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A relaxation-function theory for the dynamic spin susceptibility in the t-J model is presented. By a sum-rule-conserving generalized mean-field approximation (GMFA), the two-spin correlation functions of arbitrary range, the staggered magnetization, the uniform static susceptibility, and the antiferromagnetic correlation length are calculated in a wide region of hole doping and temperaturs. A good agreement with available exact diagonalization (ED) data is found. The correlation length is in reasonable agreement with neutron-scattering experiments on La 2−δ Sr δ CuO4. Going beyond the GMFA, the self-energy is calculated in the mode-coupling approximation. The spin dynamics at arbitrary frequencies and wave vectors is studied for various temperatures and hole doping. At low doping a spin-wave-type behavior is found as in the Heisenberg model, while at higher doping a strong damping caused by hole hopping occurs, and a relaxation-type spin dynamics is observed in agreement with the ED results. The local spin susceptibility and its ω/T scaling behavior are calculated in a reasonable agreement with experimental and ED data.

Subsoils are known to harbor large amounts of soil organic carbon (SOC) and may represent key global carbon (C) sinks given appropriate management. Although rhizodeposition is a major input pathway of organic matter to subsoils, little knowledge exists on C dynamics, particularly stabilization mechanisms, such as soil aggregation, in the rhizosphere of different soil depths. The aim of this study was to investigate the influence of natural and elevated root exudation on C allocation and aggregation in the topsoil and subsoil of a mature European beech (Fagus sylvatica L.) forest. We experimentally added model root exudates to soil at two different concentrations using artificial roots and analyzed how these affect SOC, nitrogen, microbial community composition, and size distribution of water-stable aggregates. Based on the experimental data, a mathematical model was developed to describe the spatial distribution of the formation of soil aggregates and their binding strength. Our results demonstrate that greater exudate additions affect the microbial community composition in favor of fungi which promote the formation of macroaggregates. This effect was most pronounced in the C-poor subsoil, where macroaggregation increased by 86% and SOC content by 10%. Our modeling exercise reproduced the observed increase in subsoil SOC at high exudate additions. We conclude that elevated root exudation has the potential to increase biotic macroaggregation and thus the C sink strength in the rhizosphere of forest subsoils.

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