\documentclass{ian-presentation}

\usepackage[hidelinks]{hyperref}
\usepackage{graphicx}
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\usepackage{xcolor}


\definecolor{ipurple}{HTML}{4B0082}
\definecolor{iyellow}{HTML}{DAA520}
\definecolor{igreen}{HTML}{32CD32}
\definecolor{iblue}{HTML}{4169E1}
\definecolor{ired}{HTML}{DC143C}

\definecolor{highlight}{HTML}{328932}
\definecolor{highlight}{HTML}{981414}

\begin{document}
\pagestyle{empty}
\hbox{}\vfil
\bf\Large
\hfil An effective equation to study Bose gases\par
\smallskip
\hfil at all densities\par
\vfil
\large
\hfil Ian Jauslin
\normalsize
\vfil
\hfil\rm joint with {\bf Eric A. Carlen}, {\bf Markus Holzmann}, {\bf Elliott H. Lieb}\par
\vfil
arXiv:{\tt\ \parbox[b]{3cm}{
  \href{https://arxiv.org/abs/1912.04987}{1912.04987}\par
  \href{https://arxiv.org/abs/2010.13882}{2010.13882}\par
  \href{https://arxiv.org/abs/2011.10869}{2011.10869}
}}
\hfill{\tt \href{http://ian.jauslin.org}{http://ian.jauslin.org}}
\eject

\setcounter{page}1
\pagestyle{plain}

\title{Bose-Einstein condensation}
\begin{itemize}
\item System of many Bosons, e.g. {\color{highlight}Helium} atoms, {\color{highlight}Rubidium} atoms, etc...
\item {\color{highlight}Bose-Einstein condensate}: most particles are in the same quantum state.
\item Related to the phenomena of {\color{highlight}superfluidity} (flow with zero viscocity) and {\color{highlight}superconductivity} (currents with zero resistance).
\item Predicted theoretically in {\color{highlight}1924-1925}, experimentally observed in {\color{highlight}1995}.
\item Mathematical understanding: still {\color{highlight}no proof} of the existence of a condensate (at finite density, in the presence of interactions and in the continuum).
\end{itemize}
\vfill
\eject

\title{Repulsive Bose gas}
\begin{itemize}
  \item Potential: {\color{highlight}$v(r)\geqslant 0$} and {\color{highlight}$v\in L_1(\mathbb R^3)$}, Hamiltonian:
  $$
    H_N:=
    -\frac12\sum_{i=1}^N\Delta_i
    +\sum_{1\leqslant i<j\leqslant N}v(|x_i-x_j|)
  $$
  \vskip-5pt
  \item Ground state: $\psi_0$, energy $E_0$.
  \item Observables in the {\color{highlight}thermodynamic limit}: ground state energy per particle and condensate fraction: $P_i$: projection onto condensate state
  $$
    e_0:=\lim_{\displaystyle\mathop{\scriptstyle V,N\to\infty}_{{\color{highlight}\frac NV=\rho}}}\frac{E_0}N
    ,\quad
    \eta_0:=\lim_{\displaystyle\mathop{\scriptstyle V,N\to\infty}_{{\color{highlight}\frac NV=\rho}}}\frac1N\sum_{i=1}^N\left<\psi_0\right|P_i\left|\psi_0\right>
    .
  $$
\end{itemize}
\vfill
\eject

\title{Low density}
\begin{itemize}
  \item Bogolyubov theory: {\color{highlight}approximation scheme} that reduces the problem to an effective {\color{highlight}1-particle problem}.
  \item Predictions \href{https://doi.org/10.1103/PhysRev.106.1135}{[Lee, Huang, Yang, 1957]}:
  \begin{itemize}
    \item Energy:
    $$
      {\color{highlight}e_0=2\pi\rho a\left(1+\frac{128}{15\sqrt\pi}\sqrt{\rho a^3}+o(\sqrt\rho)\right)}
    $$
    \vskip-10pt
    \item Condensate fraction:
    $$
      {\color{highlight}1-\eta_0\sim\frac{8\sqrt{\rho a^3}}{3\sqrt\pi}}
    $$
  \end{itemize}
\end{itemize}
\vfill
\eject

\title{Low density}
\begin{itemize}
  \item Energy asymptotics: {\color{highlight} proved}:
  \href{https://doi.org/10.1103/PhysRevLett.80.2504}{[Lieb, Yngvason, 1998]},
  \href{https://doi.org/10.1007/s10955-009-9792-3}{[Yau, Yin, 2009]},
  \href{https://doi.org/10.4007/annals.2020.192.3.5}{[Fournais, Solovej, 2020]}.

  \item Condensate fraction: {\color{highlight}still open} in the theormodynamic limit, but there are proofs of condensation in the Gross-Pitaevskii regime (ultra-dilute):
  \href{https://doi.org/10.1103/PhysRevLett.88.170409}{[Lieb, Seiringer, 2002]},
  \href{https://doi.org/10.1007/s00220-017-3016-5}{[Boccato, Brennecke, Cenatiempo, Schlein, 2018]}.

\end{itemize}
\vfill
\eject

\title{High density}
\begin{itemize}
  \item [Bogolyubov, 1947]: if $\hat v\geqslant 0$.
  $$
    {\color{highlight}e_0\sim\frac\rho2\int v}
  $$
  {\color{highlight}Hartree} (mean field) energy.
  \item {\color{highlight}Proved} in \href{https://doi.org/10.1103/PhysRev.130.2518}{[Lieb, 1963]}.
  \item Condensate fraction
  $$
    \eta\to1
  $$
  {\color{highlight}open}.
\end{itemize}
\vfill
\eject

\title{Energy as a function of density}
For $v(x)=e^{-|x|}$:

\hfil\includegraphics[height=5.5cm]{erho_lowhigh.pdf}
\vfill
\eject

\addtocounter{page}{-1}
\title{Energy as a function of density}
For $v(x)=e^{-|x|}$:

\hfil\includegraphics[height=5.5cm]{erho_effective.pdf}
\vfill
\eject

\title{Derivation of the equation}
\begin{itemize}
  \item \href{https://doi.org/10.1103/PhysRev.130.2518}{[Lieb, 1963]}.
  \item Integrate $H_N\psi_0=E_0\psi_0$:
  $$
    \int dx_1\cdots dx_N\ 
    \left(
      -\frac12\sum_{i=1}^N\Delta_i\psi_0
      +\sum_{1\leqslant i<j\leqslant N} v(x_i-x_j)\psi_0
    \right)
    =E_0\int dx_1\cdots dx_N\ \psi_0
  $$
  \item Therefore,
  $$
    \frac{N(N-1)}2\int dx_1dx_2\ v(x_1-x_2)\frac{\int dx_3\cdots dx_N\ \psi_0}{\int dx_1\cdots dx_N\ \psi_0}
    =E_0
  $$
\end{itemize}
\vfill
\eject

\title{Derivation of the equation}
\begin{itemize}
  \item Thus,
  $$
    \frac{E_0}N=\frac{N-1}{2V}\int dx\ v(x)g_2(0,x)
  $$
  \item {\color{highlight}$\psi_0\geqslant 0$}, so it can be thought of as a probability distribution.
  \item $g_n$: {\color{highlight}correlation functions} of $V^{-N}\psi_0$
  $$
    g_n(x_1,\cdots,x_n):=\frac{V^n\int dx_{n+1}\cdots dx_N\ \psi_0(x_1,\cdots,x_N)}{\int dx_1\cdots dx_N\ \psi_0(x_1,\cdots,x_N)}
  $$
\end{itemize}
\vfill
\eject

\title{Hierarchy}
\vskip-10pt
\begin{itemize}
  \item Equation for $g_2$: integrate $H_N\psi_0=E_0\psi_0$ with respect to $x_3,\cdots,x_N$:
  $$
    \begin{array}{>\displaystyle l}
      -\frac12(\Delta_x+\Delta_y) g_2(x,y)
      +\frac{N-2}V\int dz\ (v(x-z)+v(y-z))g_3(x,y,z)
      \\[0.5cm]\hfill
      +v(x-y)g_2(x,y)
      +\frac{(N-2)(N-3)}{2V^2}\int dzdt\ v(z-t)g_4(x,y,z,t)
      =E_0g_2(x,y)
    \end{array}
  $$
  \item Factorization {\color{highlight}assumption}:
  $$
    g_3(x_1,x_2,x_3)=g_2(x_1,x_2)g_2(x_1,x_3)g_2(x_2,x_3)
  $$
  $$
    g_4(x_1,x_2,x_3,x_4)=\prod_{i<j}(g_2(x_i,x_j)+O(V^{-1}))
  $$
\end{itemize}
\vfill
\eject

\title{{\color{ipurple}Big equation}}
\begin{itemize}
\item In the thermodynamic limit, if ${\color{highlight}u(x):=1-g_2(0,x)}$,
  $$
    -\Delta u(x)
    =
    (1-u(x))\left(v(x)-2\rho K(x)+\rho^2 L(x)\right)
  $$
  $$
    K:=
    u\ast S
    ,\quad
    S(y):=(1-u(y))v(y)
  $$
  $$
    L:=
    u\ast u\ast S
    -2u\ast(u(u\ast S))
    +\frac12
    \int dydz\ u(y)u(z-x)u(z)u(y-x)S(z-y)
    .
  $$

  \item {\color{ipurple}``Big'' equation}:
  $$
    L\approx
    u\ast u\ast S
    -2u\ast (u(u\ast S))
  $$
\end{itemize}
\vfill
\eject
  
\title{{\color{iblue}Simple equation}}
\vskip-10pt
\begin{itemize}
  \item Further approximate $S(x)\approx\frac{2e}\rho\delta(x)$ and $u\ll 1$.
  \item Simple equation
  $$
    {\color{iblue}-\Delta u(x)=(1-u(x))v(x)- 4eu(x)+2e\rho\ u\ast u(x)}
  $$
  $$
    {\color{iblue}e=\frac\rho2\int dx\ (1-u(x))v(x)}
  $$
  \item {\bf Theorem 1}:
  If $v(x)\geqslant 0$ and $v\in L_1\cap L_2(\mathbb R^3)$, then the {\color{iblue}simple equation} has an {\color{highlight}integrable solution} (proved constructively), with $0\leqslant u\leqslant 1$.
\end{itemize}
\vfill
\eject

\title{Energy for the {\color{iblue}simple equation}}
\vskip-10pt
\begin{itemize}
  \item {\bf Theorem 2}:
  $$
    \frac{e}{\rho}\mathop{\longrightarrow}_{\rho\to\infty}\frac12\int dx\ v(x)
  $$
  (note that there is no condition that $\hat v\geqslant 0$).
  This coincides with the {\color{highlight}Hartree energy}.
  \item {\bf Theorem 3}:
  $$
    e=2\pi\rho a\left(1+\frac{128}{15\sqrt\pi}\sqrt{\rho a^3}+o(\sqrt\rho)\right)
  $$
  This coincides with the {\color{highlight}Lee-Huang-Yang prediction}.
\end{itemize}
\vfill
\eject

\title{Energy}
$v(x)=e^{-|x|}$, Blue: {\color{iblue}simple equation}; purple: {\color{ipurple}big equation}; red: {\color{ired}Monte Carlo}

\hfil\includegraphics[height=5.5cm]{erho_fulleq.pdf}
\vfill
\eject

\title{Energy}
$v(x)=e^{-|x|}$, Blue: {\color{iblue}simple equation}; red: {\color{ired}Jastrow}; purple: {\color{ipurple}big equation}

\hfil\includegraphics[height=5.5cm]{cmp_energy1.pdf}
\vfill
\eject

\title{Condensate fraction}
\begin{itemize}
  \item Add a parameter $\mu$ to the Hamiltonian:
  $$
    H_N(\mu):=
    -\frac12\sum_{i=1}^N\Delta_i
    +\sum_{1\leqslant i<j\leqslant N}v(x_i-x_j)
    {\color{highlight}-\mu\sum_{i=1}^NP_i}
  $$
  \item Projection onto condensate wavefunction: $P_i$.
  \item Condensate fraction:
  $$
    {\color{highlight}\eta_0}:=\frac1N\left<\psi_0\right|\sum_{i=1}^NP_i\left|\psi_0\right>
    =-\frac1N\partial_\mu \left<\psi_0\right|H_N(\mu)\left|\psi_0\right>|_{\mu_0}
    \equiv
    {\color{highlight}-\partial_\mu e_0(\mu)|_{\mu=0}}
  $$
\end{itemize}
\vfill
\eject

\title{Condensate fraction}
\begin{itemize}
  \item {\bf Theorem 4}:
  For the {\color{iblue}simple equation}, as $\rho\to0$
  $$
    1-\eta\sim\frac{8\sqrt{\rho a^3}}{3\sqrt\pi}
  $$
  which coincides with {\color{highlight}Bogolyubov's prediction}.
\end{itemize}
\vfill
\eject

\title{Condensate fraction}
$v(x)=e^{-|x|}$, Blue: {\color{iblue}simple equation}; purple: {\color{ipurple}big equation}; red: {\color{ired}Monte Carlo}

\hfil\includegraphics[height=5.5cm]{condensate.pdf}
\vfill
\eject

\title{Two point correlation function}
$v(x)=16e^{-|x|}$, Blue: {\color{iblue}simple equation}; purple: {\color{ipurple}big equation}; red: {\color{ired}Monte Carlo}

\hfil\includegraphics[height=5.5cm]{correlation.pdf}
\vfill
\eject

\title{Summary and outlook}
\begin{itemize}
  \item Two {\color{highlight}effective equations}: the {\color{ipurple}big equation} and the {\color{iblue}simple equation}, which are {\color{highlight}non-linear 1-particle equations}.
  \item Reproduce the known results for both {\color{highlight}small and large densities}.
  \item Their derivation is {\color{highlight}different from Bogolyubov theory}, so they may give new insights onto studying the Bose gas in these asymptotic regimes.
  \item The {\color{ipurple}big equation} is {\color{highlight}quantitatively accurate} at intermediate densities.
  \item This opens up the possibility of studying the physics of the {\color{highlight}Bose gas at intermediate densities}.
\end{itemize}
\vfill
\eject

\title{Open problems}
\vfill
\begin{itemize}
  \item {\color{highlight}Uniqueness} of the solution of the {\color{iblue}simple equation} (done for small and large $\rho$).
  \vfill

  \item LHY as an {\color{highlight}upper bound} at low density using the {\color{iblue}simple equation} to construct an Ansatz.
  \vfill

  \item {\color{highlight}Existence} (and uniqueness) of the solution of the {\color{ipurple}big equation}.

\end{itemize}
\vfill
\eject

\title{The uniqueness problem}
  $$
    -\Delta u(x)=(1-u(x))v(x)- 4eu(x)+2e\rho\ u\ast u(x)
    ,\quad
    e=\frac\rho2\int dx\ (1-u(x))v(x)
  $$
\begin{itemize}
  \item Change the point of view: {\color{highlight}fix $e>0$}, and compute $\rho$ and $u$.

  \item {\color{highlight}Iteration}: $u_0=0$,
  $$
    (-\Delta+4e+v)u_n=v+2e\rho_{n-1}u_{n-1}\ast u_{n-1}
    ,\quad
    \rho_n:=\frac{2e}{\int dx\ (1-u_n(x))v(x)}
    .
  $$
\end{itemize}
\vfill
\eject

\title{The uniqueness problem}
\begin{itemize}
  \item {\bf Lemma}: $u_n(x)$ is an {\color{highlight}increasing} sequence, and is {\color{highlight}bounded} $u_n(x)\leqslant 1$.
  It converges to a function $u$, which is the {\color{highlight}unique} integrable solution of the equation {\color{highlight}with $e$ fixed}.

  \item {\bf Lemma}: $e\mapsto\rho(e)$ is {\color{highlight}continuous}, and $\rho(0)=0$ and $\rho(\infty)=\infty$, which allows us to  compute solutions for the problem at fixed $\rho$.

  \item We thus have a {\color{highlight}restricted} notion of uniqueness.
  The full uniqueness would follow from a proof that $e\mapsto\rho(r)$ is {\color{highlight}monotone increasing} (which must be true for the physics to make sense).
\end{itemize}
\vfill
\eject

\title{Upper bound at low density}
\begin{itemize}
  \item \href{https://doi.org/10.1007/s10955-009-9792-3}{[Yau, Yin, 2009]}: proof for {\color{highlight}weak, smooth}, rapidly decaying potentials.
  \item \href{https://arxiv.org/abs/2101.06222}{[Basti, Cenatiempo, Schlein, 2021]}: extended for {\color{highlight}$L_3$} (and compactly supported) potentials (excludes hard-core interactions).
  \item {\color{iblue}Simple Equation}: our analysis holds for the hard-core, so if one could find a good {\color{highlight}Ansatz} from it, one might get an upper bound for the energy in this case.
  \item Idea for Ansatz? (Jastrow, Dyson-Jastrow?).
\end{itemize}
\vfill
\eject

\title{Upper bound at low density: Jastrow function}
\begin{itemize}
  \item Idea:
  $$
    \psi=\prod_{i<j}e^{-u(x_i-x_j)}
  $$

  \item Why this: $\rho\ll1$, and {\color{highlight}if $\rho\|u\|_1\ll1$},
  $$
    g_2\sim1-u
    .
  $$

  \item Again, {\color{highlight}if $\rho\|u\|_1\ll1$}, we would be able to compute the energy of $\psi$ using a cluster expansion!

  \item However, {\color{highlight}$\|u\|_1=\frac1\rho$}!
\end{itemize}
\vfill
\eject

\title{Existence for the {\color{ipurple}Big Equation}}
\begin{itemize}
  \item Numerical method: {\color{highlight}Newton algorithm}.

  \item For the existence of a solution, it would suffice to prove that the Newton algorithm has a {\color{highlight}Basin of attraction}. (Kantorovich-like theorem?)

  \item Such a result, applied to the {\color{iblue}Simple Equation}, would imply the {\color{highlight}uniqueness} of a solution (provided we have convergence in an appropriate norm).
\end{itemize}

\end{document}