T-9 - Revision history

Ros: /* References */

2026-03-11T10:38:06Z

References

← Older revision		Revision as of 12:38, 11 March 2026
Line 66:		Line 66:
	Linearization and stability analysis, critical disorder, mobility edge.		Linearization and stability analysis, critical disorder, mobility edge.

	== ~~References~~ ==		== To know more ==
	* Abou-Chacra, Thouless, Anderson. A selfconsistent theory of localization. Journal of Physics C: Solid State Physics 6.10 (1973)

Ros: /* Problem 9: an estimate of the mobility edge */

2026-02-14T19:09:11Z

Problem 9: an estimate of the mobility edge

← Older revision		Revision as of 21:09, 14 February 2026
Line 30:		Line 30:

	<li><em> The stability analysis. </em> We now assume to be in the localized phase, when for <math> \eta \to 0 </math> the distribution <math> P_\Gamma(\Gamma) \to \delta (\Gamma)</math>. We wish to check the stability of our assumption. This is done by controlling the tails of the distribution <math> P_\Gamma(\Gamma)</math> for finite <math> \eta </math>.		<li><em> The stability analysis. </em> We now assume to be in the localized phase, when for <math> \eta \to 0 </math> the distribution <math> P_\Gamma(\Gamma) \to \delta (\Gamma)</math>. We wish to check the stability of our assumption. This is done by controlling the tails of the distribution <math> P_\Gamma(\Gamma)</math> for finite <math> \eta </math>.

	<ul>		<ul>
	<li>		<li>
Line 38:		Line 37:
	<li> Assume more generally that <math>P_\Gamma(\Gamma) \sim \Gamma^{-\alpha}</math> for large <math> \Gamma </math> and-->		<li> Assume more generally that <math>P_\Gamma(\Gamma) \sim \Gamma^{-\alpha}</math> for large <math> \Gamma </math> and-->
	Show, using a dimensional analysis argument, that this corresponds to a non-analytic behaviour of the Laplace transform, <math> \Phi(s) \sim 1- A \|s\|^\beta </math> for <math> s </math> small, with <math> \beta= \alpha-1 </math>. </li>		Show, using a dimensional analysis argument, that this corresponds to a non-analytic behaviour of the Laplace transform, <math> \Phi(s) \sim 1- A \|s\|^\beta </math> for <math> s </math> small, with <math> \beta= \alpha-1 </math>. </li>
	~~<!-- \in (0, 1/2]-->~~

	<li> Show that the equation for <math> \Phi(s) </math> gives for <math> s </math> small <math>1- A s^\beta =1- A k \int dV \, p(V) \frac{s^\beta \tau^{2 \beta}}{V^{2 \beta}}+ o(s^\beta) </math>, and therefore this is consistent provided that there exists a <math> \beta </math> solving		<li> Show that the equation for <math> \Phi(s) </math> gives for <math> s </math> small <math>1- A s^\beta =1- A k \int dV \, p(V) \frac{s^\beta \tau^{2 \beta}}{V^{2 \beta}}+ o(s^\beta) </math>, and therefore this is consistent provided that there exists a <math> \beta </math> solving
Line 46:		Line 44:
	</li>		</li>
	</ul>		</ul>
	~~<br>~~

Ros: /* Problem 9: an estimate of the mobility edge */

2026-02-14T19:06:40Z

Problem 9: an estimate of the mobility edge

← Older revision		Revision as of 21:06, 14 February 2026
Line 52:		Line 52:

	<li><em> The critical disorder. </em> Consider now local fields <math> V_x </math> taken from a uniform distribution in <math> [-1/2, 1/2] </math>. Compute <math> I(\beta) </math> and show that it is non monotonic, with a local minimum <math> \beta^* </math> in the interval <math> [0, 1/2]</math>. Show that <math> I(\beta^) </math> increases as the disorder is made weaker and weaker, and thus the transition to delocalisation occurs at the critical value of disorder when <math> I(\beta^)=k^{-1} </math>. Show that this gives the following estimate for the critical disorder <math>(W/t)_c=1/\tau_c </math> at which the transition to delocalisation occurs:		<li><em> The critical disorder. </em> Consider now local fields <math> V_x </math> taken from a uniform distribution in <math> [-1/2, 1/2] </math>. Compute <math> I(\beta) </math> and show that it is non monotonic, with a local minimum <math> \beta^* </math> in the interval <math> [0, 1/2]</math>. Show that <math> I(\beta^) </math> increases as the disorder is made weaker and weaker, and thus the transition to delocalisation occurs at the critical value of disorder when <math> I(\beta^)=k^{-1} </math>. Show that this gives the following estimate for the critical disorder <math>(W/t)_c=1/\tau_c </math> at which the transition to delocalisation occurs:
	~~<center>~~		<math display="block">
	<math>
	\frac{1}{\tau_c} = \, 2 k e \log \left( \frac{1}{2 \tau_c}\right) \sim \, 2 e \, k \log \left(k\right)		\frac{1}{\tau_c} = \, 2 k e \log \left( \frac{1}{2 \tau_c}\right) \sim \, 2 e \, k \log \left(k\right)
	</math>		</math>
	~~</center>~~
	Why the critical disorder increases with <math> k </math>?		Why the critical disorder increases with <math> k </math>?
	</li>		</li>

Ros: /* Problems */

2026-02-14T19:06:20Z

Problems

← Older revision		Revision as of 21:06, 14 February 2026
Line 9:		Line 9:


	~~<center>~~		<math display="block">
	<math>
	\Gamma_a= \sum_{b \in \partial a} t_{ab}^2 \frac{\Gamma_b + \eta}{(E- W\, V_b - R_b)^2+ (\Gamma_b +\eta)^2}, \quad \quad R_a = \sum_{b \in \partial a} t_{ab}^2 \frac{E- W\, V_b - R_b}{(E- W\, V_b - R_b)^2+ (\Gamma_b +\eta)^2}		\Gamma_a= \sum_{b \in \partial a} t_{ab}^2 \frac{\Gamma_b + \eta}{(E- W\, V_b - R_b)^2+ (\Gamma_b +\eta)^2}, \quad \quad R_a = \sum_{b \in \partial a} t_{ab}^2 \frac{E- W\, V_b - R_b}{(E- W\, V_b - R_b)^2+ (\Gamma_b +\eta)^2}
	</math>		</math>
	~~</center>~~

	These equations admit the solution <math> \Gamma_a=\Gamma_b=0</math> when <math>\eta=0 </math>, which corresponds to localization. We now determine when this solution becomes unstable.		These equations admit the solution <math> \Gamma_a=\Gamma_b=0</math> when <math>\eta=0 </math>, which corresponds to localization. We now determine when this solution becomes unstable.
Line 22:		Line 20:

	<li><em> Imaginary approximation and distributional equation. </em> We consider the equations for <math> \Gamma_a </math> and neglect the terms <math> R_b </math> in the denominators, which couple the equations to those for the real parts of the self energies (“imaginary” approximation). Moreover, we assume to be in the localized phase, where <math> \Gamma_a \sim \eta \ll 1 </math>. Finally, we set <math> t_{ab} \equiv t </math> and <math> E=0 </math> for simplicity. Show that under these assumptions the probability density for the imaginary part, <math> P_\Gamma(\Gamma)</math>, satisfies for <math> \tau=t/W</math>		<li><em> Imaginary approximation and distributional equation. </em> We consider the equations for <math> \Gamma_a </math> and neglect the terms <math> R_b </math> in the denominators, which couple the equations to those for the real parts of the self energies (“imaginary” approximation). Moreover, we assume to be in the localized phase, where <math> \Gamma_a \sim \eta \ll 1 </math>. Finally, we set <math> t_{ab} \equiv t </math> and <math> E=0 </math> for simplicity. Show that under these assumptions the probability density for the imaginary part, <math> P_\Gamma(\Gamma)</math>, satisfies for <math> \tau=t/W</math>
	~~<center>~~		<math display="block">
	<math>
	P_\Gamma(\Gamma)= \int \prod_{b=1}^k dV_b\,p(V_b)\int \prod_{b=1}^k d\Gamma_b \, P_\Gamma(\Gamma_b) \delta \left(\Gamma - \tau^2 \sum_{b \in \partial a} \frac{\Gamma_b + \eta}{ V_b^2} \right)		P_\Gamma(\Gamma)= \int \prod_{b=1}^k dV_b\,p(V_b)\int \prod_{b=1}^k d\Gamma_b \, P_\Gamma(\Gamma_b) \delta \left(\Gamma - \tau^2 \sum_{b \in \partial a} \frac{\Gamma_b + \eta}{ V_b^2} \right)
	</math>		</math>
	~~</center>~~
	Show that the Laplace transform of this distribution, <math> \Phi(s)=\int_0^\infty d\Gamma e^{-s \Gamma} P_\Gamma(\Gamma) </math>, satisfies		Show that the Laplace transform of this distribution, <math> \Phi(s)=\int_0^\infty d\Gamma e^{-s \Gamma} P_\Gamma(\Gamma) </math>, satisfies
	~~<center>~~		<math display="block">
	<math>
	\Phi(s)= \left[ \int dV\, p(V) e^{-\frac{s \tau^2 \eta}{V^2}} \Phi \left(\frac{s \tau^2 }{ V^2} \right) \right]^k		\Phi(s)= \left[ \int dV\, p(V) e^{-\frac{s \tau^2 \eta}{V^2}} \Phi \left(\frac{s \tau^2 }{ V^2} \right) \right]^k
	</math>		</math>
	~~</center>~~
	</li><br>		</li><br>

Line 47:		Line 41:

	<li> Show that the equation for <math> \Phi(s) </math> gives for <math> s </math> small <math>1- A s^\beta =1- A k \int dV \, p(V) \frac{s^\beta \tau^{2 \beta}}{V^{2 \beta}}+ o(s^\beta) </math>, and therefore this is consistent provided that there exists a <math> \beta </math> solving		<li> Show that the equation for <math> \Phi(s) </math> gives for <math> s </math> small <math>1- A s^\beta =1- A k \int dV \, p(V) \frac{s^\beta \tau^{2 \beta}}{V^{2 \beta}}+ o(s^\beta) </math>, and therefore this is consistent provided that there exists a <math> \beta </math> solving
	~~<center>~~		<math display="block">
	<math>
	1=k \int dV \, p(V) \left(\frac{\tau}{\|V\|}\right)^{2 \beta} \equiv k I(\beta).		1=k \int dV \, p(V) \left(\frac{\tau}{\|V\|}\right)^{2 \beta} \equiv k I(\beta).
	</math>		</math>
	~~</center>~~ </li>		</li>
	</ul>		</ul>
	~~</li>~~<br>		<br>

Ros: /* Problems */

2025-03-23T13:42:02Z

Problems

← Older revision		Revision as of 15:42, 23 March 2025
Line 71:		Line 71:

	<div style="font-size:89%">		<div style="font-size:89%">
	: <small>[*]</small> - Why do we expect power law tails? Recall that in first approximation <math> \Gamma \sim 1/\|V\|^2</math>. If <math> V</math> is uniformly distributed, then <math>P(\Gamma) \sim \Gamma^{-3/2}</math>.		: <small>[*]</small> - Why do we expect power law tails? Recall that in first approximation <math> \Gamma \sim 1/\|V\|^2</math>. If <math> V</math> is uniformly distributed, then <math>P_\Gamma(\Gamma) \sim \Gamma^{-3/2}</math>.
	</div>		</div>

Ros: /* Problem 9: an estimate of the mobility edge */

2025-03-23T13:41:44Z

Problem 9: an estimate of the mobility edge

← Older revision		Revision as of 15:41, 23 March 2025
Line 68:		Line 68:
	</ol>		</ol>
	<br>		<br>


			<div style="font-size:89%">
			: <small>[*]</small> - Why do we expect power law tails? Recall that in first approximation <math> \Gamma \sim 1/\|V\|^2</math>. If <math> V</math> is uniformly distributed, then <math>P(\Gamma) \sim \Gamma^{-3/2}</math>.
			</div>

	== Check out: key concepts ==		== Check out: key concepts ==

Ros: /* Problem 9: an estimate of the mobility edge */

2025-03-23T13:39:01Z

Problem 9: an estimate of the mobility edge

← Older revision		Revision as of 15:39, 23 March 2025
Line 39:		Line 39:
	<ul>		<ul>
	<li>		<li>
	For finite <math> \eta</math>, we expect that typically <math> \Gamma_a \sim \eta \ll 1 </math>, and thus <math> P_\Gamma(\Gamma)</math> should have a peak at this scale; however, we also expect some power law decay <math>P_\Gamma(\Gamma)\sim \Gamma^{-\alpha} </math> for large <math> \Gamma </math>.		For finite <math> \eta</math>, we expect that typically <math> \Gamma_a \sim \eta \ll 1 </math>, and thus <math> P_\Gamma(\Gamma)</math> should have a peak at this scale; however, we also expect <sup>[[#Notes\|[*] ]]</sup> some power law decay <math>P_\Gamma(\Gamma)\sim \Gamma^{-\alpha} </math> for large <math> \Gamma </math>.
	<!--and <math> \alpha \in (1, 3/2]</math>. These tails are contributed by the events in which one of the descendants <math> b </math> is such that <math> \epsilon_b </math> is very small, in which case <math> \Gamma_a \sim 1/ \epsilon_b^2 \gg 1 </math>.		<!--and <math> \alpha \in (1, 3/2]</math>. These tails are contributed by the events in which one of the descendants <math> b </math> is such that <math> \epsilon_b </math> is very small, in which case <math> \Gamma_a \sim 1/ \epsilon_b^2 \gg 1 </math>.
	Show that if <math> \Gamma \sim 1/ \epsilon^2 </math> and <math>p(\epsilon)</math> is not gapped around zero, then <math>P_\Gamma(\Gamma) \sim \Gamma^{-3/2}</math>, i.e. the distribution has tails contributed by these events in which the local random potential happen to be very small. </li>		Show that if <math> \Gamma \sim 1/ \epsilon^2 </math> and <math>p(\epsilon)</math> is not gapped around zero, then <math>P_\Gamma(\Gamma) \sim \Gamma^{-3/2}</math>, i.e. the distribution has tails contributed by these events in which the local random potential happen to be very small. </li>

Ros: /* Problem 9: an estimate of the mobility edge */

2025-03-23T13:18:26Z

Problem 9: an estimate of the mobility edge

← Older revision		Revision as of 15:18, 23 March 2025
Line 58:		Line 58:
	[[File:Bethe I(beta).png\|thumb\|left\|x140px\|Behaviour of the integral <math> I(\beta)</math> in the case of uniformily distributed disorder, for <math>W< W_c </math> .]]		[[File:Bethe I(beta).png\|thumb\|left\|x140px\|Behaviour of the integral <math> I(\beta)</math> in the case of uniformily distributed disorder, for <math>W< W_c </math> .]]

	<li><em> The critical disorder. </em> Consider now local fields <math> V_x </math> taken from a uniform distribution in <math> [-1/2, 1/2] </math>. Compute <math> I(\beta) </math> and show that it is non monotonic, with a local minimum <math> \beta^* </math> in the interval <math> [0, 1/2]</math>. Show that <math> I(\beta^) </math> increases as the disorder is made weaker and weaker, and thus the transition to delocalisation occurs at the critical value of disorder when <math> I(\beta^)=k^{-1} </math>. Show that this gives the following estimate for the critical disorder <math>(W/t)_c=1/tau_c </math> at which the transition to delocalisation occurs:		<li><em> The critical disorder. </em> Consider now local fields <math> V_x </math> taken from a uniform distribution in <math> [-1/2, 1/2] </math>. Compute <math> I(\beta) </math> and show that it is non monotonic, with a local minimum <math> \beta^* </math> in the interval <math> [0, 1/2]</math>. Show that <math> I(\beta^) </math> increases as the disorder is made weaker and weaker, and thus the transition to delocalisation occurs at the critical value of disorder when <math> I(\beta^)=k^{-1} </math>. Show that this gives the following estimate for the critical disorder <math>(W/t)_c=1/\tau_c </math> at which the transition to delocalisation occurs:
	<center>		<center>
	<math>		<math>

Ros: /* Problem 9: an estimate of the mobility edge */

2025-03-23T13:17:49Z

Problem 9: an estimate of the mobility edge

← Older revision		Revision as of 15:17, 23 March 2025
Line 46:		Line 46:
	<!-- \in (0, 1/2]-->		<!-- \in (0, 1/2]-->

	<li> Show that the equation for <math> \Phi(s) </math> gives for <math> s </math> small <math>1- A s^\beta =1- A k \int dV \, p(V) \frac{s^\beta \tau^{2 \beta}}{V^{2 \beta}}+ o(s^\beta) </math>, and therefore this is consistent provided that there exists a <math> \beta ~~\in (0, 1/2]~~ </math> solving		<li> Show that the equation for <math> \Phi(s) </math> gives for <math> s </math> small <math>1- A s^\beta =1- A k \int dV \, p(V) \frac{s^\beta \tau^{2 \beta}}{V^{2 \beta}}+ o(s^\beta) </math>, and therefore this is consistent provided that there exists a <math> \beta </math> solving
	<center>		<center>
	<math>		<math>

Ros: /* Problem 9: an estimate of the mobility edge */

2025-03-23T13:17:28Z

Problem 9: an estimate of the mobility edge

← Older revision		Revision as of 15:17, 23 March 2025
Line 43:		Line 43:
	Show that if <math> \Gamma \sim 1/ \epsilon^2 </math> and <math>p(\epsilon)</math> is not gapped around zero, then <math>P_\Gamma(\Gamma) \sim \Gamma^{-3/2}</math>, i.e. the distribution has tails contributed by these events in which the local random potential happen to be very small. </li>		Show that if <math> \Gamma \sim 1/ \epsilon^2 </math> and <math>p(\epsilon)</math> is not gapped around zero, then <math>P_\Gamma(\Gamma) \sim \Gamma^{-3/2}</math>, i.e. the distribution has tails contributed by these events in which the local random potential happen to be very small. </li>
	<li> Assume more generally that <math>P_\Gamma(\Gamma) \sim \Gamma^{-\alpha}</math> for large <math> \Gamma </math> and-->		<li> Assume more generally that <math>P_\Gamma(\Gamma) \sim \Gamma^{-\alpha}</math> for large <math> \Gamma </math> and-->
	Show, using a dimensional analysis argument, that this corresponds to a non-analytic behaviour of the Laplace transform, <math> \Phi(s) \sim 1- A \|s\|^\beta </math> for <math> s </math> small, with <math> \beta= \alpha-1 \in (0, 1/2] ~~</math>. </li~~>		Show, using a dimensional analysis argument, that this corresponds to a non-analytic behaviour of the Laplace transform, <math> \Phi(s) \sim 1- A \|s\|^\beta </math> for <math> s </math> small, with <math> \beta= \alpha-1 </math>. </li>
			<!-- \in (0, 1/2]-->

	<li> Show that the equation for <math> \Phi(s) </math> gives for <math> s </math> small <math>1- A s^\beta =1- A k \int dV \, p(V) \frac{s^\beta \tau^{2 \beta}}{V^{2 \beta}}+ o(s^\beta) </math>, and therefore this is consistent provided that there exists a <math> \beta \in (0, 1/2] </math> solving		<li> Show that the equation for <math> \Phi(s) </math> gives for <math> s </math> small <math>1- A s^\beta =1- A k \int dV \, p(V) \frac{s^\beta \tau^{2 \beta}}{V^{2 \beta}}+ o(s^\beta) </math>, and therefore this is consistent provided that there exists a <math> \beta \in (0, 1/2] </math> solving