From 6086bbc8826d558bedab6933913c6793850470fb Mon Sep 17 00:00:00 2001
From: Ian Jauslin <ian.jauslin@rutgers.edu>
Date: Thu, 9 Mar 2023 11:48:20 -0500
Subject: As presented at Rutgers on 2023-03-09

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+\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}
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