Quantum Ising model 1D

In the present repository, we describe the study of the one-dimensional quantum Ising model through numerical simulation. In particular, by implementing various diagonalization techniques, the energy of the ground state and the behavior of the energy gaps between the ground state and the first two excited levels around the quantum phase transition are obtained. Furthermore, the behavior of certain physical quantities of the system around the transition is analyzed, and the effect of finite-size scaling is examined. Finally, the integrability of the transverse field Ising model is verified by examining the statistics of the level spacing.

Repository Structure

The structure of the repository is as follows:

• class_lapacke.h and class_Eigen.h:

  These headers, located in the include directory, contain the definition of a spin Hamiltonian operator class based on the vector and Eigen library respectively. The first implements the ZHEEV subroutine for the exact diagonalization, while the second relies on Eigen::matriX's methods. In both cases, sparse diagonalization is implemented via the lambda_lanczos library.

• ising_main.cpp:

  This program calls the simulation subroutine for all the values of the transverse field and sides, collecting energy gaps, magnetization, and susceptibility measures in the Data_Ising folder. It uses the class_lapacke.h Hamiltonian, which has been proven to be faster.

• level_spacing.cpp:

  This program executes the exact diagonalization of the desired spin model and saves the Eigenvalues in the Data_Ising folder. It uses the class_Eigen.h Hamiltonian, which has been proven to be more suitable for large matrix allocation and implements a random field on the longitudinal axis to break some symmetries.

• ising_plot_all.py and level_histogram.py:

  These programs utilize the data in the Data_Ising folder to plot the physical quantities of interest.

• Heisenberg:

  This folder contains materials that may be used to simulate the Heisenberg model properties.

• Plots_and_fit:

  All produced plots are stored in this folder. Here we can find the plots of the energy gaps and the istograms of the level spacing statistics.

• Tests:

  This directory contains easy-to-use examples for testing the classes and verifying that different diagonalization methods return the same results.

Analysis Results

Here are some of the plots generated from the analysis:

• Order parameter of the phase transition:

  

• First energy gap:

  

• Level spacing statistics:

  

Feel free to explore the repository and use the provided programs for further analysis and investigation.

License

This repository is licensed under the GNU General Public License v3.0 (GPL-3.0).

See the LICENSE file for more information.