Research interest
I am a physicist, especially working on Quantum Field Theories (QFTs). We, physicists, are trying to understand the fundamental law of nature: What is the equation of motion? What interesting behavior appears from such equations?
My personal interest is to describe how the quantum matter behaves when tiny particles get together and interact with each other. Quite often, such behaviors are controlled by universality, and QFT is a powerful technique to understand such universal behaviors.
Nonperturbative Quantum Field Theory
QFT has a long history. It is developed to understand the interaction of photons and matters in a consistent manner with the theory of relativity, and this becomes quite successful. If the system is weakly coupled, perturbation theory with the mean-field approximation fairly works well. In this sense, physicists have a good understanding on perturbative aspects of QFTs.
There are, however, still many things that need to be understood for strongly-coupled QFTs! After the success of perturbative QFTs, physicists try to understand more about QFTs and notice the underlying principle of its success: Universality. Under general setup of physical requirements, such as locality, unitarity, and symmetries, QFTs provide a ubiquitous tool for the low-energy effective description of nature. Many interesting collective phenomena have QFT descriptions, but we now must go beyond the perturbation theory to understand them. I'm interested in strongly interacting many-body systems from the viewpoint of quantum field theories, and the problems originate both from high-energy and condensed matter physics.
Symmetry, Topology, and Anomaly
For strongly-coupled QFTs, we have a very limited power to compute them. Nevertheless, we still have a chance to derive rigorous properties of such systems, especially when paying attention to symmetries.
In classical statistical physics, symmetry is used to classify phases of matters. For example, let us consider water. Ice is its solid phase, and breaks translational symmetry spontaneously, while the liquid or vapor phase do not break it. When breaking pattern of symmetry is different, the phase transition completely separate those phases. This beautiful idea is summarized as Ginzburg-Landau theory, and in many cases it is very powerful even for quantum theories.
In quantum theory, the states are described by rays of the Hilbert space, and this fact allows the projective realization of the symmetry. When this happens, we have to extend the Ginzburg-Landau treatment, and this is important to understand the quantum properties of matters. An important keyword for it is 't Hooft anomaly. When we have a QFT with symmetry, then we can try to select only symmetric states. However, we sometimes encounter the topological obstruction for it, saying that there are no symmetric states. Identifying such obstructions, we cay give a rigorous constraint on possible phases of QFTs by anomaly matching.
Path Integral via Lefschetz Thimbles
Recently, Picard-Lefschetz theory attracts theoretical attention as a new approach to nonperturbative aspects of quantum mechanics. Its usefulness is found through its application to Chern-Simons theory, and it might provide a theoretical foundation of the resurgent trans-series expansion of quantum mechanics.
This technique rewrites an oscillatory multiple integration as a sum of non-oscillatory integrals in an exact way, and the saddle-point analysis is available in a systematic manner. Therefore, this method is also regarded as a new possible way to solve the sign problem. The origin of the sign problem is appearance of an oscillatory integral in the path-integral expression, because it forbids us from using the importance sampling of the Monte Carlo integration, and it also induces some difficulties in the mean-field approximation.
I am quite interested in this new technology, and it is indeed found to be useful for some practical applications to physical systems. Although its theoretical foundation is still missing when we would like to apply it to quantum systems in general, we can still be optimistic and should study its various aspects.
Computational methods of QFTs
In the strongly coupled regime, the perturbation theory breaks down and new computational methods becomes required. Roughly speaking, those methods can be classified into two categories: One is numerical computations of the lattice quantum field theory using Monte Carlo integration. This has been powerful, and succeed to reveal nonperturbative aspects of quantum gauge theories. Another one is called functional methods, which establishes nonperturbative relations between the set of Green functions just by using properties of functional integrals: Schwinger-Dyson equation, 2PI formalism, and functional renormalization group (FRG) are famous strategies belonging to this category.
These methods are complementary to each other, and thus it is very important to develop and apply these various formalisms to deepen our understandings of strongly interacting systems.