L-9 - Revision history

Rosso: /* Larkin model */

2026-03-23T20:07:03Z

Larkin model

← Older revision		Revision as of 22:07, 23 March 2026
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	By contrast, in a multifractal state the amplitudes are broadly distributed and many values of <math>\alpha</math> contribute. The function <math>f(\alpha)</math> then becomes a smooth curve describing the distribution of scaling exponents of the wavefunction.		By contrast, in a multifractal state the amplitudes are broadly distributed and many values of <math>\alpha</math> contribute. The function <math>f(\alpha)</math> then becomes a smooth curve describing the distribution of scaling exponents of the wavefunction.

	= Larkin model =		<!-- = Larkin model =

	In your homework you solved a toy model for the interface		In your homework you solved a toy model for the interface
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	\ell_L=\left(\frac{r_f}{\sigma}\right)^{2/3}.		\ell_L=\left(\frac{r_f}{\sigma}\right)^{2/3}.
	</math>		</math>
			-->

Rosso: /* Delocalized eigenstates */

2026-03-17T14:20:22Z

Delocalized eigenstates

← Older revision		Revision as of 16:20, 17 March 2026
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	<math display="block">		<math display="block">
	\tau_q=d(1-q).		\tau_q=d(q-1).
	</math>		</math>

Rosso: /* Larkin model */

2026-03-15T16:33:04Z

Larkin model

← Older revision		Revision as of 18:33, 15 March 2026
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	= Larkin model =		= Larkin model =

	In your homework you solved a toy model for the interface:		In your homework you solved a toy model for the interface

	<math display="block">		<math display="block">
	\partial_t h(r,t) = \nabla^2 h(r,t) + F(r).		\partial_t h(r,t) = \nabla^2 h(r,t) + F(r).
	</math>		</math>
	For simplicity, we assume Gaussian disorder <math>\overline{F(r)}=0~~</math>~~, ~~<math>~~\overline{F(r)F(r')}=\sigma^2 \delta^d(r-r')</math>.
			For simplicity, we assume Gaussian disorder

			<math display="block">
			\overline{F(r)}=0,
			\qquad
			\overline{F(r)F(r')}=\sigma^2 \delta^d(r-r').
			</math>

	You proved that:		You proved that:
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	* the force per unit length acting on the center of the interface is <math>f= \sigma/\sqrt{L^d}</math>		* the force per unit length acting on the center of the interface is <math>f= \sigma/\sqrt{L^d}</math>
	* at long times the interface shape is		* at long times the interface shape is

	<math display="block">		<math display="block">
	\overline{h(q)h(-q)} = \frac{\sigma^2}{q^{d+2\zeta_L}}.		\overline{h(q)h(-q)} = \frac{\sigma^2}{q^{d+2\zeta_L}}.
	</math>		</math>

	In the real depinning model the disorder is, however, a non-linear function of <math>h</math>. The idea of Larkin is that this linearization is ~~correct~~ up to <math>r_f</math>, the correlation length of the disorder ~~<Strong>~~along the <math>h</math> direction~~</Strong>~~. This defines a Larkin length.		In the real depinning model the disorder is, however, a non-linear function of <math>h</math>. The idea of Larkin is that this linearization is valid only up to a scale <math>r_f</math>, the correlation length of the disorder along the <math>h</math> direction. This defines the '''Larkin length'''.

			Indeed, starting from

	~~Indeed, from~~
	<math display="block">		<math display="block">
	\overline{(h(r)-h(0))^2}		\overline{(h(r)-h(0))^2}
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	\sim \sigma^2 r^{2\zeta_L},		\sim \sigma^2 r^{2\zeta_L},
	</math>		</math>
	~~you get~~
			we obtain

	<math display="block">		<math display="block">
	\overline{(h(\ell_L)-h(0))^2}= r_f^2,		\overline{(h(\ell_L)-h(0))^2}= r_f^2,
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	</math>		</math>

	Above this scale, roughness changes and pinning ~~starts~~ with a critical force		Above this scale the roughness changes and pinning sets in with a critical force

	<math display="block">		<math display="block">
	f_c= \frac{\sigma}{\ell_L^{d/(2 \zeta_L)}}.		f_c= \frac{\sigma}{\ell_L^{d/(2 \zeta_L)}}.
	</math>		</math>

	In <math>d=1</math> we ~~have~~ <math>\ell_L=\left(\frac{r_f}{\sigma}\right)^{2/3}</math>.		In <math>d=1</math> we therefore obtain

			<math display="block">
			\ell_L=\left(\frac{r_f}{\sigma}\right)^{2/3}.
			</math>

Rosso: /* Eigenstates */

2026-03-15T16:31:35Z

Eigenstates

← Older revision		Revision as of 18:31, 15 March 2026
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	* '''Multifractal eigenstates''', occurring at the '''mobility edge''' of the Anderson transition, where the wavefunction exhibits a complex, scale–dependent structure.		* '''Multifractal eigenstates''', occurring at the '''mobility edge''' of the Anderson transition, where the wavefunction exhibits a complex, scale–dependent structure.

	In three dimensions, increasing the disorder strength leads to a transition between a metallic phase with delocalized eigenstates and an insulating phase with localized eigenstates. The critical energy separating these two regimes is called the ~~'''~~mobility edge~~'''~~. Eigenstates at the mobility edge display multifractal statistics.		In three dimensions, increasing the disorder strength leads to a transition between a metallic phase with delocalized eigenstates and an insulating phase with localized eigenstates. The critical energy separating these two regimes is called the mobility edge. Eigenstates at the mobility edge display multifractal statistics.

	To characterize these different regimes it is useful to introduce the ~~'''~~inverse participation ratio~~'''~~ (IPR)		To characterize these different regimes it is useful to introduce the inverse participation ratio (IPR)

	<math display="block">		<math display="block">
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	</math>		</math>

	The corresponding ~~'''~~multifractal dimensions~~'''~~ are defined as		The corresponding multifractal dimensions are defined as

	<math display="block">		<math display="block">
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	</math>		</math>

	It is useful to contrast ~~'''~~fractal~~'''~~ and ~~'''~~multifractal~~'''~~ scaling.		It is useful to contrast fractal and multifractal scaling.

	* In a '''fractal object''', a single exponent describes the scaling of the measure. If the wavefunction amplitudes scale with a single exponent <math>D</math>, then		* In a '''fractal object''', a single exponent describes the scaling of the measure. If the wavefunction amplitudes scale with a single exponent <math>D</math>, then
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	</math>		</math>

	Thus <math>\tau(q)</math> and <math>f(\alpha)</math> are related by a ~~'''~~Legendre transform~~'''~~.		Thus <math>\tau(q)</math> and <math>f(\alpha)</math> are related by a Legendre transform.

	If <math>\alpha^*(q)</math> satisfies		If <math>\alpha^*(q)</math> satisfies
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	It is useful to contrast this situation with the case of a simple fractal.		It is useful to contrast this situation with the case of a simple fractal.

	In a ~~'''~~fractal object~~'''~~, the measure has a single scaling exponent. In our notation this means that the wavefunction amplitudes scale with a single value <math>\alpha_0</math> on a set of fractal dimension <math>D</math>:		In a fractal object, the measure has a single scaling exponent. In our notation this means that the wavefunction amplitudes scale with a single value <math>\alpha_0</math> on a set of fractal dimension <math>D</math>:

	<math display="block">		<math display="block">
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	</math>		</math>

	By contrast, in a ~~'''~~multifractal state~~'''~~ the amplitudes are broadly distributed and many values of <math>\alpha</math> contribute. The function <math>f(\alpha)</math> then becomes a smooth curve describing the distribution of scaling exponents of the wavefunction.		By contrast, in a multifractal state the amplitudes are broadly distributed and many values of <math>\alpha</math> contribute. The function <math>f(\alpha)</math> then becomes a smooth curve describing the distribution of scaling exponents of the wavefunction.

	= Larkin model =		= Larkin model =

Rosso: /* Eigenstates */

2026-03-15T16:28:52Z

Eigenstates

← Older revision		Revision as of 18:28, 15 March 2026
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	At the mobility edge the eigenstates are neither extended nor localized. Instead, the amplitudes fluctuate strongly across the system.		At the mobility edge the eigenstates are neither extended nor localized. Instead, the amplitudes fluctuate strongly across the system.


	The scaling of the IPR is characterized by a nonlinear function		The scaling of the IPR is characterized by a nonlinear function
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	It is useful to contrast '''fractal''' and '''multifractal''' scaling.		It is useful to contrast '''fractal''' and '''multifractal''' scaling.

	* In a '''fractal object''', a single exponent describes the scaling of the measure. ~~For instance, if a probability measure is uniformly distributed on~~ a ~~fractal set of dimension~~ <math>D</math>, ~~one has~~		* In a '''fractal object''', a single exponent describes the scaling of the measure. If the wavefunction amplitudes scale with a single exponent <math>D</math>, then

	<math display="block">		<math display="block">
	\mathrm{IPR}(q)\sim L^{-D(q-1)} .		\mathrm{IPR}(q)\sim L^{-D(q-1)} .
	</math>		</math>

	All moments are controlled by the same dimension <math>D</math>.		All moments are controlled by the same dimension <math>D</math>.

	* In a '''multifractal object''', different regions of the system scale with different exponents.		* In a '''multifractal object''', different regions of the system scale with different exponents. In this case there is no single fractal dimension: different moments probe different effective dimensions of the wavefunction.
	In this case there is no single fractal dimension: different moments probe different effective dimensions of the wavefunction.

	== Multifractal spectrum ==		== Multifractal spectrum ==
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	sites.		sites.

	The function <math>f(\alpha)</math> describes the fractal dimension of the set of sites where the wavefunction has exponent <math>\alpha</math>.		The function <math>f(\alpha)</math> therefore describes the fractal dimension of the set of sites where the wavefunction has exponent <math>\alpha</math>.

	Using this representation,		Using this representation,
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	f(\alpha_0)=d,		f(\alpha_0)=d,
	</math>		</math>

			which corresponds to the most typical amplitude of the wavefunction.

	It is useful to contrast this situation with the case of a simple fractal.		It is useful to contrast this situation with the case of a simple fractal.
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	</math>		</math>

	By contrast, in a '''multifractal state''' the amplitudes are broadly distributed and many values of <math>\alpha</math> contribute. The function <math>f(\alpha)</math> then becomes a smooth curve.		By contrast, in a '''multifractal state''' the amplitudes are broadly distributed and many values of <math>\alpha</math> contribute. The function <math>f(\alpha)</math> then becomes a smooth curve describing the distribution of scaling exponents of the wavefunction.
	~~indicating that~~ the wavefunction ~~explores the entire system but with strong spatial fluctuations~~.

	= Larkin model =		= Larkin model =

Rosso: /* Multifractal spectrum */

2026-03-15T16:25:49Z

Multifractal spectrum

← Older revision		Revision as of 18:25, 15 March 2026
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	</math>		</math>

			It is useful to contrast this situation with the case of a simple fractal.

			In a '''fractal object''', the measure has a single scaling exponent. In our notation this means that the wavefunction amplitudes scale with a single value <math>\alpha_0</math> on a set of fractal dimension <math>D</math>:

			<math display="block">
			\|\psi_n\|^2 \sim L^{-\alpha_0}
			</math>

			on approximately

			<math display="block">
			L^{D}
			</math>

			sites, and are essentially negligible elsewhere.

			In terms of the multifractal spectrum this corresponds to a trivial spectrum

			<math display="block">
			f(\alpha_0)=D,
			\qquad
			f(\alpha\neq\alpha_0)=-\infty.
			</math>

			By contrast, in a '''multifractal state''' the amplitudes are broadly distributed and many values of <math>\alpha</math> contribute. The function <math>f(\alpha)</math> then becomes a smooth curve.
	indicating that the wavefunction explores the entire system but with strong spatial fluctuations.		indicating that the wavefunction explores the entire system but with strong spatial fluctuations.

Rosso: /* Multifractal eigenstates */

2026-03-15T16:19:01Z

Multifractal eigenstates

← Older revision		Revision as of 18:19, 15 March 2026
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	At the mobility edge the eigenstates are neither extended nor localized. Instead, the amplitudes fluctuate strongly across the system.		At the mobility edge the eigenstates are neither extended nor localized. Instead, the amplitudes fluctuate strongly across the system.

	The scaling of the IPR is characterized by a ~~'''~~nonlinear function~~'''~~
			The scaling of the IPR is characterized by a nonlinear function

	<math display="block">		<math display="block">
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	</math>		</math>

	~~Multifractality means that~~ different ~~moments~~ of the ~~wavefunction~~ probe different effective dimensions.		It is useful to contrast '''fractal''' and '''multifractal''' scaling.

			* In a '''fractal object''', a single exponent describes the scaling of the measure. For instance, if a probability measure is uniformly distributed on a fractal set of dimension <math>D</math>, one has
			<math display="block">
			\mathrm{IPR}(q)\sim L^{-D(q-1)} .
			</math>
			All moments are controlled by the same dimension <math>D</math>.

			* In a '''multifractal object''', different regions of the system scale with different exponents.
			In this case there is no single fractal dimension: different moments probe different effective dimensions of the wavefunction.

	== Multifractal spectrum ==		== Multifractal spectrum ==

Rosso: /* Eigenstates */

2026-03-15T16:08:22Z

Eigenstates

Show changes

Rosso at 17:01, 3 March 2026

2026-03-03T17:01:36Z

Show changes

Rosso: /* Multifractal eigenstates */

2025-03-23T14:38:01Z

Multifractal eigenstates

← Older revision		Revision as of 16:38, 23 March 2025
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	'''Multifractal wavefunctions''' exhibit a smoother dependence, leading to a continuous spectrum with a maximum at <math>\alpha_0</math>, where <math>f(\alpha_0) = d</math>. At <math>q = 1</math>, we have <math>f'(\alpha_1) = 1</math> and <math>f(\alpha_1) = \alpha_1</math>.		'''Multifractal wavefunctions''' exhibit a smoother dependence, leading to a continuous spectrum with a maximum at <math>\alpha_0</math>, where <math>f(\alpha_0) = d</math>. At <math>q = 1</math>, we have <math>f'(\alpha_1) = 1</math> and <math>f(\alpha_1) = \alpha_1</math>.

	~~Sometimes one writes:~~
	~~<center><math>~~
	~~IPR(q)=L^{D_q(1-q)} \quad \tau_q=D_q(1-q)~~
	~~</math></center>~~
	~~Here <math>D_q</math> is q-dependent multifractal dimension, smaller than <math>d</math> and larger than zero.~~

	=Larkin model=		=Larkin model=