Accurate Numerical Computation Of Frustrated Magnet And Twisted Bilayer Graphene

Exotic quantum states,such as spin ice or spin liquid,have been found in geo-metrically frustrated magnets both theoretically and experimentally.However,it is still a big challenge to make a direct and accurate comparison between theoretical calcu-lation and experimental measurement for such quantum frustrated systems.Recently,a novel quantum frustrated magnet,Tm Mg Ga O₄(TMGO),has been successfully syn-thesized in the laboratory.The unique geometrical structure and atomic properties of TMGO make it possible to describe its magnetic properties with a quantum Ising model on a triangular lattice,which provides an opportunity to realize the direct comparison between theory and experiment.We use exponential tensor renormalization group nu-merical method to fit the experimental measurements of TMGO,then obtain a set of quantum Ising model parameters.From these parameters,we further calculate the spin spectra with quantum Monte Carlo and stochastic analytic continuation method,and found that the numerical spectra is very well consistent with inelastic neutron scatter-ing results of TMGO,which means the quantum Ising model parameters we fitting is correct and accurate.Furthermore,we predict that there is most probably a Kosterlitz-Thouless phase transition in TMGO,and our prediction is verified by nuclear magnetic resonance experiment measurements that give a plateau-like spin-lattice relaxation rate as function of temperature.In addition,our numerical calculation result of spin-lattice relaxation rate obtaining with quantum Monte Carlo and stochastic analytic continua-tion method is consistent with experimental results.The plateau-like shape means the magnetic fluctuation is very strong in this phase,which is expected as unique property of Kosterlitz-Thouless phase transition.The Kosterlitz-Thouless mechanism,building upon vortex pair excitation in a two-dimensional spin system,is the first example of phase transition beyond the Landau-Ginzburg paradigm of symmetry breaking.Such a mechanism had never been discovered directly in quantum magnetic crystal materi-als before our work was published.Our works provide an ideal platform for studying the mechanism of Kosterlitz-Thouless in magnetic crystal materials.In addition,we further use stochastic series expansion quantum Monte Carlo and density matrix renor-malization group numerical method to investigate the phase diagram of the quantum Ising model on a triangular lattice spanned by temperature and external magnetic field.We find that there is a two-dimensional Potts phase transition in the model,and the con-tinuous phase transition will be transformed into a weak first order through a tricritical point.This work may provide guidance for future experimental studies on TMGO.The discovery of strongly-correlated insulator with topological properties and un-conventional superconductor in magic angle twisted bilayer graphene provides a unique platform for the study of strongly-correlated electronic properties and topological be-havior in two-dimensional systems.Theoretical progress on magic angle twisted bilayer graphene have derived two kinds of extended interaction effective lattice model in real space,which are predicted to capture the low-energy physical properties of the novel material approximately.Following these progress,we use the unbiased large-scale pro-jector quantum Monte Carlo numerical method to study the ground state properties of the two kinds of lattice model and give their ground state phase diagrams accurately.For the first effective lattice model,our numerical results show that the ground state phase diagram exhibits-by gradually increasing the interaction strength-a Dirac semi-metal,a plaquette valence bond solid and a columnar valence bond solid phases in turn.The phase transition from Dirac semi-metal to plaquette valence bond solid is continuous and belongs to the 3D=4 Gross-Nevue chiral XY universal class.On the contrary,the plaquette valence bond solid and the columnar valence bond solid phases are in-sulators,and the phase transition between them is first order.For the second effective lattice model,in the case of small interaction,the ground state is a quantum valley Hall phase,which is two copies of Haldane model,leading to a Chern number of±1 for the two different valleys.With the increase of interaction strength,an intervalley coherent state and columnar valence bond solid state appear successively,and phase diagram fi-nally returns to intervalley coherent state at strong-coupling limit.Intervalley coherent state is ferromagnetic-like with translational symmetry.The three ground states are all insulators,and the phase transitions between them are all first order.Our numerical results have direct relevance with theoretical and experimental results of magic angle twisted bilayer graphene and are expected to indicate the electronic structure of the cor-related insulator state in low-energy region.Our works may also help to understand the perplexing electronic behaviors of magic angle twisted bilayer graphene.