lack sqrt operation in tensornetwork/matrixproductstates/infinite_mps.py

[veya2ztn]

the code location:
tensornetwork/matrixproductstates/infinite_mps.py
line 287 to 298

    U, singvals, V, _ = self.backend.svd(
        tmp,
        pivot_axis=1,
        max_singular_values=D,
        max_truncation_error=truncation_threshold,
        relative=True)
    lam = self.backend.diagflat(singvals)
    self.tensors[0] = ncon([lam, V, inv_sqrtr, self.tensors[0]],
                           [[-1, 1], [1, 2], [2, 3], [3, -2, -3]],
                           backend=self.backend.name)

    # absorb connector * inv_sqrtl * U * lam into the right-most tensor
    # Note that lam is absorbed here, which means that the state
    # is in the parallel decomposition
    # Note that we absorb connector_matrix here
    self.tensors[-1] = ncon([self.get_tensor(len(self) - 1), inv_sqrtl, U, lam],
                            [[-1, -2, 1], [1, 2], [2, 3], [3, -3]],
                            backend=self.backend.name)

The lam is the singular value after SVD.
When it contracte into the first tensor (i.e. self.tensor[0]) and the last tensor (i.e. self.tensor[1]), it should be sqrt(lam) not lam to keep the whole MPS invariant. or as the commend said, only absorb connector to left or right tensor.

so the right code is

self.tensors[0] = ncon([self.backend(lam), V, inv_sqrtr, self.tensors[0]],
                           [[-1, 1], [1, 2], [2, 3], [3, -2, -3]],

self.tensors[-1] = ncon([self.get_tensor(len(self) - 1), inv_sqrtl, U, self.backend(lam)],
                            [[-1, -2, 1], [1, 2], [2, 3], [3, -3]],
                            backend=self.backend.name)

so the contraction between self.tensors[-1] and self.tensors[0] keep same
new self.tensors[-1] <-> new self.tensors[0] =
self.get_tensor(len(self) - 1)<->inv_sqrtl <->U<->self.backend(lam)<->self.backend(lam)<->V<-> inv_sqrtr<->self.tensors[0]

I am now learning the canonical form of iMPS, in case I make mistake, I also check the invariant:

X,U,S,V,Y = inv_sqrtl, U, lam,V,inv_sqrtr
print(np.einsum('ea,ab,bc,cd,df->ef',X,U,S,V,Y).real)
[[ 1.00000000e+00 -5.85469173e-18  3.96384314e-16 -9.84021892e-16]
 [ 1.24032729e-16  1.00000000e+00 -4.77916318e-16  4.44089210e-16]
 [ 9.77950360e-16 -6.96925156e-16  1.00000000e+00 -6.01081684e-16]
 [ 8.32667268e-16  2.81458884e-16  6.97358837e-16  1.00000000e+00]]

However, if I double multiply S

print(np.einsum('ea,ab,bc,cd,df,fg->eg',X,U,S,S,V,Y).real)
[[ 0.22848829 -0.03708758 -0.0173063   0.06811959]
 [-0.05568507  0.33938397  0.05829849  0.00706872]
 [-0.04525413  0.09879068  0.303025    0.03182943]
 [-0.04647557  0.00931772 -0.00282994  0.39147297]]