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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…
The issue of probing the pseudogap regime of the cuprate superconductors, specifically with regard to the existence and nature of superconducting pairing correlations of d-wave symmetry, is explored theoretically. It is shown that if the…
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…
An unidentified quantum fluid designated as the pseudogap (PG) phase is produced by electron-density depletion in the CuO$_2$ antiferromagnetic insulator. Current theories suggest that the PG phase may be a pair density wave (PDW) state…
The defining characteristic of hole-doped cuprates is $d$-wave high temperature superconductivity. However, intense theoretical interest is now focused on whether a pair density wave state (PDW) could coexist with cuprate superconductivity…
We review the physics of pair density wave (PDW) superconductors. We begin with a macroscopic description that emphasizes order induced by PDW states, such as charge density wave, and discuss related vestigial states that emerge as a…
A recent discovery of the pair density wave (PDW) order creates a stir in understanding an underlying physics in the pseudogap states of the cuprate high temperature superconductors. We have performed a Spectroscopic Imaging Scanning…
To capture various experimental results in the pseudogap regime of the underdoped cuprate superconductors for temperature $T<T^{*}$, we propose a four-component pair density wave (PDW) state, in which all components compete with each other.…
Extensive research into high temperature superconducting cuprates is now focused upon identifying the relationship between the classic 'pseudogap' phenomenon$^{1,2}$ and the more recently investigated density wave state$^{3-13}$. This state…
We analyze the anomalies of superconducting state (s and d-wave pairing) in a simple model of pseudogap state, induced by fluctuations of short - range order (e.g. antiferromagnetic), based on the model Fermi surface with "hot patches". We…
A recent scanning tunneling microscopy (STM) experiment reports the observation of charge density wave (CDW) with period of approximately 8a in the halo region surrounding the vortex core, in striking contrast to the approximately period 4a…
The Hubbard model and its extensions are canonical theoretical frameworks for understanding correlated electronic states, including those in high-$T_c$ cuprates. Here, we use state-of-the-art thermal tensor network method to map out the…
We introduce and study an XY-type model of thermal and quantum phase fluctuations in a two-dimensional correlated lattice d-wave superconductor based on the QED3 effective theory of high temperature superconductors. General features of and…
We consider the id-density wave (DDW) order, representing the pseudo-gap (PG) state, and the d-wave superconductivity (DSC)within the BCS framework for the two-dimensional (2D) fermion system on a square lattice starting with a mean-field…
The Chiral d-density wave (CDDW) order, at the anti-ferromagnetic wave vector (pi,pi), is assumed to represent the pseudo-gap (PG) state of a hole-doped cuprate superconductor. The pairing interaction required for the PG phase corresponds…
When very high magnetic fields suppress the superconductivity in underdoped cuprates, an exceptional new electronic phase appears. It supports remarkable and unexplained quantum oscillations and exhibits an unidentified density wave (DW)…
A possible explanation for the existence of the cuprate "pseudogap" state is that it is a d-wave superconductor without quantum phase rigidity. Transport and thermodynamic studies provide compelling evidence that supports this proposal, but…
Pair density waves (PDWs) are a inhomogeneous superconducting states whose Cooper pairs possess a finite momentum resulting in a oscillatory gap in space, even in the absence of an external magnetic field. There is growing evidence for the…
In weakly coupled BCS superconductors, only electrons within a tiny energy window around the Fermi energy, $E_F$, form Cooper pairs. This may not be the case in strong coupling superconductors such as cuprates, FeSe, SrTiO$_3$ or cold atom…
We present a theoretical model to show that the transition from an antiferromagnetic (AFM) phase to the $d_{x^2-y^2}$-wave superconductivity occurs through a robust companion of a finite momentum pairing between $({\bf k},\uparrow)$ and…