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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} |