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IFUM 891-FT
Lecture notes on classical integrability and on the
algebraic Bethe ansatz for the XXX12
Heisenberg
spin chain
Christoph Sieg1
Universita degli Studi di MilanoVia Celoria 16, 20133 Milano, Italy
Contents
1.1 Introductory comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Small review of Hamiltonian mechanics . . . . . . . . . . . . . . . . . . . 3
1.2.1 Hamilton canonical equations and canonical transformations . . . 31.2.2 Symplectic geometry . . . . . . . . . . . . . . . . . . . . . . . . . 61.2.3 Hamilton-Jacobi theory . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 The isospectral deformation method . . . . . . . . . . . . . . . . . . . . . 121.3.1 Lax pair . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.3.2 Yang-Baxter equation . . . . . . . . . . . . . . . . . . . . . . . . 13
1.4 Algebraic Bethe ansatz for the XXX 12
Heisenberg spin chain . . . . . . . 161.4.1 Definition and some first considerations . . . . . . . . . . . . . . . 161.4.2 Lax operator, transfer matrix and Yang-Baxter equation . . . . . 181.4.3 Conserved charges . . . . . . . . . . . . . . . . . . . . . . . . . . 211.4.4 Algebraic Bethe ansatz equations, eigenstates and eigenvalues . . 23
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30A Matrix representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30B The coefficients Ok(λ) in (??) . . . . . . . . . . . . . . . . . . . . . . . 32Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
1
1.1 Introductory comments
These lecture notes provide an elemetary introduction to integrable systems. Basicknowledge in classical analytical mechanics and quantum mechanics should suffice tofollow all arguments. We motivate and define integrability for classical dynamical sys-tems. We introduce the concept of the Lax pair and of the Yang-Baxter equation for theR-matrix. At the example of the XXX 1
2Heisenberg spin chain, we will then introduce
the algebraic Bethe ansatz.These notes start with a small review of Hamiltonian mechanics with special at-
tention put on those aspects which are usually not discussed in introductory courses.The formulation of Hamiltonian mechanics in the language of symplectic geometry isemphasized. The N Poincare integral invariants, which imply that volumina of regionsin certain subspaces of the phase space remain constant, are derived. The well knownLiouville theorem, in which the invariant volume extends into the full phase space, isequivalent to the statement of invariance of one of these integrals. For the preparationof this part, we have used [1, 2].
As an important approach to the study of the properties of classical dynamicalsystems, the Hamilton-Jacobi theory is reviewed. This part is based on [1]. In thiscontext (completely) separable problems are defined, and the possible motions in phasespace are presented. The angle-action variables for compact motions are introduced.The concept of integrability by quadratures then appears natural in the completelyseparable case. Finally, the definition of Liouville integrability is presented. We refrainfrom presenting the various definitions and theorems which have been formulated in theclassical case. We refer the interested reader to the literature [2,3], which we have usedfor the preparation of this part.
To describe and solve integrable systems, the isospectral deformation method ofLax and the concept of a spectral parameter is introduced. It is shown how the in-tegrals of motion are found in this formalism. To prove that the integrals of motion(Poisson-)commute, the concept of theR-matrix which obeys the (classical) Yang-Baxterequation is introduced, and it is shown how the classical and quantum R-matrices areconnected. An interpretation in terms of factorized scattering is then given. We do notdiscuss the projection method of Perelmov, which the interested reader can find in [3–5].These references were also sources for the preparation of this part.
As an example of a quantum integrable system, the XXX 12
Heisenberg spin chain is
discussed. This part is very similar to the first four sections in the review [6], supportedby [7] which presents some calculations in more detail. After the Bethe equations arecompleted and the spectrum can in principle be calculated, this review stops withoutapplying the formalism to extract explicit results for the XXX 1
2Heisenberg spin chain,
2
which can be found in the review [6]. Instead, the derivation has been extended bysome explicit calculations and interpretations which are not part of the used literature.They supplement the considerations. This concerns the material presented in the twoappendices and some of its interpretations in the main text. To be more precise, the fullset of commutation relations (1.4.34), and the interpretation of the found symmetry asa symmetry that exchanges the role of spin up and spin down is not part of the usedliterature. Also, the proof for the vanishing of the coefficients Ok(λ) in (1.4.57) andtheir derivation cannot be found in [6].
I would like to end this summary with an invitation to make any kind of criticalremarks that help to improve the presentation of the selected material.
1.2 Small review of Hamiltonian mechanics
1.2.1 Hamilton canonical equations and canonical transforma-tions
A dynamical system of N particles with generalized coordinates q = (q1, . . . , qN) can bedescribed by its action S, which is given by
S =
∫L(q, q, t) dt , (1.2.1)
where L is the Lagrange function that depends on the generalized coordinates q, ve-locities q and in general also explicitly on the time t. The equations of motion followfrom the variation of the action w.r.t. the coordinates according to q → q + δq, whereδq = 0 vanishes at the integration boundaries. This yields the Euler-Lagrange equationsof motion
d
dt
∂L
∂qi− ∂L
∂qi= 0 , i = 1, . . . N . (1.2.2)
Instead of using the set q, q as variables, one can introduce a set of independent variablesgiven by the coordinates q and the momenta pi = ∂L
∂qi . The Lagrange function L is thentransformed via a Legendre transformation into the Hamilton function H as
H(q, p, t) = pq − L(q, q, t) . (1.2.3)
Inserting this result into the action, and varying q → q + δq, p→ p+ δp which are nowindependent, one obtains the set of 2N canonical Hamilton equations
q =∂H
∂p, p = −∂H
∂q. (1.2.4)
3
Their solution is a trajectory, parameterized by t, in the 2N -dimensional phase space inwhich a point is labeled by (q1, . . . qN , p1, . . . , pN).
The advantage of introducing the independent variables q and p is that one can usecanonical transformations, i.e. transformations that keep the form of (1.2.4) invariant,to arrive at a set of more convenient coordinates and momenta to solve a given problem.The canonical transformations are of the form
q → Q(q, p, t) , p→ P (q, p, t) . (1.2.5)
In particular, they mix coordinates and momenta and are therefore more general thanthe transformations q → Q(q, t) of the Lagrange formulation. The transformations(1.2.5) also transform the Hamiltonian according to
H(q, p, t)→ K(Q,P, t) . (1.2.6)
To find the explicit expressions for the transformed quantities, it is most convenientto introduce a generating function F for the transformations. This ensures that thetransformations are canonical, i.e. that they leave invariant the form of (1.2.5). This isthe case if the two Lagrangians expressed in the old and new variables, only differ by atotal derivative. The corresponding actions (1.2.1) then only differ by a boundary term.In terms of the Hamilton functions, we find
pq −H(q, p, t) = PQ−K(Q,P, t) +dF
dt. (1.2.7)
Since we have 2N old variables, 2N new variables and 2N equations that relate the oldand new variables, F should depend on the 2N independent old and new variables. Thisleaves four possibilities to combine the old and new coordinates and momenta.
It is, however, also possible and for an explicit demonstration of the invariance of(1.2.4) particularly useful, to choose F = F (q, p, t). Inserting
Q =∂Q
∂qq +
∂Q
∂pp+
∂Q
∂t(1.2.8)
into (1.2.7), one finds the relations
∂F
∂q= p− P ∂Q
∂q,
∂F
∂p= −P ∂Q
∂p, K(Q,P, t) = H(q, p, t)− P ∂F
∂t+ P
∂Q
∂t.
(1.2.9)
4
It is now easy to demonstrate the invariance of the equations of motion (1.2.4) underthe above transformation. We first rewrite (1.2.4) as(
qp
)= J
(∂H∂q∂H∂p
), J =
(0 1
−1 0
). (1.2.10)
We then introduce Q and P as variables which depend on q, p and t
∂H
∂q=∂K
∂Q
∂Q
∂q+∂K
∂P
∂P
∂q− ∂2F
∂q∂t− ∂
∂q
(P∂Q
∂t
),
∂H
∂p=∂K
∂Q
∂Q
∂p+∂K
∂P
∂P
∂p− ∂2F
∂p∂t− ∂
∂p
(P∂Q
∂t
).
(1.2.11)
From (1.2.9) we find with ∂p∂q
= 0
∂2F
∂q∂t= − ∂
∂t
(P∂Q
∂q
),
∂2F
∂p∂t= − ∂
∂t
(P∂Q
∂p
). (1.2.12)
The last two terms on the r.h.s. in each line of (1.2.11) thus combine to
∂2F
∂q∂t− ∂
∂q
(P∂Q
∂t
)=∂P
∂t
∂Q
∂q− ∂P
∂q
∂Q
∂t,
∂2F
∂p∂t− ∂
∂p
(P∂Q
∂t
)=∂P
∂t
∂Q
∂p− ∂P
∂p
∂Q
∂t.
(1.2.13)
The vector of partial derivatives on the r.h.s. of (1.2.10) can therfore be rewritten as(∂H∂q∂H∂p
)= At
[(∂K∂Q∂K∂P
)+
(∂P∂t
−∂Q∂t
)], A =
(∂Q∂q
∂Q∂p
∂P∂q
∂P∂p
), (1.2.14)
where At is the transpose of the matrix A. Furthermore, total time derivatives of Q andP can also be rewritten with the help of the matrix A. We find(
Q
P
)= A
(qp
)+
(∂Q∂t∂P∂t
)(1.2.15)
Inserting (1.2.14) and (1.2.15) into (1.2.10), we obtain(Q
P
)= AtJA
[(∂K∂Q∂K∂P
)+
(∂P∂t
−∂Q∂t
)]+
(∂Q∂t∂P∂t
)= J
(∂K∂Q∂K∂P
), (1.2.16)
since J = AJAt.The antisymmetric matrix J , which is explicitly given (1.2.10) in the coordinates
(q, p) or (Q,P ) locally gives an explicit expression for a symplectic structure on thephase space. It is left invariant by the canonical transformations. It is hence useful torecall some facts about symplectic geometry in the following.
5
1.2.2 Symplectic geometry
A manifold M is called symplectic, if a closed (dω = 0) non-degenerate (detωab 6= 0)2-form ω exists globally. Locally, one finds on M a coordinate system (qi, pj) in whichω = − dθ = dpi ∧ dqj (Darboux’s theorem).
E.g. in the phase space (RN ,T∗RN) ∼ R2N θ, ω are realized with coordinatesxa = (qi, pj) as
θ =1
2Jabx
a dxb , ω = − dθ = −1
2Jab dxa ∧ dxb . (1.2.17)
Thereby, the phase space has been interpreted as the cotangent bundle E over RN .This appears natural if one remembers that pi = ∂L
∂qi transform as the components of a1-form.
Since ω and hence Jab are non-degenerate, we can define the inverse of J as
Jab = (Jab)−1 = −Jab . (1.2.18)
On E a 2-tensor P called Poisson structure is then globally defined. For example onehas
P ( · , λ) = Vλ , λ ∈ T∗E, Vλ ∈ TE . (1.2.19)
We also find
P ( · , ω(V, · )) = Jab(−Jcb)V c ∂
∂xa= V a ∂
∂xa= V ,
ω(P ( · , λ), · ) = −JabJacλc dxb = λb dxb = λ ,(1.2.20)
i.e. −P is the inverse of ω.If λ, µ are exact 1-forms, i.e. λ = df , µ = dg with f , g functions on E, the Poisson
bracket is defined as
f , g = P (df, dg) = Jab∂f
∂xa∂g
∂xb=∂f
∂qi∂g
∂pi− ∂f
∂pi
∂g
∂qj. (1.2.21)
Recall that the Poisson bracket fulfills
f , g = −g , f antisymmetry ,
f , g , h+ g , h , f+ h , f , g = 0 Jacobi identity ,
f , gh = f , gh+ f , h g Leibnitz rule .
(1.2.22)
Some special brackets areqi , qj
=pi , pj
= 0 ,
qi , pj
= δij . (1.2.23)
6
Furthermore, the total time derivative of a function f = f(p, q, t) can be rewritten as
f = f , H+∂f
∂t, (1.2.24)
where H is the Hamiltonian of the system. The corresponding canonical equations ofmotion (1.2.4) become
qi =qi , H
= X i
H ,
pj = pj , H = XjH ,
(1.2.25)
where
XH = P ( · , dH) =∂H
∂pi
∂
∂qi− ∂H
∂qi∂
∂pi(1.2.26)
is called a Hamiltonian vector field. It is the vector field of tangent vectors to the curvexa(t) = (qi(t), pj(t)), i.e. xa = Xa
H . Such a curve is called a flow. It defines an abeliangroup for the curve parameter (here time t).
Inserting XH into ω yields an exact 1-form
ω(XH , · ) = ω(P ( · , dH), · ) = dH . (1.2.27)
The Hamilton canonical equations (1.2.4) can therefore also be expressed as
ω(XH , · ) = dH , XH = xa∂
∂xa. (1.2.28)
We want to derive constants of motion from the above results. To measure the changeof an element of TE or T∗E or of tensor products thereof into a direction X ∈ TE, andin particular along a flow, we recall that the Lie derivatives LX along X for a functionf , a vector Y , an r-form ωr are given by
LX f = X(f) = Xa ∂f
∂xa,
LX Y = −LY X = [X , Y ] ,
LX ωr = (diX + iX d)ωr ,
(1.2.29)
where iX is the inner product operator which with Xi ∈ TE acts as
(iX ωr)(X1, . . . , Xr−1) = ωr(X,X1, . . . , Xr−1) . (1.2.30)
For the closed symplectic 2-form ω one finds that the Lie derivative into the directionof the Hamiltonian vector field XH is given by
LXHω = diXH
ω = dω(XH , · ) = d2H = 0 . (1.2.31)
7
Therefore, ω is constant along XH , i.e. it is constant under time evolution driven by theHamiltonian H. This is not surprising, if we remind ourselves that ω is invariant undercanonical transformations, and that time evolution itself can be seen as a canonicaltransformation with generating function H. We have hence found a first constant ofmotion. Additional ones immediately follow from
LXH(ω ∧ · · · ∧ ω︸ ︷︷ ︸
k
) = k ω ∧ · · · ∧ ω︸ ︷︷ ︸k−1
LXHω = 0 . (1.2.32)
We have therefore found N differential forms which are invariants under Hamiltonianflows. They are the N non-vanishing powers of ω. Integration of these differential formsyields the integral invariants of Poincare defined by
I1 =
∫Σ2
ω = −∫∂Σ2
θ ,
I2 =
∫Σ4
ω ∧ ω ,
...
IN =
∫Σ2N
ωN ,
(1.2.33)
where Σ2k denote 2k-dimensional subvolumina in phase space. In particular, the invari-ance of IN , i.e. the constancy of the volume of an arbitrary 2N dimensional region inphase space, is well known under the name Liouville theorem.
Let us reinterpret canonical transformations. Any canonical transformation shouldleave invariant the symplectic structure ω = − dθ and hence all Ik. Given ω = − dθ inthe variables (qi, pj) and ω = − dΘ in the variables (Qi, Pj), a canonical transforma-tion relates θ − Θ = dF , where F is the generating function of the transformation asintroduced before.
1.2.3 Hamilton-Jacobi theory
The Hamilton Jacobi theory is useful to study the generic properties of dynamical sys-tems. Here, we will use it to motivate one possible definition of classical integrability.
Recall that a cyclic coordinate is a coordinate qi that does not appear in H, i.e.pi = ∂H
∂qi= 0 such that the conjugate momentum pi is constant.
Canonical transformations allow us to transform to new coordinates in which morecoordinates are cyclic. In the Hamilton-Jacobi theory, one searches for a canonical
8
transformation which transforms all variables to cyclic variables. This is clearly thecase if the Hamiltonian in the new coordinates vanishes identically.
We therefore search for the canonical transformation with generating function S =S(q, P, t) that fulfills
K(Q,P, t) = H(q, p, t) +∂
∂tS(q, P, t) = 0 , p =
∂S
∂q, Q =
∂S
∂P. (1.2.34)
This yields a partial differential equation for S which reads
H(q, ∂S∂q, t) +
∂S
∂t= 0 . (1.2.35)
The new momenta Pj are then constant.In general one cannot find a solution of (1.2.35). Instead, one can investigate in
which special cases a solution can be found in closed form. First of all, if the originalHamiltonian does not explicitly depend on time, i.e. H = H(q, p), time can be separatedaccording to the ansatz
S(q, P, t) = W (q, P )− Et . (1.2.36)
The so called action W (q, P ) then has to fulfill the time independent Hamilton-Jacobiequation
H(q, ∂W∂q, t) = E . (1.2.37)
Remember that the momenta Pj are constant. We thus have
dW =∂W
∂qidqi = pi dq
i , (1.2.38)
which has the dimension of angular momentum, like the action S in (1.2.1).The Hamilton-Jacobi equation is called completely separable if it is solved by the
ansatz
S(q, P, t) = W (q, P )− Et , W (q, P ) =N∑i=1
W(i)(qi, P ) , (1.2.39)
where W(i) only depends on a single coordinate qi. The original momenta pi thereforefulfill
pi =∂W(i)
∂qi= pi(q
i, P ) , (1.2.40)
9
qi
pi
(a) noncompact motion
qi
pi
(b) libration
0qi
pi
L 2L
(c) rotation
Figure 1.1:
i.e. they only depend on the corresponding coordinate qi and on the constant momentaPj which fix the initial conditions.
The motion in phase space therefore decomposes into independent motions in Ntwo-dimensional planes spanned by (qi, pi). In each plane, there exist three qualitativelydifferent motions with pi being always bounded.
1. Non-compact motion, in which qi is unbounded from above and/or below.
2. Compact motion, in which qi is also bounded:
(a) Libration: qi oscillates, pi changes its sign (existence of turning points)
(b) Rotation: qi is unbounded, but is periodically identified qi ∼ qi + L withsome period L and pi remains either positive or negative.
In the directions of compact motion, the phase space has the topology of an S1 whichcorresponds to the circular motion in case of the libration or to the periodical identifi-cation of the spatial coordinate in case of the rotation. In these cases, one very oftenintroduces the so called angle-action variables (φi, Ji) that describe the angle φi andthe angular momentum Ji around the circle, i.e. (Qi, Pj) are replaced by (φi, Ji). Jithen specifies the individual torus, i.e. the compact path Ci ∼ S1, while φi specifies theposition on Ci and is normalized such that its periodicity is 2π.
The periodicity condition is expressed with the first Poincare invariant (1.2.33) as
2π =
∮Ci
dφi =
∮Ci
d∂W
∂Ji=
∮Ci
∂2W
∂qj∂Jidqj =
∂
∂Ji
∮Ci
pj dqj =∂
∂JiIi . (1.2.41)
10
Therefore, the action variables are found as
Ji =1
2π
∫Σi
ω . (1.2.42)
The angle variables are then determined from the constancy of Ji, that implies K(φ, J) =K(J). We then find
φi =∂K
∂Ji= νi , (1.2.43)
such that φi = νit+ φ0.A system for which the Hamilton-Jacobi equation is completely separable is called
completely integrable. This is obvious if one defines integrability as follows.A system of differential equations is called integrable by quadratures if its solution
can be obtained by finitely many algebraic operations (including inverting functions)and quadratures (integrals of given functions).
The definition and theorem of Liouville integrability is as follows.Let R2N = (qi, pj) be the phase space of a dynamical system with Hamiltonian
H(q, p, t) and with , being the standard Poisson bracket. If there exist N integralsof motion F1, . . . FN in involution, i.e. the Fi fulfill
∂Fi∂t
+ Fi , H = 0 , Fi , Fj = 0 , (1.2.44)
and F1, . . . FN are independent on M = (q, p, t) = R2N × R : Fi = ai , i = 1, . . . , N,then qi = ∂H
∂pi, pi = −∂H
∂qi is integrable by quadratures.There exist a lot of further definitions and theorems about classical integrability. We
refer the interested reader to [3, 4].In brief one could say that to solve an integrable system one should find sufficiently
many independent integrals of motion which are expressed in terms of the coordinates ofthe problem. Then, solving for the coordinates in dependence of the constants of motion,one has solved the dynamical problem. One very powerful method which immediatelyyields integrals of motion, is the so called isospectral deformation method of Lax, whichwe summarize in the following.
11
1.3 The isospectral deformation method
1.3.1 Lax pair
Suppose that for a dynamical system with N degrees of freedom we have found a socalled Lax pair (L,M) of n× n matrices such that the Lax equation
L = [M , L] (1.3.1)
is equivalent to the equations of motion for the system.The time dependence of L then has to be described by
L(t) = U(t)L0U−1(t) , M = U(t)U−1(t) , (1.3.2)
where the result for M follows when inserting L(t) into the Lax equation (1.3.1). ThatL(t) has to be of the above form becomes obvious if we define
Ik =1
ktrLk , k = 1, . . . , n (1.3.3)
which allows us to extract all eigenvalues of L. From (1.3.1) it follows immediately that
Ik = trLk−1 [L , M ] = 0 , (1.3.4)
i.e. the Ik are integrals of motion, and the eigenvalues of L are time independent. Lundergoes an isospectral deformation which is in accord with (1.3.2).
If we finds sufficiently many integrals of motion Ik, i.e. if the matrices (L,M) havesufficiently high rank n ≥ N , and if we can show that the integrals Ik are in involution,we have solved the problem.
In particular, for systems with a large number of degrees of freedom, e.g. in a contin-uum limit, it can be impossible or impractical to find a Lax pair of a large rank. In [3] itis also argued that the time dependence of L is restricted to that of a rational functionwhich depends on t explicitly and/or on exponents of the form eat with some constantsa. Introducing a spectral parameter into the Lax equation ensures the existence ofsufficiently many integrals of motion, and it generalizes the possible time dependence.Involutivity of the integrals of motion follows from the so called Yang-Baxter equation,which we will discuss afterwards.
The idea of the extension of the Lax equation (1.3.1) by a spectral parameter is verysimple. We introduce an additional parameter λ, such that (1.3.1) becomes
L(λ) = [M(λ) , L(λ)] , (1.3.5)
12
where time dependence is implicitly understood on both sides of the equations, and thedot still indicates the derivative w.r.t time. Since the above equation is fulfilled for anyvalue of the λ, the integrals of motion Ik,m automatically arise from an expansion ofIk(λ) into a power series in λ.
Ik(λ) =∑m
Ik,mλm . (1.3.6)
In addition, since the complete dynamics is independent of λ, the spectral curve
Γ = (λ, µ) ∈ C2, det(µ1− L(λ)) = 0 (1.3.7)
is time independent and only depends on the integrals of motion.
1.3.2 Yang-Baxter equation
To prove the involutivity of the integrals of motion, it is sufficient to construct a so-calledr-matrix which fulfills the Yang-Baxter equation. Let (L(λ),M(λ)) be given in somerepresentation R(g) of some Lie algebra g. If we can find an r-matrix r ∈ R(g)⊗R(g)which relates
L(λ) ⊗, L(µ) = [r(λ, µ) , L(λ)⊗ 1]− [P r(λ, µ) P , 1⊗ L(µ)] , (1.3.8)
where λ and µ are spectral parameters, then the traces Ik = 1k
trLk are in involution.Before showing this, we should better explain the notation used in the above equa-
tions. We write down components in R(g) ⊗ R(g) to make it more clear. Let A,B ∈R(g), we then have
(A⊗B)ij,kl = AikBjl . (1.3.9)
It immediately follows
(A⊗B)(C ⊗D) = AC ⊗BD ,
A⊗B + A⊗ C = A⊗ (B + C) ,
tr(A⊗B) = trA trB .
(1.3.10)
The tensor product Poisson bracket on the l.h.s. of (1.3.8) expands in components as
L(λ) ⊗, L(µ)ij,kl = L(λ)ik , L(µ)jl . (1.3.11)
The r.h.s. of (1.3.8) contains the permutation operator P in the representation R(g)⊗R(g). It acts as
P(A⊗B) P = B ⊗ A , P2 = 1⊗ 1 . (1.3.12)
13
The second commutator on the r.h.s. of (1.3.8) therefore becomes in components
[P r(λ− µ) P , 1⊗ L(µ)]ij,kl = [r(λ− µ) , L(µ)⊗ 1]ji,lk . (1.3.13)
To show the involutivity of the integrals of motion, we use (1.3.8) to evaluate a tensorproduct of powers of L. We write
L(λ)k ⊗, L(µ)l
=
k−1∑r=0
l−1∑s=0
L(λ)r ⊗ L(µ)s L(λ) ⊗, L(µ)L(λ)k−r−1 ⊗ L(µ)l−s−1
=k−1∑r=0
l−1∑s=0
L(λ)r ⊗ L(µ)s([r(λ− µ) , L(λ)⊗ 1]− [P r(λ− µ) P , 1⊗ L(µ)])
L(λ)k−r−1 ⊗ L(µ)l−s−1 .(1.3.14)
Taking the trace, and performing cyclic permutations under the trace, we findtrL(λ)k ⊗, trL(µ)l
= kl tr
(L(λ)k−1 ⊗ L(µ)l−1([r(λ− µ) , L(λ)⊗ 1]
− [P r(λ− µ) P , 1⊗ L(µ)]))
= 0 .(1.3.15)
This shows that Ik(λ) , Il(µ) = 0.The r-matrix itself has to fulfill a relation, which follows from the tensor version of
the Jacobi identity for (1.3.8). To find it, we first recast (1.3.11) as follows
L(λ) ⊗, L(µ) = L(λ)⊗ 1 , 1⊗ L(µ) . (1.3.16)
The tensor version of the Jacobi identity then reads
L(λ1)⊗ 1⊗ 1 , 1⊗ L(λ2)⊗ 1 , 1⊗ 1⊗ L(λ3)+ cycl. perm = 0 . (1.3.17)
With the definition of of matrices rij ∈ R(g) ⊗R(g) ⊗R(g), which in componentsbecome
rij aiajak,a′ia
′ja
′k
= r(λij)aiaj ,a′ia
′jδaka
′k, (1.3.18)
where λij = λi − λj, the Jacobi identity (1.3.17) holds if rij fulfill[r12 , r13
]+[r12 , r23
]+[r13 , r23
]= 0 , (1.3.19)
which is the representation of the classical Yang-Baxter equation in the threefold tensorproduct space of R(g). This equation is found as the O(~2) contribution to the quantumYang-Baxter equation in R(g)⊗R(g)⊗R(g)
R12R13R23 = R23R13R12 , (1.3.20)
14
a1 a′1
a2 a′2
a3 a′3
b1
b2 b3
a1 a′1
a2 a′2
a3 a′3
b1b2
b3
Figure 1.2:
if we identifyRij(λ) = κ(λ)(1⊗ 1⊗ 1+ ~ rij(λ)) . (1.3.21)
The quantum R-matrix in R(g)⊗R(g) is related to Rij as
Rij(λ)aiajak,a′ia
′ja
′k
= Raiaj ,a′ia
′jδaka
′k, (1.3.22)
i.e. the classical and quantum R-matrices are related as
R(λ) = κ(λ)(1⊗ 1+ ~ r(λ)) . (1.3.23)
Inserting (1.3.22) into (1.3.20) we find that the quantum Yang-Baxter equations incomponents reads
R(λ12)a1a2,bcR(λ13)ba3,a′1dR(λ23)cd,a′
2a′3
= R(λ23)a2a3,cdR(λ13)a1d,ba′3R(λ12)bc,a′
1a′2. (1.3.24)
The quantum Yang-Baxter equation has an interpretation in terms of factorizedscattering, as we will motivate in the following. Consider a relativistic 1+1-dimensionalquantum field theory. A scattering process of two on-shell particles of masses m1 and m2
with momenta pi = mi(cosh θi, sinh θi) is described by the S-matrix Sa′1a
′2
a1a2 (θ12), which canonly depend on a single scalar invariant of the scattering process, given by the differenceof the two rapidities θ12 = θ1 − θ2. In particular, this means that the kinematics isconstrained to only two possible scattering processes, given by the the identity, wherethe particle and their momenta leaving the scattering are unaffected, and the reflection,where the outgoing particles exchange their momenta w.r.t. to the incoming ones. Thisrestriction leads to an important aspect for the scattering of multiple particles. Itfactorizes into pairwise scattering processes, in which the order of the pairwise scattering
15
= | 〉
Figure 1.3: Cyclic spin 12
spin chain state and its representation as quantum mechanicalstate vector. The representation is only unique up to cyclic permutations.
is of no importance. For example, the independence of the order in a three particlescattering process is described by the equation
Sb1b2a1a2(θ12)S
b3a′3
b2a3(θ13)S
a′1a
′2
b1b3(θ23) = Sb2b3a2a3
(θ23)Sa′1b1a1b2
(θ13)Sa′1a
′3
b1b3(θ12) , (1.3.25)
where θij = θi − θj. This is precisely the Yang-Baxter equation (1.3.24), if we identifythe R-matrix with the S-matrix.
1.4 Algebraic Bethe ansatz for the XXX12
Heisen-
berg spin chain
1.4.1 Definition and some first considerations
If we directly transfer integrabilty from classical mechanics to quantum mechanics, itappears reasonable that we should denote a quantum mechanical system as integrable,if we are able to find sufficiently many integrals of motion Ik which commute with eachother, i.e. [Ik , Il] = 0.
As an example, we discuss the closed so called XXX 12
Heisenberg spin chain. Thethree X denote that the corresponding coupling constants of the spin-spin interactionsin the three spatial directions are identical, and the index 1
2indicates that we have a spin
12
degree of freedom at each site of the chain. At each site, we keep s3 diagonal, wheresi = 1
2σi, i = 1, 2, 3 with the three Pauli matrices σi (A.4) are the spin 1
2representation
matrices of the underlying su(2) Lie algebra. The eigenvalues of s3 are either +12
or −12
at each site of the chain, which we denote by up- and downwards pointing arrows ateach site. A generic state of the chain is depicted in figure 1.3.
The Hilbert space of a spin chain of length N is given as the tensor product space
HN =N⊗n=1
hn , (1.4.1)
16
where hn is the Hilbert space of the individual spin at lattice site n.The total spin operator Si for a chain of length N is defined by
Si =1
2
N∑n=1
σin , (1.4.2)
where σin denotes the ith pauli matrix acting in the space hn of the chain. S3 is obviouslydiagonal for a single state as the one in figure 1.3, and also for any linear combinationof states with the equal number of up and down spins. We do not have to considera dynamical changing of the site number N here. This becomes obvious if we lookat the XXX 1
2Hamiltonian, which only contains exchange interactions between spins
at different sites, but no annihilation or creation of spin chain sites themselves. TheHamiltonian is given by
H =1
2
N∑n=1
(Pnn+1−1nn+1) , (1.4.3)
where for a cyclic spin chain we have to identify n ∼ n+ kN , k ∈ Z. The permutationoperator Pnn+1 exchanges the two spin states at sites n and n + 1. In the spin 1
2
representation, it assumes the form
Pnn+1 =1
2(~σn~σn+1 + 1nn+1) , (1.4.4)
where ~σ = (σ1, σ2, σ3) denotes the vector formed by the three Pauli matrices σi. Fur-thermore, 1nn+1 denotes the identity operator acting in hn ⊗ hn+1.
Later on, we will require some properties of the permutation operator. The productof two permutation operators Pma and Pna, which acts in the space hm ⊗ hn ⊗ ha withboth factors having site a in common, obeys the permutation relation
Pna Pma = hm ⊗ hn ⊗ ha = hm ⊗ hn ⊗ ha = Pmn Pna
= hm ⊗ hn ⊗ ha = Pma Pmn .
(1.4.5)
Thereby, we have presented a permutation by an underbracing of the correspondingspaces, where the ordering from up to down corresponds to the ordering from right toleft. It then becomes immediately obvious that two permutations commute as long asthey do not act on exactly one common site. We summarize some obvious identities asfollows
Pna = Pan , Pnn+1 Pn+2n+3 = Pn+2n+3 Pnn+1 , Pna Pna = 1na . (1.4.6)
17
Furthermore, if we take the trace over one of the spaces in the spin 12
representation ofthe permutation operator (1.4.4), we find that it reduces to the identity in the secondspace as
tra Pna = 1n . (1.4.7)
It is futhermore obvious that a pure exchange interaction like in (1.4.3) does notchange the total number nor of up spins neither of down spins. In fact, the Hamiltoniancommutes with all spin operators [
Si , H]
= 0 . (1.4.8)
An eigenstate of S3 as the one in figure 1.3 is not automatically an eigenstate of H.Since H does not alter the length of the chain and commutes with S3, the eigenstatesof H have to be linear combinations of the states with a fixed number of up spins anddown spins, which differ only by the distribution of these spins onto the sites of thechain.
Before describing the algebraic Bethe ansatz which allows us to find all eigenstatesand the spectrum of H, we identify some of its obvious eigenstates. Consider the statesin which all spins are either up spins or down spins. This state is an eigenstate underaction of the permutation operator, and it is thus also an eigenstate of H. Hence, wefind
H | . . . 〉 = 0 , H | . . . 〉 = 0 . (1.4.9)
Furthermore, since the spin chain is invariant under cyclic permutations of its sites, thestates with a single spin directed into the opposite direction w.r.t. the other spins arealso eigenstates of H. We find
H | . . . 〉 = 0 , H | . . . 〉 = 0 . (1.4.10)
While the eigenvalue problem for H can be solved by hand for a small number ofsites N , it increases dramatically in complexity for growing N . Furthermore, the directanalysis of the limit of infinite chain length N → ∞ is hopeless. With the algebraicBethe ansatz, described in the following, the diagonalization problem can be solved forall N , which includes also the possibility to take the limit N →∞.
1.4.2 Lax operator, transfer matrix and Yang-Baxter equation
For the XXX 12
Heisenberg spin chain, a Lax operator at the individual lattice site nwith spectral parameter λ is defined as
Lna(λ) = (λ− i2)1na + iPna , (1.4.11)
18
where a labels an auxiliary space Va = C2, such that Lna acts as Lna : hn⊗Va → hn⊗Va.Later, we will need the matrix representation of Lna in Va which is given by
Lna =
(λ1+ i
2σ3 i
4σ−
i4σ+ λ1− i
2σ3
), (1.4.12)
where σ± = σ1 ± iσ2.The introduction of the auxiliary space has a reason, which can be understood al-
ready from the proof of the involutivity of the integrals of motion Ik = 1k
trLk in theclassical case. Remember that the proof was based on the relation (1.3.8) which includesthe tensor product Poisson bracket of two Lax matrices. That two integrals of motionPoisson-commute then is a direct consequence of the fact that the trace of a tensorproduct is identical to the product of the traces in the individual factors as in (1.3.10).Furthermore, the classical r-matrix in (1.3.8) thereby acts beween the two factors in thetensor product. As in the classical case, taking the trace over the auxiliary space willgive us the integrals of motion, and we will need two auxiliary spaces between whichthe quantum R-matrix acts, to proof that the integrals of motion commute. To findthe integrals of motion thereby requires that we first combine the Lax matrices at theindividual lattice sites to an operator which acts on the full chain. Before doing this, weintroduce a second auxiliary space Vb = C2 and define the R-matrix R : Va⊗Vb → Va⊗Vbas
Rab(λ) = Lab(λ+ i2) , (1.4.13)
i.e. the R-matrix has the same form as the Lax matrix up to a shift in the spectralparameter. The Yang-Baxter equation at an individual lattice site n, which with λ =λ+ i
2, µ = µ+ i
2is given by
Rab(λ− µ)Rna(λ)Rnb(µ) = Rnb(µ)Rna(λ)Rab(λ− µ) (1.4.14)
is then identical to the so called fundamental commutation relation for L
Rab(λ− µ)Lna(λ)Lnb(µ) = Lnb(µ)Lna(λ)Rab(λ− µ) . (1.4.15)
We graphically represent it as
hn ⊗ Va ⊗ Vb = hn ⊗ Va ⊗ Vb ,(1.4.16)
where in the ordering from up to down we represent the matrices Lna and Rab from rightto left respectively as hn ⊗ Va and Va ⊗ Vb.
19
The operator, which acts on the full chain, is the so called monodromy matrix givenby the product of all individual Lax matrices as
Ta(λ) = LNa(λ) . . . L1a(λ) . (1.4.17)
Ta(λ) describes the transport of the spin around the circular chain. We have to provefrom (1.4.15) that the monodromy matrix also fulfills a fundamental commutation rela-tion of the form
Rab(λ− µ)Ta(λ)Tb(µ) = Tb(µ)Ta(λ)Rab(λ− µ) , (1.4.18)
since the commuting integrals of motion are operators that act on the full chain andhence should follow from Ta(λ). For the proof it is enough to consider a product ofLax matrices at neighboured sites. We hence consider the permutation of the followingproduct
Rab(λ− µ)Ln+1 a(λ)Lna(λ)Ln+1 b(µ)Lnb(µ)
= hn ⊗ hn+1 ⊗ Va ⊗ Vb = hn ⊗ hn+1 ⊗ Va ⊗ Vb = hn ⊗ hn+1 ⊗ Va ⊗ Vb
= hn ⊗ hn+1 ⊗ Va ⊗ Vb = hn ⊗ hn+1 ⊗ Va ⊗ Vb
= Ln+1 b(µ)Lnb(µ)Ln+1 a(λ)Lna(λ)Rab(λ− µ) ,(1.4.19)
where in the second and last equality we have used that two Lax matrixes commute, ifthey act in different subspaces of the tensor product. In the third and fourth equality,the fundamental commutation relation (1.4.15) with its graphical representation (1.4.16)has been applied.
As mentioned above, we take the trace of (1.4.18) in the tensor product Va ⊗ Vb ofthe two auxiliary spaces and find
tr(Ta(λ)Tb(µ)
)= tr
(R−1ab (λ− µ)Tb(µ)Ta(λ)Rab(λ− µ)
). (1.4.20)
After a cyclic permutation of the matrices under the trace, we find that the tracescommute at different values of the spectral parameter
[T (λ) , T (µ)] . (1.4.21)
20
We have thereby defined the so called transfer matrix as
T (λ) = tra Ta(λ) = A(λ) +D(λ) , (1.4.22)
where the last equality follows if we introduce an explict representation of Ta(λ) in theauxiliary space Va as
Ta(λ) =
(A(λ) B(λ)C(λ) D(λ)
), (1.4.23)
where A(λ), B(λ), C(λ), D(λ) are operators which act in the Hilbet space HN (1.4.1).
1.4.3 Conserved charges
Let us analyze the operator Ta(λ) and its trace T (λ) in more detail. Inserting the explicitexpressions for Lna (1.4.11) into (1.4.17), and collecting terms with the same power ofλ, we find
Ta(λ) =N∏n=1
((λ− i2)1na + iPna) = λN1a − λN−1 i
2
N∑n=1
(1na − 2 Pna) + . . .
= λN1a + iλN−1~S~σa + . . . ,
(1.4.24)
where 1a without indices denotes the identity in HN ⊗ Va, and in the second line wehave inserted (1.4.4) with ~S defined in (1.4.2). Taking the trace in the auxiliary spaceVa of the above expression, the first term in the above expansion survives, while thesecond one does not contribute due to the tracelessness of the Pauli matrices. We henceobtain a polynomial of degree N in λ of the form
T (λ) = 2λN1+N−2∑n=0
λnQn , (1.4.25)
where 1 is the identity in HN , and the Qn are the non-trivial ones of the exactly Ncommuting conserved charges Qn which appear in an integrable system with N degreesof freedom.
Instead of expanding around λ = 0, in the following we will expand around λ = i2.
The lowest order terms in this expansion then contain expressions which allow us toextract the momentum operator and the Hamiltonian (1.4.3) for the spin chain.
21
The first contribution to the series comes from the monodromy matrix itself, evalu-ated at λ = i
2. We find
Ta(i2) = iN PNa PN−1 a . . .P1a
= inh1 ⊗ h2 ⊗ · · · ⊗ hN−1 ⊗ hN ⊗ Va = inh1 ⊗ h2 ⊗ · · · ⊗ hN−1 ⊗ hN ⊗ Va
= iN P12 P23 . . .PN−1N PNa .(1.4.26)
Taking the trace over the auxiliary space, thereby using (1.4.7), we find that the transfermatrix at λ = i
2is given by
T ( i2) = iNU , U = P12 . . .PN−1N , (1.4.27)
where U has the interpretation as shift operator that simultaneously shifts all spins byone site, and hence corresponds to a rotation of the whole chain by one site. Its inverseis given by U−1 = U † = PN−1N · · ·P12 such that U is unitary. We can hence define theassociated momentum operator P , the generator of shifts n→ n+ 1 as
eiP = U . (1.4.28)
The second order contribution to the series requires the result for the first derivativeof the monodromy matrix, evaluated at λ = i
2. We find
d
dλTa(λ)
∣∣∣λ= i
2
= iN−1
N∑n=1
PNa PN−1 a . . .Pn−1 a 1na Pn+1 a . . .P1a
= iN−1
N∑n=1
P12 P23 . . .Pn−2n−1 Pn−1n+1 Pn+1n+2 . . .PNa .
(1.4.29)
Taking the trace tra, thereby using (1.4.7), multiplying by T ( i2)−1 from the left, and
using the properties (1.4.6) to transform
Pnn+1 Pn−1n Pn−1n+1 = Pn−1n (1.4.30)
we find
T (λ)−1 d
dλT (λ)
∣∣∣λ= i
2
=d
dλlnT (λ)
∣∣∣λ= i
2
=1
i
N∑n=1
Pn−1n =1
i(2H +N) , (1.4.31)
22
where in the last equality we have used (1.4.3). The Hamiltonian is hence expressed interms of the transfer matrix as
H =1
2
(i
d
dλlnT (λ)
∣∣∣λ= i
2
−N)
(1.4.32)
The first two coefficients in the expansion of the transfer matrix around λ = i2
henceread
T (λ) = iNU + iN−1(λ− i2)U(2H +N) +O((λ− i
2)2) . (1.4.33)
Since the Hamiltonian appears in this expansion, the eigenvalue problem for H is solvedby diagonalizing the transfer matrix, i.e. the sum A(λ) +D(λ) in (1.4.22).
1.4.4 Algebraic Bethe ansatz equations, eigenstates and eigen-values
From the fundamental commutation relation (1.4.18) we find the algebra for the op-erators in the matrix representation (1.4.23). The explicit calculation can be found inappendix A. The relevant expressions taken from the full set, which is found from (A.9),read
[B(λ) , B(µ)] = 0 , [C(λ) , C(µ)] = 0 ,
A(λ)B(µ) = f(λ− µ)B(µ)A(λ) + g(λ− µ)B(λ)A(µ) ,
D(λ)B(µ) = h(λ− µ)B(µ)D(λ) + k(λ− µ)B(λ)D(µ) ,
A(λ)C(µ) = h(λ− µ)B(µ)A(λ) + k(λ− µ)B(λ)A(µ) ,
D(λ)C(µ) = f(λ− µ)B(µ)D(λ) + g(λ− µ)B(λ)D(µ) ,
(1.4.34)
where the functions f , g, h, k are explicitly given by
f(λ) =λ− iλ
, g(λ) =i
λ, h(λ) =
λ+ i
λ, k(λ) = − i
λ. (1.4.35)
To obtain the above equations from (1.4.34), we have exchanged λ↔ µ to make contactwith the results of [6].
The transfer matrix (1.4.22) should be diagonal on the states to be constructed.Therefore, at least A(λ) + D(λ) has to be diagonal on these states, and B(λ) andC(λ) have to be the rising and lowering operators. Thereby, as discussed at the endof appendix A, the symmetry of the full set of commutation relations (1.4.34) underthe simultaneous exchange A↔ D and B ↔ C switches spin up and spin down states,such that the rising operators in one convention become the lowering operators in theother one where all spins are flipped, and vice versa. We can therefore choose e.g. C as
23
the rising operators and define the highest weight state as the state Ω ∈ HN which isannihilated by acting with C, i.e.
C(λ)Ω = 0 . (1.4.36)
The monodromy matrix (1.4.23) thus is an upper triagonal matrix when acting on Ω.This is certainly the case, if all Lax matrices Lna n = 1, . . . N , which multiply to Tain (1.4.17), become upper triagonal matrices in the auxiliary space Va, when acting onthe corresponding state ωn in the tensor product Ω =
⊗Nn=1 ωn. If σ+ωn = 0, i.e. for
ωn = | 〉, the matrix representation for Lna (1.4.12) becomes an upper triagonal matrixwhen acting on ωn. Since this has to hold at all sites n = 1, . . . , N , Ω is finally given by
Ω =N⊗n=1
| 〉 = | . . . 〉 . (1.4.37)
Acting with the monodromy matrix on Ω, yields with (1.4.12)
Ta(λ)Ω =
(αN(λ) ∗
0 δN(λ)
)Ω , (1.4.38)
where
α(λ) = λ+i
2, δ(λ) = λ− i
2, (1.4.39)
and ∗ denotes an operator which is of no relevance for our considerations. The heigh-est weight state Ω therefore is an eigenstate of A(λ) and D(λ) and hence also of thetransfer matrix T (λ) in (1.4.22). This result is in accord with our previous findings in(1.4.9) which already identfied the state with all spins up as an eigenstate of the Hamil-tonian. As already mentioned before, the simultaneous exchange A ↔ D and B ↔ Ccorresponds to an exchange of up and down spins and hence, introducing B as creationoperator, yields the state with all spins down as the highest weight state Ω.
All eigenstates are obtained from Ω by acting with the corresponding annihilationoperator, which is B in our case. The eigenvectors at level l thus have the form
Φ(λ) = B(λ1) . . . B(λl)Ω , (1.4.40)
where λ = (λ1, . . . , λl) denotes the set of λi. Thereby, according to the commutativityof B(λ) and B(µ) also at λ 6= µ in (1.4.34), different orderings of the B(λi) do not yielddifferent eigenstates. The state Φ(λ) is an eigenstate of H only for certain values forλi which are the solutions to a set of l so called Bethe Ansatz equations, which we will
24
derive in the following. We act with A(λ) or D(λ) on (1.4.40), and use the commutationrelations (1.4.34) to commute these operators through the product of B. The result hasthe form(
A(λ)D(λ)
)B(λ1) . . . B(λl)Ω =
(∏lk=1 f(λ− λk)αN(λ)∏lk=1 h(λ− λk)δN(λ)
)B(λ1) . . . B(λl)Ω
+l∑
k=1
(Mk(λ, λ)Nk(λ, λ)
)B(λ1) . . . B(λk) . . . B(λl)B(λ)Ω .
(1.4.41)It is easy to determine M1 and N1 from a direct application of the commutation relations(1.4.34). We start from the r.h.s. of the above expression and apply the commutationrelations (1.4.34) l times. To avoid the appearance of B(λ1) in the final result andhence to isolate the terms that only contribute to M1, we have to take the first term inthe sum on the r.h.s. of the commutation relation for A(λ) and B(λ1), which generatesA(λ1). For the remaining l− 1 permutations, we then have to consider the second termin the commutation relations to keep λ1 as the argument of A. For the first equation in(1.4.41) we hence obtain
A(λ)B(λ1) . . . B(λl)Ω
= (f(λ− λ1)B(λ1)A(λ) + g(λ− λ1)B(λ)A(λ1))B(λ2) . . . B(λl)Ω
= g(λ− λ1)l∏
j=2
f(λ1 − λj)αN(λ1)B(λ2) . . . B(λl)B(λ)Ω + terms containing B(λ1) ,
(1.4.42)from which we can directly read off M1. The procedure for N1 is identical. The generalcoeffiencients then follow immediately from the commutativity of B(µ) and B(λ) for allµ and λ to
Mj(λ, λ) = g(λ− λj)l∏
k 6=j
f(λj − λk)αN(λ) ,
Nj(λ, λ) = k(λ− λj)l∏
k 6=j
h(λj − λk)δN(λ) ,
(1.4.43)
where according to (1.4.35) the first factors in both products fulfill
g(λ− λj) = −k(λ− λj) . (1.4.44)
The state Φ(λ) (1.4.40) hence becomes an eigenstate at least of A(λ) +D(λ)
(A(λ) +D(λ))Φ(λ) = Λ(λ, λ)Φ(λ) (1.4.45)
25
with eigenvalue
Λ(λ, λ) = αN(λ)l∏
j=1
f(λ− λj) + δN(λ)l∏
j=1
h(λ− λj) , (1.4.46)
if the set λ satisfies the j = 1, . . . , l equations
l∏k 6=j
f(λj − λk)αN(λj) =l∏
k 6=j
h(λj − λk)δN(λj) . (1.4.47)
With the explicit expressions taken from (1.4.35) and (1.4.39), we find(λj + i
2
λj − i2
)N=
l∏k 6=j
λj − λk + i
λj − λk − i. (1.4.48)
These equations, which fix the set λ and hence determine the eigenvectors Φ(λ)(1.4.40) of the transfer matrix (1.4.22), are the so called Bethe ansatz equations. Afterhaving found the set λ, which are called the Bethe roots, we can compute the spectrumfrom (1.4.32) by replacing T (λ) by its eigenvalue Λ(λ, λ) (1.4.46).
Let us examine more properties of the states Φ(λ) (1.4.40). To this purpose, weexpand the monodromy matrix Tb(µ) as in (1.4.24) and insert it into the fundamentalcommutation relation (1.4.18), thereby keeping only the highest powers in µ. With theexplit expression for the R-matrix, taken from (1.4.13) and (1.4.11) with (1.4.4) inserted,we find for the r.h.s.
µN−1((λ− µ+
i
2
)1ab +
i
2~σa~σb
)Ta(λ)(µ1b + i~S~σb)
= −µN+1Ta(λ) + µN((λ+
i
2
)Ta(λ) +
i
2~σa~σbTa(λ)− iTa(λ)~S~σb
)+ . . . ,
(1.4.49)
and for the l.h.s.
µN−1(µ1b + i~S~σb)Ta(λ)((λ− µ+
i
2
)1ab +
i
2~σa~σb
)= −µN+1Ta(λ) + µN
((λ+
i
2
)Ta(λ) +
i
2Ta(λ)~σa~σb − i~S~σbTa(λ)
)+ . . . .
(1.4.50)
Equating both sides, yields the commutation relations[Ta(λ) ,
1
2~σa + ~S1a
]= 0 . (1.4.51)
26
In particular, we find for the third component of ~S([A(λ) , S3] [B(λ) , S3][C(λ) , S3] [D(λ) , S3]
)=
(0 B(λ)
−C(λ) 0
). (1.4.52)
The diagonal equations simply ensure that the total spin operator is part of the com-muting charges found from T (λ) = A(λ)+D(λ). The two off-diagonal equations confirmthat B(λ) and C(λ) respectively decreases and increase the total spin by 1
2. This can
be seen if we act with S3 on the state Φ(λ) at level l (1.4.40). We then find
S3Φ(λ) =(N
2− l)
Φ(λ) , (1.4.53)
where with S+ = S1 + iS2 we have also used the relations
S+Ω = 0 , S3Ω =N
2Ω , (1.4.54)
which follows immediately from (1.4.37).We can also use the commutator relations (1.4.51) for the component S+ to show
that all states Φ(λ) are highest weight states, if the Bethe ansatz equations (1.4.48)are fulfilled. We first find from (1.4.51)(
[A(λ) , S+] [B(λ) , S+][C(λ) , S+] [D(λ) , S+]
)=
(C(λ) D(λ)− A(λ)
0 −C(λ)
). (1.4.55)
(1.4.56)
Acting with S+ on a state Φ(λ), and using the commutation relation with B(λ), wehence obtain
S+Φ(λ) =l∑
j=1
B(λ1) . . . B(λj−1)(A(λj)−D(λj))B(λj+1) . . . B(λl)Ω
=l∑
k=1
Ok(λ)B(λ1) . . . B(λk) . . . B(λl)Ω = 0 .
(1.4.57)
In appendix (B) we prove the last equality, i.e. we show that the coefficients Ok(λ)vanish if the Bethe ansatz equations (1.4.48) are fulfilled. Therefore, all eigenstatesΦ(λ) are also highest weight states, since they are annihilated by the spin creationoperator S+
S+Φ(λ) = 0 . (1.4.58)
27
In particular, if we demand that the highest weight state has positive spin eigenvalue,it follow from (1.4.53) that l is restricted to l ≤ N
2. In other words, the spin chain
state with the least difference between the number of spin up and spin down states isthe last state to be considered as a highest weight state. The spectrum is degenerateunder exchanging all spin up and spin down states. This is directly visible from theHamiltonian (1.4.3), since it does not fix the convention of what we call spin up or spindown. We have also seen that this symmetry appears in the commutation relations(A.9) for the operator matrix elements in the monodromy matrix (1.4.23).
We can also find the momentum P of the state Φ(λ). For this, we first have to actwith the shift operator U and replace T ( i
2) by the eigenvalues (1.4.46) at λ = i
2, which
are given by
Λ( i2, λ) = iN
l∏j=1
f( i2− λj) (1.4.59)
Then, we immediately find with (1.4.27)
UΦ(λ) =l∏
j=1
λj + i2
λj − i2
Φ(λ) . (1.4.60)
According to (1.4.28), the momentum P is proportional to the logarithm of the eigen-value of U . We thus obtain
PΦ(λ) =l∑
j=1
p(λj)Φ(λ) , p(λ) =1
ilnλ+ i
2
λ− i2
. (1.4.61)
We therefore see, that the momentum is additive, and that to each λi the momentum piis associated. With the above results, the Bethe ansatz equations (1.4.48) can be castinto the form
eiNp(λj) =l∏
j=1
S(λj − λk) , (1.4.62)
where the S-matrix is given by
S(λj − λk) =λj − λk + i
λj − λk − i. (1.4.63)
The interpretation of the Bethe equations at level l is hence as follows. The momentump(λj) corresponds to jth of the l quasi-particles, called magnons, which are moving alongthe spin chain. The Bethe roots λj are also called magnon rapidities. The phase shift
28
for the jth magnon, given by the exponential of its momentum on the l.h.s. of the Betheansatz equations, is then determined by the scattering with all other l − 1 magnons.This scattering process factorizes into l− 1 scattering processes of the jth magnon withone of the remaining l − 1 magnons, where the S-matrix that appears on the r.h.s. ofthe Bethe ansatz equations describes the scattering of two magnons.
As the momentum P , also the energy E is given as a sum over the correspondingcontribution from each individual magnon. The energy is determined by inserting
d
dλln Λ( i
2, λ)
∣∣∣λ= i
2
= −iN + il∑
j=1
1
λ2j + 1
4
(1.4.64)
into (1.4.32), which yields
HΦ(λ) =l∑
j=1
ε(λj)Φ(λ) , ε(λ) = −1
2
1
λ2 + 14
. (1.4.65)
Since at λ = i2
the first contribution in (1.4.64) coming from the factor αN(λ) in (1.4.46)cancels with the shift of−N in (1.4.32), and since the remaining terms in the sum dependonly on the difference λ− λj, we can rewrite the derivative w.r.t. λ as derivatives w.r.t.λi. This allows us to set λ = i
2before taking the derivatives, and hence to express the
individual magnon energies as derivatives of the magnon momenta as
ε(λ) =1
2
d
dλp(λ) . (1.4.66)
The individual magnon energy can also directly be expressed as a function of the indi-vidual magnum momentum as
ε(p) = cos p− 1 . (1.4.67)
Since all individual energies are negative, Ω is not the ground state in this case. Theground state is given by the state with maximum level l = N
2or l = N−1
2for respectively
even or odd N , i.e. by the state in which the number of neighboured spins with oppositeorientation is maximized. This situation with the Hamiltonian given by (1.4.3) describesthe antiferromagnetic spin chain. If instead we flip the sign of H, all energies becomepositive and the state Ω becomes the ground state with zero energy, in which all spinsare parallel oriented. This situation clearly describes the ferromagnetic spin chain.
29
Appendix
A Matrix representations
For explicit computations we introduce explicit matrix representations for the operatorsin the tensor product Va ⊗ Vb of both auxiliary spaces Va = C2 and Vb = C2. In each ofthe spaces, a spin state is represented as
| 〉 =
(10
), | 〉 =
(01
), (A.1)
the four tensor product states hence become
| 〉⊗ | 〉 =
1000
, | 〉⊗ | 〉 =
0100
, | 〉⊗ | 〉 =
0010
, | 〉⊗ | 〉 =
0001
,
(A.2)The tensor product of two matrices A and B, which multiply vectors respectively in Vaand Vb, then reads
A⊗B =
A11B11 A11B12 A12B11 A12B12
A11B21 A11B22 A12B21 A12B22
A21B11 A21B12 A22B11 A22B12
A21B21 A21B22 A22B21 A22B22
. (A.3)
With the Pauli matrices given by
σ1 =
(0 11 0
), σ2 =
(0 −ii 0
), σ3 =
(1 00 −1
), (A.4)
we find that the permutation operator as in (1.4.4) which acts in Va ⊗ Vb has matrixrepresentation
Pab =
1 0 0 00 0 1 00 1 0 00 0 0 1
. (A.5)
The R-matrix defined in (1.4.13) is hence given by
Rab(λ) =
λ+ i 0 0 0
0 λ i 00 i λ 00 0 0 λ+ i
. (A.6)
30
The monodromy matrices Ta(λ) and Tb(µ) in the tensor product space Va⊗ Vb are thenrespectively given by
Ta(λ)⊗1 =
A(λ) 0 B(λ) 0
0 A(λ) 0 B(λ)C(λ) 0 D(λ) 0
0 C(λ) 0 D(λ)
, 1⊗Tb(µ) =
A(µ) B(µ) 0 0C(µ) D(µ) 0 0
0 0 A(µ) B(µ)0 0 C(µ) D(µ)
.
(A.7)The two sides of the fundamental commutation relation (1.4.18) read with δ = λ− µ
Rab(δ)Ta(λ)Tb(µ)
=
0BB@(δ + i)A(λ)A(µ) (δ + i)A(λ)B(µ) (δ + i)B(λ)A(µ) (δ + i)B(λ)B(µ)
δA(λ)C(µ) + iC(λ)A(µ) δA(λ)D(µ) + iC(λ)B(µ) δB(λ)C(µ) + iD(λ)A(µ) δB(λ)D(µ) + iD(λ)B(µ)δC(λ)A(µ) + iA(λ)C(µ) δC(λ)B(µ) + iA(λ)D(µ) δD(λ)A(µ) + iB(λ)C(µ) δD(λ)B(µ) + iB(λ)D(µ)
(δ + i)C(λ)C(µ) (δ + i)C(λ)D(µ) (δ + i)D(λ)C(µ) (δ + i)D(λ)D(µ)
1CCA ,
Tb(µ)Ta(µ)Rab(δ)
=
0BB@(δ + i)A(µ)A(λ) δB(µ)A(λ) + iA(µ)B(λ) δA(µ)B(λ) + iB(µ)A(λ) (δ + i)B(µ)B(λ)(δ + i)C(µ)A(λ) δD(µ)A(λ) + iC(µ)B(λ) δC(µ)B(λ) + iD(µ)A(λ) (δ + i)D(µ)B(λ)(δ + i)A(µ)C(λ) δB(µ)C(λ) + iA(µ)D(λ) δA(µ)D(λ) + iB(µ)C(λ) (δ + i)B(µ)D(λ)(δ + i)C(µ)C(λ) δD(µ)C(λ) + iC(µ)D(λ) δC(µ)D(λ) + iD(µ)C(λ) (δ + i)D(µ)D(λ)
1CCA .
(A.8)From the equality of both sides, we hence find the operator algebra
[A(λ) , A(µ)] = 0 , [B(λ) , B(µ)] = 0 , [C(λ) , C(µ)] = 0 , [D(λ) , D(µ)] = 0 ,
[A(λ) , B(µ)] =i
λ− µ(A(µ)B(λ)− A(λ)B(µ)) ,
[B(λ) , A(µ)] =i
λ− µ(B(µ)A(λ)−B(λ)A(µ)) ,
[A(λ) , C(µ)] =i
λ− µ(C(µ)A(λ)− C(λ)A(µ)) ,
[C(λ) , A(µ)] =i
λ− µ(A(µ)C(λ)− A(λ)C(µ)) ,
[B(λ) , C(µ)] =i
λ− µ(D(µ)A(λ)−D(λ)A(µ)) ,
[A(λ) , D(µ)] =i
λ− µ(C(µ)B(λ)− C(λ)B(µ)) ,
(A.9)where the remaining six relations follow from the last six one by exchanging A↔ D andB ↔ C. This corresponds to exchanging the role of spin up and spin down. The fullset of relations is also symmetric under the simultaneous exchange A↔ C and B ↔ D.
31
B The coefficients Ok(λ) in (1.4.57)
To find the explicit form of the coefficients Ok(λ) in (1.4.57), we modify both equationsin (1.4.41) as follows(
A(λj)D(λj)
)B(λj+1) . . . B(λl)Ω =
(∏lk=j+1 f(λj − λk)αN(λj)∏lk=j+1 h(λj − λk)δN(λj)
)B(λj+1) . . . B(λl)Ω
+l∑
k=j+1
(Mj,k(λ)Nj,k(λ)
)B(λj) . . . B(λk) . . . B(λl)Ω ,
(B.1)where the coefficients, which are similar to the ones in (1.4.43), become
Mj,k(λ) = g(λj − λk)l∏
r=j+1r 6=k
f(λk − λr)αN(λk) ,
Nj,k(λ) = k(λj − λk)l∏
r=j+1r 6=k
h(λk − λr)δN(λk) .
(B.2)
We then find with (B.1) that a substitution of B(λj) in the definition for Φ(λ) (1.4.40)by either A(λj) or B(λj) yields
l∑j=1
B(λ1) . . .
(A(λj)D(λj)
). . . B(λl)Ω
=l∑
j=1
(∏lk=j+1 f(λj − λk)αN(λj) +
∑j−1k=1Mk,j(λ)∏l
k=j+1 h(λj − λk)δN(λj) +∑j−1
k=1 Nk,j(λ)
)B(λ1) . . . B(λj) . . . B(λl)Ω ,
(B.3)where we have exchanged the order of the two summations for the terms that dependon Mj,k and Nj,k and then we have relabeled j ↔ k to collect the expressions that aremultiplied by the same state. The coefficients Ok(λ) in (1.4.57) are obtained as thedifference of the terms in the first and in the second row in the above equation.
Inserting the explicit expressions (B.2) into these terms, we can rewrite them as
32
follows
l∏k=j+1
f(λj − λk)αN(λj) +
j−1∑k=1
Mk,j(λ)
=l∏
k 6=j
f(λj − λk)αN(λj)( j−1∏k=1
f−1(λj − λk) +
j−1∑r=1
g(λr − λj)r∏
k=1
f−1(λj − λk))
l∏k=j+1
h(λj − λk)δN(λj) +
j−1∑k=1
Nk,j(λ)
=l∏
k 6=j
h(λj − λk)δN(λj)( j−1∏k=1
h−1(λj − λk) +
j−1∑r=1
k(λr − λj)r∏
k=1
h−1(λj − λk)),
(B.4)where in the two expressions we have pulled out the factors which respectively correspondto the r.h.s. and l.h.s. of the Bethe ansatz equations (1.4.47).
If (1.4.47) are fulfilled, these factors are identical for both terms, and the differenceof both results and thus the coefficients Ok(λ) are zero if the following relation holds
1−j−1∏k=1
f(λj − λk)h(λj − λk)
−j−1∑r=1
g(λj − λr)j−1∏
k=r+1
f(λj − λk)(
1 +r∏
k=1
f(λj − λk)h(λj − λk)
)= 0 . (B.5)
We have thereby also used that, according to (1.4.35), g(λ) = −g(−λ) = −k(λ). Fur-thermore, we have f(λ) = 1 − g(λ), h(λ) = 1 + g(λ). This allows us to rephrase theabove relation. It is true, if for arbitrary elements gk the relation
1−L∏k=1
1− gk1 + gk
−L∑r=1
gr
L∏k=r+1
(1− gk)(
1 +r∏
k=1
1− gk1 + gk
)= 0 (B.6)
is fulfilled. With hk = 1−gk
1+gk,i.e. gk = 1−hk
1+hk, we can simplify the above expression even
further. We have to prove that
1−L∏k=1
hk −L∑r=1
1− hr1 + hr
L∏k=r+1
2hr1 + hr
(1 +
r∏k=1
hk
)= 0 (B.7)
is true.We proof this relation by induction. First of all, for L = 1 we find
1− h1 −1− h1
1 + h1
(1 + h1) = 0 , (B.8)
33
i.e. the above relation is true for L = 1. Then, we show that, if the above expressionis true for generic L, this induces that it must also hold for L + 1. We start from theexpression for L+ 1 and transform it into
1−L+1∏k=1
hk −1− hL+1
1 + hL+1
(1 +
L+1∏k=1
hk
)− 2hL+1
1 + hL+1
L∑r=1
1− hr1 + hr
L∏k=r+1
2hr1 + hr
(1 +
r∏k=1
hk
)= 1−
L+1∏k=1
hk −1− hL+1
1 + hL+1
(1 +
L+1∏k=1
hk
)− 2hL+1
1 + hL+1
(1−
L∏k=1
hk
)= 0 ,
(B.9)where in the second line we have used the assumption that the expression is true forL. We then find zero, i.e. that the expression then also holds for L + 1. The proofis complete, and we have shown the vanishing of all Ok(λ) if the set λ fulfills theBethe ansatz equations (1.4.47).
34
Bibliography
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[2] D. Finley Lecture notes to Physics 573: nonlinear systems, 2004.
[3] A. M. Perelomov, Integrable systems in classical mechanics and Lie algebras.
[4] J. Hoppe, “Lectures on integrable systems,” Lect. Notes Phys. M10 (1992) 1–111.
[5] E. D’Hoker and D. H. Phong, “Lectures on supersymmetric Yang-Mills theory and integrablesystems,” hep-th/9912271.
[6] L. D. Faddeev, “How algebraic Bethe ansatz works for integrable model,” hep-th/9605187.
[7] R. I. Nepomechie, “A spin chain primer,” Int. J. Mod. Phys. B13 (1999) 2973–2986,hep-th/9810032.
35
