Periodic Reporting for period 1 - FAIM (Finding All Integrable Models)
Période du rapport: 2023-09-01 au 2026-02-28
Symmetry plays an important role in our current understanding of nature. For instance, in the development of the Standard Model of particle physics the understanding of the gauge group of symmetries was crucial. There is a class of models, called integrable systems, which have so many symmetries that they are exactly solvable. Such models have the exciting possibility to be understood in all aspects and thus give valuable insights into physical phenomena. In this way integrable models offer a unique approach to tackling open problems in physics, such as, for instance, describing strongly coupled systems.
The aim of this proposal is to develop a new method to find and classify new integrable systems. Our approach is based on a new framework which was very recently put forward by the PI and his group. This new approach was applied to models that are closely related to regular integrable systems from string theory, quantum field theory and condensed matter physics. Several new models were discovered in this way but their physical and mathematical properties still remain to be understood.
I will be particularly focussed on models that have long-range interactions. These models are crucial in understanding strong coupling behaviour in for instance integrable models that appear in quantum field theory and string theory. Understanding long-range interactions is paramount to the computation of correlation functions in these models. Long-range interactions are also important for quantum systems in condensed matter such as cellular automatons.
More generally, integrable structures appear in all areas of physics from quantum computing to theories of gravity. For this reason, finding new integrable models and classifying them will have a large multidisciplinary impact, with exciting applications ranging from condensed matter to string theory. This will potentially help us understand physical phenomena in various different fields.
The aim of this proposal is to develop a new method to find and classify new integrable systems. Our approach is based on a new framework which was very recently put forward by the PI and his group. This new approach was applied to models that are closely related to regular integrable systems from string theory, quantum field theory and condensed matter physics. Several new models were discovered in this way but their physical and mathematical properties still remain to be understood.
I will be particularly focussed on models that have long-range interactions. These models are crucial in understanding strong coupling behaviour in for instance integrable models that appear in quantum field theory and string theory. Understanding long-range interactions is paramount to the computation of correlation functions in these models. Long-range interactions are also important for quantum systems in condensed matter such as cellular automatons.
More generally, integrable structures appear in all areas of physics from quantum computing to theories of gravity. For this reason, finding new integrable models and classifying them will have a large multidisciplinary impact, with exciting applications ranging from condensed matter to string theory. This will potentially help us understand physical phenomena in various different fields.
The first objective of the proposal was to extend the boost method to long-range interactions, non-regular solutions and non standard integrable models. We have made considerable progress in this regard. In particular, we finished a preprint where we develop a method to derive non-regular solutions of the Yang-Baxter equation and we apply this to matrices of size 4x4. We were able to carry this out and finish the classification that was started with the regular solutions in precious work (e-Print: 2411.18685). We are currently trying to derive additional relations and results to extend our framework to the general case. We are also trying to extend the boost formalism to the continuous case. Some first results where recently published in a preprint (e-Print: 2506.13598).
The second objective focusses on integrable systems with long-range interactions. We have made important progress in this direction beyond the current state of the art. We adapted our integrability framework to systems with constraints, such as the Rydberg constraint, and to systems that exhibit generalised symmetries. Generalised symmetries are more general than the usual symmetry groups and appear in areas such as topological defects, anyons and conformal field theories. Recently there has been a surge of interest in these models and they typically contain medium range interactions. In recent works (e-Print: 2405.15848 2410.16356 2510.19902) we successfully set up a method to deal with integrability in these cases, found new models and addressed some open questions in the literature. We are also working on some follow-up research.
The third direction mentioned in the ERC proposal deals with solving the new integrable models that are found. We have made some preliminary results in this direction by studying the fusion relations of solutions of the Yang-Baxter equation and the corresponding transfer matrices. The fusion procedure matches singular points of the matrix to the existence of composite particles. We are trying to extend some earlier results obtained by the PI to general solutions and to higher orders. We have made some important progress in this direction, but do not have the general picture yet. A different approach we are taking is trying to extend the Algebraic Bethe Ansatz method to systems with medium range interactions, but more work needs to be done in this direction as well.
The topic of the fourth objective is to study the algebraic properties of integrable models. We have made important progress in understanding the underlying symmetry algebra of models with longer range interactions. We found a new link with Drinfeld twists and associators. We are applying our findings to the su(2) sector in N=4 super Yang-Mills theory which is an important model where long range interactions are important, but the algebra remains elusive. We plan to publish our findings in a few months and are on schedule. There is also a strong relation with the third research direction since the spectrum of the composite particles is closely related to the representations of the algebra.
The second objective focusses on integrable systems with long-range interactions. We have made important progress in this direction beyond the current state of the art. We adapted our integrability framework to systems with constraints, such as the Rydberg constraint, and to systems that exhibit generalised symmetries. Generalised symmetries are more general than the usual symmetry groups and appear in areas such as topological defects, anyons and conformal field theories. Recently there has been a surge of interest in these models and they typically contain medium range interactions. In recent works (e-Print: 2405.15848 2410.16356 2510.19902) we successfully set up a method to deal with integrability in these cases, found new models and addressed some open questions in the literature. We are also working on some follow-up research.
The third direction mentioned in the ERC proposal deals with solving the new integrable models that are found. We have made some preliminary results in this direction by studying the fusion relations of solutions of the Yang-Baxter equation and the corresponding transfer matrices. The fusion procedure matches singular points of the matrix to the existence of composite particles. We are trying to extend some earlier results obtained by the PI to general solutions and to higher orders. We have made some important progress in this direction, but do not have the general picture yet. A different approach we are taking is trying to extend the Algebraic Bethe Ansatz method to systems with medium range interactions, but more work needs to be done in this direction as well.
The topic of the fourth objective is to study the algebraic properties of integrable models. We have made important progress in understanding the underlying symmetry algebra of models with longer range interactions. We found a new link with Drinfeld twists and associators. We are applying our findings to the su(2) sector in N=4 super Yang-Mills theory which is an important model where long range interactions are important, but the algebra remains elusive. We plan to publish our findings in a few months and are on schedule. There is also a strong relation with the third research direction since the spectrum of the composite particles is closely related to the representations of the algebra.
The novel methodologies that are developed are mainly focussed on the extension of integrability to non-regular cases and medium range interactions. For non-regular solutions we use the observation that at the point of coinciding spectral parameters, we should recover one of the constant solutions. We then expand the Yang-Baxter equation around this point in a controlled way. This structured approach allowed us to find the non-regular solutions to the Yang-Baxter equation.
We looked at system with constrained Hilbert spaces and generalised symmetries. The key to make extend the boost operator formalism to this cases is to dress the Lax operators and R-matrices with projectors in the auxiliary space. These projectors eliminate non-physical states from the Hilbert space. Doing this, we loose invertibility, but it turns out that this is not required for the method to work. We are currently developing other new methodologies but these are still under development. Finally, there is interest in our methods from several sides and I would like to highlight possible applications to quantum computing. I have started a collaboration with IBM where we are starting to explore potential applications of integrability to quantum computing.
We looked at system with constrained Hilbert spaces and generalised symmetries. The key to make extend the boost operator formalism to this cases is to dress the Lax operators and R-matrices with projectors in the auxiliary space. These projectors eliminate non-physical states from the Hilbert space. Doing this, we loose invertibility, but it turns out that this is not required for the method to work. We are currently developing other new methodologies but these are still under development. Finally, there is interest in our methods from several sides and I would like to highlight possible applications to quantum computing. I have started a collaboration with IBM where we are starting to explore potential applications of integrability to quantum computing.