image: Schematic phase diagram of BCVO with transverse magnetic field applied along the [010] direction. The light blue regions illustrate the three-dimensional antiferromagnetic phase. The dark blue region shows emergence of E8 particles near the putative 1D QCP, respectively. The right figure shows structure of the 1D CoO6 helical chain in BCVO.
Credit: ©Science China Press
Recently, Professor Jianda Wu’s research team from the Tsung-Dao Lee Institute at Shanghai Jiao Tong University, in collaboration with Professor Jie Ma from the School of Physics and Astronomy and Professor Bella Lake's neutron scattering team from Germany, made significant progress in studying exotic collective excitations in low-dimensional quantum magnetic materials. Their findings, titled "Spin Dynamics of the E8 Particles," have been published in Science Bulletin.
The low-energy physics of the Transverse Field Ising Chain (TFIC) at its quantum critical point (QCP) can be described by a conformal field theory (CFT) with central charge being 1/2. At the QCP, when the model is perturbed by a longitudinal magnetic field, a quantum integrable system known as the quantum E8 integrable model emerges. The physics in the model can be precisely described by the exceptional E8 Lie algebra. It includes eight different types of massive quasiparticles, with their mass spectrum determined by the roots of the Cartan matrix of the E8 exceptional Lie algebra. Notably, the mass ratio of the second particle to the first one satisfies the golden ratio m2/m1=2cos(π/5)≈1.618 . Given the simplicity, elegance, and rich physics embodied by the quantum E8 integrable model, understanding its excitations and identifying a material that can host the E8 physics not only significant advances in the fundamental research of quantum integrable models and statistical field theory, but also hold promise for the design of novel quantum magnetic devices.
BaCo2V2O8 (BCVO), a typical quasi-one-dimensional antiferromagnet, sparked significant research interest in the field over the past decade, and has led to the discovery of a series of exotic physical phenomena within the material. For instance, under an applied transverse magnetic field, the physical phenomena near the low-temperature critical transverse field of BCVO can be described by the universality class of TFIC. Specifically, when a magnetic field is applied along the [100] or [010] direction, the critical magnetic field corresponding to the QCP of TFIC is around 5T, much lower than the three-dimensional quantum critical field of 10T of the material. Within the three-dimensional order, the weak antiferromagnetic interactions between chains in the material can be effectively treated as a staggered longitudinal magnetic field perturbation. This led to the theoretical prediction that the magnetic excitations near the QCP in BCVO will exhibit the E8 physics. Previous studies have confirmed the presence of the TFIC universality class in the material within the relevant experimental parameter range. Additionally, they also observed the exotic E8 excitation modes at the Brillouin zone center [PRB(R) (2020), PRB (2021), PRL (2021), etc.].
However, to fully confirm the emergence of complete E8 physics within the material in the relevant parameter range, it is crucial to accurately determine the dispersion relations of the quantum E8 particles. In this study, the collaborative team utilized inelastic neutron scattering techniques to measure the spin dynamics structure factor of BCVO under an applied transverse magnetic field of approximately 4.7T along the [010] direction. The measurement results aligned almost perfectly with analytical calculations and numerical simulations, thereby experimentally confirming the presence of quantum E8 particles in BCVO.
The E8 particles, as exotic particles emerging from the quantum E8 integrable field theory, with energy-momentum dispersion relationship following the relativistic form E2=p2c2+m2c4 , where E , p and m represent the energy, momentum, and rest mass of the E 8 particles, respectively, and c represents the speed of light. In our context, the speed of light is not the speed of light in vacuum, but rather to the maximum speed at which information propagates within the relativistic quantum field theory of the quantum E8 integrable model. In reality it depends on the specific microscopic details of the system. In BCVO, due to the four-periodicity of the CoO6 helical chain structure, the excitation spectrum folds twice. Through numerical simulations using the infinite Time Evolving Block Decimation algorithm, the research teams identified and excluded the effects of Brillouin zone folding, revealing the relativistic excitation spectrum of the three lightest quantum E8 particles. The observed excitation spectrum matched almost perfectly with the predictions of the quantum E8 integrable field theory. These findings confirm the existence of quantum E8 particles in the quasi-one-dimensional antiferromagnetic material BCVO. Furthermore, the team utilized experimental data to fit the upper limit of the information propagation speed (i.e., the effective speed of light) in low-energy magnetic excitations within BCVO under critical transverse magnetic fields, estimating it to be about 1600 m/s .
The results from the collaboration expand research across multiple fields, including integrable systems, statistical field theory, and strongly correlated many-body physics. They also lay a solid foundation for a systematic study of the exotic physics of E8 particles and the potential to achieve physical manipulation of E8 particles in the future.
The corresponding authors of this research paper include Prof. Jianda Wu from the Tsung-Dao Lee Institute, Prof. Jie Ma from the School of Physics and Astronomy, and Prof. Bella Lake from the Helmholtz-Zentrum Berlin for Energy and Materials in Germany. The first author of the paper is Xiao Wang, a PhD student under the supervision of Prof. Jianda Wu. In this project, Prof. Jianda Wu received funding support from the National Natural Science Foundation of China, the Major Project of Science and Technology Innovation 2030 -- “Quantum Communication and Quantum Computer”, the Shanghai Natural Science Foundation, and the Shanghai Pujiang Talent Program.
See the article:
Xiao Wang, Konrad Puzniak, Karin Schmalzl, C. Balz, M. Matsuda, Akira Okutani,M. Hagiwara,Jie Ma,Jianda Wu,Bella Lake.Spin dynamics of the E8 particles. Science Bulletin 2024;69(19):
https://doi.org/10.1016/j.scib.2024.07.040
Journal
Science Bulletin