The complexity for describing strongly correlated quantum many-body systems scales exponentially with the particle number. This is the famous "exponential-wall" problem pointed out by Nobel laureate Prof. Walter Kohn in his Nobel Prize speech in 1999 and it remains one of the biggest challenges in the study of strongly correlated systems.

A novel numerical method developed to solve quantum lattice problem, named the tensor renormalization group (TRG) technique, has drawn increasing attention in recent years. This method represents a wave function or an action as a tensor network and probes its physical properties by utilizing the renormalization group transformation. It is a method of general applicability that satisfies the variational principle and the area law of quantum entanglement entropy, and does not suffer from the negative-probability problem encountered in quantum Monte Carlo simulations.

Prof. XIANG Tao from Group T06 at the Institute of Physics, Chinese Academy of Sciences and his collaborators have been working on the TRG for several years and have achieved significant progress. Their introduction of the entanglement mean-field approximation, the second renormalization group, and the  coarse-graining TRG using higher-order singular value decomposition have provided an efficient and accurate numerical tool for studying low-dimensional quantum lattice models and also for statistical models in three dimensions. Recently, they have made further progress on the development of a novel TRG method that can be used to handle problems of quantum frustration.

Frustrated quantum magnetism is a fundamental but extremely difficult problem in condensed matter theory. The intense interest in this problem arises because a number of novel quantum states, including the quantum spin liquid, are believed to exist in frustrated quantum systems. A thorough understanding of this type of state may shed light on many key topics, including entanglement, topology, and the pairing mechanism of high temperature superconductivity. Frustrated systems are characterized by strong quantum fluctuations and strong competition between different types of order, and thus cannot be studied reliably using the well-established quantum field theory. Similarly, conventional numerical methods such as quantum Monte Carlo simulations and the density matrix renormalization group also have intrinsic  limitations.

TRG methods have been applied to quantum frustration problems in numerous studies, but  very little progress, or even insight, has been obtained. Recently, Prof. XIANG and his group, in collaboration with Prof. Bruce Normand from Renmin University of China, found that the problem results from the almost complete cancelation of two-body quantum entanglement in the ground state. Two-body entanglement is the most important effect considered in the conventional representation of tensor-network states, but the nature of a frustrated system is that two-body terms cancel: indeed the team found that it is three- or more-body entanglement within a simplex which plays the most important role. A simplex is the building block of the two or three-dimensional frustrated lattice, for example a triangle is the basic simplex of the kagome (Figure 1) or the triangular lattice. To capture the key features of the many-body simplex entanglement, they proposed a new class of tensor-network state to describe the ground-state wave function for frustrated systems. This kind of state, the projected entangled simplex state, is based on the simplex and is defined on a frustration-free lattice (Figure 1), providing a natural way to eliminate the complexity of quantum frustration.

Based on this new tensor-network wave function, they calculated the ground state for the Heisenberg model on the kagome lattice, a key model in the study of quantum spin liquids. Their preliminary result obtained with a local tensor dimension as small as 13 already reaches the accuracy of the best results currently obtained by the variational Monte Carlo or by any density matrix renormalization group methods (Figure 2). This result not only constitutes remarkable progress in the study of the kagome Heisenberg model but also lays a solid foundation for a complete solution of the quantum frustration problem.

This work was published recently as Physical Review X 4, 011025 (2014). It was supported by the National Natural Science Foundation and the Ministry of Science and Technology of China.

CONTACT:
Prof. XIANG Tao
Institute of Physics
Chinese Academy of Sciences
Email: txiang@iphy.ac.cn

Reference：Z. Y. Xie, J. Chen, J. F. Yu, X. Kong, B. Normand, and T. Xiang, Tensor Renormalization of Quantum Many-Body Systems Using Projected Entangled Simplex States, Phys. Rev. X 4, 011025 (2014). http://journals.aps.org/prx/abstract/10.1103/PhysRevX.4.011025