U1cMPO

Performing a tensor network simulation of the nonlinear sigma model (NLSM) with $\theta=\pi$ topological term with Julia.

This repository includes

• a package that implements $U(1)$ symmetry in the continuous matrix product operator (cMPO) method (see repo for cMPO)
• Simulation and measurements for the NLSM with $\theta=\pi$.

Installation

• Download the repository by

   git clone https://github.com/TensorBFS/U1cMPO.git

• Enter the repository, and open Julia's interactive session (known as REPL). Press ] to enter the pkg mode, and then type the following command

   pkg> activate .
   pkg> instantiate

  then exit the REPL.

Usage

To simulate the NLSM with $\theta=\pi$ term with the cMPO method, we map it to a modified quantum rotor model $$ a \hat{H} = \sum_j \frac{(\hat{\boldsymbol{L}}'j)^2}{2 K} + K \sum{\langle i, j \rangle} \hat{\boldsymbol{n}}_i \cdot \hat{\boldsymbol{n}}_j ,. $$ where $\hat{\boldsymbol{L}}'_j$ is the angular momentum operator decorated by a magnetic monopole, $\hat{\boldsymbol{n}}$ is the rotor operator, $a$ is the lattice spacing, and $K>0$ is a constant. In the low-energy and long-wavelength limit, the field theoretical description of this rotor model is the $O(3)$ NLSM with $\theta=\pi$ and $1/g^2=K$. The tensor network representation of this modified quantum rotor model is derived using the monopole harmonics basis. More details can be found in this manuscript.

To run the simulation, enter the repository and type the following command

julia --project=. o3nlsm/o3nlsm_u1cmps.jl --beta 8 --K 2 --doublelmax 3 --chi 8

which will perform the simulation for $K=2.0$ and inverse temperature $\beta=8$. Moreover, the monopole harmonics basis will be truncated at $l_{\mathrm{max}}=3/2$, and the bond dimension of the boundary cMPS is $\chi=8$. The results (including the optimized cMPS, the free energy, and the bipartite entanglement entropy of the boundary cMPS) will be saved in a hdf5-format file named as rawdata_o3pi_K=2.0_beta=8.00_lmax=1.5_chi=8.jld. With this file, we can either perform further measurements like

julia --project=. o3nlsm/o3nlsm_measurement.jl --dataname rawdata_o3pi_K=2.0_beta=8.00_lmax=1.5_chi=8.jld

or use this datafile to provide the initial guess for the simulations with larger bond dimensions, such as

julia --project=. o3nlsm/o3nlsm_u1cmps.jl --beta 8 --K 2 --doublelmax 3 --chi 12 --init rawdata_o3pi_K=2.0_beta=8.00_lmax=1.5_chi=8.jld

Alternatively, the U1cMPO package can also be used to calculate the finite-temperature properties of other models with $U(1)$ symmetry.

To cite

@article{PhysRevD.104.114513,
  title = {Tensor network simulation of the ($1+1$)-dimensional $O(3)$ nonlinear $\ensuremath{\sigma}$-model with $\ensuremath{\theta}=\ensuremath{\pi}$ term},
  author = {Tang, Wei and Xie, X. C. and Wang, Lei and Tu, Hong-Hao},
  journal = {Phys. Rev. D},
  volume = {104},
  issue = {11},
  pages = {114513},
  numpages = {13},
  year = {2021},
  month = {Dec},
  publisher = {American Physical Society},
  doi = {10.1103/PhysRevD.104.114513},
  url = {https://link.aps.org/doi/10.1103/PhysRevD.104.114513}
}