DensityMatrixMixer

Inheritance Diagram

Inheritance diagram of tenpy.algorithms.dmrg.DensityMatrixMixer

Methods

DensityMatrixMixer.__init__(options, ...)

DensityMatrixMixer.mix_and_decompose_1site(...)

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

DensityMatrixMixer.mix_and_decompose_2site(...)

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

DensityMatrixMixer.mix_rho(engine, theta, ...)

Calculate the (possibly mixed) reduced density matrices.

DensityMatrixMixer.mixed_svd_2site(engine, ...)

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

DensityMatrixMixer.perturb_svd(engine, ...)

DensityMatrixMixer.svd_from_rho(engine, ...)

Diagonalize rho_L, rho_R to rewrite theta as U S V with isometric U/V.

DensityMatrixMixer.update_amplitude(sweeps)

Update the amplitude, possibly disable the mixer.

Class Attributes and Properties

DensityMatrixMixer.can_decompose_1site

DensityMatrixMixer.deprecated

DensityMatrixMixer.update_sites

class tenpy.algorithms.dmrg.DensityMatrixMixer(options, sweep_activated)[source]

Bases: Mixer, DensityMatrixMixer

Deprecated.

Deprecated since version 1.0.0: Use DensityMatrixMixer instead. Note the changed function names and signatures

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 mive) 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.

mix_rho(engine: Sweep, theta: Array, i0: int, mix_left: bool, mix_right: bool)[source]

Calculate the (possibly mixed) reduced density matrices.

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 by perturbing rho_L.

  • mix_right (bool) – If the virtual index right of S should be expanded by perturbing rho_R.

Returns

  • rho_L (Array) – Reduced density matrix on the left site or a perturbation thereof. Hermitian square array with labels '(vL.p0)', '(vL*.p0*)'.

  • rho_R (Array) – Reduced density matrix on the right site or a perturbation thereof. Hermitian square array with labels '(p1.vR)', '(p1*.vR*)'.

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 avaible.

See also

mix_and_decompose_2site

svd_from_rho(engine: Sweep, rho_L: Array, rho_R: Array, theta: Array, qtotal_LR)[source]

Diagonalize rho_L, rho_R to rewrite theta as U S V with isometric U/V.

If rho_L and rho_R were the actual density matrices of theta, this function just performs an SVD by diagonalizing rho_L with U and rho_R with VH and then rewriting theta == U (U^dagger theta VH^dagger VH) = U S V`. Since the actual rho_L and rho_R passed as arguments are perturbed by mix_rho, we get a similar decomposition but S is a general (non-diagoanl) bond matrix.

Returns

As defined in mixed_svd_2site().

Return type

U, S, VH, err, S_approx

update_amplitude(sweeps)[source]

Update the amplitude, possibly disable the mixer.

Parameters

sweeps (int) – The total number of performed sweeps, to check if we need to disable the mixer.

Returns

mixer – Returns self if we should continue mixing, or None, if the mixer should be disabled.

Return type

Mixer | None