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We study the charge-density-wave order and its competition with superconductivity in the one-band Hubbard model for high-$T_c$ superconducting cuprates. We use cluster dynamical mean field theory (CDMFT) at $T=0$. The one-band Hubbard model…
For decades, superconductivity in high-Tc cuprates has been viewed as a competitor that suppresses charge-density-wave (CDW) order by reducing its amplitude and spatial extent. Here, we show that this picture is incomplete, as…
Typical BCS superconductors are microscopically homogeneous in real space governed by the coherent Cooper pairs with high phase stiffness of superfluid density, which is characterized by a coherence length. However, a periodic oscillation…
Transition-metal dichalcogenides (TMDs) $MX_2$ ($M$ = Ti, Nb, Ta; $X$ = S, Se, Te) exhibit a rich set of charge density wave (CDW) orders, which usually coexist and/or compete with superconductivity. The mechanisms of CDWs and…
Electron-lattice coupling effects in low dimensional materials give rise to charge density wave (CDW) order and phase transitions. These phenomena are critical ingredients for superconductivity and predominantly occur in metallic model…
Charge order is commonly believed to compete with superconducting order. An intertwined form of superconducting wave function, known as pair-density-wave (PDW) order, has been proposed; however, there has been no direct evidence,…
Suggestive, but indirect evidence of the existence of pair-density wave (PDW) order in several high-$T_{\rm c}$ cuprates has been reported. As this constitutes a new quantum phase of matter, it is important to {\it establish} its existence…
A composite pairing structure of superconducting state is revealed by density matrix renormalization group study in a two-leg $t$-$J$ model. The pairing order parameter is composed of a pairing amplitude and a phase factor, in which the…
A spatially modulated superconducting state, known as pair density wave (PDW), is a tantalizing state of matter with unique properties. Recent scanning tunneling microscopy (STM) studies revealed that spin-triplet superconductor UTe$_2$…
We provide a quasi-one-dimensional model which can support a pair-density-wave (PDW) state, in which the superconducting (SC) order parameter modulates periodically in space, with gapless Bogoliubov quasiparticle excitations. The model…
Superconducting states that break space-group symmetries of the underlying crystal can exhibit nontrivial spatial modulation of the order parameter. Previously, such remarkable states were intimately associated with the breaking of…
Kagome superconductors AV$_3$Sb$_5$ (A = K, Rb, Cs) provide a fertile playground for studying intriguing phenomena, including non-trivial band topology, superconductivity, giant anomalous Hall effect and charge density wave (CDW). Recently,…
We explore the interplay of a charge density wave (CDW) order and s-wave superconductivity (sSC) in a disordered system. Recent experiments on 1T-TiSe_2, where the pristine sample has a commensurate CDW order and the superconductivity…
Charge density wave (CDW) correlations feature prominently in the phase diagram of the cuprates, motivating competing theories of whether fluctuating CDW correlations aid superconductivity or whether static CDW order coexists with…
We present a theory for charge-$4e$ superconductivity as a leading low-temperature instability with a nontrivial $d$-wave symmetry. We show that in several microscopic models for the pair-density-wave (PDW) state, when the PDW wave vectors…
The recent experimental observations of high temperature superconductivity in bilayer nickelate have attracted lots of attentions. Previous studies have assumed a mirror symmetry $\mathcal M$ between the two layers and focused on uniform…
Two-dimensional (2D) strongly correlated electron systems underlie many of the most important phenomena in contemporary condensed matter physics, including the Quantum Hall Effect (QHE), ``high T_c'' superconductivity, and possible exotic…
Real-space modulated Charge Density Waves (CDW) are an ubiquituous feature in many families of superconductors. In particular, how CDW relates to superconductivity is an active and open question that has recently gathered much interest…
With the discovery of charge density waves (CDW) in most members of the cuprate high temperature superconductors, the interplay between superconductivity and CDW has become a key point in the debate on the origin of high temperature…
In the $\textit{A}$V$_{3}$Sb$_{5}$ ($\textit{A}$ $=$ K, Rb, Cs) kagome materials, superconductivity coexists with a charge density wave (CDW), constituting a new platform to study the interplay of these two orders. Despite extensive…