Cumulant expansion in renormalization-group transformations on Ising spin systems: A third-order calculation

1976 ◽  
Vol 13 (3) ◽  
pp. 980-982 ◽  
Author(s):  
Aa. S. Sudbø ◽  
P. C. Hemmer
Keyword(s):  
Renormalization Group ◽  
Spin Systems ◽  
Cumulant Expansion ◽  
Third Order ◽  
Ising Spin ◽  
Order Calculation

Monte Carlo renormalization-group study of Ising spin glasses

1988 ◽  
Vol 37 (13) ◽  
pp. 7745-7750 ◽  
Author(s):  
Jian-Sheng Wang ◽  
Robert H. Swendsen
Keyword(s):  
Monte Carlo ◽  
Renormalization Group ◽  
Spin Glasses ◽  
Ising Spin ◽  
Monte Carlo Renormalization Group

Entanglement in quantum dissipative Ising spin systems

2007 ◽  
Vol 398 (2) ◽  
pp. 455-459
Author(s):  
G.S. Lozano ◽  
H.F. Lozza ◽  
D. Pérez Daroca
Keyword(s):  
Spin Systems ◽  
Ising Spin

A Property of Zeros of the Partition Function for Ising Spin Systems

1972 ◽  
Vol 13 (5) ◽  
pp. 781-784 ◽  
Author(s):  
Elliott H. Lieb ◽  
David Ruelle
Keyword(s):  
Partition Function ◽  
Spin Systems ◽  
Ising Spin

scholarly journals Residual dipolar line width in magic-angle spinning proton solid-state NMR

2021 ◽  
Vol 2 (1) ◽  
pp. 499-509
Author(s):  
Matías Chávez ◽  
Thomas Wiegand ◽  
Alexander A. Malär ◽  
Beat H. Meier ◽  
Matthias Ernst
Keyword(s):  
Chemical Shift ◽  
Line Width ◽  
Magic Angle Spinning ◽  
Spin Systems ◽  
Second Order ◽  
Magic Angle ◽  
Line Broadening ◽  
Third Order ◽  
Angle Spinning ◽  
Effective Hamiltonians

Abstract. Magic-angle spinning is routinely used to average anisotropic interactions in solid-state nuclear magnetic resonance (NMR). Due to the fact that the homonuclear dipolar Hamiltonian of a strongly coupled spin system does not commute with itself at different time points during the rotation, second-order and higher-order terms lead to a residual dipolar line broadening in the observed resonances. Additional truncation of the residual broadening due to isotropic chemical-shift differences can be observed. We analyze the residual line broadening in coupled proton spin systems based on theoretical calculations of effective Hamiltonians up to third order using Floquet theory and compare these results to numerically obtained effective Hamiltonians in small spin systems. We show that at spinning frequencies beyond 75 kHz, second-order terms dominate the residual line width, leading to a 1/ωr dependence of the second moment which we use to characterize the line width. However, chemical-shift truncation leads to a partial ωr-2 dependence of the line width which looks as if third-order effective Hamiltonian terms are contributing significantly. At slower spinning frequencies, cross terms between the chemical shift and the dipolar coupling can contribute in third-order effective Hamiltonians. We show that second-order contributions not only broaden the line, but also lead to a shift of the center of gravity of the line. Experimental data reveal such spinning-frequency-dependent line shifts in proton spectra in model substances that can be explained by line shifts induced by the second-order dipolar Hamiltonian.

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