Research Topics

My research goal has been to uncover universal/interesting structures behind various non-equilibrium physical phenomena through the concepts such as symmetry and topology. 

Machine learning & physics

Understanding Diffusion Models by Feynman's Path Integral

Diffusion models for image generation have shown remarkable effectiveness and have become widely adopted. In [arXiv:2403.11262], we proposed a new approach to understanding diffusion models through Feynman's path integral developed in quantum physics. This approach allows us to describe various aspects of score-based diffusion models, facilitating the derivation of backward stochastic differential equations and loss functions. One can introduce a parameter that interpolates stochastic and deterministic sampling, and this parameter plays an analogous role to Planck's constant in quantum mechanics. Leveraging this analogy, we employ the Wentzel-Kramers-Brillouin (WKB) method from quantum physics to analyze the negative log-likelihood, shedding light on the performance disparity between stochastic and deterministic sampling strategies.

Neural network representation of quantum systems

In [arXiv:2403.11420], we provide a novel map with which a wide class of quantum mechanical systems can be cast into the form of a neural network with a statistical summation over network parameters. The basic idea is to approximate the paths of a particle in Feynman's path integral via a neural network. The map can be applied to interacting quantum systems / field theories, even away from the Gaussian limit. Our findings bring machine learning closer to the quantum world.

Biochemical reaction networks

Inside a cell of biological systems, numerous chemical reactions form a web of complex networks. So that cells can survive and thrive in an environment with fluctuations and various disturbances, they implement various regulatory mechanisms. Unraveling these mechanisms presents an intriguing  challenge. 

Robust perfect adaptation in biochemical networks

Maintaining stability in an uncertain environment is essential for proper functioning of living systems. Robust perfect adaptation (RPA) is a property of a system that generates an output at a fixed level even after fluctuations in input stimulus without fine-tuning parameters. Although there have been a number of biological implementation of RPA in living systems, identifying RPA and its associated control actions in generic high-dimensional biochemical networks has been considered a formidable task due to the complexity of the system. 

In [2307.07444], we developed a method to characterize and enumerate all the RPA properties in general chemical reaction networks. We also provide an open-source code RPAFinder that performs the enumeration of RPA for a given reaction network. We also gave a recipe of constructing the corresponding controller for each RPA property. The method would be useful for uncovering control mechanisms operating  within biochemical networks for maintaining homeostasis. 

Reduction of complex reaction networks

In biological systems, chemical reactions form complex networks. Even if ample computational resources were available, faithful modeling of such a system is challenging, because our knowledge about the reaction kinetics and parameters is limited. It is thus desirable to obtain a network that is simpler, has fewer parameters, and yet captures the essential feature of the original network. 

We examine this problem through the lens of topology [Phys. Rev. Research 3, 043123]. In a collaboration among physicists, a mathematician and a biologist, we developed a topological method of simplifying a complex chemical reaction network. In particular, we proved that, if a subnetwork satisfies certain topological conditions, it can be eliminated while preserving the original steady state exactly. Therefore, the details of such a subnetwork are irrelevant, as far as the remaining part of the network is concerned, and the system can be studied efficiently with a simpler network. In deriving the results, the methods of algebraic topology play an essential role. 

The method sheds new light on the analysis of biochemical systems from an algebraic-topological perspective. The tools developed here will be useful for studying complex network systems efficiently and identifying relevant network components for biological functionalities.

Robust perfect adaptation of reaction fluxes

We found structural conditions on reaction networks under which all the reaction fluxes exhibit RPA against the perturbation of system parameters [arXiv:2302.01270]. Based on this finding, we gave a recipe for obtaining a simpler reaction network, from which we can fully recover the steady-state reaction fluxes of the original system. This helps us identify key parameters that determine the fluxes and study the properties of complex reaction networks using a smaller one without losing any information about steady-state reaction fluxes.

Quantum phases of matter

Nambu-Goldstone modes for higher-form symmetries

In recent years, the notion of symmetry has been generalized in several direction. One such generalization is the so-called higher-form symmetries. The defining feature of higher-form symmetries is that the corresponding charged objects are extended: for a p-form symmetry, charged objects are p-dimensional. From this point of view, an ordinary symmetry can be regarded as a 0-form symmetry, whose charged objects are point-like. Many concepts discussed for ordinary symmetries are also generalized to higher-form symmetries. For example, higher-form symmetries can be spontaneously broken: Indeed, photons can be regarded as Nambu-Goldstone modes associated with the breaking of a  U(1) 1-form symmetry. 

In [Phys. Rev. Lett. 126, 071601], we derived effective field theories for broken higher-form symmetries by extending the coset construction. Based on the effective theories, we derived a formula for counting the number of physical gapless Nambu-Goldstone modes for spontaneously broken higher-form symmetries in systems without Lorentz invariance. This is a universal result determined solely by symmetry breaking patterns. In the special case when the system has only 0-form symmetries, it reduces to the counting formula for ordinary symmetries in non-relativistic systems. The number of physical gapless modes is a model-independent  result  based only on the symmetry breaking patterns. 

Gapless fracton phases from spontaneous breaking of higher-form symmetries

More recently, we showed that phases with fractons, which are excitations with restricted mobility, are realized as a result of spontaneously broken higher-form symmetries, whose charges do not commute with spatial translations [arXiv:2207.00854]. Based on this understanding, we developed a method to engineer gapless fracton phases with desired mobility restrictions. 

Phases of QCD matter

On hadron-quark continuity 

Phases of matter has been conventionally classified by Ginzburg-Landau (GL) theory based on the symmetry breaking patterns, which are diagnosed by local order parameters. It is known by now that the GL description is not sufficient to classify quantum phases; there are orders that can only be detected using non-local order parameters, and they are called topological orders. Some of topologically ordered phases appear as a result of spontaneously broken higher-form symmetries, which are a generalization of the conventional symmetries. Indeed, the topological order of the toric code can be understood as a consequence of a spontaneously broken  Z2 1-form symmetry.
We gave a constraint on the phase structure of dense QCD using higher-form symmetries [Phys. Rev. Lett. 122, 212001,  J. High Energ. Phys. 2019, 62 (2019)]. There is a proposal on the phase diagram of dense QCD called quark-hadron continuity scenario, which states that the color superconducting phase can be continuously connected with the superfluid phase of nucleons without phase transitions. Recently, there appeared a proposal that those two states are different, based on the fractional statistics of vortices and colored particles in a color superconductor. We examined this proposal from the view point of generalized symmetries. We found that those two phases have the same symmetries including higher-form ones, and consequently, we showed that the continuity scenario extends as a continuity of quantum phases. 

Quantum anomaly and macroscopic transport phenomena

It was theoretically pointed out in 2004 that the quantum anomaly induces macroscopic dissipationless transport effects. One of those is the Chiral Magnetic Effect (CME), which is a phenomenon that an electric current is generated by an applied magnetic field, in the presence of chirality imbalance. These effects are expected to be present in heavy-ion collisions, and it is experimentally discovered in other systems like Dirac/Weyl semimetals. The magnetic fields generate CME currents, and in turn, they create electromagnetic fields and modify the field configurations. On such a coupled system, we found the following: 

• An inverse energy cascade and self-similar time evolutions of magnetic fields at late times [Phys. Rev. D 92, 125031]. 

• A new mechanism for the CME current generation by the change of topology of the magnetic fields [Phys. Rev. Lett. 117, 172301].

• For the interacting system of a chiral plasma and electromagnetic fields, we constructed the low-energy effective theory, the chiral magnetohydrodynamics [Phys. Rev. D 100, 065023].  

• Integrable dynamics of magnetic field lines (or vortices) in a medium with a chirality imbalance [Phys. Rev. Lett. 121, 142301].