Topological and magnetic properties of the interacting Bernevig-Hughes-Zhang model

Rahul Soni, Harini Radhakrishnan, Bernd Rosenow, Gonzalo Alvarez, and Adrian Del Maestro

Abstract

We investigate the effects of electronic correlations on the Bernevig-Hughes-Zhang model using the real-space density matrix renormalization group (DMRG) algorithm. We introduce a method to probe topological phase transitions in systems with strong correlations using DMRG, substantiated by an unsupervised machine learning methodology that analyzes the orbital structure of the real-space edges. Including the full multi-orbital Hubbard interaction term, we construct a phase diagram as a function of a gap parameter ($m$) and the Hubbard interaction strength ($U$) via exact DMRG simulations on $N\times 4$ cylinders. Our analysis confirms that the topological phase persists in the presence of interactions, consistent with previous studies, but it also reveals an intriguing phase transition from a paramagnetic to an antiferromagnetic topological insulator. The combination of the magnetic structure factor, strength of magnetic moments, and the orbitally resolved density, provides real-space information on both topology and magnetism in a strongly correlated system.

Description

This repository includes data files, scripts, codes and analysis used to generate the figures in this paper.

Requirements

The data in this project is generated using two different methods:

1. The data for the interacting BHZ model was generated via DMRG++ software developed by Dr. Gonzalo Alvarez. Here we provide detailed instructions on how to reproduce the DMRG results used in the main text. The results reported in this work were obtained with the DMRG++ version 6.05 and PsimagLite version 3.04. The documentation for the same is provided here, for compilation follow the steps below:
2. Dependencies include the BOOST, HDF5 and OpenBLAS libraries

Obtaining the Code and Compilation

git clone https://code.ornl.gov/gonzalo_3/PsimagLite.git
cd PsimagLite/lib
perl configure.pl
make
cd ../../
git clone https://code.ornl.gov/gonzalo_3/dmrgpp.git
cd dmrgpp/src
perl configure.pl
make

Running the Code

This will generate dmrg and observe executables. Run the dmrg executable first to save the ground state and then use the observe executable to evaluate all the necessary observables. See the steps below:

./dmrg -f input.inp
./observe -f input.inp ss,nn >> ss_nn.dat (this will provide all the spin-spin and charge-charge corrlations)
./observe -f input.inp '<gs|n?0|gs>' >>nl_0.dat

This will provide the local charge density for orbital-0 with spin-up. Similarly, one can observe the charge density for spin-down by replacing n?0 with n?1. For spin-up and spin-dn charge density of orbital-1 do observe n?2 and n?3.)

To generate input file, go to the 'input_files' folder and run the following command :
python Create_input_dmrg.py M_val

2. The data for the non-interacting BHZ model was generated via exact diagonalization. The code can be found here. Detail instructions are provided in this repo regarding compilations, executions and more.

Support

This work was supported by the U. S. Department of Energy, Office of Science, Office of Basic Energy Sciences,under Award No. DE-SC0022311.

Figures

Figure 1: $6\times 4$ Cylindrical Geometry

Figure 2: Nearest-Neighbor Hopping Connections

Figure 3(a,b): Topological Phase Transition at $U=0$

Figure 3(c): Numbering and Bulk-Sites in $6\times 4$ Geometry

Figure 4: Magnetic Properties at $m=3.0$

Figure 5: Magnetic Properties at $m=1.75$

Figure 6: Topological Properties at $m=3.0$

Figure 7: Topological Properties at $m=1.75$

Figure 8: Topological and Magnetic phase diagram using DMRG

Appendix Figure 1: Single Site Picture

Appendix Figure 2: Finite Size Scaling-A

Appendix Figure 3: Finite Size Scaling-B