Competing pair density wave orders in the square lattice t-J model

Over the last two decades, the competing orders in high- T c cuprates have been intensely studied, such as charge density waves (CDW) and pair density waves (PDW), which are thought to play a crucial role in high-temperature superconductivity. Using the t - J model on a square lattice as the simplest model for high- T c cuprates, we employed the fermionic tensor product state (fTPS) method for numerical investigations. Here we show, to the best of our knowledge, several types of PDW states within the low-energy subspace of the t - J model are unveiled thanks to the fermionic cluster update method, alongside the well-known d -wave state. In particular, a kind of C 4 symmetric plaquette PDW state is most likely to survive at low dopings δ < 5%. We believe that the competition among these states in the underdoped region suggests the potential existence of a fluctuating quantum liquid of PDW states, which could serve as a natural starting point for exploring the pseudogap phase and understanding the key mechanisms of d-wave superconductivity. Furthermore, we discuss the potential experimental implications of our discovery.