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%%%%%%%%%%%%%%%% Figures %%%%%%%%%%%%%%%%%%
\useexternalfigure[fig:statKD][statKD1][type=jpg]
\useexternalfigure[fig:gs][gs][type=jpg]
\useexternalfigure[fig:stringDemo][stringDemo][type=jpg]
\useexternalfigure[fig:stringCors][stringCors][type=jpg]
\useexternalfigure[fig:cuplayers][cuprateplanes][type=jpg]
\useexternalfigure[fig:cuoplanes][cuoplanes][type=jpg]
\useexternalfigure[fig:stripes1][stripes1][type=jpg]
\useexternalfigure[fig:htcphase_a][htcphase_a][type=jpg]

\useexternalfigure[fig:rigidKDw][rigidKDw][type=jpg]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%% Presentation %%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\starttext

\TitlePage{
Effects of disorder in fluctuating one-dimensional interacting systems}
{by Uri London\\
supervisor: Dr. Dror Orgad}

%\Topic[introduction]{Introduction}
%\blank[5*big]

%Motivation:
%\startitemize[broad]
%\item How time-dependence of disorder affects \goto{Anderson localization}[urilecExtra::andersonModel].
%\blank[2*big]

%\item Relevance to fluctuative \goto{quasi one-dimensional systems}[urilecExtra::1d]

%(In particular regarding the stripes in the cuprates).
%\stopitemize
%\blank[2*big]
%Outline:
%\startitemize
%\item Stripes
%\item Non-interacting disordered systems
%\item Using RG for 1D interacting disordered systems
%\item Introducing fluctuations with string dynamics
%\item Results
%\stopitemize


\Topic[stripes]{ Motivation  - Stripes}
\placefigure[here,force][]{}
{\startcombination[3*1]
{\externalfigure[fig:cuplayers][width=6cm]}{Cuprates has qusi-2d structure}
{\externalfigure[fig:cuoplanes][width=4cm]}{AF Planes}
{\externalfigure[fig:stripes1][width=4cm]}{Doped holes segregates to 1d channels - stripes}
\stopcombination}

\placefigure[here,force][]{Tc drops where stripes are static}
{\externalfigure[fig:htcphase_a][width=13cm]}


\Topic[mit]{Metal-Insulator Transition}
\blank[4*big]
\placetable[here,none][tab:mit]{}
\starttable[|c|c|]
\NC Metal \NC Insulator\NC\MR
\NC $\sigma(T=0)>0$\NC $\sigma(T=0)=0$\NC\MR
\stoptable
\blank[4*big]

\startitemize
\item \goto{Anderson transition}[urilecExtra::andersonModel] - localization induced by disordered.

\item Mott transition - localization induced by interactions.
\stopitemize
\blank[3*big]

\placetable[here,none][tab:strLength]{}
\starttable[|l|l|]
\NC 1D non-interacting systems: \NC 3D non-interacting systems: \NC\MR
\NC \startitemize \item exact solution.\stopitemize\NC\startitemize \item scaling theory. \stopitemize\NC\MR
\NC \startitemize \item always insulating.\stopitemize\NC\startitemize \item Metal-Insulator transiotion depends on\stopitemize\NC\MR
\NC \startitemize \item localization length $\sim$ mean free path. \quad\quad  \stopitemize\NC disorder strength\NC\MR
\NC\NC\startitemize \item long localization length .\stopitemize\NC\MR
\NC 2D non-interacting systems:\NC \NC\MR
\NC \startitemize \item scaling theory.\stopitemize\NC \NC\MR
\NC \startitemize \item always insulating.\stopitemize\NC \NC\MR
\NC \startitemize \item long localization length .\stopitemize\NC \NC\MR
\stoptable





%Scaling theory: A non-interacting system of dimension $\leq$ 2 is insulating.





%%%%%%%%%%%%%%%% luttinger %%%%%%%%%%%%%%%%%%%%%
\Topic[luttinger1]{One-Dimensional Interacting Systems}
\blank[4*big]
\startitemize
\item \goto{Luttinger Model}[urilecExtra::luttingerLiquid]: low energy physics

\placeformula[eq:gsHam]\startformula\eqalignno{
H &= H_{0}+H_{e-e,f}+H_{e-e,b}+H_{dis}
}\stopformula
\stopitemize

\Topic[luttinger2]{One-Dimensional Interacting Systems}
\blank[4*big]
\startitemize
\item \goto{Luttinger Model}[urilecExtra::luttingerLiquid]: low energy physics

\placeformula[eq:gsHam]\startformula\eqalignno{
H &= H_{0}+H_{e-e,f}+H_{e-e,b}+H_{dis}
}\stopformula

\item bosonization:
\placeformula[eq:bos1]\startformula\eqalignno{
\psi_{\pm 1,\uparrow\downarrow}(x) &\to \phi_{c,s}(x),\theta_{c,s}(x)\cr
v_{F},g &\to v_{c,s}\; ,K_{c,s}}\stopformula
\stopitemize

\Topic[luttinger3]{One-Dimensional Interacting Systems}
\blank[4*big]
\startitemize
\item \goto{Luttinger Model}[urilecExtra::luttingerLiquid]: low energy physics

\placeformula[eq:gsHam]\startformula\eqalignno{
H &= H_{0}+H_{e-e,f}+H_{e-e,b}+H_{dis}
}\stopformula

\item bosonization:
\placeformula[eq:bos1]\startformula\eqalignno{
\psi_{\pm 1,\uparrow\downarrow}(x) &\to \phi_{c,s}(x),\theta_{c,s}(x)\cr
v_{F},g &\to v_{c,s}\; ,K_{c,s}}\stopformula
\blank[3*big]
\startformula\eqalignno{
H &= H_{KG}(\phi,\theta)+\int dx\Bigl[g\cos\phi_{s}+
\bigl(\xi(x) e^{i\phi_{c}(x)}\cos\phi_{s}(x)+h.c\bigr)\Bigr]
}\stopformula
\stopitemize

\Topic[luttinger4]{One-Dimensional Interacting Systems}
\blank[4*big]
\startitemize
\item \goto{Luttinger Model}[urilecExtra::luttingerLiquid]: low energy physics

\placeformula[eq:gsHam]\startformula\eqalignno{
H &= \hpos{X-11}{$H_{0}+H_{e-e,f}$}+H_{e-e,b}+H_{dis}
}\stopformula

\item bosonization:
\placeformula[eq:bos1]\startformula\eqalignno{
\psi_{\pm 1,\uparrow\downarrow}(x) &\to \phi_{c,s}(x),\theta_{c,s}(x)\cr
v_{F},g &\to v_{c,s}\; ,K_{c,s}}\stopformula
\blank[3*big]
\startformula\eqalignno{
H &= \hpos{X-21}{$H_{KG}(\phi,\theta)$}+\int dx\Bigl[g\cos\phi_{s}+
\bigl(\xi(x) e^{i\phi_{c}(x)}\cos\phi_{s}(x)+h.c\bigr)\Bigr]
}\stopformula
\stopitemize

\Topic[luttinger5]{One-Dimensional Interacting Systems}
\blank[4*big]
\startitemize
\item \goto{Luttinger Model}[urilecExtra::luttingerLiquid]: low energy physics

\placeformula[eq:gsHam]\startformula\eqalignno{
H &= \hpos{X-12}{$H_{0}+H_{e-e,f}$}+\hpos{X-12a}{$H_{e-e,b}$}+H_{dis}
}\stopformula

\item bosonization:
\placeformula[eq:bos1]\startformula\eqalignno{
\psi_{\pm 1,\uparrow\downarrow}(x) &\to \phi_{c,s}(x),\theta_{c,s}(x)\cr
v_{F},g &\to v_{c,s}\; ,K_{c,s}}\stopformula
\blank[3*big]
\startformula\eqalignno{
H &= \hpos{X-22}{$H_{KG}(\phi,\theta)$}+\int dx\Bigl[\hpos{X-22a}{$g\cos\phi_{s}$}+
\bigl(\xi(x) e^{i\phi_{c}(x)}\cos\phi_{s}(x)+h.c\bigr)\Bigr]
}\stopformula
\stopitemize

\Topic[luttinger6]{One-Dimensional Interacting Systems}
\blank[4*big]
\startitemize
\item \goto{Luttinger Model}[urilecExtra::luttingerLiquid]: low energy physics

\placeformula[eq:gsHam]\startformula\eqalignno{
H &= \hpos{X-13}{$H_{0}+H_{e-e,f}$}+\hpos{X-13a}{$H_{e-e,b}$}+\hpos{X-13b}{$H_{dis}$}
}\stopformula

\item bosonization:
\placeformula[eq:bos1]\startformula\eqalignno{
\psi_{\pm 1,\uparrow\downarrow}(x) &\to \phi_{c,s}(x),\theta_{c,s}(x)\cr
v_{F},g &\to v_{c,s}\; ,K_{c,s}}\stopformula
\blank[3*big]
\startformula\eqalignno{
H &= \hpos{X-23}{$H_{KG}(\phi,\theta)$}+\int dx\Bigl[\hpos{X-23a}{$g\cos\phi_{s}$}+
\hpos{X-23b}{$\bigl(\xi(x) e^{i\phi_{c}(x)}\cos\phi_{s}(x)+h.c\bigr)$}\Bigr]
}\stopformula
\stopitemize

\Topic[luttinger]{One-Dimensional Interacting Systems}
\blank[4*big]
\startitemize
\item \goto{Luttinger Model}[urilecExtra::luttingerLiquid]: low energy physics

\placeformula[eq:gsHam]\startformula\eqalignno{
H &= \hpos{X-1}{$H_{0}+H_{e-e,f}$}+\hpos{X-1a}{$H_{e-e,b}$}+\hpos{X-1b}{$H_{dis}$}
}\stopformula

\item bosonization:
\placeformula[eq:bos1]\startformula\eqalignno{
\psi_{\pm 1,\uparrow\downarrow}(x) &\to \phi_{c,s}(x),\theta_{c,s}(x)\cr
v_{F},g &\to v_{c,s}\; ,K_{c,s}}\stopformula
\blank[3*big]
\startformula\eqalignno{
H &= \hpos{X-2}{$H_{KG}(\phi,\theta)$}+\int dx\Bigl[\hpos{X-2a}{$g\cos\phi_{s}$}+
\hpos{X-2b}{$\bigl(\xi(x) e^{i\phi_{c}(x)}\cos\phi_{s}(x)+h.c\bigr)$}\Bigr]
}\stopformula
\blank[6*big]

\item weak random potential $\xi(x)$ uncorrelated in space
\placeformula[eq:disorder]\startformula\eqalignno{
\overline{\xi^{*}(x)\xi(x')} &= D\delta(x-x')\cr
}\stopformula
\stopitemize
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\Topic[gs]{Renormalization}
perturbation in  $g$ (e-e backward spin-flip scattering) and $D$ (backward disorder)

real-space renormalization group

\placetable[here,none][tab:strLength]{}
\starttable[|c|c|c|]
\NC \THREE \JustCenter{electron interactions:} \NC\MR
\NC attractive \NC non-interactive \NC repulsive \NC\MR
\NC $K>1$ \NC $K=1$ \NC $K<1$ \NC\MR
\stoptable
\placefigure[here,force][]{T. Giamarchi, H. J. Schulz, PRB 37:325, 1988}{
\externalfigure[fig:statKD][width=12cm]}

\Topic[string1]{Fluctuations}
Three models of string fluctuations
\startfiguretext[left][]{none}
{\externalfigure[fig:stringDemo][width=8cm]}
$L_{u}=\frac{M}{2}\left(\partial_{\tau}u\right)^{2}+\frac{M\omega_{0}^{2}}{2}u^{2}$
\blank[big]
$\left\langle u^2\right\rangle = \lambda^2$
\blank[big]
$\lambda=\frac{1}{\sqrt{2M\omega_{0}}}$
\blank[4*big]
$L_{u}=\int_{0}^{L_{x}}dx\left[\frac{\rho}{2}\left(\partial_{\tau}u\right)^{2}+\frac{\sigma}{2}\left(\partial_{x}u\right)^{2}\right]$
\blank[big]
$\left\langle\left(\frac{\partial u}{\partial x}\right)^{2}\right\rangle = \frac{\lambda}{2\pi\alpha}$
\blank[big]
$\lambda=\frac{1}{(\sigma\rho)^{\frac{1}{4}}}$
\blank[2*big]
$L_{u}=\int_{0}^{L}ds\left[\frac{\rho}{2}\left(\partial_{\tau}u\right)^{2}+\frac{\gamma}{2}\left(\partial_{s}^{2}u\right)^{2}\right]$
\blank[big]
$\left\langle\left(\frac{\partial u}{\partial x}\right)^{2}\right\rangle =  \sqrt{\frac{\lambda}{2\pi^{2}\alpha}}$
\blank[big]
$\lambda=\frac{1}{(\gamma\rho)^{\frac{1}{2}}}$
\stopfiguretext

\Topic[string]{Fluctuations}
Three models of string fluctuations
\placefigure[here,force][fig:strings]{none}
{\startcombination[2*1]
{\externalfigure[fig:stringDemo][width=8cm]}{}
{\externalfigure[fig:stringCors][width=7.7cm]}{}
\stopcombination}


%%%%%%%%%%%%%%%%%%%% Results %%%%%%%%%%%%%%%%%%%
\Topic[results1]{Results}
\blank[4*big]
\startitemize
\item The bare disorder is decreased by fluctuations $D\to\frac{D}{\lambda}$.

Fluctuations thus tend to delocalize.
\stopitemize

\Topic[results2]{Results}
\blank[4*big]
\startitemize
\item The bare disorder is decreased by fluctuations $D\to\frac{D}{\lambda}$.

Fluctuations thus tend to delocalize.
\blank[3*big]

\item \goto{Rigid string}[urilecExtra::rgRigidSu2] - the critical point is $K^{*}=2$ (same as in the
static case)
\stopitemize
\placefigure[here,force][]{}{
\externalfigure[fig:rigidKDw][width=12cm]}

\Topic[results3]{Results}
\blank[4*big]
\startitemize
\item The bare disorder is decreased by fluctuations $D\to\frac{D}{\lambda}$.

Fluctuations thus tend to delocalize.
\blank[3*big]

\item \goto{Rigid string}[urilecExtra::rgRigidSu2] - the critical point is $K^{*}=2$ (same as in the
static case)
\blank[3*big]

\item \goto{Elastic string}[urilecExtra::rgGausSu2] - the critical point is $K^{*}=2$ (same as in the
static case)

for white noise disorder average.

The critical point can be shifted to $K^{*}\lesssim2$ for different types of disorder average.
\stopitemize


\Topic[results4]{Results}
\blank[4*big]
\startitemize
\item The bare disorder is decreased by fluctuations $D\to\frac{D}{\lambda}$.

Fluctuations thus tend to delocalize.
\blank[3*big]

\item \goto{Rigid string}[urilecExtra::rgRigidSu2] - the critical point is $K^{*}=2$ (same as in the
static case)
\blank[3*big]

\item \goto{Elastic string}[urilecExtra::rgGausSu2] - the critical point is $K^{*}=2$ (same as in the
static case)

for white noise disorder average.

The critical point can be shifted to $K^{*}\lesssim2$ for different types of disorder average.
\blank[3*big]

\item \goto{Floppy string}[urilecExtra::rgFlopSu2] - the critical point is shifted to $K^{*}=7/4$.
\blank[3*big]
\stopitemize


\Topic[results]{Results}
\blank[4*big]
\startitemize
\item The bare disorder is decreased by fluctuations $D\to\frac{D}{\lambda}$.

Fluctuations thus tend to delocalize.
\blank[3*big]

\item \goto{Rigid string}[urilecExtra::rgRigidSu2] - the critical point is $K^{*}=2$ (same as in the
static case)
\blank[3*big]

\item \goto{Elastic string}[urilecExtra::rgGausSu2] - the critical point is $K^{*}=2$ (same as in the
static case)

for white noise disorder average.

The critical point can be shifted to $K^{*}\lesssim2$ for different types of disorder average.
\blank[3*big]

\item \goto{Floppy string}[urilecExtra::rgFlopSu2] - the critical point is shifted to $K^{*}=7/4$.
\blank[3*big]

\item Extension of the floppy model: $\text{Elastic Energy}=\int ds\gamma\partial_{s}^{n}u$, results in shift of

the critical point to $K^{*}=3/2$ when $n\to\infty$.
\stopitemize
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\Topic[conclusions1]{Conclusions}
\blank[4*big]
\startitemize
\item Fluctuations tend to delocalize, as expected.
\stopitemize

\Topic[conclusions2]{Conclusions}
\blank[4*big]
\startitemize
\item Fluctuations tend to delocalize, as expected.
\blank[3*big]
\item The effect is small - even when the critical point changes it

resides at strong attractive intreactions regime.
\stopitemize

\Topic[conclusions]{Conclusions}
\blank[4*big]
\startitemize
\item Fluctuations tend to delocalize, as expected.
\blank[3*big]
\item The effect is small - even when the critical point changes it

resides at strong attractive intreactions regime.
\blank[3*big]
\item Further investigation - coupled fluctuating strings

(imitating better quasi-one dimensionality)
\stopitemize
\blank[10*big]

\rightaligned{\externalfigure[bpp]}

\stoptext

