PurificationTEBD2¶
full name: tenpy.algorithms.purification.PurificationTEBD2
parent module:
tenpy.algorithms.purification
type: class
Inheritance Diagram
Methods
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Initialize self. |
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see |
Disentangle theta before splitting with svd. |
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Try global disentangling by determining the maximally entangled pairs of sites. |
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Perform a sweep through the system and disentangle with |
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Generalization of |
(Real-)time evolution with TEBD (time evolving block decimation). |
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TEBD algorithm in imaginary time to find the ground state. |
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Run imaginary time evolution to cool down to the given beta. |
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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 bonds in unit cell. |
Class Attributes and Properties
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For each bond the total number of iterations performed in any |
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truncation error introduced on each non-trivial bond. |
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class
tenpy.algorithms.purification.
PurificationTEBD2
(psi, model, options)[source]¶ Bases:
tenpy.algorithms.purification.PurificationTEBD
Similar as PurificationTEBD, but perform sweeps instead of brickwall.
Instead of the A-B pattern of even/odd bonds used in TEBD, perform sweeps similar as in DMRG for real-time evolution (similar as
update_imag()
does for imaginary 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 bonds in unit cell.
Depending on the choice of odd, perform a sweep to the left or right, updating once per site with a time step given by U_idx_dt.
- 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
-
property
disent_iterations
¶ For each bond the total number of iterations performed in any
Disentangler
.
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disentangle
(theta)[source]¶ Disentangle theta before splitting with svd.
For the purification we write \(\rho_P = Tr_Q{|\psi_{P,Q}><\psi_{P,Q}|}\). Thus, we can actually apply any unitary to the auxiliar Q space of \(|\psi>\) without changing the result.
Note
We have to apply the same unitary to the ‘bra’ and ‘ket’ used for expectation values / correlation functions!
The behaviour of this function is set by
used_disentangler
, which in turn is obtained fromget_disentangler(options['disentangle'])
, seeget_disentangler()
for details on the syntax.- Parameters
theta (
Array
) – Wave function to disentangle, with legs'vL', 'vR', 'p0', 'p1', 'q0', 'q1'
.- Returns
theta_disentangled (
Array
) – Disentangled theta;npc.tensordot(U, theta, axes=[['q0*', 'q1*'], ['q0', 'q1']])
.U (
Array
) – The unitary used to disentangle theta, with labels'q0', 'q1', 'q0*', 'q1*'
. If no unitary was found/applied, it might also beNone
.
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disentangle_global
(pair=None)[source]¶ Try global disentangling by determining the maximally entangled pairs of sites.
Caclulate the mutual information (in the auxiliar space) between two sites and determine where it is maximal. Disentangle these two sites with
disentangle()
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disentangle_global_nsite
(n=2)[source]¶ Perform a sweep through the system and disentangle with
disentangle_n_site()
.- Parameters
n (int) – maximal number of sites to disentangle at once.
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disentangle_n_site
(i, n, theta)[source]¶ Generalization of
disentangle()
to n sites.Simply group left and right n/2 physical legs, adjust labels, and apply
disentangle()
to disentangle the central bond. Recursively proceed to disentangle left and right parts afterwards. Scales (for even n) as \(O(\chi^3 d^n d^{n/2})\).
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run
()[source]¶ (Real-)time evolution with TEBD (time evolving block decimation).
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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.
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option
TEBD
.
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.
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option
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run_imaginary
(beta)[source]¶ Run imaginary time evolution to cool down to the given beta.
Note that we don’t change the norm attribute of the MPS, i.e. normalization is preserved.
- Parameters
beta (float) – The inverse temperature beta = 1/T, by which we should cool down. We evolve to the closest multiple of
options['dt']
, see alsoevolved_time
.
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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
-
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
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property
trunc_err_bonds
¶ truncation error introduced on each non-trivial bond.
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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. This would look something like:
| | | | ... - B1 - s - B2 - ... | | | | |-------------| | | U | | |-------------| | | |
- Parameters
- Returns
trunc_err – The error of the represented state which is introduced by the truncation during this update step.
- 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
-
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
-