Engine¶

full name: tenpy.algorithms.tebd.Engine
parent module: tenpy.algorithms.tebd
type: class

class tenpy.algorithms.tebd.Engine(psi, model, TEBD_params)[source]¶

Bases: object

Time Evolving Block Decimation (TEBD) ‘engine’.

Parameters

psiMPS: Initial state to be time evolved. Modified in place.
modelNearestNeighborModel: The model representing the Hamiltonian for which we want to find the ground state.
TEBD_paramsdict: Further optional parameters as described in the tables in run() and run_GS() for more details. Use verbose=1 to print the used parameters during runtime.

Attributes

verboseint: Level of verbosity (i.e. how much status information to print); higher=more output.
evolved_timefloat | complex: Indicating how long psi has been evolved, psi = exp(-i * evolved_time * H) psi(t=0).
trunc_errTruncationError: The error of the represented state which is introduced due to the truncation during the sequence of update steps.
psiMPS: The MPS, time evolved in-place.
modelNearestNeighborModel: The model defining the Hamiltonian.
TEBD_params: dict: Optional parameters, see run() and run_GS() for more details.
_Ulist of list of Array: Exponentiated H_bond (bond Hamiltonians), i.e. roughly exp(-i H_bond dt_i). First list for different dt_i as necessary for the chosen order, second list for the L different bonds.
_U_paramdict: A dictionary containing the information of the latest created _U. We don’t recalculate _U if those parameters didn’t change.
_trunc_err_bondslist of TruncationError: The local truncation error introduced at each bond, ignoring the errors at other bonds. The i-th entry is left of site i.
_update_indexNone | (int, int): The indices i_dt,i_bond of U_bond = self._U[i_dt][i_bond] during update_step.

Methods

`calc_U`(self, order, delta_t[, type_evo, …])	Calculate `self.U_bond` from `self.bond_eig_{vals,vecs}`.
`run`(self)	(Real-)time evolution with TEBD (time evolving block decimation).
`run_GS`(self)	TEBD algorithm in imaginary time to find the ground state.
`suzuki_trotter_decomposition`(order, N_steps)	Returns list of necessary steps for the suzuki trotter decomposition.
`suzuki_trotter_time_steps`(order)	Return time steps of U for the Suzuki Trotter decomposition of desired order.
`update`(self, N_steps)	Evolve by `N_steps * U_param['dt']`.
`update_bond`(self, i, U_bond)	Updates the B matrices on a given bond.
`update_bond_imag`(self, i, U_bond)	Update a bond with a (possibly non-unitary) U_bond.
`update_imag`(self, N_steps)	Perform an update suitable for imaginary time evolution.
`update_step`(self, U_idx_dt, odd)	Updates either even or odd bonds in unit cell.

property trunc_err_bonds¶: truncation error introduced on each non-trivial bond.

run(self)[source]¶

(Real-)time evolution with TEBD (time evolving block decimation).

The following (optional) parameters are read out from the TEBD_params.

key	type	description
dt	float	Time step.
order	int	Order of the algorithm. The total error scales as O(t, dt^order).
N_steps	int	Number of time steps dt to evolve. (The Trotter decompositions of order > 1 are slightly more efficient if more than one step is performed at once.)
trunc_params	dict	Truncation parameters as described in `truncate()`.

run_GS(self)[source]¶

TEBD algorithm in imaginary time to find the ground state.

Note

It is almost always more efficient (and hence advisable) to use DMRG. This algorithms can nonetheless be used quite well as a benchmark and for comparison.

The following (optional) parameters are read out from the TEBD_params. Use verbose=1 to print the used parameters during runtime.

key	type	description
delta_tau_list	list	A list of floats: the timesteps to be used. Choosing a large timestep delta_tau introduces large (Trotter) errors, but a too small time step requires a lot of steps to reach `exp(-tau H) --> \|psi0><psi0\|`. Therefore, we start with fairly large time steps for a quick time evolution until convergence, and the gradually decrease the time step.
order	int	Order of the Suzuki-Trotter decomposition.
N_steps	int	Number of steps before measurement can be performed
trunc_params	dict	Truncation parameters as described in `truncate()`

static suzuki_trotter_time_steps(order)[source]¶

Return time steps of U for the Suzuki Trotter decomposition of desired order.

See suzuki_trotter_decomposition() for details.

Parameters

orderint: The desired order of the Suzuki-Trotter decomposition.

Returns

time_stepslist of float: We need U = exp(-i H_{even/odd} delta_t * dt) for the dt returned in this list.

static suzuki_trotter_decomposition(order, N_steps)[source]¶

Returns list of necessary steps for the suzuki trotter decomposition.

We split the Hamiltonian as \(H = H_{even} + H_{odd} = H[0] + H[1]\). The Suzuki-Trotter decomposition is an approximation \(\exp(t H) \approx prod_{(j, k) \in ST} \exp(d[j] t H[k]) + O(t^{order+1 })\).

Parameters

orderint: The desired order of the Suzuki-Trotter decomposition.

Returns

ST_decompositionlist of (int, int): Indices j, k of the time-steps d = suzuki_trotter_time_step(order) and the decomposition of H. They are chosen such that a subsequent application of exp(d[j] t H[k]) to a given state |psi> yields (exp(N_steps t H[k]) + O(N_steps t^{order+1}))|psi>.

calc_U(self, order, delta_t, type_evo='real', E_offset=None)[source]¶

Calculate self.U_bond from self.bond_eig_{vals,vecs}.

This function calculates

U_bond = exp(-i dt (H_bond-E_offset_bond)) for type_evo='real', or
U_bond = exp(- dt H_bond) for type_evo='imag'.

For first order (in delta_t), we need just one dt=delta_t. Higher order requires smaller dt steps, as given by suzuki_trotter_time_steps().

Parameters

orderint: Trotter order calculated U_bond. See update for more information.
delta_tfloat: Size of the time-step used in calculating U_bond
type_evo'imag' | 'real': Determines whether we perform real or imaginary time-evolution.
E_offsetNone | list of float: Possible offset added to H_bond for real-time evolution.

update(self, N_steps)[source]¶

Evolve by N_steps * U_param['dt'].

Parameters

N_stepsint: The number of steps for which the whole lattice should be updated.

Returns

trunc_errTruncationError: The error of the represented state which is introduced due to the truncation during this sequence of update steps.

update_step(self, U_idx_dt, odd)[source]¶

Updates either even or odd bonds in unit cell.

Depending on the choice of p, this function updates all even (E, odd=False,0) or odd (O) (odd=True,1) bonds:

|     - B0 - B1 - B2 - B3 - B4 - B5 - B6 -
|       |    |    |    |    |    |    |
|       |    |----|    |----|    |----|
|       |    |  E |    |  E |    |  E |
|       |    |----|    |----|    |----|
|       |----|    |----|    |----|    |
|       |  O |    |  O |    |  O |    |
|       |----|    |----|    |----|    |

Note that finite boundary conditions are taken care of by having Us[0] = None.

Parameters

U_idx_dtint: Time step index in self._U, evolve with Us[i] = self.U[U_idx_dt][i] at bond (i-1,i).
oddbool/int: Indication of whether to update even (odd=False,0) or even (odd=True,1) sites

Returns

trunc_errTruncationError: The error of the represented state which is introduced due to the truncation during this sequence of update steps.

update_bond(self, i, U_bond)[source]¶

Updates the B matrices on a given bond.

Function that updates the B matrices, the bond matrix s between and the bond dimension chi for bond i. The correponding tensor networks look like this:

|           --S--B1--B2--           --B1--B2--
|                |   |                |   |
|     theta:     U_bond        C:     U_bond
|                |   |                |   |

Parameters

iint: Bond index; we update the matrices at sites i-1, i.
U_bondArray: The bond operator which we apply to the wave function. We expect labels 'p0', 'p1', 'p0*', 'p1*'.

Returns

trunc_errTruncationError: The error of the represented state which is introduced by the truncation during this update step.

update_imag(self, N_steps)[source]¶

Perform an update suitable for imaginary time evolution.

Instead of the even/odd brick structure used for ordinary TEBD, we ‘sweep’ from left to right and right to left, similar as DMRG. Thanks to that, we are actually able to preserve the canonical form.

Parameters

N_stepsint: The number of steps for which the whole lattice should be updated.

Returns

trunc_errTruncationError: The error of the represented state which is introduced due to the truncation during this sequence of update steps.

update_bond_imag(self, i, U_bond)[source]¶

Update a bond with a (possibly non-unitary) U_bond.

Similar as update_bond(); but after the SVD just keep the A, S, B canonical form. In that way, one can sweep left or right without using old singular values, thus preserving the canonical form during imaginary time evolution.

Parameters

iint: Bond index; we update the matrices at sites i-1, i.
U_bondArray: The bond operator which we apply to the wave function. We expect labels 'p0', 'p1', 'p0*', 'p1*'.

Returns

trunc_errTruncationError: The error of the represented state which is introduced by the truncation during this update step.