SRQ seminar – October 8th, 2018

Notes from a talk in the SRQ series.

INI Seminar 20181010 Toninelli (revised version 20181012)

Interacting dimer model

$M$ matchings. $Π_{L, \underset{―}{t}} \propto \prod_{e \in M} t_{e}$ the non-interacting measure has an infinite volume limit $Π_{\underset{―}{t}}$ which is determinantal. Under the limit measure $Π_{\underset{―}{t}}$ behaves like the massless Gaussian free field. It is true when $\underset{―}{t} = (1, 1, 1, 1)$ and in an open region of weights.

One can consider $φ \in C_{c}^{\infty}$ and $\int φ = 0$ and $ε^{2} \sum_{x} h (x) φ (ε x) \xRightarrow d N (0, \int d x d y φ (x) φ (y) G (x - y))$ when $\underset{―}{t} = (1, 1, 1, 1)$ .

Propagator of the non-interacting model

g (x, y) = \int_{[- π, π]^{2}} \frac{d p}{(2 π)^{2}} \frac{e^{i p (x - y)}}{μ (p)},

with $μ (p) = t_{1} + i t_{2} e^{i p_{1}} + \dots$ .

G (x) = - \frac{1}{2 π^{2}} \log | ϕ_{+} (x) |

where $ϕ_{+}$ is related to the gradient of $μ$ at its two simple zeros $p^{+}$ and $p^{-}$ :

ϕ_{+} (x) = (β_{+} x_{1} - α_{+} x_{2}), β_{+} = \partial_{p_{2}} μ (p^{+}), α_{+} = \partial_{p_{1}} μ (p^{+}) .

and $ϕ_{+}^{*} = - ϕ_{-}$ . When $\underset{―}{t} = (1, 1, 1, 1)$ we have $p^{+} = (0, 0)$ and $p^{-} = (π, π)$ .

Interacting model

Π_{L, \underset{―}{t}, λ} (M) \propto (\prod_{e \in M} t_{e}) \exp (λ W (M))

where $W (M)$ is a local function of dimer configuration summed over all translations.

Model 1

Model 2

W (M) = \sum_{f \in all faces} (I_{\scriptsize{\begin{array}{l} f \end{array}}} + I_{\scriptsize{\begin{array}{l} f \end{array}}})

W (M) = \sum_{f \in even faces} (I_{\scriptsize{\begin{array}{l} f \end{array}}} + I_{\scriptsize{\begin{array}{l} f \end{array}}})

(which is linked to the $6$ -vertex models with $Δ = 1 - e^{λ}$ ) where we say (arbitrarily) that even faces are those having black vertices on the upper left and lower right corners.

We ask $λ \in R$ being (very) small and $W$ to be local.

Recall the definition of $K_{r}$ depending on the type of edge:

Theorem 1. [AIHP '15] $\underset{―}{t} = (1, 1, 1, 1)$ and $| λ | ⩽ λ_{0}$ then

$Π_{L, \underset{―}{t}, λ} \to Π_{λ}$

If $e$ is an edge of type $r = 1, 2, 3, 4$ and $b (e) = x$ , and $e^{'}$ is of type $r^{'}$ and $b (e^{'}) = x^{'}$ then
$Π_{λ} (I_{e}; I_{e^{'}}) = \frac{A (λ)}{2 π^{2}} Re [\frac{K_{r^{'}} K_{r}}{ϕ_{+} (x - x^{'})^{2}}] +$ $+ (- 1)^{x_{1} + x_{2} - (x_{1}^{'} + x_{2}^{'})} \frac{C_{r, r^{'}}}{4 π^{2}} B (λ) \frac{1}{| ϕ_{+} (x - x^{'}) |^{2 ν}} + O (\frac{1}{| \dots |^{2 + θ}})$
as $| x - x^{'} | \to \infty$ , where $ν = ν (λ)$ and $A, B$ are analytic functions of $λ$ .

The height field converges to a log correlated Gaussian field, for example we have, under $Π_{λ}$ ,
$Var [\frac{h (f^{'}) - h (f)}{(\log | f - f^{'} |)^{1 / 2}}] \to \frac{1}{π^{2}} A (λ), as | f - f^{'} | \to \infty,$
and in the same limit
$\frac{h (f^{'}) - h (f)}{(Var (\dots))^{1 / 2}} \to N (0, 1) .$
Note that the oscillating term disappear in the variance.

Remark 2. In Model 1, we worked out $ν, A$ which turns out to be

ν (λ) = 1 - \frac{4}{π} λ + O (λ^{2}), A (λ) = \dots

and $ν (λ)$ depends non–trivially on the weight $\underset{―}{t}$ but recall that $ν (0) = 1$ for any $\underset{―}{t}$ .

Theorem 3. (J.Stat.Mech.) It holds (analog of Haldane's relation for Luttinger liquids)

$ν (λ) = A (λ)$

for any $| λ | ⩽ λ_{0}$ . (Proven for $\underset{―}{t} = (1, 1, 1, 1)$ , but work in progress for more general parameters).

Now we want to show how to represent the correlation functions and partition function as a Grassmann integral.

Kasteleyn theory

$G \subset Z^{2}$ bipartite, admitting perfect matchings like a box $2 m \times 2 n$ .

Z_{G, \underset{―}{t}} = | det K |

where the matrix $K$ has row indexed by black and colums indexed by white sites and

K (b, w) = {\begin{cases} 0 & if b ≁ w \\ K_{r} t_{r} & if b \sim w and the edgetype is r \end{cases}

On the torus one need to modify the method. One need to define other three matrices $K_{θ_{1}, θ_{2}}$ obtained by multiplying the edges which go out of the torus in direction $e_{1}$ by $(- 1)^{θ_{1}}$ and in direction $e_{2}$ by $(- 1)^{θ_{2}}$ , then

Z_{G, \underset{―}{t}} = \frac{1}{2} \sum_{θ_{1}, θ_{2} = \pm 1} c_{θ_{1}, θ_{2}} det K_{θ_{1}, θ_{2}}

for certain coefficients $c_{θ_{1}, θ_{2}}$ .

From the formula for the partition function one can derive all the formulas for edge edge correlations.

I will pretend from now on that the partition function is given by a single determinant, namely

Z_{L, \underset{―}{t}} = det K_{1, 1}

because $K_{1, 1}$ is invertible. Note that $K_{0, 0} = 0$ . These matrices are translation invariant so the can be diagonalized in Fourier basis:

f_{p} (w_{x}) = \frac{1}{L} e^{- i p \cdot x}

for $p \in D = {(p_{1}, p_{2}) : p_{i} = \frac{2 π}{L} (n_{i} + \frac{1}{2}), 0 ⩽ n_{i} ⩽ L - 1}$ and then

\sum_{w} K (b, w) f_{p} (w) = f_{p} (b) μ (p) .

Therefore from this is easy to deduce that the free energy is

F (\underset{―}{t}) = lim_{L \to \infty} \frac{1}{L^{2}} \log Z_{L, \underset{―}{t}} = \int \frac{d p}{(2 π)^{2}} \log μ (p)

and

K^{- 1} (w_{x}, b_{y}) = \frac{1}{L^{2}} \sum_{p \in D} \frac{e^{- i p \cdot (x - y)}}{μ (p)} = g (x, y) .

We assign Grassmann variables ${ψ_{x}^{+}, ψ_{x}^{-}}_{x \in T_{L}}$ where $ψ_{x}^{+}$ is on black vertex $b_{x}$ and $ψ_{x}^{-}$ on the white vertex $w_{x}$ . Define the integral of a non–commuting polynomial of the ${ψ_{x}^{+}, ψ_{x}^{-}}_{x \in T_{L}}$

\int D ψ \prod_{x \in Λ} ψ_{x}^{-} ψ_{x}^{+} = 1,

and the integral change sign when we exchange two variables (so variables cannot appear more than linearly), moreover when any variable is missing the integral is zero.

For example

\int D ψ e^{ψ_{1} ψ_{2}} = \int D ψ (1 + ψ_{1} ψ_{2}) .

A few consequences:

we can rewrite the determinant of a matrix as a Grassmann integral. If $A$ is an $n \times n$ matrix, then
$det A = \int D ψ e^{- \sum_{x, y} A_{x, y} ψ_{x}^{+} ψ_{y}^{-}} .$
Which recall the Gaussian integral
$(det Σ)^{1 / 2} = \int \frac{d x_{1} \dots d x_{n}}{(2 π)^{n / 2}} e^{- (x, Σ^{- 1} x) / 2} .$
Moreover
$A^{- 1} (x, y) = \frac{\int D ψ e^{- \sum_{x, y} A_{x, y} ψ_{x}^{+} ψ_{y}^{-}} ψ_{x}^{-} ψ_{y}^{+}}{\int D ψ e^{- \sum_{x, y} A_{x, y} ψ_{x}^{+} ψ_{y}^{-}}}$
again similar to the Gaussian formula for the covariance.
Fermionic Wick's rule. Let us denote (with $A^{- 1} = G$ )
$E_{G} (f) = \frac{\int D ψ e^{- \sum_{x, y} G_{x, y}^{- 1} ψ_{x}^{+} ψ_{y}^{-}} f (ψ)}{\int D ψ e^{- \sum_{x, y} G_{x, y}^{- 1} ψ_{x}^{+} ψ_{y}^{-}}}$
then
$E_{G} (ψ_{x_{1}}^{-} ψ_{y_{1}}^{+} \dots ψ_{x_{n}}^{-} ψ_{x_{n}}^{+}) = det [G_{n} (\underset{―}{x}, \underset{―}{y})]$
where we introduce the $n \times n$ matrix $G_{n} (\underset{―}{x}, \underset{―}{y})$ as $G_{n} (\underset{―}{x}, \underset{―}{y})_{i, j} = G (x_{i}, y_{j})$ for $\underset{―}{x} = (x_{1}, \dots, x_{n})$ and $\underset{―}{y} = (y_{1}, \dots, y_{n})$ and $i, j = 1, \dots, n$ .

For the dimer model we obtain

Z = det K_{1, 1} = \int D ψ e^{S (ψ)}, S (ψ) = - \sum_{x, y} K_{1, 1} (x, y) ψ_{x}^{+} ψ_{y}^{-} .

The partition function of the interacting model can be also expressed as a fermionic integral. Define

e^{W_{Λ} (A)} = \sum_{M} w (M) e^{\sum_{edges} A_{e} I_{e \in M}}

where $A_{e} \in R$ . This is equivalent to the replacement $t_{e} \to t_{e} e^{A_{e}}$ therefore we consider

e^{W_{Λ} (A)} = \int D ψ e^{- \sum_{x, y} e^{A_{(b_{x}, w_{y})}} K (b_{x}, w_{y}) ψ_{x}^{+} ψ_{y}^{-}} = \int D ψ e^{S_{A} (ψ)}

Proposition 4. When $W (M) = \sum_{f \in even faces} (I_{\scriptsize{\begin{array}{l} f \end{array}}} + I_{\scriptsize{\begin{array}{l} f \end{array}}})$ then

$Z_{Λ, λ} = \sum_{M} w (M) e^{\scriptsize{λ \sum_{f \in even faces} (I_{\scriptsize{\begin{array}{l} f \end{array}}} + I_{\scriptsize{\begin{array}{l} f \end{array}}})}} = \int D ψ e^{S (ψ) + α \sum_{γ} \prod_{e \in γ} E_{e}}$

where $γ = \begin{array}{l} f \end{array}, \begin{array}{l} f \end{array}$ and $E_{e} = K (e) ψ_{x}^{+} ψ_{x}^{-}$ if $e = (b_{x}, w_{y})$ and $α = e^{λ} - 1 \approx λ$ .

If we are interested in the generating function of the interacting model we replace $S (ψ) \to S_{A} (ψ)$ and $E_{e} \to E_{e} e^{A_{e}}$ .