TEBDEngine¶
full name: tenpy.algorithms.tebd.TEBDEngine
parent module:
tenpy.algorithms.tebd
type: class
Inheritance Diagram
Methods
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Initialize self. |
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Calculate |
Return necessary data to resume a |
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Resume a run that was interrupted. |
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Run TEBD real time evolution by N_steps`*`dt. |
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TEBD algorithm in imaginary time to find the ground state. |
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Returns list of necessary steps for the suzuki trotter decomposition. |
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Return time steps of U for the Suzuki Trotter decomposition of desired order. |
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Evolve by |
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Updates the B matrices on a given bond. |
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Update a bond with a (possibly non-unitary) U_bond. |
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Perform an update suitable for imaginary time evolution. |
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Updates either even or odd bonds in unit cell. |
Class Attributes and Properties
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truncation error introduced on each non-trivial bond. |
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class
tenpy.algorithms.tebd.
TEBDEngine
(psi, model, options, **kwargs)[source]¶ Bases:
tenpy.algorithms.algorithm.TimeEvolutionAlgorithm
Time Evolving Block Decimation (TEBD) algorithm.
Deprecated since version 0.6.0: Renamed parameter/attribute TEBD_params to
options
.- Parameters
psi (
MPS
) – Initial state to be time evolved. Modified in place.model (
NearestNeighborModel
) – The model representing the Hamiltonian for which we want to find the ground state.options (dict) – Further optional parameters as described below.
Options
-
config
TEBDEngine
¶ option summary delta_tau_list in PurificationTEBD.run_GS
A list of floats: the timesteps to be used. [...]
dt (from TimeEvolutionAlgorithm) in TimeEvolutionAlgorithm
Minimal time step by which to evolve.
N_steps in PurificationTEBD.run_GS
Number of steps before measurement can be performed
order in PurificationTEBD.run_GS
Order of the Suzuki-Trotter decomposition.
start_time (from TimeEvolutionAlgorithm) in TimeEvolutionAlgorithm
Initial value for :attr:`evolved_time`.
Initial truncation error for :attr:`trunc_err`.
trunc_params (from Algorithm) in Algorithm
Truncation parameters as described in :cfg:config:`truncation`.
-
option
start_trunc_err
:TruncationError
¶ Initial truncation error for
trunc_err
.
-
option
order
: int¶ Order of the algorithm. The total error for evolution up to a fixed time t scales as
O(t*dt^order)
.
-
option
-
trunc_err
¶ The error of the represented state which is introduced due to the truncation during the sequence of update steps.
- Type
-
model
¶ The model defining the Hamiltonian.
- Type
-
_U
¶ 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.- Type
list of list of
Array
-
_U_param
¶ A dictionary containing the information of the latest created _U. We don’t recalculate _U if those parameters didn’t change.
- Type
-
_trunc_err_bonds
¶ The local truncation error introduced at each bond, ignoring the errors at other bonds. The i-th entry is left of site i.
- Type
list of
TruncationError
-
_update_index
¶ The indices
i_dt,i_bond
ofU_bond = self._U[i_dt][i_bond]
during update_step.
-
property
trunc_err_bonds
¶ truncation error introduced on each non-trivial bond.
-
run_GS
()[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.
-
option
TEBDEngine
.
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.
-
option
TEBDEngine
.
order
: int¶ Order of the Suzuki-Trotter decomposition.
-
option
TEBDEngine
.
N_steps
: int¶ Number of steps before measurement can be performed
-
option
-
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
order (int) – The desired order of the Suzuki-Trotter decomposition.
- Returns
time_steps – We need
U = exp(-i H_{even/odd} delta_t * dt)
for the dt returned in this list.- Return type
list of float
-
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
order (
1, 2, 4, '4_opt'
) – The desired order of the Suzuki-Trotter decomposition. Order1
approximation is simply \(e^A a^B\). Order2
is the “leapfrog” e^{A/2} e^B e^{A/2}. Order4
is the fourth-order from [suzuki1991] (also referenced in [schollwoeck2011]), and'4_opt'
gives the optmized version of Equ. (30a) in [barthel2020].- Returns
ST_decomposition – Indices
j, k
of the time-stepsd = suzuki_trotter_time_step(order)
and the decomposition of H. They are chosen such that a subsequent application ofexp(d[j] t H[k])
to a given state|psi>
yields(exp(N_steps t H[k]) + O(N_steps t^{order+1}))|psi>
.- Return type
-
calc_U
(order, delta_t, type_evo='real', E_offset=None)[source]¶ Calculate
self.U_bond
fromself.bond_eig_{vals,vecs}
.This function calculates
U_bond = exp(-i dt (H_bond-E_offset_bond))
fortype_evo='real'
, orU_bond = exp(- dt H_bond)
fortype_evo='imag'
.
For first order (in delta_t), we need just one
dt=delta_t
. Higher order requires smaller dt steps, as given bysuzuki_trotter_time_steps()
.- Parameters
order (int) – Trotter order calculated U_bond. See update for more information.
delta_t (float) – Size of the time-step used in calculating U_bond
type_evo (
'imag' | 'real'
) – Determines whether we perform real or imaginary time-evolution.E_offset (None | list of float) – Possible offset added to H_bond for real-time evolution.
-
update
(N_steps)[source]¶ Evolve by
N_steps * U_param['dt']
.- Parameters
N_steps (int) – The number of steps for which the whole lattice should be updated.
- Returns
trunc_err – The error of the represented state which is introduced due to the truncation during this sequence of update steps.
- Return type
-
update_step
(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_dt (int) – Time step index in
self._U
, evolve withUs[i] = self.U[U_idx_dt][i]
at bond(i-1,i)
.odd (bool/int) – Indication of whether to update even (
odd=False,0
) or even (odd=True,1
) sites
- Returns
trunc_err – The error of the represented state which is introduced due to the truncation during this sequence of update steps.
- Return type
-
update_bond
(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
- Returns
trunc_err – The error of the represented state which is introduced by the truncation during this update step.
- Return type
-
update_imag
(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_steps (int) – The number of steps for which the whole lattice should be updated.
- Returns
trunc_err – The error of the represented state which is introduced due to the truncation during this sequence of update steps.
- Return type
-
update_bond_imag
(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
- Returns
trunc_err – The error of the represented state which is introduced by the truncation during this update step.
- Return type
-
get_resume_data
()[source]¶ Return necessary data to resume a
run()
interrupted at a checkpoint.At a
checkpoint
, you can savepsi
,model
andoptions
along with the data returned by this function. When the simulation aborts, you can resume it using this saved data with:eng = AlgorithmClass(psi, model, options, resume_data=resume_data) eng.resume_run(resume_data)
An algorithm which doesn’t support this should override resume_run to raise an Error.
- Returns
resume_data – Dictionary with necessary data (apart from copies of psi, model, options) that allows to continue the simulation from where we are now.
- Return type
-
resume_run
()[source]¶ Resume a run that was interrupted.
In case we saved an intermediate result at a
checkpoint
, this function allows to resume therun()
of the algorithm (after re-initialization with the resume_data). Since most algorithms just have a while loop with break conditions, the default behaviour implemented here is to just callrun()
.