Multi-label classification" algorithm solves the challenging problem of one-dimensional strong correlation

Since the discovery of the Mott insulator-metal phase transition, quantum strongly correlated systems have become one of the core research areas in condensed matter physics due to their rich and novel physical phenomena. With the breakthrough advancements in ultracold atomic experimental techniques, theoretical and experimental studies of low-dimensional strongly correlated systems have entered a new surge of development. A key tool in understanding quantum many-body systems lies in the precise characterization of their correlation functions. However, strongly correlated systems face fundamental challenges for traditional perturbative theoretical methods due to the lack of well-defined single-particle pictures. In this context, quantum integrable systems, which have strict analytical solutions, have become an important theoretical framework for understanding generic strong correlation physics.

Recently, a joint research team composed of institutions including Northwestern University, the University of Hong Kong, the Institute of Precision Measurement Science and Technology Innovation of the Chinese Academy of Sciences, the University of Houston, and Zhejiang University achieved a significant breakthrough. The team innovatively developed a multi-label classification algorithm, achieving the exact solution of the spectral function of one-dimensional Bose gases at arbitrary interaction strengths for the first time. They successfully observed its singular behavior at the spectral thresholds, thereby validating the theoretical predictions of the nonlinear Luttinger liquid theory (NTLL) at a large scale of 4000 particles and establishing a new approach to the one-dimensional strongly correlated quantum systems.

Based on the Bethe ansatz tenique, the research team, creatively proposed the algorithm of "relative excitations," thereby successfully incorporating the calculations of the properties for the ground state, finite-temperature equilibrium state, and non-equilibrium steady state into a unified framework. By introducing four quantum numbers (Pm, Np, Pl, Nl) with clear physical meanings as labels for the subspaces of the Hilbert space, the infinite-dimensional Hilbert space is partitioned into a series of finite-dimensional subspaces, greatly enhancing computational efficiency. Using this method, the team first accurately calculated the complete spectral function of a one-dimensional Bose gas in the full  momentum-energy plane, as shown in Figure 1. Notably, the research successfully captured the power-law behavior of the spectral function at the spectral thresholds by observing its line shape in a huge system size of 4000 particles, confirming the prediction of NTLL theory in large systems for the first time. As shown in the fitting results in Figure 2, the four critical exponents (-0.465, 1.141, -0.529, 0.977) are in high agreement with the predictions of NTLL (-0.422, 0.934, -0.501, 1.043). Additionally, the research team also calculated the momentum distribution function of the one-dimensional Bose gas, observing the power-law behavior governed by the linear TLL theory and Tan's Contact in the small and large momentum regions, respectively, as shown in Figure 3.

Their findings on the exact spectral function of one-dimensional Bose gases undoubtedly input new vitality into the research of quantum many-body systems and open up new directions for future experimental observations and theoretical explorations. This research was supported by the National Natural Science Foundation of China, the Hong Kong Research Grants Council, and the Chinese Academy of Sciences.

---

---