Criticality in two-dimensional quantum systems: Tensor network approach
Shi-Ju Ran,¹ Cheng Peng,² Wei Li,³ Maciej Lewenstein,^1,4 and Gang Su^2,5,*
1ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels, Barcelona, Spain
²CAS Key Laboratory for Vacuum Physics, School of Physics, University of Chinese Academy of Sciences,
P.O. Box 4588, Beijing 100049, China
³Department of Physics, Beihang University, Beijing 100191, China
⁴ICREA-Institució Catalana de Recerca i Estudis Avançats, Lluis Companys 23, 08010 Barcelona, Spain
⁵Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China
(Received 29 September 2016; revised manuscript received 15 February 2017; published 10 April 2017)

Determination and characterization of criticality in two-dimensional (2D) quantum many-body systems is one of the most important challenges and problems in modern physics. Based on the scaling theory in one-dimensional (1D) systems and tensor networks, we propose an efficient scheme to access the criticality of 2D quantum states. We demonstrate that the boundary state of a critical infinite projected entangled pair state can be described by 1D conformal field theory, which provides a solid indicator that allows one to identify and characterize the criticality of 2D states. Our scheme is verified on the resonating valence bond (RVB) states on kagome and square lattices, where the boundary state of the honeycomb RVB is found to be described by a c = 1 conformal field theory. We apply our scheme also to the ground state of the spin-1/2 XXZ model on honeycomb lattice, illustrating the difficulties of standard variational tensor network approaches to study the critical ground states. Our scheme is of high versatility and flexibility, and can be further applied to investigate the quantum criticality in many other phenomena, such as finite-temperature and topological phase transitions.