Green's function of a finite chain and the discrete Fourier transform

Green’s function of a finite chain and the discrete Fourier transform S. Cojocaru

arXiv:0704.2898v1 [math-ph] 22 Apr 2007

Dipartimento di Fisica “E.R. Caianiello”, Universit`a degli Studi di Salerno, Italy Institute of Applied Physics, Chi¸sin˘au, Moldova

Abstract A new expression for the Green’s function of a finite one-dimensional lattice with nearest neighbor interaction is derived via discrete Fourier transform. Solution of the Heisenberg spin chain with periodic and open boundary conditions is considered as an example. Comparison to Bethe ansatz clarifies the relation between the two approaches.

1

I.

INTRODUCTION

Lattice Green functions have a broad spectrum of applications in physics and allow to study dynamic and statistical properties. [1, 2, 3, 4] Usually the lattice is assumed to be infinite while distances remain finite and discrete. [5, 6, 7] Often the range of relevant excitations allows to consider the system as continuous and apply field theoretic methods. [8] Accordingly, the integral Fourier transform or infinite Fourier series are commonly used as powerful tools for such problems. It therefore appears natural to employ the discrete, or finite, Fourier transform (DFT) when the finite size is relevant. However, DFT has mainly served for numerical analysis or in fast Fourier transform calculations [9] and much less as an analytic method. Most of the known exact results on finite systems have been obtained in the Bethe ansatz approach [10] (see, e.g., [11, 12] for recent review) that is based on a different strategy. In [3] the Bethe ansatz solution is compared to the solution by Fourier series in the thermodynamic limit of the periodic quantum spin chain. However it is noted that the relation between them remains obscure. For instance, the bound two-magnon excitation is characterized by the same eigenenergy, but only one eigenstate is found in the Fourier approach, instead of the two solutions in Bethe ansatz. This discrepancy has been resolved in [13] by an approximate treatment of finite size corrections. In the present paper it is shown that the DFT approach recovers the exact solution. The derivation presents interest in itself, in view of further extensions. As another example, some results on open boundary conditions obtained by Bethe ansatz [14] are recovered. The finite lattice sums are first expressed in terms of infinite Fourier series. For the nearest neighbor interaction the series represent an expansion in modified Bessel functions. Its Laplace transform results in analytic expressions for the finite lattice Green functions and allows to determine the corresponding discrete Fourier transform. The standard definitions of the Fourier series for a function periodically continued from the interval x ∈ [0; 2π) are f (z; x) =

∞ X

cn exp (inx) ,

(1)

n=−∞

cn =

1 2π

Z

2π

f (z; y) exp (−iny) dy,

0

where the parameters z and δ (below) take real values. The Fourier expansion of the lattice

2

sum

N −1 1 X  m f z; δ + 2π N m=0 N

can then be written as

# N −1 ∞  X n 1 X exp i2πm = ckN exp (iδkN) . cn exp (inδ) N m=0 N n=−∞ ∞ X

"

k=−∞

It has been taken into account that the sum in the square brackets above gives either 1 or 0, provided n/N is an integer (k) or not. We note that the same property is essential for the definition of the DFT of a function a (X) defined on a set of N consecutive integers (0, 1, ..N − 1 ):

 N −1  1 X 2π a (X) = b q= m exp (iqX) . N m=0 N

(2)

Z 2π N −1 ∞ X 1 m 1 X  f z; δ + 2π = f (z; y) exp (ikN (δ − y)) dy. N m=0 N 2π 0 k=−∞

(3)

Then, with the expression of the Fourier coefficients c in (1), our lattice sum becomes

Now the problem of finite summation has been reduced to integration and series summation. To motivate a particular choice of f (z; y) in (3) we note that dynamics of different models on a finite d-dimensional lattice can be described in terms of Green functions of a generic form G (p; X, ..., Y ) =

1 X X exp (iQ1 X1 + ... + iQd Yd ) ... , Nd Q p − ε (Qd , ..., Qd ) Q 1

(4)

d

where p is the spectral parameter, ε is the lattice dispersion and (X1 , ..., Xd ) are the discrete coordinates on a periodic lattice (X ∈ [0, 1, ..., N − 1]). The conjugate ”quasimomenta” (Q1 , ..., Qd ) defined as Qi =

∆ 2π mi + ; mi = 0, ..., N − 1, N N

(5)

are already present in the l.h.s. of (3) and their definition is explained in the example below. For simple lattices the dispersion is a sum of cosines ε = α1 cos Q1 + ... + αd cos Qd .

(6)

Therefore the Green function can be related to a Laplace transform of f (z; y) = exp (z (α1 cos y1 + ... + αd cos yd )) . 3

(7)

II.

LATTICE GREEN’S FUNCTION

Let us consider the consequences of (3) for the choice (7) of f (z; y) corresponding to the chain with nearest neighbor interaction. Integration gives the modified Bessel function IkN (z) and (3) becomes ∞ ∞ N −1 X X 1 X IkN (z) cos (k∆) . IkN (z) exp (ik∆) = I0 (z) + 2 exp (z cos Qm ) = N m=0

(8)

k=0

k=−∞

The parameters z and ∆ can now be continued to complex values, e.g., rewriting Eq. (8) in an equivalent form, we obtain a generalization of the Jacobi expansion [15]:       N −1 ∞ X 2πm 1 z 2πm 1 X exp w exp i + exp −i = IkN (z) w kN . N m=0 2 N w N k=−∞

(9)

One can check that the expressions 9.6.33-9.6.40 in [16] are particular cases of (8) or (9) for N = 1, 2. For instance, taking ∆ = θ +

π 2

in (8) one finds

     N −1 ∞ X πk θ π m 1 X exp z cos = IkN (z) exp i exp (iθk) + + 2π N m=0 N 2N N 2 k=−∞ = I0 (z) + 2

∞ X

(−1)k I2kN (z) cos (2kθ) + 2

∞ X

(10)

(−1)k I(2k+1)N (z) sin ((2k + 1) θ) .

k=0

k=1

It is also not difficult to obtain a further generalization of these results by including the space dependent factor of the Green function (4) and a dispersion of the form: ε = b cos Q + c sin Q,

(11)

so that the propagator can be written as N −1 1 X exp (iQX) G (p; X) = . N m=0 p − cos (Q + η)

It corresponds to the Laplace transform of the lattice sum        ∆ 1 X m m ∆ exp z cos X . L= + 2π + η exp i + 2π N m N N N N By denoting ξ ≡ ∆ + ηN we have        N −1 1 X ξ ξ m m L = exp (−iηX) exp z cos exp i X . + 2π + 2π N m=0 N N N N 4

(12)

(13)

(14)

Then substitute (14) into (3) ∞ X

1 L = exp (−iηX) exp (ikξ) 2π k=−∞

Z

2π 0

exp (z cos y) exp (−i (kN − X) y) dy,

and finally obtain        N −1 ∆ m m ∆ 1 X exp z cos X + 2π + η exp i + 2π N m=0 N N N N = exp (−iηX)

∞ X

IkN +X (z) exp (−ik (∆ + ηN)) .

(15) (16)

k=−∞

The above expressions demonstrate a direct relation of the generalized Jacobi expansion to the finite lattice problem. However their main advantage becomes clear by realizing that the increasing orders of the Bessel function in these series correspond to the consecutive terms of the asymptotic N−expansion of the lattice propagators. I.e., the term with k = 0 describes the thermodynamic limit ( N −→ ∞ ) and the first finite-N correction is contained in the term with k = 1. One can also evaluate the convergence of these series, that depends on lattice and on the spectral parameter p. We will provide specific examples and explain the choice of ”quasimomentum” Q and its apparent conflict with periodicity of the lattice. The lattice Green function (12) can now be obtained from the Laplace transform of (16) with respect to z [17] Z

∞

exp (−pz) In (z) dz =

0

exp (− |n| v) , sinh v

where n is an integer and we have introduced the notation p ≡ cosh v ( convergence of the Laplace transform requires Re p > 1 ). Thus

=

∞ X

k=−∞



N −1 ∆ + 2π m X exp i N sinh v X N
m ∆ N m=0 cosh v − cos N + 2π N + η exp (−v |kN + X|) exp (−ik (∆ + ηN) − iηX) ,

leads to the following expression for the propagator (12):

N −1 ∆ + 2π m X exp i N exp (iη (N/2 − XM ) + i (M + 1/2) ∆) 1 X N
= m ∆ N m=0 cosh v − cos N + 2π N + η 2 sinh v ×



exp (−v (N/2 − XM )) exp (v (N/2 − XM )) + sinh ((vN + i (∆ + ηN)) /2) sinh ((vN − i (∆ + ηN)) /2) 5



.

(17)

The integer M in the (17) defines a translation relating an arbitrary X to XM from the main interval: XM = 0, 1, ...N − 1,

(18)

X = XM + MN. We note the transformation properties following from the above definition, X →N +X : X → −X :

X M → XM ,

X M → N − XM ,

X →N −X :

M → M + 1; M → − (M + 1) ;

X M → N − XM ,

M → −M.

(19)

After the r.h.s. of (17) has been found, it is now easy to verify that the inverse DFT (i.e. taking the summation in X ) indeed reproduces the function in the l.h.s.With (19) one also finds another useful form of DFT (17):

N −1 m ∆ + 2π N X cos N cosh (v + iη) 2 sinh v X
= m ∆ N m=0 cosh v − cos N + 2π N + η sinh

N 2 N 2




− XM + i M + 21 ∆
(v + iη) + i ∆2




cosh (v − iη) N2 − XM − i M + 21 ∆
+ . sinh N2 (v − iη) − i ∆2




N −1 ∆ N 1 sin N sinh (v + iη) + 2π m X − X ∆ + i M + 2i sinh v X M N 2
=
2 m ∆ N N cosh v − cos N + η + 2π N sinh (v + iη) 2 + i∆/2 m=0


sinh (v − iη) N2 − XM − i M + 21 ∆
− . (20) sinh (v − iη) N2 − i∆/2

The condition cosh v > 1 in our derivation was due to the use of the Laplace transform. It

is however important to mention that the above expressions can be analytically continued to arbitrary complex values of the parameters ( v, η, ∆ ), since the resonance conditions (zeros of the denominator in the l.h.s.) correspond to simple poles. For instance, a physically important possibility is the region Re p < 1 (p 6= cos Q), where respective solutions of the Schr¨odinger equation with Im p = 0 are known as scattered states.

III.

THE HEISENBERG RING

The model of a cyclic chain of N quantum spins S = 1/2 with ferromagnetic nearest neighbor interaction was the first to be solved by Bethe ansatz [10] and it offers a clear 6

test for the above formal results. The two-magnon subspace of the eigenfunctions of the Hamiltonian H = −J

N X hiji

Si · Sj ,

(21)

contains the main features of the interaction problem and is described by the amplitude of two spins flipped on the sites n1 and n2 : |ψi =

X

1≤n1