Triangular Lattice Hubbard Model Research Articles

Doping the Mott insulating state of the triangular-lattice Hubbard model reveals the Sordi transition

Doping the Mott insulating state of the triangular-lattice Hubbard model reveals the Sordi transition

Triangular lattice Hubbard model physics at intermediate temperatures

Moire systems offer an exciting playground to study many-body effects of strongly correlated electrons in regimes that are not easily accessible in conventional material settings. Motivated by a recent experiment on $\text{WSe}_2/\text{WS}_2$ moire bilayers [Y. Tang et al., Nature 579, 353-358 (2020)], which realizes 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 $t \lesssim T < U$. 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 $\Theta$ with filling, using the reported interaction strength $U/t=20$. We focus on the large increase of $|\Theta|$ on decreasing the density below half filling and the sign change of $\Theta$ at higher fillings, which signals the onset of ferromagnetism. The increase in $|\Theta|$ is also seen in the $t$-$J$ 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 $\Theta$ seen in the experiment.

Mott transition, Widom line, and pseudogap in the half-filled triangular lattice Hubbard model

The Mott transition is observed experimentally in materials that are magnetically frustrated so that long-range order does not hide the Mott transition at finite temperature. The Hubbard model on the triangular lattice at half-filling is a paradigmatic model to study the interplay of interactions and frustration on the normal-state phase diagram. We use the dynamical cluster approximation with continuous time auxiliary field quantum Monte Carlo to solve this model for 1, 4, 6, 12, and 16 site clusters with detailed analysis performed for the 6 site cluster. We show that a) for every cluster there is an inflection point in the double occupancy as a function of interaction, defining a Widom line that extends above the critical point of the first-order Mott transition; b) the presence of this line and the cluster size dependence argue for the observability of the Mott transition at finite temperature in the thermodynamic limit; c) the loss of spectral weight in the metal to Mott insulator transition as a function of temperature and for strong interactions is momentum dependent, the hallmark of a pseudogap. That pseudogap spans a large region of the phase diagram near the Mott transition.

Magnetic excitations, phase diagram, and order-by-disorder in the extended triangular-lattice Hubbard model

The dynamical structure factor is an important observable of quantum magnets but due to numerical and theoretical limitations, it remains a challenge to make predictions for Hubbard-like models beyond one dimension. In this work, we study the magnetic excitations of the triangular lattice Hubbard model including next-nearest-neighbor hopping. Starting from the expected ${120}^{\ensuremath{\circ}}$ and stripe magnetic orders, we compute the magnon spectra within a self-consistent random phase approximation. In the stripe phase, we generically find accidental zero modes related to a classical degeneracy known from the corresponding ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model. We extend the order-by-disorder mechanism to Hubbard systems and show how quantum fluctuations stabilize the stripe order. In addition, the frustration-induced condensation of magnon modes allows us to map out the entire phase diagram which is in remarkable agreement with recent numerical works. We discuss connections to experiments on triangular lattice compounds and the relation of our results to the proposed chiral spin liquid phase.

Tuning Quantum Phase Transitions at Half Filling in 3L−MoTe2/WSe2 Moiré Superlattices

Many sought-after exotic states of matter are known to emerge close to quantum phase transitions, such as quantum spin liquids and unconventional superconductivity. It is, thus, desirable to experimentally explore systems that can be continuously tuned across these transitions. Here, we demonstrate such tunability and the electronic correlation effects in triangular moiré superlattices formed between trilayer MoTe2 and monolayer WSe2 (3L−MoTe2/WSe2). Through transport measurements, we observe an electronic analog of the Pomeranchuk effect at half filling of the first moiré subband, where increasing temperature paradoxically enhances charge localization. At low temperatures the system exhibits the characteristic of a Fermi liquid with strongly renormalized effective mass, suggesting a correlated metal state. The state is highly susceptible to out-of-plane electric and magnetic fields, which induce a Lifshitz transition and a metal-insulator transition (MIT), respectively. It enables identification of a tricritical point in the quantum phase diagram at the base temperature. We explain the Lifshitz transition in terms of interlayer charge transfer under the vertical electric field, which leads to the emergence of a new Fermi surface and immediate suppression of the Pomeranchuk effect. The existence of quantum criticality in the magnetic-field-induced MIT is supported by scaling behaviors of the resistance. Our work shows the 3L−MoTe2/WSe2 lies in the vicinity to the MIT point of the triangular lattice Hubbard model, rendering it a unique system to manifest the rich correlation effects at an intermediate interaction strength.Received 27 February 2022Revised 10 August 2022Accepted 14 September 2022DOI:https://doi.org/10.1103/PhysRevX.12.041015Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasCritical phenomenaDensity of statesElectrical conductivityFermi surfaceHall effectLandau levelsMagnetotransportPhase diagramsQuantum phase transitionsQuantum transportTransport phenomenaCondensed Matter, Materials & Applied Physics

Magnetic excitations, phase diagram and order-by-disorder in the extended triangular-lattice Hubbard model

Magnetic excitations, phase diagram and order-by-disorder in the extended triangular-lattice Hubbard model

Skyrmion and vortex crystals in the Hubbard model

A mutual interplay between the charge and spin degrees of freedom in itinerant magnets leads to a plethora of topological spin textures, such as magnetic skyrmion and vortex crystals, in both centrosymmetric and noncentrosymmetric hosts. Meanwhile, their stabilization has been extensively studied in the system including the classical localized spins. We here study a realization of the skyrmion crystal in the centrosymmetric triangular-lattice Hubbard model, where the itinerant nature of electrons plays a more significant role. By performing the self-consistent mean-field calculations, we find that two types of skyrmion crystals with spatially nonuniform charge modulations appear in the ground state at a zero magnetic field. Moreover, we obtain another noncoplanar vortex crystal phase without a net scalar chirality in the vicinity of the skyrmion crystal phase. We show that the latter vortex crystal exhibits the topological phase transition to a different skyrmion phase in an applied magnetic field. Our results provide a possibility of the skyrmion and vortex crystals in itinerant magnets without the localized moments.

Competing instabilities of the extended Hubbard model on the triangular lattice: Truncated-unity functional renormalization group and application to moiré materials

A simple yet paradigmatic model for the interplay of strong electronic correlations and geometric frustration is the triangular lattice Hubbard model. Recently it was proposed that moir\'e structures of transition metal dichalcogenides can be used to simulate extended versions that include non-local density-density interactions. We study competing instabilities of interacting electrons in such an extended Hubbard model on the triangular lattice near a filling where the density of states has a Van Hove singularity. We employ a truncated-unity functional renormalization group approach to investigate two cases: a paradigmatic minimally extended Hubbard model and a specific model with parameters that are applicable to hetero-bilayers of transition metal dichalcogenides. We unravel rich phase diagrams, including tendencies to spin-density-wave order and unconventional pairing, which can give rise to topological superconductivity. We classify the symmetry of the superconducting instabilities according to their irreducible representations and show that higher lattice harmonics are dominant when the nearest-neighbor interaction is sizable indicating pair formation between second-nearest neighbors. The phenomenological consequences can be enhanced spin and thermal quantum Hall responses in a topological superconductor.

Quantum spin liquid with emergent chiral order in the triangular-lattice Hubbard model

The interplay between spin frustration and charge fluctuation gives rise to an exotic quantum state in the intermediate-interaction regime of the half-filled triangular-lattice Hubbard model, while the nature of the state is under debate. Using the density matrix renormalization group with SU(2)$_{\rm{spin}} \otimes $U(1)$_{\rm{charge}}$ symmetries implemented, we study the triangular-lattice Hubbard model defined on the long cylinder geometry up to circumference $W=6$. A gapped quantum spin liquid, with on-site interaction $9 \lesssim U / t \lesssim 10.75$, is identified between the metallic and the antiferromagnetic Mott insulating phases. In particular, we find that this spin liquid develops a robust long-range spin scalar-chiral correlation as the system length $L$ increases, which unambiguously unveils the spontaneous time-reversal symmetry breaking. In addition, the degeneracy of the entanglement spectrum supports symmetry fractionalization and spinon edge modes in the obtained ground state. The possible origin of chiral order in this intermediate spin liquid and its relation to the rotonlike excitations have also been discussed.

Functional renormalization of spinless triangular-lattice fermions: N-patch vs. truncated-unity scheme

We study competing orders of spinless fermions in the triangular-lattice Hubbard model with nearest-neighbor interaction. We calculate the effective, momentum-resolved two-particle vertex in an unbiased way in terms of the functional renormalization group method and compare two different schemes for the momentum discretization, one based on dividing the Fermi surface into patches and one based on a channel decomposition. We study attractive and repulsive nearest-neighbor interaction and find a competition of pairing and charge instabilities. In the attractive case, a Pomeranchuk instability occurs at Van Hove filling and f-wave and p-wave pairing emerge when the filling is reduced. In the repulsive case, we obtain a charge density wave at Van Hove filling and extended p-wave pairing with reduced filling. The p-wave pairing solution is doubly degenerate and can realize chiral p+ip superconductivity with different Chern numbers in the ground state. We discuss implications for strongly correlated spin-orbit coupled hexagonal electron systems such as moiré heterostructures.Graphic

Site-resolved observables in the doped spin-imbalanced triangular Hubbard model

The suppression of antiferromagnetic ordering in geometrically frustrated Hubbard models leads to a variety of exotic quantum phases including quantum spin liquids and chiral states. Here, we focus on the Hubbard model on one of the simplest frustrated lattice geometries, a triangular lattice. Motivated by the recent realization of ultracold fermionic atoms in triangular optical lattices, we study the properties of the triangular-lattice Hubbard model through a Numerical Linked-Cluster Expansion algorithm. We investigate the Mott insulator transition finding a critical interaction $U_c/t = 7.0(2)$ and use spatial two- and three-point correlation functions to explore doped and imbalanced systems. Our results demonstrate that many interesting features occur at temperatures previously obtained for ultracold fermions in optical lattices and are accessible by upcoming experiments. Our calculations will be helpful for thermometry in ultracold atom quantum simulators and can guide experimental searches for exotic quantum phases in atomic triangular Hubbard quantum simulators.

Doped Mott insulators in the triangular-lattice Hubbard model

We investigate the evolution of the Mott insulators in the triangular lattice Hubbard Model, as a function of hole doping $\delta$ in both the strong and intermediate coupling limits. Using the advanced density matrix renormalization group (DMRG) method, at light hole doping $\delta\lesssim 10\%$, 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, $\delta \approx 10\%\sim 20\%$ SDW order and coherent hole Fermi pockets are found at both strong and intermediate couplings. At even higher doping $\delta \gtrsim 20\%$, 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.

Effect of hole doping on the 120 degree order in the triangular lattice Hubbard model: a Hartree–Fock revisit

We revisit the unrestricted Hartree Fock study on the evolution of the ground state of the Hubbard model on the triangular lattice with hole doping. At half-filling, it is known that the ground state of the Hubbard model on triangular lattice develops a 120 degree coplanar order at half-filling in the strong interaction limit, i.e., in the spin 1/2 anti-ferromagnetic Heisenberg model on the triangular lattice. The ground state property in the doped case is still in controversy even though extensive studies were performed in the past. Within Hartree Fock theory, we find that the 120 degree order persists from zero doping to about 0.3 hole doping. At 1/3 hole doping, a three-sublattice collinear order emerges in which the doped hole is concentrated on one of the three sublattices with antiferromagnetic Neel order on the remaining two sublattices, which forms a honeycomb lattice. Between the 120 degree order and 1/3 doping region, a phase separation occurs in which the 120 degree order coexists with the collinear anti-ferromagnetic order in different regions of the system. The collinear phase extends from 1/3 doping to about 0.41 doping, beyond which the ground state is paramagnetic with uniform electron density. The phase diagram from Hartree Fock could provide guidance for the future study of the doped Hubbard model on triangular lattice with more sophisticated many-body approaches.

Hole Spectral Function of a Chiral Spin Liquid in the Triangular Lattice Hubbard Model

Quantum spin liquids are fascinating phases of matter, hosting fractionalized spin excitations and unconventional long-range quantum entanglement. These exotic properties, however, also render their experimental characterization challenging and finding ways to diagnose quantum spin liquids is therefore a pertinent challenge. Here, we numerically compute the spectral function of a single hole doped into the half-filled Hubbard model on the triangular lattice using techniques based on matrix product states. At half filling the system has been proposed to realize a chiral spin liquid at intermediate interaction strength, surrounded by a magnetically ordered phase at strong interactions and a superconducting/metallic phase at weak interactions. We find that the spectra of these phases exhibit distinct signatures. By developing appropriate parton mean-field descriptions, we gain insight into the relevant low energy features. While the magnetic phase is characterized by a dressed hole moving through the ordered spin background, we find indications of spinon dynamics in the chiral spin liquid. Our results suggest that the hole spectral function, as measured by Angle-Resolved Photoemission Spectroscopy (ARPES), provides a useful tool to characterize quantum spin liquids.

Mott Insulating States with Competing Orders in the Triangular Lattice Hubbard Model

Interacting electrons on triangular lattice structures arrange themselves in various competing magnetic and possible topological orders, leading to paradoxical electron localization as temperature increases.

Four-Spin Terms and the Origin of the Chiral Spin Liquid in Mott Insulators on the Triangular Lattice.

At strong repulsion, the triangular-lattice Hubbard model is described by s=1/2 spins with nearest-neighbor antiferromagnetic Heisenberg interactions and exhibits conventional 120° order. Using the infinite density matrix renormalization group and exact diagonalization, we study the effect of the additional four-spin interactions naturally generated from the underlying Mott-insulator physics of electrons as the repulsion decreases. Although these interactions have historically been connected with a gapless ground state with emergent spinon Fermi surface, we find that, at physically relevant parameters, they stabilize a chiral spin liquid (CSL) of Kalmeyer-Laughlin (KL) type, clarifying observations in recent studies of the Hubbard model. We then present a self-consistent solution based on a mean-field rewriting of the interaction to obtain a Hamiltonian with similarities to the parent Hamiltonian of the KL state, providing a physical understanding for the origin of the CSL.

Bosonic resonating valence bond theory of the possible chiral spin-liquid state in the triangular-lattice Hubbard model

Recent numerical simulations find a possible chiral spin-liquid state in the intermediate coupling regime of the triangular lattice Hubbard model. Here we provide a simple picture for its origin in terms of a bosonic resonating valence bond (RVB) description. More specifically, we show that such a chiral spin-liquid state can be understood as a quantum disordered tetrahedral spin state stabilized by the four spin ring exchange coupling, which suppresses the order-by-disorder effect toward a stripy spin state. Such a chiral spin-liquid state features a spin Berry phase of $\frac{\ensuremath{\pi}}{2}$ per triangle. However, we show that the topological property of such a state is totally missed in the Schwinger boson mean-field description as a result of the lack of boson rigidity caused by the no double occupancy constraint.

Erratum: Magnetic and charge susceptibilities in the half-filled triangular lattice Hubbard model [Phys. Rev. Research 2 , 013295 (2020)

Received 13 July 2021DOI:https://doi.org/10.1103/PhysRevResearch.3.039001Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasFrustrated magnetismMagnetic phase transitionsMagnetic susceptibilityMagnonsPhase transitions by orderQuantum phase transitionsSpin dynamicsTechniquesHubbard modelCondensed Matter, Materials & Applied PhysicsGeneral Physics

Phase diagram of the anisotropic triangular lattice Hubbard model

In a recent study [Phys. Rev. X 10, 021042 (2020)], we showed using large-scale density matrix renormalization group (DMRG) simulations on infinite cylinders that the triangular lattice Hubbard model has a chiral spin liquid phase. In this work, we introduce hopping anisotropy in the model, making one of the three distinct bonds on the lattice stronger or weaker compared with the other two. We implement the anisotropy in two inequivalent ways, one which respects the mirror symmetry of the cylinder and one which breaks this symmetry. In the full range of anisotropy, from the square lattice to weakly coupled one-dimensional chains, we find a variety of phases. Near the isotropic limit we find the three phases identified in our previous work: metal, chiral spin liquid, and 120$^\circ$ spiral order; we note that a recent paper suggests the apparently metallic phase may actually be a Luther-Emery liquid, which would also be in agreement with our results. When one bond is weakened by a relatively small amount, the ground state quickly becomes the square lattice N\'{e}el order. When one bond is strengthened, the story is much less clear, with the phases that we find depending on the orientation of the anisotropy and on the cylinder circumference. While our work is to our knowledge the first DMRG study of the anisotropic triangular lattice Hubbard model, the overall phase diagram we find is broadly consistent with that found previously using other methods, such as variational Monte Carlo and dynamical mean field theory.

Light-induced switching of magnetic order in the anisotropic triangular-lattice Hubbard model

The time-dependent exact-diagonalization method is used to study the light-induced phase transition of magnetic orders in the anisotropic triangular-lattice Hubbard model. Calculating the spin correlation function, we confirm that the phase transition from the 120$^{\circ}$ order to the N\'{e}el order can take place due to high-frequency periodic fields. We show that the effective Heisenberg-model Hamiltonian derived from the high-frequency expansion by the Floquet theory describes the present system very well and that the ratio of the exchange interactions expressed in terms of the frequency and amplitude of the external field determines the type of the magnetic orders. Our results demonstrate the controllability of the magnetic orders by tuning the external field.