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arXiv:2305.04035 · Physical review. B./Physical review. B

Supervised learning of an interacting two-dimensional hardcore boson model of a weak topological insulator using correlation functions

Amrita Ghosh, Mugdha Sarkar

Published Oct 15, 2024• 2 citations
arXivHTMLPDF

Abstract

We study a system of hard-core bosons on a two-dimensional periodic honeycomb lattice subjected to an on-site potential with alternating signs along $y$ direction, using machine learning (ML) techniques. The model hosts a rich phase diagram consisting of six different phases including a charge density wave, a superfluid phase and two dimer insulator phases, one of which is also a weak topological insulator with zero Chern number but a nontrivial Berry phase [Ghosh et al., SciPost Phys. 10, 059 (2021)]. Using two distinct correlation functions computed via quantum Monte Carlo method, a relatively simple ML model is able to learn information from the various phases simultaneously and accurately predict their phase boundaries. By employing our ML model trained on the data set obtained from the Hamiltonian without explicit interactions, we determine the phase structure of the system in the presence of nearest-neighbor interactions. Additionally, we investigate the robustness of the weak topological insulator phase against interactions by predicting the topological invariant, which is otherwise difficult to obtain.

DOI
10.1103/physrevb.110.165134

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We study a system of hard-core bosons on a two-dimensional periodic honeycomb lattice subjected to an on-site potential with alternating signs along yyy direction, using machine learning (ML) techniques. The model hosts a rich phase diagram consisting of six different phases including a charge density wave, a superfluid phase and two dimer insulator phases, one of which is also a weak topological insulator with zero Chern number but a nontrivial Berry phase [Ghosh et al., SciPost Phys. 10, 059 (2021)]. Using two distinct correlation functions computed via quantum Monte Carlo method, a relatively simple ML model is able to learn information from the various phases simultaneously and accurately predict their phase boundaries. By employing our ML model trained on the data set obtained from the Hamiltonian without explicit interactions, we determine the phase structure of the system in the presence of nearest-neighbor interactions. Additionally, we investigate the robustness of the weak topological insulator phase against interactions by predicting the topological invariant, which is otherwise difficult to obtain.
We study the phases of hard-core bosons on a two-dimensional periodic honeycomb lattice in the presence of an on-site potential with alternating sign along the different y-layers of the lattice. Using quantum Monte Carlo simulations supported by analytical calculations, we identify a weak topological insulator, characterized by a zero Chern number but non-zero Berry phase, which is manifested at either density 1/4 or 3/4, as determined by the potential pattern. Additionally, a charge-density-wave insulator is observed at 1/2-filling, whereas the phase diagram at intermediate densities is occupied by a superfluid phase. The weak topological insulator is further shown to be robust against any amount of nearest-neighbor repulsion, as well as weak next-nearest-neighbor repulsion. The experimental realization of our model is feasible in an optical lattice setup.
The topological properties of hardcore bosons on ribbons of honeycomb lattice are studied using quantum Monte Carlo simulations. We map out a rich phase diagram with the superfluid and insulator phases at various fillings. Particularly, it is revealed that the insulator state at half filling is a topological bosonic state, which is characterized by a nontrivial Berry phase and a pair of end states. We provide an intuitive picture to understand this topological bosonic insulator state by showing that it can be adiabatically mapped to a topological fermionic model. It is also shown that the topological end states are robust against weak interactions beyond the hardcore repulsion. Our results can be simulated by using bosonic cold atoms trapped in designed optical lattices.
We study hard-core bosons on the honeycomb lattice in the presence of anisotropic nearest-neighbor repulsive interactions. Using a quantum Monte Carlo (QMC) technique, we extract the phase diagram of the model in terms of the filling and the anisotropy. At half-filling we find a dimer insulator phase near maximum anisotropy that is characterized by a finite topological entanglement entropy ln(2)/2ln(2)/2ln(2)/2, indicative of a fractional quantum Hall state for bosons. We identify the presence of edge states and derive a QMC-based method to extract and verify their chirality. Remarkably, this phase arises in the absence of magnetic flux and without explicit lattice frustration.
The recent advances in machine learning algorithms have boosted the application of these techniques to the field of condensed matter physics, in order e.g. to classify the phases of matter at equilibrium or to predict the real-time dynamics of a large class of physical models. Typically in these works, a machine learning algorithm is trained and tested on data coming from the same physical model. Here we demonstrate that unsupervised and supervised machine learning techniques are able to predict phases of a non-exactly solvable model when trained on data of a solvable model. In particular, we employ a training set made by single-particle correlation functions of a non-interacting quantum wire and by using principal component analysis, k-means clustering, t-distributed stochastic neighbor embedding and convolutional neural networks we reconstruct the phase diagram of an interacting superconductor. We show that both the principal component analysis and the convolutional neural networks trained on the data of the non-interacting model can identify the topological phases of the interacting model. Our findings indicate that non-trivial phases of matter emerging from the presence of interactions can be identified by means of unsupervised and supervised techniques applied to data of non-interacting systems.
We study hard-core bosons on the honeycomb lattice subjected to anisotropic nearest-neighbor hopping along with anisotropic nearest-neighbor repulsion, using a quantum Monte Carlo technique. At half filling, we find a transition from strong topological interacting order to weak topological interacting order as a function of the hopping anisotropy. The strong topological phase is characterized by a finite topological entanglement entropy, while the weak topological order is identified with a nontrivial value of the bipartite entanglement entropy. Some of the order parameters and their derivatives demonstrate abrupt changes when varying the parameters controlling the lattice anisotropies, thus revealing the nature of this topological interacting phase transition.
We study strong correlation effects in topological insulators via the Lanczos algorithm, which we utilize to calculate the exact many-particle ground-state wave function and its topological properties. We analyze the simple, noninteracting Haldane model on a honeycomb lattice with known topological properties and demonstrate that these properties are already evident in small clusters. Next, we consider interacting fermions by introducing repulsive nearest-neighbor interactions. A first-order quantum phase transition was discovered at finite interaction strength between the topological band insulator and a topologically trivial Mott insulating phase by use of the fidelity metric and the charge-density-wave structure factor. We construct the phase diagram at T=0T=0T=0 as a function of the interaction strength and the complex phase for the next-nearest-neighbor hoppings. Finally, we consider the Haldane model with interacting hard-core bosons, where no evidence for a topological phase is observed. An important general conclusion of our work is that despite the intrinsic nonlocality of topological phases their key topological properties manifest themselves already in small systems and therefore can be studied numerically via exact diagonalization and observed experimentally, e.g., with trapped ions and cold atoms in optical lattices.