# Automorphic Symmetries, String integrable structures and Deformations

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###### Abstract

We address the novel structures arising in quantum and string integrable
theories, as well as construct methods to obtain them and provide further
analysis. Specifically, we implement the automorphic symmetries on periodic
lattice systems for obtaining integrable hierarchies, whose commutativity
along with integrable transformations induces a generating structure of
integrable classes. This prescription is first applied to 2-dim and 4-dim
setups, where we find the new $\mathfrak{sl}_{2}$ sector,
$\mathfrak{su}(2)\oplus\mathfrak{su}(2)$ with superconductive modes,
Generalised Hubbard type classes and more. The corresponding 2- and 4-dim $R$
matrices are resolved through perturbation theory, that allows to recover an
exact result. We then construct a boost recursion that allows to address the
systems, whose $R$-/$S$-matrices exhibit arbitrary spectral dependence, that
also is an apparent property of the scattering operators in $AdS$
integrability. It is then possible to implement the last for Hamiltonian
Ansätze in $D=2,3,4$, which leads to new models in all dimensions. We also
provide a method based on a coupled differential system that allows to resolve
for $R$ matrices exactly. Importantly, one can isolate a special class of
models of non-difference form in 2-dim case (6vB/8vB), which provides a new
structure consistently arising in $AdS_{3}$ and $AdS_{2}$ string backgrounds.
We prove that these classes can be represented as deformations of the
$AdS_{\\{2,3\\}}$ models. We also work out that the latter satisfy free
fermion constraint, braiding unitarity, crossing and exhibit deformed
algebraic structure that shares certain properties with $AdS_{3}\times
S^{3}\times\mathcal{M}^{4}$ and $AdS_{2}\times S^{2}\times T^{6}$ models. The
embedding and mappings of known $AdS_{\\{2,3\\}}$ models to 6vB/8vB
deformations are demonstrated, along with a discussion on the associated
candidates of sigma models.

School of Mathematics & Hamilton Mathematics Institute
Trinity College Dublin
Dublin, Ireland

##  1 Summary

The structure of present dissertation is given as follows, we start by
introducing the necessary objects, apparatus and properties that will underlie
the essence of the questions we raise, e.g. finding the novel integrable
structures in various dimensions, the obstacles arising in the classification
program, development of a novel method for the latter, identification of the
properties, relations to other integrable sectors and many more 2.

First of all, we develop a method based on automorphic boost symmetries, which
allows to investigate the existence of a new integrable sector in two-
dimensional lattice systems. To achieve that, we consider a generic
Hamiltonian Ansatz for a spin chain with 2-dim local space $\mathbb{C}^{2}$.
We implement the boost automorphism for closed spin chains and generate the
local charges, whose commutativity guarantees integrability. The solution
space that follows from resolving a polynomial system of the commutativity
constraint requires a separate treatment for inducing a structure on it. This
can be achieved by the integrable identification transformations, which
isolate associated solution generators, where every such generator defines a
class. Then the corresponding $R$ matrices are worked out by a novel
perturbative bottom-up approach along with all quantum consistency checks. In
this base setting, not only a full known model classification is achieved, but
also novel integrable structures are found in the $\mathfrak{sl}_{2}$ sector,
whose underlying quantum algebra still remains to be resolved. In addition,
the applied technique provided completion for the $\mathbb{C}^{1|1}$ sector
and established graded mapping, Sec. 3.

The four-dimensional analysis followed from the boost method discovered in the
first part. The Hubbard symmetry algebra has been taken as a base for
constructing Hamiltonian Ansätze in 4-dim. By appropriate extensions of the
approach, 8 new integrable classes have been found, the five of which followed
from distinct representations of $\mathfrak{su}(2)\oplus\mathfrak{su}(2)$ and
the rest three came from the so called Generalised Hubbard Models. The GHM
sector was obtained from the separate construction, that contained Hubbard
kinetic part, generilised hopping, flipping and potential terms up to
quadruple order in number operators. A separate method based on
$\mathcal{H}$-$R$ differential system was proposed and proven to find all the
underlying $R$ matrices satisfying all consistency conditions. The discussion
on resolution of the new models through ABA and QSC is provided, Sec. 4.

At the next stage, a generalisation to higher dimensions and arbitrary
spectral dependence is considered, Sec. 5. Namely, one can address the charge
$\mathcal{Q}$ automorphic recursion for the systems, whose $R$-/$S$-matrix
exhibits non-difference (non-additive) spectral dependence, which is
characteristic for an integrable model classes arising in $AdS_{n}$
backgrounds. In what we derive the boost generator that respects arbitrary
spectral dependence and seek for the new structures in $D=2,3,4$. In two
dimensions four new models found that are relevant for the $AdS_{\\{2,3\\}}$
integrable string backgrounds. The 3-dim sector contains nontrivial pseudo-
difference models exceptionally in the sub-15-vertex classes. In dimension
four, we find a perturbative support for a new structure to exist in the
q-deformed sector of $AdS_{5}\times S^{5}$. It have been also noted that some
of the difference form models do not induce the non-difference form analogues.
A comparison analysis of formalism with both spectral dependence types is
performed. The $R$-sector is resolved by constructing an approach of coupled
Sutherland type system, that leads to non-linear differential system, solution
of which establishes exact $R$ matrices.

Consequently the 2-dim sector of arbitrary spectral form (or pseudo-
difference) is addressed for its properties and algebraic structure.
Specifically, it is 6- and 8-vertex models, that are identified as B class
that are relevant for $AdS_{3}$ and $AdS_{2}$ integrable deformations, Sec. 6,
7. In the first part we prove that these models satisfy the free-fermion
condition, braiding unitarity and crossing symmetry. It is shown that free
fermions provide useful constraints for modified ABA of $AdS$ models with
background fluxes. In the last part we prove an algebra of 6vB and 8vB arise
in deformed form w.r.t. $AdS_{3}\times S^{3}\times\mathcal{M}^{4}$ and
$AdS_{2}\times S^{2}\times T^{6}$ models. The 6vB/8vB were established as
$AdS_{\\{2,3\\}}$ deformations, in this respect, the corresponding mappings
and embeddings of known $AdS$ models along with their deformations have been
resolved.

###### Contents

  1. 1 Summary
  2. 2 Introduction
    1. 2.1 Automorphism generators and charges
    2. 2.2 Quantum system on the lattice and Master Symmetries
    3. 2.3 $AdS_{n}$ integrability
  3. 3 $\mathbb{V}^{2}$-difference sector
    1. 3.1 Method
      1. 3.1.1 Automorphisms and $\mathcal{B}$-operator
      2. 3.1.2 Ansatz and Algebraic system
      3. 3.1.3 Integrable transformations
      4. 3.1.4 $H$ $\Leftrightarrow$ $R$-matrix and consistency
    2. 3.2 2-dim space
      1. 3.2.1 Solution space
      2. 3.2.2 The 8-vertex and subspace
      3. 3.2.3 The new classes
      4. 3.2.4 Structure analysis of the new classes
      5. 3.2.5 Graded mapping
  4. 4 $\mathbb{V}^{4}$-difference sector: New superconductive integrability and GHM
    1. 4.1 4-dim setup
      1. 4.1.1 $\mathcal{B}$ and $\mathcal{H}$
      2. 4.1.2 Oscillator basis
      3. 4.1.3 Integrable identifications
    2. 4.2 Hubbard type: $2\oplus 1\oplus 1$ sector
      1. 4.2.1 $\mathcal{H}_{\text{Hub}}$ Ansatz
      2. 4.2.2 Hamiltonian solution space
      3. 4.2.3 $R$ matrices
      4. 4.2.4 The novel models: XXX deformations and Coupled $\psi\psi$-models
      5. 4.2.5 Completeness and comparison analysis
    3. 4.3 Chains: $2\oplus 2$ sector
    4. 4.4 Generalised Hubbard sector
  5. 5 Non-difference sector and generalisation for strings
    1. 5.1 Method generalisation
      1. 5.1.1 Setting
      2. 5.1.2 Identification transformations
    2. 5.2 Two dimensions
    3. 5.3 Three dimensions
      1. 5.3.1 Densities
      2. 5.3.2 $R$-matrices
    4. 5.4 Four dimensions
      1. 5.4.1 $\mathfrak{su}(2)\oplus\mathfrak{su}(2)$
  6. 6 Free fermions in $AdS_{2,3}$
    1. 6.1 $AdS_{\\{2,3\\}}$ backgrounds
    2. 6.2 $AdS$ 8v classes and Free fermions
    3. 6.3 $AdS_{3}$ with RR-flux: $R$ and $t$ analysis
  7. 7 $AdS_{\\{2,3\\}}$ integrable deformations
    1. 7.1 $AdS$ string integrable structure
    2. 7.2 Blocks: Structure of vertex models
    3. 7.3 $AdS$ Deformations
      1. 7.3.1 Prescription
      2. 7.3.2 6vB as $AdS_{3}$ deformation
      3. 7.3.3 6vB sector: Symmetries of $AdS_{3}$ deformation
      4. 7.3.4 $AdS_{\\{2,3\\}}$ deformations in 8vB
      5. 7.3.5 8vB sector: Symmetries of $AdS_{\\{2,3\\}}$ deformations
  8. 8 Conclusions and Remarks
  9. 9 Acknowledgments
  10. A Spin chain operators and algebras
    1. A.1 $\mathbb{V}=\mathbb{C}^{2}$
    2. A.2 Graded spaces and algebra
    3. A.3 Hopf algebras and tensor algebraic properties
  11. B Hubbard type: $\mathfrak{su}(2)\oplus\mathfrak{su}(2)$ 4-dim representations
  12. C (Non-)difference apparatus and computations
    1. C.1 Difference and Non-difference $\mathcal{B}$oost construction
    2. C.2 Elliptic Functions and properties
    3. C.3 6v computation
    4. C.4 8vB limits
  13. D AdS integrable backgrounds
    1. D.1 Sigma models and $q$-deformed $R$-matrix
      1. D.1.1 $\mathfrak{psu}(1,1|2)_{\text{c.e.}}$ superalgebra
      2. D.1.2 $q$-deformed $AdS_{3}\times S^{3}\times\mathcal{M}^{4}$
    2. D.2 Free fermion properties: Recursion for $AdS_{3}$ $T$-matrix
    3. D.3 $AdS_{\\{2,3\\}}$ crossing symmetry
      1. D.3.1 6v $AdS_{3}$
      2. D.3.2 8vB $\mathbf{AdS_{3}}$
      3. D.3.3 8vB $\mathbf{AdS_{2}}$
    4. D.4 $AdS_{3}\times S^{3}\times\mathcal{M}^{4}$ string sector
      1. D.4.1 $AdS_{3}\times S^{3}\times T^{4}$
      2. D.4.2 $AdS_{3}\times S^{3}\times S^{3}\times S^{1}$
      3. D.4.3 $AdS_{3}\times S^{3}\times S^{3}\times S^{1}\mapsto AdS_{3}\times S^{3}\times T^{4}$

##  2 Introduction

It is known that many interacting field theoretic setups exhibit number
important physical properties, where at the same time computation of their
specific observables involves resolution of infinities a.k.a. different
regularisation schemes and additional technical tools. One of the ways to
regulate classes of field theoretic models is the spatial discretisation. This
procedure allows to bring field (continuous) models to finite-volume models
with finite degrees of freedom (dof). Another important characteristic of a
class of field theory models is integrability, which initially stemmed from
the classical notion of being (infinite-dimensional) Hamiltonian integrable
system [1], and later quantum which exhibited (in)finite conserved quantum
symmetries [2], that in turn, define complete exact solvability of such models
through a variety of techniques. Combining both notions of integrability and
space discreteness we obtain a very rich class of integrable models and
techniques deriving from them.

###  2.1 Automorphism generators and charges

It is well known there exists a variety of quantum field and statistical
theories that respect factorised scattering in one or the other form. The
fundamental characteristic of the last is an infinite tower of conservation
laws (quantities), which makes them exactly solvable. It can be also
established that if an $R$\- or $S$-matrix accordingly satisfy YB constraint
(2.23)

$S_{12}S_{13}S_{23}=S_{23}S_{13}S_{12}\qquad
S_{ij}\in\text{End}\left(\mathbb{V}\otimes\mathbb{V}\otimes\mathbb{V}\right)$
(2.1)

then the corresponding transfer matrix $t(x)$

$t(u)=\text{Tr}_{0}\prod_{i=1}^{L}S_{0i}\qquad
t(u)t(v)=\text{Tr}_{a}\text{Tr}_{b}\,\prod_{i=1}^{L}\left[S_{ai}(u)S_{bi}(v)\right],$
(2.2)

As indicated in (2.2), the trace is taken over the product of $S$-operators
($S$-monodromy), which describe scattering of particles of various types Fig.
1. The commutativity of $t$ implies its diagonalisation through known
integrability methods [3, 4].

Figure 1: Auxiliary null-space tracing of $S$ with particle species $m_{k}$
and $n_{k}$

In general for the quantum transfer matrix $t$ to exist, there must exist a
conserved quantum monodromy based on asymptotic particle states, which
satisfies $RTT$-factorisation and the discrete analogues of
$\mathcal{P}\mathcal{T}$ can be realised in the underlying quantum theory.
Given prescription along with unitarity relations proposed in [5, 6, 7, 8, 9]
is taken to be conventional, since it also permits to construct the
corresponding conserved charges by perturbing (2.2) and obtain induced
commutativity [3, 10].

Remarkably, from YB relations one can establish a group of transformations
$\mathcal{G}_{\bar{L}}$ on the hierarchy of conserved charges, which can be
proven to constitute the discrete analogue of the Lorentz symmetry. Indeed
this group of continuous transformations analogous to the Lorentzian one has
direct implications for integrable systems on the lattice. In fact, the
Lorentz boosts identify the set of transformations on the pairwise commuting
integrals of motion (conserved charges, including Hamiltonian). It is also
possible to consider a continuous limit of this lattice Lorentz group, in
which case it can decomposed into Lorentz group and a group of transformations
of higher integrals of motion [11]. Specifically, one can also consider
parametric series of the given lattice group, where infinite tower of boost
generators is mapped to an infinite set of motion integrals.

We can also note this from the continuous setting. If we consider the Lorentz
transformation of a certain generating and conserved observable, like a
current $\mathcal{J}^{\mu}(x)$ of the underlying quantum theory

$\mathcal{U}\mathcal{J}^{\mu}(x)\mathcal{U}^{-1}=\Lambda_{\nu}^{\mu}\,\mathcal{J}^{\nu}(\Lambda
x)$ (2.3)

where $\mathcal{U}\left[\Lambda(u)\right]$ and $\Lambda_{\mu}^{\nu}$ is
Lorentz operator with fugacity u. In addition, the Lorentzian limit defines
the boost $\mathcal{B}$ and its commutation properties provide an (integrable)
algebraic structure

$\mathcal{B}=\operatorname{d\\!}_{\,u}\Lambda(u)\big{|}_{u=0}\qquad\left[\mathcal{B},\mathcal{J}^{\mu}_{a}(x)\right]=\epsilon_{\alpha\beta}\,x^{\alpha}\partial^{\beta}\mathcal{J}^{\mu}_{a}(x)+\epsilon^{\mu\beta}\mathcal{J}_{a,\,\beta}^{\mu}$
(2.4)

where $a$ indicates symmetry indices. By looking at the commutation (2.4), we
can get commutativity with the charge [12]

$\left[\mathcal{B},Q_{a}\right]=0\qquad\text{with}\quad\mathcal{J}_{a}(\pm\infty)\equiv
0,\quad
Q_{a}=\int_{-\infty}^{+\infty}\operatorname{d\\!}\,x\mathcal{J}_{a}^{0}(x)$
(2.5)

In fact one could achieve an equivalent result with higher currents
(accounting for quantum corrections) by looking at commutativity with the
stress-energy tensor $T^{\mu\nu}$ of the system [13]. It can be immediately
seen that the above mentioned decomposition and higher symmetries are
reflected not only at the level of the Poincaré algebra, but would also admit
infinite-dimensional extension of internal symmetries, where the boost
manifests its automorphic nature, i.e.

$\text{Poincar\'{e}:}\qquad\left[\mathcal{P}_{-},\mathcal{P}_{+}\right]=0,\quad\left[\mathcal{B},\mathcal{P}_{\pm}\right]=\pm\mathcal{P}_{\pm}$
(2.6) $\text{Yangian supplement }\mathcal{Y}\left[\mathfrak{g}\right]\text{:
}\quad\begin{cases}\quad\left[\mathcal{P}_{\pm},Q_{a}\right]=0\\\
\quad\left[\mathcal{B},Q_{a}\right]=0\\\
\end{cases}\quad\begin{array}[]{l}\quad\left[\mathcal{P}_{\pm},\,^{(1)}Q_{a}\right]=0\\\
\quad\left[\mathcal{B},\,^{(1)}Q_{a}\right]=-\mathfrak{C}\dfrac{\hbar}{4\pi
i}Q_{a}\\\ \end{array}$ (2.7)

where $\mathcal{P}_{\pm}$ are lightcone translations, $\mathfrak{C}$ is the
Casimir of the enveloping algebra and by ${}^{(n)}Q_{a}$ we shall indicate
level-$n$ Yangian generators (internal generating symmetry). We can notice
that the boost $\mathcal{B}$ generates nontrivial internal symmetry extension
(2.34)-(2.37).

As we shall see further, this formalism can be translated into discrete case
upon appropriate structure resolutions and conditions. However it is also
important to note that boost construction (2.4) is characteristic for
continuous systems and is distinctive from the boost automorphism for
integrable systems in discretised space. Moreover for a number of integrable
models it is not clear if there exist mapping limits (e.g. irrelevantly
deformed 2-dim CFTs [14, 15, 16, 17, 18]). Particularly, it would allow to
identify novel integrable structures and provide full control over symmetries.
Let us now analyse in greater detail how to implement the automorphic
apparatus and what are implications of the above conserved quantities for
lattice systems.

###  2.2 Quantum system on the lattice and Master Symmetries

Despite the fact that field theories exhibit very distinct structure,
properties and interpretation, it is important to acquire maps from continuous
symmetries to their discrete analogues in $1+1$-dim. Indeed, by construction
the symmetry algebra above along with $\mathcal{Y}\left[\mathfrak{g}\right]$
internal extension, possesses compatible discretised structure in the Hopf
realisation (algebraic actions, quotients) [19, 20, 21]. Moreover all
operations can be straightforwardly replaced by their discrete counterparts
(space stratification, locality, summation etc). Importantly as we shall show
below, these discontinuous objects must also close to a symmetry algebra.

Such integrable systems are defined on the discretised spaces or lattices,
hence the name of the class - lattice integrable models. In particular, by
lattice models we shall mean Integrable Spin Chains (ISC), which are quantum
spin models of diverse group-theoretic and physical properties. The canonical
definition of the spin chain includes the collection of lattice sites
assembled into spin chain of length $L$, with local quantum space
$\mathfrak{h}_{i}$ assigned to each site, hence the Hilbert space
$\mathfrak{H}_{L}$ of the system

$\mathfrak{H}_{L}=\bigotimes_{i=1}^{L}\mathfrak{h}_{i}$ (2.8)

where a variety of spin chain systems can include different dimensionality and
representations.

Figure 2: Heisenberg spin chain (magnet class) with Hilbert space
$\mathfrak{H}$ formed by local quantum space $\mathfrak{h}$ (spins) at site
$i$.

Specifically at each local space $\mathfrak{h}_{i}$ there is some spin
orientation $\mathfrak{s}_{i}$, which forms a state in the system

$\ket{\psi_{s}}=\ket{\cdots\mathfrak{s}_{i}\,\mathfrak{s}_{i+1}\,\mathfrak{s}_{i+2}\cdots}\quad\qquad\ket{\psi_{s}}\in\mathcal{H},\,\,\mathfrak{s}_{k}\in\mathfrak{h}_{k}$
(2.9)

where $\mathfrak{s}_{k}$ corresponds to a local basis vector assigned to
$\mathfrak{h}_{k}$. It can be clearly illustrated by the example of a spin
chain with two-dimensional local quantum space
$\mathfrak{h}_{i}=\mathbb{C}^{2}$. Specifically, one can define generic
spin-$\dfrac{1}{2}$ Heisenberg Hamiltonian of the system in an external
magnetic field

$H_{XYZ}^{\varkappa}=-\dfrac{1}{2}\sum_{i=1}^{L}J_{x}\sigma_{i}^{x}\sigma_{i+1}^{x}+J_{y}\sigma_{i}^{y}\sigma_{i+1}^{y}+J_{z}\sigma_{i}^{z}\sigma_{i+1}^{z}+\varkappa\sigma_{i}^{z}$
(2.10)

where $J_{i}$ are components of a coupling constant, Pauli matrices
$\sigma_{i}$ and external field $\varkappa$. In fact, we shall be considering
L-site closed spin chains with quantum spin-operators $S_{i}$ that obey
periodicity and algebra on it A.1

$S_{L+1}^{\mu}\equiv
S_{1}^{\mu}\qquad\left[S_{j}^{\mu},S_{k}^{\nu}\right]=i\delta_{jk}\varepsilon_{\mu\nu\rho}S_{j}^{\rho}$
(2.11)

where for spin-$\dfrac{1}{2}$ case $S_{j}^{\mu}$ are Pauli operators
$S_{j}^{\mu}=\dfrac{\sigma_{j}^{\mu}}{2}$. One can further proceed with the
set of reduced models, which are obtained through identifications of the
coupling of the so called XYZ-model from above. Namely, for $J_{y}=J_{x}$ one
obtains XXZ-model or the XXX-model for $J_{x}=J_{y}=J_{z}=J$. XYZ, XXZ and
XXX-model are characterised by Jacobi elliptic, trigonometric and rational
function, hence the name of the functional sectors they define. It is possible
to reduce even further by removing interaction along specific directions, for
instance one can get XY-model for $J_{z}=0$ or quantum Ising chain in the
transversal field with $J_{y}=J_{z}=0$.

Figure 3: Distinct boundary conditions correspond to different types of
integrable spin chains. Here $\partial\mathrm{\Sigma}_{i}$ constitute specific
boundary and $V_{i}\in\mathbb{C}^{n}$ is $n$-dimensional local quantum space
at site $i$.

It is also important to distinguish between types of the spin chains depending
on their boundary condition Fig. 3. Since different boundary conditions
apriori lead to a variety of integrable systems that significantly differ in
the spectrum and properties. One can classify 5 main boundary types for the
spin chain

  * •

Infinite spin chain: $-\infty\leftarrow\partial\mathrm{\Sigma}_{1}$,
$\partial\mathrm{\Sigma}_{2}\rightarrow+\infty$

  * •

Semi-infinite spin chain: $\partial\mathrm{\Sigma}_{1}=V_{0}$,
$\partial\mathrm{\Sigma}_{2}\rightarrow+\infty$

  * •

Open spin chain: $\partial\mathrm{\Sigma}_{1}=V_{0}$,
$\partial\mathrm{\Sigma}_{2}=V_{L+1}$

  * •

Closed spin chain:
$(0)\rightarrow(L+1),\,\,\partial\mathrm{\Sigma}_{1}=\partial\mathrm{\Sigma}_{2}=V_{L+1}$

  * •

Cyclic spin chain:
$\partial\mathrm{\Sigma}_{1}=\partial\mathrm{\Sigma}_{2}=V_{L+1}$ along with
shift symmetry111in general the level-one Yangian does not commute with shift
operator $\left[\mathcal{U},^{(1)}\mathcal{Y}[J_{a}]\right]\neq 0$
$i\rightarrow i\pm 1$, and the total momentum of the spin chain excitations
vanishes $P=0$.

As it was stated above, the central characteristics of an integrable spin
chain are symmetries and associated conserved charges, which can be
represented in the hierarchy set
$\\{\mathbb{Q}_{1},\mathbb{Q}_{2}\dots\mathbb{Q}_{r}\dots\mathbb{Q}_{n}\\}$
and completely define the underlying integrable system. In the context of spin
chains the first two charges are usually assigned to translations/momentum
$\mathbb{Q}_{1}\equiv P$ and Hamiltonian $\mathbb{Q}_{2}\equiv\mathbb{H}$. For
the case of a spin chain with interacting adjacent sites, it is identified as
a chain with the nearest-neighbour (NN) interaction. In the present framework,
$\mathbb{H}$ is exactly an operator that defines NN-interactions. So the
charge $\mathbb{Q}_{r}$ that spans $r$-sites constitutes an interaction range
$r$. It is sometimes referred to as higher charge Fig. 4 (or higher rank
Hamiltonians as it will become clear below). It means that on the spin chain
we can define local densities of the appropriate operators, the sum of which
will correspond to the full operator on the given spin chain

Figure 4: Local charge densities acting on the spin chain

$\mathbb{Q}_{2}=\mathbb{H}=\sum_{i=1}^{L}\mathcal{H}_{i,i+1}\qquad\qquad\mathcal{H}_{L,L+1}=\mathcal{H}_{L,1}$
(2.12)

with Hamiltonian density $\mathcal{H}$ at $\left(i,i+1\right)$-sites and the
spin chain periodicity (closure condition). So in general, the core integrable
structure is encoded in mutual commutativity of the above

$\left[\mathbb{Q}_{r},\mathbb{Q}_{s}\right]=0\qquad\mathbb{Q}_{x}\equiv\sum_{k=1}^{L}\mathcal{Q}_{k,\,k+1,\,\dots\,,\,k+x-1}$
(2.13)

where $\mathbb{Q}_{x}$ indicate range $x$ charge. Although we need to remark
that such labelling is not always possible. It is a class of spin chains that
produces such consistent mapping. It will be explained that automorphism
generating property and periodic boundary conditions are important for this.
Characteristic properties of these charges are locality and homogeneity.
Namely, the $\mathcal{Q}_{x}$ densities act locally on a given number of spins
$x$, interaction is isotropic w.r.t. all sites and all charges respect a group
$G$ symmetry with an underlying algebra $\mathfrak{g}$. However by
implementing the boost construction from above, one still needs an additional
object to strictly prove integrability and uniqueness of the novel class.

Such a fundamental object turns out to carry scattering data, braiding
symmetry and subsequently the spectrum generating information for the quantum
integrable model – quantum $R$-matrix. For the construction, it is necessary
to consider a vector space $\mathbb{V}$ over field $\mathfrak{F}$ and a linear
map

$R:\mathbb{V}\otimes\mathbb{V}\rightarrow\mathbb{V}\otimes\mathbb{V}$ (2.14)

with endomorphic structure on
$R_{ij}\in\text{End}\left(\mathbb{V}\otimes\mathbb{V}\otimes\mathbb{V}\right)$,
where appropriate continuation on identities is assumed

$R_{12}=R\otimes\mathds{1}_{\mathbb{V}}\qquad
R_{23}=\mathds{1}_{\mathbb{V}}\otimes R\qquad
R_{13}=\left(\mathds{1}_{\mathbb{V}}\otimes\mathfrak{t}_{\mathbb{V}\mathbb{V}}\right).\left(R\otimes\mathds{1}_{\mathbb{V}}\right).\left(\mathds{1}_{\mathbb{V}}\otimes\mathfrak{t}_{\mathbb{V}\mathbb{V}}\right)$
(2.15)

with twist map222which in the integrable lattice systems corresponds to
permutation operator $P_{ij}$ $\mathfrak{t}_{\mathbb{V}\mathbb{V}}$ defined as
follows

$\mathfrak{t}_{\mathbb{V}\mathbb{V}}:\mathbb{V}\otimes\mathbb{V}\rightarrow\mathbb{V}\otimes\mathbb{V}\qquad\mathfrak{t}_{\mathbb{V}\mathbb{V}}\left[\mathbb{W}_{1}\otimes\mathbb{W}_{2}\right]\equiv\mathbb{W}_{2}\otimes\mathbb{W}_{1}$
(2.16)

Structure and properties of the quantum $R$-matrix are completely identified
by the main quantum constraint (or more specifically, characterised by the
underlying quantum group structure). The study of braid groups, quantum
symmetries and algebra of quantum operators leads us to the main constraint –
Quantum Yang-Baxter equation (QYBE) 5

$R_{12}\left(z_{1},z_{2}\right)R_{13}\left(z_{1},z_{3}\right)R_{23}\left(z_{2},z_{3}\right)=R_{23}\left(z_{2},z_{3}\right)R_{13}\left(z_{1},z_{3}\right)R_{12}\left(z_{1},z_{2}\right)$
(2.17)

with $z_{i}$ to be a spectral parameter associated to space $i$. In addition,
the $R$-matrix possesses special spectral points, and in particular, reduces
to the permutation operator for coinciding parameters

$R_{12}(z,z)=P_{12}$ (2.18)

which corresponds to the regularity condition. It can be also shown that the
Hamiltonian is obtained from $\log$-derivative of the $R$-matrix A.1

$\mathbb{H}_{12}=R_{12}\left(z_{1},z_{2}\right)^{-1}\partial_{z_{1}}R_{12}\left(z_{1},z_{2}\right)\big{|}_{z_{2}=z_{1}=z}=P_{12}\partial_{z_{1}}R_{12}\left(z_{1},z_{2}\right)\big{|}_{z_{2}=z_{1}=z}$
(2.19)

It is important that for a large class of integrable models it is possible to
obtain reduced spectral dependence of the underlying $R$-matrix. As we shall
see, that will result in significant computational reductions, analytic
properties and form of the derived operators of the new models. Generically,
one can implement the quantum $R$-matrix in the fundamental
$\mathbf{GL}\left(N,\mathbb{C}\right)$ representation
$R_{12}^{*}\left(z\right)\in\text{Mat}\left(N,\mathbb{C}\right)^{\otimes 2}$

$R_{12}^{*}\left(z\right)=\sum_{i,j,k,l=1}^{N}R_{ijkl}^{*}\left(z\right)E_{ij}\otimes
E_{kl}$ (2.20)

where $R_{12}^{*}\left(z\right)$ is $N^{2}\times N^{2}$ matrix, asterisk
indicates presence of a quantum or other characteristic parameter (e.g.
$\hbar$), $z$ is a spectral parameter and $E_{ij}$ constitutes a basis in
$\text{Mat}\left(N,\mathbb{C}\right)$.

Figure 5: QYBE structure graphical representation, which underlies
factorisation of the pairwise consequential scattering.

Moreover we can impose333It is derived independently for several of the novel
classes below additional conditions on the $R$-matrix, one of them respects
braiding and unitarity symmetric property (or unitarity)

$R_{12}^{*}\left(z_{1}-z_{2}\right)R_{21}^{*}\left(z_{2}-z_{1}\right)\simeq{\mathds{1}}\otimes{\mathds{1}}$
(2.21)

with $R_{21}^{*}\left(u\right)=P_{12}R_{12}^{*}\left(u\right)P_{12}$ and
tensor product component permutation $P_{12}=\sum E_{ij}\otimes E_{ji}$. In
the provided setup, one can also generically obtain antisymmetry

$R_{12}^{q_{h}}\left(z_{1}-z_{2}\right)=-R_{21}^{-q_{h}}\left(z_{2}-z_{1}\right)$
(2.22)

with some prescribed quantum parameter $q_{h}$444It would also hold for
associative YBE in classical integrable systems. That would lead to the
special spectral dependence of 2.17, i.e. YBE of the difference form555The
quantum parameter indication on $R$-matrix will be omitted for the rest of the
discussion

$R_{12}R_{13}R_{23}=R_{23}R_{13}R_{12}\qquad
R_{mn}=R_{mn}\left(z_{m}-z_{n}\right)$ (2.23)

or reassociating $R_{13}\left(u\right)$ and $R_{23}\left(v\right)$

$R_{12}(u-v)R_{13}(u)R_{23}(v)=R_{23}(v)R_{13}(u)R_{12}(u-v)$ (2.24)

One of the questions that will be addressed throughout this work is the
classification of the quantum integrable models, which from many perspectives
is a complex problem on its own. For example, a number of attempts for new
integrable models, were based on the resolvability of the Yang-Baxter equation
(2.17) or (2.24), which apriori is not a straightforward task. Moreover, it is
known to be directly solvable only for certain subsectors [22]. A subsector of
integrable lattice vertex models and their relation with Zamolodchikov
tetrahedral algebra, as well as constant YBE solutions were shown in [23, 24,
25, 26]. More recently a classification of $\\{4\text{-}8\\}$-vertex via
differential approach was given in [27, 28].

Figure 6: Monodromy construction scheme

Other crucial constraint that will appear more efficient for derivation of
properties and algebra of models in the non-additive sector 5, 6, 7 – is the
braiding algebra realised by intertwining the monodromy $T(u)$ and the
$R$-matrix,

$R_{0\tilde{0}}(u-v)T_{0}(u)T_{\tilde{0}}(v)=T_{\tilde{0}}(v)T_{0}(u)R_{0\tilde{0}}(u-v)$
(2.25)

where the monodromy $T_{0,\\{1,\dots,L\\}}(u)$ acts on $L$ physical spaces and
one extra space is auxiliary, which we can set through the sequence of the Lax
matrices $\mathcal{L}$ as can noticed from Fig. 6

$T_{m_{1}m_{2}\dots m_{L},\,\nu}^{n_{1}n_{2}\dots
n_{L},\,\mu}=\mathcal{L}_{s_{2}m_{1}}^{\mu\,n_{1}}(u-\theta_{1})\mathcal{L}_{s_{3}m_{2}}^{s_{2}n_{2}}(u-\theta_{2})\dots\mathcal{L}_{\nu
m_{L}}^{s_{L}n_{L}}(u-\theta_{L})$ (2.26)

with inhomogeneities $\theta_{i}$ at site $i$ and $m_{i},n_{j},s_{k}$
correspond to the space position free indices. The notion of
$\mathcal{L}_{i,a}$ as a linear operator involved in auxiliary spectral
problem comes from resolution of integrable systems through Classical Inverse
Scattering Method. Specifically, $\mathcal{L}$ structure is defined on local
quantum $\mathfrak{h}_{i}$ and auxiliary $\mathbb{V}_{a}$ spaces, i.e. its
action is defined on $\mathfrak{h}_{i}\otimes\mathbb{V}_{a}$. In the case of
isotropic XXX${}_{\frac{1}{2}}$ spin chain it takes the form

$\mathcal{L}_{i,a}(u)=u{\mathds{1}}_{i}\otimes{\mathds{1}}_{a}+i\sum_{\mu}S_{i}^{\mu}\otimes\sigma^{\mu}$
(2.27)

where $u$ is a spectral parameter, ${\mathds{1}}_{i}$ and spin operator
$S_{i}$ act in $\mathfrak{h}_{i}$, whereas ${\mathds{1}}_{a}$ and Pauli
operators in $\mathbb{V}_{a}$.

The $RTT$-relation also constitutes one of the main constraints in derivation
of the algebraic structures (Bethe Ansätze, QISM). After all the monodromic
tracing over auxiliary space would result in the transfer matrix $t(u)$, with

$\text{Tr}\,T_{m_{1}m_{2}\dots m_{L},\,\nu}^{n_{1}n_{2}\dots
n_{L},\,\mu}(u)=t_{m_{1}m_{2}\dots m_{L},\,\mu}^{n_{1}n_{2}\dots
n_{L},\,\mu}(u)\qquad
t(u):\mathbb{V}_{1}\otimes\mathbb{V}_{2}\otimes\dots\otimes\mathbb{V}_{L}\rightarrow\mathbb{V}_{1}\otimes\mathbb{V}_{2}\otimes\dots\otimes\mathbb{V}_{L}$
(2.28)

As we shall see below it will become crucial for us to identify the conserved
charges following from the conservation laws, whose origin can be read off the
aforementioned commutativity and perturbation [10]

$\left[t(u),t(v)\right]=0$ (2.29)

along with

$t(u+u^{*})=\mathcal{U}+iu\,\mathcal{U}\sum_{i}\mathcal{Q}_{i,i+1}+\mathcal{O}(u^{2})\equiv\mathcal{U}\exp\left[i\sum_{r=2}^{L}\left(u-\dfrac{i}{2}\right)^{r-1}\mathbb{Q}_{r}\right]$
(2.30)

(a) $\mathcal{U}$ (b) $\mathcal{U}\,\mathbb{Q}_{2}$

Figure 7: First and second order of the expansion (2.30), corresponding to the
shift operator $\mathcal{U}$ and product of the shift and Hamiltonian of the
system

where the $t$ is expanded around special spectral point $\frac{i}{2}$ (one can
also consider expansion around $u^{*}=0$, depending on existence of the
distinguished points and properties of the corresponding $R$-matrix),
$\mathbb{Q}_{r}$ is a range $r$ conserved charge Fig. 4 and $\mathcal{U}$ is a
shift operator (recursive product of permutations) Fig. 7

$\mathcal{U}\ket{\psi_{1},\psi_{2},\dots,\psi_{L}}=\ket{\psi_{2},\dots,\psi_{L},\psi_{1}}\qquad\mathcal{U}\equiv
t\left(\frac{i}{2}\right)$ (2.31)

Hence from (2.30) one can derive a generating expression for range $r$ local
charge ($r\geq 2$)

$\mathbb{Q}_{r}=-\dfrac{i}{(r-1)!}\dfrac{\operatorname{d\\!}^{\,\,r-1}}{\operatorname{d\\!}u^{r-1}}\log\left[t(u)\right]\big{|}_{u=\frac{i}{2}}$
(2.32)

In this respect, if one follows the $RTT$ or $R\mathcal{L}\mathcal{L}$
analysis along with the local charge properties (2.25)-(2.30) and appropriate
boundary limits, one derives an automorphic condition on monodromy $T(u)$

$\left[T,\mathcal{B}\right]=i\,\dot{T}$ (2.33)

where $\dot{T}(u)\equiv\operatorname{d\\!}_{\,\,u}T(u)$ and the boost operator
is associated with the weighted generating density sums

$\mathcal{B}=\sum_{m=-\infty}^{+\infty}m\,\mathcal{H}_{m,m+1}$ (2.34)

Based on the definition of the boost, we consider infinite spin chain setting
with NN-interaction. From above one can see that there is an emergent symmetry
that not only leaves invariant the commutativity of hierarchy, but also
implies a generating automorphism property. These two properties characterise
the master symmetry, which does not affect the commutators after recursive
application and generates higher conserved charges by its action on the lower
ones (2.37), hence automorphically producing infinite commutative hierarchy
[29, 30]. By taking into account (2.30) we can rewrite (2.33) for the transfer
matrix

$\left[t,\mathcal{B}\right]=i\,\dot{t}$ (2.35)

with $\dot{t}(u)=\operatorname{d\\!}_{\,\,u}t(u)$ and along with (2.32), one
can straightforwardly identify the generating scheme

$\begin{array}[]{c}\mathcal{Q}_{2}=\mathcal{U}^{-1}\left[\mathcal{B},\mathcal{U}\right]\\\\[8.61108pt]
\mathcal{Q}_{3}=\frac{i}{2}\left[\,\mathcal{U}^{-1}\left[\mathcal{B},\mathcal{U}\mathcal{Q}_{2}\right]-\mathcal{Q}_{2}\mathcal{Q}_{2}\,\right]=\frac{i}{2}\big{[}\,\underbrace{\left(\mathcal{U}^{-1}\,\mathcal{B}\,\mathcal{U}-\mathcal{Q}_{2}\right)}_{\mathcal{B}}\mathcal{Q}_{2}-\mathcal{Q}_{2}\mathcal{B}\,\big{]}=\frac{i}{2}\left[\mathcal{B},\mathcal{Q}_{2}\right]\\\\[8.61108pt]
\mathcal{Q}_{4}=\frac{i}{3}\left[\mathcal{B},\mathcal{Q}_{3}\right]\\\\[4.30554pt]
\dots\\\\[4.30554pt]
\mathcal{Q}_{r+1}=\frac{i}{r}\left[\mathcal{B},\mathcal{Q}_{r}\right]\\\\[8.61108pt]
\dots\\\\[4.30554pt] \end{array}$ (2.36)

so from the first moment $\mathcal{B}[\mathcal{Q}_{2}]$ it possible to
generate the complete hierarchy,

$\begin{array}[]{l}\qquad\qquad\qquad\quad\mathbf{\mathcal{B}}\text{{oost
automorphism scheme}}\\\\[8.61108pt]
\left[\,\mathcal{Q}_{i}\,,\,\mathcal{Q}_{j}\,\right]=0:\quad\mathcal{Q}_{1}\xrightarrow[]{\quad\mathcal{B}\quad}\mathcal{Q}_{2}\xrightarrow[]{\quad\mathcal{B}\quad}\mathcal{Q}_{3}\xrightarrow[]{\quad\mathcal{B}\quad}\dots\xrightarrow[]{\quad\mathcal{B}\quad}\mathcal{Q}_{r+1}\xrightarrow[]{\quad\mathcal{B}\quad}\end{array}$
(2.37)

which can be noted to be reminiscent of the Lorentz boost for the space and
time translations of the two-dimensional Poincaré algebra.

##### Remark

It is important to note, that strictly speaking the above boost lattice
formalism have been defined on the infinite spin chains. However for the
periodic integrable spin chains, which will be crucial throughout the work the
short-range operators are equivalent to the ones on the infinite spin chain,
as well as any boundary contributions vanish in both cases. Indeed, that is
another natural restriction that charge range must not exceed the length of
the spin chain666Such prescriptions can occur for systems different boundary
conditions, where the spectrum admits an asymptotic analysis and in the
integrable QFT [31, 32] or string system can be resolved by TBA [33, 34, 35,
36]. Assuming this, one can develop generating recursion scheme for the
integrable charges for spin chain with periodic boundary conditions.

In the context of 2.1, it can be noted that this discrete generating
automorphism also implies $\left[\mathcal{B},\,^{(1)}Q_{a}\right]\eqsim
Q_{a}$, which also for closed spin chains is analogous to level-1 Yangian
symmetry generators. Later we shall make it clear on how it is implemented for
the cases considered in the present work (more detailed on the automorphic
structure and implications for generating Yangian symmetries can be found in
C.1).

###  2.3 $AdS_{n}$ integrability

Recently significant progress has been achieved in studying integrable systems
[37, 38, 39, 40, 33] in the context of the Gauge/Gravity duality [41, 42, 43,
44, 45, 46]. Especially important results were in performing various
integrability checks of the holographic duality and developing novel
integrable structures [47, 48, 49, 50] along with their properties. In
particular, the main discoveries lie in the planar limit of $\mathcal{N}=4$
superconformal Yang-Mills theory and free type IIB string on $AdS_{5}\times
S^{5}$. In fact, as in any field theory, one of the most crucial questions is
the computation of the spectrum of $\mathcal{N}=4$ SYM. Since it is a SCFT,
there is an analogue to mass spectrum computation, which corresponds to
obtaining local operators (fundamental field composites) and their scaling
dimension spectrum. Generically, one also needs to account for quantum
corrections stemming from interactions, which results contributing to the
anomalous dimensions of the theory. The core point is that it becomes possible
to determine the spectrum in $\mathcal{N}=4$ planar limit by a variety of
techniques exactly, e.g. Asymptotic Bethe Ansatz, Thermodynamic Bethe Ansatz
[51, 52, 53, 54, 35], Y-systems [55, 56], Quantum Spectral Curve [57, 58, 59]
and other.

At the level of symmetries, the Gauge/Gravity duality states that global
symmetries on both sides must agree, meaning that $\mathcal{N}=4$
superconformal symmetry and superspace isometries of $AdS_{5}\times S^{5}$ are
governed by the covering group $\widetilde{PSU}(2,2|4)$. For that reason, the
Lie superalgebra $\mathfrak{psu}(2,2|4)$ as well as its deformed counterparts
play a central role in both super-CFT and string integrability. The
$\mathfrak{psu}$ superalgebra has dimension $30|32$, which accordingly
constitutes even|odd sectors. It can be represented by the
$\mathbb{C}$-supermatrices

$\mathfrak{M}=\left(\begin{array}[]{ c | c}A&B\\\ \hline\cr C&D\end{array}\right)$ (2.38)

where A,D are even and B,C are odd $4\times 4$ Non-Graßmann
$\mathbb{C}$-blocks. In turn, the supermatrices must also satisfy

$\displaystyle\left.\left[\mathfrak{M},\mathfrak{R}\right.\right\\}=\mathfrak{M}\mathfrak{R}-(-1)^{\mathfrak{M}\mathfrak{R}}\mathfrak{R}\mathfrak{M},\quad
S\text{Tr}\;\mathfrak{M}=\text{Tr}A-\text{Tr}D,\quad
S\text{Tr}\,\left.\left[\mathfrak{M},\mathfrak{R}\right.\right\\}=0$ (2.39)
$\displaystyle(-1)^{\mathfrak{M}\mathfrak{T}}\left.\left[\left.\left[\mathfrak{M},\mathfrak{R}\right.\right\\},\mathfrak{T}\right.\right\\}+(-1)^{\mathfrak{R}\mathfrak{M}}\left.\left[\left.\left[\mathfrak{R},\mathfrak{T}\right.\right\\},\mathfrak{M}\right.\right\\}+(-1)^{\mathfrak{T}\mathfrak{R}}\left.\left[\left.\left[\mathfrak{T},\mathfrak{M}\right.\right\\},\mathfrak{R}\right.\right\\}=0$
(2.40)

The complete holographic existence/consistency checks include different
background geometries and theories. This manifests in dimensionality, amount
of preserved supersymmetry, gauging schemes and many more. The most developed
integrable backgrounds include $AdS_{5}\times S^{5}$ [40, 33],
$AdS_{4}\times\mathbb{CP}^{3}$ [60, 61], $AdS_{3}\times
S^{3}\times\mathcal{M}^{4}$ [50, 62, 63, 64] and $AdS_{2}\times S^{2}\times
T^{6}$ [65, 66, 67]. The last two cases are of particular interest as their
investigation is still incomplete, especially, on the field theoretic side.
These two backgrounds will be analysed in detail in Sec. 7, where a set of
novel $AdS_{\\{3,2\\}}$ deformations will be discussed.

For the $AdS_{3}/CFT_{2}$ integrability one needs to start with the
prescription of Non-Linear Sigma Models in order to get an insight into
analytic structure and symmetries of the model. In particular, onee can
address the bosonic NLSM on $AdS_{3}\times S^{3}\times T^{4}$.

In such framework with additional constraining (e.g. mass sector) and gauge-
fixed functional of BNLSM it is more convenient to identify symmetries (for
more details [68, 69]). For the purpose of underlying integrable structures it
is useful to consider the corresponding superstring action as a coset action
[70]

$\dfrac{PSU(1,1|2)\times PSU(1,1|2)}{SO(1,2)\times SO(3)}\times U(1)^{4}$
(2.41)

One could also start from the Green-Schwarz action, but in this setting the
superisometries are more obscure [71, 72]. On the other hand one can start
with the coset formulation, but a $\kappa$-gauge fixing is necessary
(identification of the physical region for the fermions). To achieve that, one
can consider the closed string bosonic NLSM functional777Fradkin-Tseytlin term
in this prescription is suppressed $R_{(2)}\rightarrow 0$

$S=-\dfrac{h}{2}\int\limits_{-\frac{L}{2}}\limits^{\frac{L}{2}}\operatorname{d\\!}\sigma\operatorname{d\\!}\tau\gamma^{\alpha\beta}\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}G_{\mu\nu}(X)$
(2.42)

along with canonical definition of conjugate momenta

$p_{\mu}=\dfrac{\delta
S}{\delta\dot{X}^{\mu}}=-h\gamma^{0\beta}\partial_{\beta}X^{\nu}G_{\mu\nu}(X),$
(2.43)

which for $AdS_{3}\times S^{3}\times T^{4}$ result in conserved charges
associated with spacetime isometries, i.e. time direction in $AdS_{3}$ and
angle $\phi$ in $S^{3}$. The last two correspond to translations and rotations
along these directions, hence we acquire conserved energy and angular momentum

$H=-\int\limits_{-\frac{L}{2}}\limits^{\frac{L}{2}}\operatorname{d\\!}\sigma
p_{t},\qquad
J=\int\limits_{-\frac{L}{2}}\limits^{\frac{L}{2}}\operatorname{d\\!}\sigma
p_{\phi}$ (2.44)

where $p_{i}$ are conjugate momenta, $H$ is a target space energy (spectra
contribution after quantisation) and $J$ is a distinguished angular momentum.
It can be found that the corresponding Noether charge $\mathcal{P}_{+}$ along
with worldsheet energy (Hamiltonian) $\mathfrak{H}$

$\mathcal{P}_{+}=\dfrac{1}{2}\left(H+J\right),\qquad\mathfrak{H}=H-J.$ (2.45)

Since it is not explicit on how to directly quantise $\mathfrak{H}$ to obtain
the spectrum, it is necessary to redefine the fields and coordinates to allow
for large $h$ perturbation for Hamiltonian density

$\mathcal{H}=\mathcal{H}_{2}+\dfrac{1}{h}\mathcal{H}^{4}+\dfrac{1}{h^{2}}\mathcal{H}_{6}+\mathcal{O}(h^{-3})$
(2.46)

and one can consider the BMN limit [73]

$P_{+}\rightarrow\infty,\quad
h\rightarrow\infty,\quad\dfrac{P_{+}}{h}\sim\text{fixed}$ (2.47)

it is then possible to expand action in $\frac{1}{P_{+}}$, where $P_{+}$ is an
eigenvalue of $\mathcal{P}_{+}$

$P_{+}=\int\limits_{-\frac{L}{2}}\limits^{\frac{L}{2}}\operatorname{d\\!}\sigma
p_{+}=L,\qquad\text{in light-cone gauge}\,\,x_{+}=\tau,\text{ and }p_{+}=1$
(2.48)

where it is meant to light-cone gauge fix the corresponding coordinates. Or
one can proceed with the decompactifying limit

$P_{+}\rightarrow\infty,\quad h\sim\text{fixed}$ (2.49)

which admits asymptotic states and construction of the $S$-matrix. In fact one
can similarly proceed for other related geometries and find out that
$AdS_{3}\times S^{3}\times\mathcal{M}^{4}$ contains two non-trivial integrable
backgrounds with 16 preserved supercharges (more detailed in Sec. 7)

$\begin{cases}\mathcal{M}^{4}=T^{4},\text{with }\mathfrak{psu}(1,1|2)^{2}\\\
\mathcal{M}^{4}=S^{3}\times S^{1},\text{with
}\mathfrak{d}(2,1;\alpha)^{2}\sim\mathfrak{d}(2,1;\alpha)_{L}\oplus\mathfrak{d}(2,1;\alpha)_{R}\oplus\mathfrak{u}(1)\\\
\end{cases}$ (2.50)

where $\alpha$ is proportional to ratio of sphere radii and bosonic subalgebra
of the latter decomposes to left and right sector

$\left[\mathfrak{su}(1,1)\oplus\mathfrak{su}(2)\oplus\mathfrak{su}^{\prime}(2)\right]_{L}\oplus\left[\mathfrak{su}(1,1)\oplus\mathfrak{su}(2)\oplus\mathfrak{su}^{\prime}(2)\right]_{R}\oplus\mathfrak{u}(1)$
(2.51)

The $AdS_{2}/CFT_{1}$ possesses even less symmetry, a quarter of the total
amount of supersymmetry. Still this background has an interesting underlying
worldsheet sigma model description. The supercoset formalism is also
applicable here, it is the $AdS_{2}\times S^{2}$ part that can be described by
the Metsaev-Tseytlin [74] type supercoset

$AdS_{2}\times S^{2}:\,\dfrac{PSU(1,1|2)}{SO(1,1)\times SO(2)}$ (2.52)
$S_{MT}=\int
d^{2}x\;\text{STr}\left[\left(\text{Pr}_{+}\mathcal{J}_{+}\right)\mathcal{J}_{-}\right]$
(2.53)

where $\mathcal{J}_{\pm}$ are group $\mathcal{G}$ valued currents,
$\text{Pr}_{+}$ are specific projectors. The $\mathfrak{psu}(1,1|2)$
superalgebra D.1.1 contains a $\mathbb{Z}_{4}$ automorphism, which implies
classical integrability for the same reason as $AdS_{5}\times S^{5}$ [75]. As
mentioned, also here $AdS_{2}\times S^{2}\times T^{6}$ geometry has suitably
truncated form of the Green-Schwarz action [71] for $AdS_{2}$ supercoset.
However in this case one cannot decouple $AdS_{2}$ dof from the GS fermions,
due to the absence of suitable $\kappa$ gauge choice. So this class is
addressed from the light-cone decompactifying gauge, where the $S$-matrix
would describe the worldsheet excitations above BMN vacuum.

The overall supercoset parts for three backgrounds can be given by generic
construction of the factorised supergroup

$\dfrac{\hat{\mathfrak{F}}}{\mathfrak{f}}=\dfrac{\hat{G}\times\hat{G}}{\mathfrak{f}}$
(2.54)

along with

$AdS_{n}\times S^{n}=\hat{G}/H$ (2.55)

where $\mathfrak{f}$ constitutes a bosonic diagonal subgroup of the factorised
supergroup $\mathfrak{F}$. By the given construction, one can get the
following supercoset models

  * •

$AdS_{5}\times S^{5}=\dfrac{PSU(2,2|4)}{SO(1,4)\times SO(5)}$

  * •

$AdS_{3}\times S^{3}=\dfrac{PSU(1,1|2)\times PSU(1,1|2)}{SO(1,2)\times SO(3)}$

  * •

$AdS_{2}\times S^{2}=\dfrac{PSU(1,1|2)}{SO(1,1)\times SO(2)}$

All of the setups described above will have a deeper investigation in 7, when
we shall come to the novel 6- and 8-vertex deformations found by our method,
developed in Sec. 3.1, 5. We shall explicitly elaborate on how 6vB/8vB models
given as $AdS_{\\{2,3\\}}$ deformations admit the models above and provide the
novel structure.

In the next section we shall start our discussion by implementing an approach,
which exploits automorphic symmetries of the conserved charges. Firstly, the
last will arise in two-dimensional integrable models, however the method will
be proven to be universal. In our considerations there will be spin chain
Hamiltonians with the nearest-neighbour interaction and YBE with regular
solution sector. We shall find novel Hamiltonians that are constant and by
bottom-up technique obtain the corresponding $R$ matrices. Each integrable
class will be characterised by a certain generating Hamiltonian, which will
not only result in the compact presentation, but also will reproduce all the
relevant information of the class by virtue of generic integrable symmetry
constraints.

Among a number of advantages of the approach, is that one obtains a coupled
polynomial system of equations instead of a high degree functional system,
which might appear non-solvable in general. One of the aims of the current
thesis is to demonstrate a new approach of constructing a novel integrable
structure and resolve the classification problem, which usually is not
accessible by standard techniques or requires significant restrictions. This
method is rather universal and can be applied to any sector of integrable
models, since it is based on universal integrability constraints that of
course require minor attention when passing to specific subfield (statistical
models, integrable spin chains, string integrability, its continuous
counterparts for integrable system with irrelevant deformations is currently
under investigation). In addition we have developed a coding system, which
allows to make number of analytic derivations autonomous, with the requirement
that initial system Ansatz is provided. Now it can be stated that for
$D=2,3,4$ dimensions the Ansatz can be completely generic (technical
implementation allows to use it even in high complexity case). Its structure
is somewhat reminiscent of analytic bootstrapping, where a set of restricting
conditions is applied with an recursive manifestations at the intermediate
level.

The current work presentation is structured as follows, first of all, we
address the necessary objects involved in the method and show the core steps
necessary for its implementation, which will be reflected in finding a new
integrable structure and completing classification for 2-dim systems
($\mathfrak{h}^{2}$), Sec. 3. In Sec. 4, we ask for new integrability and
implementation of our technique in the four dimensional sector
($\mathfrak{h}^{4}$), where a variety of symmetry algebras and representations
are studied. The Sec. 5 provides a complete unification of the approach
structure ($\mathcal{B}$ and $\mathcal{H}/R$ resolution schemes) for
integrable systems with arbitrary spectral dependence in $D=2,3,4$, whose
structure appear relevant for AdS/CFT integrability. Consequently an
investigation of string type classes that come from 2-dim setups of Sec. 5 is
followed. Special properties, deformations, symmetries and proposed
correspondences of these models are discussed in Sec. 6, 7. The algebraic
apparatus, examples, further proofs and derivations can be found in the
corresponding Sections and Appendices.

##  3 $\mathbb{V}^{2}$-difference sector

###  3.1 Method

####  3.1.1 Automorphisms and $\mathcal{B}$-operator

##### Notation

We begin by looking into integrable spin chains equipped with two-dimensional
local quantum spaces $\mathfrak{h}=\mathbb{V}\equiv\mathbb{C}^{2}$ and possess
a quantum $R$-matrix 2.14 satisfying the quantum-YBE

$R_{12}(u-v)R_{13}(u)R_{23}(v)=R_{23}(v)R_{13}(u)R_{12}(u-v)$ (3.1)

in addition $R(u)$ obeys the regularity condition $R(0)\equiv P$ and behaves
analytically in the $0$-neighborhood. This minimal setting identifies the
structure for NN-interacting spin chains of a given length $L$ and total
Hilbert space $\mathfrak{H}=\left(\mathbb{C}^{2}\right)^{\otimes L}$. In
general, the integrable structure and properties of such system can be
obtained from the transfer matrix $t(u)$ through the $\mathcal{R}$-monodromy

$t=\text{Tr}_{0}\mathcal{R}_{0i}\qquad\mathcal{R}_{0i}\equiv\prod_{i=L}^{1}R_{0i}(u)$
(3.2)

where $t=t(u)$ is purely quantum and the trace is over the auxiliary null
space [10], we shall be considering closed spin chains with periodicity
$L+1\equiv 1$.

It is known from an algebraic structure of integrable systems, that one can
generate conserved charges [19, 10] via $t(u)=A(u)+D(u)$888$A$, $D$ are
monodromy diag-operators obeying fundamental commutation relations

$\left[t(u),t(v)\right]=0\qquad
t(u)=2u^{n}+\sum_{i=0}^{n-2}\mathbb{Q}_{i}u^{i}$ (3.3)

that generates the commuting hierarchy of $n-1$ charges
$\\{\mathbb{Q}_{r}\\}$, where the range of these operators rises as $r$
increases. It can be proven that $\mathbb{Q}_{1}$, which can be associated
with momentum $P$ and $\mathbb{Q}_{2}$ with Hamiltonian $\mathbb{H}$ are part
of this hierarchy. The permutation shift and log-derivative of the $R$-matrix
provide accordingly momentum and Hamiltonian

$\displaystyle P_{1,2}P_{2,3}\dots
P_{L-1,L}=e^{iP}\qquad\qquad\mathbb{Q}_{1}\eqsim P$ (3.4)
$\displaystyle\mathbb{Q}_{2}\equiv\sum_{n=1}^{L}R_{n,n+1}^{-1}\dfrac{dR_{n,n+1}}{du}\Bigg{|}_{u\rightarrow
0}=\sum_{n=1}^{L}\mathcal{Q}_{n,n+1}=\mathbb{H}$ (3.5)

where $\mathcal{Q}_{n,n+1}$ are $\left(n,n+1\right)$-site local Hamiltonian
densities (in general $\mathbb{H}$ can be of higher interaction range, but in
the present construction $\mathbb{Q}_{2}$ will be NN Hamiltonian). Hence along
with 3.3 one gets the aforementioned commuting hierarchy. It can be seen from
3.4 that Hamiltonian density $\mathcal{Q}_{n,n+1}$ is an operator of range two
or the nearest-neighbor. Following the algebraic pattern, one can notice for
the third charge

$\mathbb{Q}_{3}\equiv\sum_{n}\mathcal{Q}_{n,n+1,n+2}\qquad\mathcal{Q}_{n,n+1,n+2}=\left[\mathcal{Q}_{n,n+1},\mathcal{Q}_{n+1,n+2}\right]$
(3.6)

which exhibits exactly range three action. Analogously it is possible to
continue the derivation of all higher charges, where one can eventually find
the automorphic symmetry structure, as we have indicated in 2. That can be
compactly formulated by means of the boost operator $\mathcal{B}$ (2.35) [11,
12].

$\mathcal{B}\left[\mathbb{Q}_{2}\right]=\sum_{k=-\infty}^{+\infty}k\mathcal{Q}_{k,k+1}$
(3.7)

The advantage of such automorphism is reflected in its recursive generating
property for all higher charges

$\mathbb{Q}_{r+1}=\left[\mathcal{B}\left[\mathbb{Q}_{2}\right],\mathbb{Q}_{r}\right]$
(3.8)

it follows that the higher part of the hierarchy can be constructed from the
Hamiltonian $\mathbb{H}$ [12]. One can also note that by construction, 3.8
contains local densities of range/length $r+1$, which under appropriate finite
length closed spin chain prescription defines a consistent $(r+1)$-site
operator. In general, that provides a closed set of commutative constraints on
the charges that follow intertwined boost recursion. More detailed
argumentation C.1, proofs, further restrictions and extensions will be
discussed in the following sections.

####  3.1.2 Ansatz and Algebraic system

To initiate the implementation of the method, we can start with integrable
spin chains with two-dimensional local quantum spaces. To achieve that, one
can write a completely generic Ansatz for the Hamiltonian density using some
basis. In particular, we can consider the $2\times 2$ extended Pauli set
$\sigma^{b}$ in the $b=\\{0,+,-,z\\}$ basis, which is given by

$\sigma^{b}\equiv\mathds{1}_{2}\text{,
}\quad\sigma^{\pm}\equiv\dfrac{\sigma_{x}\pm i\sigma_{y}}{2}\text{,
}\quad\sigma^{z}=\sigma^{z}$ (3.9)

$\mathds{1}_{2}$ is the two dimensional identity and the Ansatz for density is
formed by the corresponding range tensor product

$\mathcal{Q}_{ij}=\mathcal{A}_{ab}\,\sigma^{a}\otimes\sigma^{b}$ (3.10)

which in this case contains 16 free parameters $\mathcal{A}_{ab}$, however
already at this level one could try reducing the number of free parameters by
virtue of some generic symmetry transformations. For example, any Hamiltonian
normalization does not break integrability, as well as local basis transform

$\mathcal{Q}^{\prime}=\mathfrak{V}\mathcal{Q}\mathfrak{V}^{-1}\qquad\mathfrak{V}=\mathcal{V}\otimes\mathcal{V}$
(3.11)

with unimodular restriction on $\mathcal{V}$

$\mathcal{V}=\begin{pmatrix}\mathfrak{a}&\mathfrak{b}\\\
\mathfrak{g}&\mathfrak{d}\\\ \end{pmatrix}\qquad\text{det}\,\mathcal{V}=1$
(3.12)

Without loss of generality we can demonstrate this by demanding vanishing of
7- and 8-vertex entries, i.e. when $\mathcal{A}_{bb}=0$ with $b=\pm$, so that
transformed $\mathcal{Q}^{\prime}$ coefficients at
$\sigma^{b}\otimes\sigma^{b}$ are

$\begin{array}[]{c}\mathfrak{a}^{4}\mathcal{A}_{++}+\mathfrak{b}^{4}\mathcal{A}_{--}-2\mathfrak{a}^{3}\mathfrak{b}(\mathcal{A}_{+z}+\mathcal{A}_{z+})-\mathfrak{a}^{2}\mathfrak{b}^{2}(\mathcal{A}_{+-}+\mathcal{A}_{-+}-4\mathcal{A}_{zz})\\\
+2\mathfrak{a}\mathfrak{b}^{3}(\mathcal{A}_{-z}+\mathcal{A}_{z-})=\mathcal{A}^{\prime}_{++}\\\
\mathfrak{g}^{4}\mathcal{A}_{++}+\mathfrak{d}^{4}\mathcal{A}_{--}-2\mathfrak{g}^{3}\mathfrak{d}(\mathcal{A}_{+z}+\mathcal{A}_{z+})-\mathfrak{g}^{2}\mathfrak{d}^{2}(\mathcal{A}_{+-}+\mathcal{A}_{-+}-4\mathcal{A}_{zz})\\\
+2\mathfrak{g}\mathfrak{d}^{3}(\mathcal{A}_{-z}+\mathcal{A}_{z-})=\mathcal{A}^{\prime}_{--}\\\
\end{array}$ (3.13)

By looking more closely, we can notice that this argument holds in both ways,
i.e. if we take $\mathcal{A}_{++}=\mathcal{A}_{--}\equiv 0$ the Ansatz is
trivially of the desired form. Another possibility is if one allows either of
$\mathcal{A}_{bb}$ to vanish, then there is a nontrivial solution map to
$\mathcal{A}^{\prime}_{++}=\mathcal{A}^{\prime}_{--}=0$ if and only if
$\mathcal{A}_{\pm z}\neq-\mathcal{A}_{z\pm}$ and
$\mathcal{A}_{+-}+\mathcal{A}_{-+}\neq 4\mathcal{A}_{zz}$, and the case
$\mathcal{A}_{++}=\mathcal{A}_{--}\neq 0$ is related to the latter case. This
is only an illustration of the possible coefficient reduction scheme and can
be implemented in any other form upon demand.

##### Commutative constraints

Next one would need to impose constraints on the obtained Ansatz. As we have
mentioned above these constraints would come from the commutation properties
characteristic for the given integrable system, hence
$\left[\mathbb{Q}_{p},\mathbb{Q}_{r}\right]=0$ will restrict the underlying
densities of range p and r defined on the spin chain of an appropriate length
(which will be always assumed from now on). In other words commutation of
densities produces an algebraic system to be solved on the
$\mathcal{A}$-space.

There is a number of questions related to the type and number of the
commutators, their analytic structure and sufficiency, but it will be
explained that in all studied cases the first nontrivial commutator
$\left[\mathbb{Q}_{2},\mathbb{Q}_{3}\right]$ will appear to be necessary and
sufficient [76, 77]. In this respect, let us perform an analysis on the
structure and dof properties following from it. As it has been stated, range 3
and 2 charges follow the prescribed Ansatz

$\begin{array}[]{c}\begin{split}\left[\mathbb{Q}_{2},\mathbb{Q}_{3}\right]&=\sum_{m,n}\mathcal{A}_{ab}\mathcal{A}_{efg}\left[\dots\sigma_{m}^{a}\sigma_{m+1}^{b}\dots,\dots\sigma_{n}^{e}\sigma_{n+1}^{f}\sigma_{n+2}^{g}\dots\right]\\\
&\equiv\mathcal{C}_{abef}\sum_{m}\dots\sigma_{m}^{a}\sigma_{m+1}^{b}\sigma_{m+2}^{e}\sigma_{m+3}^{f}\dots\end{split}\end{array}$
(3.14)

with $\mathcal{C}_{abef}$ to include combinations of $\mathcal{A}$ and algebra
structure constants 999dots represent appropriate tensor embedding on the
chain, such that the overlaps in the commutator are not trivial, i.e.
$\sum_{m,n}\mathcal{A}_{ab}\mathcal{A}_{efg}\left[\delta_{m+1,n}+\delta_{m,n}+\delta_{m,n+1}+\delta_{m,n+2}\right]\left[\dots\sigma_{m}^{a}\sigma_{m+1}^{b}\dots,\dots\sigma_{n}^{e}\sigma_{n+1}^{f}\sigma_{n+2}^{g}\dots\right]$.

In the general case for the closed spin chain hierarchy, it can be shown that
$\left[\mathbb{Q}_{r},\mathbb{Q}_{s}\right]$ generates a polynomial system of
$\dfrac{1}{2}\left(3^{r+s-1}-1\right)$ equations of degree $r+s-2$. A
significant part of these equations will be trivially satisfied. Again, it is
important to note, that $\left[\mathbb{Q}_{2},\mathbb{Q}_{3}\right]$ is
sufficient and for two-dimensional case higher commutators do not impose more
constraints on the found solutions - the first commutator appears to contain
all non-degenerate unique information on integrable models in two dimensions
[77]. It will be also shown later that a generalisation of this statement can
be made.

Another important remark on the boost symmetry includes differences between
the initial boost $\mathcal{B}\left[\cdot\right]$ formulation and the closed
spin chain setup considered in the current work. Specifically, one needs to
pay attention to the fact that the boost formulation on the infinite open spin
chains does not apriori admit addition of terms which do vanish on the
periodic systems. In particular, densities taking the form of

$\delta\mathcal{Q}_{m}=\mathcal{Q}\otimes\mathds{1}-\mathds{1}\otimes\mathcal{Q}$
(3.15)

obviously become vanishing on the closed chains due to periodicity
$\sum_{m}\delta\mathcal{Q}_{m}=0$, hence such symmetry transformations could
be performed only for the final result (Hamiltonains), otherwise precaution
required, since $\mathcal{B}\left[\mathbb{Q}_{2}\right]$ will affect the
structure of the higher charges in the hierarchy.

##### Generating commutators

By observing the commutative structure, it is possible to see that only part
of the commutators are actually required in order to understand recursive
structure of the whole hierarchy. It means that there must be commutative
degeneracy present and not all commutators are independent. For instance, let
us consider $\left[\mathbb{Q}_{2},\mathbb{Q}_{4}\right]$

$\displaystyle\left[\mathbb{Q}_{2},\mathbb{Q}_{4}\right]=\left[\mathbb{Q}_{2},\left[\mathcal{B}\left[\mathbb{Q}_{2}\right],\mathbb{Q}_{3}\right]\right]=-\left[\left[\mathcal{B}\left[\mathbb{Q}_{2}\right],\left[\mathbb{Q}_{3},\mathbb{Q}_{2}\right]\right]+\left[\mathbb{Q}_{3},\left[\mathbb{Q}_{2},\mathcal{B}\left[\mathbb{Q}_{2}\right]\right]\right]\right]=$
(3.16)
$\displaystyle[\underbrace{\left[\mathbb{Q}_{3},\mathbb{Q}_{2}\right]}_{\equiv
0},\mathcal{B}\left[\mathbb{Q}_{2}\right]]+[\mathbb{Q}_{3},\underbrace{\left[\mathcal{B}\left[\mathbb{Q}_{2}\right],\mathbb{Q}_{2}\right]}_{\mathbb{Q}_{3}}]=0$

with Jacobi identity for intermediate step. One can analyze recursion and
prove by induction that commutators could expressed by means of
$\left[\mathbb{Q}_{r},\mathbb{Q}_{2r+1}\right]$ and
$\left[\mathbb{Q}_{2},\mathbb{Q}_{2r+2}\right]$ $r\geq 1$. However it is
required to have $\left[\mathbb{Q}_{2},\mathbb{Q}_{3}\right]=0$ and
$\left[\mathbb{Q}_{3},\mathbb{Q}_{4}\right]=0$ as a commutativity initial
conditions.

####  3.1.3 Integrable transformations

Once an algebraic system that follows from 3.14 is resolved, one arrives at a
significantly large space of solutions. In this respect, it necessary to
address if such set of solutions can be reduced to a finite independent subset
due to potentially present symmetries. It is indeed true, since there are
integrable continuous and discrete transformations (part of YB algebra) whose
action on a solution of the YBE will generate another set of solutions
satisfying it. It is preferable to proceed with such generators, as for now we
are interested in the novel independent set of solutions, which could be
presented in a more compact form. We shall discuss the minimal set of allowed
transformations in the current section.

##### Special cases

The most basic possibility is the reduction from the general case to the sub-
cases, a conventional illustration of this can be the XYZ $\rightarrow$ XXZ
$\rightarrow$ XXX, which is achieved by consecutively setting coefficients to
zero. In this context, the variety of all special cases of XYZ will be given
by the generic 8-vertex model.

##### Shifts and norms

It is consistent to allow suitable normalisation of the given Hamiltonian $H$
or perform diagonal shift by adding terms of $\mathfrak{C}\cdot\mathds{1}$
type.

##### Local basis transformation

As it was noticed above, an integrable spin chain is characterised by the
corresponding Hamiltonian $\mathbb{H}$ and $R$-matrix admits a freedom of
local basis change 3.11 in the form
$\mathcal{O}^{\prime}=\mathfrak{V}\mathcal{O}\mathfrak{V}^{-1}$,
$\mathfrak{V}\in\text{End}\left(\mathbb{V}\right)$, which also produces an
equivalence class of integrable spin chains. As it follows from the
definition, $\mathcal{V}$ assigned to each local space defines a
transformation for range $r$ density $\mathcal{Q}_{m_{1},\dots,m_{r}}$
according to

$\mathcal{Q}^{\prime}_{m_{1},\dots,m_{r}}=\mathfrak{V}\mathcal{Q}^{\prime}_{m_{1},\dots,m_{r}}\mathfrak{V}^{-1}\qquad\mathfrak{V}=\bigotimes_{l=1}^{r}\mathcal{V}_{l}\qquad\det\,\mathcal{V}=1$
(3.17)

where $\mathcal{V}_{l}$ is $2\times 2$ operator for $2$-dim local spaces. In
this form one can map all other models of the specific class, or obtain a
necessary form of the Ansatz, which can noticeably lower algebraic system
complexity (e.g. removing off-diagonal elements as stated in 3.1.2).

##### Discrete maps

Further mapping within the classes can be achieved by the discrete
transformations for the $R$ matrices, since it is known that consistent
application of transposition-permutation will again lead to the solution of
the YBE, analogous arguments can be demonstrated for the underlying
Hamiltonians. More explicit reasoning could be seen from perturbation theory
of the YBE, which will be presented in the next section, whereas evident
transformations are

$\begin{matrix}R(u)&\qquad\mathcal{H}\\\ R(u)^{T}&\qquad P\mathcal{H}^{T}P\\\
PR(u)P&\qquad P\mathcal{H}P\\\ PR(u)^{T}P&\qquad\mathcal{H}^{T}\\\
\end{matrix}$ (3.18)

where presentation above implies an agreement at the level of operatorial
entries (as two objects are on distinct spaces, satisfy properties etc). It
appears that with this minimal amount of identification transformations it
becomes possible to find generating representative for each equivalence class.

####  3.1.4 $H$ $\Leftrightarrow$ $R$-matrix and consistency

As we have mentioned, all of the above transformations are implemented at the
final stage, however strictly speaking solutions extracted following
methodology of 3.1.2 and 3.1.3 could be considered as a necessary step for
establishing the space of two-dimensional integrable models. Along the known
claims of the sufficiency of the
$\left[\mathbb{Q}_{2},\mathbb{Q}_{3}\right]=0$ condition, it can be noted that
relevant checks of the commutators of densities up to range 6 have not further
constrained the resulting system. Nevertheless, to be completely exact, one
needs to accomplish true quantum integrable checks aka YBE, that require the
associated quantum $R$ matrices.

In order to achieve that from our setting, it is necessary to exploit a
bottom-up approach, i.e. obtain the $R$ matrices from the Hamiltonians
$\mathcal{H}$. For that we need to recall quantum YBE solution existence in
the form of formal series

$R(u,q_{h})={\mathds{1}}+q_{h}\left(\rho\otimes\rho\right)r(u)+\sum_{k=2}^{\infty}\mathfrak{A}_{k}(u)q_{h}^{k}$
(3.19)

where the first two terms constitute the classical YBE part. As known the
solution also exists in the form [19]

$R(u)={\mathds{1}}+\rho(\mathcal{I}_{\mu})\otimes\rho(\mathcal{I}_{\mu})\,u^{-1}+\sum_{k=2}^{\infty}R_{k}u^{-k}\qquad
R_{k}\in\text{End}\left(\mathbb{V}\otimes\mathbb{V}\right)$ (3.20)

with an orthonormal basis $\\{\mathcal{I}_{\mu}\\}$ in algebra $\mathfrak{g}$
with a defined representation
$\rho:\mathfrak{g}\rightarrow\text{End}(\mathbb{V})$. Since the Hamiltonian
has units of energy and from ABA arguments the $R$-matrix known to be
dimensionless, the expansion can also have inverse reformulation

$R(u)=P+P\mathcal{H}u+\sum_{k=2}^{\infty}R_{k}u^{k}$ (3.21)

Important to note that for the XYZ class all the $R_{k}$ coefficients take
specific perturbative form $R_{k}=Pf_{k}(\mathcal{H})$, where $f_{k}$ is
degree $k$ polynomial in $\mathcal{H}$. It can be also noted, that in two-
dimensions for $\mathcal{H}$ one can get a perturbative reduction, since
higher terms of the expansion 3.21 can be re-expressed through lower orders
$\\{{\mathds{1}},\mathcal{H},\mathcal{H}^{2},\mathcal{H}^{3}\\}$ due to
Hamilton-Cayley corollary. Taking into account the above statements, one can
proceed with the following prescription

$R(u)=P\sum_{k=0}^{3}f_{k}(u)\mathcal{H}^{k}u^{k}$ (3.22)

with $f_{k}$ certain functions on the spectral parameter. It must be
emphasized, that such Ansatz might not be applied to any $R$-matrix, due to
the dimensional limitations of the Hamilton-Cayley argument, analytic
properties of the $R$-matrix itself and $f_{k}$ functions. However such
construction has proven to work for the current two-dimensional classification
problem (including the novel models). Specific constraining of $f_{k}$ can be
drawn immediately from allowed gauging of the $R$-matrix by setting either
$f_{0}(u)$ or $f_{1}(u)$ to be unity. Also from dimensional analysis and
expansion structure consistency we can set $f_{0}(0)=f_{1}(0)=1$ and
$f_{0}^{\prime}(0)=0$.

By such minimal perturbative implementation one is able to obtain the $R$
matrices that underlie all models with $2$-dim local spaces. Consistency and
existence of the associated $R$ matrices under YBE101010From the modified
quantum YBE and of 3.22 Ansatz one can obtain nontrivial system on intertwined
$\mathcal{H}$ and $f_{k}$, instead of order by order perturbation for $R$,
which can be analytically resolved by appropriate convergence only in the very
limited number of cases leads to their quantum integrability proof. In fact
there is much more universal procedure for constructing $R$ matrices with
reverse approach from the $\mathcal{H}$’s, which is distinct from conventional
methods, more details are provided in Sec. 5.

###  3.2 2-dim space

####  3.2.1 Solution space

In the setting described above [78], the system constrained by the
$\left[\mathbb{Q}_{2},\mathbb{Q}_{3}\right]$ commutator has proven to be
sufficient [77]. Initially, the current method was checked on the $2$-dim
model setup and provided a substrate for further investigation. From the
nontrivial results in 2-dim, it becomes clear that it can be also exploited in
other frameworks, including string background integrability.

By resolving the first commutator, one obtains a nontrivial coupled cubic
system, which after eliminating trivial subsector results in $\sim 250$
solutions111111the choice of a suitable algebraic basis leads to the complete
solvability. At the next stage, one needs to address if they could be grouped
into certain classes. As it was shown in 3.1.3, indeed, by the set of
integrable transformations, one can identify operatorial equivalence classes
(although it is a matter of presentation). We can illustrate this on the
example, which containts XYZ/XXZ reductions

$\displaystyle\mathcal{H}=\begin{pmatrix}a&b&-b&0\\\ 0&c-a&2a-c&0\\\
0&2a+c&-a-c&0\\\ 0&0&0&a\end{pmatrix}$ (3.23)

it is possible to notice that it has a limit to the XXZ model

$\displaystyle\mathcal{H}=\begin{pmatrix}a&0&0&0\\\ 0&c-a&2a-c&0\\\
0&2a+c&-a-c&0\\\ 0&0&0&a\end{pmatrix}$ (3.24)

which is obvious and no transformation needed. However in order to include
generic XYZ content, one needs to apply local basis transformation taking the
form

$\displaystyle\mathcal{V}=\begin{pmatrix}-\frac{2\beta c}{b}&\beta\\\
0&-\frac{b}{2\beta c}\end{pmatrix}$ (3.25)

Separate analysis is necessary, since when $b$ or $c\rightarrow 0$ due to the
apparent singularities given solution can not be brought to the XYZ, and one
finds an independent model 3.41. We shall now briefly discuss the structure of
a generic elliptic class for the purposes of $2$-dim space completion and
demonstration of further grouping schemes. An important note on the grading-
related solutions will be provided below.

####  3.2.2 The 8-vertex and subspace

The consistency of the method must show completeness of the solution space
found. In this regard, one must expect an elliptic and its sub-sectors to be
present. The full classification of $\leq 8$-vertex models has been provided
in [27]. In general 8-vertex Ansatz can be given as follows

$\displaystyle\mathcal{H}^{XYZ}=\begin{pmatrix}a_{1}&0&0&d_{1}\\\
0&b_{1}&c_{1}&0\\\ 0&c_{2}&b_{2}&0\\\ d_{2}&0&0&a_{2}\end{pmatrix}.$ (3.26)

In this category, we can establish 8 independent generating solutions, which
include the known $2$-dim solutions of the YBE

##### Diagonal (4-vertex)

Any main diagonal $\mathcal{H}$ is an integrable solution

$\displaystyle\mathcal{H}^{XYZ}_{1}=\begin{pmatrix}a_{1}&0&0&0\\\
0&b_{1}&0&0\\\ 0&0&b_{2}&0\\\ 0&0&0&a_{2}\end{pmatrix}$ (3.27)

##### XXZ

There are two families in the 6-vertex sector, which agrees with [79]

$\displaystyle\mathcal{H}^{XYZ}_{2}=\begin{pmatrix}a_{1}&0&0&0\\\
0&b_{1}&c_{1}&0\\\ 0&c_{2}&b_{2}&0\\\ 0&0&0&a_{1}\end{pmatrix}$
$\displaystyle\mathcal{H}^{XYZ}_{3}=\begin{pmatrix}a_{1}&0&0&0\\\
0&b_{1}&c_{1}&0\\\ 0&c_{2}&b_{2}&0\\\ 0&0&0&-a_{1}-b_{1}-b_{2}\end{pmatrix}$
(3.28)

##### 7-vertex

There are two generators of the 7-vertex form

$\displaystyle\mathcal{H}^{XYZ}_{4}=\begin{pmatrix}a_{1}&0&0&d_{1}\\\
0&a_{1}+b_{1}&c_{1}&0\\\ 0&-c_{1}&a_{1}-b_{1}&0\\\ 0&0&0&a_{1}\end{pmatrix}$
$\displaystyle\mathcal{H}^{XYZ}_{5}=\begin{pmatrix}a_{1}&0&0&d_{1}\\\
0&a_{1}-c_{2}&c_{1}&0\\\ 0&c_{2}&a_{1}-c_{1}&0\\\
0&0&0&a_{1}-c_{1}-c_{2}\end{pmatrix}$ (3.29)

##### 8-vertex

The elliptic sector contains three distinct generators in the following form

$\displaystyle\mathcal{H}^{XYZ}_{6}=\begin{pmatrix}a_{1}&0&0&d_{1}\\\
0&b_{1}&c_{1}&0\\\ 0&c_{1}&b_{1}&0\\\ d_{2}&0&0&a_{1}\end{pmatrix}$
$\displaystyle\mathcal{H}^{XYZ}_{7}=\begin{pmatrix}a_{1}&0&0&d_{1}\\\
0&b_{1}&c_{1}&0\\\ 0&c_{1}&b_{1}&0\\\ d_{2}&0&0&2b_{1}-a_{1}\end{pmatrix}$
(3.30) $\displaystyle\mathcal{H}^{XYZ}_{8}=\begin{pmatrix}a_{1}&0&0&d_{1}\\\
0&a_{1}&b_{1}&0\\\ 0&-b_{1}&a_{1}&0\\\ d_{2}&0&0&a_{1}\end{pmatrix}$ (3.31)

The corresponding $R$ matrices can be found elsewhere, including
classification performed in [27]. The approach 3.1.4 was cross-checked for XYZ
and all subclasses. In addition, we make it explicit on a more nontrivial
novel classes presented below.

####  3.2.3 The new classes

##### Class I

The first novel generator $\mathcal{H}$ obtained by virtue of the method 3.1

$\displaystyle\mathcal{H}_{\text{{I}}}=\begin{pmatrix}0&a_{1}&a_{2}&0\\\
0&a_{5}&0&a_{3}\\\ 0&0&-a_{5}&a_{4}\\\ 0&0&0&0\end{pmatrix}$ (3.32)

with $a_{1}a_{3}-a_{2}a_{4}=0$. Along with its associated $R$-matrix to be

$\displaystyle
R_{\text{{I}}}(u)=\begin{pmatrix}1&\frac{a_{1}(e^{a_{5}u}-1)}{a_{5}}&\frac{a_{2}(1-e^{-a_{5}u})}{a_{5}}&\frac{a_{1}a_{3}+a_{2}a_{4}}{a_{5}^{2}}(\cosh(a_{5}u)-1)\\\
0&0&e^{-a_{5}u}&\frac{a_{4}(1-e^{-a_{5}u})}{a_{5}}\\\
0&e^{a_{5}u}&0&\frac{a_{3}(e^{a_{5}u}-1)}{a_{5}}\\\ 0&0&0&1\end{pmatrix}$
(3.33)

It is straightforward that (3.33) satisfies YBE, regularity and underlying
braiding unitarity $R_{ij}(u)R_{ji}(-u)\sim 1$. In what follow the novel
nontrivial classes below

##### Class II

The second class constitutes reduced upper-triangular form

$\mathcal{H}_{\text{{II}}}=\left(\begin{array}[]{cccc}0&a_{2}&a_{3}-a_{2}&a_{5}\\\
0&a_{1}&0&a_{4}\\\ 0&0&-a_{1}&a_{3}-a_{4}\\\ 0&0&0&0\\\
\end{array}\right)\qquad
R_{\text{{II}}}(u)=uP\Big{[}\,\frac{a_{1}}{\sinh(a_{1}u)}+\mathcal{H}_{{II}}+\frac{\tanh(\frac{a_{1}u}{2})}{a_{1}}\mathcal{H}^{2}_{{II}}\Big{]}$
(3.34)

which is also regular and satisfies braiding unitarity.

##### Class III

The third class is associated to the following generating pair

$\displaystyle\mathcal{H}_{\text{{III}}}=\begin{pmatrix}-a_{1}&\left(2a_{1}-a_{2}\right)a_{3}&\left(2a_{1}+a_{2}\right)a_{3}&0\\\
0&a_{1}-a_{2}&0&0\\\ 0&0&a_{1}+a_{2}&0\\\ 0&0&0&-a_{1}\\\ \end{pmatrix}$
(3.35)
$R_{\text{{III}}}(u)=\left(\begin{array}[]{cccc}e^{-a_{1}u}&a_{3}\left(e^{(a_{1}-a_{2})u}-e^{-a_{1}u}\right)&a_{3}\left(e^{(a_{1}+a_{2})u}-e^{-a_{1}u}\right)&0\\\
0&0&e^{(a_{1}+a_{2})u}&0\\\ 0&e^{(a_{1}-a_{2})u}&0&0\\\
0&0&0&e^{-a_{1}u}\end{array}\right)$ (3.36)

it can be taken as a 4-vertex deformation by $a_{3}$ parameter, which
reproduces it in the $a_{3}\rightarrow 0$ limit

$\displaystyle\mathcal{H}^{\prime}_{4v}=\begin{pmatrix}-a_{1}&0&0&0\\\
0&a_{1}-a_{2}&0&0\\\ 0&0&a_{1}+a_{2}&0\\\ 0&0&0&-a_{1}\end{pmatrix}$ (3.37)

The corresponding $R$-matrix was also provided in the [27] classification
problem and appears of the form

$R^{\prime}_{4v}(u)=P_{12}\left(f_{0}(u)+uf_{1}(u)\mathcal{H}+u^{2}f_{2}(u)\mathcal{H}^{2}\right)$
(3.38)

where $f_{i}\equiv f_{i}\left(u,a_{1},a_{2}\right)$, and the $R$-matrix has
equivalent $\mathcal{H}$-functional representation in both cases of $a_{3}=0$
and $a_{3}\neq 0$, since $a_{3}$ is present only through the
$\mathcal{H}_{4v}$.

##### Class IV

The fourth class again respects braiding unitarity and is identified as
follows

$\displaystyle\mathcal{H}_{\text{{IV}}}=\begin{pmatrix}a_{1}&a_{2}&a_{2}&a_{3}\\\
0&-a_{1}&0&a_{4}\\\ 0&0&-a_{1}&a_{4}\\\ 0&0&0&a_{1}\end{pmatrix}$ (3.39)
$R_{\text{{IV}}}(u)=\left(\begin{array}[]{cccc}e^{a_{1}u}&\frac{a_{2}\sinh(a_{1}u)}{a_{1}}&\frac{a_{2}\sinh(a_{1}u)}{a_{1}}&\frac{e^{a_{1}u}(a_{2}a_{4}+a_{1}a_{3}\coth(a_{1}u))\sinh^{2}(a_{1}u)}{a_{1}^{2}}\\\
0&0&e^{-a_{1}u}&\frac{a_{4}\sinh(a_{1}u)}{a_{1}}\\\
0&e^{-a_{1}u}&0&\frac{a_{4}\sinh(a_{1}u)}{a_{1}}\\\ 0&0&0&e^{a_{1}u}\\\
\end{array}\right)$ (3.40)

##### Class V

The fifth obeys braiding unitarity, regularity and shows distinct non-
removable off-diagonal configuration

$\mathcal{H}_{\text{{V}}}=\left(\begin{array}[]{cccc}a_{1}&a_{2}&-a_{2}&0\\\
0&-a_{1}&2a_{1}&a_{3}\\\ 0&2a_{1}&-a_{1}&-a_{3}\\\
0&0&0&a_{1}\end{array}\right)$ (3.41) $\displaystyle
R_{\text{{V}}}=(1-a_{1}u)\left(\begin{array}[]{cccc}2a_{1}u+1&a_{2}u&-a_{2}u&a_{2}a_{3}u^{2}\\\
0&2a_{1}u&1&-a_{3}u\\\ 0&1&2a_{1}u&a_{3}u\\\ 0&0&0&2a_{1}u+1\\\
\end{array}\right)$ (3.46)

##### Class VI

The last class appears as

$\displaystyle\mathcal{H}_{\text{{VI}}}=\begin{pmatrix}a_{1}&a_{2}&a_{2}&0\\\
0&-a_{1}&2a_{1}&-a_{2}\\\ 0&2a_{1}&-a_{1}&-a_{2}\\\ 0&0&0&a_{1}\end{pmatrix}$
(3.47)
$R_{\text{{VI}}}(u)=(1-a_{1}u)(1+2a_{1}u)\begin{pmatrix}1&a_{2}u&a_{2}u&-a_{2}^{2}u^{2}(2a_{1}u+1)\\\
0&\frac{2a_{1}u}{2a_{1}u+1}&\frac{1}{2a_{1}u+1}&-a_{2}u\\\
0&\frac{1}{2a_{1}u+1}&\frac{2a_{1}u}{2a_{1}u+1}&-a_{2}u\\\
0&0&0&1\end{pmatrix}$ (3.48)

which satisfies braiding unitarity.

####  3.2.4 Structure analysis of the new classes

As we have been able to notice above, the proposed approach for models with
two-dimensional local space $\mathbb{C}^{2}$ and generic Ansatz for
$\mathcal{H}$ turned not only to show full agreement with the set of
integrable models that are found from the conventional YBE resolution (i.e.
Heisenberg class, *-magnets, multivertex etc), but also finds new higher
parametric integrable models 3.2.3. Some of these new classes exhibit non-
diagonalisability and nilpotency of the $\mathcal{H}$, but others develop
conserved charges with non-trivial Jordan blocks, which leads to important
results and corollaries. It also came to our knowledge that some of these
classes represent high parametric deformations in the $\mathfrak{sl}_{2}$
sector, which also appears to be a generalisation121212Known one-parameter
family of $\mathfrak{sl}_{2}$ models, which also could be related to [80]. In
the past it was conjectured, that higher-parametric generalisations might
exist, however it appeared non-resolvable by $RTT$-approach, and in our
computation we prove their existence and relations. of [81]. It is important
to find all underlying Yangian deformations
$\mathcal{Y}^{*}[\mathfrak{sl}_{2}]$ and associated quantum groups for these
models, which could also bring novel quantum group structure [82, 83]. It is
also known that the Belavin-Drinfeld cohomological classification of quantum
symmetries for novel models is an important question on its own. Additionally,
it might appear of interest for non-unitary theories, which are studied in the
context of conformal fishnet theories [84, 85] or other systems that develop
Jordanian structure [86, 87]

##### Class I and II

The conserved charges of the first two classes possess nilpotency, similar
property have been observed [85].

##### Class III-VI

It is possible to note that the rest of the classes, in general appear to be
non-diagonalisable, however for a set of appropriate parametric restrictions
they can be diagonalised. The diagonalisability holds demanding the following

  * •

for $a_{3}=0$ Class III reduces to 4-vertex model (3.37)

  * •

$a_{2}=a_{4}$ and $a_{3}=\dfrac{a_{4}^{2}}{a_{1}}$ for Class IV

  * •

$a_{2}=-a_{3}$ for class V

  * •

$a_{2}=0$ for class VI

It is important to mention that since the eigenspectrum seems to depend on
$a_{1}$ in such restricted cases, it means that eigenvalues of classes III and
IV can be related to the $z$-component density
$\mathcal{H}=\sigma^{z}\otimes\sigma^{z}$ and $\mathcal{H}=1-2P$ (also part of
classes V and VI).

####  3.2.5 Graded mapping

Now we shall discuss the graded structure which initially was implicit in our
investigation, in what we provide apparatus relevant for $\mathbb{C}^{1|1}$
emerging in our solution space. Let us consider a $\mathfrak{G}$-graded vector
space $\mathbb{V}$ over the complex field $\mathfrak{F}=\mathbb{C}$, which
exhibits the direct decomposition

$\mathbb{V}=\bigoplus_{\mathfrak{a}\in\mathfrak{G}}\mathbb{V}_{\mathfrak{a}}$
(3.49)

If $\exists\,\mathfrak{G}^{\prime}$ being nontrivial subgroup of the grading
group $\mathfrak{G}$, then there is a quotient group
$\mathfrak{G}/\mathfrak{G}^{\prime}$, with a quotient map defined

$\pi:\mathfrak{G}\rightarrow\mathfrak{G}/\mathfrak{G}^{\prime}$ (3.50)

then the established grading is odd or even. More specifically, if in the
space $\mathbb{V}$ we introduce the homogeneous basis
$\\{\mathfrak{e}_{i}:\,i\in\mathcal{I}\\}$, then
$\mathfrak{e}_{i}\in\mathbb{V}_{\mathfrak{a}}$ has a grade
$p(i)=\mathfrak{a}$, with grading map

$p:\mathcal{I}\rightarrow\mathfrak{G}\qquad\qquad\quad\mathcal{I}\equiv\\{1,\dots
N\\}\text{ and }\dim\mathbb{V}=N$ (3.51)

which can be unified to the basis
$\\{\mathfrak{e}_{i_{\mathfrak{a}},\mathfrak{a}}:\,i\in\mathcal{I}_{\mathfrak{a}}\\}$,
with $\mathcal{I}_{\mathfrak{a}}=\\{i\in\mathcal{I}:p(i)=\mathfrak{a}\\}$, and
one can define commutation factor $\mathcal{E}_{\mathfrak{a}}$ of
$\mathfrak{a}$

$\mathcal{E}_{\mathfrak{a}}=\left(-1\right)^{\pi(\mathfrak{a})}\qquad\qquad\pi(\mathfrak{a})=\begin{cases}+1\\\
0\end{cases}$ (3.52)

where if $\mathcal{E}_{\mathfrak{a}}=+1\,\forall\mathfrak{a}\in\mathfrak{G}$
then grading is even, and if $\exists\,\mathfrak{a}\in\mathfrak{G}$ s.t.
$\mathcal{E}_{\mathfrak{a}}=-1$ the grading is odd. More specifically, looking
at the quotient isomorphic to $\mathbb{Z}_{2}$ with
$\mathbb{V}_{0}\oplus\mathbb{V}_{1}$ and homogeneous elements $\mathfrak{v}$
(possessing zero projection onto either of subspaces), we have parity $p$

$\begin{cases}p(\mathfrak{v})=0,\,\,\mathfrak{v}\in\mathbb{V}_{0},\,\,\dim\mathbb{V}_{0}=d_{0}\quad\text{{even}}\\\
p(\mathfrak{v})=1,\,\,\mathfrak{v}\in\mathbb{V}_{1},\,\,\dim\mathbb{V}_{1}=d_{1}\quad\text{{odd}}\end{cases}$
(3.53)

with $\\{\mathfrak{e}_{1}\dots\mathfrak{e}_{d_{0}}\\}\in\mathbb{V}_{0}$,
$\\{\mathfrak{e}_{d_{0}+1}\dots\mathfrak{e}_{d_{1}}\\}\in\mathbb{V}_{1}$ and
total space dimension $\dim\mathbb{V}=\left(d_{0},d_{1}\right)$. The linear
operators131313Strictly speaking right operators having the right module in
the space $\mathbb{V}$ in the chosen basis can be provided by

$\mathcal{O}=\mathcal{O}(\mathfrak{e}_{i}\mathfrak{v}_{i})=\mathfrak{e}_{j}\mathcal{O}_{ij}\mathfrak{v}_{i}$
(3.54)

since grading is assigned to the space of $\mathcal{O}_{ij}$, one accordingly
equips rows and columns with parity and get parity for $\mathcal{O}$

$p(\mathcal{O})=p(i)+p(j)+p(\mathcal{O}_{ij})$ (3.55)

in what follows we shall be considering even operators, which implies

$p(\mathcal{O}_{ij})=p(i)+p(j)$ (3.56)

which is also a central property of graded braid-commutative structures. In
this respect, we can see that $RTT$ relation

$T_{a}(u)T_{b}(v)=R_{ab}(u-v)T_{b}(v)T_{a}(u)R_{ab}(u-v)^{-1}$ (3.57)

would imply consistent commutativity for the underlying transfer matrices, if
and only if we shall consider even $R$ matrices as an operator, i.e.

$\left[t(u),t(v)\right]=0\quad\text{along with
}t(u)=\text{STr}_{\mathfrak{s}}T_{\mathfrak{s}}(u)\quad\text{STr}\left(\mathcal{O}_{1}\mathcal{O}_{2}\right)=(-1)^{p(\mathcal{O}_{1})p(\mathcal{O}_{2})}\text{STr}\left(\mathcal{O}_{2}\mathcal{O}_{1}\right)$
(3.58)

where $\mathfrak{s}$ indicates the appropriate space over which the supertrace
is taken. On the other hand, it is known that one can derive a one-to-one
correspondence (based on the Zamolodchikov graded algebra and its triple
monoid consistency)

$\tilde{R}_{ij}(u)=(-1)^{p(m)p(n)}R_{ij}(u)$ (3.59)

where $\\{m,n\\}$ are basis vector indices associated to the action of the
$R(u)$ operator, which in general can act in the tensor product
$\mathbb{C}^{N}\otimes\mathbb{C}^{N}$. Namely, by this one can establish 1-1
correspondence between solutions of the qYBE ($R_{ij}$) and its graded
($\tilde{R}_{ij}$) analogue [22, 88, 89, 90]. More on graded structure (3.59),
algebra and other relevant properties can be found A.2. From here we clearly
see that our interest includes only even $R$ operator and the corresponding
YBE solutions. Also from the physical integrable perspective odd structure
does not appear to be relevant141414non-mixed structure only, although
bosonisation is admissible, whereas consistent commutativity of the generated
charges comes from the even sector.

##### Mappings

As stated above, we can generalise our tensor algebra to the graded one. By
looking at the additional subset of solutions obtained along with 3.2.2, one
can notice the ones that differ up to a sign in front of the $E_{21}\otimes
E_{12}$ term, which appears in the conventional graded representation

$\mathcal{A}=\sum(-1)^{p(k)\left(p(i)+p(j)\right)}\mathcal{A}_{ijkl}E_{ij}\otimes
E_{kl}\quad\rightarrow\quad\left(\begin{array}[]{cccc}\mathcal{A}_{1111}&\mathcal{A}_{1112}&\mathcal{A}_{1211}&\mathcal{A}_{1212}\\\
\mathcal{A}_{1121}&\mathcal{A}_{1122}&\mathcal{A}_{1221}&\mathcal{A}_{1222}\\\
\mathcal{A}_{2111}&\mathcal{A}_{2112}&\mathcal{A}_{2211}&\mathcal{A}_{2212}\\\
\mathcal{A}_{2121}&\mathcal{A}_{2122}&\mathcal{A}_{2221}&\mathcal{A}_{2222}\end{array}\right)$
(3.60)

where after certain redefinitions the graded prescription results in the
equivalent set of generators. This fact could be seen in two independent ways,
first by applying method 3.1 for 3.60 and finding the corresponding
$\mathcal{H}$, or the second, where one can exploit qYBE and its graded form
relation between solution spaces [22]. More concrete, having regular XYZ
$R$-matrix (all free functions assumed to depend on the spectral parameter
$u$)

$R(u)=\left(\begin{array}[]{cccc}a_{1}&0&0&d_{1}\\\ 0&b_{1}&c_{1}&0\\\
0&c_{2}&b_{2}&0\\\ d_{2}&0&0&a_{2}\end{array}\right)$ (3.61)

allows also to have an associated regular $R$ of the graded-YBE in the
modified form

$R(u)=\left(\begin{array}[]{cccc}a_{1}&0&0&\sigma_{1}d_{1}\\\
0&\sigma_{2}b_{1}&c_{1}&0\\\ 0&c_{2}&\sigma_{2}b_{2}&0\\\
-\sigma_{1}d_{2}&0&0&-a_{2}\end{array}\right)$ (3.62)

with $\sigma_{i}=\pm$. From here it is possible to note, that we can establish
a bijection mapping between regular non- and graded YBE solutions

$\left(\begin{array}[]{cccc}a_{1}&0&0&d_{1}\\\ 0&b_{1}&c_{1}&0\\\
0&c_{2}&b_{2}&0\\\
d_{2}&0&0&a_{2}\end{array}\right)\quad\rightarrow\quad\left(\begin{array}[]{cccc}a_{1}&0&0&d_{1}\\\
0&b_{1}&c_{1}&0\\\ 0&c_{2}&b_{2}&0\\\ -d_{2}&0&0&-a_{2}\end{array}\right),$
(3.63)

This also completes the classification problem for models with
$\mathbb{C}^{2}$ local quantum space in terms generating solutions, which are
characterised by their Hamiltonians $\mathcal{H}$ and associated $R$ matrices.
In the next section we generalise our approach, investigate the models with
four-dimensional local space and discover novel models in several sectors,
including the Hubbard-Shastry type.

##  4 $\mathbb{V}^{4}$-difference sector: New superconductive integrability
and GHM

Another possible extension of integrable models is the study of classes with
higher dimensional local spaces. Enlarging the quantum spaces leads to
interesting and important physical implications. It can be illustrated by the
Hubbard model, which has a 4-dim space and represent a strongly coupled
fermionic system. Namely, a site equipped with the 4-dim space admits vacuum,
single electron of either spin up or down, or an occupation of two electrons.
By such combination it is clear that there is a possibility of superconducting
modes or electron pair formation. It can be described by the Hubbard
Hamiltonian [91, 92]

$\mathbb{H}^{\text{Hubbard}}=\sum_{i}\sum_{\sigma=\\{\uparrow,\downarrow\\}}(\mathbf{c}^{\dagger}_{\sigma,i}\mathbf{c}_{\sigma,i+1}+\mathbf{c}^{\dagger}_{\sigma,i+1}\mathbf{c}_{\sigma,i})+\mathfrak{u}\,\mathbf{n}_{\uparrow,i}\mathbf{n}_{\downarrow,i}.$
(4.1)

where from now we shall work in the oscillator basis
$\\{\mathbf{c},\mathbf{c}^{\dagger}\\}$, which appears in the kinetic part
that provides electron dynamics between neighbouring sites (hopping term). The
potential part counts the number $\mathbf{n}$ of electron pairs for each site
and coupling $\mathfrak{u}$ provides the scale in the theory.

It is important that systems with higher local space dimensionality can be
also put on a lattice and proven to be integrable. Meaning one needs to deal
with 4-dim setup, which results in the $16\times 16$ $R$-matrix that satisfies

$R_{12}(u,v)R_{13}(u,w)R_{23}(v,w)=R_{23}(v,w)R_{13}(u,w)R_{12}(u,v)$ (4.2)

The integrability for Hubbard and related models was first proven in [93, 94,
95], which automatically implies the existence of integrable hierarchy of
conserved charges. However the construction and investigation of the latter
followed very different approaches. As it was indicated in 3.1 the
commutativity and braiding that is present in such hierarchies is highly
constraining. It is important to ask for a broader family of integrable
theories in this sector.

As we know, the Hubbard and Shastry class manifest different spectral
dependence, symmetry properties and more. This question can be approached by
the method above 3.1, which would allow to solve for potential new integrable
structures without actually solving qYBE. Moreover the quantum YBE will be
only used for quantum consistency checks of the found models (sufficiency
condition). At first, we shall consider classes, which are apriori different
from the Shastry sector and whose generalisation will be discussed in Sec. 5.

It has been already shown in [78], one could address this problem from an
appropriate Ansatz for the $\mathcal{H}$ and eventually obtain the underlying
$R$-matrix151515There was analogous attempt resolving YBE for nineteen-vertex
model [96] by means of perturbative $R$-matrix according to the scheme 8. Once
this procedure is (if) completed, one ends up with the new integrable
structure, and one can further attempt their detailed study by virtue of
different techniques, e.g. Quantum Inverse Scattering, Bethe Ansätze, Quantum
Spectral Curve and other. For example a similar classification problem was
addressed in [97, 98, 99] by applying the Coordinate Bethe Ansatz. It is
important to note, that not for all models it is possible to apply the above
procedures. We shall discuss this later for the novel models.

$\mathcal{H}$ Ansatz

$\mathcal{B}oost$ and Commutative hierarchy
$\left[\mathbb{Q}_{2},\mathbb{Q}_{3}\right]=\dots=0$

Algebraic system and Solution space

Integrable identifications

$R$-perturbation and YBE

Bethe Ansätze and further analysis

Figure 8: The boost automorphism method scheme for complete resolution of the
integrable structure

In what follows, we shall solve for novel models with 4-dim local space, where
$R$ matrices exhibit difference $R_{ij}(z_{i},z_{j})=R_{ij}(z_{i}-z_{j})$
spectral dependence (2.23). Generically, it is a highly nontrivial problem,
since an Ansatz for $\mathcal{H}$ can contain up to 256 free parameters. In
order to make computationally approachable, we shall constrain such generic
Ansatz to share some structure with the Hubbard model. Specifically, the first
part of the analysis will include the models with
$\mathfrak{su}(2)\oplus\mathfrak{su}(2)$ symmetry and the second will have
kinetic part in common with the Hubbard model. For the latter the algebra is
represented by charge-spin symmetry
$\mathfrak{su}_{\mathcal{C}}(2)\times\mathfrak{su}_{\eta}(2)$ [100, 101]. On
the other hand, the Hubbard model itself will not be considered as the part of
our solution space, since its spectral structure is different [94, 102]. We
shall return to this fact during the discussion of an arbitrary spectral
dependence and integrable structures arising in Gauge/Gravity duality, where
one deals with the central extensions leading to superalgebra
$\mathfrak{su}(2|2)$ [38, 39, 103].

We shall perform analysis for all (sub)sectors isomorphic to
$\mathfrak{su}(2)\oplus\mathfrak{su}(2)$, among them will include spin chain
structure of $\mathfrak{su}(4)$, $\mathfrak{so}(4)$, $\mathfrak{sp}(4)$ and
$\mathfrak{su}(2|2)$. Beyond the sectors of conventional magnetics, we find
novel structure, where only fermionic pairs have dynamics and their individual
dof are not present (nevertheless they do affect the spectrum).

Another class constitutes free Hubbard deformations, which we call Generalised
Hubbard Models. For GHM we consider the Hubbard type kinetic part and a
generic potential up to quartic order in number operators. This setup is then
extended by specific flipping and pair hopping terms. For physical consistency
we demand the conservation of the fermion number, which eventually results in
spin-up/spin-down decoupled models and a nontrivial model that contains two
free parameters. The last one contains spin flipping behaviour (mixing of up-
up and down-down states), which resembles elliptic dynamics. Specifically due
to the presence of the XYZ-type terms, the inhomogeneous161616also other types
of CBA coordinate Bethe Ansatz is not applicable. It is still in progress to
solve this type of models by Generalised ABA or QSC. In what follows we shall
now develop the structure [78, 104] of an approach and constraints necessary
for the resolution of both groups of models.

###  4.1 4-dim setup

In this section we define the structure necessary for the search of novel
models and their properties in four dimensions. For this reason we briefly
provide core steps of 3.1 [78] and details appropriate for the extended 4-dim
setting. In this respect, along with restrictions required for the ansatz, we
shall proceed in two-particle representation to have $16\times 16$ embedding
for the YBE.

####  4.1.1 $\mathcal{B}$ and $\mathcal{H}$

As indicated before we require boost symmetry [11, 30, 12] as a sum of local
densities

$\mathcal{B}\left[\mathbb{Q}_{2}\right]=\sum_{k=-\infty}^{+\infty}k\mathcal{Q}_{k,k+1}$
(4.3)

where the second charge is the Hamiltonian, which now becomes a $16\times 16$
operator

$\mathbb{Q}_{2}\equiv\mathbb{H}=\sum_{k=-\infty}^{+\infty}\mathcal{H}_{k,k+1}\qquad\mathbb{Q}_{r+1}\sim\left[\mathcal{B}\left[\mathbb{Q}_{2}\right],\mathbb{Q}_{r}\right]$
(4.4)

In this case, one can also implement recursion for generating the charge of
range $r$. In general, it is still a requirement to have infinite commuting
hierarchy $\left[\mathbb{Q}_{2},\mathbb{Q}_{3}\right]=\dots=0$. However it can
be proven at least for the classes under investigation that it is sufficient
to provide the first commutator171717Although it is known that in general that
does not need to be true, there is a number of examples in QFT and integrable
deformations when higher charges are necessary, e.g. Zhiber-Shabat-Mikhailov
model [2], so that along with the found $R$-matrix the YBE is satisfied.

####  4.1.2 Oscillator basis

The mentioned physical setup can be conveniently formulated in terms of
oscillators, which on quantum level are nothing but creation and annihilation
operators that satisfy an algebra. As we have already mentioned the Hubbard
kinetic and potential parts appear as

$\mathbb{H}^{\text{Hubbard}}=\sum_{i}\sum_{\sigma=\\{\uparrow,\downarrow\\}}(\mathbf{c}^{\dagger}_{\sigma,i}\mathbf{c}_{\sigma,i+1}+\mathbf{c}^{\dagger}_{\sigma,i+1}\mathbf{c}_{\sigma,i})+\mathfrak{u}\,\mathbf{n}_{\uparrow,i}\mathbf{n}_{\downarrow,i}.$
(4.5)

where fermionic operators
$\\{\mathbf{c}_{\sigma,i},\mathbf{c}_{\sigma,i}^{\dagger}\\}$ act on $i$th
lattice site with up/down spins $\sigma$. Accordingly the pair counting is
captured by the number operators arising in the potential
$\mathbf{n}_{\sigma,i}=\mathbf{c}_{\sigma,i}^{\dagger}\mathbf{c}_{\sigma,i}$.
The local quantum space $\mathfrak{h}_{i}$ spanned by four states

$\begin{cases}\ket{\phi_{1}}=\ket{0}\\\
\ket{\phi_{2}}=\mathbf{c}_{\uparrow,i}^{\dagger}\mathbf{c}_{\downarrow,i}^{\dagger}\ket{0}\end{cases}\qquad\begin{cases}\ket{\psi_{1}}=\mathbf{c}_{\uparrow,i}^{\dagger}\ket{0}\\\
\ket{\psi_{2}}=\mathbf{c}_{\downarrow,i}^{\dagger}\ket{0}\end{cases}$ (4.6)

with conventional vacuum annihilation $\mathbf{c}_{\sigma,i}\ket{0}=0$.
Because of the fermionic nature of the oscillators, we also consider the
anticommutative algebra

$\\{\mathbf{c}_{\rho,i}^{\dagger},\mathbf{c}_{\sigma,j}\\}=\delta_{\rho\sigma}\delta_{ij}\quad\text{and}\quad\\{\mathbf{c}_{\rho,i}^{\dagger},\mathbf{c}_{\sigma,j}^{\dagger}\\}=\\{\mathbf{c}_{\rho,i},\mathbf{c}_{\sigma,j}\\}=0$
(4.7)

It could be also noted from (4.6) and algebraic structure (4.7), that one can
develop graded structure with such a prescription, i.e. if one allocates
$\\{\phi_{1},\phi_{2}\\}$ as an even sector and has fermions
$\\{\psi_{1},\psi_{2}\\}$ for the odd. With this established, one can identify
boson and fermion subspaces, and the total space becomes graded
$\mathbb{C}^{2|2}$. For that we have graded YBE structure also in the present
case [22, 92, 90], more on the graded algebra and properties is discussed in
A.2.

####  4.1.3 Integrable identifications

Since the solution space in four dimensional case can be even larger, we need
to implement an identification mechanism through allowed transformations. Such
a procedure will again allow to create classes in the solution space and
present them in concise form. Here we mention the necessary set of these
transformations.

  * •

Normalisation and Reparametrisation: A YBE solution multiplied by a scalar
function, or some solutions that contain free parameters can be mapped to
other solutions by redefining these parameters, which clearly does not produce
new integrable structures.

  * •

Local Basis Transform:

$R\xRightarrow{\text{LBT}}R^{\prime}=\mathfrak{V}\mathcal{R}\mathfrak{V}^{-1}\qquad\mathfrak{V}=\mathcal{V}\otimes\mathcal{V}$
(4.8)

where $\mathcal{V}$ acts in 4-dim
$\mathcal{V}:\mathbb{C}^{4}\rightarrow\mathbb{C}^{4}$.

  * •

Discrete Transform Transposition and permutation (3.18) establish other maps
between solutions.

  * •

Twisting

$R\xRightarrow{\mathcal{W}_{1}\otimes\mathcal{W}_{2}}R^{\prime}=\mathfrak{W}\mathcal{R}\mathfrak{W}^{-1}\qquad\mathfrak{W}=\mathcal{W}_{1}\otimes\mathcal{W}_{2}\quad\mathfrak{W}^{-1}=\mathcal{W}_{2}\otimes\mathcal{W}_{1}$
(4.9) $\text{along with
}\left[R,\mathcal{W}_{i}\otimes\mathcal{W}_{i}\right]=0\qquad\mathcal{W}_{i}:\mathbb{C}^{4}\rightarrow\mathbb{C}^{4}$
(4.10)

over both subspaces $i=1,2$. It is important to remark, that in general the
twists do not need to respect $R$-matrix symmetries, hence they break them.
Moreover by means of twists one can obtain models with different spectrum
(change of physical properties). However for now we shall restrict to
integrable configurations that can be related by the transformations above

###  4.2 Hubbard type: $2\oplus 1\oplus 1$ sector

Before starting with our prescription, we need to find not only an appropriate
parametric restriction due to reasons stated above, but we should also fix a
suitable representation to work with. In this respect it is natural to
consider that our ansatz respects $\mathfrak{su}(2)\oplus\mathfrak{su}(2)$
symmetry algebra similar to Hubbard’s spin-charge
$\mathfrak{su}_{\mathcal{C}}(2)\otimes\mathfrak{su}_{\eta}(2)$ symmetry [92].
It is known that there exist two representations of interest for the current
symmetry algebra.

####  4.2.1 $\mathcal{H}_{\text{Hub}}$ Ansatz

We shall use the standard notation for nontrivial representations, which
exhibit direct decomposition
$\mathfrak{R}=\bigoplus_{\mathfrak{n}}\mathfrak{r}_{\mathfrak{n}}$. In
particular, $\mathfrak{su}(2)\oplus\mathfrak{su}(2)$ possess a representation
in which both copies $\mathfrak{su}(2)$ are two dimensional, i.e.

$\mathfrak{r}_{2\oplus 1\oplus
1}=\begin{pmatrix}\mathfrak{r}_{2}(t_{i}^{L})&0\\\
0&\mathfrak{r}_{2}(t_{i}^{R})\\\ \end{pmatrix}$ (4.11)

where $t_{i}$ are generators and in the index we reflect the form of
decomposition, which results in the associated block-diagonal form (more
detailed on relevant structure and direct decompositions can be found in B).
In this framework, one can consider a two-state representation for the
$\mathcal{H}$ of the model

$\displaystyle\mathcal{H}|\phi_{a}\phi_{b}\rangle=A|\phi_{a}\phi_{b}\rangle+B|\phi_{b}\phi_{a}\rangle+C\epsilon_{ab}\epsilon^{\alpha\beta}|\psi_{\alpha}\psi_{\beta}\rangle$
(4.12)
$\displaystyle\mathcal{H}|\phi_{a}\psi_{\beta}\rangle=G|\phi_{a}\psi_{\beta}\rangle+H|\psi_{\beta}\phi_{a}\rangle,$
(4.13)
$\displaystyle\mathcal{H}|\psi_{\alpha}\phi_{b}\rangle=K|\psi_{\alpha}\phi_{b}\rangle+L|\phi_{b}\psi_{\alpha}\rangle$
(4.14)
$\displaystyle\mathcal{H}|\psi_{\alpha}\psi_{\beta}\rangle=D|\psi_{\alpha}\psi_{\beta}\rangle+E|\psi_{\beta}\psi_{\alpha}\rangle+F\epsilon^{ab}\epsilon_{\alpha\beta}|\phi_{a}\phi_{b}\rangle.$
(4.15)

which results in the conventional form

$\displaystyle\mathcal{H}=\tiny{\setcounter{MaxMatrixCols}{16}\begin{pmatrix}A+B&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
0&A&0&0&B&0&0&0&0&0&0&F&0&0&-F&0\\\ 0&0&G&0&0&0&0&0&L&0&0&0&0&0&0&0\\\
0&0&0&G&0&0&0&0&0&0&0&0&L&0&0&0\\\ 0&B&0&0&A&0&0&0&0&0&0&-F&0&0&F&0\\\
0&0&0&0&0&A+B&0&0&0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&G&0&0&L&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&G&0&0&0&0&0&L&0&0\\\ 0&0&H&0&0&0&0&0&K&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&H&0&0&K&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0&0&0&D+E&0&0&0&0&0\\\
0&C&0&0&-C&0&0&0&0&0&0&D&0&0&E&0\\\ 0&0&0&H&0&0&0&0&0&0&0&0&K&0&0&0\\\
0&0&0&0&0&0&0&H&0&0&0&0&0&K&0&0\\\ 0&-C&0&0&C&0&0&0&0&0&0&E&0&0&D&0\\\
0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&D+E\end{pmatrix}}$ (4.16)

where $\phi_{i}$ and $\psi_{i}$ are in their fundamental representations of
$\mathfrak{su}(2)$. That is also relevant for superstring model in
$AdS_{5}\times S^{5}$, Hubbard model, Heisenberg $\mathfrak{su}(4)$ and other.
From the oscillator representation of $\mathfrak{su}(2)\oplus\mathfrak{su}(2)$
B, its algebra and commutativity of generic nearest-neighbour operator
$\mathcal{H}_{ij}$ with each copy of the $\mathfrak{su}(2)$, we can obtain
10-parametric expression for it

$\displaystyle\mathcal{H}_{ij}=$ $\displaystyle\leavevmode\nobreak\
\sum_{\rho\neq\sigma}\Big{[}(\mathbf{c}^{\dagger}_{\rho,1}\mathbf{c}_{\rho,2}+\mathbf{c}_{\rho,1}\mathbf{c}^{\dagger}_{\rho,2})\left(\mathcal{A}_{1}+\mathcal{A}_{2}(\mathbf{n}_{\sigma,1}-\mathbf{n}_{\sigma,2})^{2}\right)$
$\displaystyle\qquad+(\mathbf{c}^{\dagger}_{\rho,1}\mathbf{c}_{\rho,2}-\mathbf{c}_{\rho,1}\mathbf{c}^{\dagger}_{\rho,2})\left(\mathcal{A}_{3}(\mathbf{n}_{\sigma,1}-\frac{1}{2})+\mathcal{A}_{4}(\mathbf{n}_{\sigma,2}-\frac{1}{2})\right)\Big{]}$
$\displaystyle\leavevmode\nobreak\
+\mathcal{A}_{0}+(\mathbf{c}^{\dagger}_{\uparrow,1}\mathbf{c}^{\dagger}_{\downarrow,1}\mathbf{c}_{\uparrow,2}\mathbf{c}_{\downarrow,2}+\mathbf{c}_{\uparrow,1}\mathbf{c}_{\downarrow,1}\mathbf{c}^{\dagger}_{\uparrow,2}\mathbf{c}^{\dagger}_{\downarrow,2})\mathcal{A}_{5}+(\mathbf{c}^{\dagger}_{\uparrow,1}\mathbf{c}_{\downarrow,1}\mathbf{c}^{\dagger}_{\downarrow,2}\mathbf{c}_{\uparrow,2}+\mathbf{c}^{\dagger}_{\downarrow,1}\mathbf{c}_{\uparrow,1}\mathbf{c}^{\dagger}_{\uparrow,2}\mathbf{c}_{\downarrow,2})\mathcal{A}_{6}$
$\displaystyle\leavevmode\nobreak\
+\mathcal{A}_{7}(\mathbf{n}_{\uparrow,1}-\frac{1}{2})(\mathbf{n}_{\downarrow,1}-\frac{1}{2})+\mathcal{A}_{8}(\mathbf{n}_{\uparrow,2}-\frac{1}{2})(\mathbf{n}_{\downarrow,2}-\frac{1}{2})+\mathcal{A}_{9}(\mathbf{n}_{\uparrow,1}-\mathbf{n}_{\downarrow,1})^{2}(\mathbf{n}_{\uparrow,2}-\mathbf{n}_{\downarrow,2})^{2}$
$\displaystyle\leavevmode\nobreak\
+(\mathcal{A}_{5}-\mathcal{A}_{6})(\mathbf{n}_{\uparrow,1}\mathbf{n}_{\downarrow,1}+\mathbf{n}_{\uparrow,2}\mathbf{n}_{\downarrow,2}-1)(\mathbf{n}_{\uparrow,1}-\mathbf{n}_{\uparrow,2})(\mathbf{n}_{\downarrow,1}-\mathbf{n}_{\downarrow,2})$
$\displaystyle\leavevmode\nobreak\
+\frac{1}{2}\mathcal{A}_{5}\left[(\mathbf{n}_{\uparrow,1}-\mathbf{n}_{\downarrow,2})^{2}+(\mathbf{n}_{\downarrow,1}-\mathbf{n}_{\uparrow,2})^{2}\right]$
(4.17)

where the free parameters $\mathcal{A}_{n}$ are defined through 2-state
coefficients in the following way

$\displaystyle\mathcal{A}_{0}=\frac{1}{2}(B+G+K),\,\mathcal{A}_{1}=\frac{1}{2}(L-H),\,\mathcal{A}_{2}=\frac{1}{2}(C-F+H-L),$
$\displaystyle\mathcal{A}_{3}=\frac{1}{2}(H+L-C-F),\,\mathcal{A}_{4}=\frac{1}{2}(C+F+H+L),\,\mathcal{A}_{5}=-B,\,\mathcal{A}_{6}=E,$
$\displaystyle\mathcal{A}_{7}=2A+B-2K,\,\mathcal{A}_{8}=2A+B-2G,\,\mathcal{A}_{9}=A+B+D+E-G-K.$
(4.18)

In what follows we shall present results of the Hubbard type solution space,
and emphasize meaning of the new models.

####  4.2.2 Hamiltonian solution space

We start by implementing the steps of the scheme 8, which are modified
accordingly for the current system properties. Specifically, we impose Ansatz
(4.16) and demand commuting hierarchy of the charges through recursion [78].
After solving the system, which leads to 45 solutions, one needs to apply
transformations 4.1.3 to extract 12 generators at the first stage [104]

Model | A | B | C | D | E | F | G | H | K | L
---|---|---|---|---|---|---|---|---|---|---
I | $\rho$ | $-\rho$ | 0 | 0 | 0 | 0 | $a$ | $\rho e^{-\phi}$ | $2\rho-a$ | $\rho e^{\phi}$
II | 0 | 0 | 0 | $\rho$ | $\rho$ | 0 | $a$ | $\rho e^{-\phi}$ | $2\rho-a$ | $\rho e^{\phi}$
III | $\rho$ | $-\rho$ | $\rho e^{-\phi}$ | $-\rho$ | $\rho$ | $-\rho e^{\phi}$ | 0 | 0 | 0 | 0
IV | $\rho$ | $-\rho$ | $\rho e^{-\phi}$ | $\rho$ | $-\rho$ | $\rho e^{\phi}$ | 0 | 0 | 0 | 0
V | $\frac{7}{4}\rho$ | $-\rho$ | $\frac{1}{2}\rho e^{-\phi}$ | $\frac{7}{4}\rho$ | $-\rho$ | $\frac{1}{2}\rho e^{\phi}$ | 0 | 0 | 0 | 0
VI | 0 | 0 | 0 | $a$ | 0 | 0 | $b$ | 0 | $c$ | 0
VII | 0 | 0 | 0 | 0 | 0 | 0 | $a$ | $b$ | $c$ | $d$
VIII | 0 | 0 | 0 | $a+c$ | 0 | 0 | $a$ | $b$ | $c$ | $d$
IX | $\rho$ | $-\rho$ | 0 | $\rho$ | $-\rho$ | 0 | $a$ | $\rho e^{-\phi}$ | $2\rho-a$ | $\rho e^{\phi}$
X | $\rho$ | $-\rho$ | 0 | $\rho$ | $\rho$ | 0 | $a$ | $\rho e^{-\phi}$ | $2\rho-a$ | $\rho e^{\phi}$
XI | $\rho$ | $-\rho$ | $\frac{1}{2}\rho e^{-\phi}$ | $\rho$ | $-\rho$ | $\frac{1}{2}\rho e^{\phi}$ | $\frac{3}{2}\rho$ | $-\frac{3}{2}\rho$ | $\frac{3}{2}\rho$ | $-\frac{3}{2}\rho$
XII | 0 | 0 | $-\rho e^{-\phi}$ | 0 | 0 | $\rho e^{\phi}$ | 0 | $\rho$ | 0 | $-\rho$
Table 1: Hubbard type $\mathcal{H}$ solution generators in the non-graded
sector

Five of the generating solutions above appear to be novel with nontrivial
properties. We shall construct all the underlying $R$ matrices, perform
quantum checks, as well as comparison and interpretation for each of the found
models.

####  4.2.3 $R$ matrices

To avoid misconception, the form of the $R$-matrix that follows from the
$\mathfrak{su}(2)\oplus\mathfrak{su}(2)$ algebra in the fixed basis
$\\{\phi_{a},\psi_{\alpha}\\}$ can be presented as follows

${\tiny
R_{ij}=\left(\begin{array}[]{cccccccccccccccc}r_{1}+r_{2}&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
0&r_{1}&0&0&r_{2}&0&0&0&0&0&0&-r_{8}&0&0&r_{8}&0\\\
0&0&r_{4}&0&0&0&0&0&r_{10}&0&0&0&0&0&0&0\\\
0&0&0&r_{4}&0&0&0&0&0&0&0&0&r_{10}&0&0&0\\\
0&r_{2}&0&0&r_{1}&0&0&0&0&0&0&r_{8}&0&0&-r_{8}&0\\\
0&0&0&0&0&r_{1}+r_{2}&0&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&r_{4}&0&0&r_{10}&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&r_{4}&0&0&0&0&0&r_{10}&0&0\\\
0&0&r_{7}&0&0&0&0&0&r_{3}&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&r_{7}&0&0&r_{3}&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0&0&0&r_{5}+r_{6}&0&0&0&0&0\\\
0&-r_{9}&0&0&r_{9}&0&0&0&0&0&0&r_{5}&0&0&r_{6}&0\\\
0&0&0&r_{7}&0&0&0&0&0&0&0&0&r_{3}&0&0&0\\\
0&0&0&0&0&0&0&r_{7}&0&0&0&0&0&r_{3}&0&0\\\
0&r_{9}&0&0&-r_{9}&0&0&0&0&0&0&r_{6}&0&0&r_{5}&0\\\
0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&r_{5}+r_{6}\\\ \end{array}\right)}$ (4.19)

We shall also use this form to present all the corresponding $R$ matrices for
Table 1. As was already stated above, the spectral dependence $r\equiv r(u)$
is assumed.

We should also note that resolution for the $R$-matrix in 4-dim case
technically differs from the analytic procedure introduced in 3.1.4.
Concretely, the first two orders can be fixed by regularity and $\mathcal{H}$,
whereas the rest can follow from the perturbative solution of the YBE. Also in
our case, the second order perturbation can provide initial restrictions, i.e.
certain relations on $r$ and mapping up to a sign. In the next step, with such
pre-conditioned ansatz for $R$ and differentiated YBE with respect to one of
the spectral parameters (along with $u\rightarrow 0$), it becomes possible to
solve for the $R$-matrix. The existence of a regular point, braiding unitarity
and initial constraints following from perturbation

$R(0)=P\qquad R_{ij}(u)R_{ji}(-u)\sim\mathds{1}\qquad
R(u)=P\left(1+\mathcal{H}u+\mathcal{H}^{2}\dfrac{u^{2}}{2}+\mathcal{O}(u^{3})\right)$
(4.20)

provides the full quantum consistency [96]. First of all we present the novel
models, new deformations, comparisons, and eventually briefly discuss the
produced known models to confirm completeness of the approach.

####  4.2.4 The novel models: XXX deformations and Coupled $\psi\psi$-models

##### Fixing parametrisation for $\mathcal{H}/R$

Before presenting the analysis of the models obtained in 1, we need to note
that it is useful to make suitable parametric changes in $\mathcal{H}$ and $R$
by allowed integrable transforms. By doing so, we shall benefit in its further
treatment and comparison analysis, it is especially relevant for 4.2.5. In
particular for models I,II,IX andX, the $a$=dependence can be decomposed into
to the term that vanishes on periodic systems

$\mathcal{H}=\mathcal{H}_{1}+\left(\mathfrak{a}\otimes\mathds{1}-\mathds{1}\otimes\mathfrak{a}\right)$
(4.21)

where $\mathcal{H}_{1}$ does not contain $a$ and
$\mathfrak{a}=\text{diag}\left(1,1,1,-a,1,-a\right)$. Moreover, we can recover
the $a$-dependence by twist like transformation for the $R$-matrix

$R_{ij}=\left(\mathcal{T}(u)\otimes\mathds{1}\right)\left(R_{ij,\,a=0}(u)\right)\left(\mathcal{T}(-u)\otimes\mathds{1}\right)\qquad\mathcal{T}(u)=\text{diag}\left(1,1,e^{au},e^{au}\right)$
(4.22)

analogously one could treat the $\phi$-dependence of the Hamiltonians
$\mathcal{H}_{\text{{I},{II},{IX},{X}}}$, i.e.

$\mathcal{H}\left(\phi\right)=\left(\mathcal{W}\otimes\mathds{1}\right)\left(\mathcal{H}_{\phi=0}\right)\left(\mathcal{W}\otimes\mathds{1}\right)\qquad\mathcal{W}(\phi)=\text{diag}\left(1,1,e^{-\phi},e^{-\phi}\right)$
(4.23)

We can see that in this way, we can suitably fix $\\{a,\phi\\}$ and
reconstruct their dependence upon demand without loss of generality. It could
be also added that for models III,IV,V,XI and XII we can also isolate the
$\phi$ through similarity

$\mathcal{H}\left(\phi\right)=\left(\mathcal{V}\otimes\mathcal{V}\right)\left(\mathcal{H}_{\phi=0}\right)\left(\mathcal{V}\otimes\mathcal{V}\right)^{-1}\qquad\mathcal{V}(\phi)=\text{diag}\left(1,1,e^{-\phi},1\right)$
(4.24)

as we have already indicated, (4.22) and (4.23) can leave the impact on the
spectrum and physical interpretation, whereas basis transformation (4.24) does
not.

We shall analyse classes I-V and highlight their properties here. In this
respect, one can notice that classes I and II represent XXX-type deformations
and classes III-V model $\psi\psi$-pair models.

##### Model I

The first class $R$-matrix can be given according to 4.2.3 as follows

$\begin{array}[]{l}r_{1}=-u\rho\ r_{2}\\\ r_{2}=(1-u\rho)^{-1}\\\
r_{3}=-e^{\phi}\ r_{1}\\\ r_{6}=1\\\
\end{array},\quad\begin{array}[]{l}r_{4}=-e^{-\phi}\ r_{1}\\\
r_{7}=e^{u(a-\rho)}\ r_{2}\\\ r_{10}=e^{u(\rho-a)}\ r_{2}\\\ \end{array}$
(4.25)

as it could be noticed it is characterised by $\\{\rho,a,\phi\\}$ and in the
limit $\\{-1,-1,i\pi\\}$ it becomes a version of $\mathfrak{su}(4)$. It can be
seen as follows

$\mathcal{H}_{\text{{I}}}=1-P+\sum_{\\{i,j,k,l\\}=1,2}\epsilon_{ij}\epsilon^{kl}E_{k}^{i}\otimes
E_{l}^{j}=1-P+\sum_{\\{i,j\\}=1,2}\left[E_{i}^{i}\otimes
E_{j}^{j}-E_{i}^{j}\otimes E_{j}^{i}\right]$ (4.26)

where by $E_{a}^{b}$ is a $4\times 4$ matrix with unit at $(a,b)$ (the
expression above is achieved by applying LBT, that brings basis vectors
$E^{a}$ to $E^{5-a}$). One can also see that the second term can interpreted
as XXX spin chain itself, namely

$E_{i}^{i}\otimes E_{j}^{j}=\mathds{1}_{2}\qquad E_{i}^{j}\otimes
E_{j}^{i}=P_{2}$ (4.27)

where it obviously restricts to the first two basis vectors.

##### Model II

For the $R_{\text{{II}}}$ we get

$\begin{array}[]{l}r_{2}=1\\\ r_{6}=(1-u\rho)^{-1}\\\ r_{5}=u\rho\ r_{6}\\\
r_{3}=e^{\phi}\ r_{5}\end{array},\quad\begin{array}[]{l}r_{4}=e^{-\phi}\
r_{5}\\\ r_{7}=e^{u(a-\rho)}\ r_{6}\\\ r_{10}=e^{u(\rho-a)}\ r_{6}\end{array}$
(4.28)

where in the limit $\\{1,1,i\pi\\}$, the model I can be related to II by
grading transform. It could be written as

$\mathcal{H}_{\text{{II}}}=1-P_{g}+\sum_{\\{i,j,k,l\\}=1,2}\epsilon_{ij}\epsilon^{kl}E_{k}^{i}\otimes
E_{l}^{j}$ (4.29)

with graded permutation $P_{g}$. As a periodic systems, both models I and II
possess ferrovacuum $\ket{\phi_{i}\dots\phi_{i}}=\ket{0}$ and degenerate
ground state. We can also note, that $\mathfrak{su}(3)$ chain with a basis
$\\{\phi_{1},\psi_{1},\psi_{2}\\}$ appears as a subsystem for these models. In
general the spectrum depends on the twist parameters, however assuming
different $\\{\rho,a,\phi\\}$ limits, one obtains distinct eigenspectra, e.g.
including similar to $\mathfrak{su}(4)$.

#####  $\psi\psi$ models

The classes III-V correspond to models, where no isolated fermionic fields are
propagating, as it can be seen from (4.12) and Table 1 $G=H=K=L=0$. Their $R$
matrices are found to be

##### Model III

$\begin{array}[]{l}r_{1}=-r_{5}=-{\rm tan}(u\ \rho)\\\ r_{2}=1-r_{1}\\\
r_{6}=1+r_{1}\end{array},\quad\begin{array}[]{l}r_{7}=r_{10}=1\\\
r_{8}=e^{\phi}\ r_{1}\\\ r_{9}=-e^{-\phi}\ r_{1}\\\ \end{array}$ (4.30)

##### Model IV

$r_{1}=r_{5}=2+\sqrt{3}\coth\left(\sqrt{3}\rho
u+\log\left(2-\sqrt{3}\right)\right)$
$\begin{array}[]{l}r_{2}=r_{6}=1-r_{1}\\\
r_{7}=r_{10}=1\end{array}\quad\begin{array}[]{l}r_{8}=-e^{\phi}\ r_{1}\\\
r_{9}=-e^{-\phi}\ r_{1}\end{array}$ (4.31)

##### Model V

$\begin{array}[]{l}r_{1}=r_{5}=\frac{2(e^{\frac{3\rho
u}{2}}-1)}{e^{\frac{3\rho u}{2}}-4}\\\ r_{2}=r_{6}=-\frac{e^{\frac{3\rho
u}{2}}+2}{e^{\frac{3\rho u}{2}}-4}\\\
\end{array}\quad\begin{array}[]{l}r_{7}=r_{10}=e^{-\frac{1}{4}(3\rho u)}\\\
r_{8}=e^{-2\phi}r_{9}=-\frac{1}{2}e^{\frac{3\rho u}{4}+\phi}\
r_{1}\end{array}$ (4.32)

So all boson-fermion scattering is suppressed, nevertheless coupled fermionic
dynamics is possible. The problematics that comes up in these models, is that
the magnon-form picture is not evident here, hence the issues with Bethe
Ansätze. For example, appropriately adopted inhomogeneous nested CBA already
fails at level 2, but it is important to note that not all possibilities have
been tried, including generalised ABA [105] or QSC. On the other hand,
analysis on short spin chains or small number of excitations indicates the
highly nontrivial eigenspectrum.

The class III appears as combination of Hermitian and anti-Hermitian
operators, whereas IV and V are Hermitian. Also IV ground state is a
ferromagnetic vacuum, on contrary, the V has a nontrivial ground state energy
and degeneration pattern. The classes I-V are nontrivially interesting in both
their generality and complete analytic resolution.

####  4.2.5 Completeness and comparison analysis

The rest of the models VI-XII and their modified forms already appeared in the
literature before. Here we only want to comment on the consistency and
completeness of the method for the representation (4.11) under discussion. The
model list, their structure and properties follow.

##### Model VI

This class is diagonal, hence the integrability is trivially implied. The
corresponding $R$-matrix

$\begin{array}[]{l}r_{2}=1\\\ r_{6}=e^{au}\\\
\end{array}\quad\begin{array}[]{l}r_{7}=e^{bu}\\\ r_{10}=e^{cu}\\\
\end{array}$ (4.33)

##### Model VII

This class can be viewed as XXZ $16\times 16$ embedding (or quadruple
embedding - a single vector for each of the doublets
$\\{\phi_{a},\psi_{\beta}\\}_{\mathfrak{su}(2)}$), where the Hamiltonian
arises accordingly to

$\mathcal{H}_{\text{{VII}}}=\begin{pmatrix}0&0&0&0\\\ 0&a&b&0\\\ 0&d&c&0\\\
0&0&0&0\end{pmatrix}$ (4.34)

it can be shown that the spectrum corresponds to XXZ (with the corresponding
quadruple eigen-degeneracy). The $R$-matrix appears as follows

$\begin{array}[]{l}r_{2}=r_{6}=1\\\
r_{3}=\frac{a+c}{2b}(\cot(\eta)\cot(g(u))-1)\\\
r_{4}=\frac{2b}{a+c}\sin(\eta)\csc(g(u))\cos(g(u)+\eta)\end{array},\quad\begin{array}[]{l}r_{7}=e^{\frac{1}{2}u(a-c)}\cos(\eta)\csc(g(u))\\\
r_{10}=e^{-u(a-c)}\ r_{7}\end{array}$ (4.35)

where $g(u)={\rm arccot}(\tan(\eta))-\frac{1}{2}u(a+c)\cot(\eta)$. We can also
notice that there are two singularities present, when $b\rightarrow 0$, then

$\begin{array}[]{l}r_{2}=r_{6}=1\\\
r_{3}=\frac{d\left(e^{u(a+c)}-1\right)}{a+c}\\\
\end{array}\quad\begin{array}[]{l}r_{7}=e^{au}\\\ r_{10}=e^{cu}\end{array}$
(4.36)

with introduced $d$

$d=\frac{(a+c)^{2}\csc^{2}(\eta)}{4b}$ (4.37)

or when $(a+c)\rightarrow 0$, then $R$-matrix elements reduce to

$\begin{array}[]{l}r_{2}=r_{6}=1\\\
r_{3}=\frac{d}{b}r_{4}=\sqrt{\frac{d}{b}}\tan\left(\sqrt{b\ d}\ u\right)\\\
\end{array},\quad\begin{array}[]{l}r_{7}=e^{au}\sec\left(\sqrt{b\ d}\
u\right)\\\ r_{10}=e^{-2au}\ r_{7}\end{array}$ (4.38)

In principle, along $\eta$ parameter, this analysis allows to control special
points and model subcases.

##### Model VIII

In class VIII, we have staggered XXZ structure, with the
$\mathcal{H}_{{VIII}}$

$\displaystyle\mathcal{H}_{\text{{VIII}}}^{\text{St}}=\begin{pmatrix}0&0&0&0\\\
0&a&b&0\\\ 0&d&c&0\\\ 0&0&0&a+c\end{pmatrix}$ (4.39)

and the associated $R$-matrix

$\begin{array}[]{c}\begin{split}&r_{2}=1\\\
&r_{3}=\frac{2b}{a+c}g(u)h(u)\text{sech}(\eta)\\\
&r_{4}=\frac{a+c}{2b}g(u)h(u)\cosh(\eta)\\\
\end{split}\end{array}\quad\begin{array}[]{c}\begin{split}&r_{6}=g(u)\sinh\left(\frac{1}{2}u(a+c)\tanh(\eta)+\eta\right)\\\
&r_{7}=e^{\frac{1}{2}u(a-c)}g(u)\sinh(\eta)\\\
&r_{10}=e^{\frac{1}{2}u(c-a)}g(u)\sinh(\eta)\\\ \end{split}\end{array}$ (4.40)

where by $g(u)$ and $h(u)$ we denote

$\begin{cases}g(u)=\text{csch}\left(\eta-\frac{1}{2}u(a+c)\tanh(\eta)\right)\\\
h(u)=\sinh\left(\frac{1}{2}u(a+c)\tanh(\eta)\right)\end{cases}$ (4.41)

Analogously here we observe special points, hence for $b\rightarrow 0$ the
system (4.40) becomes

$\begin{array}[]{c}\begin{split}&r_{2}=1\\\
&r_{3}=\frac{d}{a+c}\left(e^{u(a+c)}-1\right)\\\ &r_{6}=e^{(a+c)u}\\\
\end{split}\end{array}\quad\begin{array}[]{c}\begin{split}&r_{7}=e^{au}\\\
&r_{10}=e^{cu}\end{split}\end{array}$ (4.42)

with $d=\frac{(a+c)^{2}}{4b}\text{sech}^{2}(\eta)$, whereas for
$(a+c)\rightarrow 0$ the $d\rightarrow 0$ and we conclude behaviour equivalent
to (4.38).

##### Model IX

This class is nothing but the $\mathfrak{su}(4)$ spin chain that contains an
additional twist. As we have indicated in 4.2.4, when parameters acquire
$\\{-1,-1,i\pi\\}$ values, we exactly obtain reduction to $\mathfrak{su}(4)$

$\mathcal{H}\rightarrow 1-P$ (4.43)

The underlying $R$-matrix corresponds to

$\begin{array}[]{l}r_{1}=r_{5}=-u\rho\ r_{2}\\\ r_{2}=r_{6}=(1-u\rho)^{-1}\\\
r_{3}=-e^{\phi}\ r_{1}\end{array}\quad\begin{array}[]{l}r_{4}=-e^{-\phi}\
r_{1}\\\ r_{7}=e^{u(a-\rho)}\ r_{2}\\\ r_{10}=e^{u(\rho-a)}\ r_{2}\\\
\end{array}$ (4.44)

##### Model X

Finds the corresponding twisted graded version of the model IX, i.e. a similar
reduction exists under $\\{1,1,i\pi\\}$

$\mathcal{H}\rightarrow 1-P_{g}$ (4.45)

a detailed investigation can be found in [92, 38] and the $R$-matrix can be
presented in the following form

$\begin{array}[]{l}r_{1}=-r_{5}=-u\rho\ r_{2}\\\ r_{2}=r_{6}=(1-u\rho)^{-1}\\\
r_{7}=e^{u(a-\rho)}r_{2}\end{array}\quad\begin{array}[]{l}r_{3}=-e^{\phi}\
r_{1}\\\ r_{4}=-e^{-\phi}\ r_{1}\\\ r_{10}=e^{u(\rho-a)}r_{2}\end{array}$
(4.46)

##### Model XI

In model XI, one could again identify well-known integrable model, namely the
correspondence to the twisted $\mathfrak{sp}(4)$ spin chain [106, 107, 108,
109]. It can be given as follows

$\mathcal{H}_{\text{{XI}}}=\frac{3\rho}{2}1-(\mathcal{T}\otimes\mathcal{T})\mathcal{H}_{\mathfrak{sp}(4)}(\mathcal{T}\otimes\mathcal{T})^{-1}\qquad\mathcal{T}(\phi)=\begin{pmatrix}1&0&0&0\\\
0&0&0&e^{\phi/2}\\\ 0&0&1&0\\\ 0&e^{-\phi/2}&0&0\end{pmatrix}$ (4.47)

where we can again obtain distinct family by twisting. The $R$-matrix appears
as

$\begin{array}[]{l}r_{1}=r_{5}=\rho u(3\rho u-4)\ f(u)\\\ r_{2}=r_{6}=4(1-\rho
u)\ f(u)\\\ r_{3}=r_{4}=-\frac{3}{2}u\rho\
r_{7}\end{array},\quad\begin{array}[]{l}r_{7}=r_{10}=-2(3\rho u-2)^{-1}\\\
e^{2\phi}r_{9}=r_{8}=2\rho ue^{\phi}\ f(u)\\\ f(u)^{-1}=(\rho u-2)(3\rho
u-2)\end{array}$ (4.48)

and $R_{\mathfrak{sp}(4)}$ in the graded shifted basis

$\displaystyle
R_{\mathfrak{sp}(4)}=u\mathds{1}+P-\frac{u}{u+3}(-1)^{p(i)+p(k)}E^{i}_{j}\otimes
E^{5-i}_{5-j}.$ (4.49)

complete analysis and resolution of $\mathfrak{so}$\- and
$\mathfrak{sp}$-magnetics through BA was provided in [109, 110].

##### Model XII

This class reflects the kinetic part of the Hubbard model (free theory), to
establish mapping we need to perform twisting. For Hamiltonian one would find

$\mathcal{H}_{\text{{XII}}}=-i\mathcal{V}\mathcal{H}\mathcal{V}^{-1}\qquad\mathcal{V}=\mathcal{V}_{0}\otimes\mathds{1}\text{,
}\quad\mathcal{V}_{0}=\text{diag}\\{1,-1,i,i\\}$ (4.50)

and upon appropriate grading and factoring twisting out, one can relate it to
the Hubbard model

$H^{\text{Hub}}\simeq\left(\mathfrak{T}\otimes\mathfrak{T}\right)\mathcal{V}\mathcal{H}\mathcal{V}^{-1}\left(\mathfrak{T}\otimes\mathfrak{T}\right)^{-1}\qquad\mathfrak{T}=\begin{pmatrix}0&e^{-\frac{\phi}{2}}&0&0\\\
e^{-\frac{\phi}{2}}&0&0&0\\\ 0&0&i&0\\\ 0&0&0&i\end{pmatrix}$ (4.51)

one can also note here a possible separation into two XX spin chains. The
$R$-matrix can be presented in the following form

$\begin{array}[]{l}r_{1}=r_{5}=r_{4}^{2}\\\ r_{2}=r_{6}=r_{7}^{2}\\\
r_{3}=-r_{4}=-{\rm
tanh}(u\rho)\end{array}\quad\begin{array}[]{l}r_{7}=r_{10}={\rm
sech}(u\rho)\\\ r_{8}=e^{\phi}\ r_{4}\ r_{7}\\\ r_{9}=-e^{-\phi}\ r_{4}\
r_{7}\end{array}$ (4.52)

###  4.3 Chains: $2\oplus 2$ sector

By completing detailed analysis of 1, one can immediately notice that the
$\mathfrak{so}(4)$ spin has not appeared so far on our list, or its form might
have been obscure. To make it manifest, one needs to look at all nontrivial
representations that arise for the $\mathfrak{su}(2)\oplus\mathfrak{su}(2)$
algebra. Indeed, by looking at $\mathfrak{r}_{2\oplus 2}:$
$\mathfrak{su}(2)\oplus\mathfrak{su}(2)$ and applying our Ansatz accordingly
to it, one explicitly finds $\mathfrak{so}(4)$ spin chain. We can characterise
the representation $\mathfrak{r}_{2\oplus 2}$ by $L$-$R$ generators

$\begin{cases}\mathfrak{r}_{2\oplus
2}(t_{i}^{L})=\mathds{1}\otimes\mathfrak{r}_{2\oplus 2}(t_{i})\\\
\mathfrak{r}_{2\oplus 2}(t_{i}^{R})=\mathfrak{r}_{2\oplus
2}(t_{i})\otimes\mathds{1}\end{cases}\qquad t_{i}^{L}\times
t_{i}^{R}\in{\text{{$\mathfrak{su}(2)\oplus\mathfrak{su}(2)$}}}$ (4.53)

we can further find out that associated $\mathfrak{r}_{2\oplus 2}$ gives rise
to the following

$\mathcal{H}=A+B\underbrace{E_{i}^{j}\otimes
E_{j}^{i}}_{P}+C\underbrace{E_{i}^{j}\otimes
E_{i}^{j}}_{Q}+D\epsilon_{ijkl}E_{k}^{i}\otimes E_{l}^{j}$ (4.54)

with permutation $P$, trace operator $Q$ and $E_{i}^{j}$ unit-matrices as in
4.2.4. Implementing accordingly modified procedure 4.1, we find two invariant
models

$\begin{array}[]{l}\mathcal{H}_{\text{{XIII}}}=AQ\\\
R_{\text{{XIII}}}=(1+Au)\left[\frac{\sqrt{3}\coth\left(\sqrt{3}Bu\right)-2}{\sqrt{3}\coth\left(\sqrt{3}Bu\right)-1}P+\frac{1}{\sqrt{3}\coth\left(\sqrt{3}Bu\right)-1}Q\right]\end{array}$
(4.55)

and

$\begin{array}[]{l}\mathcal{H}_{\text{{XIV}}}=A-BP+BQ+C\epsilon_{ijkl}E_{k}^{i}\otimes
E_{l}^{j}\\\
R_{\text{{XIV}}}=\frac{(1+Au)u}{1-Bu}\left[(u\left(B^{2}-C^{2}\right)-B)1+\frac{1-Bu}{u}P+BQ+C\epsilon_{ijkl}E^{i}_{k}\otimes
E^{j}_{l}\right]\end{array}$ (4.56)

The model XIII implies an integrable tracing on its own, however the model XIV
has a $\mathfrak{so}-$-subsector [109, 111]. More specifically, if we suppress
one extra integrable antisymmetric contraction by $C\rightarrow 0$, we shall
obtain regular $\mathfrak{so}(4)$ spin chain [108, 109]. However, generically
the dependence of the spectrum on $C$ appears nontrivial. Due to the
aforementioned
$\text{{$\mathfrak{su}(2)\oplus\mathfrak{su}(2)$}}\sim\mathfrak{so}(4)$
isomorphism containing two independent Casimirs, we can establish
decomposition also for XIV, i.e.

$\mathcal{H}_{\text{{XIV}}}=A+\mathcal{C}_{+}\mathfrak{r}_{2\oplus
2}(t_{i}^{L})\otimes\mathfrak{r}_{2\oplus
2}(t_{i}^{L})+\mathcal{C}_{-}\mathfrak{r}_{2\oplus
2}(t_{i}^{R})\otimes\mathfrak{r}_{2\oplus 2}(t_{i}^{R})$ (4.57)

where summation over repeated indices is assumed and
$\mathcal{C}_{\pm}=2\left(B\pm C\right)$. Indeed, it can be noted that
analogously to the regular XXX with
$\mathcal{H}\sim\sigma^{i}\otimes\sigma^{i}$, the $\mathfrak{so}(4)$ chain
does possess XXX-decomposition with the corresponding each copy contribution
to the spectrum.

###  4.4 Generalised Hubbard sector

We shall discuss novel deformation and generlisations of the Hubbard type
chains. As we have already discussed, the Hubbard model consists of the
kinetic and potential parts (4.5). On the other hand, one think of variety of
deformations/generalisations when perturbing the free Hubbard (kinetic), as
observed for model XII.

$\mathbb{H}_{\text{Hub}}=\sum_{i,\,\sigma=\\{\uparrow,\downarrow\\}}(\mathbf{c}^{\dagger}_{\sigma,i}\mathbf{c}_{\sigma,i+1}+\mathbf{c}^{\dagger}_{\sigma,i+1}\mathbf{c}_{\sigma,i})$
(4.58)

It is obvious that such family does not obey
$\mathfrak{su}(2)\oplus\mathfrak{su}(2)$, but our question would include a
class that admits certain Hubbard properties, hence Hubbard type. Physically
that would be one-dimensional systems of electrons that propagate on the
lattice (or conduction band).

In this respect, we can create a generic ansatz with additional allowed terms,
and ask if there is a new emerging integrable structure after imposing 4.1.
The first possibility is the same footing action on pair of electrons, i.e.
hopping terms. This kinetic sector would include hopping of an fermionic pair

$\mathcal{H}_{\text{PH}}=A_{1}\mathbf{c}^{\dagger}_{\uparrow,1}\mathbf{c}^{\dagger}_{\downarrow,1}\mathbf{c}_{\uparrow,2}\mathbf{c}_{\downarrow,2}+A_{2}\mathbf{c}^{\dagger}_{\uparrow,2}\mathbf{c}^{\dagger}_{\downarrow,2}\mathbf{c}_{\uparrow,1}\mathbf{c}_{\downarrow,1}$
(4.59)

and the one that performs flipping of spins of adjacent fermions

$\mathcal{H}_{\text{Flip}}=A_{3}\mathbf{c}^{\dagger}_{\uparrow,1}\mathbf{c}^{\dagger}_{\downarrow,2}\mathbf{c}_{\downarrow,1}\mathbf{c}_{\uparrow,2}+A_{4}\mathbf{c}^{\dagger}_{\downarrow,1}\mathbf{c}^{\dagger}_{\uparrow,2}\mathbf{c}_{\uparrow,1}\mathbf{c}_{\downarrow,2}+A_{5}\mathbf{c}^{\dagger}_{\uparrow,1}\mathbf{c}^{\dagger}_{\uparrow,2}\mathbf{c}_{\downarrow,1}\mathbf{c}_{\downarrow,2}+A_{6}\mathbf{c}^{\dagger}_{\downarrow,1}\mathbf{c}^{\dagger}_{\downarrow,2}\mathbf{c}_{\uparrow,1}\mathbf{c}_{\uparrow,2}$
(4.60)

although we are considering fermion number conserving setting, the flipping
does violate spin conservation since it can mutually map
$\ket{\uparrow\uparrow}\leftrightarrow\ket{\downarrow\downarrow}$. After all
we provide prescription for the generic potential of the form

$\begin{array}[]{l}V=B_{1}+B_{2}\,\mathbf{n}_{\uparrow,1}+B_{3}\,\mathbf{n}_{\downarrow,1}+B_{4}\,\mathbf{n}_{\uparrow,1}\mathbf{n}_{\downarrow,1}+\\\