Abstract 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.
Abstract 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.
The physical nature of pseudogap phase is one of the most important and intriguing problems towards understanding the key mechanism of high temperature superconductivity in cuprates. Theoretically, the square-lattice t-J model is widely believed to be the simplest toy model that captures the essential physics of cuprate superconductors. We employ the Grassmann tensor product state approach to investigate uniform states in the underdoped (δ≲0.1) region. In addition to the previously known uniform d-wave state, we discover a strongly fluctuating pair density wave (PDW) state with wave vector Q=(π,π). This fluctuating PDW state weakly breaks the C4 rotational symmetry of the square lattice and has a lower or comparable energy to the d-wave state (depending on doping and the t/J ratio), making it a promising candidate state for describing the pseudogap phase.
In light of the new experimental and theoretical important developments in high-Tc superconductivity, we revisit the fermionic hot-spot model relevant to the phenomenology of the cuprates. We extend previous results by means of a complete two-loop order renormalization group (RG) framework. Here, we explicitly study the effect of the charge-density-wave (CDW) order parameter with a d-wave form factor with the experimentally observed modulation $(\ifmmode\pm\else\textpm\fi{}{Q}_{0},0)$ and $(0,\ifmmode\pm\else\textpm\fi{}{Q}_{0})$ at the infrared-stable nontrivial fixed point obtained previously for this model. Additionally, we proceed to investigate also the so-called pair-density-wave (PDW) order that was recently proposed in the literature as a possible candidate for the ``hidden'' order to describe the pseudogap phase observed in underdoped cuprates. We confirm that although the above two ordering tendencies are also found to be nearly degenerate both at one-loop and two-loop RG orders and linked by an emergent SU(2) pseudospin symmetry, they turn out to be subleading for weaker couplings in the present model to antiferromagnetism, d-wave bond-density wave (BDW) order with modulation along Brillouin zone diagonals $(\ifmmode\pm\else\textpm\fi{}{Q}_{0},\ifmmode\pm\else\textpm\fi{}{Q}_{0})$, and d-wave singlet superconductivity (SSC). However, as we increase the strength of the initial coupling towards moderate values, we do capture a tendency for the entangled PDW/CDW order to become leading compared to BDW/SSC in the model, which suggests that the former composite order might be indeed a viable concept to describe some cuprate superconductors at high temperatures in the underdoped regime, as has been recently alluded to by many authors in the literature.
Fluctuating Pair Density Wave in Finite-temperature Phase Diagram of the t-t′ Hubbard Model
The Hubbard model and its extensions are canonical theoretical frameworks for understanding correlated electronic states, including those in high-Tc cuprates. Here, we use state-of-the-art thermal tensor network method to map out the temperature-doping phase diagram of the t-t′ Hubbard model. On the electron-doped side, we find a d-wave superconducting (dSC) regime, supporting the scenario of high-Tc superconductivity. In contrast, on the hole-doped side, no robust dSC phase is detected. Instead, a finite-temperature regime dominated by strong pair-density-wave (PDW) fluctuations emerges, which may eventually give way to charge density wave order upon further cooling. The PDW state exhibits inter-arc pairing with net momentum near (0,π), distinct from the zero-momentum pairing in conventional dSC. Furthermore, these fluctuating PDW states occupy the lower portion of the pseudogap regime on the hole-doped side. We provide a comprehensive finite-temperature perspective consistent with previous ground-state studies, shedding new light on pairing instabilities and exotic electronic states in high-Tc superconductors.
The square-lattice Hubbard and closely related t-J models are considered as basic paradigms for understanding strong correlation effects and unconventional superconductivity (SC). Recent large-scale density matrix renormalization group simulations on the extended t-J model have identified d-wave SC on the electron-doped side (with the next-nearest-neighbor hopping t_{2}>0) but a dominant charge density wave (CDW) order on the hole-doped side (t_{2}<0), which is inconsistent with the SC of hole-doped cuprate compounds. We re-examine the ground-state phase diagram of the extended t-J model by employing the state-of-the-art density matrix renormalization group calculations with much enhanced bond dimensions, allowing more accurate determination of the ground state. On six-leg cylinders, while different CDW phases are identified on the hole-doped side for the doping range δ=1/16-1/8, a SC phase emerges at a lower doping regime, with algebraically decaying pairing correlations and d-wave symmetry. On the wider eight-leg systems, the d-wave SC also emerges on the hole-doped side at the optimal 1/8 doping, demonstrating the winning of SC over CDW by increasing the system width. Our results not only suggest a new path to SC in general t-J model through weakening the competing charge orders, but also provide a unified understanding on the SC of both hole- and electron-doped cuprate superconductors.
Motivated by exploring superconductivity in multi-orbital systems, we study two orbital models of spinful fermions representing (px,py) or (dxz,dyz) orbitals on the square lattice. For minimal interorbital t-J or t-V on-site interactions, a random phase approximation uncovers regimes of instability towards incommensurate dxy pair density wave (d-PDW) superconductivity with driven by interband pairing. We study the competition of PDW order with uniform nodal dxy pairing states and magnetic and charge density wave (CDW) instabilities. At strong coupling, we derive an effective hard-core Cooper pair Hamiltonian which we study using a bosonic Gutzwiller ansatz to reveal a period-2 PDW over a wide range of fillings as well as a checkerboard CDW at quarter-filling. Our results apply to correlated multi-orbital materials with quasi-1D bands, Hubbard models on the square-octagon lattice, and atomic fermions in p-orbitals. Our work highlights the role of the orbital content and multiband Fermi surfaces in stabilizing interband PDW states.
We investigate in underdoped cuprates possible coexistence of the superconducting order at zero momentum and pair density wave (PDW) at momentum Q = (π, π) in the presence of a Néel order. By symmetry, the d-wave uniform singlet pairing dS 0 can coexist with the d-wave triplet PDW dT Q , and the p-wave singlet PDW p Q can coexist with the p-wave uniform triplet pT 0. At half filling, we find that the novel pS Q + pT 0 state is energetically more favorable than the dS 0 + dT Q state. At finite doping, however, the dS 0 + dT Q state is more favorable. In both types of states, the variational triplet parameters dT Q and pT 0 are of secondary significance. Our results point to a fully symmetric Z2 quantum spin liquid with spinon Fermi surface in proximity to the Néel order at zero doping, which may not be adiabatically connected to the d-wave singlet superconductivity at finite doping with intertwining d-wave triplet PDW fluctuations and spin moment fluctuations. The results are obtained by variational quantum Monte Carlo simulations.