T-4: Difference between revisions
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<strong>Comment: </strong> the magnetic susceptibility does not diverge at the spin-glass critical temperature. One can identify a more complicated object, the <em> spin-glass susceptibility </em>, that does diverge at the transition. At variance with the magnetic susceptibility, that is related to a 2-point function (the correlation, which involves two spins), the spin-glass susceptibility is associated to a 4-point function. This is consistent with the fact that the order parameter of the spin-glass phase, <math> q_{EA} </math>, is itself a 2-point function, while the magnetization that is a 1-point function. | <strong>Comment: </strong> the magnetic susceptibility does not diverge at the spin-glass critical temperature. One can identify a more complicated object, the <em> spin-glass susceptibility </em>, that does diverge at the transition. At variance with the magnetic susceptibility, that is related to a 2-point function (the correlation, which involves two spins), the spin-glass susceptibility is associated to a 4-point function. This is consistent with the fact that the order parameter of the spin-glass phase, <math> q_{EA} </math>, is itself a 2-point function, while the magnetization that is a 1-point function. | ||
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<!--In the Ising case, the magnetization <math> m_N(\beta,h)</math> behaves as in Fig. for <math> T < T_c</math>. Using the plot, show that the limits <math> N \to \infty</math> and <math> h \to 0</math> do not commute: if <math> N \to \infty</math> is taken first as in the definition of the thermodynamics susceptibility, one gets a finite value for <math> \chi(\beta)</math>; if instead <math> h \to 0</math> is taken before, <math> \chi_N(\beta)</math> diverges. Confirm the last observation using the FDT an the fact that <math>\overline{\langle \sigma_i \sigma_j \rangle_c}= m^2</math>.---> | <!--In the Ising case, the magnetization <math> m_N(\beta,h)</math> behaves as in Fig. for <math> T < T_c</math>. Using the plot, show that the limits <math> N \to \infty</math> and <math> h \to 0</math> do not commute: if <math> N \to \infty</math> is taken first as in the definition of the thermodynamics susceptibility, one gets a finite value for <math> \chi(\beta)</math>; if instead <math> h \to 0</math> is taken before, <math> \chi_N(\beta)</math> diverges. Confirm the last observation using the FDT an the fact that <math>\overline{\langle \sigma_i \sigma_j \rangle_c}= m^2</math>.---> | ||
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==Notes== | ==Notes== | ||
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Revision as of 16:10, 19 January 2024
Goal: Understand some physical properties of mean-field spin glasses in the low-T phase: the structure of the free-energy landscape, the response of the system to applied magnetic fields.
Key concepts: full-RSB, magnetic susceptibility, linear response, ultrametricity.