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arXiv:1901.11034 · Physical review. B./Physical review. B

Density matrix renormalization group study of superconductivity in the triangular lattice Hubbard model

Jordan Venderley, Eun-Ah Kim

Published Aug 26, 2019• 38 citations
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Abstract

With the discovery of strong-coupling physics and superconductivity in moir\'e superlattices, it is essential to have an understanding of strong-coupling driven superconductivity in systems with trigonal symmetry. The simplest lattice model with trigonal symmetry is the triangular lattice Hubbard model. Although the triangular lattice spin model is a heavily studied model in the context of frustration, studies of the hole-doped triangular lattice Hubbard model are rare. Here, we use density matrix renormalization group to investigate the dominant superconducting channels in the hole-doped triangular lattice Hubbard model over a range of repulsive interaction strengths. We find a clear transition from $p$-wave superconductivity at a moderate on-site repulsion strength ($U/t=2$) at filling above 1/4 ($n\ensuremath{\sim}0.65$) to $d$-wave superconductivity at a strong on-site repulsion strength ($U/t=10$) at filling below 1/4 ($n\ensuremath{\sim}0.4$). The unusual tunability that moir\'e superlattices offer in controlling $U/t$ would open up the opportunity to realize this transition between $d$-wave and $p$-wave superconductivity.

DOI
10.1103/physrevb.100.060506

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With the discovery of strong-coupling physics and superconductivity in moir'e superlattices, it is essential to have an understanding of strong-coupling driven superconductivity in systems with trigonal symmetry. The simplest lattice model with trigonal symmetry is the triangular lattice Hubbard model. Although the triangular lattice spin model is a heavily studied model in the context of frustration, studies of the hole-doped triangular lattice Hubbard model are rare. Here, we use density matrix renormalization group to investigate the dominant superconducting channels in the hole-doped triangular lattice Hubbard model over a range of repulsive interaction strengths. We find a clear transition from ppp-wave superconductivity at a moderate on-site repulsion strength (U/t=2U/t=2U/t=2) at filling above 1/4 (n\ensuremath∼0.65n\ensuremath{\sim}0.65n\ensuremath∼0.65) to ddd-wave superconductivity at a strong on-site repulsion strength (U/t=10U/t=10U/t=10) at filling below 1/4 (n\ensuremath∼0.4n\ensuremath{\sim}0.4n\ensuremath∼0.4). The unusual tunability that moir'e superlattices offer in controlling U/tU/tU/t would open up the opportunity to realize this transition between ddd-wave and ppp-wave superconductivity.
We investigate the evolution of the Mott insulators in the triangular lattice Hubbard Model, as a function of hole doping $\ensuremath{\delta}$ in both the strong and intermediate coupling limits. Using the advanced density matrix renormalization group (DMRG) method, at light hole doping $\ensuremath{\delta}\ensuremath{\lesssim}10%$, we find a significant difference between strong and intermediate couplings. Notably, at intermediate coupling an unusual metallic state emerges, with short ranged spin correlations but long ranged spin-chirality order. Moreover, no clear Fermi surface or wave vector is observed, this chiral metal also exhibits staggered loop current, which breaks the translational symmetry. These features disappear on increasing interaction strength or on further doping. At strong coupling, the 120 degree magnetic order of the insulating magnet persists for light doping, and produces hole pockets with a well-defined Fermi surface. On further doping, $\ensuremath{\delta}\ensuremath{\approx}10%\ensuremath{\sim}20%$ SDW order and coherent hole Fermi pockets are found at both strong and intermediate couplings. At even higher doping $\ensuremath{\delta}\ensuremath{\gtrsim}20%$, the SDW order is suppressed and the spin-singlet Cooper pair correlations are simultaneously enhanced. We also briefly comment on the strong particle-hole asymmetry of the model.
Recent experimental progress has established the twisted bilayer transition metal dichalcogenide (TMD) as a highly tunable platform for studying many-body physics. Particularly, the homobilayer TMDs under displacement field are believed to be described by a generalized triangular-lattice Hubbard model with a spin-dependent hopping phase $\ensuremath{\theta}$. To explore the effects of $\ensuremath{\theta}$ on the system, we perform density matrix renormalization group calculations for the relevant triangular lattice t-J model. By changing $\ensuremath{\theta}$ at small hole doping, we obtain a region of quasi-long-range superconducting order coexisting with charge and spin density wave within $0<\ensuremath{\theta}<\ensuremath{\pi}/3$. The superconductivity is composed of a dominant spin singlet ddd-wave and a subdominant triplet ppp-wave pairing. Intriguingly, the ${S}_{z}=\ifmmode\pm\else\textpm\fi{}1$ triplet pairing components feature pair-density waves. In addition, we find a region of triplet superconductivity coexisting with charge-density wave and ferromagnetism within $\ensuremath{\pi}/3<\ensuremath{\theta}<2\ensuremath{\pi}/3$, which is related to the former phase at smaller $\ensuremath{\theta}$ by a combined operation of spin-flip and gauge transformation. Our findings provide insights and directions for experimental search for exotic superconductivity in twisted TMD systems.
We use unrestricted Hartree-Fock, density matrix renormalization group, and variational projected entangled-pair state calculations to investigate the ground-state phase diagram of the triangular-lattice Hubbard model at “half doping” relative to single occupancy, i.e., at fillings of ( 1 ± 1 2 ) electrons per site. The electron-doped case has a nested Fermi surface in the noninteracting limit, and hence a weak-coupling instability toward density-wave orders whose wave vectors are determined by Fermi-surface nesting conditions. We find that at moderate-to-strong interaction strengths, other spatially modulated orders arise, with wave vectors distinct from the nesting vectors. In particular, we identify a series of closely competing, itinerant long-wavelength magnetically ordered states, yielding to uniform ferromagnetic order at the largest interaction strengths. For half-hole doping and a similar range of interaction strengths, our data indicate that magnetic orders are most likely absent.
Inspired by recent advances in the fabrication of surface superlattices, and in particular the triangular lattice made of tin (Sn) atoms on silicon, we study an extended Hubbard model on a triangular lattice. The observations of magnetism in these systems justify the inclusion of a strong on-site repulsion, and the observation of superconductivity suggests including an effective, nearest-neighbor attractive interaction. The attractive interaction mimics the effect of strong on-site repulsion near half filling, which can be seen in strong-coupling vertex calculations such as the Eliashberg method. With this extended Hubbard model on a triangular lattice with its geometrical frustration, we find a rich phase diagram of various magnetic orders and pairing functions, within the framework of self-consistent mean field theory. We uncover the competition between magnetism and unconventional superconductivity, and their coexistence for triplet pairings. We follow the Fermi surface of the system as the system is doped away from half filling and find nesting vectors and a Lifshitz transition, which provide an intuitive understanding of the phase transitions between the many orders we consider.
The superconducting pairing symmetry is significantly constrained by lattice symmetry, yet how this pairing symmetry evolves with changes in lattice symmetry remains unclear. In this study, through variational quantum Monte Carlo simulations on the Hubbard model, we investigate the evolution of the superconducting ground state on an anisotropic triangular lattice with smoothly tunable frustration. At half-filling, we identify a hidden C6{C}_{6}C6​-like s+d+ids+d+ids+d+id-wave superconductivity, which breaks time-reversal symmetry, emerging for moderate frustrations. This occurs as the lattice symmetry transitions from C4{C}_{4}C4​ to C2{C}_{2}C2​, where the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave and s+dxys+{d}_{xy}s+dxy​-wave superconducting ground states dominate, respectively. Notably, we find an isotropic-like d+idd+idd+id-wave superconducting phase at ${t}^{\ensuremath{'}}/t\ensuremath{\approx}0.7$, much lower than the expected value ${t}^{\ensuremath{'}}/t=1$ predicted by most previous studies. At high electron doping levels, we further discover an unexpected quasi-one-dimensional ${d}_{xy}^{\ensuremath{'}}$-wave superconductivity characterized by strong pairing on the next-nearest-neighbor bond, despite the absence of direct hopping. The numerical data also reveal that pairing symmetries, belonging to distinct irreducible representations of the given point group, can mix together, beyond the conventional theories based on the linear gap equations.
Moir\'e systems offer an exciting playground to study the many-body effects of strongly correlated electrons in regimes that are not easily accessible in conventional material settings. Motivated by a recent experiment on WSe2/WS2{\text{WSe}}_{2}/{\text{WS}}_{2}WSe2​/WS2​ moir\'e bilayers [Y. Tang et al., Nature (London) 579, 353 (2020)], which realized a triangular superlattice with a small hopping (of approximately 10 Kelvin), with tunable density of holes, we explore the Hubbard model on the triangular lattice for intermediate temperatures $t\ensuremath{\lesssim}T<U$. Employing finite-temperature Lanczos calculations and closely following the fitting protocols used in the experiment, we recover the observed trends in the reported Curie-Weiss temperature $\mathrm{\ensuremath{\Theta}}$ with filling, using the reported interaction strength U/t=20U/t=20U/t=20. We focus on the large increase of $|\mathrm{\ensuremath{\Theta}}|$ on decreasing the density below half filling and the sign change of $\mathrm{\ensuremath{\Theta}}$ at higher fillings, which signals the onset of ferromagnetism. The increase in $|\mathrm{\ensuremath{\Theta}}|$ is also seen in the $t\ensuremath{-}J$ model (the low-energy limit of the Hubbard model) in the intermediate temperature range, which we clarify is opposite to the trend in its high temperature limit. Differences between the low, intermediate, and high temperature regimes are discussed. Our numerical calculations also capture the crossover between short-range antiferromagnetic to ferromagnetic order in the intermediate temperature regime, a result broadly consistent with the experimental findings. We find that this behavior is a finite-temperature remnant of the underlying zero-temperature phase transition, which we explore with ground-state density matrix renormalization group calculations. We provide evidence of ferromagnetism characterized by weak (but robust) correlations that explain the small $\mathrm{\ensuremath{\Theta}}$ seen in the experiment.