Phase transition free regions in the Ising model via the Kac-Ward operator
PPHASE TRANSITION FREE REGIONS IN THE ISINGMODEL VIA THE KAC–WARD OPERATOR
MARCIN LIS
Abstract.
We provide an upper bound on the spectral radius of theKac–Ward transition matrix for a general planar graph. Combined withthe Kac–Ward formula for the partition function of the planar Isingmodel, this allows us to identify regions in the complex plane where thefree energy density limits are analytic functions of the inverse tempera-ture. The bound turns out to be optimal in the case of isoradial graphs,i.e. it yields criticality of the self-dual Z-invariant coupling constants.
Introduction
The Ising model, proposed by Lenz [17], and solved in one dimensionby his student Ising [13], is one of the most studied models of statisticalmechanics. It was introduced as a model for ferromagnetism with the in-tention to explain spontaneous magnetization. Ising proved that the onedimensional case does not account for the existence of this phenomenon andconcluded that the same should hold in higher dimensions. This was laterdisproved by Peierls [21], whose, now classical, argument established thatin dimensions higher than one the model does exhibit a phase transition inthe magnetic behavior. The critical point, i.e., the value of the tempera-ture parameter where the phase transition occurs, for the model defined onthe two-dimensional square lattice was first identified by Kramers and Wan-nier [16] as the fixed point of a certain duality transformation. The firstrigorous proof of criticality of the self-dual point came together with theexact solution of the two-dimensional model done by Onsager [20], who ex-plicitly computed the free energy density and showed that it is not analyticonly at this particular value of the temperature.Since then, several different methods have been developed to study thetwo-dimensional Ising model. One of them is the approach of Kac andWard [14], who expressed the partition function of the model in terms ofthe determinant of what is now called the Kac–Ward operator. This combi-natorial in nature idea has been so far a source of numerous results about theplanar Ising model. The most classical are the (alternative to the solutionof Onsager and Yang [27]) analytic derivations of the free energy densityand magnetization performed by Vdovichenko [25, 26], who built on earlier
Date : October 30, 2018.2010
Mathematics Subject Classification.
Key words and phrases.
Ising model, phase transition, Kac–Ward operator. a r X i v : . [ m a t h - ph ] M a y MARCIN LIS works of Sherman [23] and Burgoyne [6]. However, most of the articles con-cerning the Kac–Ward formula left many details of the method unexplainedand even contained errors. The first completely rigorous account of this ap-proach seems to be given much later by Dolbilin et al. [10]. A more recenttreatment, presented by Kager, Meester and the author [15], concentrateson loop expansions of the Kac–Ward determinants. As a result, the au-thors not only obtain rigorous proofs of the combinatorial foundations ofthe approach, but also rederive the critical temperature of the Ising modelon the square lattice. The Kac–Ward determinants also turned out to bethe right tool for the computation of the critical point of Ising models de-fined on planar doubly periodic graphs (Cimasoni and Duminil-Copin [9]).Moreover, Cimasoni [8] showed that the Kac–Ward formula can be general-ized to Ising models defined on surfaces of higher genus. Finally, as pointedout by the author in [18], the Kac–Ward method is intrinsically connectedwith the discrete holomorphic approach to the Ising model introduced bySmirnov [24].In this paper, we continue in the spirit of [15], where the spectral radiusand operator norm of the Kac–Ward transition matrices were first consid-ered. We explicitly compute the operator norm of what we call the conju-gated transition matrix defined for a general graph in the plane, and hencewe provide an upper bound on the spectral radius of the standard Kac–Ward transition matrix. Combining this result with the Kac–Ward formulafor the high and low-temperature expansion of the partition function yieldsdomains of parameters of the model where there is no phase transition. Wewill focus only on the analytic properties of the free energy, but our bounds,together with the methods from [15], also allow to identify regions wherethere is spontaneous magnetization or exponential decay of the two-pointfunctions. The advantage of our approach is that it does not require anyform of periodicity of the underlying graph.Moreover, our results are optimal for the Ising model defined on isoradialgraphs with uniformly bounded rhombus angles (see condition (1.5)), i.e.we can conclude that the self-dual Z-invariant coupling constants, first con-sidered by Baxter [2], are indeed critical in the classical sense. To be moreprecise, after introducing the inverse temperature parameter β to the cor-responding Ising model, we show that the thermodynamic limits of the freeenergy density can have singularities only at β = 1. The isoradial graphs, orequivalently rhombic lattices, were introduced by Duffin [11] as potentiallythe largest family of graphs where one can do discrete complex analysis. Asmentioned in [7], this class of graphs seems to be the most general familyof graphs where the critical Ising model can be nicely defined, and it alsoseems to be the one, where our bounds for the spectral radius and operatornorm of the Kac–Ward transition matrix yield the critical point of the Isingmodel.The self-dual Z-invariant Ising model has been extensively studied in themathematics literature. Chelkak and Smirnov [7] proved that the associated HASE TRANSITION FREE REGIONS IN THE ISING MODEL 3 discrete holomorphic fermion has a universal, conformally invariant scalinglimit. Boutillier and de Tili`ere [4, 5] gave a complete description of thecorresponding dimer model, yielding also an alternative proof of Baxter’sformula for the critical free energy density. Mercat [19] defined a notionof criticality for discrete Riemann surfaces and investigated its connectionwith criticality in the Ising model. The self-dual Z-invariant Ising modelis commonly referred to as critical. However, criticality in the statisticalmechanics sense has been established only in the case of doubly periodicisoradial graphs (see Example 1.6 of [9] and the references therein). Asalready mentioned, we extend this result to a wide class of aperiodic isoradialgraphs.This paper is organized as follows: in Section 1, we introduce the Isingmodel and the notion of phase transition, and we state our main theorem.Section 2 defines the Kac–Ward operator and presents its connection to theIsing model. It also contains our results for the Kac–Ward transition matrix.The proof of the main theorem is postponed until Section 3.1.
Results for the Ising model
The Ising model.
Let Γ be an infinite, planar, simple graph embed-ded in the complex plane and let Γ ∗ be its planar dual. We assume that bothΓ and Γ ∗ have uniformly bounded vertex degrees. One should think of Γ asany kind of tiling or discretization of the plane. In particular, Γ can be aregular lattice, or an instance of an isoradial graph (see Section 1.4). We calla subgraph G of Γ a subtiling if there is a collection of faces of Γ, such that G is the subgraph induced by all edges forming boundaries of these faces. Wedefine the boundary ∂ G of G to be the set of vertices of G which lie on theboundary of at least one face which is not in the defining collection of faces.For a simple graph G embedded in the complex plane, we will write V ( G ) forthe set of vertices of G , which we identify with the corresponding complexnumbers. By E ( G ) we will denote the set of edges which are represented byunordered pairs of vertices.Let J = ( J e ) e ∈ E (Γ) be a system of ferromagnetic , i.e. positive, couplingconstants on the edges of Γ. For each finite subtiling G , we will consider an Ising model on G defined by J and the inverse temperature parameter β .Borrowing the notation from [15], letΩ free G = {− , +1 } V ( G ) and Ω + G = { σ ∈ Ω free G : σ z = +1 if z ∈ ∂ G} be the spaces of spin configurations with free and positive boundary con-ditions . The Ising model with (cid:3) boundary conditions ( (cid:3) ∈ { free , + } ) isdefined by a probability measure on Ω (cid:3) G given by P (cid:3) G ,β ( σ ) = 1 Z (cid:3) G ( β ) (cid:89) { z,w }∈ E ( G ) exp (cid:0) βJ { z,w } σ z σ w (cid:1) , σ ∈ Ω (cid:3) , MARCIN LIS where the normalizing factor Z (cid:3) G ( β ) = (cid:88) σ ∈ Ω (cid:3) (cid:89) { z,w }∈ E ( G ) exp (cid:0) βJ { z,w } σ z σ w (cid:1) is called the partition function .Throughout the paper, we will make a natural assumption on the couplingconstants, namely we will require that there exist numbers m and M , suchthat for all e ∈ E (Γ), 0 < m ≤ J e ≤ M < ∞ . (1.1)1.2. Phase transition.
An object of interest in statistical physics is the free energy density (or free energy per site ) defined by f (cid:3) G ( β ) = − ln Z (cid:3) G ( β ) β | V ( G ) | . It is clear that the free energy density is an analytic function of the inversetemperature β ∈ (0 , ∞ ) for every finite subtiling G . However, when G ap-proaches Γ, or more generally, some infinite subgraph of Γ (this is calledtaking a thermodynamic limit ), the limiting function can have a criticalpoint , i.e. a particular value of β where it is not analytic. The existence ofsuch a point indicates that the system undergoes a phase transition whenone varies β through the critical value. This is a universal way of looking atthe phenomenon of phase transition since it can be applied to any model ofstatistical mechanics.Another approach, and perhaps a more natural one in the setting of theIsing model, is to investigate the magnetic behavior of the system. Tothis end, one defines the spin correlation functions , i.e. the expectations ofproducts of the spin variables taken with respect to the Ising probabilitymeasure. The simplest cases are the one and two-point functions (cid:104) σ z (cid:105) (cid:3) G ,β = (cid:88) σ ∈ Ω (cid:3) G σ z P (cid:3) G ,β ( σ ) , (cid:104) σ z σ w (cid:105) (cid:3) G ,β = (cid:88) σ ∈ Ω (cid:3) G σ z σ w P (cid:3) G ,β ( σ ) , z, w ∈ V ( G ) . Since the model is ferromagnetic, and due to the effect of positive boundaryconditions, the corresponding one-point function (cid:104) σ z (cid:105) + G ,β is strictly positivefor all finite subtilings G and for all β . In other words, in finite volume,the spins prefer the +1 state at all temperatures. However, when G ap-proaches Γ, the boundary moves further and further away and, at tempera-tures high enough, its influence on a particular spin vanishes. As a result, thelimiting one-point function equals zero and the spin equally likely occupiesthe +1 and − β is sufficiently large, then (cid:104) σ z (cid:105) + G ,β stays bounded away fromzero uniformly in G . This means that the effect of positive boundary condi-tions is carried through all length scales and there is spontaneous magneti-zation . In this approach, the critical point is the value of β , which separatesthe regions with and without spontaneous magnetization. In some cases, it HASE TRANSITION FREE REGIONS IN THE ISING MODEL 5 is more convenient to investigate the behavior of the two-point functions.Here, one also discerns two different non-critical cases: either the system is disordered , i.e. the thermodynamic limits of the two point functions decayexponentially fast to zero with the graph distance between z and w going toinfinity, or the system is ordered , which means that the limiting two-pointfunctions stay bounded away from zero uniformly in z and w . For periodicIsing models, the critical point defined as the value of β which separatesthese two regimes is the same as the critical point defined via spontaneousmagnetization (see Theorem 1 in [1] and the references therein). In partic-ular, the system exhibits long-range ferromagnetic order if and only if thereis spontaneous magnetization.Property (1.1), together with the conditions we imposed on Γ and Γ ∗ ,is enough for the existence of a phase transition in terms of spontaneousmagnetization and the behavior of the two-point functions. This is a con-sequence of the classical arguments of Peierls [21] and Fisher [12]. In thispaper, we will only focus on the phase transition in the analytic behaviorof the free energy density limits, but our results for the Kac–Ward oper-ator can be also used in the setting of the magnetic phase transition (seeSection 1.5).1.3. The main result.
Let (cid:126)E ( G ) be the set of directed edges of G whichare the ordered pairs of vertices. For a directed edge (cid:126)e = ( z, w ), we define its reversion by − (cid:126)e = ( w, z ) and we obtain the undirected version by droppingthe arrow from the notation, i.e. e = { z, w } . If z is a vertex, then we writeOut G ( z ) = { ( z (cid:48) , w (cid:48) ) ∈ (cid:126)E ( G ) : z (cid:48) = z } for the set of edges emanating from z .Let (cid:126)x and x be systems of nonzero complex weights on the directed andundirected edges of G respectively. We call (cid:126)x (Kac–Ward) contractive if (cid:88) (cid:126)e ∈ Out G ( z ) arctan | (cid:126)x (cid:126)e | ≤ π z ∈ V ( G ) , (1.2)and we say that x factorizes to (cid:126)x if x e = (cid:126)x (cid:126)e (cid:126)x − (cid:126)e for all (cid:126)e ∈ (cid:126)E ( G ) . (1.3)For the origin of condition (1.2), see Corollary 2.5.In the context of the Ising model, two particular systems of edge weightswill be important, namely the so called high and low-temperature weightsgiven bytanh βJ = (cid:0) tanh βJ e (cid:1) e ∈ E (Γ) and exp( − βJ ) = (cid:0) exp( − βJ e ) (cid:1) e ∗ ∈ E (Γ ∗ ) . Definition.
We say that the coupling constants satisfy the high-temperature condition if tanh J factorizes to a contractive system of weights on the di-rected edges of Γ, and we say that they satisfy the low-temperature conditionif exp( − J ) factorizes to a contractive system of weights on the directededges of Γ ∗ . MARCIN LIS I m β ∞ Re β T high T low Figure 1.
The high and low-temperature regimes.Let Υ (cid:3) = (cid:8) f (cid:3) G : G is a finite subtiling of Γ (cid:9) be the family of all free energy densities with (cid:3) boundary conditions, andlet Υ (cid:3) be its closure in the topology of pointwise convergence on (0 , ∞ ).Note that Υ (cid:3) contains all thermodynamic limits and can also contain othertypes of accumulation points of Υ (cid:3) . Using the definition of Z (cid:3) G , it is notdifficult to prove that, under condition (1.1), Υ (cid:3) is uniformly bounded andequicontinuous on compact subsets of (0 , ∞ ). In particular, all sequencesin Υ (cid:3) which converge pointwise, converge uniformly on compact sets, andtherefore all functions in Υ (cid:3) are continuous on (0 , ∞ ). However, this is notenough to conclude analyticity of the limiting functions, and indeed, criticalpoints do arise.In this paper, we show that, if the coupling constants satisfy the high-temperature condition, then all functions in Υ free can be extended analyti-cally to a complex domain T high = (cid:110) β : 0 < Re β < , M | Im β | < π , cosh(2 m Re β )cosh(2 m ) cos(2 M Im β ) < (cid:111) which we call the high-temperature regime . Note that (0 , ⊂ T high . Sim-ilarly we prove that, if the coupling constants satisfy the low-temperaturecondition, then all functions in Υ + can be extended to analytic functions on T low = { β : 1 < Re β } which we call the low-temperature regime . Moreover, we show that Υ (cid:3) isuniformly bounded on compact subsets of the corresponding regimes.For complex analytic functions, this is enough to conclude that all point-wise limits are also complex analytic. More precisely, let D be a complexdomain and let E ⊂ D have an accumulation point in D . The Vitali-Portertheorem (see [22, § HASE TRANSITION FREE REGIONS IN THE ISING MODEL 7 defined on D converges pointwise on E , and is uniformly bounded on com-pact subsets of D , then it converges uniformly on compact subsets of D and the limiting function is holomorphic. In our context, the role of thedomain D is played by the high and low-temperature regimes, and E is theintersection of the given regime with the positive real numbers.In other words, under the high and low-temperature conditions on thecoupling constants, the high and low-temperature regimes are free of phasetransition in terms of analyticity of the thermodynamic limits of the freeenergy density. This is summarized in the following theorem: Theorem 1.1.
If the coupling constants satisfy (i) the high-temperature condition, then all functions in Υ free extendanalytically to T high , and Υ free is uniformly bounded on compact sub-sets of T high . As a consequence, all functions in Υ free are analyticon T high , and in particular on (0 , . (ii) the low-temperature condition, then all functions in Υ + extend an-alytically to T low , and Υ + is uniformly bounded on compact subsetsof T low . As a consequence, all functions in Υ + are analytic on T low ,and in particular on (1 , ∞ ) . The proof of this theorem is provided in Section 3. Its main ingredientsare the Kac–Ward formula for the partition function of the Ising model (seeTheorem 2.1) and the bound on the spectral radius of the the Kac–Wardtransition matrix given in Theorem 2.7.In most of the applications, the role of boundary conditions is immaterialfor the thermodynamic limit of the free energy density. Indeed, it is not hardto prove that whenever | ∂ G| / | V ( G ) | is small, then for β ∈ (0 , ∞ ), f free G ( β )and f + G ( β ) are close to each other (and also to any other free energy densityfunction defined for other types of boundary conditions on G ). Hence, limitsof the free energy density taken along sequences, where the above ratioapproaches zero, are the same for all boundary conditions. In this paper,we consider the free and positive boundary conditions since in these cases,the partition function of the model is given in terms of the determinant ofthe Kac–Ward operator. Thus, one can use properties of the operator itselfto derive results for the free energy density.1.4. The isoradial case.
Assume that Γ is an isoradial graph, i.e. all itsfaces can be inscribed in circles with a common radius, and all the circum-centers lie within the corresponding faces. An equivalent characterizationsays that Γ and Γ ∗ can be simultaneously embedded in the plane in such away, that each pair of mutually dual edges forms diagonals of a rhombus.The roles of Γ and Γ ∗ are therefore symmetric and the dual graph is alsoisoradial. The simplest cases of isoradial graphs are the regular lattices: thesquare, triangular and hexagonal lattice.One assigns to each edge e the interior angle θ e that e creates with anyside of the associated rhombus (see Figure 2). Note that θ e + θ e ∗ = π/ MARCIN LIS ~e ~e θ e e e ∗ θ e ∗ Figure 2.
Local geometry of an isoradial graph and its dual.The underlying rhombic lattice is drawn in pale lines. Thedirected arc marks the turning angle ∠ ( (cid:126)e , (cid:126)e ).There is a particular geometric choice of the coupling constants given bytanh J e = tan( θ e / , or equivalently, exp( − J e ) = tan( θ e ∗ / . (1.4)These coupling constants were first considered by Baxter [2]. We will referto them as the self-dual Z-invariant coupling constants since these are theonly coupling constants that make the Ising model invariant under the star-triangle transformation, and also satisfy the above generalized Kramers-Wannier self-duality (1.4). For more details on their origin, see [3, 4].Observe that in this setting, condition (1.1) is equivalent to the existenceof constants k and K , such that for all e ∈ E (Γ),0 < k ≤ θ e ≤ K < π. (1.5)This means that the associated rhombi have a positive minimal area, andalso gives a uniform bound on the maximal degree of Γ and Γ ∗ .The next corollary states that, for the Ising model defined by the abovecoupling constants, the only possible point of phase transition in the analyticbehavior of the free energy density is β = 1. Corollary 1.2.
Let Γ be an isoradial graph satisfying condition (1.5) . Con-sider Ising models defined by the self-dual Z-invariant coupling constants onfinite subtilings of Γ . Then, all functions in Υ free are analytic on (0 , , andall functions in Υ + are analytic on (1 , ∞ ) .Proof. By (1.4) and the fact that the angles θ sum up to π around each vertexof Γ and Γ ∗ , the self-dual Z-invariant coupling constants simultaneouslysatisfy the high and low-temperature condition. Indeed, the contractiveweight systems on the directed edges are given by (cid:126)x (cid:126)e = (cid:112) tan( θ e / (cid:3) Note, that in this case, the inequalities in (1.2) become equalities.
HASE TRANSITION FREE REGIONS IN THE ISING MODEL 9
Implications for the magnetic phase transition.
Recently [15],the Kac–Ward operator and the signed weights it induces on the closed non-backtracking walks in a graph were used to rederive the critical temperatureof the homogeneous Ising model on the square lattice. It was done both interms of analyticity of the free energy density limit and the change in behav-ior of the one and two-point functions. The methods used there to analysethe correlation functions work also for general planar graphs under someslight regularity constraints. To be more precise, the proof of Theorem 1.4in [15] which gives the existence of spontaneous magnetization, uses the factthat appropriate Kac–Ward transition matrices have spectral radius smallerthan one and that the dual graph (which is Γ ∗ in our setup) has subexpo-nential growth of volume, i.e. the volume of balls in graph distance growssubexponentially with the radius. This condition is, for instance, satisfied byall isoradial graphs where (1.5) holds true. On the other hand, Theorem 1.6and Corollary 1.7 from [15], which yield exponential decay of the two-pointfunctions, use the fact that the operator norm of appropriate Kac–Wardmatrices is smaller than one.The bounds that are stated in Section 2 allow to generalize the aboveresults to arbitrary planar graphs, i.e. together with the methods from [15]they provide regions of parameters J and β where there is spontaneousmagnetization or exponential decay of the two-point functions. These re-gions coincide with those in Theorem 1.1 (one can analytically extend thecorrelation functions to the high and low-temperature regime), that is, ifthe coupling constants satisfy the low-temperature condition, then there isspontaneous magnetization on T low , and if they satisfy the high-temperaturecondition, then there is exponential decay of the two-point functions on T high .In particular, our bounds together with the methods developed in [15] provethat the self-dual Z-invariant weights are critical in the sense of magneticphase transition.We would also like to point out that the arguments, which are used in [15]to conclude analyticity of the free energy density limit, do not work forgeneral graphs since they rely on periodicity of the square lattice. This iswhy, in this paper, we go into details of this aspect of phase transition andwe do not focus on the magnetic behavior of the model.2. Results for the Kac–Ward operator
The Kac–Ward operator and the Ising model.
Let G be a finitesimple graph embedded in the plane. For a directed edge (cid:126)e = ( z, w ), wedefine its tail t ( (cid:126)e ) = z and head h ( (cid:126)e ) = w . For (cid:126)e, (cid:126)g ∈ (cid:126)E ( G ), let ∠ ( (cid:126)e, (cid:126)g ) = Arg (cid:16) h ( (cid:126)g ) − t ( (cid:126)g ) h ( (cid:126)e ) − t ( (cid:126)e ) (cid:17) ∈ ( − π, π ](2.1) be the turning angle from (cid:126)e to (cid:126)g (see Figure 2). The transition matrix for G and the weight system x is given byΛ (cid:126)e,(cid:126)g ( x ) = (cid:40) x e e i ∠ ( (cid:126)e,(cid:126)g ) if h ( (cid:126)e ) = t ( (cid:126)g ) and (cid:126)g (cid:54) = − (cid:126)e ;0 otherwise , (2.2)where (cid:126)e, (cid:126)g ∈ (cid:126)E ( G ). To each (cid:126)e ∈ (cid:126)E ( G ) we attach a copy of the complexnumbers denoted by C (cid:126)e and we define a complex vector space X = (cid:89) (cid:126)e ∈ (cid:126)E ( G ) C (cid:126)e . We identify Λ( x ) with the automorphism of X it defines via matrix mul-tiplication. The Kac–Ward operator for G and the weight system x is theautomorphism of X given by T ( x ) = Id − Λ( x ) , where Id is the identity on X . When necessary, we will use subscripts toexpress the fact that the above operators depend on the underlying graph G .If G is a finite subtiling of Γ, then we will denote by G ∗ the subgraph of Γ ∗ whose edge set consists of all dual edges e ∗ , such that at least one of theendpoints of e belongs to V ( G ) \ ∂ G . One can see that G ∗ is a subtiling of Γ ∗ whose defining set of dual faces is given by the vertices from V ( G ) \ ∂ G . Wewill call it the dual subtiling of G .We say that a graph is even if all its vertices have even degree. Thereare two classical methods of representing the partition function of the Isingmodel on G as a weighted sum over all even subgraphs of G or G ∗ . Thefirst one, called the low-temperature expansion , involves a bijective mappingbetween the spin configurations with positive boundary conditions and thecollection of even subgraphs of G ∗ . The graph associated with a spin configu-ration is composed of these dual edges, whose corresponding primal edge hastwo opposite values of spins assigned to its endpoints. Hence, the resultingeven subgraph forms an interface between the clusters of positive and neg-ative spins in the configuration. In this expansion, each even graph is givena weight which is proportional to the product of the low-temperature edgeweights exp( − βJ ) taken over all edges in the graph. The second methodis called the high-temperature expansion and it is a way of expressing thepartition function of the Ising model with free boundary conditions as asum over all even subgraphs of G . Similarly, it assigns to each even sub-graph a product weight composed of factors given by the high-temperatureweight system tanh βJ . However, unlike in the low-temperature case, theeven subgraphs do not have a geometrical interpretation in terms of the spinvariables. The weighted sums arising in both of these expansions are calledthe even subgraph generating functions .The Kac–Ward formula expresses the square of an even subgraph generat-ing function as the determinant of a Kac–Ward matrix with an appropriateedge weight system. The combined result of the high and low-temperature HASE TRANSITION FREE REGIONS IN THE ISING MODEL 11 expansion together with the Kac–Ward formula is stated in the next theo-rem. Here, we assume that the edges of G (and also G ∗ ) are embedded asstraight line segments which do not intersect. For the origin of this condi-tion, a detailed account of the high and low-temperature expansion, and theproof of the following theorem, see [15]. Theorem 2.1.
For all choices of the coupling constants J and all β with Re β > , (i) (cid:0) Z free G ( β ) (cid:1) = 2 | V ( G ) | (cid:16) (cid:89) e ∈ E ( G ) cosh ( βJ e ) (cid:17) det (cid:2) T G (tanh βJ ) (cid:3) , (ii) (cid:0) Z + G ( β ) (cid:1) = exp (cid:16) β (cid:88) e ∈ E ( G ) J e (cid:17) det (cid:2) T G ∗ (exp( − βJ )) (cid:3) . Note that the condition Re β > βJ to be well defined.The determinant of the Kac–Ward matrix is the characteristic polynomialof the transition matrix evaluated at one:det T = det(Id − Λ) = n (cid:89) k =1 (1 − λ k ) , where n is the number of edges of G , and λ k , k ∈ { , , . . . , n } , are theeigenvalues of Λ. Recall that we want to extend the free energy densityfunctions to domains in the complex plane. The free energy density is givenby the logarithm of the partition function, and the square of the partitionfunction is proportional to the above product involving eigenvalues of thetransition matrix. In this situation, it is natural to use the power seriesexpansion of the logarithm around one:ln(1 − λ ) = − ∞ (cid:88) r =1 λ r /r, | λ | < . This series is convergent whenever λ stays within the unit disc, and hence weshould require that the spectral radius of the transition matrix is boundedfrom above by one. The next section is devoted to providing the necessaryestimates.2.2. Bounds on the spectral radius and operator norm.
In this paperwe will make use of transition matrices conjugated by diagonal matrices ofa certain type: if x factorizes to (cid:126)x (see (1.3)), then we define the conjugatedtransition matrix by Λ( (cid:126)x ) = D − ( (cid:126)x )Λ( x ) D ( (cid:126)x ) , where D ( (cid:126)x ) is the diagonal matrix satisfying D (cid:126)e,(cid:126)e ( (cid:126)x ) = (cid:126)x (cid:126)e for all (cid:126)e ∈ (cid:126)E ( G ).The resulting transition matrix takes the following form:Λ (cid:126)e,(cid:126)g ( (cid:126)x ) = (cid:40) (cid:126)x − (cid:126)e (cid:126)x (cid:126)g e i ∠ ( (cid:126)e,(cid:126)g ) if h ( (cid:126)e ) = t ( (cid:126)g ) and (cid:126)g (cid:54) = − (cid:126)e ;0 otherwise . (2.3)This matrix is similar to the standard transition matrix, and in particularhas the same spectrum. Moreover, it turns out that one can explicitlycompute its operator norm.To this end, let us make some additional observations. For a squarematrix A , let (cid:107) A (cid:107) be its operator norm induced by the Euclidean norm,and let ρ ( A ) be its spectral radius. Note that there is a natural involutiveautomorphism P of X induced by the map (cid:126)e (cid:55)→ − (cid:126)e , i.e. the automorphismwhich assigns to each complex number in C (cid:126)e the same complex numberin C − (cid:126)e . Fix (cid:126)x and let A = P Λ( (cid:126)x ). Observe that (cid:107) A (cid:107) = (cid:107) Λ( (cid:126)x ) (cid:107) since P is anisometry. Moreover, the operator norm of A depends only on the absolutevalues of (cid:126)x . Indeed, if B = D ( (cid:126)u ) AD ( (cid:126)u ) , where (cid:126)u (cid:126)e = | (cid:126)x (cid:126)e | /(cid:126)x (cid:126)e , then B is given by the matrix B (cid:126)e,(cid:126)g = (cid:40) | (cid:126)x (cid:126)e (cid:126)x (cid:126)g | e i ∠ ( − (cid:126)e,(cid:126)g ) if t ( (cid:126)e ) = t ( (cid:126)g ) and (cid:126)g (cid:54) = (cid:126)e ;0 otherwise , (2.4)and (cid:107) B (cid:107) = (cid:107) A (cid:107) since D ( (cid:126)u ) is an isometry.Note that X can be decomposed as X = (cid:89) z ∈ V ( G ) X z , where X z = (cid:89) (cid:126)e ∈ Out G ( z ) C (cid:126)e . One can see from (2.4) that B gives a nonzero transition weight only betweentwo edges sharing the same tail z . In other words, B maps X z to itself andtherefore is block-diagonal, that is B = (cid:89) z ∈ V ( G ) B z , where B z : X z → X z is the restriction of B to the space X z . Moreover, theangles satisfy ∠ ( − (cid:126)e, (cid:126)g ) = − ∠ ( − (cid:126)g, (cid:126)e ) for (cid:126)e (cid:54) = (cid:126)g, (2.5)and hence B is Hermitian, i.e. B (cid:126)e,(cid:126)g = B (cid:126)g,(cid:126)e . Combining these two propertiesand the fact that the operator norm of a Hermitian matrix is given by itsspectral radius, we arrive at the identity: (cid:107) B (cid:107) = ρ ( B ) = max z ∈ V ( G ) ρ ( B z ) . (2.6)It turns out that the characteristic polynomial of B z is easily expressiblein terms of the weight vector (cid:126)x : HASE TRANSITION FREE REGIONS IN THE ISING MODEL 13
Lemma 2.2.
For any real t and any vertex z , det( t Id − B z ) = Re (cid:16) (cid:89) (cid:126)e ∈ Out G ( z ) ( t + i | (cid:126)x (cid:126)e | ) (cid:17) , where Id is the identity on X z .Proof. The proof is by induction on the degree of z . One can easily checkthat the statement is true for all vertices of degree one or two. Now supposethat it is true for all vertices of degree at most n ≥
2. Let z be a vertex ofdegree n + 1 and let (cid:126)e , (cid:126)e , . . . , (cid:126)e n +1 be a counterclockwise ordering of theedges of Out G ( z ). Consider the matrix S = t Id − B z with columns and rowsordered accordingly. Note that for all (cid:126)g ∈ Out G ( z ) different from (cid:126)e and (cid:126)e , ∠ ( (cid:126)g, (cid:126)e ) + ∠ ( (cid:126)e , (cid:126)e ) + ∠ ( (cid:126)e , (cid:126)g ) = 0 (mod 2 π ) . Also observe that, for geometric reasons, at least two of the above angles arepositive. Combining this together with the fact that Arg( w ) = Arg( − w ) ± π for any complex w , and that the angles are between − π and π , yields ∠ ( − (cid:126)e , (cid:126)g ) = ∠ ( − (cid:126)e , (cid:126)g ) + ∠ ( − (cid:126)e , (cid:126)e ) + π. (2.7)This identity guarantees that every two consecutive rows and columns of S are “almost proportional” to each other.To be more precise, we first subtract from the first row of S , the secondrow multiplied by ie i ∠ ( − (cid:126)e ,(cid:126)e ) | (cid:126)x (cid:126)e | / | (cid:126)x (cid:126)e | . Then, we subtract from the firstcolumn the second one multiplied by − ie − i ∠ ( − (cid:126)e ,(cid:126)e ) | (cid:126)x (cid:126)e | / | (cid:126)x (cid:126)e | . The result-ing matrix has the same determinant as S . By the definition of B z , (2.5)and (2.7), det S = det a b · · · b t − B z(cid:126)e ,(cid:126)e − B z(cid:126)e ,(cid:126)e · · · − B z(cid:126)e ,(cid:126)e t − B z(cid:126)e ,(cid:126)e · · · − B z(cid:126)e ,(cid:126)e − B z(cid:126)e ,(cid:126)e t · · · ... ... ... ... . . . where a = t (cid:0) | (cid:126)x (cid:126)e | / | (cid:126)x (cid:126)e | (cid:1) and b = − e i ∠ ( − (cid:126)e ,(cid:126)e ) (cid:0) it | (cid:126)x (cid:126)e | / | (cid:126)x (cid:126)e | + | (cid:126)x (cid:126)e (cid:126)x (cid:126)e | (cid:1) .Let S be the matrix resulting from removing from S the first column andthe first row, and let S be the matrix, where the first two rows and thefirst two columns of S are removed. By the induction hypothesis, det S =Re (cid:0) ( t + i | (cid:126)x (cid:126)e | ) ϑ (cid:1) and S = Re ϑ , where ϑ = (cid:81) (cid:126)g ∈ Out G ( z ) \{ (cid:126)e ,(cid:126)e } ( t + i | (cid:126)x (cid:126)g | ).Expanding the determinant, we getdet S = a det S − bb det S = t (cid:0) | (cid:126)x (cid:126)e | / | (cid:126)x (cid:126)e | (cid:1) Re (cid:0) ( t + i | (cid:126)x (cid:126)e | ) ϑ (cid:1) − (cid:0) | (cid:126)x (cid:126)e | | (cid:126)x (cid:126)e | + t | (cid:126)x (cid:126)e | / | (cid:126)x (cid:126)e | (cid:1) Re ϑ = Re (cid:0) ( t + i | (cid:126)x (cid:126)e | )( t + i | (cid:126)x (cid:126)e | ) ϑ (cid:1) . The last equality follows since both sides are real linear in ϑ , and one cancheck that it holds true for ϑ = 1 , i . (cid:3) For z ∈ V ( G ), we define ξ z ( (cid:126)x ) to be the unique solution in s of theequation (cid:88) (cid:126)e ∈ Out G ( z ) arctan (cid:0) | (cid:126)x (cid:126)e | /s (cid:1) = π . (2.8)As a corollary we obtain the following result: Corollary 2.3. ρ ( B z ) = ξ z ( (cid:126)x ) .Proof. Since B z is Hermitian, it has a real spectrum. By Lemma 2.2, thecharacteristic polynomial of B z at a nonzero real number t is given by t | Out G ( z ) | (cid:16) (cid:89) (cid:126)e ∈ Out G ( z ) cos (cid:0) arctan( | (cid:126)x (cid:126)e | /t ) (cid:1)(cid:17) − cos (cid:16) (cid:88) (cid:126)e ∈ Out G ( z ) arctan( | (cid:126)x (cid:126)e | /t ) (cid:17) . This expression vanishes only when the last cosine term is zero. The largestin modulus values of t for which this happens are equal to ± ξ z G ( (cid:126)x ). (cid:3) We can now compute the operator norm of the conjugated transitionmatrix Λ( (cid:126)x ). The following result is the main tool in our considerations:
Lemma 2.4. (cid:107) Λ( (cid:126)x ) (cid:107) = max z ∈ V ( G ) ξ z ( (cid:126)x ) . Proof.
It follows from the fact that (cid:107) Λ( (cid:126)x ) (cid:107) = (cid:107) B (cid:107) , identity (2.6), and Corol-lary 2.3. (cid:3) Note that the operator norm depends only on the absolute values of (cid:126)x .One can rephrase this result as follows:
Corollary 2.5. (cid:107) Λ( (cid:126)x ) (cid:107) ≤ s if and only if (cid:88) (cid:126)e ∈ Out G ( z ) arctan (cid:0) | (cid:126)x (cid:126)e | /s (cid:1) ≤ π for all z ∈ V ( G ) . We say that an operator is a contraction if its operator norm is smaller orequal one, and hence the name of condition (1.2). Since the operator normbounds the spectral radius from above, we obtain the following corollary:
Corollary 2.6. If x factorizes to (cid:126)x , then ρ (Λ( x )) ≤ max z ∈ V ( G ) ξ z ( (cid:126)x ) . This inequality is preserved when one takes the infimum over all factor-izations of the weight system x . One can check that the spectral radius ofthe transition matrix depends not only on the moduli but also on the com-plex arguments of x . Since the above bound depends only on the absolutevalues, it is in general not sharp. Nonetheless, it is optimal for the self-dualZ-invariant Ising model on isoradial graphs. HASE TRANSITION FREE REGIONS IN THE ISING MODEL 15
Remark . Note that finiteness of G was not important in our computations.Since the transition matrix is defined locally for each vertex, we only used thefact that all vertices have finite degree. Hence, one can consider transitionmatrices and Kac–Ward operators on infinite graphs as automorphisms ofthe Hilbert space (cid:96) on the directed edges of G . The results from this sectiontranslate directly to this setting by interchanging all maxima with suprema.This is used in [18] to analyse infinitely dimensional Kac–Ward operators.2.3. High and low-temperature spectral radii.
We will now use thebounds from the previous section in a more concrete setting of the high andlow-temperature weight systems. We define R ( β ) = sup G ρ (cid:2) Λ G (tanh βJ ) (cid:3) and R ∗ ( β ) = sup G ρ (cid:2) Λ G ∗ (exp( − βJ )) (cid:3) , where the suprema are taken over all finite subtilings of Γ. The reasonfor our particular choice of the high and low-temperatures regimes in thestatement of Theorem 1.1 is the following result: Theorem 2.7.
If the coupling constants satisfy (i) the high-temperature condition, then sup β ∈ K R ( β ) < for any com-pact set K ⊂ T high . (ii) the low-temperature condition, then sup β ∈ K R ∗ ( β ) < for any com-pact set K ⊂ T low .Proof. We will prove part (i). Fix a compact set K ⊂ T high and let L ( β ) = sup j ∈ [ m,M ] | tanh βj | tanh j = sup j ∈ [ m,M ] cosh j sinh j (cid:115) cosh(2 j Re β ) − cos(2 j Im β )cosh(2 j Re β ) + cos(2 j Im β ) . By compactness of [ m, M ] and the fact that the hyperbolic tangent does notvanish and is continuous in the right half-plane, L is a continuous functionon { β : 0 < Re β } . From a simple computation, it follows that L ( β ) < j Re β ) / cosh 2 j < cos(2 j Im β ) for all j ∈ [ m, M ] . The above inequality can hold only when | Re β | < M | Im β | < π . Underthese assumptions, both sides of the inequality are decreasing functions of j .This means that the above condition is satisfied whenever 0 < Re β < M | Im β | < π and cosh(2 m Re β ) / cosh 2 m < cos(2 M Im β ). Hence, by thedefinition of T high , we have that T high ⊂ { β : L ( β ) < } and thus, bycontinuity of L , s := sup β ∈ K L ( β ) < . From the definition of L , it follows that | tanh βJ e | / tanh J e ≤ s for all e ∈ E (Γ) and β ∈ K. We assume that the coupling constants J satisfy the high-temperaturecondition, which means that the weight system tanh J factorizes to a con-tractive weight system (cid:126)x . Therefore, for β ∈ K , tanh βJ factorizes to aweight system (cid:126)x ( β ) satisfying | (cid:126)x (cid:126)e ( β ) | = (cid:112) | tanh βJ e | / tanh J e · | (cid:126)x (cid:126)e | , andhence | (cid:126)x (cid:126)e ( β ) | /s ≤ | (cid:126)x (cid:126)e | for all (cid:126)e ∈ (cid:126)E (Γ). Since arctan is increasing and (cid:126)x is contractive, we have by Corollary 2.5 that (cid:107) Λ G ( (cid:126)x ( β )) (cid:107) ≤ s for all sub-tilings G and all β ∈ K . The claim follows because the spectral radiusis bounded from above by the operator norm, and Λ G ( (cid:126)x ( β )) has the samespectral radius as Λ G (tanh βJ ).Part (ii) involves less computations and can be proved similarly afternoticing that T low = (cid:110) β : sup j ∈ [ m,M ] | exp( − βj ) | exp( − j ) < (cid:111) . (cid:3) In the light of Thoerem 2.1 and the remarks which follow it, we are nowin a position to prove our main result.3.
Proof of Theorem 1.1
Proof.
We will prove part (i). Suppose that the coupling constants satisfythe high-temperature condition and fix a compact set K ⊂ T high . We have toshow that the functions f free G extend analytically to T high and are uniformlybounded on K .First of all, since zero is not in T high , the factor 1 /β is analytic on T high and uniformly bounded on K . Thus, it is enough to consider functions ofthe form ln Z free G ( β ) / | V ( G ) | . We will use the formula from part (i) of Theo-rem 2.1. The logarithm of the partition function can therefore be expressedas a sum of three different terms. The first one is the constant | V ( G ) | ln 2,which equals ln 2 after rescaling by the number of vertices.To talk about the second term, which comes from the product of hy-perbolic cosines, one has to argue that there is a continuous branch ofln(cosh βJ e ) on T high . Indeed, one can take the principal value of the log-arithm since Re(cosh βJ e ) = cosh( J e Re β ) cos( J e Im β ) > T high . An-alyticity of this term follows since cosh βJ e is analytic. Furthermore, wehave (cid:12)(cid:12)(cid:12) ln (cid:16) (cid:89) e ∈ E ( G ) cosh βJ e (cid:17)(cid:12)(cid:12)(cid:12) ≤ (cid:88) e ∈ E ( G ) (cid:12)(cid:12) ln(cosh βJ e ) (cid:12)(cid:12) ≤ (cid:88) e ∈ E ( G ) (cid:16)(cid:12)(cid:12) ln | cosh βJ e | (cid:12)(cid:12) + | Arg(cosh βJ e ) | (cid:17) ≤ | E ( G ) | (cid:16) sup j ∈ [ m,M ] (cid:12)(cid:12) ln | cosh βj | (cid:12)(cid:12) + π/ (cid:17) . Since the hyperbolic cosine does not vanish in the right half-plane and [ m, M ]is compact, the above supremum is a continuous function of β on T high , andtherefore is bounded on K . The number of edges is bounded by the number HASE TRANSITION FREE REGIONS IN THE ISING MODEL 17 of vertices times the maximal degree of Γ, and thus, after rescaling by thevolume, this term is uniformly bounded in G .The last term is given by the logarithm of the determinant of the Kac–Ward operator. Let λ k , k ∈ { , , . . . , n } , n = | E ( G ) | , be the eigenvaluesof Λ G (tanh βJ ). By Theorem 2.7, we know that their moduli are boundedfrom above by some constant s < G and β ∈ K ). One cantherefore define the logarithm by its power series around one, i.e.ln det (cid:2) Id − Λ G (tanh βJ ) (cid:3) = ln n (cid:89) k =1 (1 − λ k ) = n (cid:88) k =1 ln(1 − λ k )= − n (cid:88) k =1 ∞ (cid:88) r =1 λ rk /r = − ∞ (cid:88) r =1 2 n (cid:88) k =1 λ rk /r = − ∞ (cid:88) r =1 tr[Λ r G (tanh βJ )] /r, where tr is the trace of a matrix. It is clear that tr[Λ r G (tanh βJ )] is ananalytic function of β . Moreover, | tr[Λ r G (tanh βJ )] | ≤ | E ( G ) | s r for any r ,and therefore the above series converges uniformly on K . It follows that theseries defines a holomorphic function on T high . Again, after rescaling by thenumber of vertices, it becomes uniformly bounded in G . This completes theproof of the first part of the theorem.The proof of part (ii) uses the second formula from Theorem 2.1 andproceeds in a similar manner. (cid:3) Acknowledgments.
The research was supported by NWO grant Vidi639.032.916.
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