TransferMatrix

• full name: tenpy.networks.mps.TransferMatrix

• parent module: tenpy.networks.mps

• type: class

class tenpy.networks.mps.TransferMatrix(bra, ket, shift_bra=0, shift_ket=None, transpose=False, charge_sector=0, form='B')

Bases: tenpy.linalg.sparse.NpcLinearOperator

Transfer matrix of two MPS (bra & ket).

For an iMPS in the thermodynamic limit, we often need to find the ‘dominant RP’ (and LP). This mean nothing else than to take the transfer matrix of the unit cell and find the (right/left) eigenvector with the largest (magnitude) eigenvalue, since it will dominate \((TM)^n RP\) (or \(LP (TM)^n\)) in the limit \(n \rightarrow \infty\) - whatever the initial RP is. This class provides exactly that functionality with eigenvectors().

Given two MPS, we define the transfer matrix as:

|    ---M[i]---M[i+1]- ... --M[i+L]---
|       |      |             |
|    ---N[j]*--N[j+1]* ... --N[j+L]*--

Here the M denotes the matrices of the bra and N the ones of the ket, respectively. To view it as a matrix, we combine the left and right indices to pipes:

|  (vL.vL*) ->-TM->- (vR.vR*)   acting on  (vL.vL*) ->-RP

Note that we keep all M and N as copies.

Parameters

braMPS

The MPS which is to be (complex) conjugated.

ketMPS

The MPS which is not (complex) conjugated.

shift_braint

We start the N of the bra at site shift_bra (i.e. the j in the above network).

shift_ketint | None

We start the M of the ket at site shift_ket (i.e. the i in the above network). None defaults to shift_bra.

transposebool

Wheter self.matvec acts on RP (False) or LP (True).

charge_sectorNone | charges | 0

Selects the charge sector of the vector onto which the Linear operator acts. None stands for all sectors, 0 stands for the zero-charge sector. Defaults to 0, i.e., assumes the dominant eigenvector is in charge sector 0.

form'B' | 'A' | 'C' | 'G' | 'Th' | None | tuple(float, float)

In which canonical form we take the M and N matrices.

Attributes

Lint

Number of physical sites involved in the transfer matrix, i.e. the least common multiple of bra.L and ket.L.

shift_braint

We start the N of the bra at site shift_bra.

shift_ketint | None

We start the M of the ket at site shift_ket. None defaults to shift_bra.

transposebool

Wheter self.matvec acts on RP (True) or LP (False).

qtotalcharges

Total charge of the transfer matrix (which is gauged away in matvec).

formtuple(float, float) | None

In which canonical form (all of) the M and N matrices are.

flat_linopFlatLinearOperator

Class lifting matvec() to ndarrays in order to use speigs().

pipeLegPipe

Pipe corresponding to '(vL.vL*)' for transpose=False or to '(vR.vR*)' for transpose=True.

label_split :

['vL', 'vL*'] if tranpose=False or ['vR', 'vR*'] if transpose=True.

_bra_Nlist of npc.Array

Complex conjugated matrices of the bra, transposed for fast matvec.

_ket_Mlist of npc.Array

The matrices of the ket, transposed for fast matvec.

_contract_legsint

Number of physical legs per site + 1.

Methods

eigenvectors(self[, num_ev, max_num_ev, …])	Find (dominant) eigenvector(s) of self using scipy.sparse.
initial_guess(self[, diag])	Return a diagonal matrix as initial guess for the eigenvector.
matvec(self, vec)	Given vec as an npc.Array, apply the transfer matrix.

matvec(self, vec)

Given vec as an npc.Array, apply the transfer matrix.

Parameters

vecArray

Vector to act on with the transfermatrix. If not transposed, vec is the right part RP of an environment, with legs '(vL.vL*)' in a pipe or splitted. If transposed, the left part LP of an environment with legs '(vR*.vR)'.

Returns

mat_vecArray

The tranfer matrix acted on vec, in the same form as given.

initial_guess(self, diag=1.0)

Return a diagonal matrix as initial guess for the eigenvector.

Parameters

diagfloat | 1D ndarray

Should be 1. for the identity or some singular values squared.

Returns

matArray

A 2D array with diag on the diagonal such that matvec() can act on it.

eigenvectors(self, num_ev=1, max_num_ev=None, max_tol=1e-12, which='LM', v0=None, **kwargs)

Find (dominant) eigenvector(s) of self using scipy.sparse.

If no charge_sector was selected, we look in all charge sectors.

Parameters

num_evint

Number of eigenvalues/vectors to look for.

max_num_evint

scipy.sparse.linalg.speigs() somtimes raises a NoConvergenceError for small num_ev, which might be avoided by increasing num_ev. As a work-around, we try it again in the case of an error, just with larger num_ev up to max_num_ev. None defaults to num_ev + 2.

max_tolfloat

After the first NoConvergenceError we increase the tol argument to that value.

whichstr

Which eigenvalues to look for, see scipy.sparse.linalg.speigs.

**kwargs :

Further keyword arguments are given to speigs().

Returns

eta1D ndarray

The eigenvalues, sorted according to which.

wlist of Array

The eigenvectors corresponding to eta, as npc.Array with LegPipe.