We investigate the interface separating two topologically distinct insulating phases of matter using extensive density-matrix renormalization group calculations to study the triangular-lattice Hofstadter-Hubbard model with a spatially varying interaction strength, chosen to realize both integer quantum Hall and chiral spin liquid states in different spatial regions. We find that the integer quantum Hall-chiral spin liquid interface hosts a spinless charged excitation that is bound to the interface. This mode at the interface is identified through charge and spin pumping, and by direct calculations of low-lying excited states. We also characterize bulk excitations in both phases, finding evidence for fractionalization in the chiral spin liquid and for spin-triplet exciton formation in the integer quantum Hall phase.
We investigate the interface separating two topologically distinct insulating phases of matter using extensive density-matrix renormalization group calculations to study the triangular-lattice Hofstadter-Hubbard model with a spatially varying interaction strength, chosen to realize both integer quantum Hall and chiral spin liquid states in different spatial regions. We find that the integer quantum Hall-chiral spin liquid interface hosts a spinless charged excitation that is bound to the interface. This mode at the interface is identified through charge and spin pumping, and by direct calculations of low-lying excited states. We also characterize bulk excitations in both phases, finding evidence for fractionalization in the chiral spin liquid and for spin-triplet exciton formation in the integer quantum Hall phase.
We investigate the weak interaction integer quantum Hall (IQH) phase, the intermediate interaction phase identified as a chiral spin liquid (CSL) and the transition between them in the triangular lattice Hofstadter-Hubbard model at a density of one electron per site in an orbital magnetic field corresponding to one-quarter flux per plaquette. Our primary tool is the finite system density matrix renormalization group (DMRG) method with both interaction-strength scan and fixed interaction techniques for cylinders of circumference 3, 5, and 7 and lengths up to 240. For the IQH phase, we use single particle exact diagonalization to clarify finite size effects, including an excess charge on the edges of our cylinders, and the limitations of entanglement spectra degeneracies on small circumference cylinders. For both phases, we use DMRG to study the entanglement spectra, the entanglement entropy, and the effect of flux insertion on charge and spin pumping, all of which show key differences between the two phases. To study the transition, we use interaction-strength scans extending between the two phases, and apply a scaling data collapse of a bond-dimerization order parameter to extract critical exponents. We also extract critical behavior from the divergence of correlation lengths on the IQH side, measuring decay away from edges of both the dimerization order parameter and transverse edge currents. The critical behavior and exponents are consistent with an Ising transition in 1+1 dimensions. Finally, we obtain excited states in various quantum number sectors finding that the gap to a charge neutral momentum π excitation corresponding to fluctuations of the dimerization order parameter closes in the vicinity of the critical point but gaps to other excitations remain large.
Topological Criticality with Charge-3e Fluctuations in the SU(3) Triangular-Lattice Hofstadter--Hubbard Model
A transition between an integer quantum Hall insulator and a chiral spin liquid can become critical without closing the single-electron gap, raising the question of which composite excitations govern the change in topological order. We investigate this question in the SU(3) triangular-lattice Hofstadter--Hubbard model at one particle per site and magnetic flux π/3 per triangle. Using infinite density matrix renormalization group calculations, we identify a weak-coupling integer quantum Hall phase and an interaction-driven SU(3)1 chiral spin liquid. Near the transition, both the neutral and charge-3e correlation lengths increase substantially, whereas the single-electron correlation length remains short, providing evidence for a Chern--Simons--Higgs critical theory, where a gauge-invariant monopole carries the quantum numbers of a fermionic SU(3)-singlet trion. We discuss the implications for doping the system and recent experiments on doping ν=−2/3 fractional quantum anomalous Hall in tMoTe2.
Motivated by experimental studies that have found signatures of a quantum spin liquid phase in organic crystals whose structure is well described by the two-dimensional triangular lattice, we study the Hubbard model on this lattice at half filling using the infinite-system density matrix renormalization group (iDMRG) method. On infinite cylinders with finite circumference, we identify an intermediate phase between observed metallic behavior at low interaction strength and Mott insulating spin-ordered behavior at strong interactions. Chiral ordering from spontaneous breaking of time-reversal symmetry, a fractionally quantized spin Hall response, and characteristic level statistics in the entanglement spectrum in the intermediate phase provide strong evidence for the existence of a chiral spin liquid in the full two-dimensional limit of the model.
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.
The interplay between topology and strong interactions gives rise to a variety of exotic quantum phases, including fractional quantum Hall states and their lattice analogs—fractional Chern insulators (FCIs). Such topologically ordered states host fractionalized excitations and, for spinful systems, are often accompanied by ferromagnetism and skyrmions. Here, we study a Hofstadter-Hubbard model of spinful fermions on a square lattice, extended by nearest-neighbor interactions. Using large-scale density matrix renormalization group simulations, we demonstrate the emergence of a spin-polarized 1 / 3 -Laughlin-like FCI phase, characterized by a quantized many-body Chern number, a finite charge gap, and hidden off-diagonal long-range order. We further investigate the quantum Hall ferromagnet at ν = 1 and its skyrmionic excitations upon doping. In particular, we find that nearest-neighbor repulsion is sufficient to stabilize both particle- and hole-skyrmions in the ground state around ν = 1 , whereas we do not find such textures around ν = 1 / 3 . The diagnostic toolbox presented in this work, based on local densities, correlation functions, and spin-resolved observables, is directly applicable in quantum gas microscopy experiments. Our results open new pathways for experimental exploration of FCIs with spin textures in both ultracold atom and electronic systems.
The Bose-Hubbard model subjected to an effective magnetic field hosts a plethora of phases with different topological orders when tuning the chemical potential. Using the density matrix renormalization group method, we identify several gapped phases near the first Mott lobe at strong interactions. They are connected by a particle-hole symmetry to a variety of quantum Hall states stabilized at low fillings. We characterize phases of both particle and hole type and identify signatures compatible with Laughlin, Moore-Read, and bosonic integer quantum Hall states by calculating the quantized Hall conductance and by extracting the topological entanglement entropy. Furthermore, we analyze the entanglement spectrum of Laughlin states of bosonic particles and holes for a range of interaction strengths, as well as the entanglement spectrum of a Moore-Read state. These results further corroborate the existence of topological states at high fillings, close to the first Mott lobe, as hole analogs of the respective low-filling states.