We investigate the phase diagram of the $t\text{\ensuremath{-}}J$ model on a triangular lattice using a variational Monte Carlo approach. We use an extended set of Gutzwiller projected fermionic trial wave functions allowing for simultaneous magnetic and superconducting order parameters. We obtain energies at zero doping for the spin-
1∕2 Heisenberg model in very good agreement with the best estimates. Upon electron doping (with a hopping integral
t<0) this phase is surprisingly stable variationally up to $n\ensuremath{\approx}1.4$, while the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}+i{d}_{xy}$ order parameter is rather weak and disappears at $n\ensuremath{\approx}1.1$. For hole doping, however, the coplanar magnetic state is almost immediately destroyed and ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}+i{d}_{xy}$ superconductivity survives down to $n\ensuremath{\approx}0.8$. For lower
n, between
0.2 and
0.8, we find saturated ferromagnetism. Moreover, there is evidence for a narrow spin density wave phase around $n\ensuremath{\approx}0.8$. Commensurate flux phases were also considered, but these turned out not to be competitive at finite doping.
It has long been proposed that doping a chiral spin liquid (CSL) or fractional quantum Hall state can give rise to topological superconductivity. Despite intensive effort, definitive evidences still remain lacking. We address this problem by studying the t-J model supplemented by time-reversal symmetry breaking chiral interaction J_{χ} on the triangular lattice using density-matrix renormalization group with a finite concentration δ of doped holes. It has been established that the undoped, i.e., δ=0, system has a CSL ground state in the parameter region 0.32≤J_{χ}/J≤0.56. Upon light doping, we find that the ground state of the system is consistent with a Luther-Emery liquid with power-law superconducting and charge-density-wave correlations but short-range spin-spin correlations. In particular, the superconducting correlations, whose pairing symmetry is consistent with d±id wave, are dominant at all hole doping concentrations. Our results provide direct evidences that doping the CSL on the triangular lattice can naturally give rise to topological superconductivity.
Two-dimensional density-matrix renormalization group method is employed to examine the ground-state phase diagram of the Hubbard model on the triangular lattice at half-filling. The calculation reveals two discontinuities in the double occupancy with increasing the repulsive Hubbard interaction
U at ${U}_{\mathrm{c}1}\ensuremath{\sim}7.8t$ and ${U}_{\mathrm{c}2}\ensuremath{\sim}9.9t$ (
t being the hopping integral), indicating that there are three phases separated by first-order transitions. The absence of any singularity in physical quantities for $0\ensuremath{\le}U<{U}_{\mathrm{c}1}$ implies a metallic phase in this regime. For
U>Uc2, the local spin density induced by an applied pinning magnetic field exhibits a three sublattice feature, which is compatible with the ${120}^{\ensuremath{\circ}}$ N\'eel-ordered state realized in the limit of $U\ensuremath{\rightarrow}\ensuremath{\infty}$. For
Uc1<U<Uc2, a response to the applied pinning magnetic field is comparable to that in the metallic phase with a relatively large spin correlation length, but showing neither valence bond nor chiral magnetic order, which therefore resembles gapless spin liquid. However, the spin structure factor for the intermediate phase exhibits the maximum at the
K and ${\mathrm{K}}^{\ensuremath{'}}$ points in the momentum space, which is not compatible to spin liquid with a large spinon Fermi surface. The calculation also finds that the pairing correlation function monotonically decreases with increasing
U and thus the superconductivity is unlikely in the intermediate phase.
Phase diagram of the one dimensional
t1−t2−J model: Ferromagnetism, triplet pairing, charge and pair density waves
We present a density matrix renormalization group (DMRG) study of an extended
t−J model with hopping to the first and second neighbors -- the one dimensional
t1−t2−J model. The full phase diagram as a function of the density
n and exchange strength
J, for both positive and negative values of
t2, is obtained. For
t2=−0.5 we observe that, in the strongly interacting region, Nagaoka Ferromagnetism (FM) is accompanied by a triplet pair density wave (PDW) upon doping. As the spin exchange
J increases, a charge density wave (CDW) phase emerges and then gives way to singlet superconductivity (SC). This phase behaves as a singlet PDW with vanishing spacial average of the order parameter and a spin gap. When
t2=0.5, the physics is basically reminiscent of the conventional
t−J model, undergoing a transition from a metallic to a SC phase as a function of
J.
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.
The triangular-lattice Fermi-Hubbard model has been extensively investigated in the literature due to its connection to chiral spin states and unconventional superconductivity. Previous simulations of the ground state of the doped system rely on quasi-one-dimensional lattices where true long-range order is forbidden. Here we simulate two-dimensional and quasi-one-dimensional triangular lattices using state-of-the-art Auxiliary-Field Quantum Monte Carlo. Upon doping a non-magnetic chiral spin state, we observe evidence of chiral superconductivity supported by long-range order in Cooper-pair correlation and a finite value of the chiral order parameter. With this aim, we first locate the transition from the metallic to the non-magnetic insulating phase and the onset of magnetic order. Our results pave the way towards a better understanding of strongly correlated lattice systems with magnetic frustration.