Multiple particle bootstrap

Abstract

Quantum field theories in lower dimensions usually have more elegant structures and are highly related to other branches of physics, like statistic physics, condensed matter systems, and even AMO physics. This research project explores various interesting aspects of quantum field theories in two dimensions and mainly focuses on two examples: 2d Ising field theory (IFT) and 2d Chiral Potts field theory (CPFT). Here we will first briefly introduce these examples, and then discuss the related methodologies and techniques. 2d Ising field theory, as the continuous limit of the 2d Ising model near the Curie critical point, is a perfect example of low dimensional quantum field theory with various interesting properties. The parameter space of 2d IFT is two-dimensional, and within the parameter space at some special points/singularities, the theory becomes exactly solvable, due to the conformal symmetry or integrability. A good example is the famous Yang-Lee point, which can be described by a conformal minimal model. Furthermore, the physical interpretation of the Ising model in ordered/disordered phases is completely different, but many thermodynamic quantities in both phases are related by the analyticity of the parameters, which is determined by those singularities and their symmetries. With the help of these solvable points together with their symmetries, it's possible to explore further properties of the IFT theory space and benefit our understanding of the theory space of quantum fields. 2d Chiral Potts field theory is defined as the continuous limit of the 2d Chiral Potts model and is very interesting because it has a multi-dimensional integrable parameter space, with the price of losing Lorentzian symmetry. The phase diagram of 2d CPFT is fascinating, due to the transition between the incommensurate phase and the massive phase. Still, people know little about many other details of the CPFT parameter space, including the existence of other singularities/critical points, the emergent of super-integrability, and the properties of spectrum and interactions. One of the goals of this project is to gain a better understanding of these phenomena from the field theory point of view. In two dimensions, many analytic computational methods are based on the philosophy of bootstrap, which is supported by the emergent symmetries at solvable points. For example, at the critical points, the conformal bootstrap method can compute most information of the theory, starting from the axiomatic assumptions. Another example is when the theory becomes massive and integrable. The S-matrix bootstrap method can restrict/solve the form of interactions, the thermodynamic Bethe ansatz is helpful for a better understanding of the spectrum, and the form-factor bootstrap method is useful for perturbations. These methods can find many applications in the two examples of this project. Besides the analytic computations, various numerical methods will also be an important part of the project, especially when away from the special points in the parameter space, where perturbation theory loses its power. The main numerical experiment we will use is the Hamiltonian truncation method. The basic idea is to truncate the infinite-dimensional Hamiltonian and do numerical diagonalization, and it can provide a good numerical approximation of the field theory spectrum on a finite geometry, from which much information is readable, including not only the thermodynamical quantities like the particle spectrum and free energy density but also some dynamical characteristics like the interaction between particles and the decay rates of resonances. With the combined strength of symmetry-based bootstrap-inspired analytic analysis, and powerful numerical experimental method, surely more interesting aspects of 2d quantum field theories will be discovered in this project, enhancing our understanding of the quantum field theories in lower dimensions.