horizontal Visibility Graphs (HVGs), a recently developed method to construct networks based on time series. Values (the nodes of the network) of the time series are linked to each other if there is no value higher between them. The network properties, such as the degree and distance distributions, reflect the nonlinear dynamics of the time series. For certain classes of stochastic processes and for periodic time series, analytic results can be obtained for some network properties. HVGs have the potential to discern between deterministic-chaotic and correlated-stochastic time series. We classify the Bramke series according to their stochastic nature, with a focus on inter-catchment comparison on one hand, on different nutrients for one catchment on the other, and conclude on possible reasons for the observed changes and their ecological interpretation

Computation of static Heisenberg-chain correlators: Control over length and temperature dependence

We communicate results on correlation functions for the spin-1/2 Heisenberg-chain in two particularly important cases: (a) for the infinite chain at arbitrary finite temperature $T$, and (b) for finite chains of arbitrary length $L$ in the ground-state. In both cases we present explicit formulas expressing the short-range correlators in a range of up to seven lattice sites in terms of a single function $\omega$ encoding the dependence of the correlators on $T$ ($L$). These formulas allow us to obtain accurate numerical values for the correlators and derived quantities like the entanglement entropy. By calculating the low $T$ (large $L$) asymptotics of $\omega$ we show that the asymptotics of the static correlation functions at any finite distance are $T^2$ ($1/L^2$) terms. We obtain exact and explicit formulas for the coefficients of the leading order terms for up to eight lattice sites.Comment: 5 pages, 3 figures, v2: text slightly shortened, typos in eqns. (16), (17) corrected, Fig. 1 replaced, v3: typo in eqn. (11) correcte

The complete conformal spectrum of a sl(2∣1)sl(2|1) invariant network model and logarithmic corrections

We investigate the low temperature asymptotics and the finite size spectrum of a class of Temperley-Lieb models. As reference system we use the spin-1/2 Heisenberg chain with anisotropy parameter $\Delta$ and twisted boundary conditions. Special emphasis is placed on the study of logarithmic corrections appearing in the case of $\Delta=1/2$ in the bulk susceptibility data and in the low-energy spectrum yielding the conformal dimensions. For the $sl(2|1)$ invariant 3-state representation of the Temperley-Lieb algebra with $\Delta=1/2$ we give the complete set of scaling dimensions which show huge degeneracies.Comment: 18 pages, 5 figure

Quantum spin chains of Temperley-Lieb type: periodic boundary conditions, spectral multiplicities and finite temperature

We determine the spectra of a class of quantum spin chains of Temperley-Lieb type by utilizing the concept of Temperley-Lieb equivalence with the S=1/2 XXZ chain as a reference system. We consider open boundary conditions and in particular periodic boundary conditions. For both types of boundaries the identification with XXZ spectra is performed within isomorphic representations of the underlying Temperley-Lieb algebra. For open boundaries the spectra of these models differ from the spectrum of the associated XXZ chain only in the multiplicities of the eigenvalues. The periodic case is rather different. Here we show how the spectrum is obtained sector-wise from the spectra of globally twisted XXZ chains. As a spin-off, we obtain a compact formula for the degeneracy of the momentum operator eigenvalues. Our representation theoretical results allow for the study of the thermodynamics by establishing a TL-equivalence at finite temperature and finite field.Comment: 29 pages, LaTeX, two references added, redundant figures remove