Competition between charge-density-wave and superconducting orders on eight-leg square Hubbard cylinders

TL;DR

This study addresses whether $d$-wave superconductivity emerges in the square-lattice Hubbard model at intermediate coupling by performing large-scale, $SU(2)$-symmetric DMRG on eight-leg square cylinders with varying $t'$ and hole doping $\delta$. The authors find a CDW-dominated ground state for $t'\le 0$, with exponentially decaying SC correlations, while $t'>0$ produces strong boundary-condition dependence that can enhance SC in some cases but does not robustly realize long-range $d$-wave order in this geometry. The work highlights intertwined competing orders and demonstrates that accurate, symmetry-respecting DMRG with very large bond dimensions is essential to distinguish CDW, SDW, and SC tendencies on quasi-2D lattices. The results suggest that achieving robust high-temperature SC may require additional model ingredients or parameter regimes beyond the studied eight-leg cylinder, guiding future explorations of cuprate-like physics in Hubbard-type models.

Abstract

The issue of whether $d$-wave superconductivity (SC) occurs in the square-lattice Hubbard model with $U$ of order of the bandwidth has been one of the most debated issues to emerge from the study of high temperature SC. Here, we report variational results on eight-leg cylinders with next-nearest-neighbor hopping in the range $-0.5 t \leq t'\leq 0.25 t$ with $U = 8t$ and $12t$ and doped hole concentrations $δ=1/12$ and $1/8$. For $t'\leq 0$, the ground-state appears to be a charge-density wave (CDW) of one sort or another with SC correlations that are extremely short-ranged. In contrast, in some cases, the local magnetic order has a correlation length greater than half the cylinder width - suggestive that magnetic order might also arise in the 2D limit. For $t'>0$, our results depend more strongly on boundary conditions (periodic vs antiperiodic), making it still harder to correctly guess whether SC or CDW correlations dominate in the 2D limit. These results were obtained employing matrix-product states with bond dimensions large enough that energy differences as small as $10^{-3}t$ per site can be resolved.

Figures (8)

• Figure 1: Ground state phase diagram of the Hubbard model on an eight-leg cylinder as a function of $t'$ and $\delta$ for $U=12t$. Everywhere except the region marked with a ?, the heirarchy of ordering tendencies (if not all details of the order) is independent of BCs. Some form of apparently long-range CDW order arises with substantial amplitude in all regions labeled CDW; where CDW appears within parenthesis, CDW order is either much weaker, or fluctuating. Regions with notable subdominant magnetic correlations (defined as a correlation length in excess of 4 lattice constants) are labeled (SDW) when it is locally incommensurate and (Néel) where commensurate. SC correlations are always locally $d$-wave-like, but decay exponentially with distance; only in the region labeled with SC in parenthesis is the SC correlation length in excess of 2 lattice constants. Shaded regions are guides to the eye. Open and filled symbols, respectively, indicate points at which the cylinder with ABCs or PBCs has the lower energy, respectively.
• Figure 2: Ground state correlations for $N=24\times 8$ Hubbard cylinders with $t'=-0.25$ (panels a & c) and $t'=+0.25$ (panels b & d), $U=12t$ and $\delta=1/12$. Panels a and b show the rescaled doped hole density, $\bar{n}_h(x)/\delta$. Panels c and d show the magnitude of the two-point correlations on a log-linear plot as a function of distance $r$ along the cylinders: $F(r)*(-1)^r$ is the staggered spin-spin correlation function and $\Phi_{yy}(r)$ is the pair-field correlation function for singlet, nearest-neighbor pairs oriented perpendicular to the cylinder, where dashed lines denote exponential fits $\Phi_{yy}(r)\sim e^{-r/\xi_{sc}}$ with correlation length $\xi_{sc}$. The sign structure of the spin correlations is indicated by the full symbols (where positive) and open symbols (where negative). The Néel-like character of the spin correlations for $t'>0$ is reflected in the absence of open symbols, while a pattern of anti-phase domain walls for $t'<0$ can be seen from the alternating regions of open and closed symbols.
• Figure 3: (Color online) Rescaled hole density profiles $dn_h(x,y)\equiv{\rm sign}[\delta n_h(x,y)]\times \sqrt{|\delta n_h(x,y)|}$ for 8-leg Hubbard cylinders with $U=12t$ and $t'=\pm 0.25t$, shown at $\delta=1/8$ (panels a - d) and $\delta=1/12$ (panels e - i). Here, $\delta n_h(x,y)=(n_h(x,y)-\overline{n_h})/\delta$, $\overline{n_h}$ is the average hole density and $A_{cdw}$ is the CDW amplitude (defined in Eq. \ref{['Eq:Acdw']}) and $\Delta_{14}$ is a measure of the SC correlations (defined in Eq. \ref{['Delta']}). Results with PBCs are exhibited in the right column, and ABCs in the right. The sites in dark red and blue indicate values outside the plotted scale.
• Figure S1: (Color online) Convergence of superconducting correlations. Superconducting correlation $\Phi_{yy}(r)$ for (a-b) an $N=24 \times 8$ cylinder at $\delta=1/12$ with $t'=-0.25t$ under PBC and ABC, and for (c–d) an $N=32\times 8$ cylinder at $\delta=1/8$ with $t'=0.25t$ under the same boundary conditions. Results are presented for a range of $SU(2)$ multiplet bond dimensions $D$, together with their extrapolation to the limit $D=\infty$. All panels are plotted on semi-logarithmic scales, where $r$ is the separation between two Cooper pairs along the $\hat{x}$ direction. The black dashed lines indicate exponential fits of the form $\Phi_{yy}(r)\sim e^{-r/\xi_{SC}}$.
• Figure S2: Ground-state energy density difference between boundary conditions. The energy density difference $\delta \varepsilon$ for 8-leg Hubbard cylinders is defined such that $\delta \varepsilon > 0$ indicates the system with ABC has a lower energy, while $\delta \varepsilon < 0$ indicates the system with PBC has a lower energy. Panel (a) presents results at doping $\delta=1/8$ for cylinder lengths $L=16 - 32$, and panel (b) presents results at $\delta=1/12$ for $L=24 - 36$. Here, $U=12t$ and dashed lines are included as visual guides.