L-1 - Revision history

Lptmswikids at 16:38, 14 January 2026

2026-01-14T16:38:17Z

Lptmswikids at 10:44, 14 January 2026

2026-01-14T10:44:08Z

← Older revision		Revision as of 12:44, 14 January 2026
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	It can be shown that the associated susceptibility corresponds to the nonlinear susceptibility:		It can be shown that the associated susceptibility corresponds to the nonlinear susceptibility:
	<center><math> \frac{M}{H} = \chi + a_3 H^2 + a_5 H^4 + \ldots </math></center> where <math>\chi</math> is the linear susceptibility, and <math>a_3, a_5, \ldots</math> are higher-order coefficients. Experiments have demonstrated that <math>a_3</math> and <math>a_5</math> exhibit singular behavior, providing experimental evidence for the existence of a thermodynamic transition at <math>T_f</math>.		<center><math> \frac{M}{H} = \chi + a_3 H^2 + a_5 H^4 + \ldots </math></center> where <math>\chi</math> is the linear susceptibility, and <math>a_3, a_5, \ldots</math> are higher-order coefficients. Experiments have demonstrated that <math>a_3</math> and <math>a_5</math> exhibit singular behavior, providing experimental evidence for the existence of a thermodynamic transition at <math>T_f</math>.

	~~[[The SK model]]~~

	<!-- == Exercise L1-A: More on extreme values ==		<!-- == Exercise L1-A: More on extreme values ==

Lptmswikids at 10:43, 14 January 2026

2026-01-14T10:43:43Z

← Older revision		Revision as of 12:43, 14 January 2026
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	[[The SK model]]		[[The SK model]]
	~~[[Random energy model]]~~

	<!-- == Exercise L1-A: More on extreme values ==		<!-- == Exercise L1-A: More on extreme values ==

Lptmswikids: /* Spin glass Transition */

2026-01-14T10:43:00Z

Spin glass Transition

← Older revision		Revision as of 12:43, 14 January 2026
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	[[The SK model]]		[[The SK model]]
	[[Random energy model]]		[[Random energy model]]
	~~[[Detour: Extreme Value Statistics]]~~


	<!-- == Exercise L1-A: More on extreme values ==		<!-- == Exercise L1-A: More on extreme values ==

Lptmswikids: /* Spin glass Transition */

2026-01-14T10:42:27Z

Spin glass Transition

← Older revision		Revision as of 12:42, 14 January 2026
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	[[Random energy model]]		[[Random energy model]]
	[[Detour: Extreme Value Statistics]]		[[Detour: Extreme Value Statistics]]
	~~[[A Concrete Example: The Gaussian Case]]~~
	~~[[The Scaling Form in the Large ''M'' Limit]]~~
	~~[[Density above the minimum]]~~
	~~[[Back to the REM]]~~
	~~[[Phase Transition in the Random Energy Model]]~~
	~~[[Behavior in Different Phases:]]~~
	~~[[More general REM and systems in Finite dimensions]]~~

	<!-- == Exercise L1-A: More on extreme values ==		<!-- == Exercise L1-A: More on extreme values ==

Lptmswikids: /* Edwards Anderson order parameter */

2026-01-14T10:42:03Z

Edwards Anderson order parameter

← Older revision		Revision as of 12:42, 14 January 2026
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	[[The SK model]]		[[The SK model]]
			[[Random energy model]]
			[[Detour: Extreme Value Statistics]]
			[[A Concrete Example: The Gaussian Case]]
			[[The Scaling Form in the Large ''M'' Limit]]
			[[Density above the minimum]]
			[[Back to the REM]]
			[[Phase Transition in the Random Energy Model]]
			[[Behavior in Different Phases:]]
			[[More general REM and systems in Finite dimensions]]

	<!-- == Exercise L1-A: More on extreme values ==		<!-- == Exercise L1-A: More on extreme values ==

Lptmswikids at 10:39, 14 January 2026

2026-01-14T10:39:17Z

← Older revision		Revision as of 12:39, 14 January 2026
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	[[The SK model]]		[[The SK model]]

			<!-- == Exercise L1-A: More on extreme values ==
			For a large set of iid random variables there are only three scaling form for the distribution of the minimum. In this lecture we studied the Gumbel case for fast decaying functions. The other two possibilities are
			* <strong>Frechet case: </strong> for a power law tail <math> p(E) \sim c/\|E\|^{\gamma+1} </math> with <math> \gamma>0 </math>
			* <strong>Weidbul case: </strong> for a bounded distribution with <math> p(E) \sim c(E-E_0)^\gamma \; \text{when } E\to E_0^+ </math>
			Compute <math>a_M, b_M</math> in both cases as well as the limiting distribution. -->

			=References=
			* ''Spin glass i-vii'', P.W. Anderson, Physics Today, 1988
			* ''Spin glasses: Experimental signatures and salient outcome'', E. Vincent and V. Dupuis, Frustrated Materials and Ferroic Glasses 31 (2018).
			* ''Theory of spin glasses'', S. F. Edwards and P. W. Anderson, J. Phys. F: Met. Phys. 5 965 (1975).
			* ''Non-linear susceptibility in spin glasses and disordered systems'', H. Bouchiat, Journal of Physics: Condensed Matter, 9, 1811 (1997).
			* ''Solvable Model of a Spin-Glass'', D. Sherrington and S. Kirkpatrick, Physical Review Letters, 35, 1792 (1975).
			* ''Random-Energy Model: An Exactly Solvable Model of Disordered Systems'', B.Derrida,Physical Review B, 24, 2613 (1980).
			* ''Extreme value statistics of correlated random variables: a pedagogical review'', S. N. Majumdar, A. Pal, and G. Schehr, Physics Reports 840, 1-32, (2020).

Lptmswikids at 10:31, 14 January 2026

2026-01-14T10:31:53Z

← Older revision		Revision as of 12:31, 14 January 2026
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	It can be shown that the associated susceptibility corresponds to the nonlinear susceptibility:		It can be shown that the associated susceptibility corresponds to the nonlinear susceptibility:
	<center><math> \frac{M}{H} = \chi + a_3 H^2 + a_5 H^4 + \ldots </math></center> where <math>\chi</math> is the linear susceptibility, and <math>a_3, a_5, \ldots</math> are higher-order coefficients. Experiments have demonstrated that <math>a_3</math> and <math>a_5</math> exhibit singular behavior, providing experimental evidence for the existence of a thermodynamic transition at <math>T_f</math>.		<center><math> \frac{M}{H} = \chi + a_3 H^2 + a_5 H^4 + \ldots </math></center> where <math>\chi</math> is the linear susceptibility, and <math>a_3, a_5, \ldots</math> are higher-order coefficients. Experiments have demonstrated that <math>a_3</math> and <math>a_5</math> exhibit singular behavior, providing experimental evidence for the existence of a thermodynamic transition at <math>T_f</math>.

			[[The SK model]]

Lptmswikids at 10:27, 14 January 2026

2026-01-14T10:27:53Z

Show changes

Lptmswikids: /* Edwards Anderson order parameter */

2026-01-14T09:52:15Z

Edwards Anderson order parameter

← Older revision		Revision as of 11:52, 14 January 2026
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	In the paramagnetic phase, <math>q_{EA} = 0</math>, while in the spin glass phase, <math>q_{EA} > 0</math>.		In the paramagnetic phase, <math>q_{EA} = 0</math>, while in the spin glass phase, <math>q_{EA} > 0</math>.

	This raises the question of whether the transition at <math>T_f</math> 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 <math>T_f</math>. The divergence of the magnetic susceptibility in ferromagnets is due to the fact that the magnetization <math>M = \sum_i \sigma_i</math> 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 <math>q_{EA}</math>.		This raises the question of whether the transition at <math>T_f</math> 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 <math>T_f</math>. The divergence of the magnetic susceptibility in ferromagnets is due to the fact that the magnetization <math>M = \sum_i \sigma_i</math> 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 <math>q_{EA}</math>.
	It can be shown that the associated susceptibility corresponds to the nonlinear susceptibility:		It can be shown that the associated susceptibility corresponds to the nonlinear susceptibility:
	<center><math> \frac{M}{H} = \chi + a_3 H^2 + a_5 H^4 + \ldots </math></center> where <math>\chi</math> is the linear susceptibility, and <math>a_3, a_5, \ldots</math> are higher-order coefficients. Experiments have demonstrated that <math>a_3</math> and <math>a_5</math> exhibit singular behavior, providing experimental evidence for the existence of a thermodynamic transition at <math>T_f</math>.		<center><math> \frac{M}{H} = \chi + a_3 H^2 + a_5 H^4 + \ldots </math></center> where <math>\chi</math> is the linear susceptibility, and <math>a_3, a_5, \ldots</math> are higher-order coefficients. Experiments have demonstrated that <math>a_3</math> and <math>a_5</math> exhibit singular behavior, providing experimental evidence for the existence of a thermodynamic transition at <math>T_f</math>.