1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
|
\documentclass{ian-presentation}
\usepackage[hidelinks]{hyperref}
\usepackage{graphicx}
\usepackage{dsfont}
\begin{document}
\pagestyle{empty}
\hbox{}\vfil
\bf\Large
\hfil Nematic liquid crystal phase\par
\smallskip
\hfil in a system of interacting dimers\par
\vfil
\large
\hfil Ian Jauslin
\normalsize
\vfil
\hfil\rm joint with {\bf Elliott H. Lieb}\par
\vfil
arXiv: {\tt \href{http://arxiv.org/abs/1709.05297}{1709.05297}}
\hfill{\tt \href{http://ian.jauslin.org}{http://ian.jauslin.org}}
\eject
\setcounter{page}1
\pagestyle{plain}
\title{Gas-liquid-crystal}
\vfill
\hfil
\includegraphics[width=3cm]{gas.png}
\includegraphics[width=3cm]{liquid.png}
\includegraphics[width=3cm]{crystal.png}
\vfill
\eject
\title{Liquid crystals}
\begin{itemize}
\item Orientational order and positional disorder.
\end{itemize}
\hfil\includegraphics[width=4.5cm]{nematic.png}
\hfil\includegraphics[width=4.5cm]{chiral.png}
\vfill
\eject
\title{Heilmann-Lieb model}
\hfil\href{http://dx.doi.org/10.1007/BF01009518}{[Heilmann, Lieb, 1979]}
\vfil
\hfil\includegraphics[width=5cm]{interaction.pdf}
\vfil\eject
\title{Heilmann-Lieb model}
\begin{itemize}
\item Grand-canonical Gibbs measure:
$$
\left<A\right>_{\mathrm v}
:=
\lim_{\Lambda\to\mathbb Z^2}
\frac1{\Xi_{\Lambda,\mathrm v}(z)}
\sum_{\underline\delta\in\Omega_{\mathrm v}(\Lambda)}A(\underline\delta)z^{|\underline\delta|}\prod_{\delta\neq \delta'\in \underline\delta}e^{\frac12J\mathds 1_{\delta\sim\delta'}}
$$
\vskip-15pt
\begin{itemize}
\item $\Lambda$: finite box.
\item $\Omega_{\mathrm v}(\Lambda)$: non-overlapping dimer configurations satisfying the boundary condition.
\item $z\geqslant 0$: fugacity.
\item $J\geqslant 0$: interaction strength.
\item $\mathds 1_{\delta\sim\delta'}$ indicator that dimers are adjacent and aligned.
\end{itemize}
\end{itemize}
\vfill
\eject
\title{Boundary condition}
\begin{itemize}
\item Fix length $\ell_0:=e^{\frac32J}\sqrt z$,
\end{itemize}
\hfil\includegraphics[width=5cm]{boundary.pdf}
\title{Heilmann-Lieb conjecture}
\begin{itemize}
\item \href{http://dx.doi.org/10.1007/BF01009518}{[Heilmann, Lieb, 1979]}: proved orientational order using reflection positivity.
\item HL Conjecture: absence of positional order.
\item \href{http://dx.doi.org/10.1007/s10955-005-8085-8}{[Ioffe, Velenik, Zahradn\'\i k, 2006]}, \href{http://dx.doi.org/10.1007/s00220-013-1767-1}{[Disertori, Giuliani, 2013]}: nematic liquid crystal phase in systems of hard rods on $\mathbb Z^2$.
\item \href{http://dx.doi.org/10.1007/s10955-015-1421-8}{[Alberici, 2016]}: different fugacities for horizontal and vertical dimers.
\item \href{http://dx.doi.org/10.1103/PhysRevB.89.035128}{[Papanikolaou, Charrier, Fradkin, 2014]}: numerics.
\end{itemize}
\vfill
\eject
\title{Theorem}
For $1\ll z\ll J$, $\|(x,y)\|_{\mathrm{HL}}:=J|x|+e^{-\frac32J}z^{-\frac12}|y|$,
\begin{itemize}
\item Given two vertical edges $e_{\mathrm v},f_{\mathrm v}$, $\left<\mathds 1_{e_{\mathrm v}}\right>_{\mathrm v}$ is {\it independent} of $e_{\mathrm v}$ and
$$
\begin{array}{c}
\left<\mathds 1_{e_{\mathrm v}}\right>_{\mathrm v}=\frac12(1+O(e^{-\frac12J}z^{-\frac12}))
\\[0.3cm]
\left<\mathds 1_{e_{\mathrm v}}\mathds 1_{f_{\mathrm v}}\right>_{\mathrm v}
-\left<\mathds 1_{e_{\mathrm v}}\right>_{\mathrm v}
\left<\mathds 1_{f_{\mathrm v}}\right>_{\mathrm v}
=O(e^{-c\ \mathrm{dist}_{\mathrm{HL}}(e_{\mathrm v},f_{\mathrm v})})
\end{array}
$$
\vskip-5pt
\item Given two horizontal edges $e_{\mathrm h},f_{\mathrm h}$, $\left<\mathds 1_{e_{\mathrm h}}\right>_{\mathrm v}$ is {\it independent} of $e_{\mathrm h}$ and
$$
\begin{array}{c}
\left<\mathds 1_{e_{\mathrm h}}\right>_{\mathrm v}=O(e^{-3J})
\\[0.3cm]
\left<\mathds 1_{e_{\mathrm h}}\mathds 1_{f_{\mathrm v}}\right>_{\mathrm v}
-\left<\mathds 1_{e_{\mathrm h}}\right>_{\mathrm v}
\left<\mathds 1_{f_{\mathrm h}}\right>_{\mathrm v}
=O(e^{-3J-c\ \mathrm{dist}_{\mathrm{HL}}(e_{\mathrm v},f_{\mathrm v})})
\end{array}
$$
\end{itemize}
\vfill
\eject
\title{1D system}
\begin{itemize}
\item {\it Only} vertical dimers: integrable.
\item Given two vertical edges $e_{\mathrm v},f_{\mathrm v}$, $\left<\mathds 1_{e_{\mathrm v}}\right>_{\mathrm v}$ is {\it independent} of $e_{\mathrm v}$ and
$$
\begin{array}{c}
\left<\mathds 1_{e_{\mathrm v}}\right>_{\mathrm v}=\frac12(1+O(e^{-\frac12J}z^{-\frac12}))
\\[0.3cm]
\left<\mathds 1_{e_{\mathrm v}}\mathds 1_{f_{\mathrm v}}\right>_{\mathrm v}
-\left<\mathds 1_{e_{\mathrm v}}\right>_{\mathrm v}
\left<\mathds 1_{f_{\mathrm v}}\right>_{\mathrm v}
=O(e^{-c\ \mathrm{dist}_{\mathrm{1D}}(e_{\mathrm v},f_{\mathrm v})})
\end{array}
$$
with $\|(x,y)\|_{\mathrm{1D}}:=e^{-\frac32J}z^{-\frac12}|y|$.
\end{itemize}
\vfill
\eject
\title{Loop model}
\vfill
\hfil\includegraphics[width=5cm]{dimer_contour.pdf}
\begin{itemize}
\item Weight of a loop of length $|l|$: $e^{-\frac12J|l|}$.
\end{itemize}
\vfill
\eject
\title{Difficulty: loops interact}
\vskip-10pt
\begin{itemize}
\item Correlated dimers induce an interaction between loops, which decays exponentially with a rate $e^{-\frac32J}z^{-\frac12}$.
\end{itemize}
\hfil\includegraphics[width=3.5cm]{segments.pdf}
\begin{itemize}
\item Vertical-to-horizontal boundaries and horizontal-to-vertical ones have different geometries.
\end{itemize}
\end{document}
|
