2.4 Matrix Product States
2.4.5 Truncation and Entanglement
We have discovered various concepts of the MPS representation of the many-body state. We outlined its properties in terms of entanglement and have seen that they are not unique, but can be gauged to have special properties. However, we have not yet discussed where the reduction of complexity is applied to the many-body state.
We introduced MPS in Equation2.56as the exact decomposition of the expansion coefficients of the many-body state, which is not related to any approximation. There-fore MPS, as described at this point, still represent the complete many-body state in Equation2.35. Now we want to discuss how to simplify the representation of the many-body state using the MPS approach.
Earlier, we discussed that the complete representation of the many-body state is impossible to manage for medium sized basis sets due to the large number of expan-sion coefficients. When using the exact MPS representation of the many-body state, the number of coefficients is still growing factorially with the system size. To gain any computational benefit compared to the complete representation, we need to reduce the number of degrees of freedom of the MPS representation, which trans-lates to a reduction of the bond dimension of the MPS tensors. For one-dimensional systems, with short-ranged interaction and a gapped ground state, we know that this reduction of dimension must be advantageous.
Next, we will discuss methods that reduce the MPS tensor dimension by using tensor decomposition based onsingular value decomposition.
In Section2.4.4we have already met the first two tensor decomposition methods, namely the LQ and the QR decomposition. In line with these tensor decomposition
methods, we can decompose any of the tensors in the MPS in terms ofsingular value decomposition(SVD)
· · · A[i] · · · ni↑ni↓
= · · · U S V† · · ·
ni↑ni↓
(2.71)
= · · · U S V† · · ·
ni↑ni↓
(2.72) where the matrixS is diagonal holding the real singular value spectrum. We are free to choose where to put the index specifying the physical stateni↑ni↓, either we put them at the matrix U or at the matrixV†. The choice where to put the indices determines if the left-orthogonalization or right-orthogonalization property is implemented at this bond of the MPS, as we knowUU†= 1andV†V = 1.
The interesting feature of SVD is now that we can use it to find the optimal approxi-mation of the tensorA[i]using a reduced dimensional tensor˜A[i]. If we remove the singular values smaller than a certain thresholdfrom the singular value spectrum and only keep theDsingular values that are larger then, we obtain the optimal approximation of the matrixA[i][104]
||A[i]−˜A[i]||2= X
λi<
λ2i, (2.73)
where|| · ||denotes the Frobenius norm and the sum is over all removed singular values. The idea is now to use SVD to truncate the tensors in Equation2.54to find a representation [124] of the entire MPS representation of the many-body state that is optimal with respect to some dimensionD. This optimal representation is then constructed from smaller tensors˜A[i]
· · · A[i] · · · ni↑ni↓
= · · · U S V† · · ·
ni↑ni↓
(2.74)
≈ · · · ˜U ˜S ˜V† · · ·
ni↑ni↓
= · · · ˜A[i] · · ·
n1↑n1↓
, (2.75) where ˜U and ˜V†are the adapted transformation matrices where the lines (rows) belonging to neglected singular values have been removed. As soon as the truncated singular values are nonzero, this truncation introduces antruncation error, i.e. the MPS from the MPS tensors represents the many-body state approximately.
When truncating singular values from the singular value spectrum, we change the dimension of the horizontal bonds in Equation2.56. From now on, we will distinguish
two types of bonds the tensors in the MPS have. There are bonds going vertically with the occupation number being the related quantityni↑ni↓. We will call these bondsphysical bondsas they have a direct physical meaning, namely the occupa-tion numbers of the orbitali. Then there are also the bonds going horizontally in Equation2.56 that we will callvirtual bonds. Those are the bonds we introduced with tensor factorization of the complete coefficient tensor. The truncation based on SVD changes the dimensions of the tensorsA[i]with respect to virtual bonds only.
The dimension of a virtual bond is called thebond dimension. The physical bonds remain unaffected by the SVD truncation, therefore, all configurations can contribute to the MPS representation of the many-body state. This is in strong contrast to other post-Hartree–Fock methods outlined above (see2.3), which all cut configurations from the many-body Hilbert space, i.e. they cut the physical degrees of freedom.
Truncating the MPS in this manner has implications on the entanglement the MPS is able to resolve. Let us divide the orbitals into two setsKandMand assume that the orbitals of the setKare on the left of site and include the orbitaliand the orbitals of the setMare on the right of orbitali. The entanglement entropy between electrons in these two orbital basis sets is given by
S(ρK) =tr{ρKlogρK}, (2.76) whereρKis thereduced density matrix. In the tensor network notation it is
ρK=
U[1]
U[1]
· · ·
· · ·
U[i]
U[i]
S S
V[k]
V[k]
· · ·
· · ·
V[L]
V[L]
sub systemM sub systemK
(2.77)
=
U[1]
U[1]
· · ·
· · ·
U[i]
U[i]
S2
sub systemK
(2.78)
assuming that the MPS|ΨiMPS is in a mixed canonical form with left-normalized tensors forσ ≤iand right-normalized tensors forσ >i. In this basis,ρKis diagonal andS2holds the eigenvalues of the reduced density matrix. If we assume a com-pletely mixed spectrum with all eigenvalues of the reduced density being D1 (this is the worst case), we can estimate the entanglement entropy as
S(ρK)∝log(D). (2.79)
Therefore, the entanglement entropy that the MPS approach is able to resolve grows logarithmically with bond dimensionD. In little entangled states, a small bond
di-mensions is sufficient to describe the state, whereas, as the entanglement grows, also the bond dimension of the MPS needs to grow to maintain precision. Depend-ing on the many-body state, it can vary from bond dimension1, which means no entanglement (Hartree–Fock) up to extremely large bond dimension, which means maximally entangled. When using MPS with small bond dimensions, we assume that the entanglement of the electrons is limited and the real entanglement does not exceed the entanglement the MPS are able to describe.
After the truncation to bond dimensionDhas been performed on all tensors in the MPS, we obtain a new, approximated MPS representation of the many-body state
|ψi ≈ |ψMPSi= X
n1↑n1↓···nL↑nL↓
˜A[1] ˜A[2] · · · ˜A[L]
n1↑n1↓ n2↑n2↓ nL↑nL↓
|n1↑n1↓· · ·nL↑nL↓i, (2.80)
where all MPS tensors have a maximum virtual bond dimensionD. For a given bond dimensionD, this is the quasi optimal approximation to the complete many-body state [124].