Ann. Physik 2 (1993) 79-91

Annalen der Physik

0 Johann Ambrosius Barth 1993

Effects of geometry on transfer matrices, spin chains and critical behaviour

Brian Davies and Ingo Peschel*

Department of Mathematics, the Faculties, Australian National University, GPO Box 4, Canberra, ACT 2601, Australia

* Fachbereich Physik, Freie Universiat Berlin, Arnimallee 14, W-1000 Berlin 33, Germany

Received 21 August 1992, accepted 1 October 1992

Abstract. We consider two-dimensional Ising models bounded by general parabolic curves and study their transfer matrices and associated quantum spin chains. We derive their eigenvalue spectra numeri- cally and analytically, both at the critical point and in its vicinity. From this we find how the geometrical form of the system is reflected in the spectrum and how it influences the critical behaviour near the tip.

Keywords: king model; Parabolic boundaries; Transfer matrices; Magnetization.

1 Introduction

When one studies phase transitions and ordering phenomena one finds an interesting interplay between the geometrical form of a system and its critical behaviour. This was first pointed out by Cardy [I, 21 for the case of wedge shaped regions in arbitrary dimensions. In this case the critical exponents become continuous functions of the opening angle of the wedge. This result has been verified in a number of calculations for two-dimensional Ising models [3 - 61 and self avoiding random walks 171 and also folows, in d = 2, from the theory of conformal invariance [2,3,8]. A more general situation has been studied recently, also for d = 2 [9]. The system was in this case bounded by a generalised parabola, i.e. y = Cxa asymptotically. For a < 1 the result was that the critical behaviour is no longer described by power laws, but by stretched exponentials of either the reduced temperature or the size of the system. The geometrical exponent a then governs the amount of stretching of the exponentials.

One of the tools in [9] was the transfer matrix of such a parabolic system, operating between its upper and lower boundaries. Such an object is a generalisation of Baxter’s corner transfer matrix (CTM) for the wedge-like geometry [lo]. This CTM has the remarkable feature that, if one writes it as erp [ -HI, the eigenvalues of the generating Hamiltonian N are equally spaced at all temperatures. Thus they follow a linear dispersion law when considered as a function of their ordering index, for both T = T, and T =k T,. For the parabolic transfer matrix (PTM), the result of a calculation treating the Ising model in the anisotropic limit was that, for a = 1/2 and T * T,, the dispersion became a square root law. The dependence of the eigenvalues on the temperature also follows this law. Thus an interesting correspondence between the geometrical form of the system and the PTM emerges.

80 Ann. Physik 2 (1993)

In the present article we study this phenomenon in more detail. We consider the PTM for an king model for arbitrary values of the parameter a. Working again in the Hamiltonian limit we are then led to study a transverse Ising chain with position dependent couplings and fields. We first find its eigenvalues numerically and determine the dispersion and scaling behavioiir from these calculations. We derive differential equations for the eigenvectors and eigenvalues, valid both at and near the critical point, and discuss their solution in both cases. The two studies, numeric and asymptotic, are complementary and in complete agreement. In this way we confirm that the spectra of the Hamiltonians depend, for their qualitative form, only on the geometrical parameter a. We show that, at the critical point, The spectrum for 0 < a < 1 is a consequence of conformal invariance, and so obeys a linear dispersion law for T = T,. But away from criticality the spectrum obeys a quite different dispersion law, with the eigenvalues varying as na, where n is the ordering index. This is the generalisation of the previously found result [9] for a = 112. We firid also that the scaling with respect to temperature and with respect to size is different in the general case. This is made possible by the fact that the boundary introduces an extra length parameter into the problem. We use our results to determine the tip magnetization for the case of a system which is closed onto itself like a paraboloid and thereby confirm previous predictions for this quantity. We also consider briefly the case a > 1 , for which the spectrum retains the dispersion law nu even at T = T,. In the concluding section we make some general observations and also a conjecture about spin chains with coefficients which are appropriate for an Ising model on a hyperlattice. Such a system has been the subject of recent investigation [ 111.

2 Transfer matrices and inhomogeneous spin chains

2.1 Formulation

Consider an Ising model on a square lattice, which, when viewed on large scales, has curved boundaries y = 2 Cxa, such as the system treated in [9]. We are interested in the transfer matrix Trelating the variablles at the upper and lower boundaries. In general this is a complicated object but it simplifies if one considers the Hamiltonian limit with strong vertical bonds Kt and weak horizontal bonds Kz. Then T = exp [-2K,*H], where Ktis the dual of the coupling constant K,, and H describes an inhomogeneous king spin chain.

So one is led to study the Hamiltonian

where A = Kz/KT. The ordered phase is given by A > 1 . The coefficients pn, vn, represent (at least in the sense of a co ntinuum limit) the number of vertical and horizontal bonds at position n from the origin 17 = 0. In this paper, we use the rather general forms

pn = (2n + p y , v, = (2n i- v)a (2.2)

where a is an arbitrary real parameter, although most of our results will in fact be for the rather natural special case Y = p + 1 . In addition we set

B. Davies, I. Peschel, Geometry effects on transfer matrices, etc. 81

This has the effect of fixing the end spins of the chain. For the two-dimensional system all spins at the right boundary therefore have the same value and this allows the tip (centre) magnetization at n = 0 to be evaluated [lo].

Such spin chains are diagonalised by fermion techniques. We omit the details which may be found in many places [ 12 - 141. Standard transformations bring Hto the diagonal form

H = C m y n fn + const n

(2.4)

with Fermi operators f',, f,,, whilst the squares of the excitation energies on are the eigenvalues of a matrix of the form M = (A + B ) (A - B ) , in the notation of [12]. The explicit form, for the Hamiltonian spin chain (2.1), is

M =

Here

The zero entries in the last row and column are a result of the fixed boundary conditions. Generally, one expects that the spectrum will be dominated by the coefficients p,, for

small A (high temperature), and by the coefficients A,, for large A (low temperature). For the normal CTM with a = 1 , p = 0, v = 1, however, an even stronger statement is true: the spectrum retains its limiting form in both regions all the way to the critical point and only the overall scale factor changes [15]. At A = 1 the changeover takes place via a collapse whereby the on assume the asymptotic form, for N + 03, [14, 16- 181

(2n - 1) InN 0" -

It is natural to expect that this logarithmic behaviour for a = 1 separates two qualitatively different regions. Our calculations, both numerical and analytic, show that this is indeed the case.

2.2 Asymptotically zero eigenvalue

The onset of disorder, as the value of A is decreased from the ordered phase, is heralded by the appearance of an exponentially small (in N ) eigenvalue, in addition to the zero eigenvalue which is present for all values of A. The mechanism for this is quite generally

82 AM. Physik 2 (1993)

seen by applying the variational principle to the matrix M. The eigenvalues are stationary values of the functional

Moreover, E( Y ) is an upper bound for the minimum eigenvalue: since the zero eigenvalue comes from the trivial fact that the la:f row and column of Mare zero, (2.8) also provides an upper bound to the next eigenvalue when used with a vector !P whose last component is zero. Now from the fact that (A -k B ) T = (A - B ) , we may write this bound as

The matrix (A - B ) has the general form

(A - B ) =

0 10 Pl

21 P2 '.. ... . .

A N - 2 l N - 1 I N - I

(2.10)

It has the doubly degenerate eigerivalue 0, but only one zero eigenvector @, with components @,, = a,.,. Now use this @ in (2.91, then the components of Yare readily obtained as

1 k-1 /

(2.11) YN = 0, WN-k = - (--"). P N - 0 < k d N AN-1 n=l \AN-n-l

and the upper bound (2.9) is

(2.12)

One sees immediately that if the magnitude of the components of Y increase geometrically with decreasing n, then amin becomes exponentially small for large system size. When the converse obtains, the bound approaches a finite value. With coefficients of the form (2.2) it is also easy to see that the ratio has the asymptotic value A so that A = 1 is the critical point of the system regardless of the values of a, p and v.

2.3 Some numerical results

The eigenvalues were computed using the Sturm sequencing property of symmetric tridiagonal matrices [19]. Quite exteinsive calculations (up to sizes N = lo4) were made, for values of a in the range 0 < a < 2. Here we report on the qualitative features of the eigenvalue spectrum which were observed in this way.

B. Davies, I. Peschel, Geometry effects on transfer matrices, etc. 83

- Dependence on the size of the system at the critical point. The value a = 1 divides two regions of different behaviour. For 0 C a < 1 , we find that (2.7) is replaced by

O < a < l , l e n e N (2.13)

so the spectrum collapses. For a > 1 , we find that the spectrum does not collapse with increasing system size. - Spacing of the spectrum at the critical point. The value a = 1 again divides two

regions of different behaviour. For 0 < a < 1 , we find that the spectrum becomes equaily spaced, i. e.,

o n - ( 2 n - 6 ) , A = l , O < a < l , l e n < N (2.14)

for all values o fp and v. But for a > 1 , the dependence of the eigenvalues on the ordering index n is the same as for the coefficients in the spin chain:

a,, - (2n - a)@, A = 1, a > 1 , 1 a n -e N (2.15)

for all values of p and v.

the spectrum, for all values of a, is - Spacing of the spectrum away from the critical point. The qualitative behaviour of

a,, - (2n - a)a, A =t= 1 , 1 e n e N (2.16)

Only for a 2 1 does the spectrum have the-same form for T = T, and T * T,. This implies that the spectrum, as a function of the temperature and the size of the system, has a mixed asymptotic behaviour when 0 < a < 1 . - We show some numerical examples for a = 1 /2 in Fig. 1 . These illustrate the fact

that the dispersion law changes its nature with increasing n. The low lying states see an essentially infinite system and so obey (2.16), whilst the higher states notice the finite size of the system and obey (2.14). This will also be investigated analytically in section 4.

Fig. 1 Low-lying eigenvalues w, of H for a spin chain of 1024 sites with a = 1/2, p = 0, v = 1 , at five temperatures T 5 T, (measured by the parameter A) . The curves are guides for the eye.

84 Ann. Physik 2 (1993)

3 Criticalcase

3.1 Continuum limit

For large systems, the eigenvectors corresponding to the lowest eigenvalues are slowly varying functions and a continuum limit may be taken to obtain a differential equation [I 8,201. One must be careful, however, with the inner boundary. Thus one considers first a spin chain with its left end at n = k instead of n = 0. It will transpire that, for 0 c a < 1 , it is possible to set k = 0 at the end of the computation. The eigenvectors Yare solutions of a linear difference equation. To obtain the continuum limit, one first changes wn -, (- 1)" wn in Y. Then the equations read

(3.1) 2 - A n - l p n w n - i + + p2n)vn - Lnpn+lWn+l = 0 Il/n

with A = 1. Writing tyn = ~ ( x ) where x = ns and s is the lattice spacing, and expanding w(x -I- s) up to second order in s, one arrives at the second order linear differential equation

- ( 2 ~ ) ~ " ~ " ( x ) - 4 a ( 2 ~ ) ~ ~ - ' w ' ( x ) + ~ ( ~ x ) ' " - ~ v / ( x ) = ( ~ ) ~ " - ~ 0 ~ , 2 w ( x ) (3.2)

where

p = a ( 2 p - 2 v + 4) + a2[Cp - v ) ~ - 41, To = US'-' (3 -3)

The boundary conditions which follow from the discrete equations are

21Y/(l ) = a(v - p - 2)1y( l ) , I = ks.. (3.4a)

w(L) = 0, L = Ns, (3.4b)

These equations are a generalisation of those used in [20].

simply Bessel's equation. Explicitely, one makes the substitutions By a suitable change of both dependent and independent variables, Eq. (3.2) becomes

u = (2x ) ' -Q , y ( x ) = (2x) '"-af$(u) (3.5)

whereupon the equation takes the form

[ ::I 1 @"(u) + -@'(u) + 2 - .- @ ( x ) = 0 , U

tc, 2(1 - a)

1 i- a(u - v ) 2(1 - a )

c = ,4=-

(3.6a)

(3.6b)

The general solution is @ ( u ) = a,.T9(cu) + a,J-,(cu). In using these solutions, the lattice spacing s will always be set to unity, so there is no need to distinguish between o and 0 .

B. Davies, I. Peschel, Geometry effects on transfer matrices, etc. 85

3.2 Validity of the continuum approximation

The basic approximation employed to obtain Eq. (3.2) is a Taylor series expansion, such as y(x k s) = y(x) k sy'(x). Note the Bessel form (3.6a) tells us about the qualitative behaviour of the solutions near the origin. For 0 < a < 1 , the substitutions (3.5) simply rescale the x-axis while retaining the orientation. So the solutions are well-behaved at the origin, having at worst fractional powers of x which come from the regular singular point. However, in the case a > 1 , the transformation (3.5) maps small x to large u and conversely. Since the asymptotic from the Bessel functions is simply periodic, this means that the solutions ~ ( x ) oscillate as sin [0x'-~/2(1- a)] as x -+ 0: they have an essential singularity. In principle this problem may be avoided by choosing the inner cut-off I to be sufficiently large. But since the rate of oscillation of y ( x ) also increases with increasing o, even this remedy is limited to only a few of the lower lying eigenstates. We conclude that the continuum approximation is of limited utility for a > 1 . This conclusion is supported by our numerical investigations.

3.3 The special case v = p k 1

In this special case, which includes the Hamiltonian for the usual CTM (3.6b) shows that q = 1/2, so the equation may be solved in terms of trigonometric functions. For a = 1 , such trigonometric solutions have already been considered in some detail in [20]. In the case 0 < a < 1 , the boundary condition on y(1) at the inner boundary is completely regular in the limit I -+ 0, and a simple analysis shows that the solution consists solely of the Bessel function J - Explicitely, the solution which satisfies the inner boundary condition when I -+ 0, is

which has the property of being singular at I = 0, reflecting the behaviour of the amplitudeof the components iyn with increasing n. Imposing the outer boundary condition, the eigenvalues are obtained as

(1 -- a)z 0, =-. (2n - 1)

(2L)I-a

which agrees with-the numerical results already for chains of quite modest size. Thus we recover the scaling behaviour of (2.13).

3.4 Argument from conformal invariance

The spectrum of H fbund above can also be derived from conformal arguments, as in the case af the CTM'[i?J]. Consider the transformation of [9]

C = 1r(z/2p)l-", Q < a < 1 (3.9)

p is the parameter inthe normal form of the parabola. A piece of a vertical strip in the C-plane (1; = 4 + ib),is thereby transformed into a section of a parabolic figure in the

86 Ann. Physik 2 (1993)

z-plane (z = x + iy), as shown in Fig. 2. In particular, a line r,~ = const becomes, for large x, the line y = C(q)x", where

(3.10)

For a = 1/2 one obtains normal parabolas and (, q are the usual parabolic coordinates.

Fig. 2 Mapping of a rectangle into a parabolic figure by the transformation (3.9). Shown is the case a = 1/2. The shaded sections indicate infinitesimal pieces described by the operators H, and H .

Consider now the infinitesimal vertical transfer matrix H, in the strip with fixed boundary spins at the left and right edges. It can be obtained by expanding the system in the vertical direction via the non-conformal transformation

where O(q - vo) is a step function. This leads to

(3.12)

where T, are the components of the stress tensor [22]. The same transformation, however, inserts a parabolic piece into the system in the z-plane, as indicated in Fig. 2. The corresponding PTM therefore is equal to H , up to a constant involving the central charge. The total PTM between (- q) and (q) is therefore

H = 2qH, + const (3 .13 )

Now the levels of Hs are given by

(3.14) n Yn = - (2n - 1)

2LS

where L, = <, - <O is the width of the strip. Assuming through the size of the parabolic system on the axis, L, = n(L/2p) ' -" , one finds

% to and expressing L,

B. Davies, I. Peschel, Geometry effects on transfer matrices, etc. 87

nC(1 - a ) (2n - 1) Ll-a 0, = (3.15)

In order to compare with the spin chain results one has to rescale the anisotropic system considered there. One then finds that H effectively describes an isotropic system with boundary given by the parameter C = (~/2)’-~. Inserting this into ( 3 . 1 9 , we recover exactly Eq. (3.8). The levels thus are strictly equidistant here whereas for the spin chain this is in general only true for larger values of n. In this way, slight differences between the discrete system with arbitraryp, v and the continuous system obtained via (3.9) show up. The prefactor and especially the dependence on L, however, are in complete agree- ment.

4 Non-critical case

4.1 Differential equation for I =?= 1, v = p + 1

The linear difference equations (3.1) may be treated exactly as in section 3, even when I =k 1. The deviation from the critical point is measured by the correlation length (including a sign)

(4.1) < = s/(L - 1)

where s is the lattice spacing. Introducing the I-dependence into (3. l), and following the same procedure of expanding up to second order in s, one obtains two additional terms in the differential Eq. (3.2). It is convenient to again make the substitutions of Eq. (3.9, and in addition to set @(u) = ~ ( u ) / u ” ~ , whereupon the new differential equation becomes a one-dimensional Schrodinger equation

- f ‘ ( ~ ) + V ( U ) X ( U ) = Ex(u) , E = c2 (4.2)

where c retains its definitions given in (3.6b) and

U2a/(l - a) au(2a-l) / ( l -a) V(u) = +

4(1 - 2(1 - a)2< (4.3)

V ( u ) is a confining potential dominated by the first term and the eigenvalues do not collapse with increasing system size away from the critical point. The second term is a repulsive barrier at the origin for a < 1/2 and a further contribution to confinement for a > 1/2. For a = 1 /2 it is just a constant. The boundary conditions are unchanged from (3 .4 ) i f I< \ < I .

4.2 Special case a = 1/2

It is known from [9] that, when a = 1/2, it is possible to diagonalise the problem exactly in terms of orthogonal polynomials. This case also displays most clearly how the continuum limit contains information about the interplay between the geometry and the size of the system. In this case, V ( u ) takes the simple form

88 Ann. Physik 2 (1993)

and the equation has the standard form for a harmonic oscillator

- f ( u ) + (do2 X ( U ) = E X ( U ) , E = a2 - l/< (4.5)

In general the eigenvalues E of this equation, for an infinite system, are (2n + 1)/ I < I. However, the boundary condition at .u = 0 excludes the levels with odd values of R . So the result, for an infinite system, is

= (4n + 1)/1<1 + I/<, n = 0, 1,2, . . . (4.6)

which coincides, to first order in r - ' , with the results of [9] and also with the numerical results of section 2. It is interesting to see how the two terms in V(u), scaled respectively by ( - 2 and (-', together give the discontinuous change in the spectrum at the critical point, where ( changes sign. It is al:;o dear how a finite size of the system affects the eigenvalue spectrum. The solutions of the differential Eq. ( 4 . 3 , with the boundary condition w(L) = 0, are no longer tlhe harmonic oscillator eigenfunctions. But so long as these eigenfunctions are already in1 their exponential decay region for x < L, they are a good approximation. At the other extreme, for sufficiently large E, the problem approximates closely a free particle bounded by x = 0 and x = L (exactly as in the critical case) and the spectrum changes its nature accordingly. This is the feature seen in Fig. 1.

4.3 General case 0 < a < 1: WKB approximation

In the case of general 0 < a < 1 the qualitative description of the eigenvalue problem is unchanged, but the differential equation no longer assumes any standard classical form. Since for large values of 1 < I the problem is essentially that of a particle in a slowly varying potential, the WKB approximation will yield the asymptotics. It is clear also from the analysis of section 4.2 that the crucial factor is the potential well

W(u) = 4(1 - C Y ) ~ < ~

(4.7)

The WKB approximation for the solution which satisfies the inner boundary condition is

1 X ( U ) - cos ( a [E - W(W)]'/2 d w (4.8)

For sufficiently large E, the integraind is approximated by a constant and the problem reverts to that of a free particle with the upper limit of integration given by x -- L. Of more interest here is the nature of the lower lying eigenvalues. Then WKB quantization gives [23]

K(E,) = (n + 1 / 4 ) x (4.9)

where K@,) is the integral in (4.8) with the upper limit at the point where the integrand becomes zero. The substitution w' == w2a/('-a)/4c2(1 - a)'(' reduces the integral to a

B. Davies, I. Peschel, Geometry effects on transfer matrices, etc. 89

Beta function, or more simply, the ratio of Gamma functions [24]. Making all of the necessary substitutions into (4.9), one finally obtains an asymptotic form for the eigenvalues

The validity of this formula has been verified numerically for a selection of large systems. It must be noted that the nature of the approximation differs according as a < 1/2 or a > 1 /2. In the former case, the neglected term is a repulsive barrier but in the latter case it contributes to the confinement. We have not determined the value of 6, because the integrals do not reduce to standard forms in general. In any case the approximations made are sufficient to verify analytically the statements which were made in section 2.

5 Magnetization

The magnetization at the tip of the parabolic system can be calculated in a simple way if one identifies upper and lower boundaries. Then one has to evaluate a trace over the transfer matrix as in the CTM calculations [ 101. This gives

mo = n tanh (+) = exp ( C In tanh n z I 2 n z l

with the appropriate single fermion eigenvalues w,. Consider first the critical case. For a < 1 the w, collapse as the size of the system increases and one can convert the sum in (5.1) to an integral. Using the on from (3.15) then gives

Thus mo is a stretched exponential in L and its detailed form is determined by the geometrical parameters a and C. This result can also be obtained purely from conformal considerations. For this one starts from a circle with radius R(in the w-plane) with variables fixed along its boundary. The order parameter in the centre then varies as R-X, where x is the bulk scaling index [25]. The transformation [9]

w = i cosh [ n($)'-I (5.3)

then maps this circle onto a parabolic system in the z-plane with spins fixed at its right boundary. Using the usual transformation formula for the order parameter [22] one then obtains

which is valid for any conformally invariant system. For the Ising model with x = 118 one recovers (5.2). We note that the exponential factor in (5-4) is the Same as inthe critical

Ann. Physik 2 (1993) 90

end-to-end correlation function in this geometry. For a + 0 one obtains the result for a strip of length L and width 2C.

For T < T, one can use the result (4.10) according to which w, - <'-'nu for the in- finite system. As < diverges the o, again collapse and one obtains near the critical point

m0 - exp [- const $'-')'@I (5.5)

The order parameter thus vanishes faster than any power of (T - T,) in this geometry. This result is in accord with the predictions in [9] based on a scaling argument. For a = 1/2 one recovers the result for the simple parabola. It is important to note that the exponent a enters in a different way here as compared to (5.2). Therefore the general form of mo near the critical point is

where the scaling function f is not symmetric in the two variables.

6 Discussion

The calculations presented here show, for the case of power-law boundaries, how the form of a two-dimensional statistical system determines the spectra of transfer matrices and the behaviour of physical quantities. We studied this problem (which is a variant of Kac's well-known question [26]) mainly for the strongly anisotropic Ising model via the associated spin chain. The results at the critical point also follow from conformal invariance and thus are rather general. But also the off-critical results, which are more remarkable, will probably be present in other systems too.

The use of the continuum limit near criticality simplified the problem considerably. It was thereby reduced to a certain one-dimensional Schrodinger equation. The simplest situation is obtained for the special case a = 112 when one ends up at a quantum- mechanical harmonic oscillator. In this sense the present treatment gives more insight than the previous solution via orthclgonal polynomials, although the Iatter is valid for all temperatures. In the region where our focus was (0 < a < 1) the problem is also rather insensitive to the inner boundary condition and thus easier to handle than the case of the wedge (a = 1).

The type of transfer matrix used here lends itself also to other situations where the width of a system increases from a centre. An example is the recently studied hyperlattice [ 111 which may be viewed as a conical system with exponentially inceasing circumference. For the Ising chain one is lead to study an operator H as in (2.1) with parameters p, = pexp(yn), v, = vexp(yn). From the conditions for an asymptotically vanishing eigen- value (section 2.2) the critical point can be located at 1, = p/v. This leads to a critical temperature higher than given by the usual self-dual condition 1 = 1 , in agreement with [l I]. The values of A , which we obtain from our simple argument differ, however, from the estimates given in [I I] by a larger margin than the error estimates given therein. This may be connected with the use of the Hamiltonian limit which is problematic in this case. In the parabolic systems which we studied, this problem is not serious. Nevertheless, it would be interesting to treat the situation without invoking the Hamiltonian limit.

BD thanks the Deutsche Forschungsgemeinschaft for support and the Freie Universitat Berlin for hospitality during the time this work was done.

B. Davies. I. Peschel, Geometry effects on transfer matrices, etc. 91

References

[ l ] J. L. Cardy, J. Phys. A.: Math. Gen. 16 (1983) 3617 [2] J. L. Cardy, Nucl. Phys. B240 (1984) 514 [3] M. N. Barber, I. Peschel, P. A. Pearce, J. Stat. Phys. 37 (1984) 497 [4] I. Peschel, J. Phys. A: Math. Gen. 21 (1988) L185; Phys. Lett. l lOA (1985) 313 [5] C. Kaiser, I. Peschel, J. Stat. Phys. 54 (1989) 567 [6] B. Davies, I. Peschel, J. Phys. A: Math. Gen. 24 (1991) 1293 [7] A. J. Guttmann, G. M. Torrie, J. Phys. A: Math. Gen. 17 (1984) 3539

[8] R. Z. Bariev, Theor. Math. Phys. 69 (1986) 1061 [9] I. Peschel, L. Turban, F. Igldi, J. Phys. A: Math. Gen. 24 (1991) L 1229

J. L. Cardy, S. Redner, J. Phys. A: Math. Gen. 17 (1984) L953

[lo] R. J. Baxter, Exactly Solved Models in Statistical Mechanics, Academic Press London 1982 [ i l l R. Rietman, B. Nienhuis, J. Oitmaa (1992) preprint [12] E. H. Lieb, T. D. Schultz, D. C. Mattis, Ann. Phys. NY 16 (1961) 407 [13] B. Davies, Physita A154 (1988) 1 [14] T. T. Ti-uong, I. Peschel, Z. Phys. B75 (1989) 119 [15] R. J. Baxter, J. Stat. Phys. 17 (1977) 1 [16] I. Peschel, T. T. Truong, Z. Phys. B69 (1987) 395 [17] B. Davies, P. A. Pearce, J. Phys. A: Math. Gen. 23 (1990) 1295 (181 I. Peschel, T. T. Truong. Ann. Physik 48 (1991) 185 [19] J. H. Wilkinson, The Algebraic Eigenvalue Problem, Clarendon Press, Oxford 1977 [20] 1. Peschel, R. Wunderling, Ann. Physik l(1992) 125 [21] J. L. Cardy, I. Peschel, Nucl. Phys. B300 (1988) 377 [22] J. L. Cardy, Phase Transitions and Critical Phenomena, vol. 1 1 , ed. C. Domb, J. Lebowitz,

[23] L. D. Landau, E. M. Lifschitz, Quantenmechanik, Akademie-Verlag, Berlin 1970 [24] M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, Dover, New York 1965 [25] T. W. Burkhardt, E. Eisenriegler, J. Phys. A: Math. Gen. 18 (1985) L83 [26] M. Kac, “Can one hear the shape of a drum?”, Amer. Math. Monthly 73 (1966) 1

Academic Press 1982