SRQ seminar – October 12th, 2018
Notes from a talk in the SRQ series.
INI Seminar 20181012 Toninelli
Interacting dimer models (lecture 3)
Partition function
\(\displaystyle Z_{\Lambda, \lambda} = e^{W_{\Lambda, \lambda}
(\underline{0})},\)
\(\displaystyle e^{W_{\Lambda, \lambda} (A)} = \sum_M \left[ \prod_{e
\in M} t_e \right]
e^{\lambda W (M) + \sum_e A_e \mathbb{I}_{e \in
M}},\)
where \(A = \{ A_e \}_{e \in \text{edges of $\Lambda$}}\) is a source
variable to generate the multi–edge correlations.
Grassmann representation
\(\displaystyle e^{W_{\Lambda, \lambda = 0} (A)} = \int \mathrm{D} \psi
e^{S_A (\psi)}, \qquad
S_A (\psi) = - \sum_{\text{edges $e$}} E_e
e^{A_e}\)
with \(E_e = \psi^+_x \psi^-_y \mathcal{K} (e)\) for an edge \(e = (b_x,
w_y)\).
Proposition 1. For the
interaction
\(\displaystyle W (M) = \sum_{\text{even faces $f$}}
\left(
\mathbb{I}_{\text{\scriptsize{$\begin{array}{l}
\hline
f\\
\hline
\end{array}$}}}
+\mathbb{I}_{\text{\scriptsize{$\begin{array}{|l|}
f
\end{array}$}}}
\right)\)
we have
\(\displaystyle e^{W_{\Lambda, \lambda} (A)} = \int \mathrm{D} \psi
e^{S_A (\psi) + \alpha
\sum_{\gamma} \prod_{e \in \gamma} E_e
e^{A_e}},\)
where \(\alpha = e^{\lambda} - 1\).
Proof. For simplicity let us fix \(A = \underline{0}\).
\(\displaystyle Z_{\Lambda, \lambda} = \sum_{\text{matchings $M$}} p
(M)
\underbrace{\prod_{\text{even $f$}} \left( 1 +
\alpha
\mathbb{I}_{\text{\scriptsize{$\begin{array}{|l|}
f
\end{array}$}}} \right) \left( 1 +
\alpha
\mathbb{I}_{\text{\scriptsize{$\begin{array}{l}
\hline
f\\
\hline
\end{array}$}}} \right)}_{\prod_{\gamma = \{ e, e' \}} (1 +
\alpha
\mathbb{I}_{e \in M} \mathbb{I}_{e' \in M})}\)
expanding the product we obtain
\(\displaystyle = \sum_{\text{matchings $M$}} p (M) \sum_{n \geqslant 0}
\sum_{\{ \gamma_1,
\ldots, \gamma_n \}} \xi (\gamma_1) \cdots \xi
(\gamma_n)\)
where \(\xi (\gamma) = \alpha \mathbb{I}_{e \in M} \mathbb{I}_{e' \in
M}\) for \(\gamma = \{ e, e' \}\). The various \(\gamma\)s do not shar
edges even if they can share vertices (this do not cause problems). All
these terms are of the type
\(\displaystyle \sum_M p (M) \mathbb{I}_{e_1 \in M} \cdots
\mathbb{I}_{e_{2 n} \in M} .\)
These expressions can be obtained by the generating function as
\(\displaystyle = \partial_{A_1} \ldots \partial_{A_n} e^{W_{\Lambda,
\lambda = 0} (A)} |_{A
= 0},\)
and since we have already an expression for this we get
\(\displaystyle = \int \mathrm{D} \psi e^{S_{A = 0} (\psi)} E_{e_1}
\cdots E_{e_{2 n}},\)
now we have just to resum to obtain the formula in the statement.\(\Box\)
Quasi–particle fields
A naive expansion of the partition functions in powers of \(\alpha\) is
divergent, so the point is to have a clever way to resum the naive
expansion in a suitable way.
We are going to make a change of variables for the Grassmann fields
introducing quasi–particle fields
\(\displaystyle g (x, y) =\mathcal{E}_0 (\psi^-_x \psi^+_y) = \int
\frac{\mathrm{d} p}{(2
\pi)^2} \frac{e^{i p \cdot (x - y)}}{\mu (p)}
(\chi^+ (p) + \chi^- (p)) =:
g_+ (x, y) + (- 1)^{x - y} g_- (x, y)\)
where \(\chi^{\pm}\) are functions concentrated near the singularities,
\(\chi^+\) around \((0, 0)\) and \(\chi^-\) is around \((\pi, \pi)\) and
they form a partition of unity: \(\chi^+ (p) + \chi^- (p) = 1\). Then
\(\displaystyle g_{\omega} (x, y) \approx \frac{1}{\phi_{\omega} (x, y)}
\qquad \text{for $| x
- y | \rightarrow \infty$}\)
where \(\phi_+ (x) = ((- i - 1) x_1 + (i + 1) x_2)\) and \(\phi_- (x) =
- \phi_+^{\ast} (x)\).
In terms of fields this splitting can be interpreted as \(\psi^{\pm}_{x,
\pm} = \psi^{\pm}_{x, +} + (- 1)^{(x_1 + x_2)} \psi^{\pm}_{x,
-}\) and
we use the addition principle for Grassmann variables. If
\(\mathcal{E}_g\) is an expectation for a field \(\psi\) which satisfies
the Wick rule for propagator \(g\) and we split \(g = g_1 + g_2\) then
we can split \(\psi = \psi_1 + \psi_2\) such that the first has
propagator \(g_1\), and \(\psi_2\) with propagator \(g_2\) and the mixed
propagator is \(0\).
\(\displaystyle \mathcal{E}_g (f (\psi))
=\mathcal{E}_{\text{\scriptsize{$\left(
\begin{array}{ll}
g_1 & 0\\
0 & g_2
\end{array} \right)$}}} (f (\psi_1 + \psi_2)) .\)
This is analogous to the same property for Gaussians. You can verify the
identity with \(f (\psi)\) being a product of two.
Suppose we want to compute the partition function
\(\displaystyle \log Z_{\Lambda, \lambda} = \log \int \mathrm{D} \psi
e^{S (\psi) + \alpha V
(\psi)},\)
with \(V (\psi) = \sum_{\gamma = (e, e')} E_e E_{e'}\) and \(\psi = \{
\psi^+_x, \psi^-_x \}_{x \in \Lambda} = \{ \psi^+_{x,
\omega},
\psi^-_{x, \omega} \}_{x \in \Lambda, \omega = \pm}\) and
\(\alpha = e^{\lambda - 1}\). By Taylor expansion (without worrying
about convergence) one gets:
\(\displaystyle \log \frac{Z_{\Lambda, \lambda}}{Z_{\Lambda, 0}} =
\sum_{n \geqslant 1}
\frac{1}{n!} \alpha^n \mathcal{E}_0 (\underbrace{V
; V ; \cdots ;
V}_{\text{$n$ times}}),\)
where
\(\displaystyle \mathcal{E}_0 (\underbrace{V ; V ; \cdots ;
V}_{\text{$n$ times}}) =
\partial_{\lambda}^n \log \mathcal{E}_0
(e^{\lambda V}) |_{\lambda = 0} .\)
Fact: \(\mathcal{E}_0 (V ; \cdots ; V)\) is computed by
performing contractions in Wick's rule for which all the blocks relating
to different \(V\)'s form a connected set of blocks. (i.e. one consider
the graph where edges are the Wick contractions and vertices are the set
of fields belonging to the same \(V\)).
Remark 2. As long as the volume
is finite all these expressions are well–defined and the expansion
is convergent. Indeed at some point fields start repeating and they give
zero. So the series is just a finite polynomial. And for \(| \alpha |
\ll 1\) one can take the log and everything is ok.
Remark 3. Single terms of this
series can give problems as \(\Lambda \rightarrow \infty\). Then there
is the problem of the convergence of the series.
For graphs of the form
\(\displaystyle
\text{\raisebox{-0.5\height}{\includegraphics[width=14.8741637150728cm,height=8.92450478814115cm]{image-1.pdf}}}
=
| \Lambda | \int \frac{\mathrm{d} p}{(2 \pi)^2} \left[ \frac{1}{\mu
(p)}
\right]^n f (p)^n\)
and \(f (p) \approx 1\) so there are singularities in the denominator as
\(n\) is not small. Here \(f (p)\) is the contribution from the subgraph
\(\displaystyle f (p)
=
\text{\raisebox{-0.5\height}{\includegraphics[width=14.8741637150728cm,height=8.92450478814115cm]{image-1.pdf}}}\)
The idea is to break the propagator in dyadic pieces to localize the
scale of the fluctuations and analyse better the behaviour near the
singularity:
\(\displaystyle g_+ (x) = \int \frac{\mathrm{d} p}{(2 \pi)^2} \frac{e^{i
p \cdot x}}{\mu (p)}
\chi_+ (p)\)
Let \(\chi_h (p) = \chi (2^{- h} p)\) and write
\(\displaystyle \chi_+ (p) = \underbrace{\chi_+ (p) - \chi_{- 1}
(p)}_{f_0 (p)} +
\underbrace{\chi_{- 1} (p) - \chi_{- 2} (p)}_{f_1 (p)}
+ \underbrace{\chi_{-
2} (p) - \chi_{- 3} (p)}_{f_2 (p)} + \cdots +
\chi_{h_L} (p)\)
where \(f_h\) is concentrated on disks where \(| p | \approx 2^h\) and
\(h \leqslant 0\). \(h_L = - \log_2 (L)\). We have now
\(\displaystyle g_+ (x) = \sum_{h_L + 1}^0 g_+^{(h)} (x) +
g_+^{(\leqslant h_L)} (x)\)
and for fixed \(h\) each \(g_+^{(h)} (x)\) decay very fast since they
are Fourier transforms of nice functions.
Proposition 4. For \(h
\leqslant 0\),
\(\displaystyle | g_+^{(h)} (x) | \leqslant C 2^h e^{- \sqrt{2^h | x
|}} .\)
Multi–scale expansion
According to this decomposition of the propagator we have a multiscale
expansion of the fermionic fields as:
\(\displaystyle \psi^{\pm}_{x, \omega} = \sum_{h = h_L}^0 \psi^{\pm,
(h)}_{x, \omega} .\)
\(\displaystyle Z =\mathcal{E} (e^{\alpha V (\psi)}) =\mathcal{E}
(e^{\alpha V (\psi^{(0)} +
\psi^{(- 1)} + \psi^{(- 2)} + \cdots)})
=\mathcal{E}_{h_L} \cdots
\mathcal{E}_0 \left( e^{\alpha V (\psi^{(0)} +
\psi^{(- 1)} + \psi^{(- 2)} +
\cdots + \psi^{(h_L)})} \right)\)
so we start by integrating out the scale \(\psi^{(0)}\) and going
forward. If we let \(V^{(0)} = V\) and recursively \(V^{(- 1)}\) as the
result of the integration
\(\displaystyle e^{\alpha V^{(- 1)} (\psi^{(- 1)} + \psi^{(- 2)} +
\cdots + \psi^{(h_L)})} :=
\mathcal{E}_0 \left( e^{\alpha V (\psi^{(0)}
+ \psi^{(- 1)} + \psi^{(- 2)} +
\cdots + \psi^{(h_L)})} \right)\)
obtaining the formula
\(\displaystyle V^{(- 1)} (\psi^{(\leqslant - 1)}) = \sum_n
\frac{\alpha^n}{n!} \mathcal{E}_0
(V^{(0)} (\psi^{(0)} +
\psi^{(\leqslant - 1)}) ; \cdots ; V^{(0)} (\psi^{(0)}
+
\psi^{(\leqslant - 1)})) .\)
Going on to expand and introducing proper notations to label the results
of the integrations as polynomials in the remaning variables one
obtains:
\(\displaystyle V^{(- 1)} (\psi^{(\leqslant - 1)}) = \sum_{n \geqslant
1} \frac{\alpha^n}{n!}
\sum_{\tau \in \mathcal{T}^{(- 1)}_n}
\sum_{P_{v_0} = \varnothing}
\mathcal{E}_0 (\psi^{(0)} (P_{v_1}) ;
\cdots ; \psi^{(0)} (P_{v_n}))\)
\(\displaystyle + \sum_{n \geqslant 1} \frac{\alpha^n}{n!} \sum_{\tau
\in \mathcal{T}^{(-
1)}_n} \sum_{P_{v_0} \neq \varnothing} \mathcal{E}_0
(\psi^{(0)} (P_{v_1}
\backslash Q_{v_1}) ; \cdots ; \psi^{(0)} (P_{v_n}
\backslash Q_{v_n}))
[\psi^{(\leqslant - 1)} (Q_{v_1}) \cdots
\psi^{(\leqslant - 1)} (Q_{v_n})]\)
\(P_{v_k}\) is the collection of indexes of the fields appearing in the
\(k\)-the leaf of the vertex \(v\)...
[but this part I didn't managed to follow...]