Quasilocal conservation laws from semicyclic irreducible representations of $U_q(\mathfrak{sl}_2)$ in $XXZ$ spin-$1/2$ chains

We construct quasilocal conserved charges in the gapless ($|\Delta| \le 1$) regime of the Heisenberg $XXZ$ spin-$1/2$ chain, using semicyclic irreducible representations of $U_q(\mathfrak{sl}_2)$. These representations are characterized by a periodic action of ladder operators, which act as generators of the aforementioned algebra. Unlike previously constructed conserved charges, the new ones do not preserve magnetization, i.e. they do not possess the $U(1)$ symmetry of the Hamiltonian. The possibility of application in relaxation dynamics resulting from $U(1)$-breaking quantum quenches is discussed.


Introduction
In this paper we shall consider the anisotropic Heisenberg XXZ spin-1/2 chain with periodic boundary conditions, from the point of view of theory of integrability. Heisenberg spin chains, providing successful theoretical description of magnetism-related phenomena in spin-chain materials [1], can be studied analytically using the so called algebraic Bethe ansatz method [2,3], resulting in an infinite family of conservation laws. These can then be applied in numerous ways to treat various aspects of equilibrium and nonequilibrium statistical physics of the model in consideration. An example of application of these results out-of-equilibrium is a rigorous derivation and evaluation [4,5,6,7] of Mazur-Suzuki lower bound [8,9] for spin transport [10]. The result of this evaluation in linear response theory implies strict ballistic property of high temperature spin transport in the thermodynamic limit in the |∆| < 1 regime of the anisotropic Heisenberg spin-1/2 Hamiltonian. This issue has been controversial in view of the fact that the thermodynamic Bethe ansatz approach to quantum spin transport allowed for different, mutually inconsistent results [11,12]. Another, closely related application of (quasi)local conserved operators is in relaxation dynamics that follows quantum quenches of integrable systems, where the precise formulation of the so-called generalized Gibbs ensemble is currently under intense investigation [13,14,15].
The success of the algebraic Bethe ansatz method, in case of Heisenberg spin chains relies heavily on the existence of quantum group U q (sl 2 ) and its universal R-matrix, satisfying the so called quantum Yang-Baxter equation [16]. Using the fundamental, two-dimensional representations of these objects, Faddeev and his Leningrad school [2] developed a general technique which generates the Heisenberg Hamiltonian, together with the full family of local conserved charges in involution, via the logarithmic derivatives of the quantum transfer matrix. However, in a recent progress [6,7,17], other highest weight irreducible representations at the root-of-unity values of q, which densely populate the entire critical interval −1 < ∆ < 1, have been implemented to construct quasi-local conserved quantities relevant for quantum spin transport and quantum relaxation [18]. In the present paper, these constructions are generalized and extended, using semicyclic irreducible representations [19,20]. Here the highest and the lowest weight vectors are coupled by the periodic action of U q (sl 2 )-generators.
In the second section we briefly review the model and the structure of cyclic and semicyclic irreducible representations of U q (sl 2 ), as given (up to module isomorphism) in Refs. [19] and [20]. Also, the problem of periodicity of the generator action is reviewed: the fundamental commutation relations described in [2], implying the conservation and involution of transfer-matrix related quantities, fail to hold in case of irreducible even-dimensional and certain odd-dimensional (semi)cyclic representations [21]. Since the anisotropy parameter is linked to the parameter q -the parameter of deformation of the quantum group and thus to the dimension of a representation, this imposes a restriction on the values of anisotropy parameter, for which this construction is valid.
In the third section we construct new quasilocal conserved quantities (31), from valid odd-dimensional irreducible representations, using the formal procedure described in [6]. Quasilocality follows from the argument stated in the latter paper. As a direct consequence of periodicity of generator actions, the total magnetization in z direction is not conserved by the newly constructed conserved operators. This is compatible with abundant degeneracies found in the spectrum of the XXZ model at the root of unity anisotropies as a consequence of the loop algebra symmetry [22,23].
Finally, we propose in section 4 some potentially interesting applications of new quasilocal conserved operators for computing generalized Drude weights and quantum relaxation dynamics resulting from U(1)-symmetry breaking quantum quenches in the regime of linear response.
2. The model and representations of U q (sl 2 ) 2.1. Anisotropic Heisenberg model and quantum group U q (sl 2 ) Denoting by H = C 2 the local physical spin-1/2 space, the Hamiltonian of the anisotropic Heisenberg spin-1/2 chain consisting of n particles can be thought of as an operator in End (H ⊗n ), with local interaction terms acting nontrivially on a pair of adjecent sites. Periodic boundary conditions are imposed by taking indices modulo n. Operator term indices j, j+1 denote local physical spaces in the chain, on which the term h j, j+1 acts nontrivially.
Thus h j, j+1 is a trivial extension of the local interaction h ∈ End (H ⊗ H), which, in our case, takes the form onto End (H ⊗n ). Here, as in standard notation, σ α , α ∈ {0, x, y, z} denote Pauli matrices, σ 0 = 1 the identity in End (H), and additionaly we have σ ± ≡ 1 2 (σ x ± iσ y ). In the gapless regime, |∆| ≤ 1, which will be the subject of our consideration, one can rewrite the parameter of anisotropy as ∆ = cos η, introducing a new parameter η ∈ [0, 2π). Setting q = e iη , we obtain an intrinsic connection between Heisenberg spin model and the quantum group U q (sl 2 ) [2,6,17,16]. The latter is a Hopf algebra, generated by elements S + , S − , q 2 S z , satisfying a set of algebraic relations 1 Here we have used common notation, allowing to write the Lax operator 2 as described in [6], where J = {+, −, 0, z} is the index set and V the auxiliary space -the representation space (module) of the Hopf algebra. The Lax components L µ (ϕ) are given by L 0 (ϕ) = sin ϕ cos(ηS z ), L z (ϕ) = cos ϕ sin(ηS z ), One can now construct continuously parametrized (ϕ ∈ C) transfer operators (transfer matrices) V n (ϕ) ∈ End (H ⊗n ), 3 V n (ϕ) = tr a (L(ϕ) ⊗ p n ).
In general their conservation and involution follows from the quantum Yang-Baxter equation, evaluated in a representation over a triple tensor product H ⊗ H ⊗ V and H ⊗ V ⊗ V, respectively [2,6]. First let us describe the explicit form of the representations used.

Irreducible representations of U q (sl 2 )
The theory of irreducible representations of U q (sl 2 ) allows for an especially rich family of representation structures at the roots of unity, where enlarged center of the algebra provides additional representation parameters [24]. Here, besides the highestweight representations, more exotic structures, e.g. (semi)cyclic representations, are encountered. For q taking values from the set of roots of unity, the anisotropy parameter ∆ densely populates the interval [−1, 1]. Let us denote by d the order of root of unity, the lowest nontrivial natural number such that q d = 1. Setting one finds that m is the highest possible dimension of irreducible representation of U q (sl 2 ) at q -root of unity [19]. We consider the following irreducible representations, preserving the form of the Lax operator (4) and up to isomoprhism equivalent to the classified ones [19,20]: The first two of these structures are isomorphic if we set The U q (sl 2 )-module isomorphism is given by the adjoint action of U : Indeed, one can easily check that for all α ∈ {+, −, z}, relation holds. Note, however, that U does not define an isomorphism for all possible values of s. A counterexample, which will be relevant in our case, is s = 0. The highest-weight representation V 1 (s, 0, 0) was used in Ref. [6]. In what follows, we will leave out the representation indices in operator definitions (S α j = S α ). Parameters α, β, γ, couple the highest-weight vector with the lowest-weight one. Therefore, they shall henceforth be referred to as coupling parameters. If one of α, β is non-zero, the representation is called semicyclic. If both are nonzero, the representation is refered to as cyclic. The representation V 3 (p, γ 0) is a priori semicyclic. Coupling of the highest and the lowest-weight vectors leads to a problem with the fundamental commutation relation used for a proof of conservation of transfer operators (6). We discuss this problem and its consequences in the next subsection.

The problem of the fundamental commutation relation
Conservation of transfer operators (6) follows from the train argument [2] used on the fundamental commutation relation (FCR), i.e. the Yang-Baxter equation evaluated on the triple tensor product H ⊗ H ⊗ V, Here x, y ∈ C \ {0} are the spectral parameters, in our case taking an explicit form x = e iϕ , y = e iϕ and operator indices 4 refer to vector spaces in the triple tensor product, on which the operators act nontrivially [2]. 5 The form of Lax operators L is given by (4) and the trigonometric R-matrix by 4 For example R 12 = R ⊗ 1, where R ∈ End (H ⊗ H) is given by (15). 5 We note that all representation modules discussed here are to be understood in the sense of so-called evaluation map representation of the infinite-dimensional loop algebra U q (Lsl 2 ).
One can prove that FCR (14) holds for R-matrix (15) and Lax operator (4) iff the following set of relations is satisfied [2]: Notice the essential difference between U q (sl 2 )-defining relations (3) and the Eqs. (16). A question that now arises is, in which case are the latter relations satisfied if the former hold. This question has been discussed in relation to the existence of intertwiner for representations H and V j , j ∈ {1, 2, 3}, in Ref. [21]. In general we can summarize the action of U q (sl 2 )-generators as for some discrete maps f + , f − , f z :  (16) and (17) result in the following set of equations: A recursion relation then follows for the index map f z , implying q m = 1, since f z is nonzero. Note that in case of a representation with both the highest and the lowest-weight vector, there is no such constraint on the module dimension. Using Eq. (7), one can conclude, that the dimension of semicyclic or cyclic irreducible representations of U q (sl 2 ), for which FCR (14) holds, is always odd. If one considers FCR as the sole source of conservation of transfer operators (6), then the construction of conservation laws from the semicyclic irreducible representations is invalid for q being a root of unity of even order. This restricts the values of the anisotropy parameter ∆ ∈ [−1, 1] one can discuss using these representations, onto the following dense countable set Indeed, explicit symbolic evaluation of commutators [V n (ϕ), H XXZ ] for small systems in case of ∆ S , ∆ = cos η, where q = e iη is a root of unity, yields non-zero results. Note that although this set densely covers the interval [−1, 1], it does not so in a symmetric fashion with respect to ∆ = 0. For example, parameter ∆ = −1/2 (η = 2π/3) is included, while ∆ = 1/2 (η = π/3) is not. This can be amended in the thermodynamic limit by noting, that the adjoint action of unitary operator U ∈ End (H ⊗n ), n ∈ 2N given by is equivalent to flipping the sign of the anisotropy parameter ∆, i.e. UH XXZ (∆)U = −H XXZ (−∆). Hence we have an equivalence i.e. if Y is a conserved charge in case of anisotropy ∆, thenỸ = UYU is a conserved charge for anisotropy −∆.

About involution
For q equal to a root of unity there is, in general, no solution to the Yang-Baxter equation in the algebraic sense [20]. From Arnaudon's discussion of validity of Yang-Baxter equation [20] and the existence of intertwiner at s = s = 0 [21], we can conclude that holds in case q m = 1, m = odd, where m is the dimension of irreducible representation. In case q m = −1 the symbolic computation for small system sizes yields non-vanishing commutators among transfer matrices.

Construction
In this section we explicitly describe the construction of new conservation laws, exhibiting an additional property called quasilocality. For completeness, let us revisit the definition of quasilocality, as stated in Ref. [6]. Definition 1. An operator sequence {Y n } n∈N , Y n ∈ End (H ⊗n ) which can be written as with operator terms q r ∈ End (H ⊗r ) satisfying for some positive constants γ, ζ > 0, is called quasilocal. 6 As has been seen in the preceding section, for ∆ ∈ S , the conservation and commutation of transfer operators (6) is guaranteed. Transfer operators inherit a dependence on parameters of the representation from the Lax operators they consist of. As was implicitly hinted above, the representation parameters of our interest are the ones, coupling the highest and the lowest-weight vectors, thus making the representation (semi)cyclic. The task is now to make transfer operators quasilocal. In order to achieve this, we will differentiate the transfer operator with respect to the complex coupling parameter β (or α or γ) and then set all the representation parameters, i.e. all parameters except ϕ: the coupling parameter β (or α or γ) and the spin parameter s (or p), to zero. The representations V 1 and V 2 are related by the map given by (12). Since for our choice of representation parameters (s = 0), this map is no longer an isomorphism, we do not expect exact equivalence of the resulting conservation laws.
Example 3. Taking the third representation with Lax components given by (28) and differentiating the corresponding transfer operator with respect to γ, we get for operator terms. It is straightforward to see that again holds. For an example of explicit form of operator terms, see Eqs. (39). Therefore, we have -up to a trivial spin flip -a single additional family of semicyclic quasilocal charges which, remarkably, do not conserve the z-magnetization,

Applications
Let us briefly discuss the possibility of applications. Conserved quantities are especially important for estimation of dynamical susceptibilities of extensive observables, by means of the Mazur-Suzuki inequality [6,7,5,9]. Let A be such an observable with a fixed parity ν, i.e. PAP = (−1) ν A. As was done in Ref. [6] for the case of magnetization (or spin) current, the dynamical susceptibility can be bounded from below in accordance with the inequality Here A denotes a formal time average, is a projection of an observable onto the conserved quantity, and f is a complex-valued holomorphic function, defined on D m = {ϕ ∈ C ; | Re ϕ − π 2 | < π 2m }, satisfying the Fredholm equation of the first kind, Since z-magnetization M z n is not conserved by our conservation laws, the observables eligible for such a consideration must not conserve it as well in order for a(ϕ) to be possibly nonvanishing. An example of such an observable is a sum of translated, local spin-flip operators, acting on an odd number of adjecent sites, k odd, which is obviously an observable of positive (or even) parity ν = +1. We can now, for example, calculate the projection of A k onto the conserved quantities from m-dimensional irreducible semicyclic representations. Note that, since our observable has positive parity, the choice of particular semicyclic representation (V 1 , V 2 , or V 3 ) is irrelevant, since (A k , X n (ϕ)) = (A k , PX n (ϕ)P). The projections (A k , q n (ϕ)) can be left out of calculation of scalar products, since we are working in the thermodynamic limit and hence, using Cauchy-Schwarz inequality and quasilocality (25) of local operator terms, we can estimate |(A k , q n (ϕ))| ≤ ||A k || HS ||q n (ϕ)|| HS ≤ ζ n e −γ n .
From the form of Lax components, it can be deduced, that the function g, contributed by the action of (k − m)/2 pairs of operators L + 0 (ϕ) and L − 0 (ϕ), must always be a homo- for some constant C, is needed. At the moment we are not able to provide general analytic solutions f (ϕ) to this equation, which however can easily be solved numerically. As a possible physical realization, one may consider a quantum quench protocol from the Hamiltonian H λ = H XXZ − λA k to H XXZ at t = 0, restoring U(1)-symmetry of the XXZ Hamiltonian (which is broken by k−spin operator A k ), starting with a thermal state ρ(t < 0) = ρ β = Z −1 exp(−βH λ ). For weak quenches (small λ), one can use linear response theory [18], to predict the after-quench steady state ρ ss = ρ β (1 + βλĀ k ).
Our generalized Drude weight D A k thus yields a high-temperature Hilbert-Schmidt norm of all the magnetization non-conserving terms in the steady state, and can be bounded from below (45) using the newly constructed quasi-local operators. This is an indicatetion that the latter have to be included in a complete generalized Gibbs ensemble of the XXZ model at roots of unity.

Conclusion
The present paper extends the construction of quasilocal conservation laws for the anisotropic Heisenberg XXZ model with periodic boundary conditions, developed in Refs. [6,17], to the case of exotic representaiton structures provided by the quantum group U q (sl 2 ) at roots of unity, namely the semicyclic representations. The conservation law property and commutativity are discussed in relation to the dimension of representation: only representations with dimension equal to the order of root of unity generate conserved quantities in involution. Newly constructed conserved quantities are quasilocal and do not conserve z-magnetization, i.e. they do not possess the U(1) symmetry of the Hamiltonian. Interesting remaining problems and questions which should be investigated in future include: (i) physical applications of newly constructed quasilocal conserved charges, for example in relation to quench dynamics and generalized thermalizaiton, and (ii) analytically solving the Fredholm equation (53).