Lattice models are powerful tools for studying strongly correlated quantum many-body systems, but their general lack of exact solutions motivates efforts to simulate them in tunable platforms. Recently, a promising new candidate has emerged for such platforms from two-dimensional materials. A subset of moiré systems can be effectively described as a two-dimensional electron gas (2D EG) subject to a moiré potential, with electron-electron interactions screened by nearby metallic gates. In this paper, we investigate the realization of lattice models in such systems. We show that, by controlling the gate separation, a 2D EG in a square moiré potential can be systematically tuned into a system whose ground state exhibits orders analogous to those of the square lattice Hubbard model, including the stripe phase. Furthermore, we study how variations in gate separation and moiré potential depth affect the ground-state orders. A number of antiferromagnetic phases, as well as a ferromagnetic phase and a paramagnetic phase, are identified. We then apply our quantitative downfolding approach to triangular moiré systems closer to current experimental conditions, compare them with the square lattice parameters studied, and outline routes for experimental realization of the phases.
Lattice models are powerful tools for studying strongly correlated quantum many-body systems, but their general lack of exact solutions motivates efforts to simulate them in tunable platforms. Recently, a promising new candidate has emerged for such platforms from two-dimensional materials. A subset of moiré systems can be effectively described as a two-dimensional electron gas (2D EG) subject to a moiré potential, with electron-electron interactions screened by nearby metallic gates. In this paper, we investigate the realization of lattice models in such systems. We show that, by controlling the gate separation, a 2D EG in a square moiré potential can be systematically tuned into a system whose ground state exhibits orders analogous to those of the square lattice Hubbard model, including the stripe phase. Furthermore, we study how variations in gate separation and moiré potential depth affect the ground-state orders. A number of antiferromagnetic phases, as well as a ferromagnetic phase and a paramagnetic phase, are identified. We then apply our quantitative downfolding approach to triangular moiré systems closer to current experimental conditions, compare them with the square lattice parameters studied, and outline routes for experimental realization of the phases.
Moiré systems provide a rich platform for studies of strong correlation physics. Recent experiments on heterobilayer transition metal dichalcogenide Moiré systems are exciting in that they manifest a relatively simple model system of an extended Hubbard model on a triangular lattice. Inspired by the prospect of the hetero-transition metal dichalcogenide Moiré system's potential as a solid-state-based quantum simulator, we explore the extended Hubbard model on the triangular lattice using the density matrix renormalization group. Specifically, we explore the two-dimensional phase space spanned by the key tuning parameters in the extended Hubbard model, namely, the kinetic energy strength and the further-range Coulomb interaction strengths. We find competition between Fermi fluid, chiral spin liquid, spin density wave, and charge order. In particular, our finding of the optimal further-range interaction for the chiral correlation presents a tantalizing possibility.
This thesis investigates quantum magnetism in two-dimensional synthetic systems, where strong correlations, geometric frustration, and enhanced quantum fluctuations produce unconventional magnetic phases. I focus on two major classes of engineered platforms: moiré materials and quantum simulators, which offer precise control over lattice geometry, interaction strengths, and dynamics, enabling access to regimes beyond conventional condensed-matter approaches. The first project studies moiré magnetism in strained multilayers of CrBr₃, where strain generates a spatial modulation of the interlayer exchange alternating between ferromagnetic and antiferromagnetic. With S=3/2 moments, the system admits a classical bilayer description with competing interactions. I develop a theoretical framework capturing the resulting non-collinear orders and reproducing experimental signatures, demonstrating how moiré engineering stabilizes non-trivial magnetic textures. The second project examines quantum magnetism in a WSe₂/WS₂ heterobilayer at three-quarters filling, where electrons form an effective kagome lattice. From the underlying Hubbard model, I derive a frustrated Heisenberg Hamiltonian with competing exchanges. Using Schwinger-boson mean-field theory, I map a phase diagram of non-collinear orders and spin-liquid phases. For realistic parameters, the ground state is a chiral spin liquid, highlighting transition-metal dichalcogenide moiré materials as platforms for topological and strongly correlated magnetism. The third project investigates quantum phase-transition dynamics with superconducting circuit quantum simulators, focusing on Hamiltonian ramps in the 2D XY model. An analogue-digital protocol drives the system from a gapped staggered state to the gapless XY phase. Analysis of correlations reveals scaling of the correlation length with ramp rate, broadly consistent with Kibble–Zurek predictions, with deviations reflecting finite-size, dimensionality, and experimental constraints. The fourth project develops a Holstein–Primakoff spin-wave description for the same XY ramps. Adapting the quantization axis to the instantaneous magnetization, this solvable quadratic theory provides magnon dispersions, time- and space-dependent correlations, and single-magnon dynamics, including decay and scattering. This framework offers a detailed picture of non-equilibrium quasiparticle behavior in a 2D magnet under time-dependent fields. Together, these projects show how moiré materials and quantum simulators enable controlled studies of frustrated and strongly correlated magnetism in two dimensions. By combining analytical modeling with experimentally relevant conditions, this thesis advances understanding of how frustration, topology, and non-equilibrium dynamics shape quantum magnetic phenomena in low-dimensional systems.
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) ≫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.
Abstract Moiré superlattices provide a new platform to engineer various many-body problems. In this work, we consider arrays of quantum dots (QD) realized on semiconductor moiré superlattices with a deep moiré potential. We diagonalize single QD with multiple electrons, and find degenerate ground states serving as local degrees of freedom (qudits) in the superlattice. With a deep moiré potential, the hopping and exchange interaction between nearby QDs become irrelevant, and the direct Coulomb interaction of the density–density type dominates. Therefore, nearby QDs must arrange the spatial densities to optimize the Coulomb energy. When the local Hilbert space has a two-fold orbital degeneracy, we find that a square superlattice realizes an anisotropic XY model, while a triangular superlattice realizes a generalized XY model with geometric frustration.
Recent experiments in moiré materials have enabled the realization of a variety of exotic quantum phases. In this context, the Hofstadter-Hubbard model has been proposed as a possible setting for hosting chiral spin liquid. Concurrently, significant progress has been recently made in the computational methods for two-dimensional many-body fermion systems, which makes numerically studying this challenging model a real possibility in genuine 2D geometry. Motivated by these advances, we investigate the putative chiral spin liquid phase in the triangular-lattice Hofstadter-Hubbard model using variational Monte Carlo with neural quantum states (NQS) and projected entangled pair states (PEPS). We observe spin-charge separation directly in real space through numerical spin-pumping simulation and real-time spin and charge motion. In addition, in the context of anyonic superconductivity conjectured in this model, we find a positive two-electron binding energy on small systems, but it decreases below our numerical resolution as the system size increases. Our work demonstrates NQS and PEPS as powerful tools, capable of cross-checking each other, for diagnosing topological order and fractionalized excitations in strongly correlated electronic systems.
Exotic phenomena in strongly correlated electron systems emerge from the interplay between spin and motional degrees of freedom. For example, doping an antiferromagnet is expected to give rise to pseudogap states and high-temperature superconductors. Quantum simulation with ultracold fermions in optical lattices offers the potential to answer open questions about the doped Hubbard Hamiltonian, and has recently been advanced by quantum gas microscopy. In order to take advantage of these possibilities, a stable, high-power, 2D square lattice system was developed and coupled with an image plane based spatial light modulator. The ability to finely tune the atomic potential has made it possible to realize an antiferromagnet in a repulsively interacting Fermi gas on a 2D square lattice. At our lowest temperatures of T/t=0.25(2), antiferromagnetic long-range order (LRO) manifests through the divergence of the correlation length that reaches the size of the system, the development of a peak in the spin structure factor and a value of the staggered magnetization approaching the ground state value. Similarly, by carefully shaping the confinement, we have produced ultra-low entropy band insulators, which promise to be a perfect starting point for more advanced cooling schemes. In addition to the production of new states, Fermi gas microscopy is superbly well suited to studies of many-body dynamics. To this end I report on preliminary measurements of the propagation of holes in a Mott insualtor. These results demonstrate that Fermi gas microscopy can address open questions on the low-temperature Hubbard model.