An Exactly Solvable Spin Chain Related to Hahn Polynomials
We study a linear spin chain which was originally introduced by Shi et al. [Phys. Rev. A 71 (2005), 032309, 5 pages], for which the coupling strength contains a parameter α and depends on the parity of the chain site. Extending the model by a second parameter β, it is shown that the single fermion e...
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| Цитувати: | An Exactly Solvable Spin Chain Related to Hahn Polynomials /N.I. Stoilova, J. Van der Jeugt // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 22 назв. — англ. |
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irk-123456789-1468022019-02-12T01:24:14Z An Exactly Solvable Spin Chain Related to Hahn Polynomials Stoilova, N.I. Van der Jeugt, J. We study a linear spin chain which was originally introduced by Shi et al. [Phys. Rev. A 71 (2005), 032309, 5 pages], for which the coupling strength contains a parameter α and depends on the parity of the chain site. Extending the model by a second parameter β, it is shown that the single fermion eigenstates of the Hamiltonian can be computed in explicit form. The components of these eigenvectors turn out to be Hahn polynomials with parameters (α,β) and (α+1,β−1). The construction of the eigenvectors relies on two new difference equations for Hahn polynomials. The explicit knowledge of the eigenstates leads to a closed form expression for the correlation function of the spin chain. We also discuss some aspects of a q-extension of this model. 2011 Article An Exactly Solvable Spin Chain Related to Hahn Polynomials /N.I. Stoilova, J. Van der Jeugt // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 22 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 81P45; 33C45 DOI:10.3842/SIGMA.2011.033 http://dspace.nbuv.gov.ua/handle/123456789/146802 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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We study a linear spin chain which was originally introduced by Shi et al. [Phys. Rev. A 71 (2005), 032309, 5 pages], for which the coupling strength contains a parameter α and depends on the parity of the chain site. Extending the model by a second parameter β, it is shown that the single fermion eigenstates of the Hamiltonian can be computed in explicit form. The components of these eigenvectors turn out to be Hahn polynomials with parameters (α,β) and (α+1,β−1). The construction of the eigenvectors relies on two new difference equations for Hahn polynomials. The explicit knowledge of the eigenstates leads to a closed form expression for the correlation function of the spin chain. We also discuss some aspects of a q-extension of this model. |
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Stoilova, N.I. Van der Jeugt, J. |
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Stoilova, N.I. Van der Jeugt, J. An Exactly Solvable Spin Chain Related to Hahn Polynomials Symmetry, Integrability and Geometry: Methods and Applications |
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Stoilova, N.I. Van der Jeugt, J. |
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Stoilova, N.I. |
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An Exactly Solvable Spin Chain Related to Hahn Polynomials |
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An Exactly Solvable Spin Chain Related to Hahn Polynomials |
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An Exactly Solvable Spin Chain Related to Hahn Polynomials |
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An Exactly Solvable Spin Chain Related to Hahn Polynomials |
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An Exactly Solvable Spin Chain Related to Hahn Polynomials |
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exactly solvable spin chain related to hahn polynomials |
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Інститут математики НАН України |
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2011 |
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An Exactly Solvable Spin Chain Related to Hahn Polynomials /N.I. Stoilova, J. Van der Jeugt // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 22 назв. — англ. |
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Symmetry, Integrability and Geometry: Methods and Applications |
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AT stoilovani anexactlysolvablespinchainrelatedtohahnpolynomials AT vanderjeugtj anexactlysolvablespinchainrelatedtohahnpolynomials AT stoilovani exactlysolvablespinchainrelatedtohahnpolynomials AT vanderjeugtj exactlysolvablespinchainrelatedtohahnpolynomials |
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2025-07-11T00:37:45Z |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 7 (2011), 033, 13 pages
An Exactly Solvable Spin Chain Related
to Hahn Polynomials
Neli I. STOILOVA †‡ and Joris VAN DER JEUGT ‡
† Institute for Nuclear Research and Nuclear Energy, Boul. Tsarigradsko Chaussee 72,
1784 Sofia, Bulgaria
‡ Department of Applied Mathematics and Computer Science, Ghent University,
Krijgslaan 281-S9, B-9000 Gent, Belgium
E-mail: Neli.Stoilova@UGent.be, Joris.VanderJeugt@UGent.be
Received January 25, 2011, in final form March 22, 2011; Published online March 29, 2011
doi:10.3842/SIGMA.2011.033
Abstract. We study a linear spin chain which was originally introduced by Shi et al. [Phys.
Rev. A 71 (2005), 032309, 5 pages], for which the coupling strength contains a parameter α
and depends on the parity of the chain site. Extending the model by a second parameter β,
it is shown that the single fermion eigenstates of the Hamiltonian can be computed in
explicit form. The components of these eigenvectors turn out to be Hahn polynomials with
parameters (α, β) and (α+ 1, β− 1). The construction of the eigenvectors relies on two new
difference equations for Hahn polynomials. The explicit knowledge of the eigenstates leads
to a closed form expression for the correlation function of the spin chain. We also discuss
some aspects of a q-extension of this model.
Key words: linear spin chain; Hahn polynomial; state transfer
2010 Mathematics Subject Classification: 81P45; 33C45
1 Introduction
Consider a linear chain of N + 1 interacting fermions described by the Hamiltonian
Ĥ =
N−1∑
k=0
Jk
(
a†kak+1 + a†k+1ak
)
. (1)
The lattice fermions {ak, a†k | k = 0, 1, . . . , N} obey the common anticommutation relations,
and Jk expresses the coupling strength between sites k and k+1. The Hamiltonian (1) describes
a system of N + 1 fermions on a chain with nearest-neighbour interaction (hopping between
adjacent sites of the chain) subject to a zero background magnetic field.
Hamiltonians of the type (1) appear in various contexts. In particular, spin chains of this
type are popular as channels for short distance quantum communication, and were introduced
by Bose [2, 3, 4]. The system then originates from a linear qubit chain with nearest neigh-
bour interaction described by a Heisenberg XY Hamiltonian, and is being mapped into (1) by
a Jordan–Wigner transformation [5, 6]. In such models, the communication is achieved by state
dynamical evolution in the spin chain, which does not require any on/off switches of the inter-
actions between the spins, nor any modulation of external fields. Many articles dealing with
such spin chains in the context of Bose’s scheme focus on perfect transmission (or perfect state
transfer) in these chains [7, 8, 9, 10, 11, 12].
By far the most elegant and simplest scheme to realize perfect state transfer (over an arbitrary
long chain) was proposed by Christandl et al. [7, 8, 9]. Their choice of the modulation of the
mailto:Neli.Stoilova@UGent.be
mailto:Joris.VanderJeugt@UGent.be
http://dx.doi.org/10.3842/SIGMA.2011.033
2 N.I. Stoilova and J. Van der Jeugt
coupling strengths is given by:
Jk = JN−k−1 =
√
(k + 1)(N − k), k = 0, 1, . . . , N − 1. (2)
The simplicity of Christandl’s model follows from the following observation. Consider first the
single-fermion states of the system: in a single-fermion basis, the Hamiltonian Ĥ takes the
matrix form
M =
0 J0 0 · · · 0
J0 0 J1 · · · 0
0 J1 0
. . .
...
...
. . .
. . . JN−1
0 0 JN−1 0
. (3)
The dynamics (time evolution) of the system is completely determined by the eigenvalues εj and
eigenvectors ϕj of this interaction matrix. It is indeed a standard technique [5, 8] to describe
the n-fermion eigenstates of Ĥ (n ≤ N) using the single-fermion eigenstates ϕj and Slater
determinants. In Christandl’s case, determined by (2), the eigenvalues and eigenvectors of M
are explicitly known. In particular, the eigenvalues are given by εj = −N + 2j (j = 0, 1, . . . , N)
and the eigenvectors are given in terms of Krawtchouk polynomials.
In the model of Christandl there is, for arbitrary N , an analytic (closed form) expression for
the eigenvalues and eigenvectors of M . Such spin chains are analytically solvable [13, 14]. It
implies, in particular, that the correlation function at time t,
fr,s(t) = (r| exp(−itĤ)|s),
where r and s are site labels belonging to {0, 1, . . . , N} and |r), |s) denote the corresponding
single spin states at the ‘receiver’ and ‘sender’ sites r and s, can be computed explicitly [13, 14].
Christandl’s spin chain model allows perfect state transfer, essential for using the spin chain
as a transmission channel. Perfect state transfer at time t = T from one end of the chain to the
other end is expressed by |fN,0(T )| = 1. The topic of perfect state transfer in spin chains has
received a lot of attention [12]. Fairly easy sufficient conditions have been formulated in order
to achieve perfect state transfer (such as mirror symmetry [8, 12]). Shi et al. showed that the
“spectrum parity matching condition” is necessary and sufficient for perfect state transfer [1].
According to this condition, they found a one-parameter extension of Christandl’s model in the
case there is an even number of fermion sites in the chain; in our notation this means that N is
odd, i.e.
N = 2m+ 1, m ∈ Z+.
The couplings in Shi’s model [1] are determined by (k = 0, 1, . . . , N − 1)
Jk =
{√
(k + 1)(N − k), if k is odd,√
(k + 2α+ 2)(N − k + 2α+ 1), if k is even.
(4)
Herein, α is a real parameter satisfying α > −1 (the case of Shi actually corresponds to half-
integer α, but for our purposes α can be any real number greater than−1). Note that for α = −1
2 ,
Shi’s model reduces to Christandl’s model (i.e. (4) reduces to (2)), at least when N = 2m+ 1 is
odd. In Shi et al. [1], the spectrum of the single fermion states (i.e. the eigenvalues of M with
data determined by (4)) was found; however no closed form expressions for the eigenvectors were
obtained.
An Exactly Solvable Spin Chain Related to Hahn Polynomials 3
In the present paper, we will show that the eigenvectors can be expressed in terms of Hahn
polynomials. In fact, we will first work with a two-parameter extension of Christandl’s model:
Jk =
{√
(k + 1)(N − k), if k is odd,√
(k + 2α+ 2)(N − k + 2β − 1), if k is even.
(5)
Now α and β are real parameters satisfying α > −1 and β > 0. The case of Shi corresponds to
β = α + 1, and the case of Christandl to α = −1
2 , β = 1
2 . For the general case (5), we obtain
in this paper an explicit form of the eigenvalues, and an explicit form of the eigenvectors. The
components of the eigenvectors are given by means of Hahn polynomials Qn(x;α, β,m) [15, 16]:
the even components are proportional to Qn(x;α, β,m) and the odd components to Qn(x;α +
1, β − 1,m). In order to prove our assertions about eigenvalues and eigenvectors, we need
some (new) difference equations for Hahn polynomials. Section 2 of this paper is devoted
to introducing the common notation for Hahn polynomials and to proving the new difference
equations. In Section 3 we obtain the main result of this paper: the explicit construction of the
spectrum of M for the values (5) and the construction of its eigenvectors in terms of the Hahn
polynomials. Section 4 returns to the model governed by the spin chain data (5). Since the spin
chain is analytically solvable, we can compute the correlation function explicitly, and determine
under which conditions perfect state transfer is possible. Finally, in Section 5 we present the
q-generalization of the results obtained (in terms of q-Hahn polynomials).
Although our paper is strongly inspired by the model introduced by Shi et al., it should be
emphasized that our results are dealing mainly with mathematical aspects of this model. In [1],
the emphasis was on quantum state transfer. Our main result is to show that the eigenstates
of the Hamiltonian (1) in the case of Shi, (4), or in the extended case, (5), can be computed in
closed form, with coefficients given as Hahn polynomial evaluations.
It should be mentioned that some completely different spin chain models related to Hahn
polynomials have been considered before. The second solution of [8] is actually related to
an interaction matrix corresponding to the Jacobi matrix of dual Hahn polynomials. In [13],
the interaction matrix corresponding to the Jacobi matrix of Hahn polynomials was studied,
following some ideas of [17]. In that case, the matrix of eigenvectors U is directly a matrix of
Hahn polynomial evaluations. However, due to the complicated coefficients in the three term
recurrence relations, the actual coefficients in the interaction matrix become quite involved,
see e.g. [13, Lemma 2]. In the present paper, the main innovation comes from “doubling” the
matrix U (hence the technique works for chains with an even length only), in a way that it
contains Hahn polynomial evaluations of two different types (one with parameters (α, β) and
one with (α+ 1, β − 1)), such that the interaction matrix coefficients (5) are very simple.
2 Hahn polynomials and new difference equations
The Hahn polynomial Qn(x;α, β,m) [15, 16] of degree n (n = 0, 1, . . . ,m) in the variable x,
with parameters α > −1 and β > −1, or α < −m and β < −m is defined by [15, 16]:
Qn(x;α, β,m) = 3F2
(
−n, n+ α+ β + 1,−x
α+ 1,−m
; 1
)
. (6)
Herein, the function 3F2 is the generalized hypergeometric series [18, 19]:
3F2
(
a, b, c
d, e
; z
)
=
∞∑
k=0
(a)k(b)k(c)k
(d)k(e)k
zk
k!
. (7)
In (6), the series is terminating because of the appearance of the negative integer −n as a nume-
rator parameter. Note that in (7) we use the common notation for Pochhammer symbols [18, 19]
4 N.I. Stoilova and J. Van der Jeugt
(a)k = a(a+1) · · · (a+k−1) for k = 1, 2, . . . and (a)0 = 1. Hahn polynomials satisfy a (discrete)
orthogonality relation [15]:
m∑
x=0
w(x;α, β,m)Ql(x;α, β,m)Qn(x;α, β,m) = hn(α, β,m)δln, (8)
where
w(x;α, β,m) =
(
α+ x
x
)(
m+ β − x
m− x
)
, x = 0, 1, . . . ,m,
hn(α, β,m) =
(−1)n(n+ α+ β + 1)m+1(β + 1)nn!
(2n+ α+ β + 1)(α+ 1)n(−m)nm!
.
Denote the orthonormal Hahn functions as follows:
Q̃n(x;α, β,m) ≡
√
w(x;α, β,m)Qn(x;α, β,m)√
hn(α, β,m)
. (9)
For our construction, the essential ingredient is a set of new difference equations for Hahn
polynomials. These relations involve Hahn polynomials of the same degree in variables x or
x+ 1, and with parameters (α, β) and (α + 1, β − 1); in this sense it could also be appropriate
to speak of “contiguous relations” rather than difference equations.
Proposition 1. The Hahn polynomials satisfy the following difference equations:
(m+ β − x)Qn(x;α, β,m)− (m− x)Qn(x+ 1;α, β,m)
=
(n+ α+ 1)(n+ β)
α+ 1
Qn(x;α+ 1, β − 1,m), (10)
(x+ 1)Qn(x;α+ 1, β − 1,m)− (α+ x+ 2)Qn(x+ 1;α+ 1, β − 1,m)
= −(α+ 1)Qn(x+ 1;α, β,m). (11)
Proof. Both equations follow from a simple computation using the hypergeometric series ex-
pression. In the case of (10), the left hand side is expanded as follows:
(m+ β − x)Qn(x;α, β,m)− (m− x)Qn(x+ 1;α, β,m)
=
n∑
k=0
(−n)k(α+ β + n+ 1)k(−x)k−1
k!(α+ 1)k(−m)k
[(β +m− x)(k − x− 1)− (m− x)(−x− 1)].
Rewriting the expression in square brackets as [k(m − k + 1) + (β + k)(k − x − 1)], the above
sum splits in two parts:
−
n∑
k=1
(−n)k(α+ β + n+ 1)k(−x)k−1
(k − 1)!(α+ 1)k(−m)k−1
+
n∑
k=0
(−n)k(α+ β + n+ 1)k(−x)k
k!(α+ 1)k(−m)k
(β + k). (12)
The first part can be brought in the following form:
−
n∑
k=1
(−n)k(α+ β + n+ 1)k(−x)k−1
(k − 1)!(α+ 1)k(−m)k−1
= −
n−1∑
j=0
(−n)j+1(α+ β + n+ 1)j+1(−x)j
j!(α+ 1)j+1(−m)j
=
n∑
k=0
(−n)k(α+ β + n+ 1)k(−x)k
k!(α+ 1)k(−m)k
(n− k)(α+ β + n+ k + 1)
(α+ k + 1)
.
An Exactly Solvable Spin Chain Related to Hahn Polynomials 5
So (12) becomes
n∑
k=0
(−n)k(α+ β + n+ 1)k(−x)k
k!(α+ 1)k(−m)k
[(n− k)(α+ β + n+ k + 1)
(α+ k + 1)
+ (β + k)
]
=
n∑
k=0
(−n)k(α+ β + n+ 1)k(−x)k
k!(α+ 1)k(−m)k
[(n+ α+ 1)(n+ β)
(α+ k + 1)
]
=
(n+ α+ 1)(n+ β)
(α+ 1)
n∑
k=0
(−n)k(α+ β + n+ 1)k(−x)k
k!(α+ 2)k(−m)k
leading to the right hand side of (10).
The second equation is even simpler to prove. The left hand side of (11) can be written as
(x+ 1)
n∑
k=0
Ak
(−x+ k − 1)
(α+ k + 1)
− (α+ x+ 2)
n∑
k=0
Ak
(−x− 1)
(α+ k + 1)
, (13)
where
Ak =
(−n)k(α+ β + n+ 1)k(−x)k−1
k!(α+ 2)k−1(−m)k
.
A simple addition of the two terms in (13) yields
−
n∑
k=0
Ak(−x− 1) = −(α+ 1)
n∑
k=0
(−n)k(α+ β + n+ 1)k(−x− 1)k
k!(α+ 1)k(−m)k
,
giving the right hand side of (11). �
The set of difference equations will turn out to be the essential ingredient for the eigenvector
construction in the next section.
3 Eigenvalues and eigenvectors of the interaction matrix
Let N = 2m + 1 be an odd integer, and consider the (N + 1) × (N + 1) interaction matrix M
of the form (3) with spin chain data Jk determined by (5), i.e.
Jk =
{√
(k + 1)(2m+ 1− k), if k is odd,√
(k + 2α+ 2)(2m+ 2β − k), if k is even.
(14)
We begin with the construction of a (N + 1)× (N + 1) matrix U = (Uij)0≤i,j≤N whose even
rows are given in terms of normalized Hahn polynomials (9) with parameters (α, β) and whose
odd rows are given in terms of normalized Hahn polynomials with parameters (α+ 1, β− 1). In
order to have positive weight functions for both, we require that α > −1 and β > 0.
Definition 1. The (N + 1)× (N + 1) matrix U with indices running from 0 to N = 2m+ 1 is
defined by
U2i,m−j = U2i,m+j+1 =
(−1)i√
2
Q̃j(i;α, β,m), (15)
U2i+1,m−j = −U2i+1,m+j+1 = −(−1)i√
2
Q̃j(i;α+ 1, β − 1,m), (16)
where i, j ∈ {0, 1, . . . ,m}.
6 N.I. Stoilova and J. Van der Jeugt
It is easy to verify that U is an orthogonal matrix. Indeed, let us compute UTU :
(UTU)jk =
2m+1∑
i=0
UijUik =
m∑
i=0
U2i,jU2i,k +
m∑
i=0
U2i+1,jU2i+1,k. (17)
For j, k ∈ {0, . . . ,m}, (17) gives
m∑
i=0
1
2
Q̃m−j(i;α, β,m)Q̃m−k(i;α, β,m)
+
m∑
i=0
1
2
Q̃m−j(i;α+ 1, β − 1,m)Q̃m−k(i;α+ 1, β − 1,m)
=
1
2
δm−j,m−k +
1
2
δm−j,m−k = δjk
using the orthogonality (8) of Hahn polynomials. For j, k ∈ {m+1, . . . , 2m+1}, the computation
is essentially the same and (17) gives again δjk. For j ∈ {0, . . . ,m} and k ∈ {m+1, . . . , 2m+1},
(17) gives
m∑
i=0
1
2
Q̃m−j(i;α, β,m)Q̃k−m−1(i;α, β,m)
−
m∑
i=0
1
2
Q̃m−j(i;α+ 1, β − 1,m)Q̃k−m−1(i;α+ 1, β − 1,m)
=
1
2
δm−j,k−m−1 −
1
2
δm−j,k−m−1 = 0,
and for j ∈ {m + 1, . . . , 2m + 1} and k ∈ {0, . . . ,m}, the result is the same. So it follows that
(UTU)jk = δjk, or UTU = I, the identity matrix. Hence UT is the inverse of U , so UUT = I
holds as well.
Now we have the main proposition.
Proposition 2. Let M be the tridiagonal (2m+ 2)× (2m+ 2)-matrix
M =
0 J0 0
J0 0 J1
. . .
0 J1 0
. . . 0
. . .
. . .
. . . J2m
0 J2m 0
,
where the Jk are given in (14), and let U be the matrix determined in Def inition 1. Then U is
an orthogonal matrix:
UUT = UTU = I.
Furthermore, the columns of U are the eigenvectors of M , i.e.
MU = UD, (18)
where D is a diagonal matrix containing the eigenvalues εj of M :
D = diag(ε0, ε1, . . . , ε2m+1), εm−k = −2
√
(α+ k + 1)(β + k),
εm+k+1 = 2
√
(α+ k + 1)(β + k), k = 0, 1, . . . ,m. (19)
An Exactly Solvable Spin Chain Related to Hahn Polynomials 7
Proof. The orthogonality of U has already been proved, so it remains to verify (18) and (19).
Now
(
MU
)
ij
=
2m+1∑
k=0
MikUkj = Ji−1Ui−1,j + JiUi+1,j . (20)
We have to consider (20) in four distinct cases, according to i even or odd, and to j belonging
to {0, 1, . . . ,m} or to {m + 1,m + 2, . . . , 2m + 1}. Let us consider the case that i is odd and
j ∈ {0, 1, . . . ,m}. Then, relabelling the indices appropriately, (14) and (15) yield:
(MU)2i+1,m−j = J2iU2i,m−j + J2i+1U2i+2,m−j
= (−1)i
√
2
√
(α+ i+ 1)(m+ β − i)Q̃j(i;α, β,m)
+ (−1)i+1
√
2
√
(i+ 1)(m− i)Q̃j(i+ 1;α, β,m)
= (−1)i
√
2
√
(α+ 1)i+1(β + 1)m−i−1
i!(m− i)!hj(α, β,m)
[(β +m− i)Qj(i;α, β,m)
− (m− i)Qj(i+ 1;α, β,m)].
Applying (10), this becomes
= (−1)i
√
2
√
(α+ 1)i+1(β + 1)m−i−1
i!(m− i)!hj(α, β,m)
(α+ j + 1)(β + j)
(α+ 1)
Qj(i;α+ 1, β − 1,m)
= −2
√
(α+ j + 1)(β + j)U2i+1,m−j = εm−jU2i+1,m−j =
(
UD
)
2i+1,m−j .
For i odd and j ∈ {m + 1,m + 2, . . . , 2m + 1}, the computation is essentially the same. For i
even (and the two cases for j), the computation is also similar, but now the second difference
equation (11) must be used. �
Note that the spectrum of M is symmetric, consisting of the values ±2
√
(α+ k + 1)(β + k)
(k = 0, 1, . . . ,m). Furthermore, when β = α+1, the spectrum consists of integers ±2(α+k+1).
This latter case corresponds to the model of Shi et al. [1].
4 Some aspects of the corresponding spin chain model
Let us consider a spin chain (1) with data determined by (5). The dynamics of this system
is described by the unitary time evolution operator exp(−itĤ). The transition amplitude of
a single spin excitation from site s to site r of the spin chain is given by the time-dependent
correlation function [4, 13]
fr,s(t) = (r| exp(−itĤ)|s).
But the (orthonormal) eigenvectors of Ĥ in the single fermion mode are now known and given
by ϕj =
N∑
k=0
Ukj | k), i.e. the columns of the matrix U constructed in (15), (16), with Ĥϕj =
Mϕj = εjϕj . Using the orthogonality of the states ϕj , one finds [13]:
fr,s(t) =
N∑
j=0
UrjUsje
−itεj .
8 N.I. Stoilova and J. Van der Jeugt
Due to the expressions (15), (16), implying Ur,m−j = (−1)rUr,m+j+1, it is appropriate to
write the correlation function in the following form:
frs(t) =
m∑
j=0
(
Ur,m−jUs,m−je
−itεm−j + Ur,m+j+1Us,m+j+1e
−itεm+j+1
)
=
m∑
j=0
Ur,m−jUs,m−j
(
e−itεm−j + (−1)r+s eitεm−j
)
. (21)
Now it is a matter of considering the different parities for r and s. In the case they are both
even, one finds
f2k,2l(t) = (−1)k+l
√
w(k;α, β,m)w(l;α, β,m)
×
m∑
j=0
Qj(k;α, β,m)Qj(l;α, β,m)
cos
(
2t
√
(α+ j + 1)(β + j)
)
hj(α, β,m)
.
In the case the first index is odd and the second even, this becomes
f2k+1,2l(t) = −i(−1)k+l
√
w(k;α+ 1, β − 1,m)w(l;α, β,m) (22)
×
m∑
j=0
Qj(k;α+ 1, β − 1,m)Qj(l;α, β,m)
sin
(
2t
√
(α+ j + 1)(β + j)
)√
hj(α+ 1, β − 1,m)hj(α, β,m)
.
The expressions for the case even/odd and odd/odd are similar, the main message being that due
to the analytic expressions for the eigenvectors, we obtain explicit expressions for the correlation
function.
Let us examine, in this context, the transition from one end of the chain (s = 0) to the final
end of the chain (r = N = 2m+ 1). Expression (22) reduces to:
f2m+1,0(t) = −i(−1)m
√
(β)m+1(α+ 1)m+1
×
m∑
j=0
(2j + α+ β + 1)(−m)j
(j + α+ β + 1)m+1j!
sin
(
2t
√
(α+ j + 1)(β + j)
)√
(α+ j + 1)(β + j)
. (23)
For general α and β, this expression cannot be simplified further. Let us now consider the
special case that
β = α+ 1.
Then (23) reduces to
f2m+1,0(t) = −2i(−1)m(α+ 1)m+1
m∑
j=0
(−m)j
(j + 2α+ 2)m+1j!
sin
(
2t(α+ j + 1)
)
. (24)
This can be rewritten as
f2m+1,0(t) = −2i(−1)m
(α+ 1)m+1
(2α+ 2)m+1
m∑
j=0
(−m)j(2α+ 2)j
j!(2α+m+ 3)j
sin
(
2t(α+ j + 1)
)
. (25)
The last sum is of hypergeometric type 2F1, and so it can be further simplified for special values
of t and/or α. In particular, for t = T = π/2, one has sin
(
π(α + j + 1)
)
= −(−1)j sin(πα).
Using then Kummer’s summation formula [18, 19]
2F1
(
−m, 2α+ 2
2α+m+ 3
;−1
)
=
(2α+ 3)m
(α+ 2)m
,
An Exactly Solvable Spin Chain Related to Hahn Polynomials 9
in the right hand side of (25), this expression reduces to
fN,0
(π
2
)
= f2m+1,0
(π
2
)
= i(−1)m sin(πα). (26)
Note, by the way, that also for t = 2T = π one can simplify (25), since now sin
(
2π(α+ j+1)
)
=
sin(2πα). Then, using Gauss’s summation formula [18, 19]
2F1
(
−m, 2α+ 2
2α+m+ 3
; 1
)
=
(m+ 1)m
(2α+m+ 3)m
,
the right hand side of (25) reduces to
fN,0(π) = f2m+1,0(π) = −2i sin(2πα)(−1)m
(α+ 1)m+1(m+ 1)m
(2α+ 2)2m+1
.
Note the importance of (26). Indeed, keeping in mind that α > −1, we have∣∣∣fN,0 (π
2
)∣∣∣ =
∣∣∣f2m+1,0
(π
2
)∣∣∣ = 1 for α = −1
2
,
1
2
,
3
2
,
5
2
, . . . . (27)
So there is perfect state transfer in the chain for α assuming one of these values, at time t = π/2.
In fact, this corresponds to the values given by Shi et al. [1]. Note that for α = −1
2 , the spin
chain data further reduces to that of Christandl [7].
The case (27) corresponds to 2α+1 = 2l with l a nonnegative integer. As a matter of fact, this
can still be extended slightly. Let 2α+ 1 = 2l
2k+1 with both l and k nonnegative integers. Then
for t = T ′ = (2k+1)π/2, the factor in (25) becomes sin(2t(α+j+1)) = −(−1)j sin((2k+1)απ) =
(−1)j+k+l, and so we have the result∣∣∣fN,0 ((2k + 1)
π
2
)∣∣∣ =
∣∣∣f2m+1,0
(
(2k + 1)
π
2
)∣∣∣ = 1 for 2α+ 1 =
2l
2k + 1
, k, l ∈ Z+.
This case appears already in the paper of Qian et al. [20], who use the “mirror mode concurrence”
to find this extension of Shi’s result.
As far as perfect state transfer is concerned, our extension of Shi’s model by an extra para-
meter β does not give rise to any new cases. In fact, just for investigating perfect state transfer,
the mathematical machinery developed here is not necessary: the verification of the spectrum
parity matching condition, using the entries in the interaction matrix and the spectrum itself,
is sufficient. The main advantage of our analysis is the explicit computation of the correlation
function. In particular, the simplicity of the expressions (24) and (26), describing the transfer
from one end of the chain to the other end, is striking. Apart from the model of Christandl [7],
where the general correlation function is given in [13, § 2], there are no other models with such
an elegant and simple correlation function.
5 On the q-generalization of the previous results
As the classical orthogonal polynomials of hypergeometric type have a generalization in terms
of basic hypergeometric series, i.e. a q-generalization, one may wonder whether the present
construction of the tridiagonal interaction matrix M can also be generalized. This is indeed the
case: we can present a matrix Mq, whose eigenvectors are given in terms of q-Hahn polynomials,
and whose eigenvalues are symmetric and take a simple form. In order to present these results,
let us first briefly recall some notation related to q-series [21].
10 N.I. Stoilova and J. Van der Jeugt
For a positive real number q (6= 1), the q-Hahn polynomial Qn(q−x;α, β,m|q) of degree n
(n = 0, 1, . . . ,m) in q−x is defined by [15, 21]:
Qn(q−x;α, β,m|q) = 3Φ2
(
q−n, αβqn+1, q−x
αq, q−m
; q, q
)
. (28)
Herein, the function 3Φ2 is the basic hypergeometric series [18, 19, 21]:
3Φ2
(
a, b, c
d, e
; q, z
)
=
∞∑
k=0
(a, b, c; q)k
(q, d, e; q)k
zk. (29)
Note that in (29) we use the common notation for q-shifted factorials and their products [21]:
(a1, a2, . . . , aA; q)k = (a1; q)k(a2; q)k · · · (aA; q)k,
(a; q)k = (1− a)(1− aq) · · ·
(
1− aqk−1
)
and (a)0 = 1.
In (28), the series is terminating because of the appearance of q−n in the numerator. q-Hahn
polynomials satisfy a (discrete) orthogonality relation [15]:
m∑
x=0
w(x;α, β,m|q)Ql
(
q−x;α, β,m|q
)
Qn
(
q−x;α, β,m|q
)
= hn(α, β,m|q)δln,
where
w(x;α, β,m|q) =
(αq, q−m; q)x
(q, β−1q−m; q)x
(αβq)−x, x = 0, 1, . . . ,m,
hn(α, β,m|q) =
(αβq2; q)m(q, αβqm+2, βq; q)n(1− αβq)(−αq)n
(βq; q)m(αq)m(αq, αβq, q−m; q)n(1− αβq2n+1)
q(
n
2
)−mn.
We shall assume that 0 < q < 1; then the weight function is positive when 0 < α < q−1 and
0 < β < q−1. Denote the orthonormal q-Hahn functions as follows:
Q̃n(q−x;α, β,m|q) ≡
√
w(x;α, β,m|q)Qn(q−x;α, β,m|q)√
hn(q;α, β,m)
. (30)
Just as in Section 2, the main result needed is a set of difference equations for q-Hahn
polynomials.
Proposition 3. q-Hahn polynomials satisfy the following q-difference equations:(
1− βqm−x
)
Qn
(
q−x;α, β,m|q
)
−
(
1− qm−x
)
Qn
(
q−x−1;α, β,m|q
)
=
(1− αqn+1)(1− βqn)qm−n−x
1− αq
Qn
(
q−x;αq, βq−1,m|q
)
, (31)(
1− qx+1
)
αqQn
(
q−x;αq, βq−1,m|q
)
−
(
1− αqx+2
)
Qn
(
q−x−1;αq, βq−1,m|q
)
= −(1− αq)Qn
(
q−x−1;α, β,m|q
)
. (32)
Proof. The proof follows the same computation as in the proof of Proposition 1, with the
replacement of Pochhammer symbols by the corresponding q-shifted factorials (and keeping
track of the appropriate powers of q). �
An Exactly Solvable Spin Chain Related to Hahn Polynomials 11
We now come to the construction of the tridiagonal matrix Mq and the matrix of eigenvec-
tors U . The polynomials that appear here will be q-Hahn polynomials with parameters (α, β)
and (αq, βq−1). So in order to have positive weight functions for both sets of polynomials, we
shall assume:
0 < q < 1, 0 < α < q−1, 0 < β < 1.
As before, let N = 2m + 1, and consider the (N + 1) × (N + 1) interaction matrix Mq of the
form (3) with non-zero matrix elements given by:
J2k+1 = 2
√
(1− qk+1)(1− qm−k)qk+1α, J2k = 2
√
(1− αqk+1)(1− βqm−k)qk, (33)
where k = 0, 1, . . . ,m. The (N+1)×(N+1) matrix U with indices running from 0 to N = 2m+1
is defined similarly as in Definition 1, but in terms of q-Hahn polynomials (30):
U2i,m−j = U2i,m+j+1 =
(−1)i√
2
Q̃j
(
q−i;α, β,m|q
)
, (34)
U2i+1,m−j = −U2i+1,m+j+1 = −(−1)i√
2
Q̃j
(
q−i;αq, βq−1,m|q
)
, (35)
where i, j ∈ {0, 1, . . . ,m}. By the same argument as in Section 3, the orthogonality of the matrix
U follows from the orthogonality of the q-Hahn polynomials and the appropriate signs.
Then the main result in the q-case reads:
Proposition 4. Let Mq be the tridiagonal (2m+ 2)× (2m+ 2)-matrix
Mq =
0 J0 0
J0 0 J1
. . .
0 J1 0
. . . 0
. . .
. . .
. . . J2m
0 J2m 0
,
where the Jk are given in (33), and let U be the matrix determined in (34), (35). Then U is
an orthogonal matrix, UUT = UTU = I. Furthermore, the columns of U are the eigenvectors
of Mq, i.e.
MqU = UD,
where D is a diagonal matrix containing the eigenvalues εj of Mq:
D = diag(ε0, ε1, . . . , ε2m+1), εm−k = −2
√
(1− αqk+1)(1− βqk)qm−k,
εm+k+1 = 2
√
(1− αqk+1)(1− βqk)qm−k, k = 0, 1, . . . ,m.
The proof of this proposition is essentially the same as that of Proposition 3, and uses the
q-difference equations (31), (32). Note that, as in the ordinary case, the spectrum of Mq is
symmetric.
Once the explicit eigenvectors of the Hamiltonian are known in the q-generalized case (33),
one can compute the correlation function, using the same expression (21). The expressions
become quite involved, so we give just one example here. This is in the case of transition from
one end (s = 0) to the other end (r = 2m+ 1) of the chain, and for β = qα:
f2m+1,0(t) = i(−1)mqm/2αm/2(αq; q)m+1
×
m∑
j=0
(−1)j sin
(
2t
(
1− αqj+1
)
q(m−j)/2
)
qj
2/2 (qm−j+1; q)j(1 + αqj+1)
(α2qj+2; q)m+1(q; q)j
.
The q-generalization does not give rise to any special cases with perfect state transfer, however.
12 N.I. Stoilova and J. Van der Jeugt
6 Conclusions and outlook
We have dealt with some mathematical aspects of a spin chain model of Shi et al. [1], which
is a one-parameter extension of the popular spin chain introduced by Christandl et al. [7]. In
Christandl’s model, the single fermion eigenvalues and eigenvectors could easily be computed, the
eigenvectors being related to Krawtchouk polynomials. In Shi’s model, with an extra parameter
α, there was so far no known expression of the eigenvectors. In the current paper, we have
shown that these eigenvectors can be expressed in terms of Hahn polynomials. As a matter of
fact, we first extend Shi’s model by introducing an extra parameter β, and then construct the
eigenvectors. In this process two types of Hahn polynomials are involved, those with parameters
(α, β) and those with (α + 1, β − 1). These Hahn polynomials are appropriately combined in
a matrix U , yielding the eigenvectors wanted. When β = α + 1, the two-parameter spin chain
reduces to that of Shi. And when α = −1
2 , β = 1
2 , the spin chain reduces to that of Christandl.
Note, by the way, that in this last case the eigenvectors (expressed in terms of Hahn polynomials)
indeed reduce to the known ones (expressed in terms of Krawtchouk polynomials). This follows
from the fact that when α = −1
2 and β = 1
2 the Hahn polynomials, which are 3F2 series, reduce
to 2F1 series according to
3F2
(
−s, s+ 1,−i
1/2,−m
; 1
)
= (−1)i
(
2m+1
2i
)(
m
i
) 2F1
(
−2i,−m− s− 1
−2m− 1
; 2
)
,
3F2
(
−s, s+ 1,−i
3/2,−m
; 1
)
= − (−1)i
(2s+ 1)
(
2m+1
2i+1
)(
m
i
) 2F1
(
−2i− 1,−m− s− 1
−2m− 1
; 2
)
.
These reductions can be obtained, e.g., from [22, (48)]. The 2F1 series in the right hand side
correspond to (symmetric) Krawtchouk polynomials (with p = 1/2).
Due to the explicit forms of the eigenvectors, the time-dependent correlation function fr,s(t)
has been computed for the spin chains under consideration. In special cases, the expression of
the correlation function is particularly simple, see e.g. (26).
In the construction of the eigenvectors, the main relations needed are two new difference equa-
tions for Hahn polynomials. We have also examined the q-generalization of these results. The
q-extension of these difference equations is more or less straightforward. Also the construction
of the corresponding eigenvectors in terms of q-Hahn polynomials has been completed.
The extension of symmetric Krawtchouk polynomials (without a parameter α) to Hahn poly-
nomials (with parameters (α, α + 1)) may also be used in other applications. In particular,
we hope to extend the finite oscillator models of [22], by introducing such an extra parame-
ter. It remains to be seen, in that case, how the underlying Lie algebra is deformed, and how
the parameter has an influence on the finite oscillator eigenstates. This topic will be treated
elsewhere.
Acknowledgements
N.I. Stoilova would like to thank Professor H.D. Doebner (Clausthal, Germany) for constructive
discussions. N.I. Stoilova was supported by project P6/02 of the Interuniversity Attraction Poles
Programme (Belgian State – Belgian Science Policy) and by the Humboldt Foundation.
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Introduction
Hahn polynomials and new difference equations
Eigenvalues and eigenvectors of the interaction matrix
Some aspects of the corresponding spin chain model
On the q-generalization of the previous results
Conclusions and outlook
References
|