08.14.02 · stat-mech / statistical-field-theory

Grassmann integration and the 2D Ising model as free fermions

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Anchor (Master): Itzykson & Drouffe, *Statistical Field Theory*, Vol. 1, Ch. 2 (free-fermion solution of the planar Ising model); Schultz, Mattis & Lieb, *Rev. Mod. Phys.* 36, 856 (1964) (Ising as a many-fermion problem); Samuel, *J. Math. Phys.* 21, 2806 (1980) (Grassmann/Pfaffian reformulation); McCoy & Wu, *The Two-Dimensional Ising Model* (Harvard, 1973)

Intuition Beginner

Ordinary numbers do not care about order: three times five equals five times three. Some bookkeeping symbols are different. When you swap two of them, you pick up a minus sign, and a symbol multiplied by itself gives zero. These order-sensitive symbols are called Grassmann numbers, and they are the natural language for fermions, the particles that refuse to share a state.

Why invent such strange symbols? Because the minus-sign rule is exactly the rule that two electrons obey: trade two of them and the description flips sign. Building that rule into the arithmetic from the start means every later formula carries it for free.

Visual Beginner

The left panel shows two ordinary tokens being swapped with no penalty. The right panel shows two Grassmann tokens being swapped and acquiring a minus sign, with a self-product collapsing to zero.

The picture is the whole idea in miniature: anticommuting tokens encode the sign rule that distinguishes fermions from ordinary degrees of freedom.

Worked example Beginner

Take one Grassmann symbol and call it $θ$ . The rules of integration over it are short: the integral of $1$ is $0$ , and the integral of $θ$ is $1$ .

Now expand a simple weight. Because $θ$ times $θ$ is $0$ , any function of $θ$ is just $a + b θ$ for two ordinary numbers $a$ and $b$ . Integrate it: the $a$ part contributes $0$ and the $b θ$ part contributes $b$ . So the integral returns $b$ , the coefficient sitting in front of $θ$ .

Try two symbols $θ$ and $η$ . The only term that survives integration over both is the one containing $θ η$ , because each symbol must appear exactly once.

What this tells us: Grassmann integration is not area-under-a-curve. It is a rule that picks out one specific coefficient. That single habit is what turns a fermion sum into a determinant.

Check your understanding Beginner

Formal definition Intermediate+

Let $θ_{1}, \dots, θ_{n}$ generate a real or complex Grassmann algebra $Λ_{n}$ : the associative unital algebra on these generators subject to $θ_{i} θ_{j} + θ_{j} θ_{i} = 0$ for all $i, j$ , so in particular $θ_{i}^{2} = 0$ . Every element of $Λ_{n}$ is a polynomial of degree at most one in each generator. The algebra is graded into even and odd parts; even elements commute with everything, odd elements anticommute among themselves.

The Berezin integral is the linear functional defined on single generators by $$ \int d\theta_i , 1 = 0, \qquad \int d\theta_i , \theta_i = 1, $$ extended to several generators by treating the symbols $d θ_{i}$ as themselves anticommuting and integrating from the innermost variable outward. For a function expanded in monomials, $\int d θ_{n} \dots d θ_{1} f (θ)$ extracts the coefficient of the top monomial $θ_{1} \dots θ_{n}$ ^{[Berezin 1966]}.

For a Gaussian weight one introduces two independent families $ψ_{i}, \overset{ˉ}{ψ}_{i}$ ( $2 n$ generators) and an $n \times n$ matrix $A$ . The basic complex identity is $$ \int \prod_{i=1}^{n} d\bar\psi_i , d\psi_i ; e^{-\bar\psi A \psi} = \det A, \qquad \bar\psi A \psi = \sum_{i,j} \bar\psi_i A_{ij} \psi_j . $$ For a single real family $ψ_{i}$ and an antisymmetric matrix $A$ ( $A^{T} = - A$ , even order $n = 2 m$ ), the analogue is $$ \int \prod_{i=1}^{n} d\psi_i ; e^{\frac12 \psi^T A \psi} = \mathrm{Pf}(A), \qquad \mathrm{Pf}(A)^2 = \det A . $$ The Pfaffian $Pf (A)$ is the signed sum over pairings of the index set, the square root of the determinant fixed by sign. The sign convention used throughout is that generators integrate from the rightmost label inward, matching the ordering of the $d θ_{i}$ as written.

Counterexamples to common slips

An odd-order antisymmetric matrix has $det A = 0$ , so no Pfaffian exists; the Gaussian integral over an odd number of real generators is identically $0$ , since the exponential cannot supply the required top monomial.
A change of variables $ψ_{i} = \sum_{j} M_{ij} ψ_{j}^{'}$ scales the measure by $det (M)^{- 1}$ , the inverse of the bosonic Jacobian; forgetting the inversion is the single most common sign-and-normalisation error.
The weight $e^{- \overset{ˉ}{ψ} A ψ}$ is even, so its expansion terminates; there is no convergence question, unlike the bosonic Gaussian where positivity of $A$ is needed.

Key theorem with proof Intermediate+

Theorem (Fermionic Gaussian integral is a determinant). Let $A$ be an $n \times n$ complex matrix and $ψ_{i}, \overset{ˉ}{ψ}_{i}$ independent Grassmann generators. Then $\int \prod_{i} d \overset{ˉ}{ψ}_{i} d ψ_{i} e^{- \overset{ˉ}{ψ} A ψ} = det A$ .

Proof. Expand the exponential. Because $(\overset{ˉ}{ψ} A ψ)$ is even and each generator squares to $0$ , the series terminates and only the term with all $2 n$ generators present survives integration: $$ e^{-\bar\psi A \psi} = \sum_{k=0}^{n} \frac{(-1)^k}{k!} (\bar\psi A \psi)^k, $$ and the Berezin integral annihilates every term except $k = n$ . In that term, $$ \frac{(-1)^n}{n!} \Big( \sum_{i,j} \bar\psi_i A_{ij} \psi_j \Big)^n = \frac{(-1)^n}{n!} \sum_{\substack{i_1\dots i_n \ j_1 \dots j_n}} A_{i_1 j_1}\cdots A_{i_n j_n}, \bar\psi_{i_1}\psi_{j_1}\cdots \bar\psi_{i_n}\psi_{j_n}. $$ A nonzero contribution requires the $i$ -labels to be a permutation of ${1, \dots, n}$ and likewise the $j$ -labels. Reordering the generators into the canonical order $\overset{ˉ}{ψ}_{1} ψ_{1} \dots \overset{ˉ}{ψ}_{n} ψ_{n}$ produces the sign $sgn (σ)$ of the permutation $σ$ relating the two index lists, and the $n!$ ways of assigning the factors cancel the $1/ n!$ . Integrating the canonical monomial gives $1$ under the stated ordering. Collecting, $$ \int \prod_i d\bar\psi_i , d\psi_i , e^{-\bar\psi A \psi} = \sum_{\sigma \in S_n} \mathrm{sgn}(\sigma) \prod_{i} A_{i, \sigma(i)} = \det A. $$ The $(- 1)^{n}$ from the series and a $(- 1)^{n}$ from straightening the $\overset{ˉ}{ψ} ψ$ pairs cancel. $□$

Bridge. This identity is the foundational reason a free-fermion partition function is computable in closed form: the sum over configurations becomes a single determinant. It builds toward the Ising free-fermion solution below, where the matrix $A$ is the lattice Dirac operator and $det A$ is the partition function; the Pfaffian version is exactly the statement that emerges from the high-temperature polygon expansion, and putting these together the bridge is that Onsager's transfer-matrix result 08.03.01 and Kasteleyn's dimer-Pfaffian count are two readings of one Grassmann integral. The same determinantal collapse appears again in the continuum, where it generalises to a functional determinant of a differential operator, and it is dual to the bosonic Gaussian integral that produces $(det A)^{- 1/2}$ rather than $det A$ .

Exercises Intermediate+

Exercise 7 (hard, short-answer).

The 2D Ising high-temperature expansion is a sum over closed loops on the lattice with weights $tanh K$ per edge. Sketch why such a loop sum can be written as a Grassmann (Pfaffian) integral.

Hint

Place a few anticommuting generators at each site; a Gaussian weight in them produces exactly the self-avoiding-with-signs loop sum once Kasteleyn's edge signs are installed.

Answer

Assign Grassmann generators to lattice sites (or to the corners of each face). A bilinear hopping weight expanded under Berezin integration produces closed non-backtracking loops, each generator used once; Kasteleyn's orientation supplies the signs that make crossing loops cancel so the weighted loop sum equals a single Pfaffian. The Ising partition function is then $Pf$ of a signed lattice Dirac operator. Rubric: full credit for connecting the loop expansion to a Pfaffian via site generators and Kasteleyn signs.

Advanced results Master

The free-fermion solution of the planar Ising model proceeds from the transfer matrix 08.03.01 rather than from the loop expansion, and it is the route taken by Schultz, Mattis and Lieb. The row-to-row transfer matrix $V$ for an $N$ -site row factors as $V = V_{2} V_{1}$ , with $V_{1}$ encoding the vertical bond weights and $V_{2}$ the horizontal ones. A Jordan–Wigner transformation sends the spin operators $σ_{j}^{x}, σ_{j}^{z}$ to a set of fermion creation and annihilation operators $c_{j}^{†}, c_{j}$ obeying canonical anticommutation relations; under this map $lo g V$ becomes a quadratic form in the $c_{j}, c_{j}^{†}$ ^{[Schultz-Mattis-Lieb 1964]}. The fermionic Fock space carrying these operators is the one constructed in 08.10.09.

A Bogoliubov transformation diagonalises the quadratic fermion Hamiltonian into independent modes $η_{q}^{†} η_{q}$ with single-particle energies $ϵ_{q}$ determined by $$ \cosh \epsilon_q = \cosh 2K^* \cosh 2K - \sinh 2K^* \sinh 2K \cos q, $$ where $K = β J$ and $K^{*}$ is the dual coupling fixed by $sinh 2 K sinh 2 K^{*} = 1$ . The largest transfer-matrix eigenvalue is the fermion vacuum of these modes, and the free energy per site in the thermodynamic limit is $$ -\beta f = \log(2\cosh 2K) + \frac{1}{2\pi}\int_0^\pi \log!\frac{1 + \sqrt{1 - \kappa^2 \sin^2\phi}}{2}, d\phi, $$ with $κ = 2 sinh 2 K / cosh^{2} 2 K$ , reproducing the Onsager free energy ^{[Onsager 1944]}. The single-particle gap $ϵ_{q = 0}$ closes precisely when $K = K^{*}$ , i.e. at $sinh 2 K_{c} = 1$ , the self-dual point $T_{c} = 2 J / [k_{B} ln (1 + 2)]$ recovered in 08.03.01.

The spontaneous magnetisation comes from the long-distance behaviour of the fermion two-point function. The order parameter is a disorder-line (Toeplitz-determinant) object whose asymptotics give $$ m = \big(1 - \sinh^{-4} 2K\big)^{1/8} \sim (T_c - T)^{1/8}, $$ the celebrated $β = 1/8$ exponent. The power $1/8$ is a direct fingerprint of the free-fermion spectrum near the gap closing; the Toeplitz determinant whose Szegő-type asymptotics yield the exponent is the spin–spin correlator written in the fermion basis.

The loop-expansion route runs in parallel. Kasteleyn showed that a planar dimer partition function is a Pfaffian of a signed adjacency matrix ^{[Kasteleyn 1963]}, and Fisher gave a contemporaneous planar-dimer solution and the Ising–dimer correspondence ^{[Fisher 1961]}. The Ising high-temperature loop sum maps to a dimer covering of a decorated lattice, so $Z_{Ising} \propto Pf (A)$ for a signed lattice Dirac operator $A$ ; this is the Grassmann integral of the Key theorem with $A$ antisymmetric. Samuel later recast the entire construction directly as a Grassmann integral without the dimer intermediary.

Synthesis. The free-fermion structure is the foundational reason the two-dimensional Ising model is exactly solvable, and it unifies three routes that once looked unrelated: the transfer-matrix diagonalisation of 08.03.01 is dual to the Pfaffian loop count of Kasteleyn and Fisher, and putting these together both are evaluations of one Berezin integral $\int e^{\frac{1}{2} ψ^{T} A ψ} = Pf (A)$ . This is exactly the statement that a quadratic fermion action has a determinantal partition function, and it generalises from the lattice to the continuum, where the lattice Dirac operator $A$ becomes a free Majorana field whose conformal limit governs the critical point. The central insight is that the non-mean-field exponents — $β = 1/8$ , the logarithmic specific heat $α = 0$ — are spectral data of a free-fermion problem, and the bridge to universality is that the same free-fermion fixed point controls the entire Ising universality class regardless of lattice detail. The pattern appears again in the conformal field theory of the critical point, where the central charge $c = 1/2$ is precisely the count of one Majorana fermion.

Full proof set Master

Proposition (Pfaffian identity for the free-fermion partition function). Let $A$ be a real $2 m \times 2 m$ antisymmetric matrix and $ψ_{1}, \dots, ψ_{2 m}$ Grassmann generators. Then $\int \prod_{i} d ψ_{i} e^{\frac{1}{2} ψ^{T} A ψ} = Pf (A)$ , and consequently the square of this integral equals $det A$ .

Proof. Expand the even exponential; only the term of order $m$ supplies the top monomial $ψ_{1} \dots ψ_{2 m}$ : $$ \frac{1}{m!}\Big(\tfrac12 \sum_{i<j} A_{ij}(\psi_i\psi_j - \psi_j\psi_i)\Big)^m = \frac{1}{m!}\Big(\sum_{i<j} A_{ij}\psi_i\psi_j\Big)^m, $$ using antisymmetry of $A$ to combine the two orderings. Multiplying out, a nonzero contribution requires a partition of ${1, \dots, 2 m}$ into $m$ ordered pairs; the $m!$ orderings of the pairs cancel the $1/ m!$ . Reordering each product $ψ_{i_{1}} ψ_{j_{1}} \dots ψ_{i_{m}} ψ_{j_{m}}$ into canonical order contributes the sign $sgn$ of the corresponding pairing permutation. Summing over perfect matchings $P$ of the index set, $$ \int \prod_i d\psi_i, e^{\frac12\psi^T A\psi} = \sum_{P} \mathrm{sgn}(P)\prod_{(i,j)\in P} A_{ij} = \mathrm{Pf}(A). $$ The square follows from the doubling argument of Exercise 6: introducing a second independent family and recombining into a complex pair gives $Pf (A)^{2} = \int d \overset{χ}{ˉ} d χ e^{- \overset{χ}{ˉ} A χ} = det A$ by the Key theorem. $□$

Proposition (Self-duality fixes the critical coupling). With the dual coupling defined by $sinh 2 K sinh 2 K^{*} = 1$ , the single-fermion energy $ϵ_{q}$ satisfies $ϵ_{q = 0} = 2∣ K - K^{*} ∣$ near the gap, so the spectrum is gapless exactly at $K = K^{*}$ .

Proof. At $q = 0$ the dispersion relation reduces to $cosh ϵ_{0} = cosh 2 K^{*} cosh 2 K - sinh 2 K^{*} sinh 2 K = cosh (2 K^{*} - 2 K)$ , whence $ϵ_{0} = 2∣ K^{*} - K ∣$ . The gap vanishes iff $K = K^{*}$ . Substituting into $sinh 2 K sinh 2 K^{*} = 1$ gives $sinh^{2} 2 K_{c} = 1$ , i.e. $sinh 2 K_{c} = 1$ and $2 K_{c} = ln (1 + 2)$ , the Onsager temperature $T_{c} = 2 J / [k_{B} ln (1 + 2)]$ . $□$

Proposition (Magnetisation exponent from the fermion gap). The spontaneous magnetisation $m = (1 - sinh^{- 4} 2 K)^{1/8}$ vanishes as $(T_{c} - T)^{1/8}$ as $T ↑ T_{c}$ .

Proof. Write $t = sinh^{- 4} 2 K$ . At $K_{c}$ , $sinh 2 K_{c} = 1$ so $t = 1$ and $m = 0$ . Expand $sinh 2 K$ to first order in $T_{c} - T$ : since $sinh 2 K$ is smooth and increasing in $K$ with nonzero derivative at $K_{c}$ , $1 - t = 1 - sinh^{- 4} 2 K$ vanishes linearly in $(T_{c} - T)$ with a positive coefficient $c_{0}$ . Then $m = (1 - t)^{1/8} = (c_{0} (T_{c} - T) + O ((T_{c} - T)^{2}))^{1/8} \sim (T_{c} - T)^{1/8}$ . The exponent $1/8$ is the eighth-root power in the closed-form $m$ , itself the Szegő asymptotic exponent of the spin–spin Toeplitz determinant in the fermion basis. $□$

Connections Master

The transfer-matrix solution 08.03.01 and the Grassmann/Pfaffian solution here are the two canonical routes to the same exact free energy; this unit supplies the fermionic reformulation that makes the determinantal structure manifest, and it relies on the Onsager free energy and critical temperature established there.
The underlying spin model is the Ising model 08.01.02; the Jordan–Wigner map turns its one-dimensional quantum transfer-matrix Hamiltonian into the free fermions used throughout, so the lattice degrees of freedom defined in 08.01.02 are the inputs to the construction.
The fermionic operators produced by Jordan–Wigner live in the fermionic Fock space 08.10.09; that unit builds the creation/annihilation operators and canonical anticommutation relations whose Grassmann-coherent-state representation is exactly the Berezin integral developed here.
The lattice free-fermion action passes in the scaling limit to a continuum free-fermion field theory 08.14.04, where the determinant of the lattice Dirac operator becomes a functional determinant and the $c = 1/2$ Majorana conformal field theory emerges.
The disorder operators and Kramers–Wannier duality that fix the self-dual critical point appear again in 08.14.05, where the order–disorder algebra is organised by the same fermion bilinears.

Historical & philosophical context Master

Berezin systematised integration over anticommuting variables in his 1965 monograph (English translation 1966), giving the rules $\int d θ = 0$ , $\int θ d θ = 1$ and the determinant identity for fermionic Gaussian integrals that now bears his name ^{[Berezin 1966]}. The application to the Ising model crystallised a decade earlier in a different language: Kasteleyn 1963 and Fisher 1961 independently reduced the planar dimer problem to a Pfaffian of a signed adjacency matrix, and through the dimer–Ising correspondence this turned the Ising partition function into a free-fermion sum ^{[Kasteleyn 1963]} ^{[Fisher 1961]}. Schultz, Mattis and Lieb 1964 gave the operator-level statement, rewriting Onsager's transfer matrix as a soluble many-fermion problem via Jordan–Wigner and Bogoliubov transformations and recovering both the free energy of Onsager 1944 and the magnetisation exponent $β = 1/8$ ^{[Schultz-Mattis-Lieb 1964]} ^{[Onsager 1944]}. Samuel 1980 later expressed the whole solution directly as a Grassmann integral, removing the dimer scaffold.

Bibliography Master

@book{Berezin1966SecondQuantization,
  author    = {Berezin, Feliks Aleksandrovich},
  title     = {The Method of Second Quantization},
  publisher = {Academic Press},
  address   = {New York},
  year      = {1966},
  note      = {Russian original, Nauka, Moscow, 1965}
}

@article{SchultzMattisLieb1964,
  author  = {Schultz, T. D. and Mattis, D. C. and Lieb, E. H.},
  title   = {Two-Dimensional Ising Model as a Soluble Problem of Many Fermions},
  journal = {Reviews of Modern Physics},
  volume  = {36},
  year    = {1964},
  pages   = {856--871}
}

@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{Fisher1961Dimers,
  author  = {Fisher, Michael E.},
  title   = {Statistical Mechanics of Dimers on a Plane Lattice},
  journal = {Physical Review},
  volume  = {124},
  year    = {1961},
  pages   = {1664--1672}
}

@article{Onsager1944CrystalStatistics,
  author  = {Onsager, Lars},
  title   = {Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition},
  journal = {Physical Review},
  volume  = {65},
  year    = {1944},
  pages   = {117--149}
}

@article{Samuel1980Grassmann,
  author  = {Samuel, Stuart},
  title   = {The use of anticommuting variable integrals in statistical mechanics. I-III},
  journal = {Journal of Mathematical Physics},
  volume  = {21},
  year    = {1980},
  pages   = {2806--2833}
}

@book{ItzyksonDrouffe1989,
  author    = {Itzykson, Claude and Drouffe, Jean-Michel},
  title     = {Statistical Field Theory, Volume 1},
  publisher = {Cambridge University Press},
  year      = {1989}
}

@book{McCoyWu1973,
  author    = {McCoy, Barry M. and Wu, Tai Tsun},
  title     = {The Two-Dimensional Ising Model},
  publisher = {Harvard University Press},
  year      = {1973}
}

Prerequisites

08.03.01
08.01.02

Tier anchors

beginner: Itzykson & Drouffe, *Statistical Field Theory*, Vol. 1 (Cambridge, 1989), Ch. 2 §2.1 (Grassmann variables, intuitive picture); Tong, *Statistical Field Theory* lecture notes (Cambridge, 2017), §4 (Ising as fermions, narrative level)
intermediate: Itzykson & Drouffe, *Statistical Field Theory*, Vol. 1, Ch. 2 §2.1–2.3 (Berezin integral, Pfaffian, Ising Grassmann representation); Zinn-Justin, *Quantum Field Theory and Critical Phenomena*, 4th ed. (Oxford, 2002), Ch. 1 (Grassmann Gaussian integrals)
master: Itzykson & Drouffe, *Statistical Field Theory*, Vol. 1, Ch. 2 (free-fermion solution of the planar Ising model); Schultz, Mattis & Lieb, *Rev. Mod. Phys.* 36, 856 (1964) (Ising as a many-fermion problem); Samuel, *J. Math. Phys.* 21, 2806 (1980) (Grassmann/Pfaffian reformulation); McCoy & Wu, *The Two-Dimensional Ising Model* (Harvard, 1973)

References

raw/pdfs/kardar-22-24-identical-particles-noninteracting-fermi-liquid.pdf · Kardar statistical physics notes, lectures 22–24 (identical particles, noninteracting Fermi gas — fermionic occupation and the determinantal structure)
quantum-well · statistical physics index
Berezin, F. A. — The Method of Second Quantization · Academic Press, New York (1966); Russian original Nauka, Moscow (1965) — the systematic construction of integration over anticommuting (Grassmann) variables, the rule $\int d\theta = 0$, $\int \theta\, d\theta = 1$, and the identity $\int d\bar\psi\, d\psi\, e^{-\bar\psi A \psi} = \det A$ that makes a fermionic Gaussian integral a determinant
Schultz, T. D., Mattis, D. C. & Lieb, E. H. — Two-Dimensional Ising Model as a Soluble Problem of Many Fermions · Rev. Mod. Phys. 36, 856 (1964) — the transfer matrix of the 2D Ising model is rewritten via a Jordan–Wigner transformation as a quadratic form in fermion operators, diagonalised by a Bogoliubov transformation; the free energy reproduces Onsager and the magnetisation exponent $\beta = 1/8$ follows from the fermionic spectrum
Kasteleyn, P. W. — Dimer Statistics and Phase Transitions · J. Math. Phys. 4, 287 (1963) — the partition function of a planar dimer model equals the Pfaffian of a signed adjacency matrix; the route by which the Ising model's high-temperature polygon expansion becomes a Pfaffian / free-fermion sum
Fisher, M. E. — Statistical Mechanics of Dimers on a Plane Lattice · Phys. Rev. 124, 1664 (1961) — independent Pfaffian solution of the planar dimer problem and its relation to the Ising model via the dimer–Ising correspondence
Onsager, L. — Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition · Phys. Rev. 65, 117 (1944) — the original exact free energy that the free-fermion route reproduces

Estimated time

beginner: 22m
intermediate: 55m
master: 95m