From f950c74e38d120e42637983d504f85843a63e95d Mon Sep 17 00:00:00 2001
From: Ian Jauslin <jauslin@ias.edu>
Date: Wed, 4 Apr 2018 06:21:21 +0000
Subject: As presented at Princeton University on 2018-04-04

---
 Jauslin_EL_2018.tex | 335 ++++++++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 335 insertions(+)
 create mode 100644 Jauslin_EL_2018.tex

(limited to 'Jauslin_EL_2018.tex')

diff --git a/Jauslin_EL_2018.tex b/Jauslin_EL_2018.tex
new file mode 100644
index 0000000..01a3a21
--- /dev/null
+++ b/Jauslin_EL_2018.tex
@@ -0,0 +1,335 @@
+\documentclass{ian-presentation}
+
+
+\usepackage[hidelinks]{hyperref}
+\usepackage{graphicx}
+\usepackage{xcolor}
+
+\definecolor  {L67}{HTML}{4169E1}
+\definecolor{SML64}{HTML}{4B0082}
+\definecolor {TL71}{HTML}{DAA520}
+\definecolor {HL72}{HTML}{DC143C}
+\definecolor {HL79}{HTML}{32CD32}
+\definecolor  {L89}{HTML}{00CCCC}
+
+\begin{document}
+\pagestyle{empty}
+\hbox{}\vfil
+\bf\Large
+\hfil Dimers, Spins and Loops\par
+\smallskip
+\large\hfil Transfer Matrices, the TL Algebra and Emerging Fermions\par
+\vfil
+\large
+\hfil Ian Jauslin
+\vfil
+\normalsize
+\hfil\tt \href{http://ian.jauslin.org}{http://ian.jauslin.org}
+\rm
+\eject
+
+\setcounter{page}1
+\pagestyle{plain}
+
+\title{Outline}
+\vfill
+\hfil\includegraphics[width=5cm]{plan.pdf}
+\hfil\parbox[b]{6.5cm}{
+  \begin{itemize}
+    {\color{L67}\item [Li67] Dimers}
+    {\color{SML64}\item [SML64] 2D Ising}
+    {\color{TL71}\item [TL71] Temperley-Lieb algebras}
+    {\color{HL72}\item [HL72] Monomers and Dimers}
+    {\color{HL79}\item [HL79] Interacting Dimers}
+    {\color{L89}\item [Li89] Hubbard model}
+  \end{itemize}
+}
+\vfill
+\eject
+
+\title{Dimers}
+\vfill
+\hfil\includegraphics[width=6cm]{graph.pdf}
+\vfill
+\eject
+
+\addtocounter{page}{-1}
+\title{Dimer covering}
+\vfill
+\hfil\includegraphics[width=6cm]{cover.pdf}
+\vfill
+\eject
+
+\addtocounter{page}{-1}
+\title{Monomer-Dimer configuration}
+\vfill
+\hfil\includegraphics[width=6cm]{mdcover.pdf}
+\vfill
+\eject
+
+\title{Counting dimer coverings}
+\begin{itemize}
+  \item On planar graphs:
+    \href{http://dx.doi.org/10.1063/1.1703953}{[Kasteleyn, 1963]},
+    \href{http://dx.doi.org/10.1080/14786436108243366}{[Temperley, Fisher, 1961]}: Pfaffian formula
+    $$Z\equiv\mathrm{number\ of\ coverings} = \mathrm{determinant}.$$
+  \item {\color{L67}\href{http://dx.doi.org/10.1063/1.1705163}{{\bf E.H. Lieb}, {\it Solution of the Dimer Problem by the Transfer Matrix Method}, Journal of Mathematical Physics, 1967}}: on $M\times N$ discrete torus:
+  $$\lim_{M,N\to\infty}\frac1{MN}\log Z=\frac1{2\pi}\int_0^\pi dq\ \log\left(\sin q+\sqrt{1+\sin^2q}\right).$$
+  \item Using a Transfer Matrix and Emergent Fermions.
+\end{itemize}
+\vfill
+\eject
+
+\title{2D Ising}
+\vskip-10pt
+\begin{itemize}
+  \item {\it Spin} on every $x\in\mathbb Z^2$. Random configuration with probability
+  $$\frac1{Z(T)}e^{\frac1T\sum_{\left<i,j\right>}\sigma_i\sigma_j}.$$
+  \vskip-10pt
+\end{itemize}
+\vfill
+\hfil\includegraphics[width=4cm]{ising.pdf}
+\vfill
+\eject
+
+\title{2D Ising}
+\begin{itemize}
+  \item At $T\ll 1$, two phases:
+\end{itemize}
+\vfill
+\hfil\includegraphics[width=4cm]{ising_blue.pdf}
+\hfil\includegraphics[width=4cm]{ising_red.pdf}
+\vfill
+\eject
+
+\title{2D Ising}
+\vfill
+\begin{itemize}
+  \item Free energy:
+  $$f(T)=-T\lim_{M,N\to\infty}\frac1{MN}\log Z(T).$$
+  \item Exact solution:
+    \href{http://dx.doi.org/10.1103/PhysRev.65.117}{[Onsager, 1944]}:
+    first example of a microscopic model with a phase transition.
+  \item {\color{SML64}\href{http://dx.doi.org/10.1103/RevModPhys.36.856}{{\bf T.D. Schultz, D.C. Mattis, E.H. Lieb}, {\it Two-Dimensional Ising Model as a Soluble Problem of Many Fermions}, Reviews of Modern Physics, 1964}}.
+\end{itemize}
+\vfill
+\eject
+
+\title{2D Ising - Transfer Matrix}
+\vfill
+\hfil\includegraphics[width=4cm]{transfer1.pdf}
+\vfill
+\eject
+
+\addtocounter{page}{-1}
+\title{2D Ising - Transfer Matrix}
+\vfill
+\hfil\includegraphics[width=4cm]{transfer2.pdf}
+\vfill
+\eject
+
+\title{2D Ising - Transfer Matrix}
+\begin{itemize}
+  \item Transfer matrix: $V$, is a $2^M\times 2^M$ real symmetric matrix, and
+  $$Z(T)=\mathrm{Tr}(V^N).$$
+  \item The free energy
+  $$f(T)=-T\lim_{N,M\to\infty}\frac1{NM}\log Z(T)=-\lim_{M\to\infty}\frac TM\log\lambda_M$$
+  where $\lambda_M$ is the {\it largest} eigenvalue of $V$.
+\end{itemize}
+\vfill
+\eject
+
+\title{2D Ising - Emergent Fermions}
+\begin{itemize}
+  \item To diagonalize $V$: turn spins into Fermions using a {\it Jordan-Wigner} transformation.
+  \item Fermions: particle excitations.
+  \item {\it Non-interacting} Fermions:
+  $$V=(2\sinh(2JT^{-1}))^{\frac M2}e^{-\sum_q\epsilon_q(c_q^\dagger c_q-\frac12)}.$$
+  \item Remark: in the {\it ice model} ({\it cf} Duminil-Copin), Fermions {\it interact}.
+  \item Remark: the Ising model with weak nearest neighbor interactions is mapped to a weakly interacting Fermion model \href{http://dx.doi.org/10.1063/1.4745910}{[Giuliani, Greenblatt, Mastropietro, 2012]}.
+\end{itemize}
+\vfill
+\eject
+
+\title{Temperley-Lieb algebras}
+\begin{itemize}
+  \vskip-12pt
+  \item {\color{TL71}\href{http://dx.doi.org/10.1098/rspa.1971.0067}{{\bf H.N.V. Temperley, E.H. Lieb}, {\it Relations between the `percolation' and `colouring' problem and other graph-theoretical problems associated with regular planar lattices: some exact results for the `percolation' problem}, Proceedings of the Royal Society of London A, 1971}}.
+  \item Compute {\it Whitney} polynomial on a square lattice
+  $$W(x,y)=\sum_Gx^{l_G-s_G}y^{s_G}.$$
+  where $l_G$ is the number of lines and $s_G$ the number of cycles.
+  \item Related to counting the number of connected components in a random graph, and to the number of ways of coloring the $\mathbb Z^2$ lattice.
+  \item Transfer Matrix technique: difficult because the setting is non-Markovian.
+\end{itemize}
+\vfill
+\eject
+
+\title{Temperley-Lieb algebras}
+\begin{itemize}
+  \item In each row, keep track of who is connected to whom.
+  \item Graphical representation of the transfer matrix:
+\end{itemize}
+\vfill
+\hfil\includegraphics[width=1cm]{tl1.pdf}
+\hfil\includegraphics[width=1cm]{tl2.pdf}
+\hfil\includegraphics[width=1cm]{tl3.pdf}
+\hfil\includegraphics[width=1cm]{tl4.pdf}
+\hfil\includegraphics[width=1cm]{tl5.pdf}
+\begin{itemize}
+  \item Algebra generated by
+\end{itemize}
+\hfil\includegraphics[width=1cm]{tl1.pdf}
+\hfil\includegraphics[width=1cm]{tl2.pdf}
+\hfil\includegraphics[width=1cm]{tl3.pdf}
+\begin{itemize}
+  \item Applications to knot theory, the Jones polynomial, braids, 2D Ising, quantum groups...
+\end{itemize}
+\vfill
+\eject
+
+\title{Counting dimer coverings}
+\begin{itemize}
+  \item {\color{L67}\href{http://dx.doi.org/10.1063/1.1705163}{{\bf E.H. Lieb}, {\it Solution of the Dimer Problem by the Transfer Matrix Method}, Journal of Mathematical Physics, 1967}}.
+  \item Similar approach to Schultz-Mattis-Lieb: Transfer Matrix/Fermions.
+  \item What if there are monomers?
+\end{itemize}
+\hfil\includegraphics[width=3cm]{cover.pdf}
+\hfil\includegraphics[width=3cm]{mdcover.pdf}
+\vfill
+\eject
+
+\title{Monomer-Dimer}
+\begin{itemize}
+  \vskip-10pt
+  \item {\color{HL72}\href{http://dx.doi.org/10.1007/BF01877590}{{\bf O.J. Heilmann, E.H. Lieb}, {\it Theory of monomer-dimer systems}, Communications in Mathematical Physics, 1972}}.
+  \item Random configuration of dimers: ($z$: {\it monomer fugacity})
+  $$\frac{z^{\#\mathrm{monomers}}}{\Xi_G(z)}.$$
+  \vskip-10pt
+  \item Free energy:
+   $$f(z):=-\lim_{\mathrm{Vol}\to\infty}\frac1{\mathrm{Vol}}\log(\Xi_G(z)).$$
+  \item There is a {\it phase transition} when $f$ is singular.
+  \item Roots of $\Xi_G(z)$: {\it Lee-Yang} zeros.
+\end{itemize}
+\vfill
+\eject
+
+\title{Monomer-Dimer}
+\begin{itemize}
+  \item Recurrence relation:
+  $$\Xi_G(z)=z\Xi_{G\setminus\{i\}}+\sum_{j:(i,j)\in G}\Xi_{G\setminus\{i,j\}}(z).$$
+  \item The Lee-Yang zeros lie in a bounded subset of the imaginary axis.
+  \item This result was recently used to solve the Kadison-Singer problem
+  \href{https://arxiv.org/abs/1408.4421}{[Marcus, Spielman, Srivastava, 2014]}.
+  \item No phase transitions in the monomer-dimer model!
+\end{itemize}
+\vfill
+\eject
+
+\title{Dimers as particles}
+\vfill
+\hfil\includegraphics[width=6cm]{dimers.pdf}
+\vfill
+\eject
+
+\title{Interacting dimers}
+\vfill
+\hfil\includegraphics[width=6cm]{interaction.pdf}
+\vfill
+\eject
+
+\title{Heilmann-Lieb model}
+\vfill
+\begin{itemize}
+  \item {\color{HL79}\href{http://dx.doi.org/10.1007/BF01009518}{{\bf O.J. Heilmann, E.H. Lieb}, {\it Lattice models for liquid crystals}, Journal of Statistical Physics, 1979}}.
+  \item Long range orientational order: dimers are either mostly vertical or mostly horizontal (if the interaction is strong enough).
+  \item There is a phase transition!
+  \item Argument uses {\it reflection positivity} and a {\it chessboard estimate}.
+\end{itemize}
+\vfill
+\eject
+
+\title{Hubbard model}
+\begin{itemize}
+  \item {\color{L89}\href{http://dx.doi.org/10.1103/PhysRevLett.62.1201}{{\bf E.H. Lieb}, {\it Two Theorems on the Hubbard Model}, Physical Review Letters, 1989}}.
+  \item Electrons on a graph, with a local interaction.
+  \item If the interaction is repulsive, the graph is bipartite, and the number of electrons is equal to the number of vertices, then the spin of the ground state is
+  $$S=\frac12||B|-|A||$$
+  where $|B|$ and $|A|$ are the numbers of vertices on the $B$- and $A$-subgraphs.
+  \item Uses reflection positivity in {\it spin space}.
+\end{itemize}
+
+\title{Lieb lattice}
+\vfill
+\hfil\includegraphics[width=5cm]{lieb_lattice.pdf}
+$$|B|=2|A|$$
+\vfill
+\eject
+
+\title{Heilmann-Lieb model}
+\vfill
+\hfil\includegraphics[width=6cm]{interaction.pdf}
+\vfill
+\eject
+
+\title{Liquid crystals}
+\vfill
+\hfil\includegraphics[width=6cm]{nematic.png}
+\vfill
+\eject
+
+\title{Liquid crystals}
+\vfill
+\begin{itemize}
+  \item Orientational order {\it and} positional disorder.
+  \item Heilmann-Lieb: orientational order.
+  \item Conjecture: positional disorder.
+  \item Previous results:
+  \begin{itemize}
+    \item \href{http://dx.doi.org/10.1007/BF00535264}{[Bricmont, Kuroda, Lebowitz, 1984]}: hard needles in $\mathbb R^2$ with a {\it finite} number of orientations.
+    \item \href{http://dx.doi.org/10.1007/s10955-005-8085-8}{[Ioffe, Velenik, Zahradn\'\i k, 2006]}: hard rods in $\mathbb Z^2$ (variable length).
+    \item \href{http://dx.doi.org/10.1007/s00220-013-1767-1}{[Disertori, Giuliani, 2013]}: hard rods in $\mathbb Z^2$.
+  \end{itemize}
+\end{itemize}
+
+\title{Nematic phase in the Heilmann-Lieb model}
+\vfill
+\begin{itemize}
+  \item Proof of positional disorder: \href{https://arxiv.org/abs/1709.05297}{[Jauslin, Lieb, 2017]} (uses Pirogov-Sinai theory).
+  \item Correlations between the positions of the dimers decay exponentially.
+  \item The rate of the decay is strongly anisotropic: in a vertical phase, the correlation length is very large in the vertical direction, and small in the horizontal.
+\end{itemize}
+\vfill
+\eject
+
+\title{Pirogov-Sinai theory}
+\vfill
+\hfil\includegraphics[width=6cm]{dimer_contour.pdf}
+\vfill
+\eject
+
+\title{Summary}
+\vfill
+\hfil\includegraphics[width=5cm]{plan.pdf}
+\hfil\parbox[b]{6.5cm}{
+  \begin{itemize}
+    {\color{L67}\item [Li67] Dimers}
+    {\color{SML64}\item [SML64] 2D Ising}
+    {\color{TL71}\item [TL71] Temperley-Lieb algebras}
+    {\color{HL72}\item [HL72] Monomers and Dimers}
+    {\color{HL79}\item [HL79] Interacting Dimers}
+    {\color{L89}\item [Li89] Hubbard model}
+  \end{itemize}
+}
+\vfill
+\eject
+
+\title{Macbeth - act V scene 8}
+\vfill
+\hfil[...] Before my body\par
+\medskip
+\hfil I throw my warlike shield. {\color{red}Lay on, Macduff},\par
+\medskip
+\hfil And damn'd be him that first cries, `Hold, enough!'\par
+
+
+\end{document}
-- 
cgit v1.3.1

