1. Introduction
In the field of quantum optics [1], models that depict the interactions between a single bosonic mode and spin degrees of freedom are crucial. These models are essential for explaining how atoms engage with the electromagnetic field, resulting in phenomena like spontaneous emissions in cavities and Rabi oscillations in two-level atoms.
Introduced by I.I. Rabi over seven decades ago [2, 3], the single-mode spin–boson model stands as a foundational mechanism in both quantum optics and solid-state physics. It serves as a vital framework for understanding the interactions of atoms with electromagnetic fields. Noteworthy is that many of these models, including the spin–boson model, are exactly solvable, providing accurate insights into complex physical interactions.
Toy model Hamiltonians that describe the interaction between a single mode and a spin s are constructed as follows:
$\begin{eqnarray}H=\omega {a}^{\dagger }a+\alpha {s}^{z}+\beta ({s}^{+}a+{s}^{-}{a}^{\dagger })+\gamma ({s}^{+}{a}^{\dagger }+{s}^{-}a),\end{eqnarray}$
where a† and a represent the bosonic creation and annihilation operators, respectively, for a single bosonic mode of frequency ω, $\begin{eqnarray}[a,{a}^{\dagger }]=1.\end{eqnarray}$
These operators commute with the spin operators sv, where v ∈ {z, ±}. The system, resembling a two-level atom with a level splitting α, utilizes spin-$\tfrac{1}{2}$ operators that adhere to the su(2) Lie algebra commutation relations: $\begin{eqnarray}[{s}^{+},{s}^{-}]=2{s}^{z}\quad {\rm{and}}\quad [{s}^{z},{s}^{\pm }]=\pm {s}^{\pm }.\end{eqnarray}$
The interactions between the spin and boson systems are governed by the coefficients β and γ. Specifically, the interaction term s+a describes the annihilation of a photon and the excitation of an atom, whereas s−a† signifies the atom returning to its ground state and emitting a photon.The model with the coupling constants γ = β was initially proposed to describe dipole-like interactions in atom–radiation systems, and it is widely known as the Rabi model. Despite its apparent simplicity, as expressed in (1.1 ), it remains a physically meaningful quantum system with interactions. Because there are no conserved quantities other than energy, it is difficult to exactly solve the Rabi model. This fact has led to the common perception that the Rabi model is a so-called non-integrable [4] yet exactly solvable system [5–8].
To avoid this difficulty, Jaynes and Cummings introduced the ‘rotating-wave approximation' (RWA) in the 1960s [9]. This approximation simplifies the original model (1.1 ) by discarding the counter-rotating terms s+a† and s−a, effectively setting γ = 0. Within this approximation, the excitation number operator C = a†a + sz commutes with the modified Hamiltonian HJC, making the Jaynes–Cummings model exactly solvable. The conservation of C reflects a continuous U(1) symmetry, which ensures the integrability of the model.
The Jaynes–Cummings scheme can be extended to accommodate N non-interacting atoms within a cavity. The total spin operators are defined by
$\begin{eqnarray}\begin{array}{l}{S}^{z}=\displaystyle \sum _{j=1}^{N}{s}_{j}^{z},\quad {S}^{\pm }=\displaystyle \sum _{j=1}^{N}{s}_{j}^{\pm },\quad [{s}_{i}^{+},{s}_{j}^{-}]=2{s}_{j}^{z}{\delta }_{{ij}},\\ \quad \left[{s}_{i}^{z},{s}_{j}^{\pm }\right]=\pm {s}_{j}^{\pm }{\delta }_{{ij}}.\end{array}\end{eqnarray}$
These operators form a representation of the su(2) algebra. The resulting Hamiltonian for this expanded scenario is given by $\begin{eqnarray}{H}_{{TC}}=\omega {a}^{\dagger }a+{\omega }_{0}{S}^{z}\,+\,g({a}^{\dagger }{S}^{-}+{{aS}}^{+}),\end{eqnarray}$
known as the Tavis–Cummings model [10]. Also, the Casimir operator S2 and the excitation number operator commute with the Hamiltonian of the model HTC.Returning back to the original Rabi model (1.1 ), the inclusion of the term s+a† + s−a breaks the U(1) symmetry down to ${{\mathbb{Z}}}_{2}$. This makes the dynamics of the system becomes complicated. When excited-state atoms enter a cavity and interact with the quantized field, the detector at the output identifies the state of the atom.
Although it may seem simple, the fully quantized form of the Rabi model was only successfully solved and certified to be integrable a decade ago. According to Braak's criterion for quantum integrability [4], integrability involves f1 discrete and f2 continuous degrees of freedom, resulting in f = f1 + f2 ‘quantum numbers' that can uniquely classify eigenstates. For the Rabi model, f1 = f2 = 1 gives a total of f = 2 quantum numbers corresponding to the global dimension labels (parity) used to identify eigenstates uniquely. Braak's criterion also requires that the number of possible values for the label of the discrete degree of freedom matches the dimension of the corresponding Hilbert space. This condition is satisfied by the Rabi model. This raises an intriguing question: if the quantum Rabi model is integrable, could it also be Yang–Baxter-integrable [5–8]?
It would be beneficial to approach this discussion from a more conservative perspective. Consequently, Batchelor and Zhou [11] addressed the question of whether the Rabi model is Yang–Baxter-integrable (YBI). They demonstrated YBI at two specific points within the parameter space of the Rabi model. However, the question of YBI at generic points in this space remains an open problem.
This paper aims to construct spin–boson interactions to approach the Rabi model Hamiltonian. Our paper is organized as follows. In section 2 , we briefly review the basic conceptions for related R-matrix and open boundary conditions as well as the general idea for the quasi-classical expansion. In section 3 , we introduce the Lax representations for spin and boson respectively. In section 4 , we construct the integrable spin–boson interactions with open boundary conditions. In section 5 , we impose the quasi-classical expansion of this system and derive the integrable Hamiltonian with spin–boson interactions. Section 6 is the conclusion.
2. Preliminaries: rational R-matrix and integrable boundary conditions
The R-matrix R(u) ∈ End(V ⨂ V) of the rational six-vertex model is defined as
$\begin{eqnarray}R(u)=\left(\begin{array}{llll}u+\eta & & & \\ & u & \eta & \\ & \eta & u & \\ & & & u+\eta \end{array}\right),\end{eqnarray}$
where u represents the spectral parameter and η the crossing parameter. This R-matrix satisfies the quantum Yang–Baxter equation (QYBE), $\begin{eqnarray}\begin{array}{l}{R}_{12}({u}_{1}-{u}_{2}){R}_{13}({u}_{1}-{u}_{3}){R}_{23}({u}_{2}-{u}_{3})\\ \quad =\,{R}_{23}({u}_{2}-{u}_{3}){R}_{13}({u}_{1}-{u}_{3}){R}_{12}({u}_{1}-{u}_{2}),\end{array}\end{eqnarray}$
which ensures integrability in the context of the model. The notation used indicates that for any matrix A in the endomorphism space of A ∈ End(V), Aj acts as A on the jth space within a tensor product space V ⨂ V ⨂ ⋯ , while acting as the identity on other spaces. Similarly, Rij(u) acts non-trivially only on the ith and jth spaces, while acting as the identity elsewhere.Moreover, the rational R-matrix can be related to the spin-$\tfrac{1}{2}$ representation of su(2), providing motivation for the introduction of quantum Lax operators that associate with higher representations. These Lax operators L also satisfy the Yang–Baxter equations,2.3 ). Different choices of R-matrix may lead to distinct algebras.
$\begin{eqnarray}\begin{array}{l}{R}_{12}({u}_{1}-{u}_{2}){L}_{1n}({u}_{1}){L}_{2n}({u}_{2})\\ \quad =\,{L}_{2n}({u}_{2}){L}_{1n}({u}_{1}){R}_{12}({u}_{1}-{u}_{2}),\end{array}\end{eqnarray}$
where the subscripts 1 and 2 denote the auxiliary spaces and n the quantum space on which the algebra generators act. The general form of $L\in {End}(V)\otimes { \mathcal A }$, where ${ \mathcal A }$ is the algebra defined by the relation (Let us introduce two ‘row-to-row' monodromy matrices T(u) and $\hat{T}(u)$, which are 2 × 2 matrices with elements being operators acting on V⨂N, 2.5 ), one must consider the double-row monodromy matrix ${\mathbb{T}}(u)$ in the open spin chain case,2.1 ), general non-diagonal solutions of the reflection equations are the K-matrices of the form [13]
$\begin{eqnarray}{T}_{0}(u)={L}_{01}(u-{\theta }_{1})\cdots {L}_{0N}(u-{\theta }_{n}),\end{eqnarray}$
$\begin{eqnarray}{\hat{T}}_{0}(u)={L}_{0N}^{-1}(u+{\theta }_{n})\cdots {L}_{01}^{-1}(u+{\theta }_{1}).\end{eqnarray}$
In this context, the R-matrix induces a realization of L. To construct an integrable open chain [12], introduce a pair of K-matrices K−(u) and K+(u). The former satisfies the reflection equation (RE), $\begin{eqnarray}\begin{array}{l}{R}_{12}({u}_{1}-{u}_{2}){K}_{1}^{-}({u}_{1}){R}_{21}({u}_{1}+{u}_{2}){K}_{2}^{-}({u}_{2})\\ \quad =\,{K}_{2}^{-}({u}_{2}){R}_{12}({u}_{1}+{u}_{2}){K}_{1}^{-}({u}_{1}){R}_{21}({u}_{1}-{u}_{2}),\end{array}\end{eqnarray}$
and the latter satisfies the dual RE, $\begin{eqnarray}\begin{array}{l}{R}_{12}({u}_{2}-{u}_{1}){K}_{1}^{+}({u}_{1}){R}_{21}(-{u}_{1}-{u}_{2}-2\eta ){K}_{2}^{+}({u}_{2})\\ \quad =\,{K}_{2}^{+}({u}_{2}){R}_{12}(-{u}_{1}-{u}_{2}-2\eta ){K}_{1}^{+}({u}_{1}){R}_{21}({u}_{2}-{u}_{1}).\end{array}\end{eqnarray}$
Rather than using the standard ‘row-to-row' monodromy matrix T(u) ( $\begin{eqnarray}{\mathbb{T}}(u)=T(u){K}^{-}(u)\hat{T}(u).\end{eqnarray}$
For the rational R-matrix ( $\begin{eqnarray}{K}_{\pm }(u)=\left(\begin{array}{cc}{\xi }_{\pm }+u & u{\mu }_{\pm }\\ u{\nu }_{\pm } & {\xi }_{\pm }-u\end{array}\right).\end{eqnarray}$
Here, ξ±, μ±, and ν± are parameters related to the boundaries. Ordinarily, μ± = ν± for the XXX spin-$\tfrac{1}{2}$ chain with parallel open boundary fields. For more detailed discussions on open boundary conditions, please refer to chapters 2 and 5 of the book [14]. The ‘row-to-row' transfer matrix, often referred to in the literature as the ‘double row' transfer matrix [12] for systems with open boundaries, is defined as follows: $\begin{eqnarray}\begin{array}{l}t(u)={{tr}}_{0}\left[{K}_{+}(u+\eta ){\mathbb{T}}(u)\right]\\ \qquad =\,{{tr}}_{0}\left[{K}_{+}(u+\eta )T(u){K}_{-}(u)\hat{T}(u)\right].\end{array}\end{eqnarray}$
This matrix acts as the generator of conservation laws, ensuring that $\begin{eqnarray}\left[t(u),t(v)\right]=0.\end{eqnarray}$
This commutation property implies the integrability of the system, allowing the matrix t(u) to generate an infinite number of conservation laws necessary for the solvability of the model in the framework of quantum integrable systems [12, 14, 15].The quasi-classical expansion of the transfer matrix t(u) in powers of the parameter η is given by
$\begin{eqnarray}t(u)=\displaystyle \sum _{j=0}^{\infty }{\eta }^{j}\,{\hat{\tau }}^{(j)}(u).\end{eqnarray}$
This expansion and the property [t(u), t(v)] = 0 guarantee the existence of a hierarchy of integrable models. The commutation relation expands as $\begin{eqnarray*}[t(u),t(v)]=\displaystyle \sum _{l=0}^{\infty }{\eta }^{l}{C}_{l}(u,v)=0,\end{eqnarray*}$
which implies that each coefficient Cl(u, v) must individually vanish: $\begin{eqnarray}\begin{array}{rcl}{C}_{0}(u,v) & = & [{\hat{\tau }}^{(0)}(u),{\hat{\tau }}^{(0)}(v)],\\ {C}_{1}(u,v) & = & [{\hat{\tau }}^{(1)}(u),{\hat{\tau }}^{(0)}(v)]+[{\hat{\tau }}^{(0)}(u),{\hat{\tau }}^{(1)}(v)],\\ {C}_{2}(u,v) & = & [{\hat{\tau }}^{(2)}(u),{\hat{\tau }}^{(0)}(v)]+[{\hat{\tau }}^{(1)}(u),{\hat{\tau }}^{(1)}(v)]\\ & & +[{\hat{\tau }}^{(0)}(u),{\hat{\tau }}^{(2)}(v)],\\ & & \cdots \cdots \end{array}\end{eqnarray}$
This structured expansion reveals that the commutativity of t(u) at different spectral parameters (u and v) extends to the commutativity of the expansion terms Each term ${\hat{\tau }}^{(l)}$ in the expansion, starting from the first non-trivial term (which is not merely a constant or an algebraic invariant), contributes to a set of commuting operators, thereby constructing an integrable hierarchy. This framework lays the groundwork for exploring deeper integrable structures within the model. Notably, when the term ${\hat{\tau }}^{(0)}$ is a constant or an identity operator, significant implications arise for the subsequent terms in the series. Specifically, the term ${\hat{\tau }}^{(1)}$ becomes critical, as its commutation relation, $\begin{eqnarray}[{\hat{\tau }}^{(1)}(u),{\hat{\tau }}^{(1)}(v)]=0,\end{eqnarray}$
indicates that ${\hat{\tau }}^{(1)}$ itself forms a family of commuting operators, usually called Gaudin operators for the spin chain case.Here, we highlight that the above represents the standard (general) quasi-classical expansion scheme of the transfer matrix. As all elements are polynomials of the spectral parameter u and the crossing parameter η, the lowest order of η in the transfer matrix t(u) is the zeroth-order term ${\hat{\tau }}^{(0)}$.
Later, we shall see that there is a negative-order term η−1 in the bosonic Lax operator representation (3.3 ). Consequently, the lowest order term of η in the transfer matrix becomes η−2 after reflection for the open boundary case. Nonetheless, the fundamental idea remains the same: the first non-trivial (neither a constant nor an identity) term in the expansion (2.13 ) is the critical term that can serve as conservation laws of the system.
3. Lax operators for spin and boson
In this section, we apply the general theoretical framework discussed earlier to the spin–boson problem. Using the R-matrix (2.1 ), we consider two types of Lax matrices that satisfy the fundamental relation (2.3 ). For the spin degree of freedom, the Lax matrix is defined as3.1 ), such as the Holstein–Primakoff or the Dyson–Maleev representation. However, for this discussion, we choose the bosonic Lax matrix as [15, 16]2.10 ), combining both spin and bosonic degrees of freedom within the integrable structure.
$\begin{eqnarray}{L}_{j(s)}(u)=\left(\begin{array}{cc}u-\eta {s}_{j}^{z} & \eta {s}_{j}^{+}\\ \eta {s}_{j}^{-} & u+\eta {s}_{j}^{z}\end{array}\right),\end{eqnarray}$
where sjz and ${s}_{j}^{\pm }$ are chosen to be irreducible representations of su(2). The inverse of this spin Lax operator is given by $\begin{eqnarray}\begin{array}{rcl}{L}_{j(s)}^{-1}(u) & = & {\det }_{q}{\left[{L}_{j(s)}\right]}^{-1}\\ & & \times \,\left(\begin{array}{cc}u-\eta +\eta {s}_{j}^{z} & -\eta {s}_{j}^{+}\\ -\eta {s}_{j}^{-} & u-\eta -\eta {s}_{j}^{z}\end{array}\right).\end{array}\end{eqnarray}$
To address the bosonic degree of freedom, one approach could involve utilizing representations of su(2) in terms of Bose operators ( $\begin{eqnarray}{L}_{j(b)}(u)=\left(\begin{array}{cc}u-{\rm{\Delta }}-{\eta }^{-1}-\eta {a}_{j}^{\dagger }{a}_{j} & {a}_{j}^{\dagger }\\ {a}_{j} & -{\eta }^{-1}\end{array}\right),\end{eqnarray}$
where $[{a}_{j},{a}_{k}^{\dagger }]={\delta }_{{jk}}$, and Δ is called the detuning parameter. The inverse of this bosonic Lax operator is $\begin{eqnarray}\begin{array}{rcl}{L}_{j(b)}^{-1}(u) & = & {\det }_{q}{\left[{L}_{j(b)}\right]}^{-1}\\ & & \times \,\left(\begin{array}{cc}-{\eta }^{-1} & -{a}_{j}^{\dagger }\\ -{a}_{j} & u-\eta -{\rm{\Delta }}-{\eta }^{-1}-\eta {a}_{j}^{\dagger }{a}_{j}\end{array}\right).\end{array}\end{eqnarray}$
Here, ${\det }_{q}[{L}_{j(s)}]$ and ${\det }_{q}[{L}_{j(b)}]$ are the quantum determinant of Lj(s) and Lj(b). These determinants are crucial for calculating inversions but can be disregarded in the following sections as they are overall scalar functions. This setup facilitates the construction of the transfer matrix (4. Open boundary conditions for the spin–boson problem
To explore a model including counter-rotating terms using the rational R-matrix, we consider open boundary conditions. This approach employs the transfer matrix (2.10 ) as defined earlier to generate integrable Hamiltonians for the spin–boson interaction. The formulation involves two ‘row-to-row' monodromy matrices, T(u) and $\hat{T}(u)$, which are 2 × 2 matrices in auxiliary space. The elements of these matrices are operators acting on the combined space Vb ⨂ Vs (where Vb and Vs represent the bosonic and spin spaces, respectively),2.10 ) with open boundary conditions that combine both spin and boson systems. The construction of the spin–boson interaction model follows from the transfer matrix derived from the monodromy matrix,2.9 ).
$\begin{eqnarray}{T}_{0}(u)={L}_{0(b)}(u+{\theta }_{b}){L}_{0(s)}(u+{\theta }_{s}),\end{eqnarray}$
$\begin{eqnarray}{\hat{T}}_{0}(u)={L}_{0(s)}^{-1}(u-{\theta }_{s}){L}_{0(b)}^{-1}(u-{\theta }_{b}).\end{eqnarray}$
These monodromy matrices allow us to consider an extended double row monodromy matrix ( $\begin{eqnarray}\begin{array}{l}t(u)={{\rm{tr}}}_{0}[{K}_{+}(u+\eta ){L}_{0(b)}(u+{\theta }_{b}){L}_{0(s)}(u+{\theta }_{s}){K}_{-}(u)\\ \,\times \,{L}_{0(s)}^{-1}(u-{\theta }_{s}){L}_{0(b)}^{-1}(u-{\theta }_{b})].\end{array}\end{eqnarray}$
Here, the boundary matrices K+ and K− are adopted from previously defined settings, ensuring the model satisfies the required integrability conditions (The resulting Hamiltonian, derived from this transfer matrix, inherently includes the interaction terms that are both rotating and counter-rotating, critical for capturing the full dynamics of the quantum system under open boundary conditions.
In the following, we discuss that the spin–boson interaction could be constructed from the above transfer matrix (4.3 ).
5. Quasi-classical expansion
The expansion of the transfer matrix t(u) in the quasi-classical η-power series is expressed as follows:3.3 ) and (3.4 ). This is similar to how the lowest order term in the general quasi-classical expansion scheme (2.13 ) is ${\hat{\tau }}^{(0)}$. Here and below, we denote ${\hat{\tau }}^{(j)}(u)$ as ${\hat{\tau }}^{(j)}$ for brevity. This abbreviation is also applied to other operators like ${K}_{+}^{(0)}(u)$ in the following text.
$\begin{eqnarray}t(u)=\displaystyle \sum _{j=-2}^{\infty }{\eta }^{j}\,{\hat{\tau }}^{(j)}(u).\end{eqnarray}$
In this case, the lowest order term is η−2. The negative order of η comes from the bosonic Lax operators (After some calculations, we found that ${\hat{\tau }}^{(-2)}$ is a constant term, while ${\hat{\tau }}^{(-1)}$ is an invariant function of u multiplied by the identity operator (proportional to the identity) when the constraint relation,5.1 ).
$\begin{eqnarray}{K}_{+}^{(0)}(u)={\left[{K}_{-}(u)\right]}^{-1},\end{eqnarray}$
is satisfied. Thus, the zeroth-order term ${\hat{\tau }}^{(0)}$ in the expansion becomes critical, as it contains interaction terms that can generate non-trivial commuting relations. Here, the superscript of ${K}_{+}^{(0)}$ in the round brackets denotes the order of η, similar to the superscript of ${\hat{\tau }}^{(j)}$ in (The boundary matrices ${K}_{+}^{(0)}$ and K− are assumed in the generic forms
$\begin{eqnarray}\begin{array}{rcl}{K}_{+}^{(0)} & = & \left(\begin{array}{cc}{\rm{a}} & {\rm{b}}\\ {\rm{c}} & {\rm{d}}\end{array}\right),\,{\rm{and}}\\ {K}_{-} & = & {\left[{K}_{+}^{(0)}\right]}^{-1}=\left(\begin{array}{cc}{\rm{d}} & -b\\ -{\rm{c}} & {\rm{a}}\end{array}\right).\end{array}\end{eqnarray}$
In our analysis of the spin–boson interactions, we focus on the zeroth-order operator ${\hat{\tau }}^{(0)}$, emphasizing the terms that contribute to the interaction. The linear terms are generally neglected except for those involving sz, which are relevant when K+ includes terms linear in η. It is noted that ${K}_{+}^{(1)}$, which does not contribute to our target Hamiltonian and is orthogonal to ${K}_{+}^{(0)}$ [14]. We simply set it to zero.In this case, there are altogether eight terms in the quasi-classical power η-expansion of the operator ${\hat{\tau }}^{(0)}$, and we list the type of each term in the following table,
Based on previous studies [17], it is understood that toroidal boundary conditions (periodic boundaries with two boundary twists) can only generate either rotating or counter-rotating terms, and they cannot simultaneously come about. The occurrence of these terms depends on the direction of the z-axis.
Table 1. The middle columns show the orders of η-expansion for each operator. The corresponding types of interactions are in the right column. |
| K+ | L(b) | L(s) | ${L}_{(s)}^{-1}$ | ${L}_{(b)}^{-1}$ | Interaction Type | |
|---|---|---|---|---|---|---|
| 0 | −1 | 0 | 0 | 1 | a†a | |
| 0 | −1 | 0 | 1 | 0 | s−a†, s+a, sz | |
| Orders of | 0 | −1 | 1 | 0 | 0 | s−a, s+a†, sz |
| η- | 0 | 0 | 0 | 0 | 0 | a2, ${\left({a}^{\dagger }\right)}^{2}$ |
| expansion | 0 | 0 | 0 | 1 | −1 | s−a, s+a†, sz |
| for each | 0 | 0 | 1 | 0 | −1 | s−a†, s+a, sz |
| factor | 0 | 1 | 0 | 0 | −1 | a†a |
| 0 | −1 | 1 | 1 | −1 | sz, ${\left({s}^{z}\right)}^{2}$, ${\left({s}^{-}\right)}^{2}$, ${\left({s}^{+}\right)}^{2}$ |
As shown above, among the four terms that can induce spin–boson interaction effects, two specifically contribute rotating interaction terms (the 2nd and 6th rows in the table):
‘${K}_{+}{\left[{L}_{(b)}\right]}^{(-1)}{\left[{L}_{(s)}\right]}^{(0)}{K}_{-}{\left[{L}_{(s)}^{-1}\right]}^{(1)}{\left[{L}_{(b)}^{-1}\right]}^{(0)}$' and ‘${K}_{+}{\left[{L}_{(b)}\right]}^{(0)}{\left[{L}_{(s)}\right]}^{(1)}{K}_{-}{\left[{L}_{(s)}^{-1}\right]}^{(0)}{\left[{L}_{(b)}^{-1}\right]}^{(-1)}$'.
The other two terms (the 3rd and 5th rows in the table),
‘${K}_{+}{\left[{L}_{(b)}\right]}^{(-1)}{\left[{L}_{(s)}\right]}^{(1)}{K}_{-}{\left[{L}_{(s)}^{-1}\right]}^{(0)}{\left[{L}_{(b)}^{-1}\right]}^{(0)}$' and ‘${K}_{+}{\left[{L}_{(b)}\right]}^{(0)}{\left[{L}_{(s)}\right]}^{(0)}{K}_{-}{\left[{L}_{(s)}^{-1}\right]}^{(1)}{\left[{L}_{(b)}^{-1}\right]}^{(-1)}$', provide counter-rotating terms. They can be related to the off-diagonal elements of the K-matrices. Here, the superscripts of ${\left[{L}_{(b)}\right]}^{(-1)}$ and ${\left[{L}_{(s)}\right]}^{(0)}$ in the round brackets also denote the order of η.
Two of the terms in the above table take the leading term of L(b) and ${L}_{(b)}^{-1}$ operators; they can contribute the bosonic particle number operator a†a. The other two terms are twisted spin or boson operators, so they give rise to higher order of spin or boson effects. Now in the spin-$\tfrac{1}{2}$ case, ${\left({s}^{z}\right)}^{2}$ is proportional to the identity operator, while ${\left({s}^{-}\right)}^{2}$ and ${\left({s}^{+}\right)}^{2}$ both vanish.
To eliminate unwanted twisted spin–boson interaction terms such as sza†, the K-matrices must satisfy the conditions a = d and b = c. Consequently, the K− matrix and the zero-order K+ matrix are configured as follows:2.9 ). For example, when μ− = ν−, letting μ− and ξ− both tend to infinity, allows us to ignore the u terms on the diagonal elements, resulting in the K− matrix above.
$\begin{eqnarray}\begin{array}{rcl}{K}_{-}(u) & = & \left(\begin{array}{cc}\xi & -u\mu \\ -u\mu & \xi \end{array}\right),\quad {\rm{and}}\\ \quad {K}_{+}^{(0)}(u) & = & \left(\begin{array}{cc}\xi & u\mu \\ u\mu & \xi \end{array}\right).\end{array}\end{eqnarray}$
We remark that these are reduced forms of the K-matrices in (By adjusting the coefficients of the K-matrices and utilizing the inhomogeneous parameters θs and θb (noting that ${\left[{L}_{(b)}(u+{\theta }_{b})\right]}^{(0)}$ is independent of θb, making θs the effective parameter), we can obtain the spin–boson interaction Hamiltonian from ${\hat{\tau }}^{(0)}$ with the configuration:
$\begin{eqnarray}{\hat{\tau }}^{(0)}{\left|{}_{\xi =0}=\tfrac{1}{2}\displaystyle \frac{{\partial }^{2}\left(t(u){\eta }^{2}\right)}{\partial {\eta }^{2}}\right|}_{\{\eta =0,\xi =0\}}={H}_{{sb}}+{\rm{const.}}\end{eqnarray}$
Finally, we derive the spin–boson interaction Hamiltonian in the following form: $\begin{eqnarray}\begin{array}{rcl}{H}_{{sb}} & = & ({u}^{2}-{\theta }_{s}^{2}){u}^{2}{\mu }^{2}[-2{a}^{\dagger }a+{a}^{2}+{a}^{\dagger }{}^{2}]\\ & & -4{u}^{3}{\mu }^{2}(u-{\rm{\Delta }}){s}^{z}\\ & & -2{u}^{3}{\mu }^{2}[{s}^{-}a+{s}^{+}{a}^{\dagger }]+2{u}^{3}{\mu }^{2}[{s}^{-}{a}^{\dagger }+{s}^{+}a].\end{array}\end{eqnarray}$
Additionally, both rotating and counter-rotating terms can be adjusted to have the same sign by redefining the operators: s− ⟶ s−eiπ/2 and a ⟶ aeiπ/2.We successfully constructed spin–boson interaction terms while preserving the boson particle number operator a†a and also the direction of the spin in the z direction. This is because we introduced open boundary conditions with off-diagonal terms. This marks an advancement beyond previous studies that constructed reduced forms of the Rabi model from Yang–Baxter algebra. We note that the proposed Hamiltonian (5.6 ) still contains ‘unwanted' quadratic boson terms a2 and a†2 compared to the Rabi Hamiltonian (1.1 ). These terms share the same coefficient as the boson particle number operator a†a. When u = ± θs, these terms vanish, reducing the model to a limiting case of operator-valued twists as discussed by Batchelor and Zhou [11].
6. Conclusions
Braak's solution [4] to the Rabi model led to the prevailing view that the model is solvable but not Yang–Baxter-integrable. In recent years, Bogoliubov et al [1], Batchelor and Zhou [11], Amico et al [17] and de Cunha et al [18] have made great efforts to identify conserved quantities that could establish the integrability of the Rabi model. Some of them hold that Yang–Baxter integrability for the Rabi model can be constructed only in some special limiting cases. Specifically, Amico, Batchelor, and Zhou discussed twisted, periodic, and open boundary conditions to construct spin–boson interaction terms, but they focused primarily on diagonal boundaries.
These arguments motivate our work to construct a spin–boson interaction Hamiltonian under more general boundary conditions and further explore the Rabi model. In our study, we successfully introduce spin–boson interaction terms while maintaining the spin's orientation in the z direction, thereby emphasizing the significance of the off-diagonal terms in the Hamiltonian. This marks an advancement beyond previous studies.
By introducing inhomogeneous parameters, we were able to eliminate higher orders of spin operators, although the presence of higher-order boson operators remains a challenge in our model (5.6 ). The ‘unwanted' quadratic boson terms a2 and a†2 in comparison to the Rabi Hamiltonian (1.1 ) have the same coefficients as the boson particle number operator a†a. Interestingly, these terms disappear when the spectral parameter u = ± θs, simplifying the model to a limiting case of operator-valued twists, a scenario previously discussed by Batchelor and Zhou [11].
In future work, we plan to investigate this question under higher representations, such as XYZ spin chain, which may provide additional flexibility to cancel higher-order boson terms and enhance our understanding of the integrability of the Rabi model.