Time-dependent variational principle (TDVP)#

TDVP algorithm is suitable to perform exponentiation of some operator \(\hat H\) acting on a state \(\Psi(t_0)\), producing \(\Psi(t)=T e^{-u\int^{t_1}_{t_0} \hat H(w) dw} \Psi(t_0)\), with \(T\) indicating time ordering. The algorithm allows to use MPO operator H, which can be either hermitian or non-hermitian, and possibly time-dependent. The state psi is given as MPS or MPO.

A high-level function organizing TDVP simulations is yastn.tn.mps.tdvp_(). See examples at Sudden quench in a free-fermionic model and Quench across a quantum critical point in a transverse Ising chain.

The TDVP splits the exponentiation into local operation acting on 1 or 2 sites depending on method='1site' or '2site' similar to DMRG, following Suzuki-Trotter decomposition of given order. The '2site', although more expensive, is one of the ways to dynamically expand the MPS virtual dimensions, controlled by opts_svd. We also provide method='12site' that performs 2-site operations only for bonds that have the potential to be expanded, based on their bond dimension and Schmidt spectrum.

yastn.tn.mps.tdvp_(psi, H, times=(0, 0.1), dt=0.1, u=1j, method='1site', order='2nd', opts_expmv=None, opts_svd=None, normalize=True, subtract_E=False, precompute=False, progressbar=False, yield_initial=False)[source]#

Iterator performing TDVP sweeps to solve \(\frac{d}{dt} |\psi(t)\rangle = -uH|\psi(t)\rangle\),

Parameters:

  • psi (Mps) – Initial state. It is updated during execution. It is first canonized to the first site, if not provided in such a form. Resulting state is also canonized to the first site.

  • H (yastn.tn.mps.MpsMpoOBC | Sequence | Callable) – Evolution generator given either as (sum of) MPO for time-independent problem or as a function returning (sum of) MPO for time-dependent problem, i.e. Callable[[float], Mpo] or Callable[[float], Sequence[Mpo]], see Env().

  • time (float64 or tuple(float64)) – Initial and final times; can also provide intermediate times for snapshots returned by the iterator. If only the final time is provided, initial time is set to 0.

  • dt (double) – Time step. It is adjusted down to have an integer number of time-steps to reach the next snapshot.

  • u (number) – for real time evolution u=1j, for imaginary time evolution u=1. The default is 1j.

  • method (str) – Algorithm to use among ‘1site’, ‘2site’, ’12site’.

  • order (str) – Order of Suzuki-Trotter decomposition, ‘2nd’ or ‘4th’. 4th order step is composed of five 2nd order steps.

  • opts_expmv (dict) – Options passed to yastn.expmv(), in particular whether H is hermitian. Unspecified options use default yastn.expmv() settings. If there is information from previous time-steps stored under the hood, the initial guess of the size of krylov space opts_expmv[‘ncv’] is overriden.

  • opts_svd (dict) – Options passed to yastn.linalg.svd() used to truncate virtual spaces in method='2site' and '12site'.

  • normalize (bool) – Whether to normalize the result to unity using 2-norm. If False, keeps track of the norm. The default is True.

  • subtract_E (bool) – Whether to subtract the instantaneous energy from the Hamiltonian while performing local updates. The default is False.

  • precompute (bool) – Controls MPS-MPO-MPS contraction order. If False, use an approach optimal for a single matrix-vector product calculation during iterative Krylov-space building, scaling as O(D^3 M d + D^2 M^2 d^2). If True, uses less optimal contraction order, scaling as O(D^3 M d^2 + D^2 M^2 d^2). However, the latter allows precomputing and reusing part of the diagram during consecutive iterations. Which one is more efficient depends on the parameters. The default is False.

  • progressbar (bool) – Whether to show the progress bar toward the next snapshot. The default is False.

  • yield_initial (bool) – Whether to yield the initial state before performing evolution. The default is False.

Returns:

NamedTuple with fields:

  • ti initial time of the time-interval.

  • tf current time.

  • time_independent whether the Hamiltonian is time-independent.

  • dt time-step used.

  • steps number of time-steps in the last time-interval.

Return type:

TDVP_out(NamedTuple)

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