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Physical review. B./Physical review. B

Phase diagram of the one-dimensional t 1 − t 2 − J model: Ferromagnetism, triplet pairing, and charge and pair density waves

Luhang Yang, Adrian E. Feiguin

Published Aug 9, 2024• 2 citations
Publisher

Abstract

We present a density matrix renormalization group study of an extended $t\text{\ensuremath{-}}J$ model with hopping to the first and second neighbors---the one-dimensional ${t}_{1}\text{\ensuremath{-}}{t}_{2}\text{\ensuremath{-}}J$ model. The full phase diagram as a function of the density $n$ and exchange strength $J$, for both positive and negative values of ${t}_{2}$, is obtained. For ${t}_{2}=\ensuremath{-}0.5$ we observe that, in the strongly interacting region, Nagaoka ferromagnetism is accompanied by a triplet pair density wave (PDW) upon doping. As the spin exchange $J$ increases, a charge density wave 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 ${t}_{2}=0.5$, the physics is basically reminiscent of the conventional $t\text{\ensuremath{-}}J$ model, undergoing a transition from a metallic to a SC phase as a function of $J$.

DOI
10.1103/physrevb.110.085122

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We present a density matrix renormalization group study of an extended t\ensuremath-Jt\text{\ensuremath{-}}Jt\ensuremath-J model with hopping to the first and second neighbors---the one-dimensional t1\ensuremath-t2\ensuremath-J{t}_{1}\text{\ensuremath{-}}{t}_{2}\text{\ensuremath{-}}Jt1​\ensuremath-t2​\ensuremath-J model. The full phase diagram as a function of the density nnn and exchange strength JJJ, for both positive and negative values of t2{t}_{2}t2​, is obtained. For t2=\ensuremath−0.5{t}_{2}=\ensuremath{-}0.5t2​=\ensuremath−0.5 we observe that, in the strongly interacting region, Nagaoka ferromagnetism is accompanied by a triplet pair density wave (PDW) upon doping. As the spin exchange JJJ increases, a charge density wave 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{t}_{2}=0.5t2​=0.5, the physics is basically reminiscent of the conventional t\ensuremath-Jt\text{\ensuremath{-}}Jt\ensuremath-J model, undergoing a transition from a metallic to a SC phase as a function of JJJ.
Phase diagram of the one dimensional t1−t2−Jt_1-t_2-Jt1​−t2​−J model: Ferromagnetism, triplet pairing, charge and pair density waves
We present a density matrix renormalization group (DMRG) study of an extended t−Jt-Jt−J model with hopping to the first and second neighbors -- the one dimensional t1−t2−Jt_1-t_2-Jt1​−t2​−J model. The full phase diagram as a function of the density nnn and exchange strength JJJ, for both positive and negative values of t2t_2t2​, is obtained. For t2=−0.5t_2=-0.5t2​=−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 JJJ 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=0.5t_2=0.5t2​=0.5, the physics is basically reminiscent of the conventional t−Jt-Jt−J model, undergoing a transition from a metallic to a SC phase as a function of JJJ.
Pair-density-wave (PDW) superconductivity is a state in which the superconducting order parameter modulates at a finite wavevector. Using large-scale density matrix renormalization group, we study the doped altermagnetic Mott insulator in the checkerboard ttt-JJJ model, where altermagnetic exchange anisotropy is encoded microscopically through anisotropic ferromagnetic next-nearest-neighbor exchange. By mapping the ground-state phase diagram as a function of doping and altermagnetic anisotropy, mainly on six-leg cylinders, we identify a transition from a uniform ddd-wave superconducting regime with charge modulation to a PDW regime coexisting with stripe order. In the PDW regime, we report an unconventional wave-vector locking QPDW≈2QStripe\mathbf Q_{\mathrm{PDW}}\approx 2\mathbf Q_{\mathrm{Stripe}}QPDW​≈2QStripe​ along the cylinder direction, in contrast to the conventional relation. Pair correlations reveal a two-scale structure, consisting of short-distance local ddd-wave pairing and long-distance finite-momentum PDW correlations. A symmetry-based Ginzburg--Landau analysis is presented for the observed locking. Our results identify altermagnetism as a strong-coupling, microscopically grounded route to finite-momentum superconductivity in doped Mott insulators.
We study the one-dimensional Hubbard model with nearest-neighbor and next-nearest-neighbor hopping integrals by using the density-matrix renormalization group method and Hartree-Fock approximation. Based on the calculated results for the spin gap, total-spin quantum number, and Tomonaga-Luttinger-liquid parameter, we determine the ground-state phase diagram of the model in the entire filling and wide parameter region. We show that, in contrast to the weak-coupling regime where a spin-gapped liquid phase is predicted in the region with four Fermi points, the spin gap vanishes in a substantial region in the strong-coupling regime. It is remarkable that a large variety of phases, such as the paramagnetic metallic phase, spin-gapped liquid phase, singlet and triplet superconducting phases, and fully polarized ferromagnetic phase, appear in such a simple model in the strong-coupling regime.
We investigate the magnetic order and related strongly correlated effects in a one-dimensional Ising-Kondo lattice with transverse field. This model is the anisotropic limit of the conventional isotropic Kondo lattice model, in the sense that the itinerant electrons interact with the localized magnetic moments via only longitudinal Kondo exchange. Adopting the numerical density-matrix renormalization group method, we map out the ground-state phase diagram in various parameter spaces. Depending on the Kondo coupling and filling number, three distinct phases, including a metallic paramagnetic, a metallic ferromagnetic, and a gapped spin-density wave phase, are obtained. The spin-density wave is characterized by an ordering wave vector, which coincides with the nesting wave vector of the Fermi surface. This makes the corresponding magnetic transition a spin analog of the Peierls transition occurring in the one-dimensional metal. Moreover, by analyzing the momentum distribution function and charge correlation function, the conduction electrons are shown to behave like free spinless fermions in the ferromagnetic phase. We finally discuss the effect of the repulsive Hubbard interaction between conduction electrons. Our paper enriches the Kondo physics and deepens the current understanding of the heavy fermion compounds.
The ground-state phase diagram of the two-dimensional t-J model is investigated in the context of the tensor network algorithm in terms of the graded Projected Entangled-Pair State representation of the ground-state wave functions. There is a line of phase separation between the Heisenberg anti-ferromagnetic state without hole and a hole-rich state. For both J=0.4t and J=0.8t, a systematic computation is performed to identify all the competing ground states for various dopings. It is found that, besides a possible Nagaoka's ferromagnetic state, the homogeneous regime consists of four different phases: one phase with charge and spin density wave order coexisting with a p_x (p_y)-wave superconducting state, one phase with the symmetry mixing of d+s-wave superconductivity in the spin-singlet channel and p_x (p_y)-wave superconductivity in the spin-triplet channel in the presence of an anti-ferromagnetic background, one superconducting phase with extended s-wave symmetry, and one superconducting phase with p_x (p_y)-wave symmetry in a ferromagnetic background.
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.