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diff --git a/Jauslin_Princeton_2023.tex b/Jauslin_Princeton_2023.tex new file mode 100644 index 0000000..ebf12ea --- /dev/null +++ b/Jauslin_Princeton_2023.tex @@ -0,0 +1,457 @@ +\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 Non-perturbative behavior of interacting Bosons\par\smallskip\hfil 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} |