The Pfaffian and the dimer model

Intuition Beginner

Imagine a checkerboard and a pile of dominoes, each domino covering exactly two neighbouring squares. A dimer covering is a way to tile the whole board with dominoes so that every square is covered once and no two dominoes overlap. The question is plain: how many such tilings are there? For a large board the answer is enormous, and writing down all tilings one by one is hopeless.

The surprise is that this purely combinatorial counting problem has a clean answer through a single number attached to a matrix. That number is the Pfaffian. With one clever choice of plus and minus signs on the board's connections, every tiling contributes the same sign, and the Pfaffian adds them all at once.

Why bother? Because the same count, dressed differently, gives the exact solution of the two-dimensional Ising magnet.

Visual Beginner

The left panel shows a small grid with one domino tiling drawn in; the middle panel shows a different tiling of the same grid; the right panel shows the grid's edges decorated with small arrows, the sign pattern that makes every tiling count with the same sign.

The picture holds the whole plan in miniature: many tilings, one sign rule, one number that counts them.

Worked example Beginner

Take the smallest interesting board: a $2\times 2$ square, four cells. A domino covers two adjacent cells. How many ways can two dominoes cover all four cells?

There are exactly two: both dominoes horizontal, or both dominoes vertical. So the count is $2$.

Now meet the Pfaffian on the matching example that matches this. Label the four cells $1,2,3,4$ around the square. A pairing of the four labels into two pairs is exactly a tiling. The pairings that use only board edges are $\{(1,2),(3,4)\}$ and $\{(1,4),(2,3)\}$ — the two tilings above. The pairing $\{(1,3),(2,4)\}$ would join diagonal cells, which no domino can do, so it is dropped.

What this tells us: counting tilings is the same as counting allowed pairings of the cells. The Pfaffian is the bookkeeping device that sums those pairings with signs, and on a flat board the signs can be arranged so the sum is the honest count $2$.

Check your understanding Beginner

Formal definition Intermediate+

Let $A=(A_{ij})$ be a $2m\times 2m$ antisymmetric matrix over a field, $A^T=-A$. The Pfaffian of $A$ is $$ \mathrm{Pf}(A) = \frac{1}{2^m m!} \sum_{\sigma \in S_{2m}} \mathrm{sgn}(\sigma) \prod_{k=1}^{m} A_{\sigma(2k-1),,\sigma(2k)} = \sum_{M \in \mathcal{M}{2m}} \mathrm{sgn}(M) \prod{(i,j) \in M} A_{ij}, $$ where ${\mathcal{M}}_{2m}$ is the set of perfect matchings of $\{1,\dots ,2m\}$ — partitions into $m$ unordered pairs $(i,j)$ with $i<j$ — and $\mathrm{sgn}(M)$ is the sign of the permutation taking $1,2,\dots ,2m$ to the matched order $i_1{j}_1\cdots i_m{j}_m$. The two forms agree because the $2^{m}m!$ reorderings within and among the pairs are absorbed by antisymmetry. The basic identity is $\mathrm{Pf}(A{)}^2=\det A$, and under a congruence $\mathrm{Pf}(BA{B}^T)=\det (B)\mathrm{Pf}(A)$.

Let $G=(V,E)$ be a finite graph with $|V|=2m$. A dimer covering (perfect matching) of $G$ is a subset $D\subseteq E$ such that every vertex meets exactly one edge of $D$. With an edge weight $w_e>0$ the dimer partition function is $$ Z_{\mathrm{dimer}}(G) = \sum_{D} \prod_{e \in D} w_e , $$ the sum over perfect matchings. A Kasteleyn orientation of a planar graph $G$ is an orientation of its edges such that, traversing the boundary of every bounded face, the number of edges oriented clockwise is odd. Given such an orientation, the Kasteleyn matrix $K$ is the antisymmetric matrix with $K_{ij}=+w_e$ if the edge $e=(i,j)$ points from $i$ to $j$, $K_{ij}=-w_e$ if it points from $j$ to $i$, and $K_{ij}=0$ when $i,j$ are not adjacent ^{[Kasteleyn 1963]}.

Counterexamples to common slips

• An antisymmetric matrix of odd order has $\det A=0$ and no Pfaffian; the matching set ${\mathcal{M}}_{2m}$ is empty when the vertex count is odd, so $Z_{\mathrm{dimer}}=0$ for any graph with an odd number of vertices.
• The Pfaffian is not the square root of $\det A$ chosen arbitrarily: it is a polynomial in the entries with a definite sign, fixed by the matching expansion. The naive choice $\sqrt{\det A}$ loses that sign and is not a polynomial.
• A planar orientation that fails the odd-clockwise condition on even one face gives a $K$ whose Pfaffian mixes signs across matchings; the count then collapses through cancellation rather than reinforcement. The orientation condition is what aligns every matching's sign.

Key theorem with proof Intermediate+

Theorem (Kasteleyn). Let $G$ be a planar graph with a Kasteleyn orientation and Kasteleyn matrix $K$. Then $$ |\mathrm{Pf}(K)| = Z_{\mathrm{dimer}}(G) = \sum_{D} \prod_{e \in D} w_e , $$ so the weighted count of perfect matchings is the absolute value of a Pfaffian, and the count of close-packed dimer coverings is $|\mathrm{Pf}(K)|$ when all $w_e=1$.

Proof. By the matching expansion, $\mathrm{Pf}(K)={\sum }_M\mathrm{sgn}(M){\prod }_{(i,j)\in M}{K}_{ij}$. A matching $M$ contributes a nonzero product only when every pair $(i,j)\in M$ is an edge of $G$, so the support of the sum is exactly the set of perfect matchings (dimer coverings) of $G$, each weighted by ${\prod }_{e\in M}{w}_e$ up to a sign $\mathrm{sgn}(M){\prod }_{(i,j)\in M}\mathrm{sgn}(K_{ij})$.

The content of the theorem is that this combined sign is the same for every matching. Fix two matchings $M$ and $M^{\prime }$. Their symmetric difference $M△M^{\prime }$ is a disjoint union of cycles, each alternating between $M$- and $M^{\prime }$-edges and hence of even length. It suffices to show the relative sign across a single such cycle is $+1$, since a sequence of cycle flips connects any matching to any other. Let $C$ be one alternating cycle of length $2\ell$. Comparing the contributions of $M$ and $M^{\prime }$, the ratio of their signed weights equals $(-1)$ raised to the number of vertices of $G$ strictly enclosed by $C$, times $(-1{)}^{\ell -1}$ coming from the orientation of $C$. The Kasteleyn condition states that each bounded face has an odd number of clockwise edges; a planar-induction count over the faces enclosed by $C$ shows the clockwise-edge parity of $C$ equals $\ell -1$ plus the enclosed-vertex count modulo $2$. These two contributions cancel, so $M$ and $M^{\prime }$ enter with equal sign. Therefore $\mathrm{Pf}(K)=\pm Z_{\mathrm{dimer}}(G)$, and taking absolute value gives the count. $\square$

Bridge. This identity is the foundational reason planar dimer enumeration is solvable in closed form: a sum over exponentially many matchings collapses to one Pfaffian, hence to one determinant via $\mathrm{Pf}(K{)}^2=\det K$. It builds toward the square-lattice free energy below, where $K$ becomes a translation-invariant operator diagonalised by Fourier modes, and it appears again in the Ising solution of 08.14.02, where the high-temperature loop sum maps to a dimer covering of a decorated lattice. The central insight is that the Kasteleyn orientation is exactly the sign datum a Grassmann (Berezin) integral supplies automatically: putting these together, the dimer Pfaffian of this theorem is the fermionic Gaussian integral $\int e^{\frac{1}{2}{\psi }^{T}K\psi }=\mathrm{Pf}(K)$, so the combinatorial count and the free-fermion partition function are one object. The bridge is that what 08.14.02 derives from anticommuting variables, Kasteleyn derives from planar topology, and the two signs agree.

Exercises Intermediate+

Advanced results Master

The square-lattice close-packed dimer model is the prototype where the Pfaffian is evaluated explicitly. Take an $m\times n$ square lattice with all weights $1$. A Kasteleyn orientation can be chosen periodic: orient all horizontal bonds in one direction and assign a factor $i$ to alternating vertical bonds, an arrangement that satisfies the odd-clockwise condition on every unit face ^{[Kasteleyn 1961]}. Because the resulting $K$ is translation-invariant under the lattice's even sublattice, it diagonalises in Fourier modes. Its squared singular values are $4({\sin }^2\theta +{\sin }^2\varphi )$, and the partition function is $$ Z_{m,n} = \big(\det K\big)^{1/2}, \qquad \frac{1}{mn}\log Z_{m,n} \xrightarrow[m,n\to\infty]{} \frac{1}{2}\cdot\frac{1}{(2\pi)^2}\int_0^{2\pi}!!\int_0^{2\pi} \log!\big(4\sin^2\theta + 4\sin^2\phi\big), d\theta, d\phi . $$ Evaluating the double integral gives the Kasteleyn–Temperley–Fisher result $$ \lim_{m,n\to\infty} \frac{1}{mn}\log Z_{m,n} = \frac{G}{\pi}, \qquad G = \sum_{k=0}^{\infty} \frac{(-1)^k}{(2k+1)^2} = 0.9159\ldots, $$ with $G$ Catalan's constant ^{[Temperley-Fisher 1961]}. The free energy per dimer is therefore an explicit transcendental constant — the cleanest closed form in lattice statistical mechanics, and historically the first exact two-dimensional result obtained by the Pfaffian method.

The Ising connection runs through a lattice decoration. Fisher's construction replaces each Ising site of a planar lattice by a finite gadget so that perfect matchings of the decorated graph correspond, weight for weight, to terms in the Ising high-temperature expansion ^{[Fisher 1966]}. The decorated graph is again planar, so Kasteleyn's theorem applies and $Z_{\mathrm{Ising}}=c|\mathrm{Pf}(\tilde{K})|$ for a signed lattice operator $\tilde{K}$ and an explicit constant $c$. This is precisely the antisymmetric matrix appearing in the Grassmann integral of 08.14.02: the Berezin integral $\int e^{\frac{1}{2}{\psi }^T\tilde{K}\psi }=\mathrm{Pf}(\tilde{K})$ and Kasteleyn's planar sign rule produce the identical signed adjacency operator, so the free-fermion solution and the dimer solution are two derivations of one Pfaffian.

The non-planar case marks the boundary of the method. On a torus the odd-clockwise condition cannot be met on a single orientation globally; the partition function becomes a signed combination of four Pfaffians, one for each spin structure (periodic/antiperiodic boundary conditions in the two directions). This four-Pfaffian sum is the lattice ancestor of the four fermion sectors of the critical Ising conformal field theory, and it is why the dimer model on a higher-genus surface is governed by $2^{2g}$ Pfaffians indexed by ${\mathbb{Z}}_2$ homology.

Synthesis. The Pfaffian is the foundational reason planar close-packed dimer and planar Ising models are exactly solvable: the matching expansion identifies $\mathrm{Pf}(K)$ with the dimer count, and the Kasteleyn orientation is dual to the sign rule that a fermionic Gaussian integral carries automatically. Putting these together, the square-lattice free energy $G/\pi$ and the Onsager free energy of 08.14.02 are two evaluations of one signed determinant, and this is exactly the statement that a quadratic Grassmann action has a Pfaffian partition function. The central insight is that planarity is what makes the signs cohere — it generalises from one orientation on a planar graph to $2^{2g}$ spin structures on a genus-$g$ surface, the combinatorial shadow of the fermion sectors of the continuum theory. The bridge to universality is that the same square-root-of-a-determinant structure controls the entire free-fermion class, dimers and Ising alike, regardless of the gadget used to decorate the lattice.

Full proof set Master

Proposition (Pfaffian matching expansion). For a $2m\times 2m$ antisymmetric matrix $A$, $\mathrm{Pf}(A)={\sum }_{M\in {\mathcal{M}}_{2m}}\mathrm{sgn}(M){\prod }_{(i,j)\in M}{A}_{ij}$, where the sum runs over perfect matchings of $\{1,\dots ,2m\}$.

Proof. Start from the permutation form $\mathrm{Pf}(A)=\frac{1}{2^{m}m!}{\sum }_{\sigma \in S_{2m}}\mathrm{sgn}(\sigma ){\prod }_{k=1}^m{A}_{\sigma (2k-1),\sigma (2k)}$. Group permutations by the unordered pair partition they induce: each matching $M=\{(i_1,j_1),\dots ,(i_m,j_m)\}$ arises from $2^{m}m!$ permutations, namely the $m!$ orderings of the pairs and the $2^m$ swaps within pairs. Swapping the two entries of a pair sends $A_{ab}\mapsto A_{ba}=-A_{ab}$ but also flips $\mathrm{sgn}(\sigma )$, so each such swap leaves the signed product unchanged; reordering the pairs permutes equal factors and is likewise sign-compensated. Thus all $2^{m}m!$ permutations of a fixed matching contribute the identical signed term $\mathrm{sgn}(M){\prod }_{(i,j)\in M}{A}_{ij}$, where $\mathrm{sgn}(M)$ is the sign of any representative permutation. The prefactor $\frac{1}{2^{m}m!}$ cancels the multiplicity, leaving one term per matching. $\square$

Proposition (Pfaffian squares to the determinant). For antisymmetric $A$ of order $2m$, $\mathrm{Pf}(A{)}^2=\det A$.

Proof. Every antisymmetric matrix over a field of characteristic $\ne 2$ is congruent to a block-diagonal form $J=BA{B}^T$ with $2\times 2$ blocks $\left(\begin{array}{cc}0& {\lambda }_k\\ -{\lambda }_k& 0\end{array}\right)$ for invertible $B$. For this normal form the matching expansion has a single surviving matching, giving $\mathrm{Pf}(J)={\prod }_k{\lambda }_k$, while $\det J={\prod }_k{\lambda }_k^2$, so $\mathrm{Pf}(J{)}^2=\det J$. Under congruence $\mathrm{Pf}(BA{B}^T)=\det (B)\mathrm{Pf}(A)$ and $\det (BA{B}^T)=\det (B{)}^2\det A$, hence $\det (B{)}^2\mathrm{Pf}(A{)}^2=\mathrm{Pf}(J{)}^2=\det J=\det (B{)}^2\det A$, and cancelling $\det (B{)}^2\ne 0$ gives $\mathrm{Pf}(A{)}^2=\det A$. The identity is polynomial in the entries of $A$, so it extends to all antisymmetric $A$ by density and to other fields by the polynomial-identity principle. $\square$

Proposition (square-lattice dimer free energy). The close-packed dimer model on the square lattice has free energy per site ${\lim }_{m,n\to \infty }\frac{1}{mn}\log Z_{m,n}=G/\pi$, with $G$ Catalan's constant.

Proof. With the periodic Kasteleyn orientation, $\frac{1}{mn}\log Z_{m,n}\to \frac{1}{2}\frac{1}{(2\pi {)}^2}{\int }_0^{2\pi }{\int }_0^{2\pi }\log (4{\sin }^2\theta +4{\sin }^2\varphi )d\theta d\varphi$. Perform the $\varphi$-integral first using $\frac{1}{2\pi }{\int }_0^{2\pi }\log (a^2+b^2{\sin }^2\varphi )d\varphi =2\log \frac{a+\sqrt{a^2+b^2}}{2}$ with $a^2=4{\sin }^2\theta$, $b^2=4$. The result reduces to $\frac{1}{\pi }{\int }_0^{\pi /2}\log (2\sin \theta +\sqrt{4{\sin }^2\theta +4})d\theta$, and a standard reduction (substituting and expanding in the Fourier series of $\log (2\cos )$) yields $\frac{1}{\pi }{\sum }_{k=0}^{\infty }\frac{(-1{)}^k}{(2k+1{)}^2}=\frac{G}{\pi }$. The convergence of the Riemann sum to the integral is uniform because the integrand is $\log$-integrable with a single integrable logarithmic singularity at $\theta =\varphi =0$. $\square$

Connections Master

The free-fermion solution of the planar Ising model 08.14.02 uses the same signed antisymmetric operator: the Berezin integral $\int e^{\frac{1}{2}{\psi }^{T}K\psi }=\mathrm{Pf}(K)$ developed there is the Grassmann face of the Kasteleyn count established here, and the high-temperature loop expansion mapped to dimers is the precise bridge between the two units.

The determinant 01.01.07 is the parent object: the Pfaffian is a signed square root of the determinant with $\mathrm{Pf}(K{)}^2=\det K$, and the permutation expansion of the determinant specialises to the matching expansion of the Pfaffian once the matrix is antisymmetric. Every Pfaffian evaluation in this unit ultimately reduces to a determinant computation.

The binomial and combinatorial counting machinery 00.12.02 supplies the enumerative side of the story: counting perfect matchings is a combinatorial enumeration problem whose generating function the Pfaffian closes in determinant form, turning an exponential sum over tilings into a single algebraic quantity.

The Brownian-motion and path-integral measure of 08.14.01 is the continuum counterpart: the lattice dimer/free-fermion determinant passes in the scaling limit to a functional determinant of a Dirac-type operator, the object that the Wiener-measure construction prepares on the bosonic side.

Historical & philosophical context Master

The Pfaffian dates to Cayley, who in 1852 named the antisymmetric-matrix invariant after Johann Friedrich Pfaff and proved $\mathrm{Pf}(A{)}^2=\det A$. Its statistical-mechanical career began in 1961, when Kasteleyn solved the close-packed dimer problem on the square lattice by writing the partition function as a Pfaffian of a signed adjacency matrix, obtaining the free energy $G/\pi$ with $G$ Catalan's constant ^{[Kasteleyn 1961]}; Temperley and Fisher reached the same square-lattice result independently and simultaneously ^{[Temperley-Fisher 1961]}. Kasteleyn 1963 generalised the method to arbitrary planar graphs, proving that a planar orientation with the odd-clockwise-per-face property always exists and that the signed Pfaffian then counts perfect matchings ^{[Kasteleyn 1963]}. Fisher 1966 completed the dimer–Ising correspondence with the decorated-lattice construction that turns the planar Ising partition function into a close-packed dimer count, giving a combinatorial route to Onsager's solution parallel to the operator route of Schultz, Mattis and Lieb ^{[Fisher 1966]}.

Bibliography Master

@article{Kasteleyn1961Dimers,
  author  = {Kasteleyn, P. W.},
  title   = {The statistics of dimers on a lattice. I. The number of dimer arrangements on a quadratic lattice},
  journal = {Physica},
  volume  = {27},
  year    = {1961},
  pages   = {1209--1225}
}

@article{Kasteleyn1963Dimer,
  author  = {Kasteleyn, P. W.},
  title   = {Dimer Statistics and Phase Transitions},
  journal = {Journal of Mathematical Physics},
  volume  = {4},
  year    = {1963},
  pages   = {287--293}
}

@article{TemperleyFisher1961,
  author  = {Temperley, H. N. V. and Fisher, M. E.},
  title   = {Dimer problem in statistical mechanics --- an exact result},
  journal = {Philosophical Magazine},
  volume  = {6},
  year    = {1961},
  pages   = {1061--1063}
}

@article{Fisher1966Dimer,
  author  = {Fisher, Michael E.},
  title   = {On the Dimer Solution of Planar Ising Models},
  journal = {Journal of Mathematical Physics},
  volume  = {7},
  year    = {1966},
  pages   = {1776--1781}
}

@article{Cayley1852Pfaffian,
  author  = {Cayley, Arthur},
  title   = {On the theory of permutants},
  journal = {Cambridge and Dublin Mathematical Journal},
  volume  = {7},
  year    = {1852},
  pages   = {40--51}
}

@book{Kenyon2009Dimers,
  author    = {Kenyon, Richard},
  title     = {Lectures on Dimers},
  publisher = {IAS/Park City Mathematics Series, American Mathematical Society},
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}

@book{ItzyksonDrouffe1989,
  author    = {Itzykson, Claude and Drouffe, Jean-Michel},
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  publisher = {Cambridge University Press},
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}