# FY2013 Annual Report

# Mathematical and Theoretical Physics Unit

**Professor Shinobu Hikami**

### Abstract

Mathematical and theoretical physics unit studied the subjects related to random matrix theory, conformal field theory and statistical field theory, which attract interests. On the basis of a random matrix theory, by the tuning the external sources, the various cases for singularities were found in the connection to N=2 supersymmetric topological field theory, and CFT minimal models. The topological invariants for such various singularities were successfully derived.

In addition to orientable Riemann surfaces, non-orientable surfaces were studied by the antisymmetric random matrix theory with an external source.

This unit studied also two dimensional surface phenomena experimentally, with equipment of SNOM and STM. With a motivation of the fermion bosonization in two dimensions, the collective excitation of a surface plasmon is one of main topics. The samples of topological insulators of Bismuth compound and graphen were prepared and they were measured by SNOM in an infrared wave length region (THz). The low temperature measurements at 4K were tried in SNOM and in TERS (tip enhanced Raman scattering).

### 1. Staff

- Dr. Shinobu Hikami, Professor
- Dr. Hirohiko Shimada, Post doc. researcher
- Dr. Hirofumi Nema, Post doc. researcher
- Dr. Susumu Komiyama, Researcher (part time)
- Dr. Ayumi Kikkawa, Research assistant (part time)
- Dr. Momoka Higa, Technical staff
- Dr. Reuven Pnini, Technical staff
- Kaori Yamashiro, Research Administrator
- (After April 1st, 2014, replaced by Miwako Tokuda)

### 2. Collaborations

**Theme: Random matrix theory project**- Type of collaboration: Joint research
- Researchers:
- Professor Edouard Brezin, Ecole Normale Superieure

**Theme: Equipment of infrared microscope**- Type of collaboration: Joint research
- Researchers:
- Dr. Susumu Komiyama, University of Tokyo

**Theme: Conformal field theory project**- Type of collaboration: Joint research
- Researchers:
- Professor Jesper Lykke Jacobsen, Ecole Normale Superieure
- Dr. Yoshitomi Kamiya, Los Alamos National Laboratory

### 3. Activities and Findings

#### 3.1 Random matrix theory (Hikami):

With the tuning an external source, topological invariants of the moduli space for r-spin curves were derived. For the negative values of r, the relation to SL(2,R)/U(1) CFT was found, which have continuum and discrete spectrum, and the relation between r-spin and spin j in SL(2,R) with level k was investigated.

#### 3.2 Conformal field theory (Shimada):

Universality in the critical point of the models with quenched disorder (impurity) has been a long-standing, fundamental issue both for experimentalists and theorists. Despite substantial efforts, non-perturbative results in this domain are few and our understanding remains quite limited. We applied conformal field theory to those problems and study in particular the renormalization-group flow in the 2D disordred O(n) loop model via the landscape of the effective central charge obtained by the transfer-matrix method based on the Zamolodchikov's C-theorem. We confirm the central charges and critical exponents along the line of the random fixed points predicted via the replica method and 1-n expansion by one of us (HS). Furthermore, in a non-perturbative regime, we found a line of multicritical points, which at the special point (n=1) reduces to the Nishimori universality class possessing the supergroup symmetry OSP(2m+1|2m). Later, we corroborated this universality by an exact Kramers-Wannier duality mapping onto the Edwards-Anderson (spin glass) model.

#### 3.3 Experiments of two dimensional collective modes (Higa, Nema, Tamashiro):

**Figure 1**: Measurements of graphen by AFM (left) and SNOM (right)

The surface plasmon of graphen and topological insulator Bi2Te3 were investigated by SNOM for various wave length up to 10um. In Fig. 1, monolayer graphen on Si were measured by 9.5um infrared laser SNOM. The image of SNOM is clear than the image of AFM.

The low temperature measurement at 4.2 K by TERS (tip enhanced Raman scattering) and SNOM were started.

### 4. Publications

#### 4.1 Journals

Shimada H., Jacobsen J. L., and Kamiya Y, “*Phase diagram and strong-coupling fixed point in the disordered O(n) loop model”*, J. Phys. A: Math. Theor. **47** (2014) 122001. (IOP select)

#### 4.2 Oral and Poster Presentations

- Shimada H., Jacobsen J. L., and Kamiya Y, “
*Supersymmetric multicritical point in the RG landscape of the disordred O(n) loop model”*, RMT2013, OIST seaside house, Nov. 28, 2013 - Shimada H., “
*g-Theorem, Replica-Dirichlet Boundary States, and the Universal Entanglement in 2+1d Critical Points”,*RIKEN-OIST mathphys workshop, OIST main campus, Feb. 28, 2013 - Shimada H., Jacobsen J. L., and Kamiya Y, “
*Generalization of the Zamolodchikov's C-theorem and the Nishimori fixed point in the two-dimensional disordered O(n) loop model”,*Tokai University, JPS 69th Annual Meeting, Mar. 28, 2014 - Hikami S.,
*“Application to Black hole density of state from random matrix theory”*, (short comment) , OIST RMT2013 workshop, Oct. 27 - Nov. 1, 2013, OIST Main Campus C209. - Hikami S., “
*Duality and Topology in Random Matrix Theory”, Riken-OIST mathphys. Workshop,*Feb 25-27, 2014, B503, Lab 1, OIST Main Campus

### 5. Intellectual Property Rights and Other Specific Achievements

Nothing to report

### 6. Meetings and Events

#### 6.1 RMT2013 (Random Matrix Theory - Fluctuation in Complex Systems)

- Date: Oct. 27, 2013-Nov. 1, 2013
- Venue: OIST Seaside House & Main Campus C209-C210
- Organizer : Shinobu Hikami (OIST)
- Speakers:
- Prof. Edouard Brezin (Ecole Normale Superieure), "Gaussian Matrix Models --- Three examples of dualities ----"
- Prof. Nivedita Deo (University of Delhi), "Random Matrix Models Inspired by Interacting RNA"
- Dr. Yacine Ikhlef (LPTHE, Paris VI/CNRS), "Non-Linear Integral Equations for the SL(2,R)/U(1) black hole sigma model"
- Prof. Jesper Lykke Jacobsen (Ecole Normale Superieure), "Logarithmic correlations in geometrical critical phenomena"
- Prof. Vladimir Kazakov, (Ecole Normale Superieure), "Hirota dynamics in matrix models, quantum spin chains and sigma models"
- Prof. Ivan Kostov (Saclay), "Three-point function of heavy fields in N=4 SYM or Overlaps of Bethe states in the isotropic Heisenberg magnet"
- Prof. Arul Lakshminarayan (Indian Institute of Technology Madras), "Some applications of random matrix theory to the study of entanglement"
- Prof. Didina Serban (Saclay), "Long range integrable spin chains and the AdS/CFT correspondence"
- Dr. Andrzej Jarosz (Holon Institute of Technology), "Quantum dots and Jack polynomials"
- Dr. Shengmao Zhu (Zhejiang University), "Some applications of the ELSV-type formulas"
- Dr. Shashi Srivastava (Variable Energy Cyclotron Centre), "Random cyclic matrices"
- Dr. Hirohiko Shimada (OIST), "Supersymmetric multicritical point in the RG landscape of the disordred O(n) loop model"
- Mr. Sergio Andraus (University of Tokyo), "Interacting particle systems on the line: steady state and limiting regimes"
- Prof. Fumihiko Sugino (Okayama Institute for Quantum Physics), "Dynamical SUSY breaking in a double-well matrix model for 2D type IIA superstrings"
- Prof. Shinsuke M. Nishigaki (Shimane University), "Individual eigenvalue distributions for chGSE-chGUE crossover and determination of low-energy constants in two-color QCD+QED"

#### 6.2 RIKEN-OIST mathphys workshop

- Date: Feb 25-27, 2014
- Venue: B503, Lab 1, OIST Main Campus
- Organizer : Shingo Torii, Koji Hashimoto (RIKEN)
- Speakers:
- Dr. Masato Taki (RIKEN), “AGT Conjecture and Gaiotto-Whittaker Vectors”
- Dr. Shingo Torii (RIKEN), “Discussion on the Weak Value”
- Dr. Masaru Hongo (RIKEN), “Quantum Anomaly, Hydrodynamics, and Quark-Gluon Plasma”
- Dr. Hiroaki Ueda (RIKEN), “Soliton on the Spin Nematic and the 120 Antiferromagnetic Order on the Triangular Lattice”
- Dr. Hirohiko Shimada (OIST), “g-Theorem, Replica-Dirichlet Boundary States, and the Universal Entanglement in 2+1d Critical Points”
- Prof. Koji Hashimoto (Osaka Univ. & RIKEN), “Ferromagnetism in Neutron Stars”
- Dr. Tomoya Hayata (RIKEN), “Quantum Time Crystal and QCD in Strong Magnetic Fields”
- Mr. Han Yan (OIST), “Emergent Gauge Field in Spin Ice”
- Dr. Takashi Oka (Tokyo Univ.), “Floquet Topological Phase Transition: Control of Quantum Matter by Laser”
- Prof. Shinobu Hikami (OIST), “Duality and Topology in Random Matrix Theory

#### 6.3 Seminar

Title: How to distinguish between itinerant and localized electrons: Mott vs Kondo

- Date: June 27, 2013
- Venue: OIST Campus Lab1 C015
- Speaker: Prof. Yoshio Kuramoto

Title: Renormalization-group landscape and the multicritical point in the disordred O(n) loop model

- Date: Aug. 28, 2013
- OIST Campus Lab1 C016
- Dr. Hirohiko Shimada (Laboratoire de Physique Theorique ENS)

Title: Introducing nano-FTIR – imaging and spectroscopy at 10 nm spatial resolution

- Date: Jan. 22, 2014
- Venue: OIST Campus Lab1 C015
- Speaker: Dr. Mariam Bohmler, (NeaSpec GmbH, Munich)