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Kac-Ward formula and its extension to order-disorder correlators

through a graph zeta function

Michael Aizenman∗ Simone Warzel†

To Uzy Smilansky on the occasion of his 75th birthday

September 19, 2017

A streamlined derivation of the Kac-Ward formula for the planar Ising model’s parti-tion funcparti-tion is presented and applied in relating the kernel of the Kac-Ward matrices’ inverse with the correlation functions of the Ising model’s order-disorder correlation functions. Used in the analysis is the formula’s minor extension beyond planarity. A shortcut for both is enabled through the Bowen-Lanford graph zeta function relation.

1. Preamble

The focus of this note is on the application of a graph zeta function for a simplified Kac-Ward style derivation of Onsager’s formula for the free energy of the planar Ising model. This approach also adds insight into the origin of some of the model’s astounding properties and emergent structures.

Two specific goals are: i) present a simple derivation of the Kac-Ward formula for the free en-ergy [23], which highlights its relation to the concept of graph zeta function, ii) reach beyond the partition function, and demonstrate that the Kac-Ward matrix’ resolvent provides the correlation function of the model’s order-disorder operators. This relation exemplifies the presence of lin-ear relations among selected correlation functions of the 2D Ising model, a fact which has been previously noted in a number of other forms.

In the first part, our analysis partially overlaps with the papers by Cimansoni [6] and Lis [24, 25]. In the second part, we complete a circle which was initiated by Kac and Ward, showing how the Kaufman spinor operators emerge very naturally from the combinatorics of this classical problem. Related statements, though not exactly in the same form, can be found in the review of Cimasoni [7]. We do not specifically discuss here the model’s critical point, on which much has been said in the literature including in the recent Chelkak-Smirnov analysis of the model’s s-holomorphicity [32, 9].

The presentation of these results requires the introduction of two rather different subjects. We start with the one whose introduction can be made here shorter.

∗_{Depts. of Physics and Mathematics, Princeton University, Princeton NJ 08544, USA.} †

Zentrum Mathematik, TU M¨unchen, Boltzmannstr. 3, 85747 Garching, Germany.

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2. Graph zeta functions

2.1. Paths on graphs with transition-based weights

For a graph, described in terms of its vertex and edge set asG = (V,E), the set of itsoriented edgeswill be denote here byE ≡b Eb(G) =E × {+1,−1}. These will be used to describe paths and loops onG. The presentation of paths in this form, rather than through sequences of visited sites, simplifies the formulation of weights which depend multiplicatively on both the edges visited and turns taken.

Definition 2.1. (Paths and loops – basic terminology and notation)

1. An oriented edgee≡ (x1, x2)is said to lead into e′ ≡(y1, y2)ify1 =x2. The relation is

denotede⊲_e′_{. The reversal of}_e_{is denoted}_e_.

2. A path on G is a finite sequence_bγ = (e0, e1, . . . , en) of oriented edges sequentially lead-ing into each other. The number of such steps, n, is denoted by|γb|. A path is said to be backtracking ifej+1=ej for somej.

3. A loop is an equivalence class, modulo cyclic reparametrization, of non-backtracking paths (e0, e1, . . . , en)withen =e0. A loop is called primitiveif it cannot be presented as some

k-fold repetition of a shorter one.

4. AE ×b EbmatrixMisnon-backtrackingifMe,e = 0for alle∈Eb. For aE ×b EbmatrixM and a path or loop_bγ = (e0, e1, . . . , en), we denote:

χM(bγ) :=

j=1

Mej−1,ej. (2.1)

(which for loops is invariant under cyclic reparametrizations).

The following general results allows a natural introduction of the Bowen-Lanford zeta function

ζM [5].

Theorem 2.2. LetGbe a finite graph andMa non-backtrackingE ×b Ebmatrix. Then for allu∈C

such that|u|−1 _>_kMk

∞ := max_e∈Eb

e′_∈_E_b|Me,e′|:

det(1−uM) = Y p

1−u|p|χM(p)

=: ζM(u)−1 (2.2) where the product ranges over the primitive oriented loops, each equivalence class contributing a single factor.

Proof. Forusmall enoughln det(1−uM)is analytic inuand satisfies:

ln det(1−uM) = tr ln(1−uM) =−

∞

n=1

n trM

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The sum is convergent since|trMn| ≤ kMkn∞. Furthermore, the assumed bound|u|<kMk−∞1 also guarantees the absolute convergence of the sums obtained by splitting:

trMn = X e1,...,en∈Eb

Men,e1 × n−_Y1

j=1

Mej,ej+1 (2.4)

= X e∈Eb

b γ∋e |bγ|=n

χM(bγ) = ∞

k=1

δ_k|p|,n |p|χM(p)k,

where the last equation is obtained by representing each loop _bγ as the suitable power (k) of a primitive oriented loop andpranges over equivalence classes of such, modulo cyclic permutation. Thus

∞

n=1

n trM

n₌ ∞

k=1

p 1

u|p|χM(p)

=−X

ln1−u|p|χM(p)

. (2.5)

Combined with (2.3) this yields (2.2).

The meromorphic functionζM defined in (2.2) is related to Ihara’s graph zeta function, whose usual definition is in terms of site-indexed paths andV × V matrices. In that case the customary non-backtracking restriction onpneeds to be added explicitly. Also the statement and proof of the corresponding result are a bit more involved. Further background on this topic, other extensions, and graph theoretic applications, can be found in, e.g., [35, 28].

In view of the simplicity of the argument, it may be worth stressing that the limitations placed on the loops in (2.2) leave the possibility of self intersection, multiple crossing of edges, and arbitrary repetitions of sub-loops. The product in (2.2) is over an infinite collection of factors, whose series expansions yield terms with arbitrary powers ofu. However on the left side is a polynomial inu. Thus contained in this zeta function relation is an infinite collection of combinatorial cancellations.

3. The Ising model

3.1. The model’s basic variables

The Ising model is a system of±1valuedspin variables(σx)attached to the verticesVof a graph

Gwith the energy function

H(σ) = − X

{x,y}∈E

Jx,yσxσy−h

x∈V

σx (3.1)

given in terms of edge couplingsJx,y. The probability distribution representing thermal equilib-rium is the Gibbs measure, referred to as theGibbs state,

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for which the expectation value functional will be denoted byh·iG_β,h(some of whose indices will occasionally be omitted). The normalizing factor is thepartition function ZG(β, h). Through it, one computes thethermodynamic pressure:

ψ(β, h) = 1

|V|logZG(β, h). (3.3)

Of particular interest is the infinite volume limit|V| → ∞for the pressure, for which singularities may develop corresponding to phase transitions. Of further interest are the infinite volume limits of the Gibbs equilibrium states. These can be viewed as tangents of the pressure functional [31], and unlikeψ(β, h)their limits may depend on the boundary conditions.

For most of the analysis which follows, the model’s coupling constants need not be constant, or ferromagnetic. The first of these restriction will be invoked only in extracting Onsager’s formula for the free energy from the Kac-Ward determinantal expression for the caseG=Z2_and_J_x,y₌_J

for all{x, y} ∈ E(Z2_).

3.2. A graphical high temperature representation

In discussing the model’s partition function it is convenient to split off a trivial factor and denote:

ZG(β, h) := ZG(β, h)



2|V| Y {x,y}⊂E

cosh(βJx,y)



. (3.4)

For any collection of edgesΓ⊂ E, we denote by∂Γthe collection of vertices of the graph which are oddly covered byΓ. Edge collections with∂Γ = ∅correspond to so calledeven subgraphsof

G. In these terms one has:

Lemma 3.1. For any finite graphG the Ising partition function at vanishing field admits the fol-lowing expansion into even subgraphs:

ZG(β, h= 0) =

Γ⊂E:∂Γ=∅

w(Γ) (3.5)

with w(Γ) :=Q_{x,y}∈_ΓW_{x,y}at the weights

W{x,y} = tanh(βJx,y). (3.6)

The corresponding spin correlation functions of any even number of verticesA⊂ Vare represented by

x∈A

σxiβ,h=0 ZeG(β, h= 0) =

Γ :∂Γ=A

w(Γ). (3.7)

This well known representation is obtained by writing

eβJx,yσxσy _{= cosh(}_βJ

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expanding _Y

{x,y}⊂E

(1 +σxσytanhβJx,y) =

{x,y}∈Γ

[σxσytanh(βJx,y)],

and then summing the resulting expression over the spin configurations(σ). Equation (3.5) results from the relationP_σQ_{x,y}∈_Γσxσy = 2|V|1[∂Γ = ∅]with1[. . .]denoting the indicator function. The second identity (3.7) follows by similar arguments; see also [17, 11].

4. Onsager’s solution and the Kac-Ward formula

The field of statistical mechanics was transformed by Onsager’s announcement of the following result.

Theorem 4.1(Onsager [27]). For the Ising model with constant nearest-neighbor couplings,Jx,y=

J >0, onG=Z2 _{the infinite-volume pressure is given by}

ψ(β,0) = 1 2

lnY(β) +Y(β)−1−2+E(k1, k2)

dk1dk2

(2π)2

+ 1

2ln cosh(2βJ) (4.1) with

Y(β) := sinh(2βJ), E(k1, k2) := 2 sin2(k1

2) + 2 sin

2₍k2

2).

The significance and impact of Onsager’s solution cannot be overstated. It demonstrated for the first time the possibility of modeling the divergence of specific heat at a phase transition; in this case asClog|β−βc|, withβc characterized byY(βc) = 1.

Onsager’s solution was algebraic in nature. Its elaboration, by Kaufman [22], and later Schultz, Mattis and Lieb [33] showed the unexpected utility of fermonic structures in this context. The crit-ical exponents which the exact solution(s) yielded have provided backing to more general theories of critical phenomena, based on unproven assumptions. Developments which followed (of which there were many) allowed to establish existence of the scaling limit of the critical model, and its conformal invariance – the latter proven by Smirnov [32] forZ2 _{and by Chelkak and Smirnov [9]}

for more general planar graphs. A recent review, with emphasis on other methods than those on which we focus here can be found in [8].

Solutions of other 2D models followed, though none inD > 2dimension. Despite the lack of exact solutions many of the essential features of the model’s critical behavior have been understood (including at the level of rigorous results, which this may not be the place to list) and more may be on its way [14].

4.1. The Kac-Ward theorem and its extension beyond planarity

We now turn to the method proposed by Kac and Ward [23] for a combinatorial solution of the Ising model. A key role in it is plaid by theE ×b Ebnon-backtracking matrixK, which for planar

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graphs of straight edges is defined as

Ke′_,e := 1[e′⊲e;e′ 6=e] exp i

2∠(e, e

′₎ _, _(4.2)

with∠₍_{e, e}′₎_∈₍₋_{π, π}₎_{the change of the edge tangent’s argument from the terminal point of}_e′ _to

the origin ofe.

To increase the method’s reach it is convenient to consider also an extension of the corresponding matrix beyond strict planarity, to faithful projections of graphs toR2_{. which we define as follows.}

(An example is depicted in Fig. 1.)

Definition 4.2. A faithful projection of a graphG (which need not be planar) toR2 _{is a graph}

drawn in the plane such that

1. the projection is graph isomorphic toG, in the sense that pairs of projected edges meet at a common end point only if so do the corresponding edges inG,

2. the projected edges are described by piecewise differentiable simple curves which except for their end points avoid the graph vertices.

Figure 1: A faithful projection onR2_{of a non-planar graph. Highlighted is a loop whose projection}

exhibits a non-vertex edge crossing, i.e., one which is not realized in the original graph. Such crossings are excluded in the planar projections which are discussed in Theorem 4.3. Their effects is presented in Theorem 4.4.

We extend the definition of the matrix (4.2) to such faithful projections by letting ∠₍_e′_{, e}₎ _∈

(−π, π]be the sum of the discrete changes in the tangents argument, from the origin ofeto that of

e′, plus the integral of the increase in the tangent’s argument increments along the edgee. Referring to planar graphs we shall by default mean graphs faithfully projected toR2_{with no-crossing edges.}

The extension beyond strict planarity will be employed here in clarifying the Kac-Ward resolvent kernel’s meaning in terms of recognizable observables. However for simplicity of the reading of the main combinatorial argument, it is recommended to first focus of the planar case:

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Theorem 4.3(Kac-Ward-Feynman-Sherman). For any finite planar graphG:

ZG(β, h= 0) =

det(1− KW) (4.3) withWtheE ×b Ebdiagonal matrix with diagonal entries given byW₍_x,y₎=W₍_y,x₎= tanh(βJx,y). The right hand side involves the principal value of the square root function.

The transition from the determinantal expression (4.3) to Onsager’s explicit formula of the par-tition function (4.1) is through a standard calculation. For completeness, it is included here in the Appendix.

Theorem 4.3 has a somewhat long and oft restated history. The formula was presented in the proposal by Kac and Ward [23] for a combinatorial solution of the Ising model which does not require the algebraic apparatus of Onsager [27] and Kaufman [22]. Subsequently, gaps of formu-lation were pointed out by Feynman, who proposed a systematic approach to an order by order derivation of this relation, in steps of increasing complexity [15]. The full proof was accomplished by Sherman [34] and later also by Burgoyne [4], who organized the combinatorial analysis to all orders. At the same time the determinantal formula was given other derivations by Hurst and Green [18], Kasterleyn [21] and Fisher [16] based on Pfaffians and relations to the dimer model. Proofs were extended to arbitrary planar graphs on orientable surfaces in which the connection to spin structures and graph zeta functions was already made; see also [6, 25] and references therein. In Section 6 we provide yet another short proof of Theorem 4.3. The combinatorial arguments used there also yield the following extension of the statement.

Theorem 4.4(Beyond planarity). For any faithful projection of a graph (not necessarily planar) and a set of associated Ising model couplings{Jx,y}

det(1− KW) = X

Γ :∂Γ=∅

(−1)n0(Γ)_w_(Γ) _(4.4)

wheren0(Γ)is the number of non-vertex edge crossings of G’s projection on R2 (an example of

which is depicted in Fig. 1).

5. Basic properties of the Kac-Ward matrices

From the invariance of the edge weightsWeunder reflection (e7→ e) it follows that the multilin-ear functiondet(1− KW) is at most a quadratic polynomial of these edge weights. A stronger statement can be deduced by taking into account also the structure ofK. A key observation is that for the product of the weights associated withK along a primitive oriented loopp imbedded the planeR2 _{yields the parity of the winding}_win(_p₎_{of the loop’s tangent. This in turn is related by}

Whitney’s theorem [36] to the parity of the loop’s self crossing numbern(p):

χ_K(p) = (−1)win(p) = −(−1)n(p). (5.1) A useful consequence is:

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Lemma 5.1. For any faithful projection of a graph toR2_{the matrix}_M₌_KW _satisfies, hδe,Mnδei =hδe,Mnδei,

hδe,Mnδei =hδe,Mnδei= 0, (5.2)

for all oriented edgese∈Eband alln∈N_.

Proof. Each of the terms may be expanded in terms of paths of lengthnon the oriented edges. The first equality follows from the invariance of loop weightsχK(ˆγ), as given by (5.1), with respect

to the reversal of orientation. In particular, invariance of the weights under the mapping which transforms the loopγˆ= (e, e1, . . . , en−1, e)into(e, en−1, . . . , e1, e).

The second inequality is based on the fact that for any path (e, e1, . . . , en−1, e), which starts

oneand ends on the reversed edgee, there is the uniquely associated path(e, en−1, . . . , e1, e)of

opposite contribution, due to the difference in the self-crossing parity between the two paths. The above has the following useful implication.

Lemma 5.2. For the Kac-Ward matrix on the oriented edge set of a faithful projection of a finite graph toR2_,_det(1_{− KW}₎_{is the square of a multilinear polynomial in the weights}₍_W_e_).

Proof. Sincedet(1− M) withM = KW is a quadratic polynomial in eachWe, it suffices by analyticity to establish the result in case all(We)are small, i.e., if

kMk∞ ≤ max degree(G) ×max

e |We| <1. (5.3)

The dependence of the matrixMon the parameterWefor a fixedeis linear with derivative

D(e) := ∂M

∂We

= KP(e) (5.4)

with the rank-two matrixP(e) := |δeihδe|+|δeihδe|, abbreviating the orthogonal projection onto the subspace spanned by the (delta) functions supported only oneande. The assumption (5.3) allows us to apply standard formulas for the derivatives of the logarithm of a determinant:

∂2

∂W2

det(1− M) = 1 2

det(1− M)

(

1 2

1 1− MD

(e)

−tr

1 1− MD

(e) 1

1− MD

(e)

)

. (5.5)

Abbreviating

A:=P(e) 1

1− MD

(e)_P(e) ₌ _P(e)

∞

n=1

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and viewing it as a2×2matrix, the braket{. . .}in (5.5) equals 1

2(trA)

2₋_tr_A2 ₌₋1

2Discriminant(A). (5.7) From Lemma 5.1 and the power series representation in (5.6) we conclude thatAis a multiple of the2×2identity matrix. Hence its discriminant is zero. This implies thatpdet(1− M)is linear in eachWeas claimed.

The last statement allows the following shortcut in the zeta-function represenation (2.2) of the determinantdet(1−KW)in terms of equivalence classes of primitive oriented loops. Since we are interested in its square root, it is advantageous to rewrite this respresentation in terms of equivalence classes ofunoriented loops. For any primitive loopγ = (e0, e1, . . . , en = e0)withej ∈ E, we associate the weight

w(γ) = n

j=1

Wej. (5.8)

Lemma 5.3. For any faithful projection of a finite graphGtoR2 _{and supposing}_(5.3):

det(1− KW) = 1 + ∞

n=1

γ1,...,γn

k{γ1,...,γn}k∞=1

j=1

(−1)n(γj)w(γj). (5.9)

The sum extends over equivalence classes of unoriented loopsγ1, . . . , γnonEin which we restrict to the case that the maximal number an edge is covered by the given collection of loops is one,

k{γ1, ..., γn}k∞:= max e∈E

j=1

1[e∈γj]. (5.10)

Proof. The zeta function relation (2.2) applied to the the Kac-Ward matrix gives for |u| small enough

det(1−uKW) = Y p

[1−u χKW(p) ] (5.11)

= Y p

1 +u|p|(−1)n(p)w(p)i = Y γ

1 +u|γ|(−1)n(γ)w(γ)i2

where we used: i) the relationχKW(p) = χK(p)w(p), ii) the Whitney relation (5.1), and iii) the

fact that both(−1)n(p)_and_w₍_p₎_{are invariant under the loop’s orientation reversal. Therefore, the}

contribution of every primitive oriented loopp equals the square of the corresponding primitive unoriented loopγ. Consequently,

det(1−uKW) = Y γ

1 +u|γ|(−1)n(γ)χW(γ)

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where γ ranges over equivalence classes of unoriented non-backtracking loops. The assump-tion (5.3) also allows to expand the product on the right side into

1 +u|γ|(−1)n(γ)w(γ)i= 1 + ∞

n=1

γ1,...,γn

j=1

u|γj|₍₋₁₎n(γj)_w₍_γ

j). (5.13)

Grouping the terms in the resulting sum according tok{γ1, ..., γn}k∞we reach a key step in the argument: By Lemma 5.2, the left side of (5.12) is a multilinear function of the edge weights(We). By the uniqueness of power series expansion in the regime of its absolute convergence, one may deduce that the sum over terms on the right side of (5.12) withk{γ1, ..., γn}k∞>1vanishes. One is therefore left with only the terms with no doubly covered edges. Since there are only finitely many terms, the relation extends by analyticity to all u, with u = 1corresponding to the case claimed in (5.9).

It should be noted that the last result represents a major reduction of combinatorial complexity: the sum in (5.9) is not only finite but it ranges over only edge disjoint collections of loops.

6. A short proof of the Kac-Ward-Feynman-Sherman theorem

The sum in (5.9) is linked a with the Ising model’s partition function through the following combi-natorial identity which, in the context of graphs of degree4, was among the insightful observations in the Kac-Ward seminal paper.

Lemma 6.1([23]). For any finite planar graphGand any even subgraphsΓ⊂ E(∂Γ =∅)

Γ=⊔{γj}

(−1)n(γj) = 1, (6.1)

where the sum is over all decompositions ofΓ =⊔{γj}into unoriented primitive loops.

1 _-1 +1

= +

1 +

Figure 2: The different resolutions of a vertex of order4, which together determine the admissible loop decompositions of an even subgraph Γ. At each vertex the corresponding (±1) crossing phase factors add up to1.

Proof. The decompositions of Γ into loops are in one-to-one correspondence with the different possible pairing of the incident edges at the vertices ofΓ, carried out independently at different vertices.

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Since pairs of loops in the plane with only transversal intersections can cross only even number of times, for each of the resulting line pattern the total parity factor in (6.1) coincides with the product over the vertices of the parity of the number of line crossings at the different sites, which for future use we denote

(−1)n(γ1,...,γn) _:= Y j

(−1)n(γj). (6.2)

For vertices of degree4, which is the only case of relevance forZ2_{, the sum is over three possible}

connection patterns. As is depicted in figure Fig. 2, one of them with a single crossing and the two other with with no crossing. The net sum of the corresponding factors is1−1 + 1 = 1. This observation completes the proof for the Ising model on planar graphs degree bounded by4. The more general case is covered by the lemma which follows.

Lemma 6.2. For the collection of pairing of an even collection of lines in the plane which meet transversally at a vertex of degree2n, the sum over the parity of the number of line crossing is1. Proof. The argument can be aided by a figure in which the intersection site is amplified into a disk, as depicted in Fig. 3.

Figure 3: An even number of lines meeting at a vertex, and one of their possible pairings, in this case of parity(−1)3 =−1.

We prove by induction. The casen= 1is trivially true.

As the induction step, we assume the statement is true fornand consider the casen+ 1. Let us classify the pairing configurations by the index, 1 < j ≤ 2n, of the line with which ℓ1 is

paired. The line obtained by linkingℓ1 andℓj splits the remaining incidental line segments into two sets, of(j−2,2n−j) elements. We now note that the induction hypothesis implies that the correspondingly restricted sum of the intersection parities is(−1)j. That is so since the parity of the other lines’ intersections with the first one is deterministically (−1)j, and the sum of the parities of the internal intersections of the rest is, by the induction hypothesis+1. Therefore, the sum in question is

j=2

(−1)j = 1 (6.3)

being an alternating sum of±1which starts and ends with +1. Hence the statement exdends to

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It may be added that the line pairing can be indexed by permutations π ∈ S2n such that

π(2j −1) < π(2j) for allj ∈ [1, ..., n]andπ(2j−1)is monotone increasing in j. The line intersection parity coincides then with the permutation’s parity. The combinatorial statement of Lemma 6.2 is thus known in a number of forms.

Lemma 6.1 dealt with the leading collection of terms of (5.9) corresponding tok{γ1, ..., γn}k∞= 1. The results stated before that provide effective tools for dealing with the rest.

Proof of Theorem 4.3. Under the condition (5.3) on the weights(We)one may start from the ex-pansion (5.9) ofpdet(1− KW)into collection of loops constrained to have no double covered edges. It is natural to organize the terms of the sum in (5.9) according to the (disjoint) union of edgesΓ :=⊔n

j=1γj. One gets in this way:

det(1− KW) = X

Γ :∂Γ=∅

w(Γ) X

loop decompositions

Γ=⊔{γj}

(−1)n(γ1,...,γn)_. _(6.4)

The remaining sum is the subject of Lemma 6.1, which therefore implies the Kac-Ward formula. This proves the Kac-Ward-Feynman-Sherman theorem in case (5.3). By analyticity of the determi-nant and the square of the (non-negative) partition function in the weights, the claimed relation (4.3) extends to all values of(We)for finite graphs.

Reviewing the proof of Theorem 4.3 one finds that the graph’s planarity, which is assumed there, plays a role only in the last step.

Proof of Theorem 4.4. Starting from (6.4) it was shown that for each even subgraph Γ the sum over loop decompositions ofΓof the parity factors at each vertex adds up to(+1). In the present context, this last statement still applies to each vertex of the faithful projections, but not to the non-vertex crossings. However, at non-vertex crossings there is no ambiguity in the local structure of the loop decompositions. The arguments used to conclude the proof of Theorem 4.3 thus leaves one with exactly the factor(−1)n0(Γ)_{, as claimed in (4.4).}

7. The emergent fermionic structure

7.1. Disorder operators

It was pointed out by Kadanoff and Ceva [20] that it is natural to consider alsodisorder operators. These are associated with vertex-avoiding lines, drawn over the graph’s planar projection. Associ-ated to each lineℓis a transformation of the Hamiltonian,H 7→ RℓH, which flips the sign of the couplings(Jx,y)for edges{x, y} ∈ E which are crossed an odd number of times byℓ.

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Extending the notion of correlation function from the mean values of spin products,hQ_jσxji

G β,0,

to the effect of more general operations on functional integrals or sums, one may define:

Rℓji

G _:=

σe−β(Rℓn◦···◦Rℓ1H)(σ)

ZG(β,0) . (7.1)

where we subsequently assume thath = 0. Under the Kramers-Wannier duality, for any planar graph and collection of lines{ℓj}

Rℓji

G ₌_hY j

y∈∂ℓj

σyiG

∗

(7.2)

where the expectation valueh·iG∗refers to an Ising system on the dual graphG∗, at the dual values ofKe ≡βJe. The duality is theQ= 2case of theQstate Potts models relation

e2Ke ₋₁ _e2K∗_e∗₋₁ ₌ _{Q ,} _(7.3)

which in terms of the weightsWe = tanhKecan be restated as:

W_e∗∗ =

We−1

We+ 1

. (7.4)

Long-range order in the dual model indicates (in the homogeneous case) that the original spin system is in its disordered phase. This, and the relation (7.2) led to the proposal to callRℓdisorder operators[20]. Adapting a convention for a canonical choice of the line trajectory given its end points, leads to a useful, though slightly deceptive, picture of the disorder variables as local opera-tors, whose correlation functions coincide with the spin correlation functions of the dual model.

The disorder variables can also be viewed as partial implementations of a gauge symmetry. This gauge symmetry perspective is useful in extending the construction to other systems, e.g. dimer covers [2]. For Ising systems ath = 0 any preselected spin flip can be viewed as gauge transformation, under which the physics is preserved but the interaction appears modified. Due to this symmetry,hQ_jRℓji = 1 for any collection of loops, i.e., ∂ℓj = ∅. For open ended lines,

the Gibbs equilibrium system changes in an essential way, but modulo gauge transformation it is a homotopy invariant function of the lineℓ, i.e., up to spin flips it depends only on ∂ℓ. More explicitly, the choice of paths by which specified boundary points ofℓjare linked do not affect the expectation values in (7.2). That is not quite true for the correlation functions of mixed operators, such as:

σxk

Rℓji

G _:=

kσxke

−β(QjRℓjHJ)(σ)

ZG(β,0) . (7.5)

However, these just change sign whenever one of the lines is deformed over one of the spin sites, while the lines’ end points are kept fixed.

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6 5

4 3

Figure 4: Order-disorder variables for a planar graph. The disorder variablesτℓjare defined through

a set of linesℓj, each linking a dual sitex∗j ∈ G∗ which is a neighbor ofxj inG × G∗ with a common dual sitex∗

0 ∈ G∗, called grand central. The disorder linesℓ1, ℓ2, . . . are

enumerated cyclicly in the order of the lines’ emergence from the grand centralx∗₀. (The figure is reprinted from [2] where it was used to demonstrate a related structure in dimer covers.)

7.2. Fermionic order-disorder operator pairs

Pairs of order and disorder variables exhibit an interesting emergent structure which, as was pointed out in [20], gives an explicit expression to Kaufman’s spinor operators. The presence of spinor algebra was recognized early on in her algebraic approach to the model’s solution [22]. Since then it has been realized to be also of great relevance for the field theoretic description of the model’s critical state.

Definition 7.1. LetG = (V,E)be a planar graph, andW :E 7→C_{a set of edge weights (}_W_x,y ₌

tanh(βJx,y)). Pairs oforder-disorderoperators consist of products

µ(xj, ℓj) = σxjRℓj (7.6)

whereσxj are Ising spins associated with sitesxj ∈ V, andRℓj are the disorder operators

associ-ated with piecewise differentiable lines in the plane, which avoid the sites ofG, such that one of the end pointsx∗_j of eachℓj is in a plaquette ofGwhose boundary includesxj, and the other is in a common plaquette ofGto which we refer as thegrand centralx∗₀.

The defining feature of fermionic variables is the change of sign of their correlation functions, including with other variables, which occurs when pairs of fermionic variables are exchanged. To see this occurring in the present context one should note that the disorder operatorsµjare not quite local: tethered to each is a disorder line. When one of these lines is made to pass over a spin site the productQ_jµj changes sign. However, the correlation functionhQ_jµji depends on the lines{ℓj}only through the combination of their end points and a collection of binary variables

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Figure 5: One of the choices for association of disorder lines with disorder sites. Under any such convention the product of disorder variables may be presented as associated with just disorder sites, and a single±1variable which changes sign whenever one of the disorder lines passes over a spin site.

To organize the information in terms of a minimal set of variables it is convenient to adapt a convention, an example of which is indicated in Fig. 5, for a ‘canonical assignment’ of a disorder lineℓj to each potential disorder site x∗j. The correlation is then a function of the configuration of the order-disorder sites (xj, x∗j), multiplied by the binary variable τ =

j,kτj,k. The latter changes sign whenever one of the disorder sites moves across a well defined set of lines in the plane in which the graph is embedded. Equivalently, one may regard the correlation function as a single-valued function defined over a two-sheeted cover of the graph. This perspective is developed further in [9].

In a sense similar to the above, a sign change occurs also when any of the order-disorder pair is rotated by2π(e.g. when the disorder site is taken around its associated spin site). In this the order-disorder operator pairs resemblespinors. As was pointed out in [20], that provides a strong hint that in the scaling limit of the critical model the corresponding operator is of conformal dimension 1/2. Exact methods confirm that to be the case.

In discussing pair correlationsµj =µ(xj, ℓj),j = 1,2, it is natural to replace the two lines by a single lineℓ1,2 with end points∂ℓ1,2 ={x1, x2}. In the setup described above, this is achieved

by linking the pairs of lines (ℓ1, ℓ2) within x∗0, possibly deforming the resulting curve to fit a

convenient convention, and linking each endpointx∗_j through a straight line toxj. In case of just two order-disorder pairs, a natural convention is to link{x∗₁, x∗₂}by a straight line adjusted so that the constructed lines would avoid the vertices ofG. Associated toℓ1,2is an angle

θ1,2 := ∆arg(ℓ1,2)∈(−π, π] (7.7)

which describes the change of the tangent’s argument when starting fromx2 and following along

ℓ1,2 tox1 thus explicitly including the turn from the straight line connecting x2 tox∗2 to the line

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The main new result in this section is that the corresponding two point function is given by the Kac-Ward matrices’ resolvent kernel

G(e1, e2) :=

1 1− KW

e1,e2

, e1, e2∈Eb. (7.8)

More explicitly:

Theorem 7.2. For an Ising model on a finite planar graphGwith pair interactions whose sign can be arbitrary and any pair of order-disorder variables:

hµ1µ2i ≡ hσx1Rℓ1σx2Rℓ2i = −Re he2iθ1,2 X

e1:o(e1)=x1 e2:t(e2)=x2

e2i∠(e1,µ1)_W

e1G(e1, e2)e

−₂i∠(e2,µ2)i_. _(7.9)

The summation extends over oriented edgese1, e2 ∈Ebwith origino(e1) =x1 and terminal point

t(e2) =x2. The angles∠(ej, µj)withj= 1,2are the change of the tangent’s argument from the straight line(x∗

j, xj)to the origin ofej.

Proof. LetGebe the graph obtained by adding to the edge set ofGthe piecewise differentiable line

ℓwith ∂ℓ = {x1, x2} which includesℓ1,2 as described above. AlthoughGeis not a planar graph,

Theorem 4.4 applies to it, and implies that

det1−KeWf= X

Γ∈Ee:∂Γ=∅

(−1)n0(Γ)

w(Γ), (7.10)

{x,y}∈Γ

W_{x,y} =w_e(Γ)

withKeWfthe natural extension of the Kac-Ward matrixKW to the oriented edges of the augmented graphGewith weightWℓon the added lineℓand its reverseℓ. Orientingℓ= (x2, x1)in the direction

towardsx1, the angle arising in the definition of the matrix (4.2) are

Kℓ,e1 = exp

2(∠(e1, µ1) +θ1,2)

, Kee2,ℓ= exp

−₂i∠₍_e₂_{, µ}₂₎_, _(7.11)

and likewise forℓ.

For a givenΓ⊂Eewithℓ∈Γ, the numbern0(Γ)in the right side of (7.10) equals the number of

crossings of edgesE ⊂ Γwith the extra edgeℓ. It is natural to split the sum in the right side into terms for whichℓ6∈Γandℓ6∈ Γand the remaining terms. According to (3.5) the former sum up toZG(β,0). The latter contribution is linear inWℓ. Taking logarithmic derivatives and evaluating atWℓ = 0we arrive at

d dWℓ

ln LHS(7.10)_Wℓ=0= 1

ZG(β,0)

Γ⊂E ∂Γ={x1,x2}

(−1)n0(Γ)_w_{(Γ) =}_h_µ

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The last equality is due to (3.7) with the factor(−1)n0(Γ)_{taking into account the reversal of the} Ising couplings’ signs on edgesEwhich are crossed by the disorder lineγ1,2.

Proceeding as in (5.5) the logarithmic derivative of the determinant with respect to the weight of the added edge (and all other couplings initially small enough) gives

d dWℓ

ln RHS(7.10)_Wℓ=0

=−1

2tr

1−KeWfKeP

(ℓ)

Wℓ=0 =−

1 2

∞

n=1

trKeWfnKeP(ℓ)

Wℓ=0

=−1

e1,e2∈Eb

Kℓ,e1We1G(e1, e2)Kee2,ℓ+Ke_ℓ,e₂We2G(e2, e1)Ke_e₁_,ℓ

. (7.13)

The second equality is a consequence of the resolvent expansion, which is valid for small enough couplings, cf. (5.3). It facilitates the proof of the last line in which Wℓ was taken to zero.The summation in the last line is non-zero unlesso(e1) =x1andt(e2) =x2. In this case, we have

K_ℓ,e₂Ke_e₁_,ℓ= exp−₂i (∠₍_e₁_{, µ}₁₎₋∠₍_e₂_{, µ}₂_{) +}_θ₁_,₂₎_.

Since alsoWe2G(e2, e1) =We1G(e1, e2), the sum in the right side of (7.13) is the real part of its first term and hence agrees with the right side of (7.9).

Since both sides of (7.9) are meromorphic functions of the couplings(We), the validity of this equality extends by analyticity from initially small to all values of the pair interactions.

7.3. Linear equations for the mixed correlation function

A characteristic property of the resolvent kernel of a local operator is that it satisfies a linear re-lation, and has a natural path expansion. Correspondingly, the resolvent of the Kac-Ward matrix, which was introduced in (7.8) satisfies the linear relation

G(e1, e2) =

e′

(KW)e1,e′G(e ′_{, e}

2) + δe1,e2, (7.14) which has some resemblance to the Dirac equation. It is reminiscent also of the one satisfied by Smirnov’s parafermionic observable, which played an important role in the analysis of the critical model’s scaling limit [9]. In fact, as has been pointed out by Lis [24], the Green function equals to the complex conjugate of the Fermionic generating functions of [9].

For small enough weights (We) (cf. (5.3)), the resolvent’s definition may be expanded into a convergent path expansion:

G(e1, e2) =

1 1− KW

e1,e2

= X b γ:e1→e2

ei∆arg(bγ)/2 Y

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1 4

2 3

a b

Figure 6: An edge with four possible disorder variables attached to it. The dashed lines indicate the location of the disorder lines.

Inserting this path expansion into (7.9) yields a path expansion for the disorder correlatorshµ1µ2i.

Such path expansions are useful in the derivation of other identities. One example are the linear relations initially derived by Dostenko [10]. They concern the four correlationsχj ≡ hµjµ0i,

j ∈ {1, . . . ,4}, obtained by fixing µ0 together with an edge e = (a, b) and its two adjacent

plaquettes in each of which we pick one pointcandd. In case the disorder lines attached toc,d

are as depicted in Figure 6, one has

We (χ1+χ4) = (χ2−χ3)

We (χ2+χ3) = (χ1−χ4) . (7.16)

Inχ1+χ4, the contribution of paths which exit atatowards the edgeecancel due to the angles’s

difference of2π when flowing into ealong the respective disorder lines. All edges other thane

which originate inahave an identical contribution in χ1 andχ4. The differenceχ2 −χ3 on the

other hand cancels all paths emerging frombin directions other thaneon which the contributions fromχ2and−χ3 coincide. Due to the non-backtracking property, the paths which emerge frome

atacontribute equally toχ1+χ4. The respective accumulated angles in these paths inχ1andχ2

also agree. This proves the first relation (7.16). The other one is proven similarly.

It may be added that the fact that certain combinations of Ising model correlation functions obey simply stated linear relations has been noted in a number of previous works albeit through rather different means, cf. [29, 8] and references therein.

7.4. Pfaffian structure of correlations

Let us mention in passing that the combinatorial methods presented here allow to prove also that in planar graphs the correlation functions of the order-disorder are not only fermionic (changing sign upon exchange, as explained above) but, furthermore, resemble the correlation functions of non-interactingfermions. This feature of the model was noted and put to use in the early algebraic approach of Schultz, Mattis and Lieb [33], and numerous works which followed, in the context of a transfer matrix approach. However it is valid also for arbitrary planar graphs.

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lines do not cross, and the pairs are labeled in a cyclic order relative toℓj’s intersections with the edge boundary ofx∗₀.

Theorem 7.4(Pfaffian correlations). For a finite planar graphGwith edge weightsK :E 7→C_,

for any collection of canonical pairs of order-disorder variablespj = (xj, ℓj),j ∈ {1, . . . ,2n}, ordered cyclicly relative to the grand central

j=1

µji =

π∈Πn

sign(π) n

j=1

hµ_π₍₂_j−₁₎µ_π₍₂_j₎i ≡ Pfn(hµjµki) . (7.17)

It is of interest to note that one can deduce from this statement that the n-point correlation functions of spin variables lying along the outer boundary of a finite domain also have the Pfaffian structure. In this case the linesℓj can be chosen so they do not cross any edge, and the operators

τℓj act as identity. Consequently, the order-disorder pairs reduce to just regular spin operators.

In case the monomers{x2j−1, x2j}are pairwise adjacent, the disorder lines may be chosen so that their actions are pairwise equivalent, and thus cancel each other. In that case the pairwise product of two order-disorder variables reduces to a an ordinary product of monomers, i.e., a dimer

τ2j−1τ2j =ηx2j−1ηx2j, so that

Y2n

j=1

τj= n

j=1

µx2j−1µx2j

. (7.18)

Related observations were made and applied in the discussion of critical exponents by Kadanoff [19, 20].

We omit here the proof of Theorem 7.4, since the statement is more thoroughly discussed in [3]. Furthermore, a derivation by combinatorial path methods of a closely related dimer-cover version of Theorem 7.4 was recently presented in [2], with which Fig. 4 is shared. Its argument applies verbatim here as well. Let us add that related statements can also be found in both the classical text of the Ising model [26] and in the review of more recent developments by Chelkak, Cimasoni and Kassel [8].

A. From the Kac-Ward formula to Onsager’s explicit solution

Following is a summary of the calculation by which Onsager’s free energy (4.1) emerges from the Kac-Ward determinantal formula (4.3), which we repeat here:

ZG(β,0) =



2|V| Y {x,y}⊂E

cosh(βJx,y)



pdet(1− KW) (A.1)

withKgiven by (4.2) andW = tanh(βJ)

Taking the graph to be the two dimensional torusG =T2

Lof linear sizeL, it is most convenient to evaluatedet(1− KW)not for the free boundary conditions operator, with which the relation

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(A.1) holds in its simplest form, but for its periodicized versionKW(per). Under this change, the model ceases to be planar and the determinant is no longer directly giving the partition function. We address this small discrepancy after an outline of the determinants calculation.

the four edges→= (0,(1,0)),↑= (0,(0,1)),←= (0,(−1,0)),↓= (0,(0,−1))which originate at the vertex0. All other edgese∈Ebresult from translation of these four and may hence be written ase= (x, α)withx∈ V andα∈ {→,↑,←,↓}. The spaceℓ2(Eb)decomposes into a direct sum of

Hk:=

      

ψ(e) =eik·x

    ψ→ ψ↑ ψ← ψ↓    

ψα∈C       

, k∈ 2π

L (Z modL)

2 _. _(A.2)

The matrixK(per)_{is block diagonal in this decomposition, and on}_H

kit acts as

K_k(per):=

   

1 0 e−iπ/4 eiπ/4

0 1 eiπ/4 e−iπ/4

eiπ/4 e−iπ/4 1 0

e−iπ/4 eiπ/4 0 1

       

e−ik1 ₀ ₀ ₀

0 eik1 ₀ ₀ 0 0 eik2 ₀ 0 0 0 e−ik2

  

 . (A.3)

Calculations of the corresponding4×4determinants, and a bit of algebraic manipulation, then yield,

|V|log det

1− K(per)W = 1

2ln[2W(1−W

2_{)] +}

+ 1

k∈2π

L(Z modL) 2

Y(β) + 1

Y(β) −2

+E(k)

. (A.4)

withY(β) := sinh(2βJ)andE(k) := 2sin2(k1/2) + sin2(k2/2).

At anyβ 6= β∗_{, the latter defined by} _Y₍_β∗_{) = 1, the argument of the logarithm in (A.4) is} positive and continuous uniformly in(k1, k2). The sum can be viewed as a Riemann approximation

of an integral, and one gets:

lim L→∞

L2 ln det

1− K(per)W[0,L]2

= 1

2ln cosh(2βJ) + + 1

lnY(β) +Y(β)−1−2+E(k1, k2)

dk1dk2

(2π)2 (A.5)

Recalling that boundary conditions have only a surface effect on the energy density, it seems natural to expect that forβ 6= β∗ this would also be the case for the effect of the change in the

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Figure 7: A graph supporting the3×3Ising spin system with periodic boundary condition. Starting from the free boundary conditions periodicity is obtained by adding two sets of handles. The circles mark their non-vertex crossings.

above calculation from free to periodic boundary conditions on K. Under this assumption the above calculation yields Onsager’s formula (4.1).

The incompleteness of this argument is highlighted by the observation that forβ =β∗the above computed determinant vanishes due to the multiplicative contribution of thek= (0,0)mode. The gap can be patched in a number of ways, of which we like to mention two.

First, a direct analysis of the free boundary condition determinant is also possible and is only a bit more involved than the periodic case. The two are related by a spectral shift of orderO(L) which forβ 6=β∗does not affect the result. An alternative proof is outlined next.

B. Interpreting the periodic determinant

As shown drastically by the vanishing of det 1− K(per)_W_{, this determinant is not a partition}

function (which cannot vanish at real couplings). What it is can be answered with the help of Theorem 4.4, from which we learn:

Lemma B.1. For the Ising model on(0, L]2∩Z2_{with periodic boundary conditions,}

det1− K(per)W = X

Γ;∂Γ∅

(−1)F1(Γ)+F2(Γ)₍₋₁₎F1(Γ)·F2(Γ)_w_(Γ)

= Zb_L(per)h(−1)F1(Γ)+F2(Γ)+F1(Γ)·F2(Γ)_i(per) _(B.1) whereFj(Γ)are the numbers of the the horizontal (j = 1) and vertical (j = 2) handles included inΓ. In the last expression, the average is with respect to the probability distribution over the even subgraphs induced by the Gibbs state with periodic boundary conditions.

Proof. Equation (B.1) follows from (4.4), for whose application it should be noted that

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and the Kac-Ward phase for each of the added handles is ei(2π)/2 = −1. The product of these factors yield the oscillatory term in the first equation. The second is just a recast of the first relation in terms of the corresponding Gibbs state average.

To shed more light on (B.1) it helps to supplement it with rigorous results which are expressed in terms of bounds. For the binary term in (B.1) to take the value(−1)it is required that at least one of theFj(Γ)be odd. However, under this conditionΓhas odd flux through each of the vertical (ifj = 1) or each of the horizontal (ifj = 2) vertex-avoiding lines. For even subgraphsΓeach of these conditions implies thatΓincludes a connected path of edges of total lengthL. However, through the non-perturbative sharpness of phase transition theorem [1, 13] for allβ < βLRO, with the latter defined by the onset of long range order (LRO), the above event’s probability decays exponentially inL. Hence asL→ ∞the last factor in (B.1) converges to1, and consequently the limit computed in (A.5) yields the Onsager formula (4.1). By duality arguments the conclusion is valid also for the dual range. Taking into account the continuity of the free energy inβ, we learn that (4.1) is valid throughout the closed setR_\₍_β_LRO_{, β}∗

LRO).

By other techniques – outgrowths of either the breakthrough work of Harris (cf. [17]), or the Russo-Seymour-Welsh theory [12] – it is known that the Ising model does not have a phase in which LRO is exhibited simultaneously by the model and its dual. This implies that

βLRO=βLRO∗ =β∗. (B.3)

Hence the above arguments prove (4.1) for allβ ∈R_.

Acknowledgements

This work was supported in part by the NSF grant DMS-1613296, the Weston Visiting Professor-ship at the Weizmann Institute (MA) and a PU Global ScholarProfessor-ship (SW). MA thanks the Faculty of Mathematics and Computer Sciences and the Faculty of Physics at WIS for the hospitality enjoyed there. We thank Hugo Duminil-Copin and Vincent Tassion for the pleasure of collaboration on topics related to this work.

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lines do not cross, and the pairs are labeled in a cyclic order relative toℓj’s intersections with the edge boundary ofx∗₀.

Theorem 7.4(Pfaffian correlations). For a finite planar graphGwith edge weightsK :E 7→C_,

for any collection of canonical pairs of order-disorder variablespj = (xj, ℓj),j ∈ {1, . . . ,2n},

ordered cyclicly relative to the grand central

j=1 µji =

π∈Πn

sign(π) n

j=1

hµ_π(2j−1)µ_π(2j)i ≡ Pfn(hµjµki) . (7.17) It is of interest to note that one can deduce from this statement that the n-point correlation functions of spin variables lying along the outer boundary of a finite domain also have the Pfaffian structure. In this case the linesℓj can be chosen so they do not cross any edge, and the operators τℓj act as identity. Consequently, the order-disorder pairs reduce to just regular spin operators.

Y2n

j=1

τj= n

j=1

µx2j−1µx2j

. (7.18)

Related observations were made and applied in the discussion of critical exponents by Kadanoff [19, 20].

A. From the Kac-Ward formula to Onsager’s explicit solution

Following is a summary of the calculation by which Onsager’s free energy (4.1) emerges from the Kac-Ward determinantal formula (4.3), which we repeat here:

ZG(β,0) =



2|V| Y {x,y}⊂E

cosh(βJx,y)



pdet(1− KW) (A.1)

withKgiven by (4.2) andW = tanh(βJ)

Taking the graph to be the two dimensional torusG =T2

Lof linear sizeL, it is most convenient to evaluatedet(1− KW)not for the free boundary conditions operator, with which the relation

(2)

The periodicised Kac-Ward operator commutes with lattice shifts. Hence its determinant fac-torizes into a product of the determinants of4×4matrices, indexed by the Fourier modes, i.e. momenta(k1, k2)∈ 2π_L(Z_{mod L}₎2_{. More explicitly, one may take as the basic building blocks of}_Eb the four edges→= (0,(1,0)),↑= (0,(0,1)),←= (0,(−1,0)),↓= (0,(0,−1))which originate at the vertex0. All other edgese∈Ebresult from translation of these four and may hence be written ase= (x, α)withx∈ V andα∈ {→,↑,←,↓}. The spaceℓ2(Eb)decomposes into a direct sum of

Hk:=

      

ψ(e) =eik·x

    ψ→ ψ↑ ψ← ψ↓    

ψα∈C       

, k∈ 2π

L (Z modL)

2 _. _(A.2)

The matrixK(per)_{is block diagonal in this decomposition, and on}_H

kit acts as

K_k(per):=

   

1 0 e−iπ/4 eiπ/4 0 1 eiπ/4 e−iπ/4 eiπ/4 e−iπ/4 1 0 e−iπ/4 eiπ/4 0 1

       

e−ik1 ₀ ₀ ₀

0 eik1 ₀ ₀

0 0 eik2 ₀

0 0 0 e−ik2

  

 . (A.3)

Calculations of the corresponding4×4determinants, and a bit of algebraic manipulation, then yield,

|V|log det

1− K(per)W = 1

2ln[2W(1−W 2_{)] +}

+ 1 L2

k∈2π

L(Z modL) 2

Y(β) + 1 Y(β) −2

+E(k)

. (A.4)

withY(β) := sinh(2βJ)andE(k) := 2sin2(k1/2) + sin2(k2/2).

At anyβ 6= β∗_{, the latter defined by} _Y_(β∗_{) = 1, the argument of the logarithm in (A.4) is} positive and continuous uniformly in(k1, k2). The sum can be viewed as a Riemann approximation of an integral, and one gets:

lim L→∞

1 L2 ln det

1− K(per)W[0,L]2

= 1

2ln cosh(2βJ) + + 1

lnY(β) +Y(β)−1−2+E(k1, k2) dk1dk2

(2π)2 (A.5) Recalling that boundary conditions have only a surface effect on the energy density, it seems natural to expect that forβ 6= β∗ this would also be the case for the effect of the change in the

(3)

above calculation from free to periodic boundary conditions on K. Under this assumption the above calculation yields Onsager’s formula (4.1).

B. Interpreting the periodic determinant

As shown drastically by the vanishing of det 1− K(per)_W_{, this determinant is not a partition} function (which cannot vanish at real couplings). What it is can be answered with the help of Theorem 4.4, from which we learn:

Lemma B.1. For the Ising model on(0, L]2∩Z2_{with periodic boundary conditions,} det1− K(per)W = X

Γ;∂Γ∅

(−1)F1(Γ)+F2(Γ)₍₋₁₎F1(Γ)·F2(Γ)_w(Γ)

= Zb_L(per)h(−1)F1(Γ)+F2(Γ)+F1(Γ)·F2(Γ)_i(per) _(B.1)

whereFj(Γ)are the numbers of the the horizontal (j = 1) and vertical (j = 2) handles included

inΓ. In the last expression, the average is with respect to the probability distribution over the even subgraphs induced by the Gibbs state with periodic boundary conditions.