SubspaceExpansion

full name: tenpy.algorithms.mps_common.SubspaceExpansion
parent module: tenpy.algorithms.mps_common
type: class

Inheritance Diagram

Methods

`SubspaceExpansion.__init__`(options[, ...])
`SubspaceExpansion.mix_and_decompose_1site`(...)	Decompose single-site wavefunction and expand/mix an adjacent bond.
`SubspaceExpansion.mix_and_decompose_2site`(...)	Decompose two-site wavefunction and expand/mix enclosed bond(s).
`SubspaceExpansion.mixed_svd_2site`(engine, ...)	Mix and SVD-like decompose a two-site wavefunction.
`SubspaceExpansion.update_amplitude`(sweeps)	Update the amplitude, possibly disable the mixer.

Class Attributes and Properties

SubspaceExpansion.can_decompose_1site

class tenpy.algorithms.mps_common.SubspaceExpansion(options, sweep_activated=0)[source]

Bases: Mixer

Mixer of a direct subspace expansion.

Performs a subspace expansion following [hubig2015]. It operates on single-site wave functions theta and is thus suitable for both single-site DMRG and two-site DMRG.

It is actually not necessary to fill the next_B with zeros as described in Hubig’s paper; rather we directly project the wR leg of VH onto the IdL index, which corresponds to taking the original theta (up to truncation).

See also

DensityMatrixMixer

Notes

Pictorially for a subspace expansion of the left U while moving right:

|  --theta---            .-theta---                                  --U---S---VH---
|     |                  |   |                                         |       |
|             =dot=>    LP---H0--mix_L--     =SVD=>                    |       .---[IdL]
|                        |   |          (vL.p0),(wR.vR)

For a left-move:

|  --theta---            --theta--.                         ---U---S---VH--
|     |                      |    |                            |       |
|            =dot=>  --mix_R-H0---RP         =SVD=>      [IdR]-.       |
|                            |    |     (vL.wL),(p0.vR)

Note that only the U during the right move (or VH during left-move) is guaranteed to be an isometry as expected in the canonical form; VH during the right-move contains a “subspace expansion” and does not fulfill the canonical VH.dot(VH.conj().T) == eye. Moreover, the U constructed from a two-site theta viewing the '(p1.vR) leg as just vR in the right-move is (mathematically) equivalent to the U returned by the DensityMatrixMixer (up to degenerate singular values).

In other words, the mix_and_decompose_2site() methods of SubspaceExpansion and DensityMatrixMixer should produce equivalent results; they only differ in the way they calculate U and V internally.

mix_and_decompose_1site(engine: Sweep, theta: Array, i0: int, move_right: bool)[source]

Decompose single-site wavefunction and expand/mix an adjacent bond.

For a right move, we decompose:

|   -- theta --   ==>   -- U === S === VH --
|        |                 |

For a left move:

|   -- theta --   ==>   -- U === S === VH --
|        |                             |

The LHS is equal to the RHS up to truncation and rescaling (we normalize to norm(S)==1). The double lines (===) indicate the mixed/expanded bonds. Only the tensor with a physical leg (e.g. U for a right move) is an isometry and is equivalent to the corresponding output of mixed_svd_2site(). It carries the qtotal of theta. The other (e.g. VH for a right move) is in general not isometric. S are the usual singular values.

The mixer can be injected in a sweeping algorithm by replacing the usual SVD of theta that shifts the canonical form with this method.

Parameters:

engine (Sweep) – The engine that is using this mixer.
theta (2D Array) – Single-site wavefunction prepared for SVD. Labels either '(vL.p0)', 'vR' for a right move, or 'vL', '(p0.vR)' for a left move.
i0 (int) – The site that theta lives on. The bond to be expanded is i0, i0 + 1 for a right move or i0 - 1, i0 for a left move.
move_right (bool | None) – Whether we move to the right (True), left (False), or dont move (None).

Returns:

U (Array) – Left part as defined above. Isometric for a right move. Labels '(vL.p)', 'vR' for a right move or 'vL', '(p.vR)' for a left move.
S (1D ndarray) – Singular values on the new bond.
VH (Array) – Right part as defined above. Isometric for a left move. Labels 'vL', '(p.vR)' for a right move or '(vL.p)', 'vR' for a left move.
err (TruncationError) – The truncation error introduced.

mix_and_decompose_2site(engine: Sweep, theta: Array, i0: int, mix_left: bool, mix_right: bool, qtotal_LR=None)[source]

Decompose two-site wavefunction and expand/mix enclosed bond(s).

This is a weaker version of mixed_svd_2site(). The decomposition is also:

|   -- theta --   ==>   -- U === S --- VH --
|      |   |               |           |

But only the tensors on mixed sites (e.g. only U for the case depicted above, i.e. mix_left=True, mix_right=False) are guaranteed to be isometric, while any non-mixed tensor (VH in this example) is in general not isometric. Other than that, parameters and returns are the same as for mixed_svd_2site().

The reason to relax the isometry condition is that the decomposition described above can be done using mix_and_decompose_1site() if mixed_svd_2site() is not implemented.

mixed_svd_2site(engine: Sweep, theta: Array, i0: int, mix_left: bool, mix_right: bool, qtotal_LR=[None, None])[source]

Mix and SVD-like decompose a two-site wavefunction.

The goal is to split theta as follows:

|   -- theta --   ==>   -- U === S --- VH --
|      |   |               |           |

The LHS is equal to the RHS up to truncation and rescaling (we normalize to norm(S)==1). The double lines (===) indicate the mixed/expanded bond, here e.g. for mix_left=True, mix_right=False. U and VH are isometries like in an SVD, but S may be a general bond-matrix and in particular not necessarily diagonal or even square. Either one (or both) of the bonds next to S can be expanded / mixed. The isometry on a non-mixed side (e.g. U if mix_left=False) could have been obtained from an SVD of theta.

Parameters:

engine (Sweep) – The engine that is using this mixer.
theta (2D Array) – Two-site wavefunction prepared for SVD. Labels '(vL.p0)', '(p1.vR)'.
i0 (int) – The site index of the left site, i.e. such that theta lives on sites i0, i0 + 1.
mix_left (bool) – If the virtual index left of S should be expanded.
mix_right (bool) – If the virtual index right of S should be expanded.
qtotal_LR ([{charges}, {charges}] | None) – The desired qtotal for U and VH, respectively. If None, the qtotal are arbitrary.

Returns:

U (Array) – Left isometry as defined above. Labels '(vL.p)', 'vR'.
S (1D ndarray | 2D Array) – Singular values (1D ndarray) or general bond matrix (2D Array, labels 'vL', 'vR').
VH (Array) – Right isometry as defined above. Labels 'vL', '(p.vR)'.
err (TruncationError) – The truncation error introduced.
S_approx (ndarray) – Approximation of the singular values of theta. Exact if available.