Theoretical and Mathematical Physics, 204(3): 1195–1200 (2020)

WRONSKIAN-TYPE FORMULA FOR INHOMOGENEOUS TQ

EQUATIONS

Rafael I. Nepomechie

∗

It is known that the transfer-matrix eigenvalues of the isotropic open Heisenberg quantum spin-1/2 chain

with nondiagonal boundary magnetic ﬁelds satisfy a TQ equation with an inhomogeneous term. We

derive a discrete Wronskian-type formula relating a solution of this inhomogeneous TQ equation to the

corresponding solution of a dual inhomogeneous TQ equation.

Keywords: Bethe ansatz, TQ equation, discrete Wronskian, boundary integrability

DOI: 10.1134/S004057792009007X

1. Introduction and summary of results

We consider the famous Baxter TQ equation for the closed periodic XXX spin chain of length N:

T (u)Q(u)=(u

)

−−

(u)+(u

−

)

(u). (1.1)

Here and hereafter, we use the brief notation f

(u)=f(u ±i/2) and f

±±

(u)=f(u ±i). It is known that

for a given transfer-matrix eigenvalue T (u), Eq. (1.1) can be regarded as a second-order ﬁnite-diﬀerence

equation for Q(u). The eigenvalue T (u) is necessarily a polynomial in u (of degree N ) because the model

is integrable. It is well known that Eq. (1.1) has two independent polynomial solutions [1]. One of them is

a polynomial Q(u)ofdegreeM ≤ N/2oftheform

Q(u)=



k=1

(u − u

), (1.2)

whose zeros {u

} are solutions of the Bethe equations that follow directly from (1.1)



+ i/2

− i/2





k=1,

k=j

− u

+ i

− u

− i

,j=1,...,M. (1.3)

The other is a polynomial P (u)ofdegreeN − M +1>N/2 corresponding to Bethe roots “on the other

side of the equator.” These two solutions are related by a discrete Wronskian (or Casoratian) formula

(u)Q

−

(u) − P

−

(u)Q

(u) ∝ u

, (1.4)

∗

Physics Department, University of Miami, Coral Gables, Florida, USA, e-mail: nep[email protected].

This research was supported in part by a Cooper fellowship.

Prepared from an English manuscript submitted by the author; for the Russian version, see Teoreticheskaya

i Matematicheskaya Fizika, Vol. 204, No. 3, pp. 430–435, September, 2020. Received January 17, 2020. Revised

March 23, 2020. Accepted March 23, 2020.

0040-5779/20/2043-1195

 2020 Pleiades Publishing, Ltd.

1195

where ∝ denotes equality up to a multiplicative constant. The existence of a second polynomial solution of

the TQ equation is equivalent to the admissibility of the Bethe roots [2], [3]. Using the Wronskian formula,

we can succinctly reformulate the Q-system for this model [4] (which provides an eﬃcient way to compute

the admissible Bethe roots) in terms of Q and P [5], [6].

A generalization of Wronskian formula (1.4)fortheopenXXX spin chain with diagonal boundary

ﬁelds was recently obtained [7]:

g(u)P

(u)Q

−

(u) − f (u)P

−

(u)Q

(u) ∝ u

2N+1

, (1.5)

where Q(u)andP (u) are the respective polynomial solutions of a TQand a dual TQequation (see Eqs. (2.7)

and (2.12) below). Moreover, the functions f(u)andg(u) are given by (diagonal case)

f(u)=(u − iα)(u + iβ),g(u)=f(− u)=(u + iα)(u − iβ), (1.6)

where α and β are boundary parameters. This result was used in [7]toformulateaQ-system for the model.

Our main result is a further generalization of the Wronskian formula to the case of nondiagonal bound-

ary ﬁelds:

g(u)P

(u)Q

−

(u) − f (u)P

−

(u)Q

(u)=μ(u)u

2N+1

, (1.7)

where Q(u)andP (u) are the respective polynomial solutions of TQ equation (2.7) and dual TQ equa-

tion (2.12), f(u)andg(u)arenowgivenby(2.9), and, most importantly, μ(u) is a polynomial that satisﬁes

the remarkably simple relation

(u) − μ

−

(u)=γu(Q(u) − P (u)). (1.8)

In other words, μ(u)isthediscrete integral of γu(Q(u) −P (u)). In the diagonal case, γ = 0; it then follows

from (1.8)thatμ(u)=const,and(1.7) hence reduces to (1.5). The appearance of the nontrivial factor μ(u)

in Wronskian-type formulas (1.7)and(1.8) is due to the presence of an inhomogeneous term in the TQ

equation for the model [9]–[11]. We expect that this Wronskian-type formula will be useful for formulating

a Q-system for this model, but this remains a challenge.

In Sec. 2, we brieﬂy review the considered model and its TQ equations and then obtain a dual TQ

equation. In Sec. 3, we derive the Wronskian-type formulas (1.7)and(1.8).

2. The model and its TQ equations

We consider the isotropic (XXX) open Heisenberg quantum spin-1/2 chain of length N with boundary

magnetic ﬁelds, whose Hamiltonian is given by

H =

N−1



k=1

σ

·σ

k+1

−

, (2.1)

where σ =(σ

,σ

) are the usual Pauli matrices and α, β,andξ are arbitrary real parameters. This

model is not U (1)-invariant in the nondiagonal case ξ =0.

To construct the corresponding transfer matrix, we use the R-matrix (solution of the Yang–Baxter

equation) given by the 4×4matrix

R(u)=



u −



I + iP (2.2)

After completing this work, we learned of a similar result for the closed XXX spin chain with a nondiagonal twist (see

Theorem 4.10 in [8].

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(where P is the permutation matrix and I is the identity matrix) and the K-matrices (solutions of boundary

Yang–Baxter equations) given by the 2×2 matrices [12], [13]

(u)=



i(α − 1/2) + u 0

0 i(α +1/2) − u



(u)=



i(β − 1/2) −u −ξ(u + i/2)

−ξ(u + i/2) i(β +1/2) + u



(2.3)

which depend on the boundary parameters α, β,andξ.

The transfer matrix T(u)=T(u; α, β, ξ)isgivenby[14]

T(u)=tr

(u)M

(u)K

(u)



(u), (2.4)

where M and



M are monodromy matrices given by

(u)=R

(u)R

(u) ···R

(u),



(u)=R

(u) ···R

(u)R

(u). (2.5)

The transfer matrix is constructed to satisfy the fundamental commutativity condition

[T(u), T(v)] = 0 (2.6)

and the relation T(−u)=T(u). Hamiltonian (2.1) is proportional to (dT(u)/du)

u=i/2

up to an additive

constant.

The eigenvalues T (u) of the transfer matrix T(u)arepolynomialsinu (as a consequence of (2.6)) and

satisfy the TQ equation [9]–[11]

− uT (u)Q(u)=g

−

(u)(u

)

2N+1

−−

(u)+f

(u)(u

−

)

2N+1

(u) − γu(u

−

)

2N+1

, (2.7)

where Q(u)isanevenpolynomialofdegree2N

Q(u)=



k=1

(u − u

)(u + u

), (2.8)

the functions f (u)andg(u)aregivenby

f(u)=(u − iα)(u



1+ξ

+ iβ),g(u)=f(−u)=(u + iα)(u



1+ξ

− iβ), (2.9)

and γ is deﬁned by

γ = −2(1 −



1+ξ

). (2.10)

We note the presence of an inhomogeneous term (proportional to γ)inTQ equation (2.7). In the diagonal

case ξ =0,from(2.10), we see that γ = 0 and the inhomogeneous term hence disappears. In this case, the

functions f (u)andg(u)in(2.9) reduce to (1.6).

The transfer matrix transforms under charge conjugation by reﬂection (negation) of all the boundary

parameters:

CT(u; α, β, ξ)C = T(u; −α, −β,−ξ), C =(σ

)

⊗N

. (2.11)

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We thus obtain a dual TQequation from (2.7)asin[7] by changing Q(u) → P (u) and reﬂecting the boundary

parameters α →−α, β →−β,andξ →−ξ, which implies that f(u)andg(u) become interchanged,

− uT (u)P (u)=f

−

(u)(u

)

2N+1

−−

(u)+g

(u)(u

−

)

2N+1

(u) − γu(u

−

)

2N+1

, (2.12)

where P (u)isalsoanevenpolynomialofdegree2N,

P (u)=



k=1

(u − ˜u

)(u +˜u

), (2.13)

whose zeros can be regarded as dual Bethe roots. We emphasize that the same eigenvalue T (u)appearsin

both (2.7)and(2.12).

3. The Wronskian-type formula

We now consider the relation between Q(u)(asolutionofTQ equation (2.7) for some transfer-matrix

eigenvalue T (u)) and the corresponding P (u) (a solution of dual TQ equation (2.12) for the same transfer-

matrix eigenvalue T (u)). For this, we use the ansatz

g(u)P

(u)Q

−

(u) − f (u)P

−

(u)Q

(u)=μ(u)u

2N+1

, (3.1)

where f(u)andg(u)aregivenby(2.9) and the function μ(u) is yet to be determined. This ansatz is

motivated by result (1.5) in the diagonal case.

We next deﬁne R(u), following the results in [1], as

R =

2N+1

−



− f

−



, (3.2)

where the second equality follows from ansatz (3.1). Dividing both sides of TQequation (2.7)byQQ

−−

we obtain

−

−−

= f

−

+ g

−

− γuQR

−

. (3.3)

Substituting R in this equation using the second equality in (3.2) and then multiplying both sides of the

resulting equation by μ

−

−−

,weobtain

−uT μ

−

= f

−

−−

(μ

− μ

−

+ γuP) − f

−

−−

(μ

+ γuP)+

+ g

−

−−

(μ

−

− γuP)+f

−

−−

Q(γu). (3.4)

Similarly, we deﬁne S(u)as

S =

2N+1

−



−

− f



(3.5)

and divide both sides of dual TQ equation (2.12)byPP

−−

.Wethusobtain

−

−−

= f

−

+ g

−

− γuPS

−

. (3.6)

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Substituting the expression for S in this equation using the second equality in (3.5) and then multiplying

both sides by μ

−

−−

,weobtain

−uT μ

−

= f

−

−−

(γu) − f

−

−−

(μ

−

+ γuQ)+

+ g

−

−−

(μ

− γuQ) − f

−

−−

(μ

− μ

−

− γuQ). (3.7)

Equating the right-hand sides of (3.4)and(3.7), we obtain the constraint

[μ

− μ

−

+ γu(P − Q)]

(gP

−

− fP

−

)

(gP

−

− fP

−

)

−

=0, (3.8)

which is obviously satisﬁed if we set

− μ

−

= γu(Q − P ) (3.9)

as required by (1.8). For given polynomials Q(u)andP (u), Eq. (3.9) can be solved for a polynomial

function μ(u) up to an arbitrary additive constant.

Result (3.9) can in fact be obtained more directly.

Let

μ(u)=

2N+1



g(u)P

(u)Q

−

(u) − f(u)P

−

(u)Q

(u)



, (3.10)

which is equivalent to (1.7). We multiply TQ equation (2.7)byP (u)/(u

−

)

2N+1

, multiply dual TQ

equation (2.12)byQ(u)/(u

−

)

2N+1

, and subtract the second equation from the ﬁrst. The obtained result

writtenintermsofμ given by (3.10) exactly coincides with (3.9).

Acknowledgments

The author thanks the organizers of the CQIS-2019 workshop in St. Petersburg for the kind invitation.

Conﬂicts of interest. The author declares no conﬂicts of interest.

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