# Mathematical and Theoretical Physics Unit (Shinobu Hikami)

## Book

E. Brezin and S. Hikami,
Random Matrix Theory with an External Source,
SpringerBriefs in Mathematical Physics 19, (2016), Springer Singapore.

## Abstract

Universal behaviors can be observed in various physical phenomena. For instance, a magnet shows a phase transition with critical indices, which are determined by the symmetry and the space dimensions. They do not depend on the details of the materials. In our unit, we study the universal phenomena of the disordered systems which are typically described by random scatterings and random potentials. The random walk in Brown motion is one of the well-known problems in this field. The aim of our study is to generalize these approaches through the random matrix model and to discuss the universal phenomena.

The random matrix model can be applied to low dimensional electronic system such as a quantum dot, and also to the string theory and gauge theory. The molecular-sequence of DNA or protein seems randomly distributed, but some of these sequences can be explained by random matrix theory. The biological system is an extremely complex system. Our research project is an approach to the biological system from the mathematical and theoretical physics by the use of the random matrix theory.

## Upcoming Events

### Seminar:

• Date: Thursday, February 6th
• Time: 14:00-15:00
• Venue: C016-Lab1, OIST campus
• Speaker: Fujihiko Sugino, Institute for Basic Science, South Korea

Title: Study of highly entangled quantum spin chains

Abstract:
Quantum entanglement is one of the most surprising features of quantum mechanics.
Ground states of quantum many-body systems with local interactions typically obey
an area law'' meaning the entanglement entropy proportional to the boundary length.
It is exceptional when the system is gapless, and the area law had been believed
to be violated by at most a logarithm for over two decades.

Recent discovery of Motzkin and Fredkin spin chain models is striking, since these
models provide significant violation of the entanglement beyond the belief,
growing as a square root of the volume in spite of local interactions.
Although importance of intensive study of the models is undoubted to reveal novel
features of quantum entanglement, it is still far from their complete understanding.

In this talk, I will explain how such violation of the area law arises mainly in the Motzkin model.
In computation of the Renyi entropy, we observe a novel phase transition never seen in any other
spin chain model studied so far.

### 2020 OIST Workshop “Quantum Math, Singularities and Applications”

• Dates: 7/20-7/24, 2020 (7/19 check in, 7/25 check out)
• Venue: New Lecture Room in Lab. 4, OIST, Okinawa  (tentative)
• Lodging: Seaside House and lodging house in campus
• Dead line of application: Dec. 31. 2019.
• Organizers:
• Shihoko Ishii (Tsinghua University/Univ. Tokyo)
• Yasuyuki Kawahigashi (The University of Tokyo)
• Shinobu Hikami ( OIST)
• Contact: workshop@oist.jp

• Tentative list of invited talkers
1. Shihoko Ishii (Tsinghua Univ. /Univ. Tokyo, Organizer)
2. Mircea Mustata (University of Michigan)
3. Kei-ichi Watanabe (Nihon University)
4. Yasuyuki Kawahigashi (The University of Tokyo, Organizer)
5. Segrey Natanzon (Higher school of economics, Moscow)
7. Christine Vespa (Univ. Strasbourg)
8. Benoit Collins (Kyoto Univ.)
9. Shunsuke Takagi (The University of Tokyo)
10. Yoshinori Gongyo (The University of Tokyo)
11. Kohsuke Shibata (The University of Tokyo)
12. Takehiko Yasuda ( Tohoku University)
13. Zenghan Wang (Microsoft Station Q)
14. Shinobu Hikami (OIST, Organizer)
15. Masahiko Yoshinaga (Hokkaido University)
16. Terry Gannon(University of Alberta)
17. Motoko Kato (Ehime Univ.)
18. Makiko Mase (Mannheim Univ.)
19. Mayuko Yamashita (Kyoto Univ. RIMS)