T-8 - Revision history

Ros: /* A criterion for localization: vanishing decay rate */

2026-03-15T20:51:47Z

A criterion for localization: vanishing decay rate

← Older revision		Revision as of 22:51, 15 March 2026
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	They are random functions, because the Hamiltonian contains randomness. The Green functions and the self-energies encode properties on the spectrum of the Hamiltonian<sup>[[#Notes\|[*] ]]</sup>, and they are related to the return probability, which can be written as (Laplace transform):		They are random functions, because the Hamiltonian contains randomness. The Green functions and the self-energies encode properties on the spectrum of the Hamiltonian<sup>[[#Notes\|[*] ]]</sup>, and they are related to the return probability, which can be written as (Laplace transform):
	<math display="block">		<math display="block">
	\mathcal{A}_{x_0}(t)= \theta(t) \langle x_0\| e^{-i t H} \| x_0 \rangle =\lim_{\eta \to 0} \~~int~~ \frac{ dz}{2 \pi i}e^{-i t z} G_{x_0 x_0}(z)=\lim_{\eta \to 0} \~~int~~ \frac{ dz}{2 \pi i}\frac{e^{-i t z}}{z- V_{x_0}-\sigma_{x_0}(z)}.		\mathcal{A}_{x_0}(t)= \theta(t) \langle x_0\| e^{-i t H} \| x_0 \rangle =\lim_{\eta \to 0} \int_{-\infty}^{\infty} \frac{ dE}{2 \pi i}e^{-i t (E + i \eta)} G_{x_0 x_0}(z)=\lim_{\eta \to 0} \int_{-\infty+ i \eta}^{\infty+ i \eta} \frac{ dz}{2 \pi i}\frac{e^{-i t z}}{z- V_{x_0}-\sigma_{x_0}(z)}.
	</math>		</math>
	</li>		</li>

Ros: /* To know more */

2026-03-11T10:37:25Z

To know more

← Older revision		Revision as of 12:37, 11 March 2026
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	== To know more ==		== To know more ==
	* The Anderson model was formulated by P. W. Anderson in 1958, in the paper <em> Absence of diffusion in certain random lattices</em>, Phys. Rev. 109, 1492.		* The Anderson model was formulated by P. W. Anderson in 1958, in the paper <em> Absence of diffusion in certain random lattices</em>, Phys. Rev. 109, 1492.
	* The Bethe lattice version is discussed in the later paper Abou-Chacra, Thouless, Anderson. A selfconsistent theory of localization. Journal of Physics C: Solid State Physics 6.10 (1973)		* The Bethe lattice version is discussed in the later paper Abou-Chacra, Thouless, Anderson. <em> A selfconsistent theory of localization</em> . Journal of Physics C: Solid State Physics 6.10 (1973)

Ros: /* References */

2026-03-11T10:37:04Z

References

← Older revision		Revision as of 12:37, 11 March 2026
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	Green functions, self-energies, return probability amplitude, decay rates, trees and cavity method, the criterion for localization.		Green functions, self-energies, return probability amplitude, decay rates, trees and cavity method, the criterion for localization.

	== ~~References~~ ==		== To know more ==
	* The Anderson model was formulated by P. W. Anderson in 1958, in the paper <em> Absence of diffusion in certain random lattices</em>, Phys. Rev. 109, 1492.		* The Anderson model was formulated by P. W. Anderson in 1958, in the paper <em> Absence of diffusion in certain random lattices</em>, Phys. Rev. 109, 1492.
			* The Bethe lattice version is discussed in the later paper Abou-Chacra, Thouless, Anderson. A selfconsistent theory of localization. Journal of Physics C: Solid State Physics 6.10 (1973)

Ros: /* References */

2026-02-14T19:08:20Z

References

← Older revision		Revision as of 21:08, 14 February 2026
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	== References ==		== References ==
	* The Anderson model was formulated by P. W. Anderson in 1958, in the paper <em> Absence of diffusion in certain random lattices </em>, Phys. Rev. 109, 1492.		* The Anderson model was formulated by P. W. Anderson in 1958, in the paper <em> Absence of diffusion in certain random lattices</em>, Phys. Rev. 109, 1492.

Ros: /* Problem 8: Bethe lattice, cavity method & recursion relations */

2026-02-14T19:01:11Z

Problem 8: Bethe lattice, cavity method & recursion relations

← Older revision		Revision as of 21:01, 14 February 2026
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	<ol>		<ol>
	<li><em> Green functions identities. </em> Consider an Hamiltonian split into two parts, <math> H= H_0 + H_1 </math>. Show that the following general relation for the Green functions holds:		<li><em> Green functions identities. </em> Consider an Hamiltonian split into two parts, <math> H= H_0 + H_1 </math>. Show that the following general relation for the Green functions holds:
	~~<center>~~		<math display="block">
	<math>
	G=G^0+ G^0 H_1 G, \quad \quad G^0 =\frac{1}{z-H_0}, \quad \quad G =\frac{1}{z-H}.		G=G^0+ G^0 H_1 G, \quad \quad G^0 =\frac{1}{z-H_0}, \quad \quad G =\frac{1}{z-H}.
	</math>		</math>
	~~</center>~~ </li><br>		</li><br>

	<li><em> Cavity equations. </em>We now apply this to a specific example: we consider a Bethe lattice, and choose one site 0 as the root. We then choose <math> H_1 </math> to be the kinetic terms connecting the root to its <math> k+1 </math> neighbors <math> a_i </math>,		<li><em> Cavity equations. </em>We now apply this to a specific example: we consider a Bethe lattice, and choose one site 0 as the root. We then choose <math> H_1 </math> to be the kinetic terms connecting the root to its <math> k+1 </math> neighbors <math> a_i </math>,
	~~<center>~~		<math display="block">
	<math>
	H_1= -\sum_{i=1}^{k+1} t_{0 a_i} \left( \|a_i \rangle \langle 0\|+ \|0 \rangle \langle a_i\|\right)		H_1= -\sum_{i=1}^{k+1} t_{0 a_i} \left( \|a_i \rangle \langle 0\|+ \|0 \rangle \langle a_i\|\right)
	</math>		</math>
	~~</center>~~
	For all the <math> a_i </math> with <math> i=1, \cdots, k+1 </math> we introduce the notation		For all the <math> a_i </math> with <math> i=1, \cdots, k+1 </math> we introduce the notation
	~~<center>~~		<math display="block">
	<math>
	G^{\text{cav}}_{a_i} \equiv G^0_{a_i a_i}, \quad \quad \sigma^{\text{cav}}_{a_i} \equiv \sigma^0_{a_i},		G^{\text{cav}}_{a_i} \equiv G^0_{a_i a_i}, \quad \quad \sigma^{\text{cav}}_{a_i} \equiv \sigma^0_{a_i},
	</math>		</math>
	~~</center>~~
	where <math> \sigma^0 </math> is the self energy associated to <math> G^0 </math>. Show that, due to the geometry of the lattice, with this choice of <math> H_1 </math> the Hamiltonian <math> H_0 </math> is decoupled and <math> G^{\text{cav}}_{a_i} </math> is the local Green function that one would have obtained removing the root 0 from the lattice, i.e., creating a “cavity” (hence the suffix). Moreover, using the Green function identities iterated twice, show that		where <math> \sigma^0 </math> is the self energy associated to <math> G^0 </math>. Show that, due to the geometry of the lattice, with this choice of <math> H_1 </math> the Hamiltonian <math> H_0 </math> is decoupled and <math> G^{\text{cav}}_{a_i} </math> is the local Green function that one would have obtained removing the root 0 from the lattice, i.e., creating a “cavity” (hence the suffix). Moreover, using the Green function identities iterated twice, show that
	~~<center>~~		<math display="block">
	<math>
	G_{00}(z)= \frac{1}{z-W \, V_0 - \sum_{i=1}^{k+1} t^2_{0 a_i}G^{\text{cav}}_{a_i}(z)}		G_{00}(z)= \frac{1}{z-W \, V_0 - \sum_{i=1}^{k+1} t^2_{0 a_i}G^{\text{cav}}_{a_i}(z)}
	</math>		</math>
	~~</center>~~
	Iterating this argument, show that if <math> \partial a_i </math> denotes the collection of “descendants" of <math> a_i</math>, i.e. sites that are nearest neighbours of <math> a_i </math> <em> except</em> the root, then		Iterating this argument, show that if <math> \partial a_i </math> denotes the collection of “descendants" of <math> a_i</math>, i.e. sites that are nearest neighbours of <math> a_i </math> <em> except</em> the root, then
	~~<center>~~		<math display="block">
	<math>
	G^{\text{cav}}_{a_i}(z)= \frac{1}{z-W \,V_{a_i} - \sum_{b \in \partial a_i}t^2_{a_i b}G^{\text{cav}}_{b}(z)}, \quad \quad \sigma^{\text{cav}}_{a_i}(z)=\sum_{b \in \partial a_i}t^2_{a_i b}G^{\text{cav}}_{b}(z)=\sum_{b \in \partial a_i} \frac{t^2_{a_i b}}{z- W \, V_b - \sigma^{\text{cav}}_{b}(z)}		G^{\text{cav}}_{a_i}(z)= \frac{1}{z-W \,V_{a_i} - \sum_{b \in \partial a_i}t^2_{a_i b}G^{\text{cav}}_{b}(z)}, \quad \quad \sigma^{\text{cav}}_{a_i}(z)=\sum_{b \in \partial a_i}t^2_{a_i b}G^{\text{cav}}_{b}(z)=\sum_{b \in \partial a_i} \frac{t^2_{a_i b}}{z- W \, V_b - \sigma^{\text{cav}}_{b}(z)}
	</math>		</math>
	~~</center>~~
	</li><br>		</li><br>

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	<li><em> The “localized" solution. </em> We set <math> z=E+ i \eta </math> and <math> \sigma^{\text{cav}}_{a}(z)= R_a(z) -i \Gamma_a(z)</math>. Show that the cavity equation for the self-energies is equivalent to the following pair of coupled equations:		<li><em> The “localized" solution. </em> We set <math> z=E+ i \eta </math> and <math> \sigma^{\text{cav}}_{a}(z)= R_a(z) -i \Gamma_a(z)</math>. Show that the cavity equation for the self-energies is equivalent to the following pair of coupled equations:
	~~<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>~~
	Justify why the solution corresponding to localization, <math> \Gamma_a=0 </math>, is always a solution when <math> \eta \to 0 </math>; moreover, in the localized phase when <math> \eta </math> is finite but small one has <math> \Gamma_a \sim O(\eta) </math>. How can one argue that this solution has to be discarded, i.e. that delocalisation occurs?		Justify why the solution corresponding to localization, <math> \Gamma_a=0 </math>, is always a solution when <math> \eta \to 0 </math>; moreover, in the localized phase when <math> \eta </math> is finite but small one has <math> \Gamma_a \sim O(\eta) </math>. How can one argue that this solution has to be discarded, i.e. that delocalisation occurs?
	</li><br>		</li><br>

Ros: /* A criterion for localization: vanishing decay rate */

2026-02-14T19:00:26Z

A criterion for localization: vanishing decay rate

← Older revision		Revision as of 21:00, 14 February 2026
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	<li> <strong> Anderson's criterion for localization. </strong> If the self-energy has a non-zero imaginary part (when <math> N \to \infty </math>):		<li> <strong> Anderson's criterion for localization. </strong> If the self-energy has a non-zero imaginary part (when <math> N \to \infty </math>):
	~~<center>~~		<math display="block">
	<math>
	\sigma_{x_0}(z)= R_{x_0}(z)-i \Gamma_{x_0}(z),		\sigma_{x_0}(z)= R_{x_0}(z)-i \Gamma_{x_0}(z),
	</math>		</math>
	~~</center>~~
	then one can show (How? [[Media:2025 Self energies, decay rates and all that.pdf\| See here for some notes]] ) that <math>\mathcal{A}_{x_0}(t)</math> decays exponentially,		then one can show (How? [[Media:2025 Self energies, decay rates and all that.pdf\| See here for some notes]] ) that <math>\mathcal{A}_{x_0}(t)</math> decays exponentially,
	~~<center>~~		<math display="block">
	<math>
	\mathcal{A}_{x_0}(t) \sim A(t)e^{-\gamma t}+ B(t), \quad \quad \quad \gamma= \Gamma_{x_0}(V_{x_0}) + O(T^4),		\mathcal{A}_{x_0}(t) \sim A(t)e^{-\gamma t}+ B(t), \quad \quad \quad \gamma= \Gamma_{x_0}(V_{x_0}) + O(T^4),
	</math>		</math>
	~~</center>~~
	and the system is not localized: the probability to find it, at <math> t \gg 1</math>, in the same configuration where it was at <math> t=0 </math> decays fast. For localization to occur, it must be that the imaginary part of <math> \sigma{x_0}(E+ i\eta)</math> goes to zero when <math> \eta \to 0</math>. Given the randomness, this criterion should however be formulated probabilistically:		and the system is not localized: the probability to find it, at <math> t \gg 1</math>, in the same configuration where it was at <math> t=0 </math> decays fast. For localization to occur, it must be that the imaginary part of <math> \sigma{x_0}(E+ i\eta)</math> goes to zero when <math> \eta \to 0</math>. Given the randomness, this criterion should however be formulated probabilistically:
	~~<center>~~		<math display="block">
	<math>
	\lim_{\eta \to 0} \lim_{N \to \infty} \mathbb{P}\left(- \Im \sigma_{x_0}(E+i \eta)>0 \right)=0 \quad \Longrightarrow \quad \text{Localization}		\lim_{\eta \to 0} \lim_{N \to \infty} \mathbb{P}\left(- \Im \sigma_{x_0}(E+i \eta)>0 \right)=0 \quad \Longrightarrow \quad \text{Localization}
	</math>		</math>
	~~</center>~~
	Showing that at strong enough disorder this condition is satisfied is the core of Anderson’s 1958 work.		Showing that at strong enough disorder this condition is satisfied is the core of Anderson’s 1958 work.
	</li>		</li>
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	<div style="font-size:89%">		<div style="font-size:89%">
	: <small>[*]</small> - For example, the local density of eigenvalues <math>\nu_{x, N}(E)</math> for an Hamiltonian on a lattice of size <math> N </math> is indeed given by		: <small>[*]</small> - For example, the local density of eigenvalues <math>\nu_{x, N}(E)</math> for an Hamiltonian on a lattice of size <math> N </math> is indeed given by
	~~<center>~~ <math>		<math display="block">
	\nu_{x,N}(E)=-\frac{1}{\pi}\lim_{\eta \to 0} \Im G_{xx}(E+ i \eta) = \sum_{\alpha=1}^N \|\langle E_\alpha\| x\rangle\|^2 \delta(E-E_\alpha),		\nu_{x,N}(E)=-\frac{1}{\pi}\lim_{\eta \to 0} \Im G_{xx}(E+ i \eta) = \sum_{\alpha=1}^N \|\langle E_\alpha\| x\rangle\|^2 \delta(E-E_\alpha),
	</math>		</math>
	~~</center>~~
	where <math> E_\alpha </math> are the eigenvalues of the full Hamiltonian <math> H </math> and <math> \|E_\alpha \rangle </math> the corresponding eigenstates. When the kinetic term <math>T </math> in the Hamiltonian vanishes, the local self-energies vanish. They encode how much the energy levels <math> V_x </math> (that are the eigenvalues when <math> V=0</math>) are shifted by the presence of the kinetic (hopping) term.		where <math> E_\alpha </math> are the eigenvalues of the full Hamiltonian <math> H </math> and <math> \|E_\alpha \rangle </math> the corresponding eigenstates. When the kinetic term <math>T </math> in the Hamiltonian vanishes, the local self-energies vanish. They encode how much the energy levels <math> V_x </math> (that are the eigenvalues when <math> V=0</math>) are shifted by the presence of the kinetic (hopping) term.
	</div>		</div>

Ros: /* A criterion for localization: vanishing decay rate */

2026-02-14T18:56:12Z

A criterion for localization: vanishing decay rate

← Older revision		Revision as of 20:56, 14 February 2026
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	<ul>		<ul>
	<li> <strong> Anderson model and return probability. </strong> Consider a lattice with <math> N </math> sites <math>x </math>. We call <math> \|x \rangle </math> the wave function completely localised in site <math> x </math>. The Anderson model has Hamiltonian:		<li> <strong> Anderson model and return probability. </strong> Consider a lattice with <math> N </math> sites <math>x </math>. We call <math> \|x \rangle </math> the wave function completely localised in site <math> x </math>. The Anderson model has Hamiltonian:
	~~<center>~~		<math display="block">
	<math>
	H= W \sum_{x} V_x \|x \rangle \langle x\| - \sum_{<x, y>} t_{xy} \left(\|x \rangle \langle y\|+ \|y \rangle \langle x\| \right)		H= W \sum_{x} V_x \|x \rangle \langle x\| - \sum_{<x, y>} t_{xy} \left(\|x \rangle \langle y\|+ \|y \rangle \langle x\| \right)
	</math>		</math>
	~~</center>~~
	where the local fields <math> V_x </math> are random variables, independent and distributed according to some distribution <math> p(V)</math>. Usually, the hopping amplitude <math> t_{xy}=t</math> is chosen to be uniform.		where the local fields <math> V_x </math> are random variables, independent and distributed according to some distribution <math> p(V)</math>. Usually, the hopping amplitude <math> t_{xy}=t</math> is chosen to be uniform.

	To see whether localization occurs, consider a quantum particle initialised on the site <math>x_0 </math> at <math> t=0 </math>. The <ins>return probability amplitude </ins>, i.e. the probability amplitude to find the particle on the same site at later time, is		To see whether localization occurs, consider a quantum particle initialised on the site <math>x_0 </math> at <math> t=0 </math>. The <ins>return probability amplitude </ins>, i.e. the probability amplitude to find the particle on the same site at later time, is
	~~<center>~~		<math display="block">
	<math>
	\mathcal{A}_{x_0}(t)= \theta(t) \langle x_0\| e^{-i t H} \| x_0 \rangle.		\mathcal{A}_{x_0}(t)= \theta(t) \langle x_0\| e^{-i t H} \| x_0 \rangle.
	</math>		</math>
	~~</center>~~
	When the system is not localized, the wave function of the particle spreads in time, and the return probability decays.		When the system is not localized, the wave function of the particle spreads in time, and the return probability decays.
	</li>		</li>
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	<li> <strong> Green functions and self-energies. </strong> We introduce two functions on the complex plane: the <ins>Green functions</ins> <math> G_{xy}(z) </math> and the <ins>local self-energies</ins> <math> \sigma_x(z)</math>, where <math>z=E+ i \eta, \quad \quad \eta>0</math> belongs to the upper half of the complex plane. The functions are defined by		<li> <strong> Green functions and self-energies. </strong> We introduce two functions on the complex plane: the <ins>Green functions</ins> <math> G_{xy}(z) </math> and the <ins>local self-energies</ins> <math> \sigma_x(z)</math>, where <math>z=E+ i \eta, \quad \quad \eta>0</math> belongs to the upper half of the complex plane. The functions are defined by
	~~<center>~~		<math display="block">
	<math>
	G_{xy}(z)= \langle x\| \frac{1}{z-H}\| y \rangle , \quad \quad G_{x x}(z)= \langle x\| \frac{1}{z-H}\| x\rangle = \frac{1}{z- V_x-\sigma_x(z)}.		G_{xy}(z)= \langle x\| \frac{1}{z-H}\| y \rangle , \quad \quad G_{x x}(z)= \langle x\| \frac{1}{z-H}\| x\rangle = \frac{1}{z- V_x-\sigma_x(z)}.
	</math>		</math>
	~~</center>~~
	They are random functions, because the Hamiltonian contains randomness. The Green functions and the self-energies encode properties on the spectrum of the Hamiltonian<sup>[[#Notes\|[*] ]]</sup>, and they are related to the return probability, which can be written as (Laplace transform):		They are random functions, because the Hamiltonian contains randomness. The Green functions and the self-energies encode properties on the spectrum of the Hamiltonian<sup>[[#Notes\|[*] ]]</sup>, and they are related to the return probability, which can be written as (Laplace transform):
	~~<center>~~		<math display="block">
	<math>
	\mathcal{A}_{x_0}(t)= \theta(t) \langle x_0\| e^{-i t H} \| x_0 \rangle =\lim_{\eta \to 0} \int \frac{ dz}{2 \pi i}e^{-i t z} G_{x_0 x_0}(z)=\lim_{\eta \to 0} \int \frac{ dz}{2 \pi i}\frac{e^{-i t z}}{z- V_{x_0}-\sigma_{x_0}(z)}.		\mathcal{A}_{x_0}(t)= \theta(t) \langle x_0\| e^{-i t H} \| x_0 \rangle =\lim_{\eta \to 0} \int \frac{ dz}{2 \pi i}e^{-i t z} G_{x_0 x_0}(z)=\lim_{\eta \to 0} \int \frac{ dz}{2 \pi i}\frac{e^{-i t z}}{z- V_{x_0}-\sigma_{x_0}(z)}.
	</math>		</math>
	~~</center>~~
	</li>		</li>
	<br>		<br>

Ros: /* A criterion for localization: vanishing decay rate */

2025-03-19T14:03:26Z

A criterion for localization: vanishing decay rate

← Older revision		Revision as of 16:03, 19 March 2025
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	</math>		</math>
	</center>		</center>
	then one can show (How? [[Media:Self energies, decay rates and all that.pdf\| See here for some notes]] ) that <math>\mathcal{A}_{x_0}(t)</math> decays exponentially,		then one can show (How? [[Media:2025 Self energies, decay rates and all that.pdf\| See here for some notes]] ) that <math>\mathcal{A}_{x_0}(t)</math> decays exponentially,
	<center>		<center>
	<math>		<math>

Ros: /* A criterion for localization: vanishing decay rate */

2025-03-19T11:36:26Z

A criterion for localization: vanishing decay rate

← Older revision		Revision as of 13:36, 19 March 2025
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	</math>		</math>
	</center>		</center>
	where <math> E_\alpha </math> are the eigenvalues of the full Hamiltonian <math> H </math> and <math> \|E_\alpha \rangle </math> the corresponding eigenstates. When the kinetic term <math>T </math> in the Hamiltonian vanishes, the local self-energies vanish. They encode how much the energy levels <math> V_x </math> (that are the eigenvalues when <math> V=0</math>) are shifted by the presence of the kinetic term ~~<math>T </math>~~.		where <math> E_\alpha </math> are the eigenvalues of the full Hamiltonian <math> H </math> and <math> \|E_\alpha \rangle </math> the corresponding eigenstates. When the kinetic term <math>T </math> in the Hamiltonian vanishes, the local self-energies vanish. They encode how much the energy levels <math> V_x </math> (that are the eigenvalues when <math> V=0</math>) are shifted by the presence of the kinetic (hopping) term.
	</div>		</div>

Ros: /* A criterion for localization: vanishing decay rate */

2025-03-19T11:36:02Z

A criterion for localization: vanishing decay rate

← Older revision		Revision as of 13:36, 19 March 2025
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	<div style="font-size:89%">		<div style="font-size:89%">
	: <small>[*]</small> - ~~for~~ example, the local density of eigenvalues <math>\~~rho_~~{x, N}(E)</math> for an Hamiltonian on a lattice of size <math> N </math> is indeed given by		: <small>[*]</small> - For example, the local density of eigenvalues <math>\nu_{x, N}(E)</math> for an Hamiltonian on a lattice of size <math> N </math> is indeed given by
	<center> <math>		<center> <math>
	\~~rho_~~{x,N}(E)=-\frac{1}{\pi}\lim_{\eta \to 0} \Im G_{xx}(E+ i \eta) = \sum_{\alpha=1}^N \|\langle E_\alpha\| x\rangle\|^2 \delta(E-E_\alpha),		\nu_{x,N}(E)=-\frac{1}{\pi}\lim_{\eta \to 0} \Im G_{xx}(E+ i \eta) = \sum_{\alpha=1}^N \|\langle E_\alpha\| x\rangle\|^2 \delta(E-E_\alpha),
	</math>		</math>
	</center>		</center>
	where <math> E_\alpha </math> are the eigenvalues of the full Hamiltonian <math> H </math> and <math> \|E_\alpha \rangle </math> the corresponding eigenstates. When the kinetic term <math>T </math> in the Hamiltonian vanishes, the local self-energies vanish. ~~they~~ encode how much the energy levels <math> V_x </math> (that are the eigenvalues when <math> V=0</math>) are shifted by the presence of the kinetic term <math>T </math>.		where <math> E_\alpha </math> are the eigenvalues of the full Hamiltonian <math> H </math> and <math> \|E_\alpha \rangle </math> the corresponding eigenstates. When the kinetic term <math>T </math> in the Hamiltonian vanishes, the local self-energies vanish. They encode how much the energy levels <math> V_x </math> (that are the eigenvalues when <math> V=0</math>) are shifted by the presence of the kinetic term <math>T </math>.
	</div>		</div>