2.4 Matrix Product States
2.4.3 Graphical Notation
the sequence of matrix products in Equation2.56. Both of these reasons forbid the application of the area law, therefore, there is no mathematical guarantee that MPS with limited bond dimension are a valid approach to represent the many-body state in quantum chemical systems.
Further is the area law limited to gapped ground states and it does not say anything about how the entanglement evolves with time. In time-dependent situations, it is possible that the many-body state leaves the area of the Hilbert space we are able to resolve efficiently using the MPS approach to represent the many-body state. This issue becomes even more problematic in long time studies. The more time evolves, the more the system may leave the area of the Hilbert space the MPS are able to resolve. This issue is called therunaway error. It is one of the reasons why, in this study, we will completely focus on the short-time behavior that takes place within a few femtoseconds after excitation. The idea is that the exact many-body state has not left the area of the Hilbert space in the time range we are looking at. There, the MPS approach is still able to describe the many-body state appropriately. Later in this thesis, we will outline a couple of concepts to reduce the runaway error, as it might be possible that the many-body state can be described by the MPS approach although the entanglement entropy has left the manifold of area law states.
Despite the bad news on long-ranged interaction and time-dependence, the MPS approach has shown striking performance for both, long-ranged systems, as well as in time-dependent situations. For many years, MPS are heavily used in quantum chemistry to predict ground states, excited states [48,146–149], avoided crossings [48, 49,150,151], and spin-splittings [152–155] for many molecules regardless the system being long-ranged and high-dimensional. Of course, the guarantee of the area law does not apply here. However, this does not forbid the use of MPS as a valid approach to represent the many-body state. The same holds for time-dependent studies, also here MPS have proven to be a very efficient way in describing the time-dependent many-body state, ranging from spin dynamics [54,156–159], transport properties[54, 160–163], to ionization potentials [60,164].
a)
a
b)
a scalar
v
i
a vector c)
M
i j
a matrix d)
T
σ1σ2σ3σ4σ5
· · · σn
a rankntensor Figure 2.5.: Graphical representation of tensors with various ranks. a) a scalar is depicted
using a box without any lines attached. b) a vector is depicted using a box with one line attached. c) a matrix is depicted using a box with two lines attached. d) a rankntensor is depicted using a box withnlines attached.
Later in this section, will we explain the algorithms to perform operations on MPS in the language of tensor networks, therefore it will be beneficial to have a good understanding of the ideas behind the tensor network concept and its notation [101].
Tensor networks are (graphical) representations of linear algebra operations. To every tensor network we can either write down a formula specifying the operation or we can draw a graphic that represents the network much more comprehensively. The large number of indices and sums occurring in tensor networks usually obscures the operation from analytic formulas, however, the graphical representation makes it easy to understand it at first sight.
In these graphical representations, the tensors are described by different boxes that are connected by lines that we callbondsin the following. These bonds represent free indices of the tensor. For example, a scalaradoes not have any free indices, therefore, it is represented by a box without any lines attached (see Figure2.5a) ) . A vectorvhas one free index, therefore it is described by a box with one line attached to its graphical representation (see Figure2.5b) ). A matrixM has two free indices, therefore it is described by a box with two lines attached to its graphical representation (see Figure 2.5c) ). We can continue this for arbitrary tensors of rankn, that will havenlines attached to its box (see Figure2.5d) ). The form of the box will not have any intrinsic meaning in this work, however, we will use different shapes to distinguish tensors belonging to different quantities.
There are two types of bonds: We have open bonds and we have closed bonds. Whereas the open bonds have only one end connected to a tensor, the closed bonds
have both ends connected to tensors. Whenever a bond is closed, it represents a sum over an index. For example, the trace of a matrix can be represented by
tr(M) =X
i
Mii = M i , (2.57)
where the result is a scalar, since it does not have any free bonds. In general, it does not matter how the bonds are attached to the box, they can be attached at the upper and the lower part, or at the left and the right of a box. Sometimes we give the index a name (such as in Equation2.57), however, in most cases it is up to the reader to keep track of which bond represents which index. We can depict any linear tensor operation using this notation, for example, matrix multiplicationC =A·Bis visualized as
Cij=X
k
AikBkj = i A k B j , (2.58)
with one closed bond (k) and two open bonds (iandj) since the result of the matrix multiplication is again a matrix.
The many-body state in its MPS representation as given in Equation2.56becomes in this graphical notation
|ψMPSi= X
n1↑n1↓···nL↑nL↓
A[1] A[2] · · · A[L]
n1↑n1↓ n2↑n2↓ nL↑nL↓
|n1↑n1↓· · ·nL↑nL↓i, (2.59) where we substituted the sequence of matrix products by its graphical representation.
From this graphical representation, it can be seen easily, that for a fixed configuration n1↑n1↓· · ·nL↑nL↓, the very left and the very right tensor need to be rank1tensors (vectors) and the remaining are rank2tensors (matrices). When incorporating the indicesniσwhich specify the configuration into the tensors, all ranks are increased by two (This is why the tensors in Equation2.59have three (four) bonds).
The state norm of a many-body state in the MPS representation hψMPS|ψMPSiis calculated as (using the orthonormality of the occupation number basis and the sum rule for closed bonds)
hψMPS|ψMPSi= X
n1↑n1↓···nL↑nL↓
A∗n1↑n1↓· · ·A∗nL↑nL↓An1↑n1↓· · ·AnL↑nL↓ (2.60)
=A[1] A[2] · · · A[L]
A[1] A[2] · · · A[L]
, (2.61)
where we used the convention of complex conjugated matrices and vectors to have their indices at the bottom (This is just a convention that does not apply in general in tensor networks). The result of the overlaphψMPS|ψMPSiis a scalar, therefore, its graphical representation in Equation2.61does not have any open bonds.