We develop a method for computing form factors of local operators in the framework of Sklyanin's separation of variables (SOV) approach to quantum integrable systems. For that purpose, we consider the sine-Gordon model on a finite lattice and in finite dimensional cyclic representations as our main example. We first build our two central tools for computing matrix elements of local operators, namely, a generic determinant formula for the scalar products of states in the SOV framework and the reconstruction of local fields in terms of the separate variables. The general form factors are then obtained as sums of determinants of finite dimensional matrices, their matrix elements being given as weighted sums running over the separate variables and involving the Baxter Q-operator eigenvalues.

In this paper, we prove the off-shell equation satisfied by the transfer matrix associated with the XXZ spin-1 2 chain on the segment with two generic integrable boundaries acting on the Bethe vector. The essential step is to prove that the expression of the action of a modified creation operator on the Bethe vector has an off-shell structure which results in an inhomogeneous term in the eigenvalues and Bethe equations of the corresponding transfer matrix. MSC: 82B23; 81R12

The most general cyclic representations of the quantum integrable τ 2 -model are analyzed. The complete characterization of the τ 2 -spectrum (eigenvalues and eigenstates) is achieved in the framework of Sklyanin's Separation of Variables (SOV) method by extending and adapting the ideas first introduced in [1, 2]: i) The determination of the τ 2 -spectrum is reduced to the classification of the solutions of a given functional equation in a class of polynomials. ii) The determination of the τ 2 -eigenstates is reduced to the classification of the solutions of an associated Baxter equation. These last solutions are proven to be polynomials for a quite general class of τ 2 -self-adjoint representations and the completeness of the associated Bethe ansatz type equations is derived. Finally, the following results are derived for the inhomogeneous chiral Potts model: i) Simplicity of the spectrum, for general representations. ii) Complete characterization of the chiral Potts spectrum (eigenvalues and eigenstates) and completeness of Bethe ansatz type equations, for the self-adjoint representations of τ 2 -model on the chiral Potts algebraic curves.

We consider general cyclic representations of the 6-vertex Yang-Baxter algebra and analyze the associated quantum integrable systems, the Bazhanov-Stroganov model and the corresponding chiral Potts model on finite size lattices. We first determine the propagator operator in terms of the chiral Potts transfer matrices and we compute the scalar product of separate states (including the transfer matrix eigenstates) as a single determinant formulae in the framework of Sklyanin's quantum separation of variables. Then, we solve the quantum inverse problem and reconstruct the local operators in terms of the separate variables. We also determine a basis of operators whose form factors are characterized by a single determinant formulae. This implies that the form factors of any local operator are expressed as finite sums of determinants. Among these form factors written in determinant form are in particular those which will reproduce the chiral Potts order parameters in the thermodynamic limit. The results presented here are the generalization to the present models associated to the most general cyclic representations of the 6-vertex Yang-Baxter algebra of those we derived for the lattice sine-Gordon model.

Rare earth elements (REEs) are now considered emerging pollutants in the environment. Phytolacca americana, an REE hyperaccumulating plant, has been proposed for the remediation of REE-contaminated soils. However, there is no REE-related information for other Phytolacca species. Here, we examined five species (P. americana, P. acinosa, P. clavigera, P. bogotensis, and P. icosandra) for their response to REEs. REE accumulation and fractionation traits both occurred on the same order of magnitude among the five species. Heavy REEs were preferentially transferred to leaves relative to light REEs. Regardless of the species, lateral root length and chlorophyll content decreased under REE exposure, and lateral roots and foliar anthocyanins increased. However, plants did not experience or only slightly experienced oxidative stress. Finally, REE exposure strongly modulated the ionome of roots and, to a lesser extent, that of leaves, with a negative correlation between REE and Mn contents. In conclusion, our study provides new data on the response of several Phytolacca species to REEs. Moreover, we highlighted that the REE accumulation trait was conserved among Phytolacca species. Thus, we provide valuable information for the phytoremediation of REE-contaminated sites since the most appropriate Phytolacca species could be selected depending on the climatic/pedological area to be remediated.

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Progressive rare earth element (REE) enrichment in aquatic environments worldwide and their resulting anthropogenic anomalies have highlighted the need for a better understanding of their biological effects, with a special emphasis on microbial cells since they play a crucial role in good ecosystem functioning. Therefore, the primary aim of this work was to achieve simultaneous characterization of the 16 REE toxicity effects on the growth kinetics of the commonly found Gram-negative bacterium E. coli (BW25113 strain). Bacterial growth curve modelling showed hormetic effects in the presence of REEs, while EC50 determination (in the mid-log phase) indicated that the four HREEs from Er to Lu in addition to Y were the most toxic metals (EC50 in the range of 8.3 to 3 µM), just after Sc (EC50 of 1.1 µM). Additional subcellular parameter assessment revealed cell membrane lipid peroxidation as well as enhanced membrane depolarization and permeability in the presence of La, Gd, or Yb as representatives of LREEs and HREEs. These subcellular effects appeared to be more intense with Gd and Yb compared with La-exposed cells, in relation to the overall higher toxicity potential reported for HREEs on bacterial growth. Also, the cellular ATP production decreased after REE exposure at their EC50. Finally, these results emphasize the importance of growth kinetic consideration as well as the complexity of REE biological effect mechanisms towards bacteria.

Environmental contextRare earth elements (REEs) are strategic metals and emerging contaminants for which plant-based remediation measures are needed. We screened a collection of hardy ferns and identified new accumulator species that preferentially transferred light REEs to their fronds. This study is an important step towards understanding the mechanisms of REE accumulation in plants. AbstractRare earth elements (REEs) include the lanthanides plus yttrium and scandium, and can be split according to their atomic mass into light (LREEs) and heavy REEs (HREEs). The increasing demand for REEs is mainly driven by new technologies, and their current low recyclability has led them to become emerging contaminants. The identification of new REE accumulators may help in determining the REE transfer mechanisms and may result in interesting candidates for phytoremediation techniques. To that end, a collection of 49 hardy fern species, grown in REE-spiked substrate, were screened for their potential in REE accumulation. REE concentrations were very low in the fronds of all Polypodium species, whereas all Athyrium species highly accumulated REEs. The REE accumulation level was more variable among the different species of Dryopteris, Blechnum, Woodwardia, Cystopteris and Polystichum. However, whatever the species, LREEs were preferentially transferred to the fronds over HREEs. This conserved trait was independent of the availability of different REEs in the substrate and instead may arise from specific transfer systems in ferns for the two groups of REEs. Furthermore, REE accumulation was correlated to Ca and Al, which suggested the existence of common uptake pathways. Altogether, these results are of great interest for phytoremediation purposes since appropriate species can be chosen according to the area to be remediated, and they also provide new insights into a more in-depth characterisation of the underlying REE accumulation mechanisms in ferns.