# Octahedral tilting in Prussian blue analogues

Hanna L. B. Boström*1,2 & William R. Brant1

###### Abstract

Octahedral tilting is key to the structure and functionality of perovskites.
Here we show how these distortions manifest in the related Prussian blue
analogues (PBAs): cyanide versions of double perovskites with formula
AxM[M′(CN)6]${}_{1-y}\Box_{y}\cdot n$H2O (A = alkali metal, M and M′ =
transition metals, $\Box$ = vacancy/defect). Tilts are favoured by high values
of $x$ if A = Na or K, whereas the transition metals play a less important
role. External hydrostatic pressure can induce tilt transitions nearly
irrespective of the stoichiometry, whereas thermal transitions are only
reported for $x>1$. Interstitial water can alter the transitions induced by a
different stimulus, but (de)hydration per se does not lead to tilts.
Implications for rational design of critical functionality—including improper
ferroelectricity and electrochemical performance—are discussed. The results
are important for a fundamental understanding of phase transitions as well as
for the development of functional materials based on PBAs.

Department of Chemistry, Ångström Laboratory, Uppsala University, P.O. Box
538, SE-75121 Uppsala, Sweden

Max Planck Institute of Solid State Research, Heisenbergstraße 1, DE-70569
Stuttgart, Germany

Prussian blue analogues (PBAs) are a versatile class of materials with a wide
range of potential applications. The functional diversity partially arises
from the large scope for compositional variation: the general formula is
AxM[M′(CN)6]${}_{1-y}\Box_{y}\cdot n$H2O, where A is an alkali metal, M and M′
are normally transition metals and $\Box$ denotes a vacancy. Porous, defective
systems of formula MII[M′III(CN)6]0.67 can be used for catalysis and gas
adsorption,[1] whereas defect-free, alkali-containing PBAs, e.g.
A2M′IIM′II(CN)6, are suitable electrode materials[2] or can show interesting
magnetic properties.[3] The stoichiometry and associated functionality is
stipulated by the charges of the transition metals, where lower charges
increase the scope for inclusion of vacancies and/or A-site cations [Fig. 1].
The crystal structure of PBAs is reminiscent of that of double perovskites,
with the parent structure (aristotype) adopting the space group $Fm\bar{3}m$,
although ordered A-site cations or vacancies may reduce the symmetry to
$F\bar{4}3m$ or $Pm\bar{3}m$ [Fig. 1].

Figure 1: The crystal structure of Prussian blue analogues. If the charges of
the transition metals ($z_{M}+z_{M{\prime}}$) sum to 6, the ordered MM′(CN)6
is formed. For lower charges, vacancies or A-site cations are included to give
the formulae M[M′(CN)6]y<1 and Ax>0M[M′(CN)6], respectively. M is shown in
dark grey, M′ in light grey and A in purple.

A crucial feature of ABO3 perovskite physics is octahedral tilting, which
involves in-phase or out-of-phase rotation of BO6 octahedra. In Glazer
notation, this is denoted by superscripted ‘+’ and ‘$-$’ and lowercase letters
indicate relative magnitudes around the three pseudo-cubic axes.[4] The
importance of tilts arises from their link to both the composition and to the
properties. For example, the relative ionic radii of the constituent metals,
as expressed by the tolerance factor,[5] dictate whether tilts are favoured,
which facilitates some degree of control. Thus, compositional modification can
be used to tune the tilt angle, which in turn affects the band gap and the
Curie temperature.[6] Moreover, the link between tilting and structure is well
established $via$ group-theoretical methods,[7, 8] and knowledge of the tilts
typically allows for the prediction of the space group symmetry. This was
exploited in so-called tilt engineering of a multiferroic perovskite
derivative, where careful activation of appropriate tilts led to polar
symmetry with coexisting weak magnetism.[9] Octahedral tilts therefore present
an attractive route for the design of technologically relevant materials.

Given the structural similarities between perovskites and PBAs, it is natural
that octahedral tilting also occurs in PBAs with functional implications. To
illustrate, the presence of octahedral tilting in Na2MnMn(CN)6 nearly triples
the magnetic transition temperature compared to the untilted Cs2MnMn(CN)6.[10]
A less favourable situation is the drastic volume reduction of 17% in
Mn[Mn(CN)6]0.93 upon Na intercalation, which is driven by a tilt transition
from cubic symmetry (i.e. untilted) to $a^{-}a^{-}a^{-}$.[11] This has a
detrimental effect on the electrode stability and compromises the application
of PBAs as energy storage materials. As a consequence, understanding and
ultimately controlling the presence or absence of octahedral tilts in PBAs is
key to their commercial development as functional materials.

In spite of its importance, octahedral tilting in PBAs is poorly understood
and there is considerable confusion regarding the structural characterisation
of these materials. To illustrate, cubic (no tilts) and monoclinic symmetries
($a^{-}a^{-}b^{+}$ tilts) were postulated for the closely related
Na1.52Ni[Fe(CN)6]0.88 and Na1.48Ni[Fe(CN)6]0.89, respectively,[12, 13] while a
third study claims NaxNiFe(CN)6]0.83 to be rhombohedral ($a^{-}a^{-}a^{-}$
tilts) at all $x$-values.[14] A systematic understanding of the factors
underlying octahedral tilting in PBAs would be beneficial to a wide range of
fields and pave the way for tilt engineering approaches. Certain principles
developed for oxide perovskites are likely transferable to PBAs, which
provides a starting point for such investigations. However, PBAs show features
that are rare or unknown in perovskites, such as vacancies and porosity, and
therefore their behaviour cannot be expected to exactly mirror that of
perovskites.

Here, we perform a metastudy to investigate the octahedral tilting in Prussian
blue analogues. Our manuscript identifies the possible driving forces for
tilting—stoichiometry, composition, hydration, and non-ambient conditions—and
discusses their effect on the propensity for and nature of tilting. The
results are compared and contrasted with the tilt behaviour of double oxide
perovskites throughout. Our results show that tilts are favoured by high
A-site cation concentrations if A = Na or K, whereas the identity of the
transition metals is considerably less relevant. The presence of interstitial
water can direct the tilt transitions induced by a different stimulus, but
(de)hydration does not appear to drive tilts in its own right. External
hydrostatic pressure can induce tilt transitions almost regardless of the
stoichiometry, whereas thermal transitions only occur for high A-site cation
concentrations. We conclude by discussing strategies for tilt engineering,
using energy storage, magnetism, and improper ferroelectricity as example
functionalities.

##  1 Results

###  1.1 Stoichiometry

As mentioned, PBAs have a variable stoichiometry described by the general
formula AxM[M′(CN)6]${}_{1-y}\Box_{y}$ ($\Box$ = vacancy). As A-site cations
($x$) and vacancies ($y$) often coexist, a continuum of intermediate
stoichiometries is accessible. Figure 2 depicts the compositional space in
terms of $x$ and $y$, populated by structures reported in literature.

Figure 2: Tilts in PBAs as a function of stoichiometry. Top panel: PBAs with
composition AxM[M′(CN)6]${}_{1-y}\Box_{y}$ ($\Box$ = vacancy) for various A,
where filled (empty) symbols denote tilted (untilted) structures. The diagonal
lines show constant total transition metal charge (+5 and +4). Representative
structures are shown for key stoichiometries, with the M and M′ ions in black
and grey, respectively, and the A-site cation in orange. Cyanide ions are
omitted for clarity. Lower panel: The above phase diagram plotted with a
separate figure for each A-site cation.

Lines corresponding to compositions
AxMII[M′III(CN)6]${}_{\frac{2+x}{3}}\Box_{\frac{1-x}{3}}$ ($0\leq x\leq 1$)
and AxMII[M′II(CN)6]${}_{\frac{2+x}{4}}\Box_{\frac{2-x}{4}}$ ($0\leq x\leq 2$)
are indicated and many structures fall close to these lines. Entries in other
areas of the phase diagram represent PBAs with mixed oxidation states of M
and/or M′, e.g.
K0.4Cu[Fe${}^{\mathrm{II}}_{0.6}$Fe${}^{\mathrm{III}}_{0.4}$(CN)6]2/3.[15]
This phase diagram highlights two common features of PBAs that rarely occur in
double oxide perovskites: partially occupied A-sites, e.g. AMM′(CN)6, and
M′(CN)6 vacancies, e.g. AM[M′(CN)6]y<1.

Fig. 2 clearly highlights that a minimum concentration of alkali metals is
required for tilting. As uncertainties from elemental analyses are rarely
reported, no error bars are included in Fig. 2, and thus it is difficult to
draw compelling quantitative conclusion. Still, nearly all tilted PBAs show
$x>1$, suggesting that this is a necessary condition for tilts. However, it is
not a sufficient condition, as several reportedly cubic PBAs with higher
A-site concentrations exist.[16, 17] While such discrepancies may partially be
attributed to suboptimal data quality of in-house diffractometers, differences
in local structure as a result of deviating synthesis conditions may also play
a role. In addition, the relatively empty region at $x=1.0$–$1.5$ and
$y=0$–0.1 in Fig. 2 suggests a miscibility gap, which agrees with the phase
separation sometimes reported in PBAs with large values of $x$.[11, 18] A
detailed understanding of the critical threshold concentration and potential
miscibility gap is crucial for the application of PBAs in secondary ion
batteries, as the tilt transition changes the transport properties and causes
a large strain.[13, 11] Consequently, further studies are needed, though it is
clear that the A-site concentration is central to the tilts.

In principle, vacancies could affect the propensity for tilting as they
increase the flexibility of the framework. However, they will also reduce the
correlation length of tilting and thereby prevent global phase transitions.
The disorder of the vacancies is complex and depends both on the metal ions
and the synthesis conditions,[19] which complicates the study of the
functional implications of missing M′(CN)6 clusters. Moreover, as vacancies
and A-site cations are interdependent, their respective roles are difficult to
decouple. However, the absence of vacancies is not sufficient to induce
tilting as PBAs with formula MM′(CN)6 are invariably undistorted at ambient
conditions.[20] Likewise, certain PBAs exhibit tilts despite a high vacancy
concentration, like Na1.10Mn[Fe(CN)6]0.77.[21] Thus, while the influence of
vacancies on the propensity of tilting is still unclear, they do not appear to
play a major role.

Finally, a few general comments can be made about Fig. 2. First, a
considerable number of systems comprise transition metals ions with charges
summing to less than 4 (area above the right diagonal line). This indicates a
partially univalent transition metal or possibly the presence of hydroxide
ions in the framework.[22, 23] Second, several structures feature an apparent
vacancy content above 33% (top left corner). However, certain transition metal
ions have been postulated to occupy both the A- and M-site, such as
Cu[Fe(CN)6]0.5,[24] which means that the actual vacancy concentration of such
structures is less than 50%. While the possibility of divalent A-site cations
and their relation to the maximum vacancy concentration is interesting and
merits further study, this will not be discussed further in this manuscript.

###  1.2 A-site composition

The tendency for tilting not only depends on the concentration of A-site
cations, but also on the A-site ionic radius [Fig. 2]. PBAs based on Na or K
feature the largest proportions of tilted structures, whereas systems with A =
Li or Rb rarely tilt and, to the best of our knowledge, distorted Cs-
containing PBAs do not exist at ambient conditions. While some degree of
observational bias cannot be excluded—Na- and K-containing systems are the
most intensely studied—intermediate alkali metal radii clearly favour tilting.
This trend partially agrees perovskites, where the driving force for tilts is
considered to be the underbonding of small A-site cations, as quantified by
the tolerance factor.[5, 25, 26] Although the tolerance factor has been
extended to molecular perovskites,[27] it is less applicable as it does not
account for defects. Yet, the same argument underlies the prevalence of
tilting in PBAs with A = Na/K relative to A = Rb/Cs, as smaller cations are
more prone to underbonding. It is also consistent with the inverse
relationship between tilt angles and A-site cation size in A2MnMn(CN)6.[28]
However, it does not rationalise the lack of tilting in Li-based PBAs, but Li
is believed to occupy the window sites of the PBA subcube, as opposed to the
centre.[29] Thereby, the bonding environment can presumably be optimised even
without tilts. Altogether, the radius of the A-site cation is arguably the
most critical factor to consider when evaluating the propensity for tilting of
a given structure.

In general, the size of the A-site cation also stipulates the type of tilt
[Fig. 3].

Figure 3: Tilt patterns in Prussian blue analogues. The distribution of tilt
patterns across PBAs with various A-site cations, where ‘A’ denotes the
absence of A-site cations. The space group refers to the symmetry that the
tilt gives rise to in the absence of further distortions. Tilts that (so far)
are only known under pressure are indicated.

A-site deficient PBAs under compression preferentially adopt $a^{-}a^{-}a^{-}$
tilts giving $R\bar{3}$ symmetry,[30] which mirrors the behaviour of the
fluoride analogues MnNbF6 and CaTiF6.[31, 32] This tilt pattern is also
preferred by dehydrated Na-containing PBAs[10, 33, 34, 35] and allows the Na
ion to displace along the cubic [111] direction towards one of the transition
metals. This displacement is sometimes considered the driving force for
cubic–rhombohedral phase transitions, which would give rhombohedral
$R\bar{3}m$ symmetry in the absence of tilts.[36, 37, 38] As the reflection
conditions of $R\bar{3}m$ and $R\bar{3}$ are identical, distinguishing between
these space groups is difficult. Yet, as framework distortions and A-site
displacements are likely to couple, we believe structures reported in
$R\bar{3}m$ are possibly better described in $R\bar{3}$ with $a^{-}a^{-}a^{-}$
tilts in conjunction with A-site displacements. Similar situations have been
noted for oxide perovskites.[39]

The $a^{-}a^{-}b^{+}$ tilt pattern is the most prevalent in oxide perovskites
and it is known for PBAs with A = Na, K, or Rb and large $x$ values.[39, 10,
40] This tilt pattern drives monoclinic $P2_{1}/n$ symmetry[8] and the A-site
cation has three translational degrees of freedom, allowing a complete
optimisation of the bonding environment.[41] In addition to PBAs with
compositions close to the ideal A2MM′(CN)6, $a^{-}a^{-}b^{+}$ tilts can also
be induced by pressure in Rb0.87Mn[Co(CN)6]0.91, which adopts cubic
$F\bar{4}3m$ symmetry at ambient conditions.[42] For A = Na, the tilt pattern
is also stipulated by the hydration state, which will be discussed in the
subsequent section. It follows that $a^{-}a^{-}b^{+}$ is the tilt system
present for the widest range of A-site cations and can be observed both at
ambient and non-ambient conditions.

Finally, PBAs with large cations (Rb/Cs) may exhibit the uniaxial
$a^{0}a^{0}b^{+}$ distortion. CsMnCo(CN)6 develops $a^{0}a^{0}b^{+}$ tilts
upon compression to 2 GPa, leading to $P\bar{4}n2$ symmetry, and this
constitutes the only known tilted Cs-based PBA to date.[42] The relatively
high transition pressure highlights how large A-site cations reduce the
tendency for tilting. Likewise, pressure- and photo-induced $a^{0}a^{0}b^{+}$
tilts are known in Rb-based PBAs,[42, 43] but also feature at ambient
conditions in the Jahn-Teller distorted RbCuM′(CN)6 (M′ = CoIII or FeIII).[44,
45] Interestingly, the $a^{0}a^{0}b^{+}$ distortion is exceedingly rare in
double oxide perovskites,[39] albeit common in perovskite halides.[46] This
illustrates how trends observed in inorganic perovskites cannot always be
directly applied to their molecular counterparts.

###  1.3 M/M′-site composition

The transition metal ions may impact the propensity for structural distortions
in PBAs by modulating the bond strength and thereby the flexibility. For
example, the M–N bond strength in MPt(CN)6 depends on the identity of M
according to the Irving–Williams series, with NiPt(CN)6 (CdPt(CN)6) as the
most (least) rigid system.[20] In general, smaller M-site cations decrease the
flexibility, which also manifests in more positive thermal expansion.[47] This
agrees with the description of NaxNiFe(CN)6 as a zero-strain material with a
low tendency for distortions upon Na intercalation.[48] To investigate this
further, the data presented in Fig. 2 were replotted to highlight the effect
of the different M-sites [Fig. 4].111Compounds with formula Cu[Fe(CN)6]<0.67
feature Cu on both the A-and M-site,[24] meaning that Fig 4 is less
appropriate for M = Cu. Also, A2CuM′(CN)6 does not adopt a PBA structure; thus
the lower right quadrant of this panel is empty. Within the scatter of the
data, no trends can be discerned regarding the role of the M-sites on tilting
and several tilted PBAs with M = Ni are reported.[13, 14] Thus, the idea of
Ni-PBAs as zero-strain materials may require reevaluation.

Figure 4: The effect of M-site transition metal ions on the propensity for
tilting. PBAs with composition AxM[M′(CN)6]${}_{1-y}\Box_{y}$ ($\Box$ =
vacancy) for various A, where filled (empty) symbols denote tilted (untilted)
structures and the different panels refer to different M-site cations. The
diagonal lines show constant total transition metal charge (+5 and +4).

However, Jahn-Teller active M-site cations can contribute to tilting in
certain Rb-containing PBAs. Rb0.85Cu[Fe(CN)6]0.95 and Rb0.90Cu[Co(CN)6]0.92
exhibit tilts,[44, 45] whereas related structures with M = ZnII or MnII are
undistorted at ambient conditions.[49, 50] Group-theoretical analysis
indicates that the observed $a^{0}a^{0}b^{+}$ tilt mode arises from a
combination of the Jahn-Teller distortion and the occupational (M-point) Rb
arrangement.[45] Rb0.58Mn[Fe(CN)6]0.86 shows a Jahn-Teller distortion, but
remains untilted due to its different (R-point) Rb order.[51] As cubic
symmetry is reported for KCuFe(CN)6 and CsCuFe(CN)6,[52, 44] this phenomenon
appears specific to Rb-containing systems. The interplay between tilts, Rb
order and Jahn-Teller distortions is interesting and may suggest that tilt
engineering may be facilitated by introducing Cu${}^{\textrm{II}}$ ions in
suitable systems.

There is less choice of metal ions for the M′-site, as they have to support
octahedral low-spin coordination. Typical M′-site metals are
Cr${}^{\textrm{III}}$, Mn${}^{\textrm{II/III}}$, Fe${}^{\textrm{II/III}}$,
Co${}^{\textrm{III}}$, and Pt${}^{\textrm{IV}}$, with the largest prevalence
of Fe due to its stability, low toxicity, and low cost. As a result, it is
difficult to evaluate the influence of the M′-site on the tilting propensity,
but this effect is likely to be minor. As the M′–C interaction is largely
covalent and features strong $\pi$-backbonding from the metal to the cyanide,
linear M′–C–N bonding is favoured. Indeed, the N typically shows larger
mobility than the C atom, highlighting how most flexibility stems from the N–M
linkage, rather than from C–M′.[20] Hence, the tilt propensity should remain
relatively unchanged as a function of M′-site substitutions.

###  1.4 Hydration

PBAs are nearly always synthesised from aqueous solution and the final
products—if porous—thus contain water. It is generally accepted that defective
PBAs contain two types of water: coordinated water completing the coordination
sphere of the M-site cation, and interstitial water, which may hydrogen bond
to the coordinated molecules.[53] Due to the porous nature of these compounds,
as much as $ca$ 14 water molecules per cell are possible.[53, 54] A-site
deficient PBAs of formula MM′(CN)6 may also support $ca$ 4 water molecules per
unit cell,[55] although these guest species are only weakly interacting with
the framework. Regarding A-site-containing PBAs, systems with A = Li, Na are
sufficiently porous to allow for interstitial H2O, whereas increased A-site
occupancy of larger alkali metals reduces the potential for solvent
incorporation.[56]

Unlike many other coordination polymers,[57] PBAs are not prone to major
structural changes upon dehydration and typically remain cubic as the solvent
is removed.[20, 54, 58] Nonetheless, a tilt transition was noted for
Na1.32Mn[Fe(CN)6]${}_{0.83}\cdot 2.2$H2O upon further hydration.[59] To the
best of our knowledge, this behaviour is a unique example of a solvent-induced
phase transition in a Prussian blue analogue and further investigations will
be valuable. While the role of water in PBAs is intriguing, it is associated
with several challenges. Thermogravimetrical analysis (TGA) shows a
considerable dependence on the experimental set-up,[60, 15] and
crystallographers are faced with the challenge of disordered solvent and the
low scattering power of light elements. However, studies into the role of
water using local probes, perhaps using neutron radiation, would be
beneficial.

While (de)hydration normally does not drive phase transitions, interstitial
water sometimes dictates the tilt pattern. The distortion in Na2MM′(CN)6
switches from monoclinic $a^{-}a^{-}b^{+}$ to rhombohedral $a^{-}a^{-}a^{-}$
upon solvent removal accompanied by increased amplitude of the distortion.[10,
34] Whether K-based PBAs also show hydration-switchable tilts is less clear:
monoclinic symmetry is reported for several compounds with high concentration
of K and low hydration states,[61, 2] and the larger radius of K leaves less
free space for water molecules. Furthermore, water alters the pressure-induced
phase transition of MnPt(CN)6, which—at least in the hydrated case—is driven
by tilts.[42] Therefore, while (de)hydration per se does not drive tilting, it
has an impact on the tilts pattern induced by a different stimulus (pressure
or sodiation). Yet, many questions remain to be answered, regarding e.g. the
mechanism and critical water content required for switching.

###  1.5 Non-ambient conditions

Changes in temperature can induce a wealth of phase transitions in oxide
perovskites,[39] and thermal tilts also occur in some PBAs with high A-site
concentrations.[62, 36, 43] For instance, tilts appear in
Rb0.97Mn[Fe(CN)6]0.99 upon cooling to 91 K,[43] whereas a similar system with
a lower Rb content, Rb0.58Mn[Fe(CN)6]0.86, remains untilted to 30 K.[51] Phase
diagrams as a function of temperature and A-site cation concentration have
been created for certain PBAs, which also highlight the possibility of thermal
transitions between different tilt systems.[63] Further such efforts would be
useful to understand the relationship between the A-site cation content and
the structural instabilities. Conversely, A-site deficient PBAs typically
remain cubic down to low temperatures[55, 64, 65] as the soft tilt modes do
not condense. This is interesting in its own right, as transverse low-energy
phonons—such as tilts—typically underlie negative thermal expansion.[66]

Many PBAs undergo phase transitions when subjected to external hydrostatic
pressure. For example, MnPt(CN)${}_{6}\cdot n$H2O and
Mn[Co(CN)6]${}_{2/3}\cdot n$H2O show $a^{-}a^{-}a^{-}$ tilts when compressed
to $\sim 1.5$ GPa, which provides rare examples of tilted PBAs without A-site
cations.[30] The latter system also demonstrates that vacancies do not
necessarily disrupt the correlations between individual metal octahedra.
However, this transition relies on the presence of interstitial water, as the
dehydrated analogue retains its $Fm\bar{3}m$ symmetry when compressed.[42]
This is presumably a result of a hydrogen-bonded network of water molecules
within the structure, which mediates correlations between individual metal
sites separated by a vacancy. Likewise, FeCo(CN)6 amorphises at 10 GPa without
prior phase transitions,[67] and FePt(CN)6 remains cubic to at least 3
GPa,[68] which shows that pressure does not universally induce tilts in
A-site-deficient PBAs. Yet, as even Cs-based PBAs can be forced to tilt by the
application of pressure,[42, 69] compression is a useful method to obtain
interesting phases and explore the tilting behaviour.

###  1.6 Tilt engineering

Tilt engineering—manipulating the octahedral tilting in order to achieve a
desired property—is an appealing concept originally coined for oxide
perovskites.[9] Like for oxides, tilts in Prussian blue analogues have clear
functional implications and there is potential to improve the performance of
PBAs by judicious manipulation of the tilts. Some examples are provided below
and more can be anticipated.

Ferroelectricity is a technologically revelent property, yet is only allowed
in compounds with polar space groups, which is a subset of the non-
centrosymmetric space groups. As tilts inevitably lower the symmetry of the
parent structure, they can be exploited to construct polar symmetries $via$
improper ferroelectricity.[70] Tilts alone cannot lift all the inversion
centres of the $Fm\bar{3}m$ parent—as required for polar symmetry and
ferroelectric properties; yet this can be achieved by combining out-of-phase
tilts ($a^{-}$) with R-point occupational A-site cation order. Such A-site
arrangements are frequently found in AMM′(CN)6, where A = Rb or Cs. For
example, $a^{-}a^{-}a^{-}$ tilts in A-site-cation-ordered PBAs yields the
polar space group $R3$, whereas $a^{-}a^{-}b^{+}$ drives $Pn$ symmetry. The
former may not be experimentally realisable as $a^{-}a^{-}a^{-}$ tilts have
not been observed for PBAs with ordered A-site cations, but the latter space
group occurs in Rb0.87Mn[Co(CN)6]0.91 under modest pressure.[42] As noted
above, the tilting behaviour largely independent of the transition metals and
thus compositional substitutions may enable multiferroic PBAs to be developed.

PBAs are well known for their intriguing magnetic properties and this is also
amenable to tilt engineering strategies. In a generic PBA, both
antiferromagnetic—t${}_{2\textrm{g}}$(M) and t${}_{2\textrm{g}}$(M′)—and
ferromagnetic—e${}_{2\textrm{g}}$(M) and t${}_{2\textrm{g}}$(M′)—interactions
are present.[3] Judicious manipulation of the relative strength of the
coupling constants can give interesting phenomena, including pressure-induced
magnetic pole reversal and high ordering temperatures.[71, 3] The strength of
the ferromagnetic coupling is a function of the M–N–C angle, i.e. the tilting
angle, and as a result, the magnetic ordering temperature can be tuned by
modifying the tilting.[28, 10] This is feasible by manipulating the
composition—as in A2MnMn(CN)6—or by applying external pressure to increase the
amplitude of the distortion.[10, 28] Consequently, tilting facilitates the
rational design of intriguing magnetic phases of PBAs.

Electrode materials in secondary ion batteries must be able to reversibly
insert and extract a high concentration of carrier ions, such as Na+. As
demonstrated, A-site cations of a specific size and beyond a certain
concentration induce tilting in PBAs, which can create a large volume strain
leading to material degradation.[72, 11] In general, $a^{-}a^{-}a^{-}$ tilts
in systems with A = Na lead to a larger tilt distortion relative to the
$a^{-}a^{-}b^{+}$ distortions observed in K-based systems.[34] Furthermore,
the magnitude of the tilt distortion in Na-based systems is reduced in the
presence of water.[33] Taken together, a strategy to minimise the magnitude of
the tilt transitions is to maximise the pore filling with species active in
the electrochemical process—e.g. the larger K+ ion—or inactive species—e.g.
water or acetonitrile.[73] As the inclusion of electrochemically inactive
fillers may reduce the specific capacity through increased weight or occupancy
of sites for mobile cations, this must be carefully weighed against improved
stability. Any material adjustments must also be compatible with the
electrochemical cell chemistry and operation conditions. For example, water
excludes non-aqueous chemistries or operation at high voltages. Nonetheless,
there are multiple possible approaches towards the same goal, opening up the
possibility for engineering A-site cation-rich structures free of tilting.

##  2 Discussion and conclusion

To summarise, octahedral tilting in PBAs is a function of the stoichiometry,
composition, and hydration state. A-site cations are key to the tilting
behaviour: high concentrations of intermediately sized A-site cations favour
tilting, whereas large ions on the A-site, or no ions at all, lead to cubic,
untilted structures. Yet, tilts can condense upon compression in the two
latter systems. Three tilt systems are known in PBAs, and again the A-site
cation size dictates which one that is observed: small A-site ions drive
$a^{-}a^{-}a^{-}$ or $a^{-}a^{-}b^{+}$ tilts, whereas larger ions favour
$a^{0}a^{0}b^{+}$ distortions. In addition, Na-containing PBAs show peculiar
hydration-switchable tilt patterns, although the underlying reasons are still
unclear.

All tilt systems described here fall in the class of conventional tilts, i.e.
the octahedra rotate in alternate directions parallel to the rotation axis.
These are the only tilts allowed in inorganic perovskites. However, by virtue
of the diatomic cyanide linker, PBAs can theoretically also support so-called
unconventional tilts and columnar shifts.[74, 75] Yet, there are no confirmed
cases of such distortions and a computational study found these modes,
particularly the unconventional tilt, to be more energetically unfavourable
compared to conventional tilts.[66] This is likely due to the strong
preference for a linear M′–C–N arrangement, as discussed above, whereas the
unconventional distortions require nonlinear binding at both ends of the
linker.[76] While this simplifies the study of distortions in PBAs, as only
conventional modes need to be considered, it also limits the diversity of
possible distortions. Naturally, it cannot be excluded that even these higher-
energy modes may condense under certain conditions.

Our results identify avenues for further research within the PBA community.
Correct tilt assignment can be challenging—tilt patterns manifest in the
position of the linkers and due to the low X-ray scattering power of light
elements, locating cyanide using X-ray diffraction is difficult. Yet, accurate
tilt determination is crucial and more structural studies using neutron
diffraction are recommended. This is particularly important in light of the
unconventional distortions mentioned above, as conventional and unconventional
tilts cannot be distinguished solely based on the space group.[45]
Furthermore, it is not known to what extent changes in local structures can
affect the tilt modes, and PBAs are known for their diverse and complex
vacancy-ordered states.[19] In addition to vacancies, this is also
particularly relevant for the water arrangements and any local cation order.
The field of PBAs would also benefit from computational studies, yet the
propensity for disorder and the presence of paramagnetic ions present
challenges. Finally, the tilt distortions will certainly influence the
electronic structure of PBAs; yet this is another field which is largely
unexplored.

Octahedral tilting has a bearing on a wide range of properties, from
ferroelectricity to magnetism to electrochemistry. As a result, there is clear
scope for functional improvement by tilt engineering. In particular, Rb-based
PBAs show a large diversity in tilt behaviour and are susceptible to many
different external stimuli [Fig. 5], which may be exploited in responsive
materials.

Figure 5: Tilted structures of Rb-based PBAs. (a) Rb0.9Cu[Co(CN)6]0.92, where
the tilting is driven by Jahn-Teller distorted Cu (blue) and the unusual Rb
arrangement;[45] (b) Rb0.97Mn[Fe(CN)6]0.99 under pressure;[43] (c)
Rb2MnMn(CN)6 at ambient conditions;[40] and (d) Rb0.87Mn[Co(CN)6]0.91 under
pressure.[42]

It remains to be seen whether this is unique to Rb-containing systems.
Likewise, there is room for further studies into Li-based PBAs, which has
potential implications for energy storage, although the low scattering power
of Li render these materials less suitable for studies using X-rays. In
addition, since the A-site cation is crucial, tunability of the tilt
transitions—whether desired or not—may be expected from A-site solid
solutions, i.e. AxA${}^{\prime}_{1-x}$M[M′(CN)6]y, which have received
relatively little attention to date. Consequently, advances within the
understanding the fundamental tilt transition behaviour of PBAs will likely
couple to improvements in functionality.

A database of PBA structures was created by manually retrieving a large number
of research articles ($>$150 articles, $>$500 compounds) [SI]. The identities
of the A, M, and M′ sites, $x$\- and $y$-values, and presence of tilting were
extracted. Only studies where the ambient structure was confirmed by
diffraction and the composition characterised by either EDX or ICP were
included. As Rietveld refinements are not routinely carried out, any deviation
from cubic symmetry—manifested by reflection splitting—was considered to be
due to tilting, except in cases where other symmetry-lowering distortion (i.e.
Jahn-Teller distortions) are obviously present. As uncertainties from
elemental analysis are rarely reported, no error bars were included in the
figures and thus the quantitative conclusions should be drawn with caution.
Compounds with mixed cations on the A-site and the minority species exceeded
5% of the total $x$-value were excluded from Fig. 2. However, it should be
remembered that most PBAs will contain a small amount of K+ from the
synthesis. Likewise, systems with mixed M-site metals were not included in
Fig. 4.

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We are grateful to A. E. Phillips (Queen Mary University London) for useful
discussions. H.L.B.B acknowledges financial support from the Alexander von
Humboldt foundation. W.R.B is grateful to the strategic research area StandUp
for Energy and Energimyndigheten project no. 45517-1 for financial support.

The study was designed by both authors; data analysis carried out by H.L.B.B
and manuscript written by H.L.B.B with contributions and input from W.R.B.

W.R.B is a co-founder of the company ALTRIS AB, which produces Prussian white
powder for sodium ion battery applications.

Correspondence and requests for materials should be addressed to
<EMAIL_ADDRESS>


Algebraic Bethe Ansatz for the XXZ Gaudin Models with Generic Boundary

Algebraic Bethe Ansatz for the XXZ Gaudin Models
with Generic Boundary

Nicolas CRAMPE

N. Crampe

Laboratoire Charles Coulomb (L2C), UMR 5221 CNRS-Université de Montpellier,
Montpellier, France<EMAIL_ADDRESS>
Received November 01, 2017, in final form December 06, 2017; Published online
December 13, 2017

We solve the XXZ Gaudin model with generic boundary using the modified
algebraic Bethe ansatz. The diagonal and triangular cases have been recovered
in this general framework. We show that the model for odd or even lengths has
two different behaviors. The corresponding Bethe equations are computed for
all the cases. For the chain with even length, inhomogeneous Bethe equations
are necessary. The higher spin Gaudin models with generic boundary is also
treated.

integrability; algebraic Bethe ansatz; Gaudin models; Bethe equations

81R12; 17B80; 37J35

##  1 Introduction

The algebraic Bethe ansatz is a powerful method to solve analytically numerous
integrable models [41]. Then, the introduction of integrable boundaries in
this framework was initiated in [39]. However, the problem to solve models
with generic integrable boundaries thanks to the algebraic Bethe ansatz has
been overcome only recently [1, 5, 6, 8, 15]. The problem lies in the fact
that there is no simple particular eigenvector, which is normally the starting
point of the method. Therefore, other approaches have been used to deal with
this problem: the functional Bethe ansatz [21, 23, 34, 35], the coordinate
Bethe ansatz [16, 17], the separation of variables [19, 22, 37], the
$q$-Onsager approach [2, 4] and the matrix ansatz [18, 25, 28, 29, 30]. The
algebraic Bethe ansatz has been also used in this context [7, 10, 33, 38, 42]
but constraints on the parameters defining the boundaries were necessary.
Then, inhomogeneous T-Q relations were introduced in [11, 12, 13, 36] where
the authors obtained the eigenvalues and the Bethe equations for the XXZ model
with generic boundaries. Finally, the modified algebraic Bethe ansatz allows
us to compute the associated eigenvectors [1, 5, 6, 8, 44]. Let us mention
also that this method has been also exploited to study twisted XXX spin chain
in [9] or the rational Gaudin magnets in arbitrarily oriented magnetic fields
in [20].

In this letter, we solve the Gaudin model (introduced in [24]) with generic
boundaries using the modified algebraic Bethe ansatz. The Gaudin model is one
of the simplest quantum integrable system and we hope that its resolution will
shed more light on the method. Let us also mention that the algebraic Bethe
ansatz was already applied to solve open Gaudin model using the vertex-IRF
correspondence of the XXZ spin chain [26, 43]. However, as pointed in [14],
the model considered in [26, 43] is obtained as a limit from the XXZ spin
chain and is different of the one studied here which is constructed directly
from a classical $r$-matrix.

This letter is organised as follows. In Section 2, we recall well-known
results about the Gaudin algebra and model to fix the notations used in the
following. Then, we provide technical results in the Sections 3.1 and 3.2 to
implement the algebraic Bethe ansatz. In Section 3.3, we recover the results
for the diagonal and triangular boundaries [27, 31, 32]. Section 3.4 contains
new results for the generic boundaries when the length of the chain is odd.
Finally, we deal with the even chain in Section 3.5 where an additional
computation is necessary. We explain also why the computation is different
depending on the parity of the length of the chain. In Section 4, we
generalize these results to solve the higher spin Gaudin models with generic
boundary.

##  2 Gaudin models and algebras

In this section, we recall different well-known notions to construct Gaudin
models and prove their integrability.

For a matrix
$r(x,y)\in\operatorname{End}(\mathbb{C}^{N}\otimes\mathbb{C}^{N})$ depending
on 2 parameters $x,y$, one defines the fundamental equation, called classical
Yang–Baxter equation, given by

$\displaystyle[{r}_{13}(x_{1},x_{3}),{r}_{23}(x_{2},x_{3})]=[{r}_{21}(x_{2},x_{1}),{r}_{13}(x_{1},x_{3})]+[{r}_{23}(x_{2},x_{3}),{r}_{12}(x_{1},x_{2})].$

In the previous relation, we have used the usual notations:
$r_{12}(x)=r(x)\otimes\mathbb{I}$, $r_{23}(x)=\mathbb{I}\otimes r(x)$, …,
where $\mathbb{I}$ is the identity matrix. A solution of the classical
Yang–Baxter equation is called a $r$-matrix. In addition, if an $r$-matrix
$r(x,y)$ depends only on the quotient $x/y$ and satisfies the supplementary
relations $r_{12}(x,y)=r_{12}(x/y)=-r_{21}(y/x)=-r_{12}(y/x)^{t_{1}t_{2}}$
then it is called a skew-symmetric $r$-matrix. For a given skew-symmetric
$r$-matrix $r(x/y)$, one defines also the reflection equation

$\displaystyle
r_{12}(x/y)k_{1}(x)k_{2}(y)-k_{1}(x)k_{2}(y)r_{21}(x/y)=k_{2}(y)r_{12}(xy)k_{1}(x)-k_{1}(x)r_{21}(xy)k_{2}(y).$

A solution $k(x)$ of the reflection equation is called $k$-matrix. From a
skew-symmetric $r$-matrix and an associated $k$-matrix, we can construct a new
$r$-matrix [40]

$\displaystyle\overline{r}_{12}(x,y)=r_{12}(x/y)-k_{1}(x)r_{12}(1/(xy))k_{1}(x)^{-1},$
(2.1)

which is not skew-symmetric.

To each $r$-matrix $r(x,y)$, on can associate a Lie algebra $\mathscr{R}$.
Indeed, if
${\mathscr{K}}(x)\in\operatorname{End}\big{(}\mathbb{C}^{N}\big{)}\otimes\mathscr{R}$,
then the following relation

$\displaystyle[{\mathscr{K}}_{1}(x),{\mathscr{K}}_{2}(y)]=[{r}_{21}(y,x),{\mathscr{K}}_{1}(x)]+[{\mathscr{K}}_{2}(y),{r}_{12}(x,y)]$
(2.2)

defines a Lie commutator for $\mathscr{R}$. Indeed, the antisymmetry of the
product is obvious and the Jacobi identity is satisfied due to the classical
Yang–Baxter equation. For any invertible matrix
$M(x)\in\operatorname{End}\big{(}\mathbb{C}^{N}\big{)}$, there is a Lie
algebra isomorphism given by

$\displaystyle\Phi_{M}\colon\quad\quad\mathscr{R}$
$\displaystyle\rightarrow\mathscr{R}_{M},$ $\displaystyle{\mathscr{K}_{0}}(x)$
$\displaystyle\mapsto M_{0}(x)^{-1}\mathscr{K}_{0}(x)M_{0}(x),$

where $\mathscr{R}_{M}$ is the Lie algebra defined by (2.2) with the
$r$-matrix

$\displaystyle M_{1}(x)^{-1}M_{2}(y)^{-1}r_{12}(x,y)M_{1}(x)M_{2}(y).$

From the defining relations (2.2), one can show that the transfer matrix

$\displaystyle
t(x)=\frac{1}{4}\operatorname{tr}_{0}\big{(}{\mathscr{K}_{0}}(x)^{2}\big{)}$
(2.3)

satisfies $[t(x),t(y)]=0$. Then, when a representation for $\mathscr{R}$ is
chosen, the coefficients of $t(x)$ provide an integrable hierarchy. Let us
remark that, for a given matrix $M(x)$, $\mathscr{R}$ and $\mathscr{R}_{M}$
give the same hierarchy since it is easy to see that
$\operatorname{tr}\big{(}{\mathscr{K}_{0}}(x)^{2}\big{)}=\operatorname{tr}\big{(}{\Phi_{M}(\mathscr{K}_{0}}(x))^{2}\big{)}$.

In this letter, we focus on the skew-symmetric $r$-matrix associated to the
affine Kac–Moody algebra $\widehat{{\mathfrak{sl}}_{2}}$:

$\displaystyle r(x)=\frac{1}{x-1}\begin{pmatrix}-\frac{1}{2}(x+1)&0&0&0\\\
0&\frac{1}{2}(x+1)&-2&0\\\ 0&-2x&\frac{1}{2}(x+1)&0\\\
0&0&0&-\frac{1}{2}(x+1)\end{pmatrix}.$ (2.4)

The most general $k$-matrix associated to this $r$-matrix is given by

$\displaystyle
k(x)=\begin{pmatrix}\beta+\gamma/x&-\frac{\alpha(\beta+\rho)}{2}(x-1/x)\\\
\frac{\beta-\rho}{2\alpha}(x-1/x)&\beta+\gamma x&\end{pmatrix},$ (2.5)

where $\alpha$, $\beta$, $\gamma$ and $\rho$ are scalar parameters. As
explained previously, the $r$-matrix and the $k$-matrix allow us to construct
a new $r$-matrix (see relation (2.1)). Without loss of generality, we can use
the Lie algebra isomorphism $\Phi_{M}$ to transform this $r$-matrix.
Therefore, we use in this paper the following $r$-matrix

$\displaystyle\widetilde{r}_{12}(x,y)=M_{1}(x)^{-1}M_{2}(y)^{-1}\big{(}r_{12}(x/y)-k_{1}(x)r_{12}(1/(xy))k_{1}(x)^{-1}\big{)}M_{1}(x)M_{2}(y),$
(2.6)

where $r(x)$ is given by (2.4), $k(x)$ is given by (2.5) and

$\displaystyle
M(x)=\begin{pmatrix}\frac{\beta+\rho}{2\rho}&\frac{\alpha}{x}\\\
\frac{(\beta-\rho)x}{2\alpha\rho}&1\end{pmatrix}.$

Explicitly, this $r$-matrix reads as

$\displaystyle\widetilde{r}(x,y)=\begin{pmatrix}-\frac{1}{2}\omega(x,y)\sigma^{z}&b(x)\sigma^{z}+f(y,x)\sigma^{-}\\\
c(x)\sigma^{z}-f(1/y,1/x)\sigma^{+}&\frac{1}{2}\omega(x,y)\sigma^{z}\end{pmatrix},$

where $\sigma^{z}=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}$,
$\sigma^{+}=\begin{pmatrix}0&1\\\ 0&0\end{pmatrix}$ and
$\sigma^{-}=\begin{pmatrix}0&0\\\ 1&0\end{pmatrix}$ and

$\displaystyle\delta(x)=\frac{((\beta-\rho)x^{2}+2\gamma
x+\rho+\beta)}{2\rho(x-1/x)},\qquad b(x)=-\frac{\alpha}{\delta(x)},\qquad
c(x)=\frac{\rho^{2}-\beta^{2}}{4\alpha\rho^{2}\delta(1/x)},$ (2.7)
$\displaystyle\omega(x,y)=\frac{x+y}{x-y}+\frac{xy+1}{xy-1}\qquad\text{and}\qquad
f(x,y)=\omega(x,y)\frac{\delta(x)}{\delta(y)}.$ (2.8)

Let us emphasize that the $r$-matrix $\widetilde{r}(x,y)$ has a simpler form
than the one without the isomorphism (i.e.,
$r_{12}(x/y)-k_{1}(x)r_{12}(1/(xy))k_{1}(x)^{-1}$). Without the isomorphism,
the 16 entries of the $r$-matrix should be non-zero whereas
$\widetilde{r}(x,y)$ has 6 vanishing entries and each two by two block
submatrices are diagonal or triangular.

As explained previously, the $r$-matrix $\widetilde{r}(x,y)$ allows one to
define a Lie algebra by using relation (2.2) with

$\displaystyle\widetilde{\mathscr{K}_{0}}(x)=\begin{pmatrix}A(x)&\widetilde{B}(x)\\\
\widetilde{C}(x)&-A(x)\end{pmatrix}.$ (2.9)

We denote by $\widetilde{\mathscr{R}}$ this Lie algebra. Let us mention that
the Lie algebra $\widetilde{\mathscr{R}}$ for special choice of the parameters
in the $k$-matrix has been identified as Onsager algebras [3]111To compare
this letter with [3], we must use the additional property
$r_{12}(x)^{t_{1}}=\sigma^{y}_{1}r_{12}(x)\sigma^{y}_{1}$ satisfied by the
$r$-matrix (2.4)..

Finally, let us introduce the following representation222We keep the same
notation $\widetilde{\mathscr{K}}(x)$ for the algebraic element or its
representation.

$\displaystyle\widetilde{\mathscr{K}}_{0}(x)=\sum_{j=1}^{L}\widetilde{r}_{0j}(x,v_{j}).$
(2.10)

By using the classical Yang–Baxter equation, it is obvious to show that (2.10)
satisfies the commutation relations of $\widetilde{\mathscr{R}}$. The
parameters $v_{i}$ are called inhomogeneous parameters. The Hamiltonians are
usually defined by

$\displaystyle\widetilde{H}_{j}=\operatorname{Res}_{x=v_{j}}t(x)=\frac{1}{4}\operatorname{Res}_{x=v_{j}}\operatorname{tr}_{0}\big{(}{\widetilde{\mathscr{K}_{0}}}(x)^{2}\big{)}$
$\displaystyle\hphantom{\widetilde{H}_{j}}{}=-v_{j}\sum_{\genfrac{}{}{0.0pt}{1}{p=1}{p\neq
j}}^{L}\widetilde{r}_{jp}(v_{j},v_{p})+Lv_{j}\left(\frac{3v_{j}^{2}}{v_{j}^{2}-1}-\nu(v_{j})\right),$

where

$\displaystyle\nu(z)=\frac{z^{2}(\rho^{2}-\beta^{2})-2\beta\gamma
z+\rho^{2}-\beta^{2}}{2\rho^{2}(z^{2}-1)\delta(z)\delta(1/z)}.$ (2.11)

These Hamiltonians are integrable due to the commutation of the transfer
matrix and $[\widetilde{H}_{j},\widetilde{H}_{k}]$ $=0$. They are related to
the Gaudin models with boundary (or also called BC Gaudin models). Indeed, by
conjugating the previous Hamilonians by $M_{1}(v_{1})\cdots M_{L}(v_{L})$, one
gets the Hamiltonians

$\displaystyle H_{j}=-v_{j}\sum_{\genfrac{}{}{0.0pt}{1}{p=1}{p\neq
j}}^{L}\left(r_{jp}\left(\frac{v_{j}}{v_{p}}\right)+k_{j}(v_{j})r_{jp}\left(\frac{1}{v_{j}v_{p}}\right)k_{j}(v_{j})^{-1}\right)+Lv_{j}\left(\frac{3v_{j}^{2}}{v_{j}^{2}-1}-\nu(v_{j})\right),$

which are the usual form for the XXZ Gaudin model with boundary.

##  3 Modified algebraic Bethe ansatz

In this section, we want to diagonalize the matrix
$\frac{1}{4}\operatorname{tr}_{0}\big{(}{\widetilde{\mathscr{K}_{0}}}(x)^{2}\big{)}$
when the 4 parameters in the $k$-matrix are generic. The diagonalisation when
the $k$-matrix is diagonal was obtained in [27] and for the triangular case in
[31, 32]. For the generic case, the situation is more complicated. Here, we
use the modified algebraic Bethe ansatz, introduced and used in different
contexts [1, 5, 6, 8, 15], to compute the eigenvalues and the eigenvectors for
the generic case. The first two subsections are technical. Then, Section 3.3
contains the known results about the diagonal or triangular boundaries
recovered here and Sections 3.4 and 3.5 contains the new results for the chain
of length odd or even.

###  3.1 Algebraic relations

In this subsection, by using the commutation relations obtained from (2.2), we
provide necessary propositions to perform the algebraic Bethe ansatz. Firstly,
let us mention that the commutation relations between $A(x)$,
$\widetilde{B}(x)$ and $\widetilde{C}(x)$ given by (2.2) with the $r$-matrix
(2.6) and $\widetilde{\mathscr{K}}(x)$ given by (2.9) are quite complicated in
comparison to the usual ones. This is why we introduce the two following
shifted generators

$\displaystyle{B}(x,n)=\widetilde{B}(x)-(2n-1)b(x)\qquad\text{and}\qquad{C}(x,n)=\widetilde{C}(x)-(2n-1)c(x),$

where $b(x)$ and $c(x)$ are defined by (2.7). For these generators and the
generators $A(x)$, the commutation relations become

$\displaystyle{A}(x){A}(y)={A}(y){A}(x),$ (3.1)
$\displaystyle{B}(x,n){B}(y,n+1)={B}(y,n){B}(x,n+1),$ (3.2)
$\displaystyle{C}(x,n){C}(y,n-1)={C}(y,n){C}(x,n-1),$ (3.3)
$\displaystyle{A}(x){B}(y,m)={B}(y,m){A}(x)+\omega(x,y){B}(y,m)-f(x,y){B}(x,m),$
(3.4)
$\displaystyle{A}(x){C}(y,m)={C}(y,m){A}(x)-\omega(x,y){C}(y,m)-f(1/x,1/y){C}(x,m),$
(3.5) $\displaystyle{C}(x,n){B}(y,n)={B}(y,n+1){C}(x,n+1)$
$\displaystyle\hphantom{{C}(x,n){B}(y,n)=}{}+2f(x,y){A}(x)-2f(1/y,1/x){A}(y)-8nb(y)c(x),$
(3.6)

where $\omega(x,y)$ and $f(x,y)$ are defined by (2.8).

We introduce also a shifted transfer matrix

$\displaystyle{t}(x,j)=\frac{1}{2}{A}(x)^{2}+\frac{1}{4}{B}(x,j+1){C}(x,j+1)+\frac{1}{4}{C}(x,j){B}(x,j)-\frac{1}{2}b(x)c(x).$
(3.7)

For $j=0$ in the previous relation, the transfer matrix (2.3) is recovered:
$t(x)=\frac{1}{4}\operatorname{tr}_{0}\widetilde{\mathscr{K}_{0}}(x)^{2}={t}(x,0)$.

Secondly, we present a lemma permitting to commute the shifted transfer matrix
with the generator $B(x,p)$. This lemma is standard in the context of the
algebraic Bethe ansatz for the Gaudin model except for the modification of the
shift for the transfer matrix.

######  Lemma 3.1.

If the algebraic relations (3.1)–(3.6) are satisfied, the following relation
holds

$\displaystyle{t}(x,p-1){B}(z,p)-{B}(z,p){t}(x,p)={B}(z,p)\big{(}\omega(x,z)(A(x)+\nu(x))+2b(x)c(x)\big{)}$
$\displaystyle\qquad{}-{B}(x,p)\big{(}f(1/z,1/x)(A(z)+\nu(z))+4(p-1)c(x)b(z)\big{)},$
(3.8)

where $\nu(z)$ is defined by (2.11).

###### Proof.

By straightforward computations using the commutation relations (3.1)–(3.6),
one gets

$\displaystyle{t}(x,p-1){B}(z,p)-{B}(z,p){t}(x,p)=\omega(x,z){B}(z,p){A}(x)-f(1/z,1/x){B}(x,p){A}(z)$
$\displaystyle\qquad{}+\frac{1}{2}\big{(}\omega(x,z)^{2}+f(z,x)f(1/z,1/x)\big{)}{B}(z,p)$
$\displaystyle\qquad{}-\frac{1}{2}\big{(}\omega(z,x)f(1/z,1/x)+\omega(x,z)f(x,z)+4(2p-1)c(x)b(z)\big{)}{B}(x,p).$

Then, by using the following functional relations

$\displaystyle\omega(x,z)^{2}+f(z,x)f(1/z,1/x)=2\nu(x)\omega(x,z)+4b(x)c(x),$
$\displaystyle\omega(z,x)f(1/z,1/x)+\omega(x,z)f(x,z)=2\nu(z)f(1/z,1/x)-4c(x)b(z)$

one gets relation (3.8). ∎

Let us define the following product

$\displaystyle\mathbb{B}(\boldsymbol{z})={B}(z_{1},1){B}(z_{2},2)\cdots{B}(z_{M},M),$

where $\boldsymbol{z}=\\{z_{1},z_{2},\dots,z_{M}\\}$. Let us remark that, due
to relation (3.2), $\mathbb{B}(\boldsymbol{z})$ is invariant under any
permutation of the $z_{i}$ which justifies the definition of $\boldsymbol{z}$
as a set. One introduces also, for $k=1,2,\dots,M$

$\displaystyle\mathbb{B}(\boldsymbol{z}_{k},x)={B}(z_{1},1)\cdots{B}(z_{k-1},k-1){B}(x,k){B}(z_{k+1},k+1)\cdots{B}(z_{M},M).$

Finally, we are in position to compute the action of the transfer matrix with
the product $\mathbb{B}(\boldsymbol{z})$:

######  Proposition 3.2.

If the algebraic relations (3.1)–(3.6) are satisfied, the following relation
holds

$\displaystyle
t(x)\mathbb{B}(\boldsymbol{z})=\frac{1}{2}\mathbb{B}(\boldsymbol{z})B(x,M+1)C(x,M+1)$
$\displaystyle\hphantom{t(x)\mathbb{B}(\boldsymbol{z})=}{}+\mathbb{B}(\boldsymbol{z})\left(\Lambda(x)+\sum_{p=1}^{M}\omega(x,z_{p})\left(A(x)+\nu(x)+\sum_{q\neq
p}\frac{(x-1/x)\omega(z_{p},z_{q})}{z_{p}-1/z_{p}}\right)\right)$
$\displaystyle\hphantom{t(x)\mathbb{B}(\boldsymbol{z})=}{}-\sum_{p=1}^{M}\mathbb{B}(\boldsymbol{z}_{p},x)f(1/z_{p},1/x)\left(A(z_{p})+\nu(z_{p})+\sum_{q\neq
p}\omega(z_{p},z_{q})\right),$

where

$\displaystyle\Lambda(x)=\frac{1}{2}A(x)^{2}+xA^{\prime}(x)+\nu(x)A(x)-\frac{x^{2}+1}{x^{2}-1}A(x)-\frac{1}{2}b(x)c(x).$

###### Proof.

One gets, using a telescopic sum, the following relation

$\displaystyle{t}(x,0)\mathbb{B}(\boldsymbol{z})-\mathbb{B}(\boldsymbol{z}){t}(x,M)=\sum_{p=1}^{M}{B}_{1}\cdots{B}_{p-1}\left({t}(x,p-1){B}_{p}-{B}_{p}{t}(x,p)\right){B}_{p+1}\cdots{B}_{M},$

where ${B}_{p}={B}(z_{p},p)$. Then, using the Lemma 3.1, we express
${t}(x,p-1){B}_{p}-{B}_{p}{t}(x,p)$ in the previous sum. Then, we use
repetitively relation (3.4) to put the operators $A$ on the right. Then, we
rearrange the terms using the following relations

$\displaystyle\omega(x,z_{p})f(x,z_{q})-f(1/z_{p},1/x)f(z_{p},z_{q})=\omega(z_{q},z_{p})f(1/z_{q},1/x)-4c(x)b(z_{q}),$
$\displaystyle\sum_{q>p}\omega(x,z_{p})\omega(x,z_{q})=\sum_{q\neq
p}\frac{x-1/x}{z_{p}-1/z_{p}}\omega(z_{p},z_{q})\omega(x,z_{p}).$

Finally, ${t}(x,M)$ is simplified using relation (3.7) and (3.6) for
$y\rightarrow x$ to prove the proposition. ∎

Let us emphasize again that this type of computation and result are the same
for the periodic or diagonal Gaudin model. The only difference lies on the
shift in the operators.

###  3.2 Explicit representation

Up to now, we have only used the commutation relations of the algebra to prove
the previous results. Now, we choose the representation for
$\widetilde{\mathscr{K}}(x)$ given by (2.10). Therefore, we get explicitly

$\displaystyle A(x)=-\frac{1}{2}\sum_{j=1}^{L}\omega(x,v_{j})\sigma^{z}_{j},$
$\displaystyle
B(x,n)=(1-2n)b(x)+b(x)\sum_{j=1}^{L}\sigma^{z}_{j}+\sum_{j=1}^{L}f(v_{j},x)\sigma^{-}_{j},$
(3.9) $\displaystyle
C(x,n)=(1-2n)c(x)+c(x)\sum_{j=1}^{L}\sigma^{z}_{j}-\sum_{j=1}^{L}f(1/v_{j},1/x)\sigma^{+}_{j}.$

Let us introduce the following vectors $\Omega=\begin{pmatrix}1\\\
0\end{pmatrix}^{\otimes L}$, called usually pseudo-vacuum. We deduce that

$\displaystyle A(x)\Omega=a(x)\Omega\qquad\text{and}\qquad
C(x,n)\Omega=(L+1-2n)c(x)\Omega$ (3.10)

with $a(x)=-\frac{1}{2}\sum\limits_{j=1}^{L}\omega(x,v_{j})$.

The Bethe vectors are defined as follows

$\displaystyle\mathbb{V}(\boldsymbol{z})=\mathbb{B}(\boldsymbol{z})\Omega.$
(3.11)

These Bethe vectors are of course invariant under any permutation of the
$z_{i}$ since $\mathbb{B}(\boldsymbol{z})$ and $\mathbb{C}(\boldsymbol{z})$
are invariant. From the previous results (Proposition 3.2 and relations
(3.10)), we get easily the action of the transfer matrix on these Bethe
vectors

$\displaystyle
t(x)\mathbb{V}(\boldsymbol{z})=\frac{L-2M-1}{2}c(x)\mathbb{B}(\boldsymbol{z})B(x,M+1)\Omega$
$\displaystyle\hphantom{t(x)\mathbb{V}(\boldsymbol{z})=}{}+\left(\lambda(x)+\sum_{p=1}^{M}\omega(x,z_{p})\left(a(x)+\nu(x)+\sum_{q\neq
p}\frac{(x-1/x)\omega(z_{p},z_{q})}{z_{p}-1/z_{p}}\right)\right)\mathbb{V}(\boldsymbol{z})$
$\displaystyle\hphantom{t(x)\mathbb{V}(\boldsymbol{z})=}{}-\sum_{p=1}^{M}f(1/z_{p},1/x)\left(a(z_{p})+\nu(z_{p})+\sum_{q\neq
p}\omega(z_{p},z_{q})\right)\mathbb{V}(\boldsymbol{z}_{p},x),$ (3.12)

where

$\displaystyle\lambda(x)=\frac{1}{2}a(x)^{2}+xa^{\prime}(x)+\nu(x)a(x)-\frac{x^{2}+1}{x^{2}-1}a(x)-\frac{1}{2}b(x)c(x).$

The second term in (3.12) is the wanted term (i.e., is proportional to the
Bethe vector) and the third term is the unwanted term. These terms are already
present for the periodic or triangular case. For the generic boundary, we must
deal with the first term in the previous relation (3.12). There are different
possibilities:

  * •

$c(x)=0$, i.e., $\rho=\pm\beta$. This case corresponds to triangular (or
diagonal) boundaries and the Bethe ansatz works as usual. We get different
sectors depending on the number $M$ of excitations (see Section 3.3).

  * •

The length of chain is odd and $M=\frac{L-1}{2}$. It corresponds to a case
where the number of excitations is fixed. From relation (3.12), one gets only
the half of the spectrum. The other half is obtained starting from
$t(x)\overline{\mathbb{V}}(\boldsymbol{z})$ (see Section 3.4).

  * •

The length of the chain is even. In this case, we must compute
$\mathbb{B}(\boldsymbol{z})B(x,M+1)\Omega$. It corresponds to the modified
algebraic Bethe ansatz (see Section 3.5).

At first sight, the difference of the behavior for the lengths odd or even
seems strange. However, we can show that, in the case of odd length, two
subspaces of $(\mathbb{C}^{2})^{\otimes L}$ are stabilized by the transfer
matrix. More precisely, if we denote by $V_{J}$ the eigenspace of the total
spin $S^{z}=\frac{1}{2}\sum\limits_{j=1}^{L}\sigma^{z}_{j}$ with eigenvalue
$J$, one gets

$\displaystyle t(x)V_{1/2}\oplus V_{3/2}\oplus\dots\oplus V_{L/2}\subset
V_{1/2}\oplus V_{3/2}\oplus\dots\oplus V_{L/2},$ $\displaystyle
t(x)V_{-1/2}\oplus V_{-3/2}\oplus\dots\oplus V_{-L/2}\subset V_{-1/2}\oplus
V_{-3/2}\oplus\dots\oplus V_{-L/2}.$ (3.13)

Then it is natural that the spectrum for odd chain splits into two sectors.
For the chain of even length, this feature disappears: there is no stable
subspace.

###  3.3 Triangular and diagonal boundaries

In this subsection, we deal with the triangular boundary (i.e.,
$\rho=\pm\beta$). In this case, we can easily see that one gets, for
$M=0,1,\dots,L$,

$\displaystyle t(x)V_{L/2-M}\oplus V_{L/2-M+1}\oplus\cdots\oplus
V_{L/2}\subset V_{L/2-M}\oplus V_{L/2-M+1}\oplus\cdots\oplus V_{L/2}.$

For $M=0,1,\dots,L$, the Bethe vectors $\mathbb{V}(\boldsymbol{z})$ defined by
(3.11) is just in the sector $V_{L/2-M}\oplus\cdots\oplus V_{L/2}$ and becomes
an eigenvector of the transfer matrix if the Bethe roots $z_{i}$ satisfy the
Bethe equations:

$\displaystyle a(z_{p})+\nu(z_{p})+\sum_{\genfrac{}{}{0.0pt}{1}{q=1}{q\neq
p}}^{M}\omega(z_{p},z_{q})=0\qquad\text{for}\quad p=1,2,\dots,M.$ (3.14)

Explicitly, they read, for $p=1,\dots,M$,

$\displaystyle-\frac{1}{2}\sum_{j=1}^{L}\left(\frac{z_{p}+v_{j}}{z_{p}-v_{j}}+\frac{z_{p}v_{j}+1}{z_{p}v_{j}-1}\right)+\frac{1}{2}\left(\frac{\gamma
z-\beta}{\gamma z+\beta}+\frac{\beta z-\gamma}{\beta z+\gamma}\right)$
$\displaystyle\qquad{}+\sum_{\genfrac{}{}{0.0pt}{1}{q=1}{q\neq
p}}^{M}\left(\frac{z_{p}+z_{q}}{z_{p}-z_{q}}+\frac{z_{p}z_{q}+1}{z_{p}z_{q}-1}\right)=0.$
(3.15)

The corresponding eigenvalue of the transfer matrix
$t(x)=\frac{1}{4}\operatorname{tr}_{0}\big{(}\widetilde{\mathscr{K}}_{0}(x)^{2}\big{)}$
is

$\displaystyle\lambda(x)+\sum_{p=1}^{M}\omega(x,z_{p})\left(a(x)+\nu(x)+\sum_{\genfrac{}{}{0.0pt}{1}{q=1}{q\neq
p}}^{M}\frac{(x-1/x)\omega(z_{p},z_{q})}{z_{p}-1/z_{p}}\right).$ (3.16)

Starting from the triangular boundary, we recover the diagonal one for
$\alpha=0$. We see that the eigenvalue (3.16) and the Bethe equations (3.15)
do not depend on $\alpha$. Then, as shown previously for the Gaudin model [31,
32] (or for other models [7, 38]), the spectrum for triangular boundaries is
similar to the one with diagonal boundaries. Evidently, the Bethe eigenvectors
depends on $\alpha$, via the function $b(x)$ present in $B(x,j)$ (see (3.9)).
In particular, the function $b(x)$ vanishes for $\alpha=0$ and the Bethe
vectors $\mathbb{V}(\boldsymbol{z})$ are now in the sector $V_{L/2-M}$ which
is consistent with

$\displaystyle t(x)V_{L/2-M}\subset V_{L/2-M}.$

The last equation means that the total spin is conserved by the transfer
matrix in the case of diagonal boundary.

###  3.4 Generic boundaries for an odd chain

If the length of the chain is odd (i.e., there is an integer $\ell$ such that
$L=2\ell+1$), we can chose the number of excitations as $M=\frac{L-1}{2}=\ell$
and the coefficient in front of $\mathbb{B}(\boldsymbol{z})B(x,M+1)\Omega$
vanishes. From the explicit expression (3.9) of $B(x,j)$, we see that

$\displaystyle\mathbb{V}(\boldsymbol{z})\in V_{1/2}\oplus
V_{3/2}\oplus\dots\oplus V_{L/2}\qquad\text{for}\quad M=\ell,$

which is one of the subspaces stabilized by the transfer matrix, as explained
previously. Then, the eigenvalues for this sector are given by (3.16) with the
Bethe equations (3.14) for $M=\ell$.

The spectrum and the associated eigenvectors for the other sector (3.13) are
obtained from the Bethe vectors

$\displaystyle\overline{\mathbb{V}}(\boldsymbol{z})={C}(z_{1},0){C}(z_{2},-1)\cdots{C}(z_{\ell},-\ell+1)\begin{pmatrix}0\\\
1\end{pmatrix}^{\otimes L}.$

It is easy to show that $\overline{\mathbb{V}}(\boldsymbol{z})$ belongs to the
second sector. A relation similar to (3.12) by replacing
${\mathbb{V}}(\boldsymbol{z})$ by $\overline{\mathbb{V}}(\boldsymbol{z})$ can
be proved. Then, we show that, starting from
$\overline{\mathbb{V}}(\boldsymbol{z})$, one gets exactly the expression
(3.16) for the eigenvalues and (3.14) for the Bethe equations. Therefore, the
spectrums in both sectors are the same.

Let us also mention that the separation into two sectors is not new in this
context. Indeed, the complete spectrum of the XXZ spin chain with certain
constraints between the parameters of the boundaries is also obtained from two
different Bethe vectors [42].

###  3.5 Generic boundaries for an even chain

For a chain with an even length, we must deal with the term
$\mathbb{B}(\boldsymbol{z})B(x,M+1)\Omega$. For a generic $M$, this vector has
no particular property but for $M=L$, special feature appears. This is the
crucial point of the modified algebraic Bethe ansatz.

Firstly, let us state the following lemma:

######  Lemma 3.3.

The entries of the matrix

$\displaystyle{\mathcal{M}}(z)=\prod_{q=1}^{\genfrac{}{}{0.0pt}{2}{\longrightarrow}{L}}\left(\frac{(z-v_{q})(z-1/v_{q})}{(z-z_{q})(z-1/z_{q})}B(z_{q},q)\right)$
$\displaystyle\hphantom{{\mathcal{M}}(z)=}{}\times\left(\frac{(z-1/z)}{(z-x)(z-1/x)}B(x,L+1)\right)\big{(}\delta(z)B(z,L+1)\big{)}^{-1}$

have only simple poles at
$z=0,\infty,z_{1},z_{2},\dots,z_{L},1/z_{1},1/z_{2},\dots,1/z_{L},x,1/x$.

###### Proof.

The proof of the existence of the poles of ${\mathcal{M}}(z)$ at
$z=z_{1},z_{2},\dots,z_{L},1/z_{1},1/z_{2},\dots,1/z_{L}$ and $x$, $1/x$ is
straightforward. One gets

$\displaystyle\widehat{B}(z)=\delta(z)B(z,L+1)=\alpha(2L+1)-\alpha\sum_{j=1}^{L}\sigma^{z}_{j}+\sum_{j=1}^{L}\frac{2z(v_{j}-1/v_{j})}{(z-v_{j})(z-1/v_{j})}\delta(v_{j})\sigma^{-}_{j}.$

This matrix $\widehat{B}(z)$ is a lower triangular matrix with the diagonal
entries in the set $\\{\alpha(L+1)$, $\alpha(L+3),\dots,\alpha(3L+1)\\}$. In
particular, this shows that it is invertible which justifies the definition of
${\mathcal{M}(z)}$. Due to the Cayley–Hamilton theorem, $\widehat{B}(z)^{-1}$
is a polynomial of $\widehat{B}(z)$. Then, by using the fact that
$(\sigma^{-})^{2}=0$, we prove that the poles at $z=v_{j}$ and $z=1/v_{j}$ of
$\widehat{B}(z)^{-1}$ remain simple. Then, ${\mathcal{M}}(z)$ has no pole at
these points. In addition, its becomes a non-singular diagonal matrix for
$z=0$ which proves the simplicity of the pole of ${\mathcal{M}}(z)$ at $z=0$.
The existence of the pole at $z=\infty$ is proven using the property
${\mathcal{M}}(z)=-1/z^{2}{\mathcal{M}}(1/z)$ which concludes the proof. ∎

Similar technical lemma has been proven and used in [15]. We are now in
position to provide the following proposition:

######  Proposition 3.4.

The following relation holds

$\displaystyle\mathbb{B}(\boldsymbol{z})B(x,L+1)\Omega=\widehat{\lambda}(x)\mathbb{V}(\boldsymbol{z})+\sum_{p=1}^{L}\frac{(z_{p}-1/z_{p})\operatorname{Res}_{x=z_{p}}\big{(}\widehat{\lambda}(x)\big{)}}{(z_{p}-x)(z_{p}-1/x)}\mathbb{V}(\boldsymbol{z}_{p},x)$

with $\boldsymbol{z}=\\{z_{1},z_{2},\dots,z_{L}\\}$ and

$\displaystyle\widehat{\lambda}(x)=\frac{\alpha(L+1)}{\delta(x)}\prod_{q=1}^{L}\left(\frac{(x-v_{q})(x-1/v_{q})}{(x-z_{q})(x-1/z_{q})}\right).$

###### Proof.

The residues of $\mathcal{M}(z)\Omega$ are obtained by straightforward
computations:

$\displaystyle\operatorname{Res}_{z=z_{p}}(\mathcal{M}(z)\Omega)=\frac{1}{\delta(z_{p})}\prod_{\genfrac{}{}{0.0pt}{2}{q=1}{q\neq
p}}^{L}\left(\frac{(z_{p}-v_{q})(z_{p}-1/v_{q})}{(z_{p}-z_{q})(z_{p}-1/z_{q})}\right)\left(\frac{(z_{p}-v_{p})(z_{p}-1/v_{p})}{(z_{p}-x)(z_{p}-1/x)}\right)\mathbb{V}(\boldsymbol{z}_{p},x),$
$\displaystyle\operatorname{Res}_{z=x}(\mathcal{M}(z)\Omega)=\frac{1}{\delta(x)}\prod_{q=1}^{L}\left(\frac{(x-v_{q})(x-1/v_{q})}{(x-z_{q})(x-1/z_{q})}\right)\mathbb{V}(\boldsymbol{z}),$
$\displaystyle\operatorname{Res}_{z=0}(\mathcal{M}(z)\Omega)=-\frac{1}{\alpha(L+1)}\mathbb{B}(\boldsymbol{z})B(x,L+1)\Omega.$

By using ${\mathcal{M}}(z)=-1/z^{2}{\mathcal{M}}(1/z)$, we deduce that
$\operatorname{Res}_{z=1/z_{0}}(\mathcal{M}(z)\Omega)=\operatorname{Res}_{z=z_{0}}(\mathcal{M}(z)\Omega)$
where $z_{0}=0$, $z_{1}$, $z_{2},\dots,z_{L}$, $x$. The proof is concluded by
using the fact that the sum over all the residues (with the point at infinity)
of a rational function vanishes. ∎

Therefore, by using relation (3.12) for $M=L$ and the result of Proposition
3.4, we prove that $\mathbb{V}(\boldsymbol{z})$ (for
$\boldsymbol{z}=\\{z_{1},z_{2},\dots,z_{L}\\}$) is an eigenvector of the
transfer matrix $t(x)$ if the Bethe roots satisfy the following Bethe
equations

$\displaystyle a(z_{p})+\nu(z_{p})+\sum_{\genfrac{}{}{0.0pt}{1}{q=1}{q\neq
p}}^{L}\omega(z_{p},z_{q})=\frac{(L+1)c(z_{p})}{4z_{p}}\operatorname{Res}_{x=z_{p}}\big{(}\widehat{\lambda}(x)\big{)},\qquad\text{for
}p=1,2,\dots,L.$ (3.17)

To get the previous relation, we have used the functional relation

$\displaystyle-\frac{(L+1)(z-1/z)c(x)}{2f(1/z,1/x)(z-x)(z-1/x)}=\frac{(L+1)c(z)}{4z}.$

Explicitly, they become

$\displaystyle-\frac{1}{2}\sum_{j=1}^{L}\left(\frac{z_{p}+v_{j}}{z_{p}-v_{j}}+\frac{z_{p}v_{j}+1}{z_{p}v_{j}-1}\right)+\nu(z_{p})+\sum_{\genfrac{}{}{0.0pt}{1}{q=1}{q\neq
p}}^{L}\left(\frac{z_{p}+z_{q}}{z_{p}-z_{q}}+\frac{z_{p}z_{q}+1}{z_{p}z_{q}-1}\right)$
$\displaystyle\qquad{}=\frac{(\rho^{2}-\beta^{2})(L+1)^{2}}{16\rho^{2}}\frac{(z_{p}-v_{p})(z_{p}-1/v_{p})}{\delta(z_{p})\delta(1/z_{p})(z_{p}^{2}-1)}\prod_{\genfrac{}{}{0.0pt}{1}{q=1}{q\neq
p}}^{L}\frac{(z_{p}-v_{q})(z_{p}-1/v_{q})}{(z_{p}-z_{q})(z_{p}-1/z_{q})},$

where $\nu(z)$ is given by (2.11). The corresponding eigenvalue is

$\displaystyle\lambda(x)-\frac{L+1}{2}c(x)\widehat{\lambda}(x)+\sum_{p=1}^{M}\omega(x,z_{p})\left(a(x)+\nu(x)+\sum_{q\neq
p}\frac{(x-1/x)\omega(z_{p},z_{q})}{z_{p}-1/z_{p}}\right).$ (3.18)

Then the l.h.s. of the Bethe equations is similar to the ones for the diagonal
or triangular boundaries. The r.h.s. corresponds to the inhomogeneous part of
the TQ relation discovered in [11, 12] to solve the XXZ spin chain with twist
or boundaries.

##  4 Higher spin Gaudin models

In this section, we generalize the previous construction and consider the
higher spin Gaudin models. Indeed, it is well-known that we can find
representations of the algebra $\mathscr{R}$ using different representations
of $\mathfrak{sl}(2)$ at each sites. Indeed, the following matrix
$\widetilde{\mathscr{K}}(x)$ satisfies the defining relation (2.2):

$\displaystyle\widetilde{\mathscr{K}}_{0}(x)=\sum_{j=1}^{L}\mathscr{L}^{(s_{j})}_{0j}(x,v_{j}),$
(4.1)

where $s_{j}=1/2,1,3/2,\dots$ is the spin of the representation at the sites
$j$ and

$\displaystyle\mathscr{L}^{(s)}(x,y)=\begin{pmatrix}-\omega(x,y)S^{z}&2b(x)S^{z}+f(y,x)S^{-}\\\
2c(x)S^{z}-f(1/y,1/x)S^{+}&\omega(x,y)S^{z}\end{pmatrix}.$ (4.2)

In (4.2), $S^{z}$, $S^{+}$ and $S^{-}$ are the usual matrices representing the
$\mathfrak{sl}(2)$ generators for spin $s$. The only necessary knowledge about
these matrices for the following constructions is the fact that there exists a
highest-weight vector $w_{s}$ such that

$\displaystyle S^{z}w_{s}=sw_{s}\qquad\text{and}\qquad
S^{+}w_{s}=0\qquad\text{for}\quad s=1/2,1,3/2,\dots$

For $s_{1},s_{2},\dots,s_{L}=\frac{1}{2}$, the representation (4.1) becomes
(2.10). Let us emphasize that the representation of $\mathfrak{sl}(2)$ may be
different at each sites.

Now, we can generalize the results of Section 3. All the results of Section
3.1 remains unchanged since it is independent of the representation.

The pseudo-vacuum becomes333In this section, we keep the same notations for
all the objects which generalize the objects of the previous sections.

$\displaystyle\Omega=w_{s_{1}}\otimes w_{s_{2}}\otimes\dots\otimes w_{s_{L}}.$

Then, relation (3.12) is formally unchanged except the definition of the
function $a(x)$ which becomes

$\displaystyle a(x)=-\sum_{j=1}^{L}\omega(x,v_{j})s_{j}$ (4.3)

and the $L$ in the first line must be replaced by
$\overline{L}=2\sum\limits_{p=1}^{L}s_{p}$.

Then, there is again three possibilities to pursue the computation:

  * •

The triangular case corresponding to $c(x)=0$. The Bethe equations and the
eigenvalues are given respectively by (3.14) and (3.16) with $a(x)$ given by
(4.3). The number $M$ ranges now from 0 to $\overline{L}$.

  * •

The odd case $M=\frac{\overline{L}-1}{2}$. The discussion of Section 3.4 is
still valid.

  * •

The even case. To deal with this case, we must generalize the Lemma 3.3. This
lemma is still valid if we define ${\mathcal{M}}(z)$ as follows

$\displaystyle{\mathcal{M}}(z)=\prod_{q=1}^{L}(z-v_{q})^{2s_{q}}(z-1/v_{q})^{2s_{q}}\prod_{q=1}^{\genfrac{}{}{0.0pt}{2}{\longrightarrow}{\overline{L}}}\frac{B(z_{q},q)}{(z-z_{q})(z-1/z_{q})}$
$\displaystyle\hphantom{{\mathcal{M}}(z)=}{}\times\frac{(z-1/z)}{(z-x)(z-1/x)}B(x,\overline{L}+1)\big{(}\delta(z)B(z,\overline{L}+1)\big{)}^{-1}.$

Let us emphasize that the first product is up to $L$, the number of sites, and
the second up to $\overline{L}$. The demonstration is very similar to the one
given previously except we must use $(S^{-})^{2s+1}=0$ for a spin $s$
representation. Then, the Bethe equations and the eigenvalues are given
respectively by (3.17) and (3.18) with $L$ replace by $\overline{L}$, $a(z)$
given by (4.3) and

$\displaystyle\widehat{\lambda}(x)=\frac{\alpha(\overline{L}+1)}{\delta(x)}\prod_{q=1}^{L}(x-v_{q})^{2s_{q}}(x-1/v_{q})^{2s_{q}}\prod_{q=1}^{\overline{L}}\frac{1}{(x-z_{q})(x-1/z_{q})}.$

### Acknowledgements

I thank P. Baseilhac, S. Belliard and V. Caudrelier for their interest. This
work has been done during the stay of the author at the “Laboratoire de
Mathématiques et Physique Théorique CNRS/UMR 7350, Université de Tours”. I
thank warmly the LMPT for hospitality.

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