{"id":20229165,"url":"https://github.com/tensorbfs/u1cmpo","last_synced_at":"2025-10-13T02:24:48.185Z","repository":{"id":115940388,"uuid":"400757713","full_name":"TensorBFS/U1cMPO","owner":"TensorBFS","description":null,"archived":false,"fork":false,"pushed_at":"2022-01-04T15:20:40.000Z","size":151,"stargazers_count":8,"open_issues_count":0,"forks_count":0,"subscribers_count":1,"default_branch":"main","last_synced_at":"2025-03-03T13:24:54.048Z","etag":null,"topics":[],"latest_commit_sha":null,"homepage":null,"language":"Julia","has_issues":true,"has_wiki":null,"has_pages":null,"mirror_url":null,"source_name":null,"license":null,"status":null,"scm":"git","pull_requests_enabled":true,"icon_url":"https://github.com/TensorBFS.png","metadata":{"files":{"readme":"README.md","changelog":null,"contributing":null,"funding":null,"license":null,"code_of_conduct":null,"threat_model":null,"audit":null,"citation":null,"codeowners":null,"security":null,"support":null,"governance":null,"roadmap":null,"authors":null,"dei":null,"publiccode":null,"codemeta":null}},"created_at":"2021-08-28T09:51:54.000Z","updated_at":"2023-10-21T14:19:00.000Z","dependencies_parsed_at":"2023-04-04T12:19:17.767Z","dependency_job_id":null,"html_url":"https://github.com/TensorBFS/U1cMPO","commit_stats":null,"previous_names":[],"tags_count":0,"template":false,"template_full_name":null,"purl":"pkg:github/TensorBFS/U1cMPO","repository_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/TensorBFS%2FU1cMPO","tags_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/TensorBFS%2FU1cMPO/tags","releases_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/TensorBFS%2FU1cMPO/releases","manifests_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/TensorBFS%2FU1cMPO/manifests","owner_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/owners/TensorBFS","download_url":"https://codeload.github.com/TensorBFS/U1cMPO/tar.gz/refs/heads/main","sbom_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/TensorBFS%2FU1cMPO/sbom","scorecard":null,"host":{"name":"GitHub","url":"https://github.com","kind":"github","repositories_count":279014029,"owners_count":26085346,"icon_url":"https://github.com/github.png","version":null,"created_at":"2022-05-30T11:31:42.601Z","updated_at":"2022-07-04T15:15:14.044Z","status":"online","status_checked_at":"2025-10-13T02:00:06.723Z","response_time":61,"last_error":null,"robots_txt_status":"success","robots_txt_updated_at":"2025-07-24T06:49:26.215Z","robots_txt_url":"https://github.com/robots.txt","online":true,"can_crawl_api":true,"host_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub","repositories_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories","repository_names_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repository_names","owners_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/owners"}},"keywords":[],"created_at":"2024-11-14T07:34:32.864Z","updated_at":"2025-10-13T02:24:48.165Z","avatar_url":"https://github.com/TensorBFS.png","language":"Julia","readme":"# U1cMPO\n\n[![CI](https://github.com/TensorBFS/U1cMPO/actions/workflows/ci.yml/badge.svg)](https://github.com/TensorBFS/U1cMPO/actions/workflows/ci.yml) [![codecov](https://codecov.io/gh/TensorBFS/U1cMPO/branch/main/graph/badge.svg?token=TTbadaxozU)](https://codecov.io/gh/TensorBFS/U1cMPO)\n\nPerforming a tensor network simulation of the nonlinear sigma model (NLSM) with $\\theta=\\pi$ topological term with Julia. \n\nThis repository includes\n\n- a package that implements $U(1)$ symmetry in the [continuous matrix product operator (cMPO)](https://arxiv.org/abs/2004.12928) method (see [repo for cMPO](https://github.com/TensorBFS/cMPO))\n- Simulation and measurements for the NLSM with $\\theta=\\pi$. \n\n## Installation\n\n- Download the repository by \n\n\t```bash\n\tgit clone https://github.com/TensorBFS/U1cMPO.git\n\t```\n\t\n- Enter the repository, and open Julia's interactive session (known as REPL). Press `]` to enter the pkg mode, and then type the following command \n\n\t```julia\n\tpkg\u003e activate .\n\tpkg\u003e instantiate\n\t```\n\t\n\tthen exit the REPL. \n\n## Usage\n\nTo simulate the NLSM with $\\theta=\\pi$ term with the cMPO method, we map it to a modified quantum rotor model \n$$\na \\hat{H} = \\sum_j \\frac{(\\hat{\\boldsymbol{L}}'_j)^2}{2 K} + K \\sum_{\\langle i, j \\rangle} \\hat{\\boldsymbol{n}}_i \\cdot \\hat{\\boldsymbol{n}}_j \\,.\n$$\nwhere $\\hat{\\boldsymbol{L}}'_j$ is the angular momentum operator decorated by a magnetic monopole, $\\hat{\\boldsymbol{n}}$ is the rotor operator, $a$ is the lattice spacing, and $K\u003e0$ 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](https://arxiv.org/abs/2109.11324).\n\nTo run the simulation, enter the repository and type the following command\n\n```bash\njulia --project=. o3nlsm/o3nlsm_u1cmps.jl --beta 8 --K 2 --doublelmax 3 --chi 8\n```\n\nwhich 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\n\n```bash\njulia --project=. o3nlsm/o3nlsm_measurement.jl --dataname rawdata_o3pi_K=2.0_beta=8.00_lmax=1.5_chi=8.jld\n```\n\nor use this datafile to provide the initial guess for the simulations with larger bond dimensions, such as\n\n```bash\njulia --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\n```\n\nAlternatively, the U1cMPO package can also be used to calculate the finite-temperature properties of other models with $U(1)$ symmetry.\n\n## To cite\n```latex\n@article{PhysRevD.104.114513,\n  title = {Tensor network simulation of the ($1+1$)-dimensional $O(3)$ nonlinear $\\ensuremath{\\sigma}$-model with $\\ensuremath{\\theta}=\\ensuremath{\\pi}$ term},\n  author = {Tang, Wei and Xie, X. C. and Wang, Lei and Tu, Hong-Hao},\n  journal = {Phys. Rev. D},\n  volume = {104},\n  issue = {11},\n  pages = {114513},\n  numpages = {13},\n  year = {2021},\n  month = {Dec},\n  publisher = {American Physical Society},\n  doi = {10.1103/PhysRevD.104.114513},\n  url = {https://link.aps.org/doi/10.1103/PhysRevD.104.114513}\n}\n```","funding_links":[],"categories":[],"sub_categories":[],"project_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Ftensorbfs%2Fu1cmpo","html_url":"https://awesome.ecosyste.ms/projects/github.com%2Ftensorbfs%2Fu1cmpo","lists_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Ftensorbfs%2Fu1cmpo/lists"}