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

Abstract

Over the last two decades, the competing orders in high-$T_{c}$ cuprates have been intensely studied, such as pseudogap phase, 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. Our study revealed new types of PDW states alongside the well-known $d$-wave state and the recently discovered fluctuating PDW state within the low-energy subspace of the $t$-$J$ model. We believe that the competition among these states in the underdoped region suggests the potential existence of a fluctuating quantum liquid of PDW states, providing direct evidence for the pseudogap phase's "cheap vortex" scenario. Furthermore, we discuss the potential experimental implication of our discovery.

Figures (17)

• Figure 1: (a) Schematic $t$-$J$ model on a square lattice. (b) A fTPS comprised of a $2 \times 2$ unit cell on a square lattice, containing four independent fermionic tensors $A, B, C, D$ on lattice sites and eight independent Schmidt fermionic matrices $\Lambda_0, ..., \Lambda_7$ on nearest neighbor bonds. Bonds of a tensor corresponding to super vector spaces (or dual spaces) are represented by outgoing (or incoming) arrows.
• Figure 2: Illustration of different uniform-amplitude PDW states obtained from fTPS simulations of the square lattice $t$-$J$ model with $t/J=3.0$. (a, b, c) correspond to P0, P1 and P2 in the main text. (d) is the conventional already-known $d$-wave state. Magenta (or cyan) bonds represent the singlet pairing $\Delta>0$ (or $<0$).
• Figure 3: (a) Per-site energy $E(\delta)$ of different competing states obtained from $t$-$J$ model with $t/J=3.0$. $D=14$. The dotted dashed line represents previous VMC results. (b) Per-hole energy $E_{h}(\delta)$ of these states.
• Figure 4: Estimation of singlet pairing magnitudes $\vert\Delta\vert(\delta)$ as $D\to\infty$ for competing states in $t$-$J$ model from a $1/D$ extrapolation. $t/J=3.0$. (a, b, c, d) correspond to P0, P1, P2 and $d$-wave, respectively. Each $\vert\Delta\vert(\delta)$ is fitted by a sixth-order polynomial and plotted as a dashed line with the same color.
• Figure 5: Estimation of magnetization $M(\delta)$ as $D \to \infty$ of competing states in $t$-$J$ model from a $1/D$ extrapolation. $t/J=3.0$. (a, b, c, d) correspond to P0, P1, P2 and $d$-wave state, respectively. Each $M(\delta)$ is fitted by a fourth-degree polynomial and plotted as a dashed line with the same color.