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| 1 | +const InfiniteState = Union{InfinitePEPS, InfinitePEPO} |
| 2 | + |
| 3 | +function MPSKit.infinite_temperature_density_matrix(H::LocalOperator) |
| 4 | + T = scalartype(H) |
| 5 | + A = map(physicalspace(H)) do Vp |
| 6 | + ψ = permute(TensorKit.id(T, Vp), (1, 2)) |
| 7 | + Vv = oneunit(Vp) # trivial (1D) virtual space |
| 8 | + virt = ones(T, domain(ψ) ← Vv ⊗ Vv ⊗ Vv' ⊗ Vv') |
| 9 | + return ψ * virt |
| 10 | + end |
| 11 | + return InfinitePEPO(cat(A; dims = 3)) |
| 12 | +end |
| 13 | + |
1 | 14 | """ |
2 | 15 | get_expham(H::LocalOperator, dt::Number) |
3 | 16 |
|
|
71 | 84 | """ |
72 | 85 | $(SIGNATURES) |
73 | 86 |
|
74 | | -Use QR decomposition on two tensors connected by a bond |
75 | | -to get the reduced tensors |
| 87 | +Use QR decomposition on two tensors `A`, `B` connected by a bond to get the reduced tensors. |
| 88 | +When `A`, `B` are PEPSTensors, |
76 | 89 | ``` |
77 | | - 2 1 |
78 | | - | | |
79 | | - 5 - A - 3 ====> 4 - X ← 2 1 ← a - 3 |
80 | | - | ↘ | ↘ |
81 | | - 4 1 3 2 |
82 | | -
|
83 | | - 2 1 |
84 | | - | | |
85 | | - 5 - B - 3 ====> 1 - b → 3 4 → Y - 2 |
86 | | - | ↘ ↘ | |
87 | | - 4 1 2 3 |
| 90 | + 2 1 1 |
| 91 | + | | | |
| 92 | + 5 -A/B- 3 ====> 4 - X ← 2 1 ← a - 3 1 - b → 3 4 → Y - 2 |
| 93 | + | ↘ | ↘ ↘ | |
| 94 | + 4 1 3 2 2 3 |
| 95 | +``` |
| 96 | +When `A`, `B` are PEPOTensors, |
| 97 | +- If `gate_ax = 1` |
| 98 | +``` |
| 99 | + 2 3 1 2 1 2 |
| 100 | + ↘ | ↘ | ↘ | |
| 101 | + 6 -A/B- 4 ====> 5 - X ← 3 1 ← a - 3 1 - b → 3 5 → Y - 3 |
| 102 | + | ↘ | ↘ ↘ | |
| 103 | + 5 1 4 2 2 4 |
| 104 | +``` |
| 105 | +- If `gate_ax = 2` |
| 106 | +``` |
| 107 | + 2 3 2 2 2 2 |
| 108 | + ↘ | | ↘ ↘ | |
| 109 | + 6 -A/B- 4 ====> 5 - X ← 3 1 ← a - 3 1 - b → 3 5 → Y - 3 |
| 110 | + | ↘ | ↘ | ↘ |
| 111 | + 5 1 4 1 4 1 |
88 | 112 | ``` |
89 | 113 | """ |
90 | | -function _qr_bond(A::PEPSTensor, B::PEPSTensor) |
91 | | - X, a = leftorth(A, ((2, 4, 5), (1, 3))) |
92 | | - Y, b = leftorth(B, ((2, 3, 4), (1, 5))) |
| 114 | +function _qr_bond(A::PT, B::PT; gate_ax::Int = 1) where {PT <: Union{PEPSTensor, PEPOTensor}} |
| 115 | + @assert 1 <= gate_ax <= numout(A) |
| 116 | + permA, permB, permX, permY = if A isa PEPSTensor |
| 117 | + ((2, 4, 5), (1, 3)), ((2, 3, 4), (1, 5)), (1, 4, 2, 3), Tuple(1:4) |
| 118 | + else |
| 119 | + if gate_ax == 1 |
| 120 | + ((2, 3, 5, 6), (1, 4)), ((2, 3, 4, 5), (1, 6)), (1, 2, 5, 3, 4), Tuple(1:5) |
| 121 | + else |
| 122 | + ((1, 3, 5, 6), (2, 4)), ((1, 3, 4, 5), (2, 6)), (1, 2, 5, 3, 4), Tuple(1:5) |
| 123 | + end |
| 124 | + end |
| 125 | + X, a = leftorth(A, permA) |
| 126 | + Y, b = leftorth(B, permB) |
93 | 127 | @assert !isdual(space(a, 1)) |
94 | 128 | @assert !isdual(space(b, 1)) |
95 | | - X = permute(X, (1, 4, 2, 3)) |
96 | | - Y = permute(Y, (1, 2, 3, 4)) |
| 129 | + X, Y = permute(X, permX), permute(Y, permY) |
97 | 130 | b = permute(b, ((3, 2), (1,))) |
98 | 131 | return X, a, b, Y |
99 | 132 | end |
100 | 133 |
|
101 | 134 | """ |
102 | 135 | $(SIGNATURES) |
103 | 136 |
|
104 | | -Reconstruct the tensors connected by a bond from their QR results |
105 | | -obtained from `_qr_bond` |
| 137 | +Reconstruct the tensors connected by a bond from their `_qr_bond` results. |
| 138 | +For PEPSTensors, |
106 | 139 | ``` |
107 | 140 | -2 -2 |
108 | 141 | | | |
109 | 142 | -5- X - 1 - a - -3 -5 - b - 1 - Y - -3 |
110 | 143 | | ↘ ↘ | |
111 | 144 | -4 -1 -1 -4 |
112 | 145 | ``` |
| 146 | +For PEPOTensors |
| 147 | +``` |
| 148 | + -2 -3 -2 -3 |
| 149 | + ↘ | ↘ | |
| 150 | + -6- X - 1 - a - -4 -6 - b - 1 - Y - -4 |
| 151 | + | ↘ ↘ | |
| 152 | + -5 -1 -1 -5 |
| 153 | +
|
| 154 | + -3 -2 -2 -3 |
| 155 | + | ↘ ↘ | |
| 156 | + -6- X - 1 - a - -4 -6 - b - 1 - Y - -4 |
| 157 | + | ↘ | ↘ |
| 158 | + -5 -1 -5 -1 |
| 159 | +``` |
113 | 160 | """ |
114 | 161 | function _qr_bond_undo(X::PEPSOrth, a::AbstractTensorMap, b::AbstractTensorMap, Y::PEPSOrth) |
115 | 162 | @tensor A[-1; -2 -3 -4 -5] := X[-2 1 -4 -5] * a[1 -1 -3] |
116 | 163 | @tensor B[-1; -2 -3 -4 -5] := b[-5 -1 1] * Y[-2 -3 -4 1] |
117 | 164 | return A, B |
118 | 165 | end |
| 166 | +function _qr_bond_undo(X::PEPOOrth, a::AbstractTensorMap, b::AbstractTensorMap, Y::PEPOOrth) |
| 167 | + if !isdual(space(a, 2)) |
| 168 | + @tensor A[-1 -2; -3 -4 -5 -6] := X[-2 -3 1 -5 -6] * a[1 -1 -4] |
| 169 | + @tensor B[-1 -2; -3 -4 -5 -6] := b[-6 -1 1] * Y[-2 -3 -4 -5 1] |
| 170 | + else |
| 171 | + @tensor A[-1 -2; -3 -4 -5 -6] := X[-1 -3 1 -5 -6] * a[1 -2 -4] |
| 172 | + @tensor B[-1 -2; -3 -4 -5 -6] := b[-6 -2 1] * Y[-1 -3 -4 -5 1] |
| 173 | + end |
| 174 | + return A, B |
| 175 | +end |
119 | 176 |
|
120 | 177 | """ |
121 | 178 | $(SIGNATURES) |
122 | 179 |
|
123 | 180 | Apply 2-site `gate` on the reduced matrices `a`, `b` |
124 | 181 | ``` |
125 | | - -1← a --- 3 --- b ← -4 |
126 | | - ↓ ↓ |
127 | | - 1 2 |
128 | | - ↓ ↓ |
129 | | - |----gate---| |
130 | | - ↓ ↓ |
131 | | - -2 -3 |
| 182 | + -1← a --- 3 --- b ← -4 -2 -3 |
| 183 | + ↓ ↓ ↓ ↓ |
| 184 | + 1 2 |----gate---| |
| 185 | + ↓ ↓ or ↓ ↓ |
| 186 | + |----gate---| 1 2 |
| 187 | + ↓ ↓ ↓ ↓ |
| 188 | + -2 -3 -1← a --- 3 --- b ← -4 |
132 | 189 | ``` |
133 | 190 | """ |
134 | 191 | function _apply_gate( |
135 | | - a::AbstractTensorMap{T, S}, |
136 | | - b::AbstractTensorMap{T, S}, |
137 | | - gate::AbstractTensorMap{T, S, 2, 2}, |
138 | | - trscheme::TruncationScheme, |
| 192 | + a::AbstractTensorMap{T, S}, b::AbstractTensorMap{T, S}, |
| 193 | + gate::AbstractTensorMap{T, S, 2, 2}, trscheme::TruncationScheme |
139 | 194 | ) where {T <: Number, S <: ElementarySpace} |
140 | 195 | V = space(b, 1) |
141 | 196 | need_flip = isdual(V) |
142 | | - @tensor a2b2[-1 -2; -3 -4] := gate[-2 -3; 1 2] * a[-1 1 3] * b[3 2 -4] |
| 197 | + if isdual(space(a, 2)) |
| 198 | + @tensor a2b2[-1 -2; -3 -4] := gate[1 2; -2 -3] * a[-1 1 3] * b[3 2 -4] |
| 199 | + else |
| 200 | + @tensor a2b2[-1 -2; -3 -4] := gate[-2 -3; 1 2] * a[-1 1 3] * b[3 2 -4] |
| 201 | + end |
143 | 202 | trunc = (trscheme isa FixedSpaceTruncation) ? truncspace(V) : trscheme |
144 | 203 | a, s, b, ϵ = tsvd!(a2b2; trunc, alg = TensorKit.SVD()) |
145 | 204 | a, b = absorb_s(a, s, b) |
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