# MPOEnvironment¶

• full name: tenpy.networks.mpo.MPOEnvironment

• parent module: tenpy.networks.mpo

• type: class

Inheritance Diagram

Methods

 MPOEnvironment.__init__(bra, H, ket[, …]) Initialize self. Delete stored part strictly to the left of site i. Delete storde part scrictly to the right of site i. MPOEnvironment.expectation_value(ops[, …]) Expectation value of (n-site) operator(s). 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). 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). Return number of physical sites in the contractions of get_RP(i). Return data for (re-)initialization. Build initial left part LP. Build initial right part RP for an MPS/MPOEnvironment. 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. Sanity check, raises ValueErrors, if something is wrong.
class tenpy.networks.mpo.MPOEnvironment(bra, H, ket, init_LP=None, init_RP=None, age_LP=0, age_RP=0)[source]

Stores partial contractions of $$<bra|H|ket>$$ for an MPO H.

The network for a contraction $$<bra|H|ket>$$ of an MPO H bewteen 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* -<-.

To avoid recalculations of the whole network e.g. in the DMRG sweeps, we store the contractions up to some site index in this class. For bc='finite','segment', the very left and right part LP[0] and RP[-1] are trivial and don’t change in the DMRG algorithm, but for iDMRG (bc='infinite') they are also updated (by inserting another unit cell to the left/right).

The MPS bra and ket have to be in canonical form. All the environments are constructed without the singular values on the open bond. In other words, we contract left-canonical A to the left parts LP and right-canonical B to the right parts RP.

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_LP (None | Array) – Initial very left part LP. If None, build trivial one with :methinit_LP.

• init_RP (None | Array) – Initial very right part RP. If None, build trivial one with init_RP().

• age_LP (int) – The number of physical sites involved into the contraction yielding firstLP.

• age_RP (int) – The number of physical sites involved into the contraction yielding lastRP.

H

The MPO sandwiched between bra and ket.

Type

MPO

test_sanity()[source]

Sanity check, raises ValueErrors, if something is wrong.

init_LP(i)[source]

Build initial left part LP.

Parameters

i (int) – Build LP left of site i.

Returns

init_LP – Identity contractible with the vL leg of .ket.get_B(i), multiplied with a unit vector nonzero in H.IdL[i], with labels 'vR*', 'wR', 'vR'.

Return type

Array

init_RP(i)[source]

Build initial right part RP for an MPS/MPOEnvironment.

Parameters

i (int) – Build RP right of site i.

Returns

init_RP – Identity contractible with the vR leg of self.get_B(i), multiplied with a unit vector nonzero in H.IdR[i], with labels 'vL*', 'wL', 'vL'.

Return type

Array

get_LP(i, store=True)[source]

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) – Wheter 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)[source]

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 rigth of site i.

• store (bool) – Wheter 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)[source]

Calculate the energy by a full contraction of the network.

The full contraction of the environments gives the value <bra|H|ket> / (norm(|bra>)*norm(|ket>)), i.e. if bra is ket and normalized, the total energy. For this purpose, this function contracts get_LP(i0+1, store=False) and get_RP(i0, store=False).

Parameters

i0 (int) – Site index.

del_LP(i)[source]

Delete stored part strictly to the left of site i.

del_RP(i)[source]

Delete storde part scrictly to the right of site i.

expectation_value(ops, sites=None, axes=None)[source]

Expectation value <bra|ops|ket> of (n-site) operator(s).

Calculates n-site expectation values of operators sandwiched between bra and ket. For examples the contraction for a two-site operator on site i would look like:

|          .--S--B[i]--B[i+1]--.
|          |     |     |       |
|          |     |-----|       |
|          LP[i] | op  |       RP[i+1]
|          |     |-----|       |
|          |     |     |       |
|          .--S--B*[i]-B*[i+1]-.

Here, the B are taken from ket, the B* from bra. The call structure is the same as for MPS.expectation_value().

Parameters
• ops ((list of) { Array | str }) – The operators, for wich the expectation value should be taken, All operators should all have the same number of legs (namely 2 n). If less than len(sites) operators are given, we repeat them periodically. Strings (like 'Id', 'Sz') are translated into single-site operators defined by sites.

• sites (list) – List of site indices. Expectation values are evaluated there. If None (default), the entire chain is taken (clipping for finite b.c.)

• axes (None | (list of str, list of str)) – Two lists of each n leg labels giving the physical legs of the operator used for contraction. The first n legs are contracted with conjugated B, the second n legs with the non-conjugated B. None defaults to (['p'], ['p*']) for single site (n=1), or (['p0', 'p1', ... 'p{n-1}'], ['p0*', 'p1*', .... 'p{n-1}*']) for n > 1.

Returns

exp_vals – Expectation values, exp_vals[i] = <bra|ops[i]|ket>, where ops[i] acts on site(s) j, j+1, ..., j+{n-1} with j=sites[i].

Return type

1D ndarray

Examples

One site examples (n=1):

>>> env.expectation_value('Sz')
[Sz0, Sz1, ..., Sz{L-1}]
>>> env.expectation_value(['Sz', 'Sx'])
[Sz0, Sx1, Sz2, Sx3, ... ]
>>> env.expectation_value('Sz', sites=[0, 3, 4])
[Sz0, Sz3, Sz4]

Two site example (n=2), assuming homogeneous sites:

>>> SzSx = npc.outer(psi.sites[0].Sz.replace_labels(['p', 'p*'], ['p0', 'p0*']),
psi.sites[1].Sx.replace_labels(['p', 'p*'], ['p1', 'p1*']))
>>> env.expectation_value(SzSx)
[Sz0Sx1, Sz1Sx2, Sz2Sx3, ... ]   # with len L-1 for finite bc, or L for infinite

Example measuring <bra|SzSx|ket> on each second site, for inhomogeneous sites:

>>> SzSx_list = [npc.outer(psi.sites[i].Sz.replace_labels(['p', 'p*'], ['p0', 'p0*']),
...                        psi.sites[i+1].Sx.replace_labels(['p', 'p*'], ['p1', 'p1*']))
...              for i in range(0, psi.L-1, 2)]
>>> env.expectation_value(SzSx_list, range(0, psi.L-1, 2))
[Sz0Sx1, Sz2Sx3, Sz4Sx5, ...]
get_LP_age(i)[source]

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

Might be None.

get_RP_age(i)[source]

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

Might be None.

get_initialization_data()[source]

Return data for (re-)initialization.

The returned parameters are collected in a dictionary with the following names.

Returns

• init_LP, init_RP (Array) – LP on the left of site 0 and RP on the right of site L-1, which can be used as init_LP and init_RP for the initialization of a new environment.

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

set_LP(i, LP, age)[source]

Store part to the left of site i.

set_RP(i, RP, age)[source]

Store part to the right of site i.