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SimpleMPS

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This is a simplest demo of Matrix product state (MPS) based density matrix renormalization group (DMRG) with detailed comments, particularly suitable for new-learners to quickly get the idea of how MPS works at code level. For sophisticated and production-ready MPS packages, I recommend pytenet and quimb.

What is SimpleMPS

Density matrix renormalization group (DMRG) is a powerful method to simulate one-dimensional strongly correlated quantum systems. During recent years, combining DMRG with matrix product state (MPS) has allowed further understanding of the DMRG method. SimpleMPS aims to provide a demo for MPS based DMRG to solve the Heisenberg model. The ground state of the Heisenberg model could be obtained iteratively (see the image below, $h=1$, $J=J_z=1$, 20 sites). Heisenberg energy profile

The phase transition of transverse field Ising (TFI) model is also provided as a further example (see the image below, 20 sites. Analytical result is based on periodic boundary condition). Ising phase transition Ising phase diagram

The implementation is largely inspired by The density-matrix renormalization group in the age of matrix product states. Understanding the first 6 parts of the article is crucial to understanding the code.

A Julia alternative to the project can be found here.

Files

  • ./SimpleMPS/mps.py Implements the MPS based DMRG method
  • ./SimpleMPS/paulimat.py Defines the Pauli matrices used in Heiserberg and Ising model
  • ./SimpleMPS/heisenberg.py Defines the hamiltonian and matrix product operator of the Heisenberg model then searches for the ground state
  • ./SimpleMPS/ising.py Defines the hamiltonian and matrix product operator of the transverse field Ising model and then studies the phase diagram and phase transition.
  • ./SimpleMPS/test_heisenberg.py Contains a simple test for the programs based on Heisenberg model.
  • ./Notebook/analyze_ising.ipynb Analyzes the result calculated by ./SimpleMPS/ising.py

How to use

  • python heisenberg.py to see the energy of the system during each iteration. There are 3 parameters hard-coded in heisenberg.py:
    • SITE_NUM which is the number of sites in the model.
    • MAX_BOND_DIMENSION which is the maximum dimension of the bond degree in matrix product state. The higher bond dimension, the higher accuracy and computational cost.
    • ERROR_THRESHOLD which is the threshold for error when compressing matrix product state by SVD. The lower the threshold, the higher accuracy and computational cost.
    • One and only one of MAX_BOND_DIMENSION or ERROR_THRESHOLD should be provided.
  • python ising.py to see the relation between $M_x$ and $h$ when $J$ is fixed to 1 and $h$ varies from 0 to 2. The parameters in ising.py is the same as those in heisenberg.py. If you want to calculate the phase diagram, call calc_phase_diagram in the file, which typically takes hours.
  • Modify the codes to explore DMRG and MPS!