Continuous Matrix Product Operator (cMPO) for Quantum Thermodynamics

This is a PyTorch implementation of the cMPO approach to finite temperature quantum states. The cMPO approach is applicable to one-dimensional quantum systems with short-range or long-range interactions. This approach is described in the paper "Continuous Matrix Product Operator Approach to Finite Temperature Quantum States" (arXiv link).

Features

• Say NO to Trotter error: thanks to the coutinuous-time formulation
• Simple yet generic: a unified interface to any Hamiltonian with an MPO representation

Bonus

• Real-frequency local spectral functions: analytic continuation is joyful, finally

Example

python power_projection.py -bondD 10 -beta 10 -Gamma 1 -J 1 -resultdir isingdata

we calculate the thermodynamics of the transverse field Ising model $$ H=-J\sum_{\langle i,j\rangle} Z_i Z_j -\Gamma\sum_i X_i, $$ with $J=1,\Gamma=1$ at temperature $\beta=10$ using cMPO-cMPS method with bond dimension $\chi=10$. The calculation results, including the free energy, internal energy, specific heat, and the local susceptibility, along with the checkpoint data files, are automatically saved in the directory isingdata.

• Before running the code, one needs to set the environment variable OMP_NUM_THREADS, namely the number of threads used by PyTorch.
• More models are defined in model.py, and one can investigate the thermodynamical properties of these models by modifying power_projection.py accordingly. To do this, in power_projection.py, find the part construct cMPO

    s = model.spin_half(dtype, device)
    ising = model.ising(Gamma=args.Gamma, J=args.J, dtype=dtype, device=device) # cmpo def
    T = ising.T
    W = ising.W
    ph_leg = ising.ph_leg

​ and simply replace ising by the model that you are interested in. You can also easily simulate your own models by adding them to model.py.

• To compute more quantities, like local observables and dynamical properties, you can use postprocess.py to access the checkpoint data files and calculate these quantities without running your simulation again.

Update since publication

• commit 06b52272 (2020.04.29): original version published in the reference below
• commit 9234604d (2021.03.22)
  • We avoid self iterations in power method, i.e., we no longer use the optimized cMPS in the previous power steps as the initialization for variational compression of the cMPS. Instead, the initialization for the variational compression is always generated by the iterative SVD update. This helps to jump out of local minima during the optimization process.
  • We stabilize the the iterative SVD update (also called MERA update) by adding line-search procedures.

Benchmark

Benchmark with dTRG in transverse field Ising model is shown in the following figure.

As one can see, after the improvement in commit 9234604d, the cMPO method shows the capability to jump out of local minima, and the results can be systematically improved by increasing the bond dimension $\chi$. At both temperatures $\beta=32$ and $\beta=128$, when bond dimensions are large, the accuracy of cMPO method is not inferior to that of dTRG.

To cite

@article{PhysRevLett.125.170604,
  title = {Continuous Matrix Product Operator Approach to Finite Temperature Quantum States},
  author = {Tang, Wei and Tu, Hong-Hao and Wang, Lei},
  journal = {Phys. Rev. Lett.},
  volume = {125},
  issue = {17},
  pages = {170604},
  numpages = {7},
  year = {2020},
  month = {Oct},
  publisher = {American Physical Society},
  doi = {10.1103/PhysRevLett.125.170604},
  url = {https://link.aps.org/doi/10.1103/PhysRevLett.125.170604}
}