Exterior integrability: Yang-Baxter form of nonequilibrium steady state density operator

A new type of quantum transfer matrix, arising as a Cholesky factor for the steady state density matrix of a dissipative Markovian process associated with the boundary-driven Lindblad equation for the isotropic spin-1/2 Heisenberg (XXX) chain, is presented. The transfer matrix forms a commuting family of non-Hermitian operators depending on the spectral parameter which is essentially the strength of dissipative coupling at the boundaries. The intertwining of the corresponding Lax and monodromy matrices is performed by an infinitely dimensional Yang-Baxter R-matrix which we construct explicitly and which is essentially different from the standard XXX R-matrix. We also discuss a possibility to construct Bethe Ansatz for the spectrum and eigenstates of the non-equilibrium steady state density operator. Furthermore, we indicate the existence of a deformed R-matrix in the infinitely-dimensional auxiliary space for the anisotropic XXZ spin-1/2 chain which in general provides a sequence of new, possibly quasi-local, conserved quantities of the bulk XXZ dynamics.


Introduction
The theory of integrable quantum systems in 1 + 1 dimensions, the so-called quantum inverse scattering, is a well developed field of mathematical physics [30,32,9,16] which pioneered important new algebraic structures in pure mathematics, such as quantum groups and their representations. The fundamental object in this theory is the R-matrix, a solution of the celebrated Yang-Baxter equation [2], which gives rise to integrable Hamiltonians possessing infinite families of conserved quantities. Furthermore, these techniques often lead to explicit methods for diagonalizing the Hamiltonian, such as Algebraic Bethe ansatz (ABA) [30,9,16] or Baxter Q-operator [3]. More recently, the theory of integrable quantum systems also found applications in classical nonequilibrium physics, namely in solving markovian stochastic many-body interacting systems such as asymmetric simple exclusion process [27,5]. There has been even an attempt to develop a non-equilibrium Bethe ansatz approach to quantum impurity problems [19], nevertheless the practical feasibility of these technique and its relation to general integrability structures such as Yang-Baxter equations remains unclear.
However, very recently explicit results appeared for driven quantum manybody systems with a strong interaction, namely a closed matrix product ansatz (MPA) for non-equilibrium steady state (NESS) density-operator of the boundarydriven Lindblad equation [22,23,24] of an anisotropic Heisenberg (XXZ) spin 1/2 chain. Lindblad equation [18,11] is the canonical model of continuous-time Markovian quantum dynamics. This solution has been later interpreted in terms of infinite-dimensional representations of Lie algebra sl (2), and its quantum-group deformation for the anisotropic spin interaction, and generalized to more general boundary dissipators/drivings [15]. Remarkably, perturbative expansion of NESS in the dissipation strength gave rise to a novel XXZ quasi-local conservation law [22] which is unrelated to previously known local conserved quantities of the XXZ chain [10] derived from the 'standard' XXZ transfer-matrix, and which has important consequences for understanding ballistic transport at high-temperatures [13].
In this paper we put these results [22,23,24,15] into the framework of the theory of integrable systems. Focusing mainly on the isotropic case (XXX model), we rigorously construct an R−matrix satisfying Yang-Baxter in an infinitely dimensional auxiliary space which carries irreducible infinitely-dimensional representation of sl (2), so that the corresponding family of commuting transfer-matrices is given by the Cholesky factor of the unnormalized NESS density operator [23]. However, the commuting transfer matrix is given as the ground-state matrix element of the monodromy matrix, and not as its trace as in standard ABA, and is neither a Hermitian nor a diagonalizable operator, which is a manifestation of far-from-equilibrium character of the problem. As the spectral parameter in our R−matrix comes from the boundary dissipative coupling we chose to call our formalism the exterior integrability. One may also provide arguments for the existence of a deformed version of the infinite dimensional exterior R−matrix in the anisotropic (XXZ) case. Two important immediate applications of the new R−matrix are proposed: (i) Construction of an infinite family of new almost-conserved [13] quantities mutually in involution which include the one discussed in [22] and which should shed further light on the understanding of finite-temperature quantum transport problem [29,28,12], and (ii) Construction of ABA for diagonalization of NESS density operator. We stress that even if the exterior integrability is defined with respect to particular integrable dissipative boundaries, it may produce interesting new results for bulk properties of the system in the thermodynamic limit, such as the quasi-local conserved quantities.
After defining the main concepts of non-equilibrium quantum integrability of the XXX model in Section 2, we write explicit expression for the corresponding infinite R−matrix in Section 3 and rigorously prove that it satisfies the Yang-Baxter equation. In Section 4 we describe some interesting properties of the R-matrix and the corresponding non-equilibrium monodromy matrix. In Section 5 we discuss potential applications and extension to an anisotropic case, and conclude. Some technical aspects of our proofs are put into appendices. While material presented in Sections 2, 3 and the appendices should be mathematically rigorous, further results discussed in Sections 4 and 5 are partly based on heuristic and empirical arguments.

Exterior integrability of the nonequilibrium steady state
We focus on the stationary Lindblad equation for the NESS density operator ρ ∞ for the Heisenberg XXX Hamiltonain of a chain of n spins 1/2 where σ ± = 1 2 (σ x ± iσ y ), σ z , σ 0 = ½ 2 are standard Pauli matrices acting over a 2−dimensional quantum spin space H s ≃ C 2 and ½ d is a d-dimensional unit matrix.
We chose the simplest solvable far-from-equilibrium dissipative driving [23] with a pair of Lindblad jump operators with dissipation-driving strength ε As it has been shown in [23], the unique NESS density operator can be written explicitly in the Cholesky factorized form where the operator S(λ) admits an elegant representation in terms of MPA: which may be -up to unitary transformations -uniquely chosen as ‡ with the complex representation parameter λ being fixed by the boundary dissipation strength Defining a λ-dependent linear operator from End(H s ⊗ H a ) the Cholesky factor can be expressed even more elegantly [15] Here and below we use the following compact and unambiguous notational convention. For operator-valued matrices, we use a symbol ⊗ s as a partial tensor product, namely it implies a tensor product with respect to the quantum spin space H s and an ordinary operator/matrix product with respect to the auxiliary space H a . Analogously, ⊗ a will denote a tensor product with respect to H a , and a matrix product in H s . For example, ) making sense if j = l. To emphasize the exterior integrability concepts we shall write in bold all symbols which are not scalars with respect to auxiliary space H a .
The key step of this work is to recognize that L(λ) can be interpreted as the Lax matrix (the so-called L−matrix) matrix of a novel integrable system. This is founded on a simple empirical observation, namely that the Cholesky factors commute for arbitrary complex values of the representation/dissipation parameters [S(λ), S(µ)] = 0, ∀λ, µ ∈ C. (2.12) This observation can be understood as a consequence of existence of an R−matrix § R(λ, µ) ∈ End(H a ⊗ H a ) for almost any λ, µ ∈ C, to be shown in Section 3, which satisfies the so-called RLL (or local intertwining) relation: Following the procedure of ABA [16] the local intertwining relation immediately implies intertwining for a product of the so-called monodromy matrices T(λ) ∈ End(H ⊗n s ⊗ H a ): Indeed, Eq. (2.13) implies Eq. (2.15) after noticing that, due to associativity of matrix multiplication: Unlike in the standard formalism of ABA where the auxiliary space is finite dimensional and the concept of a transfer matrix is usually associated to the partial trace of monodromy matrix with respect to the auxiliary space, we define here the auxiliary ground state expectation 0| T(λ) |0 = S(λ) as the transfer matrix. In order to establish the commutativity of the transfer matrix we also require, besides the RT T relations (2.15), the R-matrix to satisfy additional boundary conditions where |k, l := |k ⊗|l , k, l| := k|⊗ l|. Eq. (2.12) then follows straightforwardly, after writing the transfer-matrix product in H a ⊗ H a , S(λ)S(µ) = 0, 0| T(λ) ⊗ a T(µ) |0, 0 : Despite the boundary condition (2.17) may seem a-priori unjustified at the moment, we shall show further on, that such a property naturally follows from the so-called ice-rule property of the R-matrix.
It is perhaps remarkable that the transfer matrix of our problem S(λ) is non-Hermitian, non-normal, and even non-diagonalizable operator. Using the MPA form (2.11) we can write its matrix elements in the spin basis {|ν = |ν 1 , . . . , ν n ; ν j ∈ {0, 1}} of H ⊗n s , σ z |ν = (−1) ν |ν , as so that the rules 0| A 0 = λ 0| and 0| A − = 0, following from representation (2.6), imply the matrix of S(λ) to be upper triangular, and having a constant diagonal Consequently, all eigenvalues of S(λ) are equal to λ n , but since S(λ) is not a multiple of the identity operator it must have a non-trivial Jordan decomposition, i.e. it must be non-diagonalizable. Similarly, we can write the quantum space matrix elements of the general monodromy matrix elements following the expression (2.14) in terms of MPA Tridiagonality of operators (2.6) immediately implies a magnetization selection rule, namely (2.23) vanishes unless This in turn implies that T k ′ k (λ) changes the z−component of magnetization by 2(k ′ −k), writing magnetization operator as M := n j=1 ½ 2 j−1 ⊗ σ z ⊗ ½ 2 n−j .

Ice-rule -the particle conservation law
Let us write out the R-matrix in components We will show in the following section that the exterior R−matrix of the XXX model (and also for a more general XXZ model, see subsection 5.2) obeys a selection rule, namely R kk ′ ll ′ (λ, µ) = 0 only if k + k ′ = l + l ′ . This can be interpreted as a particular particle conservation (global U(1)) symmetry of the R−matrix, meaning that it should commute with the particle number operator Consequently, one can interpret the R-matrix as a particle-number conserving scattering matrix of a system of auxiliary quasi-particles. Decomposition (2.28) suggests a natural splitting of a tensor product of two copies of auxiliary space into a direct sum of As we see, there are α + 1 states |k, α − k within each sector H (α) a . Therefore, for any X ∈ End(H a ⊗ H a ) which commutes with N, [X, N] = 0, we shall denote with upper-bracketed index α an (α + 1) . For example, we shall often write the R−matrix in the so-called ice-rule form As elements of H

Exterior R-matrix
Here we shall write out and prove our main result, an explicit form of the infinitely dimensional exterior R−matrix which satisfies the defining RLL relations (2.13).
Theorem. A solution of the RLL (2.13) relation for Lax operator (2.10) reads for any x ∈ C\ 1 2 Z + , y ∈ C. The generator H(x) admits a block decomposition according to the ice-rule, with explicit form of the matrix elements Proof. We start by using (3.1) as an ansatz and reparametrize the RLL relation (2.13) in the center-of-mass and displacement spectral parameters, yielding the form where non-trivial dependence enters through the generator H(x), in a way which resembles a Lie group structure. Furthermore, we employ the fact that the Lax matrix L(x) has a simple linear dependence on the spectral parameter At this point we emphasize that the whole x-dependence is absorbed into zero-th degree component Λ 0 (x), whereas Λ 1,2 are matrices with constant (x independent) elements.
In particular, we can write the three orders Λ 0,1,2 (3.7) as operators over H s ⊗ H a ⊗ H a factoring out the components in the physical space After inserting proposed ansatz for the solution (3.1), we shall expand (3.5) in terms of nested commutators -(i) we multiply (3.5) by the operator exp (− y 2 H(x)) from the left and from the right, and (ii) we utilize the defining Lie-group identity exp( ad X )Y = e X Y e −X , where ad X := [X, •], which brings (3.5) to an equivalent form Expanding the hyperbolic functions we obtain a power series in y, which always exists in terms of finite matrix exponentials due to decomposition (3.2). Clearly, since the expression above is an odd function in y, we find only odd orders nonvanishing. In the first order in y we have while for an arbitrary odd order y 2l+1 with l ≥ 1: The relation (3.14) is in fact an infinitesimal RLL relation for y → dy and will be in the following referred to as HLL relation.
Next we show that an infinite sequence of operator equations (3.15) can be in fact reduced to only two equations. The first one is just the third order [(3.15) for l = 1] after substituting ad H(x) Λ 0 (x) from HLL relation (3.14): Then we subsequently use (3.14) and (3.16) to eliminate Λ 0 (x) and Λ 1 from the sequence (3.15) for any l > 1, arriving at ad 2l−1 We have thus shown that three simple y−independent equations, namely (3.14), and a pair (3.16,3.17) imply validity of Eq. (3.15) for any l, and consequently of the full RLL relation for any pair of spectral parameters x, y for which H(x) exists, i.e.
x ∈ C \ 1 2 Z + , y ∈ C. The remainder of the proof is thus to verify identities (3.14,3.16,3.17) which we formulate in two lemmas below.  .14): for any where P is a permutation operator over H a ⊗ H a , acting as Permutation operator conserves the number of auxiliary excitations, hence it satisfies the ice rule We may write shortly π a (a ⊗ b) = b ⊗ a. Then we define another map π s over operators in the quantum spin space End(H s ), by It is important to note that the operators Λ 0,1,2 and the generator H(x) are eigenoperators of the parity map, i.e. they have well defined parities [see (3.9,3.10,3.11)]: Notice that H(x) operates trivially (i.e. as a scalar) in the physical space H s . The whole expression (3.18) is then an eigenoperator of π with eigenvalue −1, Let us now introduce the components in the quantum spin space, either in Weyl or The identity (3.18) to be proven then writes whereas the symmetry relation (3.25), noting (3.24), can be rewritten as 28) This means that out of four equations in End(H a ⊗ H a ), implied by (3.27), only three are independent, say the components 00, 01 = + and 10 = −.
Furthermore, we apply the α− decomposition of the Λ k operators a ) are (α + 1) × (α + 2) and (α + 2) × (α + 1) matrices, respectively. With a bit of patience one can derive explicit expressions from Eqs. (3.9,3.10,3.11), for the constant operators Λ 1 and Λ 2 . Using a compact notation for a canonical basis of H (α) a , |k ≡ |k, α − k , the only non-vanishing blocks/components are The full set of finite matrix equations which remain to be verified then reads: For this one needs to show that for all equations residua at the possible poles, x = p/2, p = 0, . . . , α + 1, match as well as the remainders. This is done in full detail in Appendix B. Proof. Despite it might be tempting to attack the problem similarly as in the case of Lemma 1, a direct calculation reveals that one cannot avoid binomial expressions with double summation involving linear combinations of quadratic terms (products of two binomial coefficients), which are extraordinary difficult to deal with. Fortunately, as we demonstrate below, there exists an elegant algebraic recursive procedure originating from an extra symmetry of the generator H(x). Since we are dealing with quadratic expressions in H(x), whose blocks H (α) (x) are singular with one-dimensional null-space, additional information about null-vectors of (H (α) ) 2 will be required as well.

Master-symmetry of the H-matrix
Here we shall label quantum space components with the Pauli basis. According to the structure (3.9,3.10,3.11) the Eqs. Consistently with our previous notation, we will place additional subscript index α, e.g. Λ (α)+ 2 , when referring to a single α-subspace.
Next we state algebraic relations among Λ (α)s 1,2 , which are straightforwardly verified using explicit representations (3.30-3.34), namely (3.44) The idea is then to derive recursive relation in α for the operators {D  are indeed linearly independent (another simple calculation).
A crucial observation is, that for every α-sector, there exist a unique pair of (α + 1)- such that In order to prove our case (3.45,3.47), we require another set of identities, expressing the action of a squared generator α-blocks (H (α) ) 2 on the vector u (α) , and transformation of both u (α) and v (α) under the action of Λ (α)s 1,2 . For the sake of brevity, we entirely omit their justification here (it can be found in Appendix C):   An entirely analogous reasoning applies to the remaining four cases from (3.39). Using null-vector of (H (α) ) 2 we derive the action of the operators {Λ s 1,α } for s = {0, +, z}, Λ (α)0 1 is of no importance, as it cancels out regardless of its prefactor. Finally we take care of D (α)+ 1 , arriving at the following coupled operator recursion (3.58) We have nevertheless already proven that D (α)z 1 = 0 for every α, hence the recurrence becomes of the same type as the ones above.
Unlike in most often encountered cases of integrable models (e.g. in fundamental models), the R-matrix here is not of difference type, i.e. its elements do not depend on the difference of the involved spectral parameters only. Yet, the difference of spectral parameters, curiously enough, enters in a way (3.1) which is reminiscent of a Lie group structure.
(vi) The matrix R(x + 1 2 y, x − 1 2 ) is holomorphic in both x, y ∈ C except at x ∈ 1 2 Z + where it has simple poles. This follows from the fact that the generator H(x) has simple poles at x ∈ 1 2 Z + as well, property (v) which terminates the exponential series after α + 1 terms in subspace H (α) a , and a curious nilpotent algebra among its residua X (p) = Res x= p 2 H(x) (A.7), namely (4.6) The property (4.6) can be studied in each space H where U(λ) is a diagonal operator from End(H a ) (invertible for λ ∈ 1 2 Z + ), we obtain the corresponding transposal symmetry for the R-matrix Note that this is a kind of Liouvillian PT symmetry of the type proposed in Ref. [25]. The sign factor (−1) s in (4.7) is a consequence of non-canonical (real) representation of (2.8).

Properties of the monodromy matrix
Rich structure and properties of the R operator discussed above are also inherited by the corresponding exterior monodromy operator T(λ) or its matrix elements (2.22). We list some of the most remarkable properties that we have observed here, with the hope that they will find useful future applications (e.g. those discussed in Section 5). Firstly, for a given system size n, selection rule (2.24) implies that the monodromy matrix is banded, i.e.
Furthermore, we claim that only the elements T k l (λ) from a (n + 1) × (n + 1) square, namely for k ≤ n, l ≤ n are linearly dependent physical operators. Therefore, for a fixed distance from the diagonal q = |k − l|, only n − q matrix elements are linearly independent, while all others can be expressed in terms of those where c ± n,q,l,k (λ) are some rational functions of λ with integer coefficients. Secondly, we were looking for linear combinations of magnetization (particlenumber) preserving diagonal matrix elements T k k (λ) that would form a commuting family. Up to linear dependences we conjecture (based on empirical evidence) that there exists a single commuting linear combination besides S(λ) = T 0 0 (λ), namelỹ Thirdly, as in our problem we are dealing with non-normal operators one may want to understand the connection between the transposed ([T l k (λ)] T ) and original (T k l (λ)) monodromy elements. Writing the reflection parity operator in quantum spin space Q = Q −1 ∈ End(H ⊗n s ), Q |ν 1 , ν 2 , . . . , ν n = |ν n , ν n−1 , . . . , ν 1 , we find immediately [applying Eqs. (2.23,4.7)] (4.14) or more compactly ¶, writing a partial transpose with respect to H ⊗n s by superscript T s , wereŨ(λ) = diag(1, −1, 1, −1 . . .)U(λ). Furthermore, the reflected monodromy elements QT k l (λ)Q can be in turn expressed in terms of linear combination of T j l−k+j (λ). For example, we state the connection explicitly for 00 matrix element |Ω m ≡ |Ω n−m . Direct inspection using explicit representation (2.8) reveals the action on the vacuum state in terms of a shift of spectral parameter whereas, one can write similar but more general expressions for the m-particle sectors where q ∈ Z + can be interpreted as the number of quasi-particles created and l > m for the relations to be non-trivial. Remarkably, the rational functions r q,m l,k (λ), s q,m l,k (λ), again having integer coefficients, and only simple poles at 1 2 Z + , do not depend on system size n. For the purpose of treating the NESS density operator, say for developing an ABA procedure for diagonalizing it, it should be handy to control transposed matrix elements at negative spectral parameter −λ, corresponding to Hermitian conjugation at real value of dissipation ε = 2i/λ, for whichλ = −λ and S † (λ) = S T (−λ). Let us writẽ where a sign factor (−1) n is put for convenience. Then, straightforward inspection again suggests remarkable connections: T l+q l (λ) |Ω m = m k=0 g q,m l,k (λ)T k k+q (λ + (q + l + k)) |Ω m , (4.23) where relations are already non-trivial for any l, and integer coefficient rational functions f q,m l,k (λ), g q,m l,k (λ) again do not depend on size n.

Discussion
After this work has been completed, we have learned about Refs. [8,14,7] where related infinitely-dimensional R-matrices have been constructed using manifestly sl(2)symmetric form of Lax and R-matrices. It seems that such a universal sl(2) R-matrix might be useful in the context of QCD and high-energy physics whereas in condensed matter physics the non-unitarity of the general infinitely-dimensional representation seems to be only compatible with phenomena far from equlibrium which we discuss here.
In fact, our Lax matrix (2.10) becomes sl(2)-symmetric after multiplying by σ z , (2) with representation parameter λ + . The R-matrices resulting from solving RLL relations for the two forms of L-matrices, L(λ) andL(λ), are different but closely related. Nevertheless, the results presented in this paper are more explicit and detailed in connection to a different form of a transfer matrix as they are taylored for non-equilibrium condensed matter applications, and hence they are essentially nonoverlapping with those of Ref. [8]. Although the sl(2)-symmetric L-matrix generates a related transfer matrix, namely S(λ)(σ z ) ⊗n , and yields an identical NESS density operator S(λ)S † (λ), we have a good reason to use also a symmetry-broken representation of the Lax matrix. Namely, only in our representation the MPA for S(λ) generates a convergent sum of local operators [13] in the q-deformed case of anisotropic XXZ model (see discussion below, in subsect. 5.2).
We foresee two immediate interesting applications of the exterior (non-equilibrium) integrability formulated here.

Algebraic Bethe Ansatz and spectrum of the density operator
A tempting proposal following from our construction is the construction of ABA procedure for diagonalizing NESS density operator. This could be particularly interesting in the light of recent suggestions [20,26] that the spectral properties of equilibrium and non-equilibrium density operators can be used as indicators of integrability (or exact solvability) similar as in the idea of quantum chaos.
Algebraic form of Bethe ansatz allows for construction of an eigensystem for a family of mutually commuting transfer operators. The procedure is based on the quasiparticle modes created under the action of (off-diagonal) elements of the monodromy matrix T(λ) = L(λ) ⊗sn . Many-particle excitations arise as a string of monodromy elements, operating on a specially chosen reference state. The role of R-matrix is to prescribe quadratic algebraic relations among elements with different value of spectral parameter (which are interpreted as quasiparticle momenta), enabling for construction of eigenstates of the quantum transfer operator. A set of n spectral parameters {λ k } for n-particle excitations has to be chosen accordingly in order to eliminate unwanted terms (those that are not the eigenvectors) which unavoidably emerge during commutation of the elements of T. The latter condition gives rise to famous Bethe ansatz equations [4].

The anisotropic XXZ model and a new family of quasi-local conservation laws
As has been pointed out in Ref. [14], the infinitely dimensional R-matrix also exists for continuous representations of the quantum group U q (sl(2)) hence all our constructions of exterior integrability should be q−deformable and should translate to the boundarydriven anisotropic XXZ spin chain [22,23,15] where the Hamiltonian density h in (2.2) should be replaced by h = 2σ + ⊗ σ − + 2σ − ⊗ σ + + ∆σ z ⊗ σ z with the anisotropy parameter ∆.
Most interesting there is the question, whether the recently discovered quasilocal conservation law [22] can be generalized and extended to a whole family. In integrable theories, local conserved quantities are usually obtained in terms of logarithmic derivatives of transfer matrices around some trivial values of the spectral parameter. Here, the spectral parameter is non-standard and is related to coupling to the environment, hence the derived conserved quantities can have different spin-flip symmetry K = (σ x ) ⊗n as in the standard case [9,16] where due to the equivalence of quantum spin and auxilliary spaces the K symmetry is imposed to the L-and Rmatrices as well and henceforth to all so-derived families of conservation laws. In the exterior integrability problem, however, the K-symmetry is explicitly broken, resulting in (potentially quasi-local) conserved quantities which may yield non-trivial Drudeweight bounds [13] even in the absence of external magnetic field.
For example, writing for the moment the commuting transfer matrix (2.11,2.12) as a function of dissipation ε (in notation of Ref. [23]), which is polynomial for finite n, the conservation law Z which has been proposed and implemented in Ref. [22] and which is quasi-local for |∆| < 1 is simply Z = d dε S(ε)| ε=0 . We conjecture that a further tower of (quasi-local, in case |∆| < 1) conservation laws which break the K-symmetry is given by higher logarithmic derivatives Z k = d 2k−1 dε 2k−1 log S(ε)| ε=0 , k = 1, 2, . . .

(5.5)
The even order logarithmic derivatives vanish as a consequence of an interesting identity which can be easily proven. Details on these constructions shall be presented elsewhere.

Conclusion
We have provided a new link between the matrix product ansatz and Yang-Baxter integrability in the context of non-equilibrium quantum physics, which is fundamentally different than the one which exists on the level of closed quantum systems [1]. The first fundamental difference is in the role of spectral parameter of the integrable theory which is now taken by a continuous representation parameter of infinitedimensional representation of the underlying quantum symmetry of the model. The second fundamental difference is the formulation of the transfer matrix, which is here, due to infinite-dimensionality of the representation space, taken by the ground-state expectation instead of a trace. Generalizations to other quantum integrable models seem straightforward, the most obvious one being perhaps the multi-component quantum hopping model [31].
vanishing (linear in x) terms on the right, we get an additional set of conditions which are to be satisfied: = (α − l + 1)δ k,l − (l + 1)δ k,l+1 , (B.9) at E 01 (equations (B.7),(B.8)) and E 10 (equation (B.9)). However, as the latter set of expressions is rather tedious for further analytical manipulations, we decide at this point to take a different (however equivalent) strategy and rather employ the first form of the generator (3.3).
For the sake of compactness, we shall only provide explicit calculation to justify validity for the set of equations pertaining to non-singular part for physical components E 00 and E 01 , whereas an entirely equivalent procedure applies to show the identity associated with E 10 component. We start with the diagonal element, where from (B.1) we obtain H (α) (x), Λ One applies the same arguments to show the remaining case of the E 10 component.