The complete set of infinite volume ground states for Kitaev's abelian quantum double models

We study the set of infinite volume ground states of Kitaev's quantum double model on $\mathbb{Z}^2$ for an arbitrary finite abelian group $G$. It is known that these models have a unique frustration-free ground state. Here we drop the requirement of frustration freeness, and classify the full set of ground states. We show that the ground state space decomposes into $|G|^2$ different charged sectors, corresponding to the different types of abelian anyons (also known as superselection sectors). In particular, all pure ground states are equivalent to ground states that can be interpreted as describing a single excitation. Our proof proceeds by showing that each ground state can be obtained as the weak$^*$ limit of finite volume ground states of the quantum double model with suitable boundary terms. The boundary terms allow for states which represent a pair of excitations, with one excitation in the bulk and one pinned to the boundary, to be included in the ground state space.Comment: 27 pages, 6 figures. v2: minor corrections, some simplifications and clarificactions adde

On the stability of charges in infinite quantum spin systems

We consider a theory of superselection sectors for infinite quantum spin systems, describing charges that can be approximately localized in cone-like regions. The primary examples we have in mind are the anyons (or charges) in topologically ordered models such as Kitaev's quantum double models and perturbations of such models. In order to cover the case of perturbed quantum double models, the Doplicher-Haag-Roberts approach, in which strict localization is assumed, has to be amended. To this end we consider endomorphisms of the observable algebra that are almost localized in cones. Under natural conditions on the reference ground state (which plays a role analogous to the vacuum state in relativistic theories), we obtain a braided tensor $C^*$-category describing the sectors. We also introduce a superselection criterion selecting excitations with energy below a threshold. When the threshold energy falls in a gap of the spectrum of the ground state, we prove stability of the entire superselection structure under perturbations that do not close the gap. We apply our results to prove that all essential properties of the anyons in Kitaev's abelian quantum double models are stable against perturbations.Comment: v2: 40 pages. Improved presentation, some corrections. v1: 37 pages. Some results were first reported in the dissertation of MC, arXiv:1708.0503

Systematic multivariate analysis (sMVA) strategy for improved process unerstanding of inustrial biorefinery processes with applications in fatty acid production

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Converting a challenge into an opportunity : optimization of fatty acid production from variable feedstocks via process modelling

Biorefineries provide an important alternative for petroleum-based refineries to reduce CO2 emissions and increase the share of renewable feedstocks for the production of chemicals and fuels. Vegetable oils and animal fats are used as renewable raw materials for the production of oleochemicals. The oleochemical industry has changed significantly in the last decades. The rapid growth of palm oil, the rise of oleochemicals production in Southeast Asian Nations and an increased competition with the biofuel industry for feedstocks form a challenge for the traditional oleochemical players in Europe. This increased competition forces European oleochemical companies such as Oleon NV to diversify the feedstocks they process, ranging from low quality animal fat to higher quality vegetable oils. Fatty acid composition, minor compounds and quality vary greatly among these oils and fats. Operating a continuous installation with a variable feed composition poses significant challenges for process control to ensure resource efficiency, high product yields and excellent product quality. Petroleum refineries use the power of mechanistic and statistical modelling to tackle this challenge of feedstock diversity and adapt process conditions for compositional variability of the incoming crude oil. Applying these techniques to biorefineries could aid in utilizing variable bio-based feedstock streams more efficiently, enable the use of lower quality-grade feedstocks and improve adaptability toward future changes in demand and supply. Within this PhD, general methodologies for applying statistical and mechanistic modelling to biorefinery processes, able to account for a variable feedstock composition and quality, are developed. The methodologies are applied for modelling and optimizing the economic and environmental sustainability of two key steps in the production of fatty acids, fat/oil hydrolysis and fatty acid distillation, on an industrial scale. The first chapter of this PhD thesis introduces the different research topics. A short introduction is given to the oleochemical industry, data-driven modelling, mechanistic modelling and sustainability assessment. In the second chapter, a systematic multivariate analysis (sMVA) strategy for improved process understanding of industrial biorefinery processes is proposed. This sMVA strategy comprises nine steps categorized into data set organization, exploratory analysis, and regression. Different MVA techniques are used, such as principal component analysis (PCA) and partial least squares regression (PLS). Key feedstock properties and process parameters affecting the product properties in fatty acid production are identified. It is found that fatty acid quality largely depends on the type of fat or oil used, such as rapeseed or palm oil, due to the large difference in composition and quality between the oil types. However, if a single oil type is used, the variability in product quality does not always critically depend on the variability in feedstock properties. In oil hydrolysis, the feed flow rate influences the residence time and thereby directly influences the hydrolysis and side reactions. In fatty acid distillation, a better control is required of the side reflux ratio in relation to this changing flow rate. The case study shows that applying the sMVA strategy improves the understanding of a biorefinery process by identifying critical sources of variability, which allows more targeted decisions for optimization and control. In the third and fourth chapter, mechanistic modelling is applied. The complexity of lipid feedstocks and the lack of data on physical properties hinder the mechanistic modelling of oleochemical processing units. Therefore, in the third chapter, an iterative lumping approach is proposed to define an adequate number of key compounds such that diversification between lipid feedstocks becomes possible, while keeping the determination of physical properties as required for process modelling manageable. As a case study, the iterative lumping approach is used for simulation and optimization of a fatty acid distillation plant. For predicting vapour–liquid equilibria of fatty acids, the best results are acquired using the property method universal quasichemical- Hayden–O'Conell (UNIQ-HOC). Using the iterative lumping approach, eleven key compounds are selected to represent the feedstock. The acquired process model properly predicts the product composition, yield, purity and heat duty. The most important process parameters are found to be side-reflux-ratio, reboiler-outlet-temperature, and heat-duty of the pitch-distiller. The results show that, for optimization, an increase of the side-reflux-ratio and reboiler-outlet-temperature is recommended. In the fourth chapter, a comprehensive model for simulation and optimization of industrial-scale splitting towers that is able to predict the yield for the hydrolysis of bio-based triglyceride feedstocks is presented. This model includes a variable glycerol equilibrium ratio, which is a function of the composition and temperature and is calculated using the polar version of the perturbed chain statistical association fluid theory (PPC-SAFT). In addition, the model accounts for the autocatalytic effect of fatty acids in hydrolysis and the isomerization of poly-unsaturated fatty acids. Model validation is performed using process data from three real-life splitting towers covering four feedstock types, i.e., tallow, rapeseed oil, palm oil, and palm fatty acid distillate. It is shown that, due to the composition gradients of the organic phase throughout the tower, it is crucial to properly account for the changes in the water and glycerol equilibrium ratio. The importance of feedstock flow rate, water/oil ratio, and temperature profile throughout the tower is analysed and confirmed by sensitivity analysis. Our results show that modifying the temperature profile may shift the reaction equilibrium toward the fatty acid product. This knowledge is crucial for improving the energy and resource efficiency of fat/oil hydrolysis, thereby improving its economic and environmental sustainability. In the fifth chapter, a modelling framework is developed, based on process simulation, techno-economic assessment (TEA) and life cycle assessment (LCA), which can be used for increasing flexibility of feedstock use in biorefineries. The presented framework should aid decision making on valuing, procuring and processing feedstocks, by calculating the techno-economic feasibility and performance and environmental sustainability for different feedstock scenarios. The modelling framework is developed and applied to an oleochemical plant in which fatty acids, glycerol and propylene glycol are produced from different quality grades tallow under changing feedstock prices. The results show that it is economically interesting to change the current operation and use lower-grade tallow instead of higher-grade tallow if the price difference between lower-grade tallow and higher-grade tallow is above a threshold of € 30 per tonne. This illustrates that the developed framework can become a crucial decision support tool for selecting and using feedstocks in the oleochemical industry, allowing an oleochemical plant to switch to other and lower quality feedstocks and increase its adaptability toward changes in demand and supply. In the sixth and last chapter, the main conclusions of this PhD are discussed and future perspectives are formulated. It is discussed how data-driven modelling, mechanistic modelling and sustainability assessment can be combined in a holistic approach for improving the sustainability of oleochemical processes. Remaining challenges and opportunities for the further development and application of the methodologies proposed within this PhD are discussed. Overall, this work shows how process modelling can deliver a strategic advantage to companies for understanding and operating their biorefinery processes on an industrial scale. In addition, process modelling contributes to improving the environmental and economic sustainability of biorefinery processes. For this, the collaboration between academia and industry is essential, as it offers realistic use cases to academia and allows displaying the benefits of new developments and methodologies in process systems engineering to industry

The Complete Set of Infinite Volume Ground States for Kitaev’s Abelian Quantum Double Models

We study the set of infinite volume ground states of Kitaev’s quantum double model on Z 2 Z2 for an arbitrary finite abelian group G. It is known that these models have a unique frustration-free ground state. Here we drop the requirement of frustration freeness, and classify the full set of ground states. We show that the set of ground states decomposes into |G | 2 |G|2 different charged sectors, corresponding to the different types of abelian anyons (also known as superselection sectors). In particular, all pure ground states are equivalent to ground states that can be interpreted as describing a single excitation. Our proof proceeds by showing that each ground state can be obtained as the weak* limit of finite volume ground states of the quantum double model with suitable boundary terms. The boundary terms allow for states that represent a pair of excitations, with one excitation in the bulk and one pinned to the boundary, to be included in the ground state space

The complete set of infinite volume ground states for Kitaev's abelian quantum double models

We study the set of infinite volume ground states of Kitaev's quantum double model on $\mathbb{Z}^2$ for an arbitrary finite abelian group $G$. It is known that these models have a unique frustration-free ground state. Here we drop the requirement of frustration freeness, and classify the full set of ground states. We show that the ground state space decomposes into $|G|^2$ different charged sectors, corresponding to the different types of abelian anyons (also known as superselection sectors). In particular, all pure ground states are equivalent to ground states that can be interpreted as describing a single excitation. Our proof proceeds by showing that each ground state can be obtained as the weak$^*$ limit of finite volume ground states of the quantum double model with suitable boundary terms. The boundary terms allow for states which represent a pair of excitations, with one excitation in the bulk and one pinned to the boundary, to be included in the ground state space