TangentMps.jl

TangentMps.al_from_ac_and_c
TangentMps.ar_from_ac_and_c
TangentMps.ibcleft
TangentMps.ibcright
TangentMps.mixedcanonical
TangentMps.mul_matrix_from_left
TangentMps.mul_matrix_from_right
TangentMps.mul_operator_onsite
TangentMps.mul_operator_with_left
TangentMps.mul_operator_with_right
TangentMps.tdvpstep
TangentMps.transfer_from_left
TangentMps.transfer_from_left
TangentMps.transfer_from_right
TangentMps.transfer_from_right
TangentMps.twosite_variance
TangentMps.vumpsstep

TangentMps.al_from_ac_and_c — Method

al_from_ac_and_c(AC, C)

Compute AL which satisfies mul_matrix_from_right(AL, C) ≈ AC

Return values:

AL, ϵL = al_from_ac_and_c(AC, C)

where ϵL = norm(mul_matrix_from_right(AL, C) - AC)

source

TangentMps.ar_from_ac_and_c — Method

ar_from_ac_and_c(AC, C)

Compute AR which satisfies mul_matrix_from_left(AR, C) ≈ AC

Return values:

AR, ϵR = ar_from_ac_and_c(AC, C)

where ϵR = norm(mul_matrix_from_left(AR, C) - AC)

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TangentMps.ibcleft — Method

ibcleft(O, AL, C; kwargs...)

Compute left infinite boundary condition, defined by

   ┌────────┐   ┌────────┐     ┌─            ┌────────┐    ─┐ P
┌──1   AL   2───1   AL   2─── ┌┘          ───1   AL   2───  └┐  ───(2)
│  └────3───┘   └────3───┘    │   ┌┐         └────3───┘      │
│       3────────────4        │  ─┘│              │          │
│      (       O      )       │    │  ───         │          │
│       1────────────2        │    │              │          │
│  ┌────3───┐   ┌────3───┐    │  ──┴──       ┌────3───┐      │
└──1conj(AL)2───1conj(AL)2─── └┐          ───1conj(AL)2───  ┌┘  ───(1)
   └────────┘   └────────┘     └─            └────────┘    ─┘

Return values:

ibc, info = ibcleft(O, AL, C)

Arguments:

O: Hamiltonian (only two-site operator is supported)
AL: left-canonical tensor of mixed-canonical uniform MPS representation
C: center matrix of mixed-canonical uniform MPS representation

If O, AL and C is AbstractArray oblect, nothing have to be done. The following methods should be defined:

TangentMps.transfer_from_left(X, O, AL)
TangentMps.transfer_from_left(X, O, (AL, AL))
TangentMps.similar_normalized_eye(X)
Base.*(X, Y)
Base.*(x::Number, X)
Base.+(X, Y)
Base.adjoint(X)
Base.eltype(X)
Base.similar(X, [T::Type<:Number])
Base.fill!(X, α::Number)
Base.copyto!(X, Y)
LinearAlgebra.mul!(X, Y, α::Number)
LinearAlgebra.rmul!(X, α::Number)
LinearAlgebra.axpy!(α::Number, X, Y)
LinearAlgebra.axpby!(α::Number, X, β::Number, Y)
LinearAlgebra.tr(X)
LinearAlgebra.dot(X,Y)
LinearAlgebra.norm(X)

where typeof(X) == typeof(Y) == typeof(C) is satisfied.

Keyword arguments:

Keyword arguments are passed to KrylovKit.linsolve used in ibc_left.

atol::Real: the requested accuracy, i.e. absolute tolerance, on the norm of the residual.
rtol::Real: the requested accuracy on the norm of the residual, relative to the norm of the right hand side b.
tol::Real: the requested accuracy on the norm of the residual which is actually used, but which defaults to max(atol, rtol*norm(b)). So either atol and rtol or directly use tol (in which case the value of atol and rtol will be ignored).
krylovdim::Integer: the maximum dimension of the Krylov subspace that will be constructed.
maxiter::Integer: the number of times the Krylov subspace can be rebuilt; see below for further details on the algorithms.

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TangentMps.ibcright — Method

ibcright(O, AR, C; kwargs...)

Compute left infinite boundary condition, defined by

        ┌─            ┌────────┐    ─┐ P    ┌────────┐   ┌────────┐
(1)─── ┌┘          ───1   AR   2───  └┐  ───1   AR   2───1   AR   2──┐
       │   ┌┐         └────3───┘      │     └────3───┘   └────3───┘  │
       │  ─┘│              │          │          3────────────4      │
       │    │  ───         │          │         (      O       )     │
       │    │              │          │          1────────────2      │
       │  ──┴──       ┌────3───┐      │     ┌────3───┐   ┌────3───┐  │
(2)─── └┐          ───1conj(AR)2───  ┌┘  ───1conj(AR)2───1conj(AR)2──┘
        └─            └────────┘    ─┘      └────────┘   └────────┘

Return values:

ibc, info = ibcright(O, AR, C)

Arguments:

O: Hamiltonian (only two-site operator is supported)
AR: right-canonical tensor of mixed-canonical uniform MPS representation
C: center matrix of mixed-canonical uniform MPS representation

If O, AR and C is AbstractArray oblect, nothing have to be done. The following methods should be defined:

TangentMps.transfer_from_right(X, O, AR)
TangentMps.transfer_from_right(X, O, (AR, AR))
TangentMps.similar_normalized_eye(X)
Base.*(X, Y)
Base.*(x::Number, X)
Base.+(X, Y)
Base.adjoint(X)
Base.eltype(X)
Base.similar(X, [T::Type<:Number])
Base.fill!(X, α::Number)
Base.copyto!(X, Y)
LinearAlgebra.mul!(X, Y, α::Number)
LinearAlgebra.rmul!(X, α::Number)
LinearAlgebra.axpy!(α::Number, X, Y)
LinearAlgebra.axpby!(α::Number, X, β::Number, Y)
LinearAlgebra.tr(X)
LinearAlgebra.dot(X,Y)
LinearAlgebra.norm(X)

where typeof(X) == typeof(Y) == typeof(C) is satisfied.

Keyword arguments:

Keyword arguments are passed to KrylovKit.linsolve used in ibc_left.

atol::Real: the requested accuracy, i.e. absolute tolerance, on the norm of the residual.
rtol::Real: the requested accuracy on the norm of the residual, relative to the norm of the right hand side b.
tol::Real: the requested accuracy on the norm of the residual which is actually used, but which defaults to max(atol, rtol*norm(b)). So either atol and rtol or directly use tol (in which case the value of atol and rtol will be ignored).
krylovdim::Integer: the maximum dimension of the Krylov subspace that will be constructed.
maxiter::Integer: the number of times the Krylov subspace can be rebuilt; see below for further details on the algorithms.

source

TangentMps.mixedcanonical — Method

mixedcanonical(A)

convert non-canonical uniform MPS |Ψ(A)⟩ = ∑ vₗ^† (∏ᵢ A^{sᵢ}) vᵣ |{s}⟩ to mixed-canonical uniform MPS |Ψ(AL, AR, AC, C)⟩ = ∑ vₗ^† (∏ᵢ AL^{sᵢ}) AC^{sⱼ} (∏ᵢ AR^{sᵢ}) vᵣ |{s}⟩ = ∑ vₗ^† (∏ᵢ AL^{sᵢ}) C (∏ᵢ AR^{sᵢ}) vᵣ |{s}⟩

Return values:

AL, AR, AC, C = mixedcanonical(A)

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TangentMps.mul_matrix_from_left — Method

mul_matrix_from_left(A, X)

     Λ  ┌───┐
(1)─1X2─1 A 2─(2)
     V  └─3─┘
          │
         (3)

source

TangentMps.mul_matrix_from_right — Method

mul_matrix_from_right(A, X)

    ┌───┐  Λ
(1)─1 A 2─1X2─(2)
    └─3─┘  V
      │
     (3)

source

TangentMps.mul_operator_onsite — Method

mul_operator_onsite(A, O)

    ┌───┐
(1)─1 A 2─(2)
    └─3─┘
      │
     .2.
    ( O )
     `1'
      │
     (3)

source

TangentMps.mul_operator_with_left — Method

mul_operator_with_left(A, O, AL)

 ┌────────┐  ┌───┐
┌1   AL   2──1 A 2─┐
│└────3───┘  └─3─┘ │
│     │        │   │
│    .3────────4.  │
│   (      O     ) │
│    `1────────2'  │
│     │        │   │
│┌────3───┐    │   │
└1conj(AL)2─┐  │  ┌┘
 └────────┘ │  │  │
(1)─────────┘ (3) └(2)

source

TangentMps.mul_operator_with_right — Method

mul_operator_with_right(A, O, AR)

    ┌───┐  ┌────────┐
  ┌─1 A 2──1   AR   2┐
  │ └─3─┘  └───3────┘│
  │   │        │     │
  │  .3────────4.    │
  │ (     O      )   │
  │  `1────────2'    │
  │   │        │     │
  │   │    ┌───3────┐│
  └┐  │  ┌─1conj(AR)2┘
   │  │  │ └────────┘
(1)┘ (3) └───────────(2)

source

TangentMps.tdvpstep — Method

tdvpstep(O, dt, AL, AR, AC, C; kwargs...)
tdvpstep(O, dt, AL, AC, C; kwargs...) # with inversion symmetry

Return values:

`(AL, AR, AC, C), norm, (ϵL, ϵR), info = tdvpstep(O, AL, AR, AC, C; kwargs...)`
`(AL, AC, C), norm, (ϵL,), info = tdvpstep(O, AL, AC, C; kwargs...)`

where norm denotes || exp(dt*O)|Ψ(AL, AR, AC, C)⟩ || / || |Ψ(AL, AR, AC, C)⟩ ||

source

TangentMps.transfer_from_left — Function

transfer_from_left(X, [O,] (AL, AR), [(BL, BR)=(AL, AR)])

   Λ
  ╱ ╲  ┌────────┐  ┌────────┐
┌1 X 2─1   AL   2──1   AR   2─(2)
│ ╲ ╱  └────3───┘  └────3───┘
│  V        │           │
│          .3───────────4.
│         (       O       )
│          `1───────────2'
│           │           │
│      ┌────3───┐  ┌────3───┐
└──────1conj(BL)2──1conj(BR)2─┐
       └────────┘  └────────┘ │
(1)───────────────────────────┘

source

TangentMps.transfer_from_left — Function

transfer_from_left(X, [O,] A, [B=A])

   Λ
  ╱ ╲  ┌───────┐
┌1 X 2─1   A   2─(2)
│ ╲ ╱  └───3───┘
│  V       │
│         .2.
│        ( O )
│         `1'
│          │
│      ┌───3───┐
└──────1conj(B)2─┐
       └───────┘ │
(1)──────────────┘

source

TangentMps.transfer_from_right — Function

transfer_from_right(X, [O,] (AL, AR), [(BL, BR)=(AL, AR)])

                             Λ
    ┌────────┐  ┌────────┐  ╱ ╲
(1)─1   AL   2──1   AR   2─1 X 2┐
    └────3───┘  └────3───┘  ╲ ╱ │
         │           │       V  │
        .3───────────4.         │
       (       O       )        │
        `1───────────2'         │
         │           │          │
    ┌────3───┐  ┌────3───┐      │
┌───1conj(BL)2──1conj(BR)2──────┘
│   └────────┘  └────────┘
└───────────────────────────────(2)

source

TangentMps.transfer_from_right — Function

transfer_from_right(X, [O,] A, [B=A])

                Λ
    ┌───────┐  ╱ ╲
(1)─1   A   2─1 X 2─┐
    └───3───┘  ╲ ╱  │
        │       V   │
       .2.          │
      ( O )         │
       `1'          │
        │           │
    ┌───3───┐       │
┌───1conj(B)2───────┘
│   └───────┘
└───────────────────(2)

source

TangentMps.twosite_variance — Method

twosite_variance(AL, AC, VL, VR, O)

  ┌────────┐    ┌────────┐
┌─1   AL   2────1   AC   2─┐
│ └────3───┘    └───3────┘ │
│      │            │      │
│    .─3────────────4──.   │
│   (         O         )  │
│    `─1────────────2──'   │
│      │            │      │
│ ┌────3───┐    ┌───3────┐ │
└─1conj(VL)2┐  ┌1conj(VR)2─┘
  └────────┘│  │└────────┘
(1)─────────┘  └─────────(2)

source

TangentMps.vumpsstep — Method

vumpsstep(O, AL, AR, AC, C; kwargs...)
vumpsstep(O, AL, AC, C; kwargs...) # with inversion symmetry

Return values:

`(AL, AR, AC, C), (ϵL, ϵR), info = vumpsstep(O, AL, AR, AC, C; kwargs...)`
`(AL, AC, C), (ϵL,), info = vumpsstep(O, AL, AC, C; kwargs...)`

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