PHYSICAL REVIEW B 103, 075137 (2021) 8-16-4 graphyne: Square-lattice two-dimensional nodal line semimetal with a nontrivial topological Zak index Arka Bandyopadhyay , 1 Arnab Majumdar , 2 , * Suman Chowdhury , 3 Rajeev Ahuja , 2, 4, † and Debnarayan Jana 1 , ‡ 1 Department of Physics, University of Calcutta, 92 A P C Road, Kolkata 700009, India 2 Department of Physics and Astronomy, Box 516, Uppsala University, Uppsala, SE-75120, Sweden 3 Skolkovo Institute of Science and Technology, Skolkovo Innovation Center, 3 Nobel Street, Moscow 121205, Russia 4 Department of Materials Science and Engineering, Royal Institute of Technology, Stockholm, SE-10044, Sweden (Received 25 August 2020; revised 2 February 2021; accepted 3 February 2021; published 22 February 2021) An unprecedented graphyne allotrope with square symmetry and nodal line semimetallic behavior has been proposed in the two-dimensional (2D) realm. The emergence of the Dirac loop around the high-symmetry points in the presence of both the inversion and time-reversal symmetries is a predominant feature of the electronic band structure of this system. Besides, the structural stability in terms of the dynamic, thermal, and mechanical properties has been critically established for the system. Following the exact analytical model based on the real- space renormalization group scheme and tight-binding approach, we have inferred that the family of 2D nodal line semimetals with square symmetry can be reduced to a universal four-level system in the low-energy limit. This renormalized lattice indeed explains the underlying mechanism responsible for the fascinating emergence of 2D square nodal line semimetals. Besides, the analytical form of the generic dispersion relation of these systems is well supported by our density-functional theory results. Finally, the nontrivial topological properties have been explored for the predicted system without breaking the inversion and time-reversal symmetry of the lattice. We have obtained that the edge states are protected by the nonvanishing topological index, i.e., Zak phase. DOI: 10.1103/PhysRevB.103.075137 I. INTRODUCTION The rise of Dirac materials, i.e., graphene [1], topological insulators [2], d -wave superconductors [3], etc. is undoubt- edly an important aspect of fundamental science because of their unusual, unique, and robust physical properties [4–6]. In principle, the electronic behavior of any system strongly relies on the corresponding lattice symmetry and dimensionality [7]. With respect to the lattices, the von Neumann–Wigner theorem [8] ascribes a number of constrains that makes the emergence of two-dimensional (2D) Dirac materials rare. Initially, it was perceived that the Dirac fermions are the consequence of graphene’s honeycomb crystal and is robust to small external perturbations [9,10]. Likewise, the elec- tronic band structures of graphene analogous silicene and germanene [11,12] also evince the existence of Dirac cone features. But, this curiosity was not bound only to group 4 elements and was extended to numerous 2D allotropes of different elements, such as boron [13], aluminum, phospho- rus, nitrogen [14], and so on. Nevertheless, Malko et al. [15] revealed that carbon systems with distinct symmetry, α-, β -, and 6,6,12-graphynes also exhibit graphenelike Dirac cones. Among these lattices, 6,6,12-graphyne with rectangular unit cell [15] indicates that the hexagonal symmetry is not a pre- requisite for the survival of Dirac fermions. Of late, some * arnab.majumdar@physics.uu.se † rajeev.ahuja@physics.uu.se ‡ djphy@caluniv.ac.in additional graphene allotropes, viz., pha- [16], SW- [17], S-, D-, E-[18], PAI-graphene [19], OPG-Z [20], δ-[21], H 4,4,4 − [22], circumcoro-graphynes [23], etc. also join the exotic family of 2D Dirac materials. It is to be noted that the above- mentioned systems belong to either hexagonal or rectangular symmetry groups and can be spontaneously transformed into graphenelike honeycomb lattices [18,24,25]. However, conventional Dirac fermions are particularly hard to find in square lattice [26–29]. In this regard, Zhang et al. [30] first explored the coexistence of square symmetry and Dirac fermions by introducing two square graphynes (S graphynes), i.e., 4,12,2- and 4,12,4-graphynes. However, the nearest-neighbor interaction in the above-mentioned systems only allows the formation nodal rings [31] at the crossing points of the valence-band maximum (VBM) and conduction- band minimum (CBM). Besides, Jiang et al. [32,33] have recently explored that square Lieb and kagome lattices also possess Dirac fermions. Emergence of the nodal rings also ex- hibits twofold band degeneracies that disperse linearly along the high-symmetry k path of the irreducible Brillouin zone (IBZ). It is worth mentioning that similar types of nodal rings are further observed for two square lattices namely of tetragonal silicene (T silicene) [34] and tetragonal germanene (T germanene) [35]. The topological phases of material are characterized by some topological invariants. In particular, Chern numbers in 2D Chern insulators [36], Z 2 indices in 2D and 3D topological insulators [37], winding numbers in topological nodal line semimetals [38,39], and topological charges in Weyl semimet- als [40] are some well-known symmetry-protected topological 2469-9950/2021/103(7)/075137(12) 075137-1 ©2021 American Physical Society