Ian Jauslin
summaryrefslogtreecommitdiff
blob: ab64e58269dc37c036a00e36834c924f4ec7d88b (plain)
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
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
\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{History}
\begin{itemize}
  \item \href{http://dx.doi.org/10.1111/j.1749-6632.1949.tb27296.x}{[Onsager, 1949]}: mean field model for hard needles in $\mathbb R^3$.
  \item \href{http://dx.doi.org/10.1007/BF01009518}{[Heilmann, Lieb, 1979]}: interacting dimers.
\vphantom{
  \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}
\vfill
\eject


\title{Heilmann-Lieb model}
\hfil\href{http://dx.doi.org/10.1007/BF01009518}{[Heilmann, Lieb, 1979]}
\vfil
\hfil\includegraphics[width=5cm]{grid.pdf}
\vfil\eject

\addtocounter{page}{-1}
\title{Heilmann-Lieb model}
\hfil\href{http://dx.doi.org/10.1007/BF01009518}{[Heilmann, Lieb, 1979]}
\vfil
\hfil\includegraphics[width=5cm]{dimers.pdf}
\vfil\eject

\addtocounter{page}{-1}
\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 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.
\vphantom{
  \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{History}
\begin{itemize}
  \item \href{http://dx.doi.org/10.1111/j.1749-6632.1949.tb27296.x}{[Onsager, 1949]}: mean field model for hard needles in $\mathbb R^3$.
  \item \href{http://dx.doi.org/10.1007/BF01009518}{[Heilmann, Lieb, 1979]}: interacting dimers.
  \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}
\vfill
\eject

\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/s10955-015-1421-8}{[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{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{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
  $$
    \begin{array}{c}
      \left<\mathds 1_{e_{\mathrm h}}\right>_{\mathrm v}=O(e^{-3J})
      \\[0.3cm]
      \left<\mathds 1_{e_{\mathrm h}}\mathds 1_{f_{\mathrm h}}\right>_{\mathrm v}
      -\left<\mathds 1_{e_{\mathrm h}}\right>_{\mathrm v}
      \left<\mathds 1_{f_{\mathrm h}}\right>_{\mathrm v}
      =O(e^{-6J-c\ \mathrm{dist}_{\mathrm{HL}}(e_{\mathrm h},f_{\mathrm h})})
    \end{array}
  $$
\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}