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
|
\documentclass{kiss}
\usepackage{presentation}
\usepackage{header}
\usepackage{toolbox}
\begin{document}
\pagestyle{empty}
\hbox{}\vfil
\bf\Large
\hfil Ground state construction\par
\smallskip
\hfil of Bilayer Graphene\par
\vfil
\large
\hfil Ian Jauslin
\normalsize
\vfil
\hfil\rm joint with {\bf Alessandro Giuliani}\par
\vfil
arXiv:{\tt \href{http://arxiv.org/abs/1507.06024}{1507.06024}}\hfill{\tt \href{http://ian.jauslin.org}{http://ian.jauslin.org}}
\eject
\setcounter{page}1
\pagestyle{plain}
\title{Monolayer graphene}
\begin{itemize}
\item 2D crystal of carbon atoms on a honeycomb lattice.
\end{itemize}
\vfill
\hfil\hskip-25pt\includegraphics[height=150pt]{figs/monolayer.pdf}\par
\eject
\title{Bilayer graphene}
\begin{itemize}
\item 2 graphene layers in {\it AB} stacking.
\end{itemize}
\vfill
\hskip12pt\includegraphics[height=150pt]{figs/basegrid.pdf}\par
\eject
\title{Bilayer graphene}
\begin{itemize}
\item Rhombic lattice $\Lambda\equiv\mathbb Z^2$, 4 atoms per site.
\end{itemize}
\vfill
\hskip12pt\includegraphics[height=150pt]{figs/cellgraph.pdf}\par
\eject
\title{Hamiltonian}
\begin{itemize}
\item Hamiltonian:
$$
\mathcal H=\mathcal H_0+UV
$$
\item Non-interacting Hamiltonian: hoppings
\end{itemize}
\vfill
\hfil\includegraphics[width=\textwidth]{figs/hoppings4.pdf}\par
\vfill
\begin{itemize}
\item Interaction: weak, short-range (screened Coulomb).
\end{itemize}
\eject
\title{Non-interacting Hamiltonian}
\vskip-10pt
$$
\mathcal H_0=\sum_{k\in\hat\Lambda}
\left(\begin{array}c
\hat a_k^\dagger\\
\hat{\tilde b}_k^\dagger\\
\hat{\tilde a}_k^\dagger\\
\hat b_k^\dagger
\end{array}\right)^T
\hat H_0(k)
\left(\begin{array}c
\hat a_k\\
\hat{\tilde b}_k\\
\hat{\tilde a}_k\\
\hat b_k
\end{array}\right)
$$
\vfill
$$
\kern-10pt
\hat H_0(k):=
\left(\begin{array}{*{4}{c}}
0&\gamma_1&0&\gamma_0\Omega^*(k)\\
\gamma_1&0&\gamma_0\Omega(k)&0\\
0&\gamma_0\Omega^*(k)&0&\gamma_3\Omega(k)e^{3ik_x}\\
\gamma_0\Omega(k)&0&\gamma_3\Omega(k)e^{-3ik_x}
\end{array}\right)
$$
\vfill
$$
\Omega(k):=1+2e^{-\frac32ik_x}\cos({\textstyle\frac{\sqrt3}2}k_y)
$$
\eject
\title{Non-interacting Hamiltonian}
\vfill
\begin{itemize}
\item Hopping strengths:
$$
\gamma_0=1,\quad
\gamma_1=\epsilon,\quad
\gamma_3=0.33\times\epsilon
$$
\item Experimental value $\epsilon\approx0.1$, here, $\epsilon\ll1$.
\end{itemize}
\vfill
\eject
\title{Interaction}
$$
V=\sum_{x,y}v(|x-y|)\left(n_x-\frac12\right)\left(n_y-\frac12\right)
$$
\begin{itemize}
\item $\displaystyle\sum_{x,y}$: sum over pairs of atoms
\item $v(|x-y|)\leqslant e^{-c|x-y|}$, $c>0$
\item $-\frac12$: {\it half-filling}.
\end{itemize}
\eject
\title{Non-interacting Hamiltonian}
\begin{itemize}
\item Eigenvalues of $\hat H_0(k)$:
\end{itemize}
\vfill
\hfil\includegraphics[width=\textwidth]{figs/global_nog4D.pdf}\par
\vfill
\eject
\title{Non-interacting Hamiltonian}
\begin{itemize}
\item $|k|\gg\epsilon$
\end{itemize}
\vfill
\hfil\includegraphics[width=\textwidth]{figs/first_nog4D.pdf}\par
\vfill
\eject
\title{Non-interacting Hamiltonian}
\begin{itemize}
\item $\epsilon^2\ll|k|\ll\epsilon$
\end{itemize}
\vfill
\hfil\includegraphics[width=\textwidth]{figs/second_nog4D.pdf}\par
\vfill
\eject
\title{Non-interacting Hamiltonian}
\begin{itemize}
\item $|k|\ll\epsilon^2$
\end{itemize}
\vfill
\hfil\includegraphics[width=200pt]{figs/third_nog4D.pdf}\par
\vfill
\eject
\title{Theorem}
$\exists U_0,\epsilon_0>0$, independent, such that, for $\epsilon<\epsilon_0$, $|U|<U_0$,
\begin{itemize}
\item the free energy
$$
f:=-\frac1{|\Lambda|\beta}\log\mathrm Tr(e^{-\beta \mathcal H})
$$
is analytic in $U$, uniformly in $\beta$ and $|\Lambda|$,
\item the two-point Schwinger function
$$
s_2(x-y):=\frac{\mathrm Tr(e^{-\beta \mathcal H}a_xa_y^\dagger)}{\mathrm Tr(e^{-\beta \mathcal H})}
$$
is analytic in $U$, uniformly in $\beta$ and $|\Lambda|$.
\end{itemize}
\vfill
\eject
\title{Renormalization group flow}
\vfill
\includegraphics[width=250pt]{figs/flow.pdf}
\end{document}
|
