\documentclass{ian-presentation}

\usepackage[hidelinks]{hyperref}
\usepackage{graphicx}
\usepackage{array}
\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}{981414}

\begin{document}
\pagestyle{empty}
\hbox{}\vfil
\bf\Large
\hfil Interacting Bosons at intermediate densities\par
\vfil
\large
\hfil Ian Jauslin
\normalsize
\vfil
\hfil\rm joint with {\bf Eric A. Carlen}, {\bf Elliott H. Lieb}\par
\vfil
arXiv:{\tt\ \parbox[b]{6cm}{
  \href{https://arxiv.org/abs/1912.04987}{1912.04987}\ 
  \href{https://arxiv.org/abs/2010.13882}{2010.13882}\par
  \href{https://arxiv.org/abs/2011.10869}{2011.10869}\ 
  \href{https://arxiv.org/abs/2202.07637}{2202.07637}\par
  \href{https://arxiv.org/abs/2302.13446}{2302.13446}\ 
  \href{https://arxiv.org/abs/2302.13449}{2302.13449}
}}
\hfill{\tt \href{http://ian.jauslin.org}{http://ian.jauslin.org}}
\eject

\setcounter{page}1
\pagestyle{plain}

\title{Bosons}
\begin{itemize}
\item Quantum particles are either {\color{highlight}Fermions} or {\color{highlight}Bosons} (in 3D).

\item Fermions: electrons, protons, neutrinos, etc...

\item Bosons: photons, Helium atoms, Higgs particle, etc...

\item At low temperatures: inherently {\color{highlight}quantum} behavior: e.g. {\color{highlight}Bose-Einstein condensation}, superfluidity, quantized vortices, etc...

\item Difficult to handle mathematically: usual approach {\color{highlight}effective theories}.

\item The connection between the original model and the effective theory is, in most cases, poorly understood.
\end{itemize}
\vfill
\eject

\title{Repulsive Bose gas}
\begin{itemize}
  \item Potential: {\color{highlight}$v(r)\geqslant 0$}, {\color{highlight}$\hat v\geqslant 0$} and {\color{highlight}$v\in L_1(\mathbb R^3)$}, on a torus of volume $V$:
  $$
    H_N:=
    -\frac12\sum_{i=1}^N\Delta_i
    +\sum_{1\leqslant i<j\leqslant N}v(|x_i-x_j|)
  $$
  \vskip-5pt
  \item Ground state ({\color{highlight}zero temperature}): $\psi_0$, energy $E_0$.

  \item Observables in the {\color{highlight}thermodynamic limit}: for instance, ground state energy per particle
  $$
    e_0:=\lim_{\displaystyle\mathop{\scriptstyle V,N\to\infty}_{{\color{highlight}\frac NV=\rho}}}\frac{E_0}N
    .
  $$

  \item Main difficulty: dealing with the interactions.
\end{itemize}
\vfill
\eject

\title{Known results}
\begin{itemize}
  \item {\color{highlight}Low density}: \href{https://doi.org/10.1103/PhysRev.106.1135}{Lee-Huang-Yang} formula
  $$
    e_0=2\pi\rho a\left(1+\frac{128}{15\sqrt\pi}\sqrt{\rho a^3}+o(\sqrt{\rho a^3})\right)
  $$
  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]},
  \href{https://doi.org/10.1017/fms.2021.66}{[Basti, Cenatiempo, Schlein, 2021]}.
  
  \item {\color{highlight}High density}: Hartree energy:
  $$
    e_0\sim\frac\rho2\int v
  $$
  proved: \href{https://doi.org/10.1103/PhysRev.130.2518}{[Lieb, 1963]}.
\end{itemize}
\vfill
\eject

\title{The Simplified approach: proof of concept}
For $v(x)=e^{-|x|}$: {\color{ipurple}Simplified approach}, {\color{iyellow}LHY}, {\color{igreen}Hartree}, {\color{ired}Monte Carlo}

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

\title{The Simplified approach}
\begin{itemize}
  \item \href{https://doi.org/10.1103/PhysRev.130.2518}{[Lieb, 1963]},
  \href{https://arxiv.org/abs/2302.13446}{[Jauslin, 2023]}.

  \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,
  $$
    \frac{N(N-1)}2\int dx_1dx_2\ v(x_1-x_2)\frac{\int dx_3\cdots dx_N\ \psi_0}{\int dy_1\cdots dy_N\ \psi_0}
    =E_0
  $$
\end{itemize}
\vfill
\eject

\title{The Simplified approach}
\begin{itemize}
  \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 $V^{-N}\psi_0$
  $$
    g_n(x_1,\cdots,x_n):=\frac{V^n\int dx_{n+1}\cdots dx_N\ \psi_0(x_1,\cdots,x_N)}{\int dy_1\cdots dy_N\ \psi_0(y_1,\cdots,y_N)}
  $$
  \item Thus,
  $$
    \frac{E_0}N=\frac{N-1}{2V}\int dx\ v(x)g_2(0,x)
  $$
\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 {\color{highlight}Infinite hierarchy} of equations.
\end{itemize}
\vfill
\eject

\title{Factorization assumption}
\begin{itemize}
  \item Factorization {\color{highlight}assumption} (clustering property): for $n=3,4$,
  $$
    g_n(x_1,\cdots,x_n)=\prod_{1\leqslant i<j\leqslant n}(1-u_n(x_i-x_j))
    ,\quad
    u_n\in L_1(\mathbb R^3)
  $$

  \item Consistency condition:
  $$
    \int \frac{dx_3}V\ g_3(x_1,x_2,x_3)=g_2(x_1,x_2)
    ,\quad
    \int \frac{dx_3}V\frac{dx_4}V\ g_4(x_1,x_2,x_3,x_4)=g_2(x_1,x_2)
  $$

  \item Remark: in general,
  $$
    \int \frac{dx_4}V\ g_4(x_1,x_2,x_3,x_4)\neq g_3(x_1,x_2,x_3)
  $$
\end{itemize}
\vfill
\eject

\title{Factorization assumption}
\begin{itemize}
  \item {\bf Lemma} 
  \href{https://doi.org/10.1103/PhysRev.130.2518}{[Lieb, 1963]},
  \href{https://arxiv.org/abs/2302.13446}{[Jauslin, 2023]}:
  Under the Factorization assumption and the consistency condition,
  $$
    u_3(x-y)=u(x-y)+\frac1V(1-u(x-y))\int dz\ u(x-z)u(z-y)+O(V^{-2})
  $$
  $$
    u_4(x-y)=u(x-y)+\frac2V(1-u(x-y))\int dz\ u(x-z)u(z-y)+O(V^{-2})
  $$

  \item With $u(x):=1-g_2(0,x)$.
\end{itemize}
\vfill
\eject

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

  \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 {\color{iblue}Simple equation}:
  $$
    -\Delta u(x)=(1-u(x))v(x)- 4eu(x)+2e\rho\ u\ast u(x)
  $$
  $$
    e=\frac\rho2\int dx\ (1-u(x))v(x)
  $$
  \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}:
  $$
    \frac{e}{\rho}\mathop{\longrightarrow}_{\rho\to\infty}\frac12\int dx\ v(x)
    .
  $$
  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}
$v(x)=e^{-|x|}$, {\color{ipurple}Big equation}, {\color{ired}Monte Carlo}

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

\title{Radial distribution function}
\begin{itemize}
  \item
  {\color{highlight}Two-point correlation}:
  $$
    C_2(y-z)=\sum_{i,j}\left<\psi_0\right|\delta(y-x_i)\delta(z-x_j)\left|\psi_0\right>
    .
  $$
  \item
  {\color{highlight}Radial distribution}: spherical average and normalization:
  $$
    G(r):=\frac1{\rho^2}\int\frac{dy}{4\pi r^2}\ \delta(|y|-r)C_2(y)
    .
  $$

  \item
  Compute $C_2$ using
  $$
    C_2(x)=2\rho\frac{\delta e_0}{\delta v(x)}
    .
  $$
\end{itemize}
\vfill
\eject

\title{Radial distribution function}
$v(x)=16e^{-|x|}$, $\rho=0.02$ {\color{ipurple}Big equation}, {\color{ired}Monte Carlo}

\hfil\includegraphics[height=5.5cm]{2pt.pdf}
\vfill
\eject

\title{Radial distribution function}
$v(x)=8e^{-|x|}$, $\rho=10^{-5}$-$10^{-1}$

\hfil\includegraphics[height=5.5cm]{2pt_rho.pdf}
\vfill
\eject

\title{Radial distribution function}
$v(x)=8e^{-|x|}$, maximal value as a function of $\rho$:

\hfil\includegraphics[height=5.5cm]{2pt_max.pdf}
\vfill
\eject

\title{Liquid behavior}
\begin{itemize}
  \item Maximum above $1$: there is a length scale at which it is {\color{highlight} more probable} to find pairs of particles.
  \item {\color{highlight}No} long range order.
  \item {\color{highlight}Short-range order}: {\color{highlight}Liquid}-like behavior.
\end{itemize}
\vfill
\eject

\title{Structure factor}
\begin{itemize}
  \item
  {\color{highlight}Structure factor}: Fourier transform of $G$:
  $$
    S(|k|):=1+\rho\int dx\ e^{ikx}(G(|x|)-1)
    .
  $$

  \item
  Directly observable in X-ray scattering experiments.

  \item
  Sharp peaks: order.

  \item
  Large deviation from $1$: uniformity.
\end{itemize}
\vfill
\eject

\title{Structure factor}
$v(x)=8e^{-|x|}$, $\rho=10^{-5}$-$0.3$

\hfil\includegraphics[height=5.5cm]{2pt_fourier_full.pdf}
\vfill
\eject

\title{Structure factor}
$v(x)=8e^{-|x|}$, $\rho=10^{-5}$-$0.3$

\hfil\includegraphics[height=5.5cm]{2pt_fourier_peak.pdf}
\vfill
\eject

\title{Structure factor}
$v(x)=8e^{-|x|}$, maximal value as a function of $\rho$:

\hfil\includegraphics[height=5.5cm]{2pt_fourier_max.pdf}
\vfill
\eject

\title{Liquid behavior}
\begin{itemize}
  \item Sharpening of the peak: more order.
  \item Not Bragg peaks: {\color{highlight}No} long range order.
  \item Larger deviation from 1: more uniform (not even close to hyperuniform).
  \item {\color{highlight}Short-range order}: {\color{highlight}Liquid}-like behavior.
\end{itemize}
\vfill
\eject

\title{Critical densities}
\begin{itemize}
  \item
  We have found two critical densities: $\rho_*\approx 0.9\times10^{-3}$ and $\rho_{**}\approx0.2$.

  \item The radial distribution function has a maximum only for $\rho>\rho_*$.

  \item The structure factor has a maximum only for $\rho<\rho_{**}$.
\end{itemize}
\vfill
\eject

\title{Condensate fraction}
\begin{itemize}
  \item
  Proportion of particles in the condensate state:
  $$
    \eta:=\frac1N\sum_i\left<\psi_0\right|P_i\left|\psi_0\right>
  $$
  where $P_i$ is the projector onto the constant state $V^{-\frac12}$.

  \item
  $\eta>0$ in thermodynamic limit: {\color{highlight}Bose-Einstein condensation} (still not proved to occur).
\end{itemize}
\vfill
\eject

\title{Condensate fraction}
$v(x)=8e^{-|x|}$:

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

\title{Summary and outlook}
\begin{itemize}
  \item Using the {\color{highlight}Simplified approach}, we were able to probe the repulsive Bose gas {\color{highlight}beyond the dilute regime}.

  \item Evidence for {\color{highlight}non-trivial behavior} at intermediate densities $\rho_*<\rho<\rho_{**}$: {\color{highlight}short-range order}.

  \item Is there a phase transition?

  \item The intermediate density regime has not been studied much, due to the lack of tools to do so.
  As we have seen, there is non-trivial behavior there.
  This warrants further investigation, both theoretical and experimental.
\end{itemize}
\vfill
\eject

\title{Open problems on the Simplified approach}
\begin{itemize}
  \item
  Connect the Simplified approach to the many-Boson system: numerics suggests the prediction of the Simplified approach is an {\color{highlight}upper bound}, for all densities.

  \item
  Understand the factorization assumption.
  It certainly does not hold exactly.
  Does it hold approximately, in some sense?

  \item
  There are still many questions about the Bose gas with hard core interactions.
  The Simplified approach is easily defined in the hard core case.
  Can it shed some light?
\end{itemize}

\end{document}