Thermodynamics of spin chains of Haldane–Shastry type and one-dimensional vertex models

# Thermodynamics of spin chains of Haldane–Shastry type and one-dimensional vertex models

Alberto Enciso Federico Finkel Artemio González-López Instituto de Ciencias Matemáticas, Consejo Superior de Investigaciones Científicas, 28049 Madrid, Spain Departamento de Física Teórica II, Universidad Complutense de Madrid, 28040 Madrid, Spain
April 13, 2012
###### Abstract

We study the thermodynamic properties of spin chains of Haldane–Shastry type associated with the root system in the presence of a uniform external magnetic field. To this end, we exactly compute the partition function of these models for an arbitrary finite number of spins. We then show that these chains are equivalent to a suitable inhomogeneous classical Ising model in a spatially dependent magnetic field, generalizing the results of Basu-Mallick et al. for the zero magnetic field case. Using the standard transfer matrix approach, we are able to compute in closed form the free energy per site in the thermodynamic limit. We perform a detailed analysis of the chains’ thermodynamics in a unified way, with special emphasis on the zero field and zero temperature limits. Finally, we provide a novel interpretation of the thermodynamic quantities of spin chains of Haldane–Shastry type as weighted averages of the analogous quantities over an ensemble of classical Ising models.

###### keywords:
Spin chains of Haldane–Shastry type, vertex models, transfer matrix method, thermodynamic limit

## 1 Introduction

In this paper we study a class of spin chains whose Hamiltonian can be collectively written as

 H=∑1⩽i

where the ’s are real constants and the interactions are chosen as described below (see Eqs. (1.7)). In the previous formula, the operators act on a state

 |s1,…,sN⟩,si∈{1,…,m},

of the canonical spin basis by permuting the -th and -th spins:

 Sij|s1,…,si,…,sj,…,sN⟩=|s1,…,sj,…,si,…,sN⟩.

The permutation operators can be expressed in terms of the (Hermitian) spin operators with the normalization (where the subindex labels the chain sites) as Polychronakos (2006)

 Sij=1m+12m2−1∑α=1TαiTαj. (1.2)

Let denote the operator whose action on the canonical spin basis is given by

 Sαk|s1,…,sN⟩=(δαsk−δmsk)|s1,…,sN⟩,α=1,…,m−1, (1.3)

so that the operators span the standard Cartan subalgebra of . The operators are then defined by

 Sα=N∑i=1Sαi,α=1,…,m−1. (1.4)

Thus the second sum in Eq. (1.1) can be interpreted as arising from the interaction with a uniform external “magnetic” field111The most general magnetic field term is of the form , where

is a traceless Hermitian matrix acting on the internal space of the -th spin. By performing a rotation in this internal space we can diagonalize the matrix , effectively replacing it by a traceless diagonal matrix. The latter matrix can in turn be expressed in the form , which yields the last term in Eq. (1.1). of strengths . Note that in the case (i.e., for spin ) we can take and , where is a Pauli matrix acting on the -th spin’s Hilbert space. Hence Eq. (1.2) adopts the more familiar form

 Sij=12(1+σi⋅σj), (1.5)

and the Hamiltonian (1.1) reduces to

 H=∑1⩽i

with . In particular, the last term represents the interaction with a uniform magnetic field parallel to the axis with strength (proportional to) .

The three models we shall study are defined by the following choice of the interaction strengths :

The Haldane–Shastry (HS) chain Haldane (1988); Shastry (1988):

 Jij=J2sin2(ξi−ξj),ξk=kπN. (1.7a)

The Polychronakos–Frahm (PF) chain Polychronakos (1993); Frahm (1993):

 Jij=J(ξi−ξj)2, (1.7b)

where is the -th root of the Hermite polynomial of degree .

The Frahm–Inozemtsev (FI) chain Frahm and Inozemtsev (1994):

 Jij=J2sinh2(ξi−ξj), (1.7c)

where is the -th root of the generalized Laguerre polynomial with . In all three cases, is a real constant whose sign determines the model’s ferromagnetic () or antiferromagnetic () character. Note that, while the sites of the HS chain are equispaced222In fact, if is interpreted as an angular coordinate, then the HS chain describes an array of spins equispaced on the unit circle, with long-range pairwise interactions inversely proportional to the square of the chord distance between the spins., this is not the case for the PF or FI chains.

We shall denote by

 H0=∑1⩽i

the Hamiltonian of the chains (1.1) in the absence of a magnetic field. Following standard terminology, we shall collectively refer to the chains (1.7)-(1.8) as spin chains of Haldane–Shastry type. They are all associated with the root system , in the sense that the interactions depend only on the differences of the site coordinates . Although several generalizations of these chains to the and root systems have been considered in the literature Yamamoto and Tsuchiya (1996); Enciso et al. (2005); Barba et al. (2008); Basu-Mallick et al. (2009, 2011), in this paper we shall restrict ourselves to the above -type models.

Spin chains of HS type are the simplest models in condensed matter physics exhibiting fractional statistics Haldane (1991). Historically, the HS chain (1.7a)-(1.8) was introduced as a simplified version of the one-dimensional Hubbard model with long-range hopping, from which it can be obtained in the half-filling regime when the on-site interaction tends to infinity Gebhard and Ruckenstein (1992). Soon after its introduction, it was shown that this chain is completely integrable, in the sense that one can explicitly construct mutually commuting integrals of motion Fowler and Minahan (1993); Bernard et al. (1993). As first observed by Polychronakos Polychronakos (1993), these integrals of motion can be obtained from those of the dynamical spin Sutherland model Ha and Haldane (1992) by means of the so-called “freezing trick”. In fact, the freezing trick can also be applied to derive the PF and FI chains from the Calogero Minahan and Polychronakos (1993) and Inozemtsev Inozemtsev (1996) spin dynamical models. In particular, these two chains are also completely integrable. Apart from their integrable character, spin chains of HS type appear in many areas of current interest in both physics and mathematics, such as quantum chaos Barba et al. (2008, 2009), supersymmetry Basu-Mallick et al. (1999); Basu-Mallick and Bondyopadhaya (2006), conformal field theory Haldane (1991); Basu-Mallick et al. (2008); Cirac and Sierra (2010), the AdS-CFT correspondence Beisert et al. (2003); Bargheer et al. (2009), one-dimensional anyons Greiter (2009) and Yangian quantum groups Bernard et al. (1993); Hikami (1995); Basu-Mallick (1999); Beisert and Erkal (2008).

The partition functions of all three chains of HS type in the absence of a magnetic field, which have been computed in closed form using again the freezing trick Polychronakos (1994); Finkel and González-López (2005); Barba et al. (2010), can be expressed in a unified way by the formula

 Z0(q)=∑k∈PNr∏i=1(m+ki−1ki)⋅q∑r−1i=1JE(Ki)N−r∏i=1(1−qJE(K′i)). (1.9)

Here , is an element of the set of partitions of with order taken into account, and the numbers in Eq. (1.9) are positive integers defined by

 Ki=i∑j=1kj,{K′1,…,K′N−r}={1,…,N−1}∖{K1,…,Kr−1}. (1.10)

Remarkably, the partition function of the chains (1.7)-(1.8) depends on the chain under consideration only through its dispersion relation

 E(i)=⎧⎨⎩i(N−i),for the HS chaini,for the PF chaini(α+i−1),for the FI chain. (1.11)

Using the previous expression for the partition function, Basu-Mallick et al. Basu-Mallick et al. (2010) showed that the spectrum of the spin chains of HS type is given by

 E0(n)=JN−1∑i=1δ(ni,ni+1)E(i),n≡(n1,…,nN), (1.12)

where

 δ(j,k)={1,j>k0,j⩽k, (1.13)

and the quantum numbers independently take the values . The vectors with components are in fact the celebrated motifs introduced by Haldane et al. Haldane et al. (1992). As pointed out in Ref. Basu-Mallick et al. (2010), Eq. (1.12) defines a classical inhomogeneous one-dimensional vertex model with bonds each taking possible values, where the contribution of the -th vertex is given by333By Eq. (1.12), the first and last vertices do not contribute to the energy. . We shall show that this connection between spin chains of HS type and vertex models still holds in the presence of a nonzero magnetic field. This is indeed the key result on which we shall base our unified approach to the analysis of the thermodynamics of all three chains of HS type (1.1).

The study of the thermodynamic properties of spin chains of HS type was initiated by Haldane himself, who used the spinon description of the spectrum to derive an indirect expression for the entropy of the spin HS chain Haldane (1991). An explicit formula for the free energy of the PF chain in the absence of a magnetic field appeared shortly afterwards without proof in Ref. Frahm (1993). In a subsequent publication Frahm and Inozemtsev (1994), Frahm and Inozemtsev obtained an analogous expression for the FI chain using the transfer matrix method, and also computed its magnetization for arbitrary magnetic field.

In this paper we have several objectives that we shall now summarize. In the first place, we shall compute in closed form the partition function of the HS-type chains (1.1) in the presence of an arbitrary magnetic field . Secondly, we shall use the expression for the partition function to establish the equivalence of the latter chains to a suitable classical inhomogeneous vertex model. We shall then take advantage of this connection to determine the equilibrium thermodynamics of the spin 12 chains of HS type in a unified and systematic way. Finally, we shall use the previous results to uncover a novel connection between spin chains of HS type and the classical Ising model.

The paper is organized as follows. In Section 2 we compute the chains’ partition functions in closed form by means of the freezing trick. Following the approach of Ref. Basu-Mallick et al. (2010), in Section 3 we construct a generating function for the partition function in terms of complete homogeneous symmetric polynomials. We then define a similar function for the corresponding vertex models, and prove that both generating functions coincide for arbitrary values of their arguments. This shows that the energies of each of the chains (1.1) are identical to those of its associated vertex model. Exploiting this connection, in Section 4 we use the transfer matrix method to find explicit expressions for the thermodynamic functions of the chains (1.1) with spin . In the case of the PF chain, we derive an exact formula for the free energy in terms of the dilogarithm function in an arbitrary magnetic field, and show that its magnetization can be expressed in terms of elementary functions. Section 5 is devoted to a detailed study of the zero magnetic field case. In particular, we show that the susceptibility can be expressed for all three chains of HS type in terms of the error function, discuss the connection of these chains with two-level systems, and derive low-temperature asymptotic expansions the chains’ thermodynamic functions. In Section 6 we study the zero temperature limit in the presence of an arbitrary external magnetic field in both the ferromagnetic and antiferromagnetic regimes. We explicitly show that in both regimes there is a phase transition, as was to be expected on general grounds Frahm (1993). In Section 7 we present a novel interpretation of the thermodynamic quantities of spin chains of HS type as weighted averages of the analogous quantities over an ensemble of classical Ising chains. The paper ends with a short section where we compare our results with previous related work, and comment on possible extensions thereof.

## 2 Partition functions

In this section we shall derive a closed form expression for the partition functions of the HS chains (1.1) in the presence of a constant magnetic field. For definiteness, we shall deal with the PF chain, whose interactions are defined by Eq. (1.7b). As we mentioned above, this chain is obtained by applying the freezing trick to the spin Calogero model in a constant magnetic field, whose Hamiltonian we shall take as

 H=H0−2aJm−1∑α=1BαSα, (2.1)

where and

 H0=−N∑i=1∂2xi+a2N∑i=1x2i+∑1⩽i≠j⩽Na(a−Sij)(xi−xj)2. (2.2)

More precisely, let

 Hsc=−N∑i=1∂2xi+a2N∑i=1x2i+∑1⩽i≠j⩽Na(a−1)(xi−xj)2 (2.3)

denote the Hamiltonian of the scalar Calogero model Calogero (1971), and define

 ~H(x)=∑1⩽i

so that

 H=Hsc+2a~H(x),H=J~H(ξ). (2.5)

It is well known Corrigan and Sasaki (2002) that the chain sites of the PF chain are the coordinates of the unique minimum of the scalar potential

 U(x)=N∑i=1x2i+∑1⩽i≠j⩽N1(xi−xj)2 (2.6)

in the principal Weyl chamber of type , where the motion of the particles of the Calogero model can be restricted due to the singularities of its Hamiltonian. From this fact and Eq. (2.1) it follows that in the large limit the eigenfunctions of the spin Calogero model become sharply peaked around the sites of the PF chain, so that the degrees of freedom effectively decouple from the dynamical ones. By Eq. (2.5), in this limit the energies of the model (2.1) are approximately given by the formula

 Eij≃Esci+2aJej, (2.7)

where and denote two arbitrary eigenvalues of and , respectively. Although the spectra of both the scalar and the spin Calogero models can be easily computed, the previous formula cannot be directly used to compute the spectrum of the PF chain. Indeed, it is not obvious a priori which energies of and can be combined to yield an energy of the PF chain. However, from Eq. (2.7) it is immediate to obtain the following exact relation between the partition functions , and of the Hamiltonians , and :

 Z(T)=lima→∞Z(2aT/J)Zsc(2aT/J). (2.8)

This formula, first derived by Polychronakos Polychronakos (1994), is the mainstay of the freezing trick method; see Enciso et al. (2008); Enciso (2009) for a rigorous proof.

We shall next evaluate both the numerator and the denominator in the previous equation. To begin with, the partition function of the scalar Calogero model is easily computed Polychronakos (1994), with the result

 Zsc(2aT/J)=qJEg2aN∏i=1(1−qiJ)−1, (2.9)

where

 Eg=a2N(N−1)+aN (2.10)

is the ground state energy.

Let us now turn to the partition function of . Consider, to this end, the spin functions

 ψℓ,s(x)=ρ(x)Λ(N∏i=1xℓii|s1,…,sN⟩), (2.11)

where , , ,

 ρ(x)=\e−ar2/2∏1⩽i

is the ground state of the scalar Hamiltonian and is the total symmetrizer with respect to particle permutations. The above states are a (non-orthonormal) basis of the Hilbert space of provided that (for instance) for all and whenever . It was shown in Ref. Finkel et al. (2001) that acts triangularly on the latter basis, with eigenvalues

 Eg+2a|ℓ|,|ℓ|≡N∑i=1ℓi. (2.12)

On the other hand, from the identities

 SijSαi=SαjSij,SijSαj=SαiSij;SijSαk=SαkSij,k≠i,j,

it immediately follows that the operators commute with the spin permutation operators , and hence with . By Eq. (1.3) we then have

 Sαψℓ,s(x)=ρ(x)Λ(N∏i=1xℓiiSα|s1,…,sN⟩)=cα(s)ψℓ,s,

where the eigenvalue is given by

 cα(s)=N∑i=1(δαsi−δmsi)≡#{si=α}−#{si=m}.

In the previous formula, the symbol denotes the number of components of the vector equal to . It follows that the term in Eq. (2.1) is diagonal in the basis (2.11), with eigenvalues

 m−1∑α=1cα(s)Bα=m−1∑α=1Bα#{si=α}−#{si=m}m−1∑α=1Bα. (2.13)

In view of the latter equation, it is convenient to introduce the notation

 Bm=−m−1∑α=1Bα, (2.14)

so that . Using this notation we can rewrite the eigenvalue (2.13) in the more compact form

 m∑α=1Bα#{si=α}≡N∑i=1Bsi. (2.15)

From the previous considerations and Eqs. (2.12)-(2.15) it then follows that the Hamiltonian (2.1) acts triangularly on the basis (2.11), with eigenvalues

 (2.16)

Using the previous equation, it is a straightforward matter to compute the partition function of the spin Calogero model (2.1). To this end, let us represent the multiindex in Eq. (2.17) as

 ℓ=(k1λ1,…,λ1,…,krλr,…,λr), (2.17)

with and . We then have

 Z(2aT/J)=qJEg2a∑k∈PN∑λ1>⋯>λr⩾0q∑ri=1Jkiλi∑s∈ℓq−∑Ni=1Bsi, (2.18)

where the last sum is extended to all spin quantum numbers compatible with the multiindex given by Eq. (2.17), i.e., such that whenever . In fact, since the latter sum clearly depends on only through , from now on we denote it by . Clearly, by Eq. (2.17) we have

 Σ(k)=r∏i=1∑s1⩾⋯⩾skiq−∑kij=1Bsj. (2.19)

With this notation, Eq. (2.18) becomes

 Z(2aT/J)=qJEg2a∑k∈PNΣ(k)∑λ1>⋯>λr⩾0q∑ri=1Jkiλi.

The inner sum in the RHS is easily computed by performing the change of indices , where and , since the new indices independently range from to . We thus obtain

 ∑λ1>⋯>λr⩾0q∑ri=1Jkiλi=(1−qNJ)−1r−1∏i=1qJKi1−qJKi, (2.20)

and therefore

 Z(2aT/J)=qJEg2a∑k∈PNΣ(k)q∑r−1i=1JKir∏i=1(1−qJKi)−1, (2.21)

where the positive integers were defined in Eq. (1.10). From the freezing trick relation (2.8) and Eqs. (2.9)-(2.21), we immediately obtain the following explicit expression for the partition function of the PF chain (1.1)-(1.7b) in the presence of an arbitrary magnetic field:

 Z(q)=∑k∈PNΣ(k)q∑r−1i=1JKiN−r∏i=1(1−qJK′i). (2.22)

The previous argument can be applied with minor modifications to both the HS and FI chains, thereby obtaining the following general expression for the partition function of the HS-type chains (1.1)-(1.7):

 Z(q)=∑k∈PNΣ(k)q∑r−1i=1JE(Ki)N−r∏i=1(1−qJE(K′i)). (2.23)

Note that when the magnetic field vanishes, by Eq. (2.19) becomes , and the previous expression for reduces to its zero field version (1.9).

A few remarks on the function are now in order. Let us first note that the latter function can be expressed as

 Σ(k)=r∏i=1σ(ki), (2.24)

where

 σ(k)=∑s1⩾⋯⩾skq−∑kj=1Bsj. (2.25)

The function can be considerably simplified by noting that the summation indices can be expressed as

 pm1′m,…,m,pm−11′m−1,…,m−1,…,p11′1,…,1, (2.26)

with . Thus

 k∑j=1Bsj=m∑α=1pαBα,

and therefore

 σ(k)=∑p1+⋯+pm=km∏α=1q−pαBα. (2.27)

The latter sum can be easily evaluated by noting that

 σ(k)=hk(q−B1,…,q−Bm), (2.28)

where denotes the complete homogeneous symmetric polynomial of degree in variables , given by

 hk(x)=∑p1+⋯+pm=kxp11⋯xpmm. (2.29)

Recalling that is the Schur polynomial associated with the single-row partition and using Jacobi’s definition of the latter polynomials in terms of determinants Macdonald (1995) we conclude that

 σ(k)=∏1⩽i

Expanding the determinant along the first row we obtain the alternative expression

 σ(k)=m∑i=1q−(k+m−1)Bim∏j=1j≠i(q−Bi−q−Bj)−1. (2.31)

In particular, for spin 12 () we have , and either expression for easily yields

 σ(k)=q(k+1)B−q−(k+1)BqB−q−B=sinh((k+1)βB)sinh(βB). (2.32)

The RHS of the latter formula can be conveniently expressed using the -number notation. Recall, to this end, that given two real numbers and the symmetric -number is define by

 [[x]]w=wx2−w−x2w12−w−12,

which reduces to the ordinary number for . We then have (in the spin 12 case)

 σ(k)=[[k+1]]q2B,

so that the partition function of the spin 12 chains (1.1)-(1.7) can be concisely written as

 Z(q)=∑k∈PNr∏i=1[[ki+1]]q2B⋅q∑r−1i=1JE(Ki)N−r∏i=1(1−qJE(K′i)). (2.33)

Note, again, that in the absence of a magnetic field the previous expression obviously reduces to Eq. (1.9) with .

## 3 The associated vertex models

In this section we shall prove that the spectrum of the HS chains (1.1)-(1.7) coincides with that of a classical inhomogeneous vertex model, which differs from the one in Eq. (1.12) by the addition of a magnetic field term. Our approach is based on the notion of generating function for the partition function, as used in Ref. Basu-Mallick et al. (2010) for the zero magnetic field case.

To this end, following the latter reference we define the generating function of the zero-field partition function (1.9) as

 F(x)=∑k∈PNr∏i=1hki(x)⋅q∑r−1i=1JE(Ki)N−r∏i=1(1−qJE(K′i)). (3.1)

Since

 hk(1,…,1)=(m+k−1k),

by Eq. (1.9) we have . More generally, substituting the expression (2.28) for into Eqs. (2.22)-(2.24) and using the definition of we readily obtain the identity

 Z(q)=F(q−B1,…,q−Bm). (3.2)

Similarly, the generating function for the partition function of the classical vertex model (1.12) was defined in Ref. Basu-Mallick et al. (2010) by

 FV(x)=m∑n1,…,nN=0xw1(n)1⋯xwm(n)mqE0(n), (3.3)

where the nonnegative integers are given by

 wα(n)=#{nk=α}.

Again, it is obvious that the partition function of the model (1.12)-(1.13) is the value of its generating function at the point . One of the fundamental results in Ref. Basu-Mallick et al. (2010) is the fact that the generating functions (3.1) and (3.3) are identically equal, i.e.,

 F(x)=FV(x),∀x∈Rm. (3.4)

Evaluating the previous identity at the point we obtain the equality of the zero-field partition functions  and , which is in fact the main result in Ref. Basu-Mallick et al. (2010). On the other hand, if we evaluate the identity (3.4) at the point and use Eq. (3.2) we immediately obtain

 Z(q)=FV(q−B1,…,q−Bm)=m∑n1,…,nN=0q−∑mα=1Bαwα(n)qE0(n). (3.5)

Taking into account that

 m∑α=1Bαwα(n)=N∑i=1Bni,

we can rewrite Eq. (3.5) as

 Z(q)=m∑n1,…,nN=0qE0(n)−∑Ni=1Bni≡ZV(q), (3.6)

where denotes the partition function of the inhomogeneous classical vertex model with energies

 E(n)=E0(n)−N∑i=1Bni,nk∈{1,…,m}, (3.7)

with given by Eqs. (1.12)-(1.13). Therefore, as stated at the beginning of this section, the spectrum of the spin chains (1.1)-(1.7) is identical to that of the classical vertex models defined by Eq. (1.12)-(3.7).

In the spin case, the previous equation can be simplified by noting that in Eq. (1.13) can be expressed as

 δ(j,k)=(j−1)(2−k),j,k=1,2, (3.8)

and similarly

 Bj=(3−2j)B,j=1,2. (3.9)

Introducing the spin variables , Eq. (3.7) becomes

 E(σ)=J4N−1∑i=1E(i)(1−σi)(1+σi+1)−BN∑1=1σi,σk∈{±1}. (3.10)

The latter equation can be alternatively written as

 E(σ)=−J4N−1∑i=1E(i)(σiσi+1−1)−N∑i=1B(i)σi, (3.11)

where

 B(i)≡B+J4[E(i)−E(i−1)] (3.12)

and we have set . The last two equations define a classical inhomogeneous Ising model in one dimension, where the coupling between the spins and is proportional to the dispersion relation , and the external magnetic field (also inhomogeneous) is given by Eq. (3.12).

## 4 Thermodynamics of the spin 12 chains

We shall next take advantage of the representation (3.10) of the spectrum of the HS-type chains (1.1)-(1.7) with spin in order to determine their equilibrium thermodynamics in a unified way. To this end, we must first normalize the Hamiltonian (1.6) to ensure that the energy per spin is finite (and nonzero) in the thermodynamic limit. It is well known that when the mean energy of the HS-type chains is , with for the PF chain and for the HS and FI chains Finkel and González-López (2005); Barba et al. (2008, 2010). Hence from now on we shall take

 J={K/N2,for the HS and FI chainsK/N,for the PF chain, (4.1)

where the constant is independent of . With this proviso, Eq. (3.10) can be rewritten in terms of the discrete variables as

 E(σ)=K4N−1∑i=1εi(1−σi)(1+σi+1)−BN∑i=1σi, (4.2)

where

 εi=⎧⎨⎩xi(1−xi),for the HS chainxi,for the PF chainxi(γN+xi),for the FI chain, (4.3)

and . We shall further assume that has a finite limit as . With the above notation, the partition function of the chains (1.6)-(1.7) can be collectively expressed as

 Z(q)=∑σqd(σ1,σ2)ε1−B2(σ1+σ2)⋯qd(σN−1,σN)εN−1−B2(σN−1+σN)q−B2(σ1+σN), (4.4)

where

 d(σ,σ′)=K4(1−σ)(1+σ′).

Equation (4.4) can be more concisely written as

 Z(q)=tr(UT1⋯TN−1), (4.5)

where and are matrices with elements

 Uσσ′=q−B2(σ+σ′),(Ti)σσ′=qd(σ,σ′)εi−B2(σ+σ′);σ,σ′=±1, (4.6)

or equivalently,

 U=(qB11q−B),Ti=(qBqKεi1q−B). (4.7)

The transfer matrix has two distinct eigenvalues

 λi,±=cosh(βB)±√sinh2(βB)+\e−Kβεi, (4.8)

and is therefore diagonalizable. A straightforward calculation shows that

 Ti=PiDiP−1i, (4.9)

where

 Di=(λi,+00λi,−),Pi=(−a+ri−a−ri11), (4.10)

and we have set

 a=sinh(βB),ri=√sinh(βB)2+\e−Kβεi. (4.11)

We thus have

 T1⋯TN−1=P1(D1C1)⋯(DN−2CN−2)DN−1P−1N−1,

where is a symmetric matrix explicitly given by

 Ci=12ri(ri+ri+1ri−ri+1ri−ri+1ri+ri+1). (4.12)

The key observation at this point is that

 Ci=I+O(N−1) (4.13)

as . Indeed, note first of all that

 0⩽εi⩽εmax,

where

 εmax=⎧⎪⎨⎪⎩14,for the HS chain1,for the PF chainγN+1,for the FI chain, (4.14)

so that . On the other hand, since the difference

 εi+1−εi=⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩1N(1−xi−xi+1),for the HS chain1N,for the PF chain1N(γN+xi+xi+1),for the FI chain

is , and hence

 ri+1ri=1+O(N