\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}
