Correlation function of the two-dimensional Ising model on a finite lattice. II
📝 Abstract
We calculate the two-point correlation function and magnetic susceptibility in the anisotropic 2D Ising model on a lattice with one infinite and the other finite dimension, along which periodic boundary conditions are imposed. Using exact expressions for a part of lattice form factors, we propose the formulas for arbitrary spin matrix elements, thus providing a possibility to compute all multipoint correlation functions in the anisotropic Ising model on cylindrical and toroidal lattices. The scaling limit of the corresponding expressions is also analyzed.
💡 Analysis
We calculate the two-point correlation function and magnetic susceptibility in the anisotropic 2D Ising model on a lattice with one infinite and the other finite dimension, along which periodic boundary conditions are imposed. Using exact expressions for a part of lattice form factors, we propose the formulas for arbitrary spin matrix elements, thus providing a possibility to compute all multipoint correlation functions in the anisotropic Ising model on cylindrical and toroidal lattices. The scaling limit of the corresponding expressions is also analyzed.
📄 Content
1 The model Two-point correlation function and magnetic susceptibility of the isotropic Ising model on a cylinder were calculated in [1,2]. Analogous results can be obtained in the anisotropic case as well. The complications due to the presence of two different coupling constants can be overcome, since the matrix whose determinant yields the correlation function continues to have a Toeplitz form. The computation idea has already been presented in [1], and below we will often use the results of that paper, omitting detailed calculations.
The calculation of multipoint correlation functions can be reduced to the problem of finding Ising spin matrix elements (the so-called form factors) in the orthonormal basis of transfer matrix eigenstates. Although neither the first nor the second problem has been solved on a finite lattice, the language of matrix elements turns out to be more convenient when constructing the corresponding expressions. A part of form factors can be found from the formulas for the two-point correlation function. Since the structure of these expressions is relatively simple, we generalize them to the case of arbitrary matrix elements.
The hamiltonian of the anisotropic Ising model on a rectangular lattice is defined as
where the two-dimensional vector r = (r x , r y ) labels the lattice sites: r x = 1, 2, . . . , M, r y = 1, 2, . . . , N; the spins σ(r) take on the values ±1. The parameters J x and J y determine the coupling energies of adjacent spins in the horizontal and vertical direction. The operators of shifts by one lattice site, ∇ x and ∇ y , are given by ∇ x σ(r x , r y ) = σ(r x + 1, r y ), ∇ y σ(r x , r y ) = σ(r x , r y + 1), where for periodic boundary conditions one has
and for antiperiodic ones
If the lattice is periodic in both directions, the partition function of the model at the temperature β -1 Z =
[σ]
e -βH [σ] can be written as a sum of four terms
each of them being proportional to the pfaffian of the operator
with different boundary conditions for ∇ x , ∇ y . Here, we have introduced the dimensionless parameters
One can verify that when the torus degenerates into a cylinder (M ≫ N), the partition function is determined by the antiperiodic term:
Let us consider the two-point correlation function in the Ising model on the cylinder
where Z def denotes the partition function of the Ising model with a defect: the coupling parameters K x , K y should be replaced by K x -iπ/2, K y -iπ/2 along a path that connects the correlating spins (see Fig. 1 in [1], where the numbering of lattice sites and the locations of correlating spins were described; the bold line in the figure is the path along which the couplings are modified). The exponent P in the formula ( 2) is equal to the number of steps along the defect line: P = r x + r y in the case of a shortest path. When the correlating spins are located along a line parallel to the cylinder axis (i. e. r y = 0), the ratio of pfaffians in the right hand side of (2) can be expressed in terms of the determinant of a Toeplitz matrix σ(0, 0)σ(r x , 0) = det A,
) whose size |r x | × |r x | is determined by the distance between the correlating spins. Here and below the superscripts (NS) and (R) in sums and products imply that the corresponding operations are performed with respect to Neveu-Schwartz (q = 2π N j + 1 2 , j = 0, 1, . . . , N -1) or Ramond (q = 2π N j, j = 0, 1, . . . , N -1) values of quasimomenta. Our goal is to transform (3)-( 4) into a representation with an explicit dependence on the distance.
In the translationally invariant case the pfaffians Pf D can be easily calculated,
The product over any of the two quasimomentum components in (5) can be found in an explicit form. For instance, the term in (1), which corresponds to periodic boundary conditions along the x axis and antiperiodic ones along the y axis, may be written as
where the functions γ(q) and γ(p) are determined by the relations
and the conditions γ(q), γ(p) > 0.
We will call the domain of values of K x , K y , where sinh 2K x sinh 2K y > 1 (and hence γ(0) = ln t x +2K y ) the ferromagnetic region of parameters. Notice that in this case the numerator of the integrand in (4) can be represented in the following factorized form in terms of the function γ(q):
x e ip = (1t 2 y )e γ(0) (1e -γ(π) e ip )(1e -γ(0) e -ip ).
Then, using the identity
one may compute the sum over the discrete Neveu-Schwartz spectrum in (4). As a result, we can express matrix elements A kk ′ in terms of contour integrals
where
Recall that in order to calculate the determinant |A| by the Wiener-Hopf method [3] one has to represent the kernel A(z) in a factorized form
where the functions P (z) and Q(z) are analytic inside the unit circle. Setting
the determinant can be written as follows:
where
We will see that the sum (15) contains a finite number of terms, i. e. there exists l 0 such that g 2l ≡ 0 for all l > l 0 .
Let us now calculate the integral (17). To do this,
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