Edwards Anderson order parameter
Since , the model does not exhibit spatial magnetic order, such as ferromagnetic or antiferromagnetic order. Instead, the idea is to distinguish between two phases:
- Paramagnetic phase: Configurations are explored with all possible spin orientations.
- Spin glass phase: Spin orientations are random but frozen (i.e., immobile).
The glass phase is characterized by long-range correlations in time, despite the absence of long-range correlations in space. The order parameter for this phase is:
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q_{EA}} measures the overlap of the spin configuration with itself after a long time.
In the paramagnetic phase, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q_{EA} = 0} , while in the spin glass phase, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q_{EA} > 0} .
This raises the question of whether the transition at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_f} is truly thermodynamic in nature. Indeed, in the definition of the Edwards-Anderson (EA) parameter, time seems to play a role, and the magnetic susceptibility does not diverge at the freezing temperature Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_f} . The divergence of the magnetic susceptibility in ferromagnets is due to the fact that the magnetization Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M = \sum_i \sigma_i} serves as the order parameter, distinguishing the ordered and disordered phases. However, in the spin glass model, magnetization is zero in both phases and the order parameter is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q_{EA}} . It can be shown that the associated susceptibility corresponds to the nonlinear susceptibility:
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi} is the linear susceptibility, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_3, a_5, \ldots} are higher-order coefficients. Experiments have demonstrated that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_3} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_5} exhibit singular behavior, providing experimental evidence for the existence of a thermodynamic transition at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_f} .