MPOEnvironment

• full name: tenpy.networks.mpo.MPOEnvironment

• parent module: tenpy.networks.mpo

• type: class

Inheritance Diagram

Methods

MPOEnvironment.__init__(bra, H, ket[, cache])
MPOEnvironment.cache_optimize([...])	Update short_term_keys for the cache and possibly preload tensors.
MPOEnvironment.clear()	Delete all partial contractions except the left-most LP and right-most RP.
MPOEnvironment.del_LP(i)	Delete stored part strictly to the left of site i.
MPOEnvironment.del_RP(i)	Delete stored part strictly to the right of site i.
MPOEnvironment.full_contraction(i0)	Calculate the energy by a full contraction of the network.
MPOEnvironment.get_LP(i[, store])	Calculate LP at given site from nearest available one (including i).
MPOEnvironment.get_LP_age(i)	Return number of physical sites in the contractions of get_LP(i).
MPOEnvironment.get_RP(i[, store])	Calculate RP at given site from nearest available one (including i).
MPOEnvironment.get_RP_age(i)	Return number of physical sites in the contractions of get_RP(i).
MPOEnvironment.get_initialization_data([...])	Return data for (re-)initialization of the environment.
MPOEnvironment.has_LP(i)	Return True if LP left of site i is stored.
MPOEnvironment.has_RP(i)	Return True if RP right of site i is stored.
MPOEnvironment.init_LP(i[, start_env_sites])	Build an initial left part LP.
MPOEnvironment.init_RP(i[, start_env_sites])	Build initial right part RP for an MPS/MPOEnvironment.
MPOEnvironment.init_first_LP_last_RP([...])	(Re)initialize first LP and last RP from the given data.
MPOEnvironment.set_LP(i, LP, age)	Store part to the left of site i.
MPOEnvironment.set_RP(i, RP, age)	Store part to the right of site i.
MPOEnvironment.test_sanity()	Sanity check, raises ValueErrors, if something is wrong.

class tenpy.networks.mpo.MPOEnvironment(bra, H, ket, cache=None, **init_env_data)

Bases: BaseEnvironment

Stores partial contractions of \(<bra|H|ket>\) for an MPO H.

The network for a contraction \(<bra|H|ket>\) of an MPO H between two MPS looks like:

|     .------>-M[0]-->-M[1]-->-M[2]-->- ...  ->--.
|     |        |       |       |                 |
|     |        ^       ^       ^                 |
|     |        |       |       |                 |
|     LP[0] ->-W[0]-->-W[1]-->-W[2]-->- ...  ->- RP[-1]
|     |        |       |       |                 |
|     |        ^       ^       ^                 |
|     |        |       |       |                 |
|     .------<-N[0]*-<-N[1]*-<-N[2]*-<- ...  -<--.

We use the following label convention (where arrows indicate qconj):

|    .-->- vR           vL ->-.
|    |                        |
|    LP->- wR           wL ->-RP
|    |                        |
|    .--<- vR*         vL* -<-.

See BaseEnvironment for further details.

Parameters:

• bra (MPS) – The MPS to project on. Should be given in usual ‘ket’ form; we call conj() on the matrices directly.

• H (MPO) – The MPO sandwiched between bra and ket. Should have ‘IdL’ and ‘IdR’ set on the first and last bond.

• ket (MPS) – The MPS on which H acts. May be identical with bra.

• **init_env_data – Further keyword arguments with initialization data, as returned by get_initialization_data(). See init_first_LP_last_RP() for details on these parameters.

H

The MPO sandwiched between bra and ket.

Type:

MPO

init_first_LP_last_RP(init_LP=None, init_RP=None, age_LP=0, age_RP=0, start_env_sites=None)

(Re)initialize first LP and last RP from the given data.

If init_LP and init_RP are not given, we try to find sensible initial values. Dummy environments can by built with init_LP() and init_RP(), especially for finite MPS.

For infinite MPS, we try to converge the environments with one of two methods:

• If start_env_sites is given as an integer, contract that many sites into the environment from the given init_LP and init_RP or new trivial environments built with init_LP() / init_RP().

• If start_env_sites is None, and bra is ket, get init_LP and init_RP with MPOTransferMatrix.find_init_LP_RP().

Parameters:

• init_LP (None | Array) – Initial very left part LP and very right part RP. If None, try to build (and converge) them as described above.

• init_RP (None | Array) – Initial very left part LP and very right part RP. If None, try to build (and converge) them as described above.

• age_LP (int) – The number of physical sites involved into the contraction of init_LP and init_RP.

• age_RP (int) – The number of physical sites involved into the contraction of init_LP and init_RP.

• start_env_sites (int | None) – Number of sites over which to converge the environment for infinite systems. See above.

test_sanity()

Sanity check, raises ValueErrors, if something is wrong.

init_LP(i, start_env_sites=0)

Build an initial left part LP.

For start_env_sites > 0, make the assumptions that bra is the same as ket and in canonical form, and that H is a Hamiltonian with the following block-form (up to a permutation of MPO indices; this is the case for any model defined in TeNPy),

\[\begin{split}W = \begin{pmatrix} 1 & C & D \\ 0 & A & B \\ 0 & 0 & 1 \end{pmatrix}\end{split}\]

Given that, we can converge the environment even in the thermodynamic limit: LP[IdL, :, :] just contains the energy for the left part of the Hamiltonian, contributing just a constant we can ignore (since we only look at relative energies) LP[IdR, :, :] = eye(:, :) is just the MPS environment. The remaining part is the harder one: we need to converge $C + CA + CAA + CAAA + … $ sandwiched between the MPS. However, H often has finite range, which makes A nil-potent, such that we only need to contract the environment a few times from the left.

Todo

Right now, for infinite/long range it just limits the number of iterations. In general, we could calculate the exact $X = C + CA + CAA +…$ with the geometric series by solving the set of linear equation $ X(1-A) = C$ for X, (and analogously $(1-A)X = B$ for the right environment RP).

Parameters:

• i (int) – Build LP left of site i.

• start_env_sites (int) – How many sites to contract to converge the init_LP; the initial age_LP.

Returns:

init_LP – Environment left of site i with labels 'vR*', 'wR', 'vR'.

Return type:

Array

init_RP(i, start_env_sites=0)

Build initial right part RP for an MPS/MPOEnvironment.

Parameters:

• i (int) – Build RP right of site i.

• start_env_sites (int) – How many sites to contract to converge the init_RP; the initial age_RP.

Returns:

init_RP – Environment right of site i with labels 'vL*', 'wL', 'vL'.

Return type:

Array

get_LP(i, store=True)

Calculate LP at given site from nearest available one (including i).

The returned LP_i corresponds to the following contraction, where the M’s and the N’s are in the ‘A’ form:

|     .-------M[0]--- ... --M[i-1]--->-   'vR'
|     |       |             |
|     LP[0]---W[0]--- ... --W[i-1]--->-   'wR'
|     |       |             |
|     .-------N[0]*-- ... --N[i-1]*--<-   'vR*'

Parameters:

• i (int) – The returned LP will contain the contraction strictly left of site i.

• store (bool) – Whether to store the calculated LP in self (True) or discard them (False).

Returns:

LP_i – Contraction of everything left of site i, with labels 'vR*', 'wR', 'vR' for bra, H, ket.

Return type:

Array

get_RP(i, store=True)

Calculate RP at given site from nearest available one (including i).

The returned RP_i corresponds to the following contraction, where the M’s and the N’s are in the ‘B’ form:

|     'vL'  ->---M[i+1]-- ... --M[L-1]----.
|                |              |         |
|     'wL'  ->---W[i+1]-- ... --W[L-1]----RP[-1]
|                |              |         |
|     'vL*' -<---N[i+1]*- ... --N[L-1]*---.

Parameters:

• i (int) – The returned RP will contain the contraction strictly right of site i.

• store (bool) – Whether to store the calculated RP in self (True) or discard them (False).

Returns:

RP_i – Contraction of everything right of site i, with labels 'vL*', 'wL', 'vL' for bra, H, ket.

Return type:

Array

full_contraction(i0)

Calculate the energy by a full contraction of the network.

The full contraction of the environments gives the value <bra|H|ket> `` ignoring the :attr:`~tenpy.networks.mps.MPS.norm` of the `bra` and `ket`, i.e. the total energy (even if bra and ket are not normalized). For this purpose, this function contracts ``get_LP(i0+1, store=False) and get_RP(i0, store=False) with appropriate singular values in between.

Parameters:

i0 (int) – Site index.

cache_optimize(short_term_LP=[], short_term_RP=[], preload_LP=None, preload_RP=None)

Update short_term_keys for the cache and possibly preload tensors.

Parameters:

• short_term_LP (list of int) – i indices for get_LP() and get_RP(), respectively, for which a repeated look-up could happen, i.e., for which tensors should be kept in RAM until the next call to this function.

• short_term_RP (list of int) – i indices for get_LP() and get_RP(), respectively, for which a repeated look-up could happen, i.e., for which tensors should be kept in RAM until the next call to this function.

• preload_LP (int | None) – If not None, preload the tensors for the corresponding get_LP() and get_RP() call, respectively, from disk.

• preload_RP (int | None) – If not None, preload the tensors for the corresponding get_LP() and get_RP() call, respectively, from disk.

clear()

Delete all partial contractions except the left-most LP and right-most RP.

del_LP(i)

Delete stored part strictly to the left of site i.

del_RP(i)

Delete stored part strictly to the right of site i.

get_LP_age(i)

Return number of physical sites in the contractions of get_LP(i).

Might be None.

get_RP_age(i)

Return number of physical sites in the contractions of get_RP(i).

Might be None.

get_initialization_data(first=0, last=None, include_bra=False, include_ket=False)

Return data for (re-)initialization of the environment.

Parameters:

• first (int) – The first and last site, to the left and right of which we should return the environments. Defaults to 0 and L - 1.

• last (int) – The first and last site, to the left and right of which we should return the environments. Defaults to 0 and L - 1.

• include_bra (bool) – Whether to also return the bra and ket, respectively.

• include_ket (bool) – Whether to also return the bra and ket, respectively.

Returns:

init_env_data – A dictionary with the following entries.

init_LP, init_RPArray

LP on the left of site first and RP on the right of site last, which can be used as init_LP and init_RP for the initialization of a new environment.

age_LP, age_RPint

The number of physical sites involved into the contraction yielding init_LP and init_RP, respectively.

bra, ketMPS

References of bra and ket. Only included if include_bra and include_ket are True, respectively.

Return type:

dict

has_LP(i)

Return True if LP left of site i is stored.

has_RP(i)

Return True if RP right of site i is stored.

set_LP(i, LP, age)

Store part to the left of site i.

set_RP(i, RP, age)

Store part to the right of site i.