Bethe ansatz inside Calogero–Sutherland models

Gwenaël Ferrando1, Jules Lamers2$\star$, Fedor Levkovich-Maslyuk2 and Didina
Serban2

1 School of Physics and Astronomy

Tel Aviv University

Ramat Aviv 69978, Israel

2 Université Paris–Saclay, CNRS, CEA

Institut de Physique Théorique

91191 Gif-sur-Yvette, France

⋆<EMAIL_ADDRESS>
## Abstract

We study the trigonometric quantum spin-Calogero–Sutherland model, and the
Haldane–Shastry spin chain as a special case, using a Bethe-ansatz analysis.
We harness the model’s Yangian symmetry to import the standard tools of
integrability for Heisenberg spin chains into the world of integrable long-
range models with spins. From the transfer matrix with a diagonal twist we
construct Heisenberg-style symmetries (Bethe algebra) that refine the usual
hierarchy of commuting Hamiltonians (quantum determinant) of the spin-
Calogero–Sutherland model. We compute the first few of these new conserved
charges explicitly, and diagonalise them by Bethe ansatz inside each
irreducible Yangian representation. This yields a new eigenbasis for the spin-
Calogero–Sutherland model that generalises the Yangian Gelfand–Tsetlin basis
of Takemura and Uglov. The Bethe-ansatz analysis involves non-generic values
of the inhomogeneities. Our review of the inhomogeneous Heisenberg xxx chain,
with special attention to how the Bethe ansatz works in the presence of
fusion, may be of independent interest.

###### Contents

  1. 1 Introduction
  2. 2 How algebraic Bethe ansatz works for inhomogeneous models
    1. 2.1 Inhomogeneous Heisenberg xxx spin chain
    2. 2.2 Algebraic Bethe ansatz and QQ-relation
      1. 2.2.1 Inhomogeneous analogues of translation operator
      2. 2.2.2 Another family of conserved charges with inhomogeneities
      3. 2.2.3 Periodic case
      4. 2.2.4 Extreme twist and Gelfand–Tsetlin basis
    3. 2.3 Fusion
      1. 2.3.1 General description
      2. 2.3.2 Bethe ansatz for fusion into singlet
      3. 2.3.3 Bethe ansatz for fusion into triplet
      4. 2.3.4 Repeated fusion
  3. 3 Fermionic Spin-Calogero–Sutherland model
    1. 3.1 Dunkl operators and nonsymmetric Jack polynomials
    2. 3.2 Hamiltonian and monodromy matrix
    3. 3.3 Effective spin chains
  4. 4 Bethe-ansatz analysis of the spin-Calogero–Sutherland model
    1. 4.1 Heisenberg-style symmetries
    2. 4.2 Internal Bethe ansatz
    3. 4.3 Limits
      1. 4.3.1 Free-fermion limit $\beta\to 0$
      2. 4.3.2 Strong-coupling limit $\beta\to\infty$ and the Haldane–Shastry spin chain
    4. 4.4 Example: $N=4$
  5. 5 Conclusion
  6. A Fused R-matrix
  7. B On the derivation of the Bethe equations with fusion
    1. B.1 Derivation from the QQ-relation
    2. B.2 Derivation from the algebraic Bethe ansatz
  8. C Action of the B-operator at the fixed root
  9. D Examples of fusion for low length
    1. D.1 Generic case and fusion for L = 2
    2. D.2 Fusion into singlet for L = 4

##  1 Introduction

Long-range interacting spin systems naturally appear in a broad range of
physical contexts, from experiments with cold atoms to high-energy theory [1,
2]. Yet on the theoretical side they have received much less attention than
their nearest-neighbour counterparts.

In this paper we will consider three integrable long-range models which, as we
will see, are all interrelated. The first one has actually been studied for
nearly half a century, although it is usually not explicitly thought of as a
long-range model: the inhomogeneous Heisenberg spin chain. It naturally arises
from the viewpoint of the six-vertex model (cf. Baxter’s Z-invariant model
[3]) and makes an appearance in the Bethe/gauge correspondence [4], where it
corresponds to a certain $\mathcal{N}=2$ supersymmetric gauge theory (with
‘twisted masses’ that relate to the inhomogeneities). The inhomogeneous
Heisenberg chain is an important example of a quantum-integrable system, with
an underlying Yangian structure that provides its commuting charges (from the
transfer matrix, or Bethe algebra) as well as a way of diagonalising them (by
algebraic Bethe ansatz). Yet, except for some special cases, one usually does
not think of it as a bona fide spin chain, and the inhomogeneity parameters
are rather seen as a technical tool. For instance, they are crucial for the
Izergin–Korepin approach to computing the domain-wall partition functions,
Gaudin norms, and Slavnov scalar products of Bethe vectors [5, 6]. Other
applications of inhomogeneities are the proofs of completeness of the Bethe
ansatz [7, 8, 9] and of some of the Razumov–Stroganov conjectures [10]. In
addition, in special semiclassical limits, inhomogenous spin chains give rise
to the Gaudin Hamiltonians [11]. For most other physical applications one
eventually takes the homogeneous limit to restore periodicity. One can also
consider special repeating values of the inhomogeneities, e.g. staggered
(alternating) values [12, 13, 14, 15] or other periodic values [16, 17], which
give access to a broader range of conformal field theories in a suitable
scaling limit.

The second long-range model will be the main object of our study: the quantum
trigonometric Calogero–Sutherland model with particles that have spins, with
Hamiltonian [18, 19, 20]

$\displaystyle\widetilde{H}=-\frac{1}{2}\sum_{i=1}^{N}\partial_{x_{i}}^{2}+\beta\,\sum_{i<j}^{N}\frac{\beta\mp
P_{ij}}{4\,\sin^{2}[(x_{i}-x_{j})/2]}\,,$ (1.1)

where the upper (lower) sign corresponds to bosonic (respectively fermionic)
statistics, and $P_{ij}$ is the spin permutation operator. This quantum many-
body system is also integrable, has eigenvectors containing Jack polynomials
with prescribed (anti)symmetry, and a deep representation-theoretic structure
[21, 22] which in particular provides a representation of the Yangian that is
very different from the usual one for Heisenberg spin chains. The quantum
determinant, i.e. the centre of the Yangian, generates the commuting
Hamiltonians of the spin-Calogero–Sutherland model, which means that the spin
symmetry is enhanced to Yangian _symmetry_. (In contrast, for the Heisenberg
spin chain the Yangian can be used to move between eigenvectors with different
energies, as in the algebraic Bethe ansatz.) The Yangian structure was studied
in detail by Takemura and Uglov [23, 24, 25]. The spin-Calogero–Sutherland
model is superintegrable, see [26], which means that it has (even) more
commuting charges than a normal integrable system such as the Heisenberg spin
chain or scalar Calogero–Sutherland model. One additional Hamiltonians is
Haldane’s ‘rapidity’ operator, see [27]. Further symmetries were constructed
by Fowler and Minnahan [27] in a special (strong coupling) case using
Polychrankos’ exchange-operator formalism [28].

One of our main motivations for studying the spin-Calogero–Sutherland model
are its connections to the third long-range model: the Haldane–Shastry spin
chain [29, 30]

$H^{\textsc{hs}}=\sum_{i<j}^{N}\frac{1+P_{ij}}{4\,\sin^{2}\bigl{[}\tfrac{\pi}{N}(i-j)\bigr{]}}\,.$
(1.2)

It exhibits fractional (exclusion) statistics [31, 32, 33] and can be viewed
as the $\textit{SU}(2)_{k=1}$ Wess–Zumino–Witten model on the lattice [34, 35,
36, 37]. The Haldane–Shastry chain arises from (1.1) in a special ‘freezing’
limit [38, 39, 40] and inherits various special properties along the way.
Amongst others, (1.2) has very simple energy eigenvalues and very high
degeneracies [32], which are to a large extent due to its Yangian symmetry
[34, 40]. Its (Yangian highest-weight) wave functions are given by certain
symmetric Jack polynomials, which are indirectly derived from freezing [40].
Higher Hamiltonians follow from freezing too [27, 41]. Although the freezing
procedure often starts from the bosonic spin-Calogero–Sutherland model [38,
39, 40], two of us showed [42] that the _fermionic_ case naturally accounts
for the form of the Haldane–Shastry wave functions with Yangian highest
weight, and used it to prove a claim from [40] about the spin-chain
eigenvectors for higher rank. We refer to the introduction of [42] for a more
in-depth survey of different connections between the spin-Calogero–Sutherland
model and Haldane–Shastry spin chain.

In this paper we import the standard toolkit of Heisenberg integrability into
the world of the spin-Calogero–Sutherland model and Haldane–Shastry chain. We
exploit the Yangian symmetry to construct additional commuting charges of the
spin-Calogero–Sutherland model. Namely, we will use a transfer matrix to
construct symmetries that form (a representation of) a maximal abelian algebra
of the Yangian, called the Bethe algebra. Going beyond the usual spin-
Calogero–Sutherland hierarchy coming from the quantum determinant, these extra
Heisenberg-type charges are no longer Yangian invariant. This will enable us
to simultaneously diagonalise them by algebraic Bethe ansatz and obtain a new
eigenbasis for (1.1). By including a (diagonal) twist, our construction
generalises the Yangian Gelfand–Tsetlin eigenbasis constructed by Takemura and
Uglov [23]. Via the freezing procedure our results extend to the
Haldane–Shastry chain; special cases have been found by Fowler and Minahan
[27].

In more detail, the eigenspaces of the spin-Calogero–Sutherland model,
labelled by partitions with bounded multiplicities, are irreducible
representation of the Yangian studied in detail by Takemura and Uglov [23]. We
reinterpret each such eigenspace as an ‘effective’ inhomogenous Heisenberg
spin chain with special values of the inhomogeneities. The Yangian highest-
weight vector, which can be described explicitly in the coordinate basis [42],
serves as the pseudovacuum of the effective spin chain. We then use the
algebraic Bethe ansatz to construct the full spin-Calogero–Sutherland
eigenspace. Remarkably, the values of the inhomogeneities that occur force us
to consider cases with fusion. Thus we will first review how fusion works and
point out various subtleties for the algebraic Bethe ansatz in this situation.

Our new Heisenberg-type symmetries provide a setting for developing separation
of variables (SoV) for long-range spin chains that are richer than
inhomogeneous Heisenberg spin chains. In addition, while we focus on the
simplest case of $\mathfrak{sl}_{2}$ spins, our approach will extend to higher
rank as well.

Outline. This paper is organised as follows. In Section 2 we review the
algebraic Bethe ansatz framework for Heisenberg xxx spin chains. We discuss
the fusion procedure in detail, and pay special attention to nontrivial
aspects of the Bethe ansatz in this case. In Section 3 we recall the basics of
the spin-Calogero–Sutherland model, its Yangian symmetry and its eigenspaces,
reinterpreted as effective Heisenberg spin chains. Section 4 contains our main
results: the construction of a refined family of conserved charges for the
spin-Calogero–Sutherland model and their diagonalisation by algebraic Bethe
ansatz. We analyse our construction in limiting cases, in particular including
the Haldane–Shastry spin chain obtained by freezing, and illustrate it in an
explicit example. Section 5 contains our conclusions. There are four
appendices containing technical details and examples related to fusion for
small systems.

##  2 How algebraic Bethe ansatz works for inhomogeneous models

In this section we review the Bethe-ansatz solution for the inhomogeneous xxx
spin chain with a spin-$1/2$ representation at each site. We pay special
attention to the subtleties of fusion, which is relevant for the long-range
models that we focus on in the rest of the paper.

###  2.1 Inhomogeneous Heisenberg xxx spin chain

The Hamiltonian of the Heisenberg xxx spin chain for $L$ spin-1/2 sites is

$H^{\textsc{h}}=\sum_{i=1}^{L}\,(1-P_{i,i+1})\qquad\text{on}\qquad\mathcal{H}=(\mathbb{C}^{2})^{\otimes
L}\,,$ (2.1)

where $P_{ij}=(1+\vec{\sigma}_{i}\cdot\vec{\sigma}_{j})/2$ is the permutation
operator for the spins at sites $i$ and $j$. It commutes with all of the
global $\mathfrak{sl}_{2}$, which acts on the spin chain by

$\displaystyle S^{\pm}=\sum_{i=1}^{L}\sigma^{\pm}_{i}\,,\qquad$
$\displaystyle\mspace{4.0mu}S^{z}=\frac{1}{2}\sum_{i=1}^{L}\sigma^{z}_{i}\,,$
(2.2) $\displaystyle[S^{z},S^{\pm}]=\pm S^{\pm}\,,$
$\displaystyle[S^{+},S^{-}]=2\,S^{z}\,.$

We denote the coordinate basis vectors of $\mathcal{H}$ by

$\lvert
i_{1},\dots,i_{M}\rangle\\!\rangle\equiv\sigma^{-}_{i_{1}}\cdots\sigma^{-}_{i_{M}}\,\lvert\uparrow\cdots\uparrow\rangle\,.$
(2.3)

Bethe’s exact characterisation of the spectrum of (2.1) [43] is one of the
cornerstones of integrability. It admits an algebraic reformulation in the
framework of the quantum inverse-scattering method developed by Faddeev et al
[44]. One of the many benefits of this framework is that it allows for the
construction of _inhomogeneous_ generalisations of (2.1) depending on
inhomogeneity parameters $\theta_{1},\dots,\theta_{L}$ that break
translational invariance (homogeneity) without spoiling integrability. Let us
briefly review how this goes. We start from the rational R-matrix [45, 46]

$\mspace{2.0mu}\overline{\mspace{-2.0mu}R\mspace{-2.0mu}}\mspace{2.0mu}(u)=u+\mathrm{i}\,P\,,\qquad\mspace{2.0mu}\overline{\mspace{-2.0mu}R\mspace{-2.0mu}}\mspace{2.0mu}(u-v)=\
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}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-16.30879pt}{3.533pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,,$ (2.4)

acting on $\mathbb{C}^{2}\otimes\mathbb{C}^{2}$. Here and below we use a bar
to distinguish the standard conventions for Heisenberg from those that we will
use for the spin-Calogero–Sutherland model in §3. Introducing another copy of
$\mathbb{C}^{2}$ as an ‘auxiliary’ space (which we label by $0$) the monodromy
matrix on $\mathbb{C}^{2}\otimes\mathcal{H}$ is defined as 111 Like the
$\mathrm{i}$ in (2.4), the shifts in (2.5) ensure that the Bethe equations
will come out in their usual form. Also note that one may incorporate the
twist from (2.7) in the monodromy matrix instead. As we consider diagonal
twist, this does not change the commutation relations for the operators (2.6).
We choose to include the twist in (2.7) so that (2.6) are independent of the
twist, which will be convenient when we take limits of the twist.

$\qquad\quad\mspace{2.0mu}\overline{\mspace{-2.0mu}T\mspace{-2.0mu}}\mspace{2.0mu}_{0}(u)=\mspace{2.0mu}\overline{\mspace{-2.0mu}R\mspace{-2.0mu}}\mspace{2.0mu}_{01}(u-\theta_{1}-\mathrm{i}/2)\cdots\mspace{2.0mu}\overline{\mspace{-2.0mu}R\mspace{-2.0mu}}\mspace{2.0mu}_{0L}(u-\theta_{L}-\mathrm{i}/2)=\
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}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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}\hbox{\footnotesize{$\theta_{1}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
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{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,,$ (2.5)

where the subscripts indicate the spaces in which the R-matrices act, and we
included an inhomogeneity $\theta_{i}$ at each site. We can write $T_{0}$ as a
$2\times 2$ matrix whose elements act on the physical space $\mathcal{H}$,

$\mspace{2.0mu}\overline{\mspace{-2.0mu}T\mspace{-2.0mu}}\mspace{2.0mu}_{0}(u)=\begin{pmatrix}\mspace{2.0mu}\overline{\mspace{-2.0mu}A\mspace{-2.0mu}}\mspace{2.0mu}(u)&\mspace{2.0mu}\overline{\mspace{-2.0mu}B\mspace{-2.0mu}}\mspace{2.0mu}(u)\\\
\mspace{2.0mu}\overline{\mspace{-2.0mu}C\mspace{-2.0mu}}\mspace{2.0mu}(u)&\mspace{2.0mu}\overline{\mspace{-2.0mu}D\mspace{-2.0mu}}\mspace{2.0mu}(u)\end{pmatrix}_{\\!0}\,.$
(2.6)

The _twisted_ transfer matrix with twist
$\kappa=\mathrm{e}^{\mathrm{i}\varphi/2}$ is a twisted trace over the
auxiliary space,

$\qquad\quad\mspace{2.0mu}\overline{\mspace{-2.0mu}t\mspace{-2.0mu}}\mspace{2.0mu}(u;\kappa)=\mathrm{Tr}_{0}\Bigl{[}\kappa^{\sigma^{z}_{0}}\;\mspace{2.0mu}\overline{\mspace{-2.0mu}T\mspace{-2.0mu}}\mspace{2.0mu}_{0}(u)\Bigr{]}=\kappa\;\mspace{2.0mu}\overline{\mspace{-2.0mu}A\mspace{-2.0mu}}\mspace{2.0mu}(u)+\kappa^{-1}\,\mspace{2.0mu}\overline{\mspace{-2.0mu}D\mspace{-2.0mu}}\mspace{2.0mu}(u)=\
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to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{
{}{{}}{}{{}}{{}{}{}{{}}{{{{}{}{}{}}}{{}{}{}{}}}}{} {} {} {} {}{} {}{}{{}}{{}}
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{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-11.38113pt}\pgfsys@curveto{1.5714pt}{-11.38113pt}{2.84523pt}{-12.0181pt}{2.84523pt}{-12.80385pt}\pgfsys@curveto{2.84523pt}{-13.5896pt}{1.5714pt}{-14.22656pt}{0.0pt}{-14.22656pt}\pgfsys@lineto{-85.35828pt}{-14.22638pt}\pgfsys@lineto{-99.58466pt}{-14.22638pt}\pgfsys@curveto{-101.15605pt}{-14.22638pt}{-102.42989pt}{-13.58942pt}{-102.42989pt}{-12.80367pt}\pgfsys@curveto{-102.42989pt}{-12.01791pt}{-101.15605pt}{-11.38095pt}{-99.58466pt}{-11.38095pt}\pgfsys@lineto{-98.62192pt}{-11.38107pt}\pgfsys@stroke\pgfsys@invoke{
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to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{
{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-87.6629pt}{-10.69337pt}\pgfsys@invoke{
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}\hbox{\footnotesize{$\kappa$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{}}
{}{}{}{}{}{}{{}}\pgfsys@moveto{-14.22638pt}{0.0pt}\pgfsys@lineto{-14.22638pt}{-27.99277pt}\pgfsys@stroke\pgfsys@invoke{
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}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-17.6286pt}{5.06367pt}\pgfsys@invoke{
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}\hbox{\footnotesize{$\theta_{L}$}} }}\pgfsys@invoke{\lxSVG@closescope
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{}{{}}{}
{}{}{}{}{}{}{{}}\pgfsys@moveto{-28.45276pt}{0.0pt}\pgfsys@lineto{-28.45276pt}{-27.99277pt}\pgfsys@stroke\pgfsys@invoke{
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}\pgfsys@endscope}}{{}}}} {}{{}}{}{{}}
{}{}{}{}{}{}{{}}\pgfsys@moveto{-42.67914pt}{0.0pt}\pgfsys@lineto{-42.67914pt}{-27.99277pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{-42.67914pt}{-27.99277pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-48.67691pt}{5.133pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{\footnotesize{$\vphantom{\theta_{1}}\cdots$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{}{}{}{}{}{{}}\pgfsys@moveto{-56.90552pt}{0.0pt}\pgfsys@lineto{-56.90552pt}{-27.99277pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{-56.90552pt}{-27.99277pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {}{{}}{}{{}}
{}{}{}{}{}{}{{}}\pgfsys@moveto{-71.1319pt}{0.0pt}\pgfsys@lineto{-71.1319pt}{-27.99277pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{-71.1319pt}{-27.99277pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-74.12967pt}{4.97656pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{\footnotesize{$\theta_{1}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\qquad,$ (2.7)

where we focus on a diagonal twist

$\kappa^{\sigma^{z}}=\begin{pmatrix}\kappa&0\\\
0&\kappa^{-1}\end{pmatrix}=\frac{\kappa+\kappa^{-1}}{2}+\frac{\kappa-\kappa^{-1}}{2}\,\sigma^{z}\,.$
(2.8)

This operator acts on ${\cal H}$. Note that $\kappa\neq\pm 1$ breaks the
global $\mathfrak{sl}_{2}$-symmetry down to (its Cartan subalgebra) $S^{z}$.
If we view the twist as a formal parameter we further have a discrete symmetry
(Weyl group of $\mathfrak{sl}_{2}$) that flips all
$\uparrow\leftrightarrow\downarrow$ and inverts the twist
$\kappa\to\kappa^{-1}$. The transfer matrix provides a family of commuting
operators,

$\bigl{[}\mspace{2.0mu}\overline{\mspace{-2.0mu}t\mspace{-2.0mu}}\mspace{2.0mu}(u;\kappa),\mspace{2.0mu}\overline{\mspace{-2.0mu}t\mspace{-2.0mu}}\mspace{2.0mu}(v;\kappa)\bigr{]}=0\,,$
(2.9)

that can be diagonalised simultaneously. The commutative algebra generated by
the transfer matrix is maximal for generic twist [7, 8, 9], and is called the
_Bethe algebra_. We give some of its explicit elements in Sections
2.2.1–2.2.2.

###  2.2 Algebraic Bethe ansatz and QQ-relation

The aim of the algebraic Bethe ansatz is to construct the eigenvectors of the
transfer matrix using the Yangian generators (2.6). Let us first review the
case when the inhomogeneities $\theta_{i}$ and the twist $\kappa$ are in
generic position. As a starting point we take a reference or ‘(pseudo)vacuum’
state $\lvert 0\rangle$, annihilated by the C-operator from (2.6),
$C(u)\,\lvert 0\rangle=0$. Later we will encounter more complicated
(pseudo)vacua, but for now we have

$\lvert 0\rangle=\lvert\uparrow\uparrow\cdots\uparrow\rangle\,,$ (2.10)

which is an eigenstate of
$\mspace{2.0mu}\overline{\mspace{-2.0mu}t\mspace{-2.0mu}}\mspace{2.0mu}(u;q)$
since the latter preserves the number of $\downarrow$s. To build the other
states, we act on it with the B-operator, which serves as a creation operator,

$\mspace{2.0mu}\overline{\mspace{-2.0mu}B\mspace{-2.0mu}}\mspace{2.0mu}(u_{1})\cdots\mspace{2.0mu}\overline{\mspace{-2.0mu}B\mspace{-2.0mu}}\mspace{2.0mu}(u_{M})\,\lvert
0\rangle\,.$ (2.11)

This is an eigenstate of the transfer matrix with eigenvalue

$\quad\mspace{2.0mu}\overline{\mspace{-2.0mu}\tau\mspace{-2.0mu}}\mspace{2.0mu}(u;\kappa)=\mspace{2.0mu}\overline{\mspace{-2.0mu}Q\mspace{-2.0mu}}\mspace{2.0mu}_{\bm{{\theta}}}^{+}\,\frac{\mspace{2.0mu}\overline{\mspace{-2.0mu}Q\mspace{-2.0mu}}\mspace{2.0mu}^{-\mspace{2.0mu}-}}{\mspace{2.0mu}\overline{\mspace{-2.0mu}Q\mspace{-2.0mu}}\mspace{2.0mu}}+\mspace{2.0mu}\overline{\mspace{-2.0mu}Q\mspace{-2.0mu}}\mspace{2.0mu}_{\bm{{\theta}}}^{-}\,\frac{\mspace{2.0mu}\overline{\mspace{-2.0mu}Q\mspace{-2.0mu}}\mspace{2.0mu}^{++}}{\mspace{2.0mu}\overline{\mspace{-2.0mu}Q\mspace{-2.0mu}}\mspace{2.0mu}}\\\
=\kappa\,\mspace{2.0mu}\overline{\mspace{-2.0mu}Q\mspace{-2.0mu}}\mspace{2.0mu}_{\bm{{\theta}}}^{+}(u)\,\prod_{m=1}^{M}\frac{u-u_{m}-\mathrm{i}}{u-u_{m}}+\kappa^{-1}\,\mspace{2.0mu}\overline{\mspace{-2.0mu}Q\mspace{-2.0mu}}\mspace{2.0mu}_{\bm{{\theta}}}^{-}(u)\,\prod_{m=1}^{M}\frac{u-u_{m}+\mathrm{i}}{u-u_{m}}\,,$
(2.12)

provided the parameters $u_{m}$, known as Bethe roots, satisfy the Bethe-
ansatz equations

$\kappa^{2}\,\prod_{i=1}^{L}\frac{u_{m}-\theta_{i}+\mathrm{i}/2}{u_{m}-\theta_{i}-\mathrm{i}/2}=\prod_{n(\neq
m)}^{M}\frac{u_{m}-u_{n}+\mathrm{i}}{u_{m}-u_{n}-\mathrm{i}}\;,\qquad
1\leqslant m\leqslant M\,.$ (2.13)

Note that these equations ensure that
$\mspace{2.0mu}\overline{\mspace{-2.0mu}\tau\mspace{-2.0mu}}\mspace{2.0mu}$ is
a polynomial of degree $L$ in $u$, in accordance with the definition (2.5) of
the monodromy matrix, whose coefficients depend on the Bethe roots $u_{m}$ as
well as the inhomogeneities $\theta_{i}$ and twist $\kappa$. Note also that
the Bethe vectors only depend on the twist through the Bethe roots. Further
observe that the Bethe equations are _symmetric_ in the inhomogeneities. For
generic values of the parameters, i.e. $\theta_{i}\neq\theta_{j}+\mathrm{i}$
for all $(i,j)$ and $\kappa\notin\\{0,\pm 1,\infty\\}$, one should take all
the solutions $\\{u_{1},\dots,u_{M}\\}$ of (2.13) for $M\in\\{0,1,\dots,L\\}$
that do not contain any coincident Bethe roots, i.e. $u_{m}\neq u_{n}$ for all
$m\neq n$. It is known that there are precisely $\binom{L}{M}$ distinct
solutions [7, 8, 9], accounting for all $M$-magnons eigenstates of the
transfer matrix via the algebraic Bethe ansatz (2.11).

The Bethe equations admit several reformulations. We will use the standard
shorthand

$f^{\pm}(u)\equiv f(u\pm\mathrm{i}/2)\,,\qquad
f^{\pm\pm}\equiv(f^{\pm})^{\pm}\,,$ (2.14)

and define the polynomial

$\mspace{2.0mu}\overline{\mspace{-2.0mu}Q\mspace{-2.0mu}}\mspace{2.0mu}_{\bm{{\theta}}}(u)\equiv\prod_{i=1}^{L}(u-\theta_{i})\,,$
(2.15)

so that
$\mspace{2.0mu}\overline{\mspace{-2.0mu}Q\mspace{-2.0mu}}\mspace{2.0mu}_{\bm{{\theta}}}^{\pm}$
are the eigenvalues of
$\mspace{2.0mu}\overline{\mspace{-2.0mu}A\mspace{-2.0mu}}\mspace{2.0mu}$ and
$\mspace{2.0mu}\overline{\mspace{-2.0mu}D\mspace{-2.0mu}}\mspace{2.0mu}$ on
$\lvert 0\rangle$, respectively. Further introduce Baxter’s _Q-function_ as
the ‘twisted polynomial’ whose zeroes are the Bethe roots

$\mspace{2.0mu}\overline{\mspace{-2.0mu}Q\mspace{-2.0mu}}\mspace{2.0mu}(u)\equiv\kappa^{\mathrm{i}\mspace{2.0mu}u}\prod_{m=1}^{M}(u-u_{m})\,.$
(2.16)

The Bethe equations (2.13) now take the concise form

$\frac{\mspace{2.0mu}\overline{\mspace{-2.0mu}Q\mspace{-2.0mu}}\mspace{2.0mu}_{\bm{{\theta}}}^{+}}{\mspace{2.0mu}\overline{\mspace{-2.0mu}Q\mspace{-2.0mu}}\mspace{2.0mu}_{\bm{{\theta}}}^{-}}=-\frac{\mspace{2.0mu}\overline{\mspace{-2.0mu}Q\mspace{-2.0mu}}\mspace{2.0mu}^{+\mspace{2.0mu}+}}{\mspace{2.0mu}\overline{\mspace{-2.0mu}Q\mspace{-2.0mu}}\mspace{2.0mu}^{-\mspace{2.0mu}-}}\quad\text{at}\quad
u=u_{m}\,,\qquad 1\leqslant m\leqslant M\,.$ (2.17)

A convenient alternative to (2.17) is their _Wronskian form_. For spin 1/2
this is a functional equation called the _QQ-relation_ :

$\left(\kappa-\kappa^{-1}\right)\mspace{2.0mu}\overline{\mspace{-2.0mu}Q\mspace{-2.0mu}}\mspace{2.0mu}_{\bm{{\theta}}}=\mspace{2.0mu}\overline{\mspace{-2.0mu}Q\mspace{-2.0mu}}\mspace{2.0mu}^{-}\,{\widetilde{\mspace{2.0mu}\overline{\mspace{-2.0mu}Q\mspace{-2.0mu}}\mspace{2.0mu}}}{}^{+}-\mspace{2.0mu}\overline{\mspace{-2.0mu}Q\mspace{-2.0mu}}\mspace{2.0mu}^{+}\,{\widetilde{\mspace{2.0mu}\overline{\mspace{-2.0mu}Q\mspace{-2.0mu}}\mspace{2.0mu}}}{}^{-}\,.$
(2.18)

The left-hand side is the (known) function (2.15),
$\mspace{2.0mu}\overline{\mspace{-2.0mu}Q\mspace{-2.0mu}}\mspace{2.0mu}$ is
given by (2.16), and
$\widetilde{\mspace{2.0mu}\overline{\mspace{-2.0mu}Q\mspace{-2.0mu}}\mspace{2.0mu}}$
is the counterpart of
$\mspace{2.0mu}\overline{\mspace{-2.0mu}Q\mspace{-2.0mu}}\mspace{2.0mu}$
‘beyond the equator’, with opposite twist and of degree $L-M$,

$\widetilde{\mspace{2.0mu}\overline{\mspace{-2.0mu}Q\mspace{-2.0mu}}\mspace{2.0mu}}(u)\equiv\kappa^{-\mathrm{i}\mspace{2.0mu}u}\prod_{n=1}^{L-M}(u-v_{n})\,.$
(2.19)

Demanding that
$\mspace{2.0mu}\overline{\mspace{-2.0mu}Q\mspace{-2.0mu}}\mspace{2.0mu}$ and
$\widetilde{\mspace{2.0mu}}\overline{\mspace{-2.0mu}Q\mspace{-2.0mu}}\mspace{2.0mu}$
be of the form (2.16) and (2.19) and solving the QQ-relation (2.18) yields a
discrete set of solutions for the Bethe roots. One can show from (2.18) that
the roots $u_{m}$ of
$\mspace{2.0mu}\overline{\mspace{-2.0mu}Q\mspace{-2.0mu}}\mspace{2.0mu}$
satisfy the Bethe equations, see Appendix B.1.222 Likewise, the ‘dual Bethe
roots’ $v_{n}$ of
$\widetilde{\mspace{2.0mu}}\overline{\mspace{-2.0mu}Q\mspace{-2.0mu}}\mspace{2.0mu}$
satisfy the Bethe equations ‘beyond equator’, with $M\rightsquigarrow L-M$,
$\kappa\rightsquigarrow\kappa^{-1}$. Note that the right-hand side of (2.18)
depends on the twist as
$\mspace{2.0mu}\overline{\mspace{-2.0mu}Q\mspace{-2.0mu}}\mspace{2.0mu}^{\mp}\,\widetilde{\mspace{2.0mu}\overline{\mspace{-2.0mu}Q\mspace{-2.0mu}}\mspace{2.0mu}}^{\pm}\propto\kappa^{\pm
1}$ due to the exponential prefactors of
$\mspace{2.0mu}\overline{\mspace{-2.0mu}Q\mspace{-2.0mu}}\mspace{2.0mu},\widetilde{\mspace{2.0mu}\overline{\mspace{-2.0mu}Q\mspace{-2.0mu}}\mspace{2.0mu}}$.
(For the periodic case with $\kappa=1$ see Section 2.2.3.)

While for generic values of $\theta_{i},\kappa$ the Bethe equations and QQ-
relation are equivalent, this is not always the case, cf. [47, 48, 49, 50].
The QQ-relation is often more useful as its solutions are in bijection with
the transfer-matrix eigenstates. This _completeness_ of the Bethe ansatz was
proven for almost all values of the inhomogeneities (including all
$\theta_{i}=0$) [7, 8, 9]; see Section 2.3 where we study precisely the cases
where the proofs fail.

####  2.2.1 Inhomogeneous analogues of translation operator

One way to calculate elements of the Bethe algebra is the inhomogeneous
analogue of the standard approach: evaluating the transfer matrix and its
logarithmic derivative(s) at a special point $u_{*}$ where at least one of the
R-matrices in the product (2.5) simplifies. Since
$\mspace{2.0mu}\overline{\mspace{-2.0mu}R\mspace{-2.0mu}}\mspace{2.0mu}(0)=\mathrm{i}\,P$
any $u_{*}=\theta_{j}+\mathrm{i}/2$ will do the job. We compute (as in Section
2.1, ‘h’ is for ‘Heisenberg’)

$\displaystyle G_{j}^{\textsc{h}}(\kappa)$
$\displaystyle\equiv-\mathrm{i}\;\mspace{2.0mu}\overline{\mspace{-2.0mu}t\mspace{-2.0mu}}\mspace{2.0mu}(\theta_{j}+\mathrm{i}/2;\kappa)$
(2.20)
$\displaystyle=-\mathrm{i}\;\mathrm{Tr}_{0}\bigl{[}\kappa^{\sigma^{z}_{0}}\,\mspace{2.0mu}\overline{\mspace{-2.0mu}R\mspace{-2.0mu}}\mspace{2.0mu}_{01}(\theta_{j}-\theta_{1})\cdots\mspace{2.0mu}\overline{\mspace{-2.0mu}R\mspace{-2.0mu}}\mspace{2.0mu}_{0L}(\theta_{j}-\theta_{L})\bigr{]}=\
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$\displaystyle=\mspace{2.0mu}\overline{\mspace{-2.0mu}R\mspace{-2.0mu}}\mspace{2.0mu}_{j,j+1}(\theta_{j}-\theta_{j+1})\cdots\mspace{2.0mu}\overline{\mspace{-2.0mu}R\mspace{-2.0mu}}\mspace{2.0mu}_{jL}(\theta_{j}-\theta_{L})\,\kappa^{\sigma^{z}_{j}}\,\mspace{2.0mu}\overline{\mspace{-2.0mu}R\mspace{-2.0mu}}\mspace{2.0mu}_{j1}(\theta_{j}-\theta_{1})\cdots\mspace{2.0mu}\overline{\mspace{-2.0mu}R\mspace{-2.0mu}}\mspace{2.0mu}_{j,j-1}(\theta_{j}-\theta_{j-1})\,.$

These operators, which are sometimes called ‘scattering operators’ [46, 11,
51, 52], obey the relations

$[G_{i}^{\textsc{h}}(\kappa),G_{j}^{\textsc{h}}(\kappa)]=0\,,\qquad\prod_{j=1}^{L}G_{j}^{\textsc{h}}(\kappa)=\prod_{j=1}^{L}\kappa^{\sigma^{z}_{j}}=\kappa^{2\,S^{z}}\,.$
(2.21)

In the semiclassical limit these operators reduce to the Hamiltonians of the
rational Gaudin model. In the homogeneous limit all (2.20) reduce to the
(twisted) translation operator

$\theta_{i}\to\theta\colon\qquad G_{j}^{\textsc{h}}(\kappa)\to
G^{\textsc{h}}(\kappa)=\kappa^{\sigma^{z}_{1}}\,P_{12}\cdots P_{L-1,L}\,,\quad
G^{\textsc{h}}(\kappa)^{L}=\kappa^{2\,S^{z}}\,.$ (2.22)

The eigenvalues of the $G_{i}(\kappa)$ are conserved, and for
$\theta_{i}\to\theta$ give rise to the notion of (twisted) momentum on the
lattice, see e.g. §B.2 of [49]. For general inhomogeneities, however, there is
no single analogue of the momentum.

One obtains $N$ inhomogeneous deformations of the spin-chain Hamiltonian (2.1)
via the logarithmic derivative $H_{j}^{\textsc{h}}(\kappa)\equiv
G_{j}^{\textsc{h}}(\kappa)^{-1}\,\partial_{u}|_{u=\theta_{j}+\mathrm{i}/2}\;\mspace{2.0mu}\overline{\mspace{-2.0mu}t\mspace{-2.0mu}}\mspace{2.0mu}(u;\kappa)$.
The spin chain given by these commuting operators has long-range interactions
involving multiple spins at a time, with terms that resemble the interactions
of the q-deformed (xxz-type) Haldane–Shastry spin chain [53, 54]. For our
purposes, however, they are not suitable.333 The values
$u_{*}=\theta_{j}+\mathrm{i}/2$ break the symmetry between the
inhomogeneities, and are not compatible with the fermionic condition
$s_{ij}\,P_{ij}=-1$ ($i\neq j$) from Section 3.2.

####  2.2.2 Another family of conserved charges with inhomogeneities

Let us instead expand the transfer matrix at $u=0$ or $u\rightarrow\infty$. We
pick the latter option and expand it as a (formal) power series in
$(-\mathrm{i}\,u)^{-1}$. In order to remove some trivial contributions to the
charges, we find it convenient to expand (cf. the shifts in (2.5))

$\frac{\mspace{2.0mu}\overline{\mspace{-2.0mu}t\mspace{-2.0mu}}\mspace{2.0mu}(u+\mathrm{i}/2;\kappa)}{\mspace{2.0mu}\overline{\mspace{-2.0mu}Q\mspace{-2.0mu}}\mspace{2.0mu}_{\bm{{\theta}}}(u)}=\kappa+\kappa^{-1}+\sum_{n=1}^{\infty}\mspace{2.0mu}\overline{\mspace{-2.0mu}t\mspace{-2.0mu}}\mspace{2.0mu}_{n}(\kappa)\,(-\mathrm{i}\,u)^{-n}\,.$
(2.23)

Here $\kappa+\kappa^{-1}$ is the quantum dimension of the auxiliary space. The
next few coefficients are

$\displaystyle\mspace{2.0mu}\overline{\mspace{-2.0mu}t\mspace{-2.0mu}}\mspace{2.0mu}_{1}(\kappa)$
$\displaystyle=\sum_{i=1}^{L}\kappa^{\sigma_{i}^{z}}=\frac{\kappa+\kappa^{-1}}{2}\,L+(\kappa-\kappa^{-1})\,S^{z}\,,$
(2.24)
$\displaystyle\mspace{2.0mu}\overline{\mspace{-2.0mu}t\mspace{-2.0mu}}\mspace{2.0mu}_{2}(\kappa)$
$\displaystyle=\sum_{i<j}\kappa^{\sigma_{j}^{z}}\,P_{ij}-\mathrm{i}\sum_{i=1}^{L}\theta_{i}\,\kappa^{\sigma_{j}^{z}}\,,$
$\displaystyle\mspace{2.0mu}\overline{\mspace{-2.0mu}t\mspace{-2.0mu}}\mspace{2.0mu}_{3}(\kappa)$
$\displaystyle=\sum_{i<j<k}\\!\\!\kappa^{\sigma_{k}^{z}}\,P_{jk}\,P_{ij}-\mathrm{i}\sum_{i<j}\,(\theta_{i}+\theta_{j})\,\kappa^{\sigma_{j}^{z}}\,P_{ij}-\sum_{i=1}^{L}\theta^{2}_{j}\,\kappa^{\sigma_{j}^{z}}\,.$

Here and below, by $\sum_{i<j}$ we mean the sum over all $1\leqslant
i<j\leqslant L$, and similarly for $\sum_{i<j<k}$.

The eigenvalues of (2.24) are obtained by analogously expanding (2.12):

$\frac{\mspace{2.0mu}\overline{\mspace{-2.0mu}\tau\mspace{-2.0mu}}\mspace{2.0mu}(u+\mathrm{i}/2;\kappa)}{\mspace{2.0mu}\overline{\mspace{-2.0mu}Q\mspace{-2.0mu}}\mspace{2.0mu}_{\bm{{\theta}}}(u)}=\kappa+\kappa^{-1}+\sum_{n=1}^{\infty}\mspace{2.0mu}\overline{\mspace{-2.0mu}\tau\mspace{-2.0mu}}\mspace{2.0mu}_{n}(\kappa)\,(-\mathrm{i}\,u)^{-n}\,.$
(2.25)

The first two coefficients read

$\displaystyle\mspace{2.0mu}\overline{\mspace{-2.0mu}\tau\mspace{-2.0mu}}\mspace{2.0mu}_{1}(\kappa)$
$\displaystyle=\frac{\kappa+\kappa^{-1}}{2}\,L+(\kappa-\kappa^{-1})\left(\frac{L}{2}-M\right)\,,$
(2.26)
$\displaystyle\mspace{2.0mu}\overline{\mspace{-2.0mu}\tau\mspace{-2.0mu}}\mspace{2.0mu}_{2}(\kappa)$
$\displaystyle=\frac{\kappa+\kappa^{-1}}{2}\,\mspace{2.0mu}\overline{\mspace{-2.0mu}\tau\mspace{-2.0mu}}\mspace{2.0mu}_{2}(1)+\frac{\kappa-\kappa^{-1}}{2}\biggl{[}-\mathrm{i}\sum_{i=1}^{L}\theta_{i}+2\,\mathrm{i}\sum_{m=1}^{M}u_{m}+(L-1)\left(\frac{L}{2}-M\right)\biggr{]}\,,$
(2.27)

where in the untwisted case

$\mspace{2.0mu}\overline{\mspace{-2.0mu}\tau\mspace{-2.0mu}}\mspace{2.0mu}_{2}(1)=-\mathrm{i}\sum_{i=1}^{L}\theta_{i}+\left(\frac{L}{2}-M\right)\left(\frac{L}{2}-M+1\right)+\frac{1}{4}\,L(L-4)\,.$
(2.28)

####  2.2.3 Periodic case

Removing the twist by setting $\kappa=1$ enhances the $S^{z}$-symmetry of the
transfer matrix
$\mspace{2.0mu}\overline{\mspace{-2.0mu}t\mspace{-2.0mu}}\mspace{2.0mu}(u;\kappa)$
to global $\mathfrak{sl}_{2}$ symmetry under (2.2). Its eigenspaces become
degenerate, forming spin multiplets (irreducible
$\mathfrak{sl}_{2}$-representations) with the same eigenvalue of
$\mspace{2.0mu}\overline{\mspace{-2.0mu}t\mspace{-2.0mu}}\mspace{2.0mu}(u)$.
As a matter of fact,
$\mspace{2.0mu}\overline{\mspace{-2.0mu}t\mspace{-2.0mu}}\mspace{2.0mu}_{2}(1)$
is essentially the quadratic Casimir

$\vec{S}\cdot\vec{S}=\frac{1}{2}(S^{+}\,S^{-}+S^{-}\,S^{+})+S^{z}\,S^{z}=\sum_{i<j}P_{ij}\
-\frac{1}{4}\,L(L-4)\,,$ (2.29)

where the constant
$-\frac{1}{4}\,L(L-4)=\frac{L}{2}\bigl{(}\frac{L}{2}+1\bigr{)}-\frac{1}{2}\,L(L-1)$
accounts for the difference in eigenvalues on
$\lvert\uparrow\cdots\uparrow\rangle$. The first non-trivial charge is then
$\mspace{2.0mu}\overline{\mspace{-2.0mu}t\mspace{-2.0mu}}\mspace{2.0mu}_{3}(1)$,
whose eigenvalue is

$\mspace{2.0mu}\overline{\mspace{-2.0mu}\tau\mspace{-2.0mu}}\mspace{2.0mu}_{3}(1)=-\sum_{i=1}^{L}\theta^{2}_{i}-\mathrm{i}\,(L-M-1)\sum_{i=1}^{L}\theta_{i}+2\,\mathrm{i}\left(\frac{L}{2}-M+1\right)\sum_{m=1}^{M}u_{m}\\\
+\frac{1}{3}\left(\frac{L}{2}-M-1\right)\,\left[L\,(L-M-1)+M\,(M-1)\right]\,.$
(2.30)

As long as the inhomogeneities are generic (including the homogeneous case)
the Bethe-ansatz construction (2.11) of the eigenstates provides the highest-
weight vector in each multiplet, by solving the Wronskian Bethe equations up
to the equator $M\leqslant\lfloor{L/2}\rfloor$.444 At $\kappa=1$ subtleties
occur that require care. For singular solutions, containing exact strings
(i.e. $u_{m}-u_{n}=\mathrm{i}$), the eigenvector and eigenvalues need to be
regularised. This already happens at $M=2$ when $L$ is even. The descendants
in the multiplet can be obtained from it by acting with the global lowering
operator $S^{-}$. This fits in the framework of the algebraic Bethe ansatz:
the leading term of the expansion of
$\mspace{2.0mu}\overline{\mspace{-2.0mu}B\mspace{-2.0mu}}\mspace{2.0mu}(u)$ in
$u$ occurs at order $L-1$ with coefficient $S^{-}$ up to a constant, so acting
with $S^{-}$ can be viewed as adding a magnon with an infinite Bethe root.
Indeed, when $\kappa=1$ then from any solution to the Bethe equations with
given $M$ one formally obtains another solution to the Bethe equations by
adding $u_{M+1}=\infty$ to the solution. For the Q-functions the absence of a
twist means that
$\mspace{2.0mu}\overline{\mspace{-2.0mu}Q\mspace{-2.0mu}}\mspace{2.0mu}$ and
$\widetilde{\mspace{2.0mu}}\overline{\mspace{-2.0mu}Q\mspace{-2.0mu}}\mspace{2.0mu}$
become (monic) polynomials, the factor $\kappa-\kappa^{-1}$ in the QQ-relation
(2.18) is replaced by $L-2M+1$, and
$\widetilde{\mspace{2.0mu}}\overline{\mspace{-2.0mu}Q\mspace{-2.0mu}}\mspace{2.0mu}$
now has $L-M+1$ roots instead of $L-M$.

####  2.2.4 Extreme twist and Gelfand–Tsetlin basis

For extreme twist $\kappa\to\infty$ the twisted transfer matrix simplifies to
$\mspace{2.0mu}\overline{\mspace{-2.0mu}t\mspace{-2.0mu}}\mspace{2.0mu}(u;\kappa)\sim\kappa\,\mspace{2.0mu}\overline{\mspace{-2.0mu}A\mspace{-2.0mu}}\mspace{2.0mu}(u)$.
Of the global $\mathfrak{sl}_{2}$ only $S^{z}$ remains as a symmetry. Together
with the quantum determinant (which is independent of the twist), the
A-operator generates a subalgebra of the Yangian called the _Gelfand–Tsetlin
subalgebra_ , and the Bethe vectors reduce to the _Gelfand–Tsetlin basis_ for
the spin chain [55, 56].555 If $\kappa\to 0$ then
$\mspace{2.0mu}\overline{\mspace{-2.0mu}t\mspace{-2.0mu}}\mspace{2.0mu}(u;\kappa)\sim\kappa^{-1}\,\mspace{2.0mu}\overline{\mspace{-2.0mu}D\mspace{-2.0mu}}\mspace{2.0mu}(u)$
yields another Gelfand–Tsetlin subalgebra, and Bethe roots
$u_{m}=\theta_{i_{m}}+\mathrm{i}/2$. Physically, the twist is
$\kappa=\mathrm{e}^{\mathrm{i}\,\varphi/2}$, so these limits correspond to
extreme _imaginary_ twists. This limit provides a useful combinatorial model
for the Yangian representation. The QQ-relation (2.18) in this case takes the
factorised form

$\mspace{2.0mu}\overline{\mspace{-2.0mu}Q\mspace{-2.0mu}}\mspace{2.0mu}_{\bm{{\theta}}}(u)=\prod_{m=1}^{M}\left(u-u_{m}-\frac{\mathrm{i}}{2}\right)\prod_{n=1}^{L-M}\left(u-v_{n}+\frac{\mathrm{i}}{2}\right)\,,$
(2.31)

with the left-hand side given by (2.15). Comparing zeroes gives explicit
values for the Bethe roots, sticking to the inhomogeneities as
$u_{m}=\theta_{i_{m}}-\mathrm{i}/2$ for
$I=\\{i_{1},\dots,i_{M}\\}\subset\\{1,\dots,L\\}$ in the $M$-magnon sector,
and the $2^{L}$ spin-chain states are given by all possible choices of such
subsets $I$. The corresponding eigenvalues of the transfer matrix
$\mspace{2.0mu}\overline{\mspace{-2.0mu}A\mspace{-2.0mu}}\mspace{2.0mu}(u)$
factorise as well,

$\alpha_{I}(u)=\prod_{i\in
I}\left(u-\theta_{i}-\frac{\mathrm{i}}{2}\right)\prod_{j\notin
I}\left(u-\theta_{j}+\frac{\mathrm{i}}{2}\right)\,.$ (2.32)

Note that the Bethe roots do not ‘interact’: from a solution
$\\{\theta_{i}-\mathrm{i}/2\\}_{i\in I}$ we get another solution by just
adding any $\theta_{j}-\mathrm{i}/2$ with $j\notin I$. Moreover, the
corresponding eigenvectors are simply related by acting with
$\mspace{2.0mu}\overline{\mspace{-2.0mu}B\mspace{-2.0mu}}\mspace{2.0mu}(\theta_{j}-\mathrm{i}/2)$
(the eigenvalue of the A-operator then changes by exchanging the factor
$u-\theta_{j}+\mathrm{i}/2\to u-\theta_{j}-\mathrm{i}/2$ in (2.32)). This
should be contrasted with the usual situation at finite twist, where adding a
root to a solution affects all other roots (except for infinite roots,
corresponding to descendants, at $\kappa=\pm 1$), and one _first_ has to
construct the ‘off-shell’ Bethe vector
$\mspace{2.0mu}\overline{\mspace{-2.0mu}B\mspace{-2.0mu}}\mspace{2.0mu}(u_{1})\cdots\mspace{2.0mu}\overline{\mspace{-2.0mu}B\mspace{-2.0mu}}\mspace{2.0mu}(u_{M})\,\lvert
0\rangle$ and _then_ plug in a solution to get a transfer-matrix eigenvector
‘on-shell’. Let us also point out that the Bethe states in this case coincide,
up to normalisation, with the vectors of Sklyanin’s separation-of-variables
(SoV) basis for the Heisenberg spin chain with anti-periodic boundary
conditions, see equation (3.10) in [57]. The Yangian Gelfand–Tsetlin basis
also plays a central role in the SoV approach with more general twist and
higher rank [58].

###  2.3 Fusion

Now we turn to the role of inhomogeneities and fusion of Yangian
representations, developed originally by Kulish, Reshetikhin and Sklyanin
[59].

####  2.3.1 General description

While the monodromy matrix and its four operator entries depend on the
ordering of the inhomogeneities $\theta_{i}$, as long as the inhomogeneities
are generic they can be reordered between the sites of the spin chain via a
similarity transformation on the Hilbert space. To see this, note that the
Yang-Baxter equation

$\mspace{2.0mu}\overline{\mspace{-2.0mu}R\mspace{-2.0mu}}\mspace{2.0mu}_{0i}(u-v)\,\mspace{2.0mu}\overline{\mspace{-2.0mu}R\mspace{-2.0mu}}\mspace{2.0mu}_{0j}(u-w)\,\mspace{2.0mu}\overline{\mspace{-2.0mu}R\mspace{-2.0mu}}\mspace{2.0mu}_{ij}(v-w)=\mspace{2.0mu}\overline{\mspace{-2.0mu}R\mspace{-2.0mu}}\mspace{2.0mu}_{ij}(v-w)\,\mspace{2.0mu}\overline{\mspace{-2.0mu}R\mspace{-2.0mu}}\mspace{2.0mu}_{0j}(u-w)\,\mspace{2.0mu}\overline{\mspace{-2.0mu}R\mspace{-2.0mu}}\mspace{2.0mu}_{0i}(u-v)\,,$
(2.33)

implies that the operator

$\check{\mspace{2.0mu}\overline{\mspace{-2.0mu}R\mspace{-2.0mu}}\mspace{2.0mu}}(u)\equiv
P\,\mspace{2.0mu}\overline{\mspace{-2.0mu}R\mspace{-2.0mu}}\mspace{2.0mu}(u)=\mathrm{i}+u\,P\,,\qquad\leavevmode\hbox
to25.26pt{\vbox to33.27pt{\pgfpicture\makeatletter\hbox{\hskip
19.6418pt\lower-22.96228pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{}{{}}{}{{}} {}{} {}{}
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}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-16.30879pt}{3.533pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{\footnotesize{$v$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\\!=\,\check{\mspace{2.0mu}\overline{\mspace{-2.0mu}R\mspace{-2.0mu}}\mspace{2.0mu}}(u-v)\,.$
(2.34)

permutes inhomogeneities. (Note that for this operator the subscripts of the
spaces do not follow the lines, unlike for the R-matrix (2.4). The parameters
follow the lines in both cases.) Indeed, if we swap the two sides of (2.33)
for convenience, multiply by $P_{ij}$ from the left and take $i=j+1$ and
$v=\theta_{j+1}+\mathrm{i}/2,w=\theta_{j}+\mathrm{i}/2$, we obtain the
intertwining relation

$\displaystyle\check{\mspace{2.0mu}\overline{\mspace{-2.0mu}R\mspace{-2.0mu}}\mspace{2.0mu}}_{j+1,j}(\theta_{j+1}-\theta_{j})$
$\displaystyle\mspace{2.0mu}\overline{\mspace{-2.0mu}R\mspace{-2.0mu}}\mspace{2.0mu}_{0j}(u-\theta_{j}-\mathrm{i}/2)\,\mspace{2.0mu}\overline{\mspace{-2.0mu}R\mspace{-2.0mu}}\mspace{2.0mu}_{0,j+1}(u-\theta_{j+1}-\mathrm{i}/2)$
(2.35) $\displaystyle={}$
$\displaystyle\mspace{2.0mu}\overline{\mspace{-2.0mu}R\mspace{-2.0mu}}\mspace{2.0mu}_{0j}(u-\theta_{j+1}-\mathrm{i}/2)\,\mspace{2.0mu}\overline{\mspace{-2.0mu}R\mspace{-2.0mu}}\mspace{2.0mu}_{0,j+1}(u-\theta_{j}-\mathrm{i}/2)\,\check{\mspace{2.0mu}\overline{\mspace{-2.0mu}R\mspace{-2.0mu}}\mspace{2.0mu}}_{j+1,j}(\theta_{j+1}-\theta_{j})\,.$

This extends to the monodromy matrix (2.5),666 By the symmetry
$\mspace{2.0mu}\overline{\mspace{-2.0mu}R\mspace{-2.0mu}}\mspace{2.0mu}_{ji}(u)\equiv
P_{ij}\,\mspace{2.0mu}\overline{\mspace{-2.0mu}R\mspace{-2.0mu}}\mspace{2.0mu}_{ij}(u)\,P_{ij}=\mspace{2.0mu}\overline{\mspace{-2.0mu}R\mspace{-2.0mu}}\mspace{2.0mu}_{ij}(u)$
the order of the subscripts of the R-matrix in (2.36) does not matter. The
decreasing subscripts are due to the order of the physical spaces in (2.5),
cf. the graphical notation.

$\displaystyle\check{\mspace{2.0mu}\overline{\mspace{-2.0mu}R\mspace{-2.0mu}}\mspace{2.0mu}}_{j+1,j}(\theta_{j+1}-\theta_{j})\,\mspace{2.0mu}\overline{\mspace{-2.0mu}T\mspace{-2.0mu}}\mspace{2.0mu}_{0}(u;\dots,\theta_{j},\theta_{j+1},\dots)=\mspace{2.0mu}\overline{\mspace{-2.0mu}T\mspace{-2.0mu}}\mspace{2.0mu}_{0}(u;\dots,\theta_{j+1},\theta_{j},\dots)\,\check{\mspace{2.0mu}\overline{\mspace{-2.0mu}R\mspace{-2.0mu}}\mspace{2.0mu}}_{j+1,j}(\theta_{j+1}-\theta_{j})\,,$
(2.36a) that is, $\displaystyle\leavevmode\hbox to142.79pt{\vbox
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{}{}{}{}{}{}{{}}\pgfsys@moveto{-14.22638pt}{0.0pt}\pgfsys@lineto{-14.22638pt}{-27.99277pt}\pgfsys@stroke\pgfsys@invoke{
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{}{{}}{}
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\theta_{j}$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{}{}{}{}{}{{}}\pgfsys@moveto{-85.35828pt}{0.0pt}\pgfsys@lineto{-85.35828pt}{-27.99277pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{-85.35828pt}{-27.99277pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {}{{}}{}{{}}
{}{}{}{}{}{}{{}}\pgfsys@moveto{-99.58466pt}{0.0pt}\pgfsys@lineto{-99.58466pt}{-27.99277pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{-99.58466pt}{-27.99277pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-102.58243pt}{4.97656pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{\footnotesize{$\theta_{1}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\ .$ (2.36b)

This will be the central identity in what follows. There are two scenarios to
consider: the generic, irreducible case, and (two) cases with an invariant
subspace leading to fusion.

Note that

$\det\,\check{\mspace{2.0mu}\overline{\mspace{-2.0mu}R\mspace{-2.0mu}}\mspace{2.0mu}}(u)=-(u+\mathrm{i})^{3}\,(u-\mathrm{i})\,.$
(2.37)

As long as
$\check{\mspace{2.0mu}\overline{\mspace{-2.0mu}R\mspace{-2.0mu}}\mspace{2.0mu}}_{j+1,j+1}(\theta_{j}-\theta_{j+1})$
is invertible, i.e. $\theta_{j}-\theta_{j+1}\neq\pm\mathrm{i}$, (2.36) gives

$\mspace{2.0mu}\overline{\mspace{-2.0mu}T\mspace{-2.0mu}}\mspace{2.0mu}_{0}(u;\dots,\theta_{j+1},\theta_{j},\dots)=\check{\mspace{2.0mu}\overline{\mspace{-2.0mu}R\mspace{-2.0mu}}\mspace{2.0mu}}_{j+1,j}(\theta_{j+1}-\theta_{j})\,\mspace{2.0mu}\overline{\mspace{-2.0mu}T\mspace{-2.0mu}}\mspace{2.0mu}_{0}(u;\dots,\theta_{j},\theta_{j+1},\dots)\,\check{\mspace{2.0mu}\overline{\mspace{-2.0mu}R\mspace{-2.0mu}}\mspace{2.0mu}}_{j+1,j}(\theta_{j+1}-\theta_{j})^{-1}\,.$
(2.38)

Thus, any two inhomogeneities can be exchanged by a similarity transformation
on ${\cal H}$ consisting of a sequence of conjugations by R-matrices that are
all invertible provided $\theta_{i}-\theta_{j}\neq\pm\mathrm{i}$ for all
$i,j$. This property is inherited by the transfer matrix
$\mspace{2.0mu}\overline{\mspace{-2.0mu}t\mspace{-2.0mu}}\mspace{2.0mu}(u)$,
whose spectrum is independent of the order of the inhomogeneities. This is
reflected in the symmetry in the $\theta_{i}$ of the Bethe equations (2.13)
and Baxter equation (2.12). Here the algebraic Bethe ansatz can be used to
construct the whole spectrum from $\lvert
0\rangle=\lvert\uparrow\cdots\uparrow\rangle$ (Figure 1). This includes the
homogeneous limit where all $\theta_{i}=0$ are equal, yielding the ordinary
Heisenberg xxx spin chain (2.1).

In terms of representation theory, an inhomogeneity $\theta_{i}$ is the
parameter of an evaluation representation of the Yangian, and the spin-chain
Hilbert space is a tensor product of such evaluation representations. For
generic values of the inhomogeneities
($\theta_{i}-\theta_{j}\neq\pm\mathrm{i}$ for all $i,j$) the Hilbert space is
_irreducible_ , and the Yangian representation is called _tame_. In this case,
(2.38) says that
$\check{\mspace{2.0mu}\overline{\mspace{-2.0mu}R\mspace{-2.0mu}}\mspace{2.0mu}}_{j,j+1}(\theta_{j}-\theta_{j+1})$
intertwines the Yangian irreps on $\mathcal{H}$ with inhomogeneities
$(\dots,\theta_{j},\theta_{j+1},\dots)$ and
$(\dots,\theta_{j+1},\theta_{j},\dots)$. The completeness of the Bethe ansatz
for generic inhomogeneities was proven in [7, 8], see also [9]. We remark that
the exchange relation (2.38) also appears as the ‘local condition’ of the
Knizhnik–Zamolodchikov (KZ) system [51, 60].

$\underline{\,\\#\\!\downarrow\,}$$0$$\vdots$$M$$\vdots$$L$$\lvert
0\rangle$$\cdots\vphantom{\mspace{2.0mu}\overline{\mspace{-2.0mu}B\mspace{-2.0mu}}\mspace{2.0mu}}$$\mspace{2.0mu}\overline{\mspace{-2.0mu}B\mspace{-2.0mu}}\mspace{2.0mu}(u_{1})\cdots\mspace{2.0mu}\overline{\mspace{-2.0mu}B\mspace{-2.0mu}}\mspace{2.0mu}(u_{M})$
Figure 1: Structure of $\mathcal{H}$ for generic inhomogeneities
($\theta_{i}-\theta_{j}\neq\pm\mathrm{i}$ for all $i,j$). Each dot represents
an eigenstate, organised into $\mathfrak{sl}_{2}$-irreps $V_{d}$ as shown by
the black vertical lines, with vertical axis recording $M=L/2-S^{z}$. The
dotted lines indicate (the algebraic Bethe ansatz) part of the Yangian action.
The ‘off shell’ Bethe vectors
$\mspace{2.0mu}\overline{\mspace{-2.0mu}B\mspace{-2.0mu}}\mspace{2.0mu}(u_{1})\cdots\mspace{2.0mu}\overline{\mspace{-2.0mu}B\mspace{-2.0mu}}\mspace{2.0mu}(u_{M})\,\lvert
0\rangle$ span the entire $M$-paticle sector, as sketched by the gray
horizontal lines (lighter for the $\mathfrak{sl}_{2}$-descendants, obtained by
acting with
$S^{-}\sim\mspace{2.0mu}\overline{\mspace{-2.0mu}B\mspace{-2.0mu}}\mspace{2.0mu}(\infty)$
in the periodic case). The Bethe equations for $u_{1},\dots,u_{M}$ single out
the points in these subspaces that are eigenvectors of the transfer matrix.

If the R-matrix in (2.36) is not invertible we cannot write (2.38), but
instead there is an _invariant_ subspace. This can be seen as follows. When
the two inhomogeneities differ by $\pm\mathrm{i}$, the R-matrix (2.34) becomes
proportional to an (anti)symmetriser,

$\check{\mspace{2.0mu}\overline{\mspace{-2.0mu}R\mspace{-2.0mu}}\mspace{2.0mu}}(\pm\mathrm{i})=2\,\mathrm{i}\,\Pi^{\pm}\,,\qquad\Pi^{\pm}=(1\pm
P)/2\,.$ (2.39)

Let us write $V_{d}$ for the spin-$s$ irrep of $\mathfrak{sl}_{2}$, which has
dimension $d=2\,s+1$. The operators (2.39) are orthogonal projectors
decomposing two spin-1/2 sites into a triplet and singlet,

$V_{2}\otimes V_{2}\supset\Pi^{+}(V_{2}\otimes V_{2})\cong V_{3}\,,\qquad
V_{2}\otimes V_{2}\supset\Pi^{-}(V_{2}\otimes V_{2})\cong V_{1}\,,$ (2.40)

corresponding to the (Clebsch–Gordan) decomposition

$V_{2}\otimes V_{2}\cong V_{3}\,\oplus\,V_{1}\qquad\text{for
$\mathfrak{sl}_{2}$}\,.$ (2.41)

Focussing on sites $j$ and $j+1$ of the Hilbert space, this decomposition
gives two orthogonal subspaces of $\mathcal{H}$ of dimension $3\times 2^{L-2}$
and $2^{L-2}$, respectively:

$\displaystyle\mathcal{H}=V_{2}^{\otimes L}={}$
$\displaystyle\Pi^{+}_{j,j+1}(\mathcal{H})\;\oplus\;\Pi^{-}_{j,j+1}(\mathcal{H})$
(2.42) $\displaystyle\cong{}$
$\displaystyle\Bigl{(}V_{2}^{\otimes(j-1)}\otimes V_{3}\otimes
V_{2}^{\otimes(L-j-1)}\Bigr{)}\;\oplus\;\Bigl{(}V_{2}^{\otimes(j-1)}\otimes
V_{1}\otimes V_{2}^{\otimes(L-j-1)}\Bigr{)}\qquad\text{for
$\mathfrak{sl}_{2}$}\,.$

While these two subspaces are generically mixed by the monodromy matrix, in
special cases one of them is preserved. To see this we return to the relation
(2.36). When $\theta_{j+1}=\theta_{j}\mp\mathrm{i}$, the right-hand side of
(2.36) annihilates any vector in $\operatorname{ker}\Pi_{j,j+1}^{\mp}$. Yet on
the left-hand side the projector acts after the monodromy matrix, so (2.36)
implies that
$\mspace{2.0mu}\overline{\mspace{-2.0mu}T\mspace{-2.0mu}}\mspace{2.0mu}_{0}(u;\theta_{1},\dots,\theta_{j},\theta_{j}\mp\mathrm{i},\dots,\theta_{L})$
must preserve
$\operatorname{ker}\Pi_{j,j+1}^{\mp}=\Pi^{\pm}_{j,j+1}(\mathcal{H})$. Given a
choice of inhomogeneities with an adjacent pair differing by $\mp\mathrm{i}$
we can thus restrict the monodromy matrix to the subspace
$\Pi^{\pm}_{j,j+1}(\mathcal{H})$ to get a copy of an inhomogeneous spin chain
of length $L-1$, containing $L-2$ sites of spin $1/2$ plus a spin 1 (triplet)
or 0 (singlet) at site $j$, cf. (2.42). In Appendix A we show that the factors
$\mspace{2.0mu}\overline{\mspace{-2.0mu}R\mspace{-2.0mu}}\mspace{2.0mu}_{0j}(u-\theta_{j}-\mathrm{i}/2)\,\mspace{2.0mu}\overline{\mspace{-2.0mu}R\mspace{-2.0mu}}\mspace{2.0mu}_{0,j+1}(u-\theta_{j+1}-\mathrm{i}/2)$
from the monodromy matrix yield a single R-matrix acting on (the spin-1/2
auxiliary space and) site $j$ with spin 1 or 0. This construction is called
_fusion_ [59].

There is no reason for the monodromy matrix
$\mspace{2.0mu}\overline{\mspace{-2.0mu}T\mspace{-2.0mu}}\mspace{2.0mu}_{0}(u;\theta_{1},\dots,\theta_{j},\theta_{j}\mp\mathrm{i},\dots,\theta_{L})$
to preserve the complementary space $\Pi^{\mp}_{j,j+1}(\mathcal{H})$ as well —
and indeed it does not, as we will illustrate shortly. In terms of the
transfer matrix, general complex values of inhomogeneities spoil hermiticity,
so its eigenspaces are not orthogonal.

In terms of representation theory, the preceding says that if
$\theta_{j+1}-\theta_{j}=\mp\mathrm{i}$ then the Yangian representation on
$\mathcal{H}$ with inhomogeneities
$\theta_{1},\dots,\theta_{j},\theta_{j+1},\dots,\theta_{L}$ is _reducible_.
However, since the orthogonal complement is not preserved by the Yangian, this
reducible representation is _indecomposable_.777 Another situation where
reducible but indecomposable representations appear is for
$U_{q}(\mathfrak{sl}_{2})$ with $q$ a root of unity, see e.g. [61]. In the
mathematical literature this situation is often described via non-split short
exact sequences. In brief, the coimage
$\operatorname{coim}(\Pi^{\mp}_{j,j+1})\equiv\mathcal{H}/\operatorname{ker}(\Pi^{\mp}_{j,j+1})$
by definition fits in the short exact sequence
$0\longrightarrow\mathrm{ker}(\Pi^{\mp}_{j,j+1})\longrightarrow\mathcal{H}\longrightarrow\mathrm{coim}(\Pi^{\mp}_{j,j+1})\longrightarrow
0\,,$ (2.43) where _exactness_ means that the image of each map is the kernel
of the next one. As $\mathfrak{sl}_{2}$-modules, this sequence _splits_ by
(2.42). This is closely related to the fact that
$\operatorname{coim}(\Pi^{\mp}_{j,j+1})\cong\operatorname{coker}(\Pi^{\pm}_{j,j+1})\cong\operatorname{ker}(\Pi^{\pm}_{j,j+1})\cong\operatorname{im}(\Pi^{\mp}_{j,j+1})$
as $\mathfrak{sl}_{2}$-modules. For the Yangian, acting by
$\mspace{2.0mu}\overline{\mspace{-2.0mu}T\mspace{-2.0mu}}\mspace{2.0mu}_{0}(u;\theta_{1},\dots,\theta_{j},\theta_{j}\mp\mathrm{i},\dots,\theta_{L})$,
the sequence (2.43) remains exact:
$V_{\mathrm{inv}}=\operatorname{ker}(\Pi^{\mp}_{j,j+1})\subset\mathcal{H}$ is
a Yangian submodule, and the quotient $\operatorname{coim}(\Pi^{\mp}_{j,j+1})$
is also a Yangian module. However, this time (2.43) is not split:
$\operatorname{ker}(\Pi^{\mp}_{j,j+1})\oplus\operatorname{coim}(\Pi^{\mp}_{j,j+1})$
is _not_ isomorphic to $\mathcal{H}$ as a $Y\mathfrak{gl}_{2}$-module. The
above equivalence between $\operatorname{coim}(\Pi^{\mp}_{j,j+1})$ and
$\operatorname{im}(\Pi^{\mp}_{j,j+1})$ does not respect the Yangian, and
$\operatorname{im}(\Pi^{\mp}_{j,j+1})$ is not even a
$Y\mathfrak{gl}_{2}$-module. In Appendix D.1 we illustrate what this means
concretely for $L=2$. Since the dimension of the invariant subspace depends on
the sign, the Yangian representations on $\mathcal{H}$ with inhomogeneities
$(\theta_{1},\dots,\theta_{j},\theta_{j}+\mathrm{i},\dots,\dots,\theta_{L})$
versus $(\theta_{1},\dots,\theta_{j}+i,\theta_{j},\dots,\dots,\theta_{L})$ are
_not_ isomorphic.

So how does all of this affect us in practice when we want to use the Bethe
ansatz for an inhomogeneous Heisenberg xxx spin chain? In the next two
subsections we illustrate this, partially based on numerics for examples with
small length $L$.

####  2.3.2 Bethe ansatz for fusion into singlet

Consider first the case where two sites are fused into a singlet. The
inhomogeneities are generic, except for one pair that we may take to be
adjacent using the intertwiners,

$\theta_{j+1}=\theta_{j}+\mathrm{i}\,.$ (2.44)

Considering the periodic case $\kappa=1$ for simplicity, we have the following
features, as is illustrated in Appendix D for examples with low $L$.

$\underline{\,\\#\\!\downarrow\,}$$0$$1$$\vdots$$L-1$$L$$\lvert
0\rangle$$\,\lvert
0^{\prime}\rangle$$\mspace{2.0mu}\overline{\mspace{-2.0mu}B\mspace{-2.0mu}}\mspace{2.0mu}(u_{0})$$\Pi_{j,j+1}^{+}(\mathcal{H})$$\Pi_{j,j+1}^{-}(\mathcal{H})$
Figure 2: Structure of $\mathcal{H}$ (cf. Figure 1) with
$\theta_{j+1}=\theta_{j}+\mathrm{i}$ and other $\theta_{i}$ generic. As
indicated, the Yangian action may send vectors in
$\Pi_{j,j+1}^{+}(\mathcal{H})$, like $\lvert
0\rangle=\lvert\uparrow\cdots\uparrow\rangle$, to anywhere in $\mathcal{H}$,
but cannot get out of $V_{\mathrm{inv}}=\Pi_{j,j+1}^{-}(\mathcal{H})$. In
particular, $\lvert
0^{\prime}\rangle=\mspace{2.0mu}\overline{\mspace{-2.0mu}B\mspace{-2.0mu}}\mspace{2.0mu}(u_{0})\,\lvert
0\rangle$ with $u_{0}=\theta_{j}+\mathrm{i}/2$ is a good reference state for
the algebraic Bethe ansatz inside $V_{\mathrm{inv}}$.

  * •

As we have just discussed, fixing the singlet state
$\lvert\uparrow\downarrow\rangle-\lvert\downarrow\uparrow\rangle$ at sites $j$
and $j+1$ and allowing any spin configuration at the $L-2$ remaining sites
together form a Yangian-invariant subspace of $\mathcal{H}$ of dimension
$2^{L-2}$,

$V_{\mathrm{inv}}=\Pi_{j,j+1}^{-}(\mathcal{H})\ \cong\
V_{2}^{\otimes(j-1)}\otimes V_{1}\otimes V_{2}^{\otimes(L-j-1)}\,.$ (2.45)

On this invariant subspace,
$\mspace{2.0mu}\overline{\mspace{-2.0mu}T\mspace{-2.0mu}}\mspace{2.0mu}_{0}(u;\theta_{1},\dots,\theta_{j},\theta_{j}+\mathrm{i},\dots,\theta_{L})$
looks just like the monodromy matrix for an inhomogeneous spin chain with
‘effective length’ $L-2$.

  * •

The Bethe ansatz is complete in the sense that _all_ eigenstates of the
transfer matrix are given by the algebraic Bethe ansatz (2.11). It works as
follows.

  * •

Consider the Bethe equations (2.13). Due to (2.44) one numerator and
denominator cancel on the left-hand side, yielding

$\frac{u_{m}-\theta_{j}+\mathrm{i}/2}{u_{m}-\theta_{j}-3\mathrm{i}/2}\,\prod_{i(\neq
j,j+1)}^{L}\frac{u_{m}-\theta_{i}+\mathrm{i}/2}{u_{m}-\theta_{i}-\mathrm{i}/2}=\prod_{n(\neq
m)}^{M}\frac{u_{m}-u_{n}+\mathrm{i}}{u_{m}-u_{n}-\mathrm{i}}\ ,$ (2.46)

Generally, for a spin-$s$ site the product on the left-hand side of the Bethe
equations features shifts $\pm\mathrm{i}\,s$ [62, 63]. Thus the first factor
in (2.46),

$\frac{u_{m}-\theta_{j}+\mathrm{i}/2}{u_{m}-\theta_{j}-3\mathrm{i}/2}=\frac{u_{m}-(\theta_{j}+\mathrm{i}/2)+\mathrm{i}}{u_{m}-(\theta_{j}+\mathrm{i}/2)-\mathrm{i}}\
,$ (2.47)

corresponds to a site with inhomogeneity $\theta_{j}+\mathrm{i}/2$ and spin 1.
That is, the solutions to (2.46) with $M\leqslant L/2$ describe the
eigenvectors _outside_ the invariant subspace (2.45). (Since the transfer
matrix is not Hermitian, their overlap with (2.45) may or may not be zero, and
indeed both happen in practice.)

  * •

The remaining eigenstates, i.e. those inside the invariant subspace (2.45),
can be constructed using the special root

$u_{0}\equiv\theta_{j}+\mathrm{i}/2\,,$ (2.48)

to get from $\lvert 0\rangle$ into the invariant subspace. Indeed, we have
(see Appendix C)

$\lvert
0^{\prime}\rangle\equiv\mspace{2.0mu}\overline{\mspace{-2.0mu}B\mspace{-2.0mu}}\mspace{2.0mu}(u_{0})\,\lvert
0\rangle\,\propto\,\lvert j\rangle\\!\rangle-\lvert
j+1\rangle\\!\rangle=(\sigma^{-}_{j}-\sigma^{-}_{j+1})\,\lvert\uparrow\cdots\uparrow\rangle\,\in\,V_{\mathrm{inv}}\,.$
(2.49)

The result is a suitable vacuum for the algebraic Bethe ansatz inside
$V_{\mathrm{inv}}$ by invariance $\lvert 0^{\prime}\rangle$ has Yangian
highest weight, i.e. is an eigenvector of
$\mspace{2.0mu}\overline{\mspace{-2.0mu}A\mspace{-2.0mu}}\mspace{2.0mu},\mspace{2.0mu}\overline{\mspace{-2.0mu}D\mspace{-2.0mu}}\mspace{2.0mu}$
and killed by
$\mspace{2.0mu}\overline{\mspace{-2.0mu}C\mspace{-2.0mu}}\mspace{2.0mu}$. Now
take any solution of the form $(u_{0},u_{1},\dots,u_{M})$, where the
$u_{1},\dots,u_{M}$ solve the ‘reduced’ Bethe equations with
$M\leqslant(L-2)/2$

$\prod_{i(\neq
j,j+1)}^{L}\frac{u_{m}-\theta_{i}+\mathrm{i}/2}{u_{m}-\theta_{i}-\mathrm{i}/2}=\prod_{n(\neq
m)}^{M}\frac{u_{m}-u_{n}+\mathrm{i}}{u_{m}-u_{n}-\mathrm{i}}$ (2.50)

for a spin chain with $L-2$ sites to construct all vectors in the invariant
subspace as

$\mspace{2.0mu}\overline{\mspace{-2.0mu}B\mspace{-2.0mu}}\mspace{2.0mu}(u_{1})\cdots\mspace{2.0mu}\overline{\mspace{-2.0mu}B\mspace{-2.0mu}}\mspace{2.0mu}(u_{M})\,\lvert
0^{\prime}\rangle=\mspace{2.0mu}\overline{\mspace{-2.0mu}B\mspace{-2.0mu}}\mspace{2.0mu}(u_{0})\,\mspace{2.0mu}\overline{\mspace{-2.0mu}B\mspace{-2.0mu}}\mspace{2.0mu}(u_{1})\cdots\mspace{2.0mu}\overline{\mspace{-2.0mu}B\mspace{-2.0mu}}\mspace{2.0mu}(u_{M})\,\lvert
0\rangle\qquad\text{in}\quad V_{\mathrm{inv}}=\Pi_{j,j+1}^{-}(\mathcal{H})\,.$
(2.51)

The observation 888 We stress that our observation is based on numerics for
small lengths, supplemented with the partial proofs from Appendices B and C.
that all states are of the usual Bethe-ansatz form is interesting because the
proofs of completeness [7, 8, 9] do not apply to this (non-generic) case.

Let us elaborate on the special Bethe root (2.48). One way to understand its
appearance is from the algebraic Bethe ansatz, see Appendix B.2. For our
purposes another proof is more convenient, using the QQ-relation (2.18). Due
to the special values of inhomogeneities, it admits a class of solutions where
$\mspace{2.0mu}\overline{\mspace{-2.0mu}Q\mspace{-2.0mu}}\mspace{2.0mu}$ and
$\widetilde{\mspace{2.0mu}\overline{\mspace{-2.0mu}Q\mspace{-2.0mu}}\mspace{2.0mu}}$
both have $u_{0}$ as a root:

$\mspace{2.0mu}\overline{\mspace{-2.0mu}Q\mspace{-2.0mu}}\mspace{2.0mu}(u)=(u-\theta_{j}-\mathrm{i}/2)\,\mspace{2.0mu}\overline{\mspace{-2.0mu}Q\mspace{-2.0mu}}\mspace{2.0mu}_{\mathrm{red}}(u)\
,\qquad\widetilde{\mspace{2.0mu}}\overline{\mspace{-2.0mu}Q\mspace{-2.0mu}}\mspace{2.0mu}(u)=(u-\theta_{j}-\mathrm{i}/2)\,\widetilde{\mspace{2.0mu}\overline{\mspace{-2.0mu}Q\mspace{-2.0mu}}\mspace{2.0mu}}_{\mathrm{red}}(u)\,,$
(2.52)

where
$\mspace{2.0mu}\overline{\mspace{-2.0mu}Q\mspace{-2.0mu}}\mspace{2.0mu}_{\mathrm{red}}$
and
$\widetilde{\mspace{2.0mu}\overline{\mspace{-2.0mu}Q\mspace{-2.0mu}}\mspace{2.0mu}}_{\mathrm{red}}$
are of the form (2.16) and (2.19) with $\kappa=1$. All three terms of the QQ-
relation now have a factor $(u-\theta_{j})(u-\theta_{j}-\mathrm{i})$. After
removing it, we are left with

$\mspace{2.0mu}\overline{\mspace{-2.0mu}Q\mspace{-2.0mu}}\mspace{2.0mu}_{\mathrm{red}}^{+}\,\widetilde{\mspace{2.0mu}\overline{\mspace{-2.0mu}Q\mspace{-2.0mu}}\mspace{2.0mu}}_{\mathrm{red}}{}^{-}-\mspace{2.0mu}\overline{\mspace{-2.0mu}Q\mspace{-2.0mu}}\mspace{2.0mu}_{\mathrm{red}}^{-}\,\widetilde{\mspace{2.0mu}\overline{\mspace{-2.0mu}Q\mspace{-2.0mu}}\mspace{2.0mu}}{}_{\mathrm{red}}^{+}=\mspace{2.0mu}\overline{\mspace{-2.0mu}Q\mspace{-2.0mu}}\mspace{2.0mu}_{\bm{{\theta}}\\!,\mspace{2.0mu}\mathrm{red}}\,,$
(2.53)

where

$\mspace{2.0mu}\overline{\mspace{-2.0mu}Q\mspace{-2.0mu}}\mspace{2.0mu}_{\bm{{\theta}}\\!,\mspace{2.0mu}\mathrm{red}}(u)=\prod_{i(\neq
j,j+1)}^{L}\\!\\!\\!\\!(u-\theta_{i})\,.$ (2.54)

But this is just the QQ-relation of a spin chain of length $L-2$. Thus
solutions consist of the fixed root $u_{0}=\theta_{j}+\mathrm{i}/2$ together
with $u_{1},\dots,u_{M}$ solving Bethe equations for an effective spin chain
of effective length $L-2$ and inhomogeneities
$\\{\theta_{1},\dots,\theta_{L}\\}\setminus\\{\theta_{j},\theta_{j+1}\\}$.

Notice that the transfer-matrix eigenvalue factorises for states with the
special Bethe root (2.48): plugging (2.52) into (2.12) we find

$\mspace{2.0mu}\overline{\mspace{-2.0mu}\tau\mspace{-2.0mu}}\mspace{2.0mu}(u)=(u-\theta_{j}+\mathrm{i}/2)\,(u-\theta_{j}-3\mathrm{i}/2)\,\frac{\mspace{2.0mu}\overline{\mspace{-2.0mu}Q\mspace{-2.0mu}}\mspace{2.0mu}_{\mathrm{red}}^{++}\,\mspace{2.0mu}\overline{\mspace{-2.0mu}Q\mspace{-2.0mu}}\mspace{2.0mu}_{\bm{{\theta}}\\!,\mspace{2.0mu}\mathrm{red}}^{-}+\mspace{2.0mu}\overline{\mspace{-2.0mu}Q\mspace{-2.0mu}}\mspace{2.0mu}_{\mathrm{red}}^{-\mspace{2.0mu}-}\,\mspace{2.0mu}\overline{\mspace{-2.0mu}Q\mspace{-2.0mu}}\mspace{2.0mu}_{\bm{{\theta}}\\!,\mspace{2.0mu}\mathrm{red}}^{+}}{\mspace{2.0mu}\overline{\mspace{-2.0mu}Q\mspace{-2.0mu}}\mspace{2.0mu}_{\mathrm{red}}}\,.$
(2.55)

The fraction is a polynomial on shell, i.e. on solutions of the Bethe
equations. Thus for states inside $V_{\mathrm{inv}}$ the eigenvalue of the
transfer matrix consists of a simple factor, corresponding to the singlet at
two sites, times a nontrivial part coming from an ‘effective’ spin chain with
$L-2$ sites.

The reason why the fixed root is not visible in the Bethe equations (2.46) is
that it corresponds to the vanishing of the factor
$u_{m}-\theta_{j}-\mathrm{i}/2$ that we have cancelled on the left-hand side
in going from (2.13) to (2.46). In the QQ-relation, however, this root is not
missed and can be treated on equal footing with the other Bethe roots. This
discrepancy between the QQ-relation and solutions of the Bethe equations is
explained by the fact that the usual derivation of the Bethe equations from
the QQ-relation fails in this case, because
$\mspace{2.0mu}\overline{\mspace{-2.0mu}Q\mspace{-2.0mu}}\mspace{2.0mu}$ and
$\widetilde{\mspace{2.0mu}\overline{\mspace{-2.0mu}Q\mspace{-2.0mu}}\mspace{2.0mu}}$
have a common root. For more details see Appendix B, where we also illustrate
a subtlety in the proof of the construction (2.11) of the eigenstates for the
case with the fixed root $u_{0}$ — namely, the ‘unwanted’ terms in the
standard proof of the algebraic Bethe ansatz do not cancel but rather vanish
individually, providing another explanation why the explicit root is absent in
the Bethe equations (see also section 6 in [64] for related discussions).

Finally, for later use we record that the special root (2.48) admits the
symmetric expression

$u_{0}=\frac{\theta_{j}+\theta_{j+1}}{2}\,.$ (2.56)

In Appendix D we illustrate various features of fusion into a singlet in the
examples of spin chains of length $L=2,4$.

####  2.3.3 Bethe ansatz for fusion into triplet

Now we consider the case of fusion into a spin-1 (triplet) representation,
with

$\theta_{j+1}=\theta_{j}-\mathrm{i}\,.$ (2.57)

Here the situation is trickier. The main features are as follows.

  * •

The triplet in combination with the other $L-2$ sites form a Yangian-invariant
subspace of $\mathcal{H}$ of dimension $3\times 2^{L-2}$,

$V_{\mathrm{inv}}=\Pi_{j,j+1}^{+}(\mathcal{H})\ \cong\
V_{2}^{\otimes(j-1)}\otimes V_{3}\otimes V_{2}^{\otimes(L-j-1)}\,.$ (2.58)

  * •

Since the spectrum of the transfer matrix is symmetric in the inhomogeneities
(see Section 2.3.1), the eigenvalues are the same as for the case with fusion
into a singlet.999 To be precise, the sets of all eigenvalues of
$\mspace{2.0mu}\overline{\mspace{-2.0mu}t\mspace{-2.0mu}}\mspace{2.0mu}(u;\kappa;\dots,\theta_{j},\theta_{j}-\mathrm{i},\dots)$
and
$\mspace{2.0mu}\overline{\mspace{-2.0mu}t\mspace{-2.0mu}}\mspace{2.0mu}(u;\kappa;\dots,\theta_{j}-\mathrm{i},\theta_{j},\dots)$
coincide. Of course this is not true for their restrictions to the
corresponding invariant subspaces.

  * •

The eigenstates inside $V_{\mathrm{inv}}$, which contains the reference state
$\lvert 0\rangle=\lvert\uparrow\cdots\uparrow\rangle$, are given by Bethe
ansatz as usual. The Bethe equations (2.13) become

$\frac{u_{m}-\theta_{j}+3\mathrm{i}/2}{u_{m}-\theta_{j}-\mathrm{i}/2}\,\prod_{i(\neq
j,j+1)}^{L}\frac{u_{m}-\theta_{i}+\mathrm{i}/2}{u_{m}-\theta_{i}-\mathrm{i}/2}=\prod_{n(\neq
m)}^{M}\frac{u_{m}-u_{n}+\mathrm{i}}{u_{m}-u_{n}-\mathrm{i}}\,,$ (2.59)

and we consider all $M\leqslant L/2$, yielding states via the algebraic Bethe
ansatz (2.11). The prefactor

$\frac{u_{m}-\theta_{j}+3\mathrm{i}/2}{u_{m}-\theta_{j}-\mathrm{i}/2}=\frac{u_{m}-(\theta_{j}-\mathrm{i}/2)+\mathrm{i}}{u_{m}-(\theta_{j}-\mathrm{i}/2)-\mathrm{i}}$
(2.60)

corresponds to a spin $s=1$ site like before, but with ‘effective
inhomogeneity’ shifted in the other way. For both types of fusion, the
effective inhomogeneity is the average $(\theta_{j}+\theta_{j+1})/2$ of the
original inhomogeneities (cf. Appendix A).

  * •

Crucially, the eigenstates of the transfer matrix outside the invariant space
cannot be generated from $\lvert 0\rangle\in V_{\mathrm{inv}}$ by applying
B-operators. Thus the algebraic Bethe ansatz (2.11) misses $2^{L-2}$
eigenstates: unlike for fusion into a singlet, it is _not_ complete. Although
the special Bethe root (2.56) still appears among the solutions of the QQ-
relations, it is not useful this time: the B-operator at this value now kills
the vacuum (Appendix C),

$\mspace{2.0mu}\overline{\mspace{-2.0mu}B\mspace{-2.0mu}}\mspace{2.0mu}\left(\frac{\theta_{j}+\theta_{j+1}}{2}\right)\lvert
0\rangle=0\,.$ (2.61)

There does not seem to be a simple way to build the eigenstates outside
$V_{\mathrm{inv}}$.101010 One way to describe them is to form the quotient
space $\mathcal{H}/V_{\mathrm{inv}}$. As a vector space it is isomorphic to
$\Pi^{-}_{j,j+1}(\mathcal{H})$. Unlike the latter, the quotient _is_ a well-
defined Yangian representation by invariance of
$V_{\mathrm{inv}}=\Pi^{+}_{j,j+1}(\mathcal{H})$. This quotient corresponds to
a spin chain with a spin-0 site and $L-2$ spin-1/2 sites, just like the
invariant space was for fusion into a singlet. Inside the quotient one can
build all eigenstates via the algebraic Bethe ansatz as usual. These states
are in one-to-one correspondence with the remaining eigenvectors of our
original spin chain in $\mathcal{H}$, yet actually reconstructing them inside
$\mathcal{H}$ is tricky in practice. For each state from the quotient one then
needs to find an appropriate correction by a vector in $V_{\mathrm{inv}}$. It
seems rather nontrivial to do it in a systematic way. Luckily, the
applications of fusion that we will need in in Section 3 only involve vectors
in the invariant subspace, so we will never need to worry about the quotient.

In appendix D.1 we illustrate some features of the fusion into a triplet on
the simple example of a length $L=2$ spin chain.

$\underline{\,\\#\\!\downarrow\,}$$0$$1$$\vdots$$L-1$$L$$\lvert
0\rangle$$\Pi_{j,j+1}^{+}(\mathcal{H})$$\Pi_{j,j+1}^{-}(\mathcal{H})$ Figure
3: Structure of $\mathcal{H}$ (cf. Figure 1) with
$\theta_{j+1}=\theta_{j}-\mathrm{i}$ and other $\theta_{i}$ generic. The
Yangian action preserves $V_{\mathrm{inv}}=\Pi_{j,j+1}^{+}(\mathcal{H})$, in
which the algebraic Bethe ansatz works as usual, but may send vectors in
$\Pi_{j,j+1}^{-}(\mathcal{H})$ anywhere in $\mathcal{H}$.

####  2.3.4 Repeated fusion

So far we have only looked at fusion of two sites, but fusion can happen at
multiple sites. This in particular provides a way to construct an integrable
Heisenberg spin chain with (possibly varying) higher-spin sites using many
spin-1/2 representations and fusion [59, 55, 65]. More generally, fusion may
lead to rather intricate combinations of invariant subspaces. Since we will
only be interested later in some simple cases, and a general discussion would
lead us too far from our goal, we merely illustrate the various possibilities
for fusion when $\theta_{i}=\theta_{j}+\mathrm{i}$ for only two pairs $(i,j)$.
Up to a similarity transformation using (2.38), we may assume that these pairs
are $(1,2)$ and either $(2,3)$ or $(3,4)$ — provided the chain has length
$L\geqslant 4$.

In all scenarios, by the discussion in Section 2.3.1 there are at least two
Yangian-invariant subspaces: $V_{\text{inv}}^{(1)}=\Pi_{12}^{\pm}({\cal H})$
and either $V_{\text{inv}}^{(2)}=\Pi_{23}^{\pm^{\prime}}({\cal H})$ or
$V_{\text{inv}}^{(2)}=\Pi_{34}^{\pm^{\prime}}({\cal H})$ (independent signs
$\pm$ and $\pm^{\prime}$), corresponding to fusion for either pair of sites.
This time, however, these invariant spaces are not necessarily irreducible,
since one could fuse both pairs at the same time. The intersection
$V_{\text{inv}}^{(1)}\cap V_{\text{inv}}^{(2)}$ is either trivial or a
Yangian-irreducible subspace. Here are the various possibilities:

  * •

Independent fusion. When $\theta_{2}=\theta_{1}\mp\mathrm{i}$ and
$\theta_{4}=\theta_{3}\mp^{\prime}\mathrm{i}$ (independent signs), we get
fusion separately at sites $1,2$ and at sites $3,4$. Here
$V_{\text{inv}}^{(i)}$ are reducible but indecomposable, since both contain
the nontrivial subspace

$\Pi_{12}^{\pm}\,\Pi_{34}^{\pm^{\prime}}\,({\cal H})\,=\,\Pi_{12}^{\pm}({\cal
H})\cap\Pi_{34}^{\pm^{\prime}}({\cal H})\,.$ (2.62)

This Yangian-invariant subspace is irreducible and can be viewed as a spin
chain of length $L-2$ containing two sites with spin $1$ or $0$, depending on
the signs.

In particular, $\theta_{2}=\theta_{1}+\mathrm{i}$,
$\theta_{4}=\theta_{3}+\mathrm{i}$ leaves us with (two spin-0 sites and) $L-4$
spin-1/2 sites. This is essentially what will happen for the fermionic spin-
Calogero–Sutherland model in the following sections. (The case
$\theta_{2}=\theta_{1}-\mathrm{i}$, $\theta_{4}=\theta_{3}-\mathrm{i}$ would
instead appear if one were to consider the bosonic spin-Calogero–Sutherland
model, cf. [23].)

  * •

Three-site antisymmetric fusion. The case
$\theta_{3}=\theta_{2}+\mathrm{i}=\theta_{1}+2\,\mathrm{i}$ corresponds to
singlet fusion at sites $1,2$ as well as at sites $2,3$. Both
$V_{\text{inv}}^{(i)}$ are irreducible as their intersection

$\Pi_{12}^{-}({\cal H})\cap\Pi_{23}^{-}({\cal H})=\\{0\\}$ (2.63)

is trivial because there is no completely antisymmetric tensor with three
indices taking only two values.111111 This is a peculiarity of the low-rank
chain we are considering. If we were studying an $\mathfrak{sl}_{r}$ spin
chain for $r>2$, then the intersection (2.63) would be a non-trivial Yangian-
irreducible subspace. It would be the space of a spin chain of length $L-2$ in
which the first site carries the (third fundamental) $\mathfrak{sl}_{r}$-irrep
whose highest-weight vector is a completely antisymmetric 3-tensor. For $r=3$
it is the trivial representation of $\mathfrak{sl}_{3}$. Such a situation will
occur in Section 4.3.2.

  * •

Three-site symmetric fusion. The case
$\theta_{3}=\theta_{2}-\mathrm{i}=\theta_{1}-2\,\mathrm{i}$ corresponds to
triplet fusion at sites $1,2$ as well as $2,3$. Both $V_{\text{inv}}^{(i)}$
are reducible and indecomposable, since the intersection

$\Pi_{12}^{+}({\cal H})\cap\Pi_{23}^{+}({\cal H})\,\cong\,V_{4}\otimes
V_{2}^{\otimes(L-3)}$ (2.64)

is a non-trivial irreducible Yangian submodule. It is the space of a spin
chain of length $L-2$ with one spin-3/2 site. (This scenario would also show
up for the bosonic spin-Calogero–Sutherland model.)

  * •

Three-site mixed fusion. For $\theta_{3}=\theta_{2}\mp\mathrm{i}=\theta_{1}$
both $V_{\text{inv}}^{(i)}$ are irreducible, as

$\Pi_{12}^{\pm}({\cal H})\cap\Pi_{23}^{\mp}({\cal H})=\\{0\\}$ (2.65)

is again trivial, because there is no 3-tensor that is symmetric in its first
(last) two indices and antisymmetric in the last (first) two. Since
$\mathrm{dim}\bigl{(}V_{\text{inv}}^{(1)}\bigr{)}+\mathrm{dim}\bigl{(}V_{\text{inv}}^{(2)}\bigr{)}=\mathrm{dim}(\mathcal{H})$,
now $\mathcal{H}$ is actually completely reducible:
$\mathcal{H}=\Pi_{12}^{\pm}({\cal H})\oplus\Pi_{23}^{\mp}({\cal H})$ as
Yangian modules.

One can continue in this way to get more and more complicated constellations
of invariant subspaces. As the scenarios with three-site fusion illustrate,
one can use this to construct a spin chain with sites of varying spins, see
[59, 55, 65] for more details and examples. For us only the case of
independent fusion will be relevant in what follows. As the preceding
illustrates, for any number of nonoverlapping pairs of neighbouring sites one
can essentially omit the fused singlets to get a spin chain of length $L-2n$
for some $n$. As we will see, the fermionic spin-Calogero–Sutherland model
contains _infinitely_ many of these spin chains.

Equipped with these preliminaries we are ready to move on to our main subject.

##  3 Fermionic Spin-Calogero–Sutherland model

The (trigonometric, quantum) spin-Calogero–Sutherland model is a quantum many-
body system describing particles that carry a spin and move around on a circle
while interacting in pairs. It is a (quantum) integrable mode with
extraordinary properties, including extremely simple eigenvalues that are
highly degenerate because of a Yangian _symmetry_. This should be contrasted
with the Heisenberg spin, whose Yangian does not commute with the spin-chain
Hamiltonian and instead allows one to move between different eigenspaces, as
in the algebraic Bethe ansatz. The algebraic origin of the spin-
Calogero–Sutherland model and its properties lies in a family of commuting
differential-difference operators known as the Dunkl operators.

Conventions. We will clean up our notation a little from now on. Let us
summarise the changes for easy reference. To remove the factors of
$\mathrm{i}$ that were floating around in Section 2 we reparametrise the
spectral parameter as $u=\mathrm{i}\,x$ and henceforth use the (slightly
differently normalised) R-matrix

$R(x)\equiv
1+x^{-1}\,P=1+\mathrm{i}\;u^{-1}\,P=u^{-1}\,\mspace{2.0mu}\overline{\mspace{-2.0mu}R\mspace{-2.0mu}}\mspace{2.0mu}(u)\,.$
(3.1)

We reparametrise the inhomogeneities as
$\theta_{i}=-\mathrm{i}\,\delta_{\\!i}$; in practice, $\delta_{\\!i}$ will
either be a Dunkl operator or its (real) eigenvalue. The mondromy matrix thus
takes the form of a product of
$R_{0i}(x+\delta_{i}-1/2)=\mspace{2.0mu}\overline{\mspace{-2.0mu}R\mspace{-2.0mu}}\mspace{2.0mu}_{0i}(u-\theta_{i}-\mathrm{i}/2)/(u-\theta_{i}-\mathrm{i}/2)$,
cf. (3.20) below. Note that the fusion condition
$\theta_{j+1}-\theta_{j}=\mp\mathrm{i}$ from Section 2.3 now reads
$\delta_{j+1}-\delta_{j}=\pm 1$ (triplet/singlet). The Bethe roots become
$u_{m}=\mathrm{i}\,x_{m}$. We reserve $L$ for the lengths of the ‘effective
spin chains’ to appear in Section 3.3, and start with $N$ particles for our
quantum many-body system.

###  3.1 Dunkl operators and nonsymmetric Jack polynomials

Let $\mathbb{C}[z_{1}^{\pm},\dots,z_{N}^{\pm}]$ be the space of complex
Laurent polynomials in $N$ variables, which will be the coordinates of the
particles on the circle in multiplicative notation,
$z_{j}=\mathrm{e}^{\mathrm{i}x_{j}}$. We denote the operators of coordinate
permutation $z_{i}\leftrightarrow z_{j}$ by $s_{ij}$ for $i\neq j$ in
$\\{1,\dots,N\\}$. For $\beta\in\mathbb{C}\setminus\\{0\\}$, the Dunkl
operators are

$d_{i}=\frac{1}{\beta}\,z_{i}\,\partial_{z_{i}}-\sum_{j=1}^{i-1}\frac{z_{i}}{z_{ji}}\,(1-s_{ij})+\\!\sum_{j=i+1}^{N}\frac{z_{j}}{z_{ij}}\,(1-s_{ij})+\frac{N+1-2i}{2}\,,$
(3.2)

where we use the abbreviation $z_{ij}\equiv z_{i}-z_{j}$. Their key properties
are the (degenerate affine Hecke algebra) relations

$d_{i}\,d_{j}=d_{j}\,d_{i}\,,\qquad
d_{i}\,s_{i,i+1}=s_{i,i+1}\,d_{i+1}+1\,,\qquad
d_{i}\,s_{jk}=s_{jk}\,d_{i}\quad\text{for}\quad i\neq j,k\,.$ (3.3)

Dunkl’s operators have a simple joint spectrum, with simultaneous
eigenfunctions that are called nonsymmetric Jack polynomials
$E_{\bm{{\mu}}}(\bm{{z}})=E_{\bm{{\mu}}}^{(\alpha)}(\bm{{z}})$, with Jack
parameter $\alpha=1/\beta$ that we suppress. These polynomials are indexed by
‘compositions’ $\bm{{\mu}}=(\mu_{1},\dots,\mu_{N})\in\mathbb{Z}^{N}$, and
defined by the conditions

$E_{\bm{{\mu}}}(\bm{{z}})=z_{1}^{\mu_{1}}\cdots
z_{N}^{\mu_{N}}+\text{lower}\,,\qquad\begin{aligned}
d_{i}\,E_{\bm{{\mu}}}(\bm{{z}})&=\delta_{i}(\bm{{\mu}})\,E_{\bm{{\mu}}}(\bm{{z}})\,,\\\
\delta_{i}(\bm{{\mu}})&\equiv\frac{1}{\beta}\,\mu_{i}+\frac{1}{2}\bigl{(}N+1-2\,\sigma^{\bm{{\mu}}}(i)\bigr{)}\,.\end{aligned}$
(3.4)

Here ‘lower’ indicates monomials that are lower in the dominance order on
compositions (see e.g. Section 2.1 of [42]). The eigenvalues
$\delta_{j}(\bm{{\mu}})$ of the Dunkl operators contain the integers

$\sigma^{\bm{{\mu}}}(i)\equiv\;\\#\bigl{\\{}1\leqslant j\leqslant
N\bigm{|}\mu_{j}>\mu_{i}\bigr{\\}}\;+\;\\#\bigl{\\{}1\leqslant j\leqslant
i\bigm{|}\mu_{j}=\mu_{i}\bigr{\\}}\,,$ (3.5)

Note that $\sigma^{\bm{{\mu}}}(i)=i$ if
$\mu_{1}\geqslant\dots\geqslant\mu_{N}$. We will further need the property
that, for all $\bm{{\mu}}\in\mathbb{Z}^{N}$,

$\delta_{i}(\bm{{\mu}})=\delta_{i+1}(\bm{{\mu}})+1\quad\text{if}\quad\mu_{i}=\mu_{i+1}\,.$
(3.6)

The Dunkl operators give rise to the spin-Calogero–Sutherland model through
intermediate operators defined as symmetric combinations of the $d_{j}$. These
in particular include the ‘gauge-transformed (total) momentum operator’

$P^{\prime}\equiv\beta\sum_{i=1}^{N}d_{i}=\sum_{i=1}^{N}\,z_{i}\,\partial_{z_{i}}$
(3.7)

and the ‘gauge-transformed Hamiltonian’

$\displaystyle H^{\prime}\equiv{}$
$\displaystyle\frac{\beta^{2}}{2}\Biggl{(}\,\sum_{i=1}^{N}d_{i}^{\mspace{2.0mu}2}\mspace{7.0mu}-E^{0}\Biggr{)}$
(3.8) $\displaystyle={}$
$\displaystyle\frac{1}{2}\sum_{i=1}^{N}\,\bigl{(}z_{i}\,\partial_{z_{i}}\bigr{)}^{2}+\frac{\beta}{2}\sum_{i<j}\frac{z_{i}+z_{j}}{z_{i}-z_{j}}\,\bigl{(}z_{i}\,\partial_{z_{i}}-z_{j}\,\partial_{z_{j}}\bigr{)}+\beta\sum_{i<j}\frac{z_{i}\,z_{j}}{z_{ij}\,z_{ji}}\,(1-s_{ij})\,,$

where we defined the constant

$E^{0}\equiv\frac{1}{4}\sum_{i=1}^{N}\,(N-2i+1)^{2}=\frac{1}{12}\,N\,(N^{2}-1)\,.$
(3.9)

The reason for the adjective ‘gauge-transformed’ is that they are related to
the true (continuum) momentum operator and Hamiltonian by conjugation: 121212
We avoid the adjective ‘effective’ that is often used instead of ‘gauge
transformed’ to prevent any confusion with our (unrelated) term ‘effective
spin chain’ to appear in Section 3.3.

$\displaystyle P^{\prime}$
$\displaystyle=\Phi_{0}^{-1}\Biggl{(}\sum_{i=1}^{N}z_{i}\,\partial_{z_{i}}\Biggr{)}\,\Phi_{0}\,,\qquad\Phi_{0}(\bm{{z}})\equiv\prod_{i\neq
j}^{N}(1-z_{i}/z_{j})^{\beta/2}\,,$ (3.10) $\displaystyle
H^{\prime}+\frac{\beta^{2}}{2}\,E^{0}$
$\displaystyle=\Phi_{0}^{-1}\Biggl{(}\frac{1}{2}\sum_{i=1}^{N}\,\bigl{(}z_{i}\,\partial_{z_{i}}\bigr{)}^{2}+\sum_{i<j}\frac{z_{i}\,z_{j}}{z_{ij}\,z_{ji}}\,\beta\,(\beta-
s_{ij})\Biggr{)}\,\Phi_{0}\,.$

The eigenvalues of these operators only depend on the partition
${\bm{{\lambda}}}$ obtained by sorting the parts of $\bm{{\mu}}$ into (weakly)
decreasing order. From the definition (3.8) of the gauge-transformed
operators, it is clear that these eigenvalues can only be of the form

$\displaystyle P^{\prime}(\bm{{\mu}})$
$\displaystyle=\beta\sum_{i=1}^{N}\delta_{i}(\bm{{\mu}})=\sum_{i=1}^{N}\lambda_{i}\,,$
(3.11) $\displaystyle E^{\prime}(\bm{{\mu}})$
$\displaystyle=\frac{\beta^{2}}{2}\Biggl{(}\,\sum_{i=1}^{N}\delta_{i}(\bm{{\mu}})^{2}\mspace{7.0mu}-E^{0}\Biggr{)}=\frac{1}{2}\sum_{i=1}^{N}\lambda_{i}^{2}+\frac{\beta}{2}\sum_{i=1}^{N}\,(N-2\mspace{1.0mu}i+1)\,\lambda_{i}\,.$

The integers $\mu_{i}$ can be interpreted as ‘quantum numbers’ for the
(quasi)momenta of the quasiparticles.

So far we have worked at the nonsymmetric level, corresponding to _dis_
tinguishable particles. The spectrum of this model is highly degenerate: the
eigenvalues (3.11) only depend on the partition ${\bm{{\lambda}}}$. By
prescribing the symmetry of the eigenvectors one obtains _in_ distinguishable
particles. For example, for spinless bosons (or fermions) the wave functions
are completely (anti)symmetric, and on the subspaces of totally
(anti)symmetric Laurent polynomials one recovers the scalar bosonic
(fermionic) trigonometric Calogero–Sutherland model,

$\displaystyle P=\Phi_{0}\,P^{\prime}\,\Phi_{0}^{-1}$
$\displaystyle=-\mathrm{i}\sum_{i=1}^{N}\partial_{x_{i}}\,,$ $\displaystyle
z_{i}=\mathrm{e}^{\mathrm{i}\,x_{i}}\,,$ (3.12) $\displaystyle
H=\Phi_{0}\,H^{\prime}\,\Phi_{0}^{-1}$
$\displaystyle=-\frac{1}{2}\sum_{i=1}^{N}\partial_{x_{i}}^{2}+\beta\,(\beta\mp
1)\sum_{i<j}\frac{1}{4\,\sin^{2}[(x_{i}-x_{j})/2]}\,,\quad$ $\displaystyle
s_{ij}=\pm 1\,,$

where we passed to additive coordinates. The eigenfunctions can be obtained
from the nonsymmetric theory too. We return to the gauge-transformed setting,
where we can work with wave functions that are Laurent polynomials. Up to
normalisation, the total (anti)symmetrisation of $E_{\bm{{\mu}}}(\bm{{z}})$
only depends on the partition ${\bm{{\lambda}}}$ corresponding to
$\bm{{\mu}}$, yielding single wave function with momentum and energy (3.11).
For bosons the symmetrisation gives the (symmetric) Jack polynomial
$P_{\bm{{\lambda}}}(\bm{{z}})$ with parameter $\alpha=1/\beta$. For fermions
the result is only nonzero if all parts of ${\bm{{\lambda}}}$ are different,
which is because the non-symmetric Jack polynomials obey

$s_{i,i+1}\,E_{\bm{{\lambda}}}=E_{\bm{{\lambda}}}\quad\text{if}\quad\lambda_{i}=\lambda_{i+1}\,.$
(3.13)

If ${\bm{{\lambda}}}$ is a _strict_ partition, i.e. if
$\lambda_{1}>\dots>\lambda_{N}$, then it is of the form
${\bm{{\lambda}}}=\bm{{\nu}}+\bm{{\delta}}_{N}$ for some (not necessarily
strict) partition $\bm{{\nu}}$, where $\bm{{\delta}}_{N}\equiv(N-1,\dots,1,0)$
is the staircase partition. For strict partitions the result of
antisymmetrisation is a Vandermonde polynomial times a Jack polynomial with
shifted parameter,

$\mathrm{Vand}(z_{1},\dots,z_{N})\,P^{\prime}_{\bm{{\nu}}}(\bm{{z}})\,,\qquad\mathrm{Vand}(z_{1},\dots,z_{N})\equiv\\!\prod_{1\leqslant
i<j\leqslant N}\\!\\!\\!\\!(z_{i}-z_{j})\,,\quad P^{\prime}_{\bm{{\nu}}}\equiv
P_{\bm{{\nu}}}\big{|}_{\beta\mapsto\beta+1}\,.$ (3.14)

In the spinless case, then, each energy in (3.11) occurs in the bosonic case,
while in the fermionic case only strict partitions are allowed. See e.g. §2.2
in [42] and references therein for more. We will instead be interested in a
generalisation with fermions that each carry a spin as well as a coordinate.

###  3.2 Hamiltonian and monodromy matrix

The Hilbert space for $N$ spin-1/2 fermions moving on a circle is

$\mathcal{F}=\Bigl{\\{}\lvert\Psi\rangle\in\bigl{(}\mathbb{C}^{2}\bigl{)}^{\otimes
N}\otimes\,\mathbb{C}\bigl{[}z_{1}^{\pm},\dots,z_{N}^{\pm}\bigr{]}\Bigm{|}P_{ij}\,s_{ij}\,\lvert\Psi\rangle=-\lvert\Psi\rangle\Bigr{\\}}\,.$
(3.15)

It consists of the vectors that are completely antisymmetric in spin and
coordinates, and coincides with the image of the projector

$\displaystyle\Pi_{-}^{\mathrm{tot}}=\frac{1}{N!}\sum_{\sigma\in
S_{N}}\\!\mathrm{sgn}(\sigma)\,P_{\sigma}\,s_{\sigma}\,,\qquad\bigl{(}\Pi_{-}^{\mathrm{tot}}\bigr{)}^{2}=\Pi_{-}^{\mathrm{tot}}\,.$
(3.16)

On the fermionic space space, the gauge-transformed Hamiltonian (3.8) takes
the form

$\widetilde{H}^{\prime}=\frac{1}{2}\sum_{i=1}^{N}\,\bigl{(}z_{i}\,\partial_{z_{i}}\bigr{)}^{2}+\frac{\beta}{2}\sum_{i<j}\frac{z_{i}+z_{j}}{z_{i}-z_{j}}\,\bigl{(}z_{i}\,\partial_{z_{i}}-z_{j}\,\partial_{z_{j}}\bigr{)}+\beta\sum_{i<j}\frac{z_{i}\,z_{j}}{z_{ij}\,z_{ji}}\,(1+P_{ij})\,.$
(3.17)

It is related by conjugation as in (3.12) to the fermionic spin-
Calogero–Sutherland Hamiltonian (1.1). The momentum operator (3.7), on the
other hand, is the same as in the spinless case. Let us emphasise that this
operator only acts on the coordinates $z_{j}$ of the particles, not on their
spins. The cyclic translation $P_{12}\cdots P_{N-1,N}$ of the spins does not
even act on the fermionic space, so the spin-chain notion of (crystal)
momentum is irrelevant for the spin-Calogero–Sutherland model.

The spectrum of (1.1) is given by (3.11) with the restriction that
${\bm{{\lambda}}}$ be a partition with multiplicities $\leqslant 2$. We denote
the set of these allowed partitions by 131313 Such partitions are called
‘3-regular’ in representation theory – not to be confused with the different
but related meaning of that term in combinatorics, cf.
https://mathoverflow.net/q/438228.

$\mathcal{P}=\\{{\bm{{\lambda}}}\in\mathbb{Z}^{N}\mid\lambda_{1}\geqslant\dots\geqslant\lambda_{N}\,,\
\lambda_{i}>\lambda_{i+2}\\}\,.$ (3.18)

Indeed, by the property (3.13) of nonsymmetric Jack polynomials, repetitions
in ${\bm{{\lambda}}}$ require antisymmetry in the corresponding spins because
of the fermionic condition (3.15). For our case of spin 1/2 (i.e.
$\mathfrak{sl}_{2}$), this means that the multiplicities are at most 2. (For
spinless fermions the multiplicities are at most $1$.)

The fermionic space comes equipped with (an action of the Yangian of
$\mathfrak{gl}_{2}$ given by) the monodromy matrix [40]

$\displaystyle T_{0}(x)$
$\displaystyle=R_{01}\bigl{(}x+d_{1}-\tfrac{1}{2}\bigr{)}\cdots
R_{0N}\bigl{(}x+d_{N}-\tfrac{1}{2}\bigr{)}$ (3.19)
$\displaystyle=\left(1+\frac{P_{01}}{x+d_{1}-\frac{1}{2}}\right)\cdots\left(1+\frac{P_{0N}}{x+d_{N}-\frac{1}{2}}\right)\,.$

Here we use the R-matrix (3.1), and Dunkl operators play the role of
inhomogeneities: in terms of the conventions of Section 2 one has

$T_{0}(x)=\biggl{.}\frac{\mspace{2.0mu}\overline{\mspace{-2.0mu}T\mspace{-2.0mu}}\mspace{2.0mu}_{\\!0}(\mathrm{i}\,x)}{\mspace{2.0mu}\overline{\mspace{-2.0mu}Q\mspace{-2.0mu}}\mspace{2.0mu}_{\bm{{\theta}}}\left(\mathrm{i}\,x-\frac{\mathrm{i}}{2}\right)}\biggr{|}_{\bm{{\theta}}=\,-\mathrm{i}\,\bm{{d}}}\,.$
(3.20)

In [40] the term ‘quantised inhomogeneities’ was used to emphasise that the
inhomogeneities are now nontrivial operators (on polynomials). The relations
(3.3) guarantee it preserves the fermionic space (see e.g. §C of [42]) and
obeys the RTT relations. The proper representation-theoretic meaning of (3.19)
stems from affine Schur–Weyl duality [21].

There are several ways to make sense of the Dunkl operators in the
denominators in (3.19): (i) expanding as a formal power series in $x^{-1}$,
(ii) using the nonsymmetric Jack basis for the polynomial factor of the
fermionic space to replace the Dunkl operators by their eigenvalues, (iii)
removing the denominator $\prod_{j}(x+d_{j}-1/2)$, which is central and acts
in a simple way. The third point is related to the following important
property of the monodromy matrix.

The spin-Calogero–Sutherland model commutes with the Yangian action given by
(3.19). Indeed, the hierarchy of spin-Calogero–Sutherland Hamiltonians [66,
34, 41] are generated by the quantum determinant [40, 41]

$\displaystyle\Delta(x)$
$\displaystyle=\mathrm{qdet}_{0}T_{0}(x)=\prod_{i=1}^{N}\frac{x+d_{i}+\frac{1}{2}}{x+d_{i}-\frac{1}{2}}$
(3.21)
$\displaystyle=1+N\,x^{-1}+\biggl{(}\frac{N^{2}}{2}-\frac{P^{\prime}}{\beta}\biggr{)}\,x^{-2}+\biggl{(}\frac{N^{3}}{4}-\frac{N\,P^{\prime}}{\beta}+\frac{2\,H^{\prime}}{\beta^{2}}\biggr{)}\,x^{-3}+O\bigl{(}x^{-4}\bigr{)}\,,$

and the quantum determinant of the Yangian generates its centre.

Let us finally mention that for $\beta>0$ one can define a scalar product on
$\mathbb{C}[z_{1}^{\pm},\dots,z_{N}^{\pm}]$ for which the Dunkl operators are
Hermitian, see Proposition 3.8 in [67] and §2 of [68]. The natural extension
of this scalar product to the fermionic space $\mathcal{F}$ defined in (3.15)
is such that the Yangian algebra is stable under Hermitian conjugation.141414
More precisely, if we expand the A-, …, D-operators in (3.19) as
$A(x)=1+\sum_{n=1}^{+\infty}A_{n}\,x^{-n}$,
$B(x)=\sum_{n=1}^{+\infty}B_{n}\,x^{-n}$, etc., then the coefficients obey
$A_{n}^{\dagger}=A_{n}$, $D_{n}^{\dagger}=D_{n}$ and $B_{n}^{\dagger}=C_{n}$
[23]. This implies in particular that the Yangian representation on the
fermionic space is completely reducible. The decomposition of $\mathcal{F}$
into irreducible components is our next topic.

###  3.3 Effective spin chains

In [23] it was shown that the Hilbert space of the fermionic spin-
Calogero–Sutherland model decomposes into a sum of irreducible representations
of the Yangian:

$\mathcal{F}=\bigoplus_{{\bm{{\lambda}}}\in\mathcal{P}}\mathcal{F}_{\bm{{\lambda}}}\,.$
(3.22)

The summands are also eigenspaces for the spin-Calogero–Sutherland model. The
momentum operator (3.7) and gauge-transformed Hamiltonian (3.17) still have
eigenvalues (3.11).

The eigenspace $\mathcal{F}_{\bm{{\lambda}}}$ is the image by the projector
(3.16) of the subspace which, in the polynomial factor, is spanned by all
nonsymmetric Jack polynomials $E_{\bm{{\mu}}}(\bm{{z}})$ with composition
$\bm{{\mu}}\in\mathbb{Z}^{N}$ differing from the partition ${\bm{{\lambda}}}$
by reordering:

$\mathcal{F}_{\bm{{\lambda}}}=\Pi^{-}_{\mathrm{tot}}\Biggl{(}\
\bigoplus_{{\bm{{\mu}}}\in
S_{N}\cdot{\bm{{\lambda}}}}\\!\\!E_{\bm{{\mu}}}(\bm{{z}})\otimes(\mathbb{C}^{2})^{\otimes
N}\Biggr{)}\subset\mathcal{F}\,.$ (3.23)

Following [23], this subspace can be equivalently viewed as an ‘effective spin
chain’ of some length $L_{\bm{{\lambda}}}\leqslant N$ with particular (scalar)
inhomogeneities. This goes as follows.

Let us set up some notation. For each allowed partition
${\bm{{\lambda}}}\in\mathcal{P}$, we define sets $I_{\bm{{\lambda}}}$ and
$J_{\bm{{\lambda}}}$ that enumerate its unique and repeated parts,
respectively:

$I_{\bm{{\lambda}}}\equiv\bigl{\\{}1\leqslant i\leqslant
N\bigm{|}\lambda_{i-1}>\lambda_{i}>\lambda_{i+1}\bigl{\\}}\,,\qquad
J_{\bm{{\lambda}}}\equiv\bigl{\\{}1\leqslant
j<N\bigm{|}\lambda_{j}=\lambda_{j+1}\bigr{\\}}\,,$ (3.24)

with the convention that $\lambda_{0}\equiv+\infty$ and
$\lambda_{N+1}\equiv-\infty$. If ${\bm{{\lambda}}}=(7,6,6,2,2,-5,-6,-6,-8)$,
for instance, then $I_{\bm{{\lambda}}}=\\{1,6,9\\}$ and
$J_{\bm{{\lambda}}}=\\{2,4,7\\}$. The set $J_{\bm{{\lambda}}}$ is called a
_motif_. $I_{\bm{{\lambda}}}$ will label the sites of the effective chain,
while $J_{\bm{{\lambda}}}$ will record pairs of sites of the original chain
that are fused into singlets. In particular, the effective length will be

$L_{\bm{{\lambda}}}\equiv\\#I_{\bm{{\lambda}}}=N-2\,\\#J_{\bm{{\lambda}}}\,.$
(3.25)

We start with the Yangian highest-weight vector in
$\mathcal{F}_{\bm{{\lambda}}}$. It contains
$M_{\bm{{\lambda}}}\equiv\\#J_{\bm{{\lambda}}}$ magnons, cf. (3.13), and can
be written as

$\lvert