arXiv:2209.00664 · Physical review. B./Physical review. B

Triangular lattice Hubbard model physics at intermediate temperatures

Kyungmin Lee, P. A. Sharma, Oskar Vafek, Hitesh J. Changlani

Published Jun 2, 2023• 36 citations

Abstract

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 ${\text{WSe}}_{2}/{\text{WS}}_{2}$ 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 $U/t=20$. 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.

DOI: 10.1103/physrevb.107.235105

Related papers

High-temperature kinetic magnetism in triangular lattices

Ivan Morera, Márton Kanász-Nagy, Tomasz Smoleński +3 more

arXiv:2209.05398Sep 12, 20226 citations

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

Jordan Venderley, Eun-Ah Kim

arXiv:1901.11034Physical review. B./Physical review. BAug 26, 201938 citations

Ground-state phase diagram of the triangular lattice Hubbard model by the density-matrix renormalization group method

Tomonori Shirakawa, Takami Tohyama, J. Kokalj +2 more

arXiv:1606.06814Physical review. B./Physical review. BNov 15, 201788 citations

Singlet, triplet, and pair density wave superconductivity in the doped triangular-lattice moiré system

Feng Chen, D. N. Sheng

arXiv:2302.06765Physical review. B./Physical review. BNov 13, 202321 citations

Itinerant magnetism in Hubbard models with long-range interactions

Johannes Dieplinger, Rhine Samajdar, R. N. Bhatt

arXiv:2410.00955Physical review. B./Physical review. BMay 22, 20260 citations

Chiral spin liquid and chiral antiferromagnetism in half-filled moiré Hubbard model: possible applications to twisted bilayer TMDs

Chuyi Tuo, Hong Yao

arXiv:2608.14503Aug 14, 20260 citations

arXiv:2209.00664 · Physical review. B./Physical review. B

Triangular lattice Hubbard model physics at intermediate temperatures

Kyungmin Lee, P. A. Sharma, Oskar Vafek, Hitesh J. Changlani

Published Jun 2, 2023• 36 citations

arXiv HTML PDF

Abstract

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

{\text{WSe}}_{2}/{\text{WS}}_{2}

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=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.

DOI: 10.1103/physrevb.107.235105

Related papers

High-temperature kinetic magnetism in triangular lattices

Ivan Morera, Márton Kanász-Nagy, Tomasz Smoleński +3 more

We study kinetic magnetism for the Fermi-Hubbard models in triangular type lattices, including a zigzag ladder, four- and six-legged triangular cylinders and a full two-dimensional triangular lattice. We focus on the regime of strong interactions, $U\gg t$ and filling factors around one electron per site. For temperatures well above the hopping strength, the Curie-Weiss form of the magnetic susceptibility suggests effective antiferromagnetic correlations for systems that are hole doped with respect to $ν=1$ , and ferromagnetic correlations for systems with electron dopings. We show that these correlations arise from magnetic polaron dressing of charge carrier propagating in a spin incoherent Mott insulator. Effective interactions corresponding to these correlations can strongly exceed the magnetic super-exchange energy. In the case of hole doping, antiferromagnetic polarons originate from kinetic frustration of individual holes in a triangular lattice. In the case of electron doping, Nagaoka type ferromagnetic correlations are induced by propagating doublons. These results provide a theoretical explanation of recent experimental results in moire TMDC materials. To understand many-body states arising from antiferromagentic polarons at low temperatures, we study hole doped systems in finite magnetic fields. At low dopings and intermediate magnetic fields we find a magnetic polaron phase, separated from the fully polarized state by a metamagnetic transition. With decreasing magnetic field the system shows a tendency to phase separate, with hole rich regions forming antiferromagnetic spinbags. We demonstrate that direct observations of magnetic polarons in triangular lattices can be achieved in experiments with ultracold atoms, which allow measurements of three point hole-spin-spin correlations.We study kinetic magnetism for the Fermi-Hubbard models in triangular type lattices, including a zigzag ladder, four- and six-legged triangular cylinders and a full two-dimensional triangular lattice. We focus on the regime of strong interactions, $U\gg t$ and filling factors around one electron per site. For temperatures well above the hopping strength, the Curie-Weiss form of the magnetic susceptibility suggests effective antiferromagnetic correlations for systems that are hole doped with respect to $ν=1$, and ferromagnetic correlations for systems with electron dopings. We show that these correlations arise from magnetic polaron dressing of charge carrier propagating in a spin incoherent Mott insulator. Effective interactions corresponding to these correlations can strongly exceed the magnetic super-exchange energy. In the case of hole doping, antiferromagnetic polarons originate from kinetic frustration of individual holes in a triangular lattice. In the case of electron doping, Nagaoka type ferromagnetic correlations are induced by propagating doublons. These results provide a theoretical explanation of recent experimental results in moire TMDC materials. To understand many-body states arising from antiferromagentic polarons at low temperatures, we study hole doped systems in finite magnetic fields. At low dopings and intermediate magnetic fields we find a magnetic polaron phase, separated from the fully polarized state by a metamagnetic transition. With decreasing magnetic field the system shows a tendency to phase separate, with hole rich regions forming antiferromagnetic spinbags. We demonstrate that direct observations of magnetic polarons in triangular lattices can be achieved in experiments with ultracold atoms, which allow measurements of three point hole-spin-spin correlations.

arXiv:2209.05398Sep 12, 20226 citations

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

Jordan Venderley, Eun-Ah Kim

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.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.

arXiv:1901.11034Physical review. B./Physical review. BAug 26, 201938 citations

Ground-state phase diagram of the triangular lattice Hubbard model by the density-matrix renormalization group method

Tomonori Shirakawa, Takami Tohyama, J. Kokalj +2 more

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 $U>{U}_{\mathrm{c}2}$ , 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 ${U}_{\mathrm{c}1}<U<{U}_{\mathrm{c}2}$ , 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 $\mathrm{K}$ 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.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 $U>{U}_{\mathrm{c}2}$, 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 ${U}_{\mathrm{c}1}<U<{U}_{\mathrm{c}2}$, 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 $\mathrm{K}$ 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.

arXiv:1606.06814Physical review. B./Physical review. BNov 15, 201788 citations

Singlet, triplet, and pair density wave superconductivity in the doped triangular-lattice moiré system

Feng Chen, D. N. Sheng

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 $d$ -wave and a subdominant triplet $p$ -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.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 $d$-wave and a subdominant triplet $p$-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.

arXiv:2302.06765Physical review. B./Physical review. BNov 13, 202321 citations

Itinerant magnetism in Hubbard models with long-range interactions

Johannes Dieplinger, Rhine Samajdar, R. N. Bhatt

A wide variety of experimental platforms, ranging from semiconductor quantum-dot arrays to moiré materials, have recently emerged as powerful quantum simulators for studying the Hubbard model and its variants. Motivated by these developments, here, we investigate a generalization of the Hubbard model which includes the effects of long-range Coulomb interactions. Working on finite-sized two-dimensional square and triangular lattices, we use exact diagonalization and density-matrix renormalization group calculations to probe the magnetic structure of the ground state in the strong-coupling regime, where $U$ (the onsite repulsion) $\gg$ $t$ (the nearest-neighbor hopping). For small electron dopings above the half-filled antiferromagnet, we numerically uncover a rich variety of magnetically ordered states, and in conjunction with theoretical arguments, infer the phase diagram of the system as a function of doping and interaction strengths. In particular, we find that the inclusion of long-range Coulomb interactions induces an instability of high-spin states$\unicode{x2014}$such as the saturated Nagaoka ferromagnet$\unicode{x2014}$towards phase separation and stripe ordering. We also present proposals for the observation of some of our key findings in experiments that would shed further light on this paradigmatic strongly correlated system.A wide variety of experimental platforms, ranging from semiconductor quantum-dot arrays to moiré materials, have recently emerged as powerful quantum simulators for studying the Hubbard model and its variants. Motivated by these developments, here, we investigate a generalization of the Hubbard model which includes the effects of long-range Coulomb interactions. Working on finite-sized two-dimensional square and triangular lattices, we use exact diagonalization and density-matrix renormalization group calculations to probe the magnetic structure of the ground state in the strong-coupling regime, where $U$ (the onsite repulsion) $\gg$ $t$ (the nearest-neighbor hopping). For small electron dopings above the half-filled antiferromagnet, we numerically uncover a rich variety of magnetically ordered states, and in conjunction with theoretical arguments, infer the phase diagram of the system as a function of doping and interaction strengths. In particular, we find that the inclusion of long-range Coulomb interactions induces an instability of high-spin states$\unicode{x2014}$such as the saturated Nagaoka ferromagnet$\unicode{x2014}$towards phase separation and stripe ordering. We also present proposals for the observation of some of our key findings in experiments that would shed further light on this paradigmatic strongly correlated system.

arXiv:2410.00955Physical review. B./Physical review. BMay 22, 20260 citations

Chiral spin liquid and chiral antiferromagnetism in half-filled moiré Hubbard model: possible applications to twisted bilayer TMDs

Chuyi Tuo, Hong Yao

Twisted transition metal dichalcogenides offer an exceptionally tunable moiré platform for studying correlation physics beyond conventional condensed matter systems. In particular, the intriguing interplay between the displacement field and the twist angle remains to be fully resolved. In this paper, we use large-scale density matrix renormalization group simulations to study the minimal moiré Hubbard model on a triangular lattice at half-filling, where the displacement field effect is captured by a spin-dependent staggered flux. We find that the displacement field significantly enriches the triangular Hubbard phase diagram in several qualitative ways. It rapidly destabilizes the chiral spin liquid phase beyond a narrow weak-field regime, induces pronounced chiral correlations in the strong-coupling $120^\circ$ -antiferromagnetic phase, and stabilizes incommensurate spin-density wave phases at weaker coupling. We further find signatures of a continuous transition between the chiral spin liquid and chiral antiferromagnetic phases at a finite displacement field, potentially driven by spinon condensation. Our results uncover rich displacement-field-driven many-body physics and provide useful guidance for future experiments in moiré superlattice systems.Twisted transition metal dichalcogenides offer an exceptionally tunable moiré platform for studying correlation physics beyond conventional condensed matter systems. In particular, the intriguing interplay between the displacement field and the twist angle remains to be fully resolved. In this paper, we use large-scale density matrix renormalization group simulations to study the minimal moiré Hubbard model on a triangular lattice at half-filling, where the displacement field effect is captured by a spin-dependent staggered flux. We find that the displacement field significantly enriches the triangular Hubbard phase diagram in several qualitative ways. It rapidly destabilizes the chiral spin liquid phase beyond a narrow weak-field regime, induces pronounced chiral correlations in the strong-coupling $120^\circ$-antiferromagnetic phase, and stabilizes incommensurate spin-density wave phases at weaker coupling. We further find signatures of a continuous transition between the chiral spin liquid and chiral antiferromagnetic phases at a finite displacement field, potentially driven by spinon condensation. Our results uncover rich displacement-field-driven many-body physics and provide useful guidance for future experiments in moiré superlattice systems.

arXiv:2608.14503Aug 14, 20260 citations

{\text{WSe}}_{2}/{\text{WS}}_{2}

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=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

\mathrm{\ensuremath{\Theta}}

seen in the experiment.

U\gg t

and filling factors around one electron per site. For temperatures well above the hopping strength, the Curie-Weiss form of the magnetic susceptibility suggests effective antiferromagnetic correlations for systems that are hole doped with respect to

ν=1

, and ferromagnetic correlations for systems with electron dopings. We show that these correlations arise from magnetic polaron dressing of charge carrier propagating in a spin incoherent Mott insulator. Effective interactions corresponding to these correlations can strongly exceed the magnetic super-exchange energy. In the case of hole doping, antiferromagnetic polarons originate from kinetic frustration of individual holes in a triangular lattice. In the case of electron doping, Nagaoka type ferromagnetic correlations are induced by propagating doublons. These results provide a theoretical explanation of recent experimental results in moire TMDC materials. To understand many-body states arising from antiferromagentic polarons at low temperatures, we study hole doped systems in finite magnetic fields. At low dopings and intermediate magnetic fields we find a magnetic polaron phase, separated from the fully polarized state by a metamagnetic transition. With decreasing magnetic field the system shows a tendency to phase separate, with hole rich regions forming antiferromagnetic spinbags. We demonstrate that direct observations of magnetic polarons in triangular lattices can be achieved in experiments with ultracold atoms, which allow measurements of three point hole-spin-spin correlations.

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.

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>{U}_{\mathrm{c}2}

, 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

{U}_{\mathrm{c}1}<U<{U}_{\mathrm{c}2}

, 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

\mathrm{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.

d

-wave and a subdominant triplet

p

-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.

U

(the onsite repulsion)

\gg

t

(the nearest-neighbor hopping). For small electron dopings above the half-filled antiferromagnet, we numerically uncover a rich variety of magnetically ordered states, and in conjunction with theoretical arguments, infer the phase diagram of the system as a function of doping and interaction strengths. In particular, we find that the inclusion of long-range Coulomb interactions induces an instability of high-spin states$\unicode{x2014}$such as the saturated Nagaoka ferromagnet$\unicode{x2014}$towards phase separation and stripe ordering. We also present proposals for the observation of some of our key findings in experiments that would shed further light on this paradigmatic strongly correlated system.

120^\circ

-antiferromagnetic phase, and stabilizes incommensurate spin-density wave phases at weaker coupling. We further find signatures of a continuous transition between the chiral spin liquid and chiral antiferromagnetic phases at a finite displacement field, potentially driven by spinon condensation. Our results uncover rich displacement-field-driven many-body physics and provide useful guidance for future experiments in moiré superlattice systems.