Coexistence of superconductivity and charge density wave in a correlated regime

TL;DR

The paper addresses how CDW and superconductivity can coexist in a correlated electron system. It develops a square-lattice, one-band model with repulsive $U$ treated via the Hubbard-I approximation within a Green's-function framework and a BCS-like description of CDW and SC, while varying the second-nearest-neighbor hopping $t_1$ and temperature. Key findings show that $t_1$ weakens perfect nesting and enables SC-CDW coexistence by relocating gaps in momentum space, whereas $U$ generally suppresses the CDW but can stabilize it at larger values, yielding a finite coexistence region at intermediate $V$. The results offer a qualitative framework for SC-CDW interplay in correlated materials and highlight $t_1$ as a tunable parameter, with potential relevance to transition metal dichalcogenides and nickelates.

Abstract

To investigate the coexistence of superconductivity and charge density wave (CDW) in a correlated regime, we employ the Green's functions formalism, as well as the Hubbard-I approximation, as a way to introduce the correlations into the problem, in the form of a repulsive Coulomb interaction $U$. In addition, we investigate the effects of second-nearest neighbor hopping $t_1$ on a pure CDW state. The analysis of the results show that, for small values of $t_1$, both CDW and superconducting gaps compete for the same region on the Fermi surface. The increase of $t_1$ decreases the competition and may lead the system to a coexistence regime. Effects of temperature in the coexistence regime, are also investigated.

Figures (7)

• Figure 1: Panels a) and b) show the spectral function $A(\vec{k},\omega=0)$ in the normal state, for two different values of $t_1$. The black lines serve as a guide to the eye and indicate the locus of the perfectly nested Fermi surface. Panels c) and d) show $A(\vec{k},\omega)$ along principal directions. The horizontal dashed line indicates the position of the Fermi energy $\varepsilon_F$. The temperature and the Coulomb interaction are $T=0$ and $U=0$, respectively.
• Figure 2: a) The CDW ($W$) order parameter as a function of the attractive interaction $V$, for $U=0$ and different values of $t_1$. b) $W$ as a function of the Coulomb interaction $U$, for $t_1=0.10|t_0|$ and different values of $V$. c) Function $F_{CDW}(\omega)$ for $V=5.0|t_0|$, $t_1=0.10|t_0|$ and different values of $U$. d) The total occupation $n_T$ versus $U$ for the same values of $V$ and $t_1$ as in b). The inset shows the density of states for $t_1=0.1|t_0|$. The temperature is fixed at $T = 0$.
• Figure 3: The spectral function $A(\omega,\vec{k})$ for $T = 0$ and $V=5.0|t_0|$. The color map indicates the intensity of the spectral weight. The solid red lines show the quasiparticle bands along principal directions. The horizontal dashed line, indicates the position of the Fermi energy $\varepsilon_F$.
• Figure 4: The CDW ($W$) and SC ($\Delta$) order parameters as a function of the attractive interaction $V$, for different values of $U$ and $t_1$. The temperature is fixed at $T=0$.
• Figure 5: (a) Phase diagram for $U/|t_0|$ versus the attractive interaction $V$ with $t_1=0.10|t_0|$. (b) Phase diagram for $t_1/|t_0|$ versus $V$ with $U=1.0|t_0|$. The temperature is fixed at $T=0$.