Two-Dimensional Tensor Networks and Contraction Algorithms

Ran, Shi-Ju; Tirrito, Emanuele; Peng, Cheng; Chen, Xi; Tagliacozzo, Luca; Su, Gang; Lewenstein, Maciej

doi:10.1007/978-3-030-34489-4_3

Shi-Ju Ran²³,
Emanuele Tirrito²⁴,
Cheng Peng²⁵,
Xi Chen²⁶,
Luca Tagliacozzo²⁷,
Gang Su²⁸ &
…
Maciej Lewenstein²⁴

Part of the book series: Lecture Notes in Physics ((LNP,volume 964))

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Abstract

In this section, we will first demonstrate in Sect. 3.1 that many important physical problems can be transformed to 2D TNs , and the central tasks become to compute the corresponding TN contractions. From Sects. 3.2 to 3.5, we will then present several paradigm contraction algorithms of 2D TNs including TRG , TEBD , and CTMRG . Relations to other distinguished algorithms and the exactly contractible TNs will also be discussed.

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3.1 From Physical Problems to Two-Dimensional Tensor Networks

3.1.1 Classical Partition Functions

Partition function, which is a function of the variables of a thermodynamic state such as temperature, volume, and etc., contains the statistical information of a thermodynamic equilibrium system. From its derivatives of different orders, we can calculate the energy, free energy, entropy, and so on. Levin and Nave pointed out in Ref. [1] that the partition functions of statistical lattice models (such as Ising and Potts models) with local interactions can be written in the form of TN. Without losing generality, we take square lattice as an example.

Let us start from the simplest case: the classical Ising model on a single square with only four sites. The four Ising spins denoted by s _i (i = 1, 2, 3, 4) locate on the four corners of the square, as shown in Fig. 3.1a; each spin can be up or down, represented by s _i = 0 and 1, respectively. The classical Hamiltonian of such a system reads

$$\displaystyle \begin{aligned} \begin{array}{rcl} H_{s_1s_2s_3s_4}=J(s_1s_2 + s_2s_3 + s_3s_4 + s_4s_1) - h (s_1+s_2+s_3+s_4), \end{array} \end{aligned} $$

(3.1)

with J the coupling constant and h the magnetic field.

When the model reaches the equilibrium at temperature T, the probability of each possible spin configuration is determined by the Maxwell–Boltzmann factor

$$\displaystyle \begin{aligned} \begin{array}{rcl} T_{s_1s_2s_3s_4}=e^{-\beta H_{s_1s_2s_3s_4}}, {} \end{array} \end{aligned} $$

(3.2)

with the inverse temperature β = 1∕T.^{Footnote 1} Obviously, Eq. (3.2) is a fourth-order tensor T, where each element gives the probability of the corresponding configuration.

The partition function is defined as the summation of the probability of all configurations. In the language of tensor, it is obtained by simply summing over all indexes as

$$\displaystyle \begin{aligned} \begin{array}{rcl} Z=\sum_{s_1s_2s_3s_4} T_{s_1s_2s_3s_4}. \end{array} \end{aligned} $$

(3.3)

Let us proceed a little bit further by considering four squares, whose partition function can be written in a TN with four tensors (Fig. 3.1b) as

$$\displaystyle \begin{aligned} \begin{array}{rcl} Z = \sum_{\{ss'\}} T_{s_1s_2s_2^{\prime}s_1^{\prime}} T_{s_2^{\prime}s_3s_4s_3^{\prime}} T_{s_4^{\prime}s_3^{\prime}s_5s_6} T_{s_8s_1^{\prime}s_4^{\prime}s_7}. \end{array} \end{aligned} $$

(3.4)

Each of the indexes {s′} inside the TN is shared by two tensors, representing the spin that appears in both of the squares. The partition function is obtained by summing over all indexes.

For the infinite square lattice, the probability of a certain spin configuration (s ₁, s ₂, ⋯) is given by the product of infinite number of tensor elements as

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} e^{-\beta H_{\{s\}}}=e^{-\beta H_{s_1s_2s_3s_4}} e^{-\beta H_{s_4s_5s_6s_7}} \cdots =T_{s_1s_2s_3s_4}T_{s_4s_5s_6s_7} \cdots \end{array} \end{aligned} $$

(3.5)

Then the partition function is given by the contraction of an infinite TN formed by the copies of T (Eq. (3.2)) as

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} Z=\sum_{\{s\}} \prod_{n} T_{s^n_1 s^n_2 s^n_3 s^n_4}, {} \end{array} \end{aligned} $$

(3.6)

where two indexes satisfy $s^n_j = s^m_k$ if they refer to the same Ising spin. The graphic representation of Eq. (3.6) is shown in Fig. 3.1c. One can see that on square lattice, the TN still has the geometry of a square lattice. In fact, such a way will give a TN that has a geometry of the dual lattice of the system, and the dual of the square lattice is itself.

For the Q-state Potts model on square lattice, the partition function has the same TN representation as that of the Ising model, except that the elements of the tensor are given by the Boltzmann weight of the Potts model and the dimension of each index is Q. Note that the Potts model with q = 2 is equivalent to the Ising model.

Another example is the eight-vertex model proposed by Baxter in 1971 [2]. It is one of the “ice-type” statistic lattice model, and can be considered as the classical correspondence of the Z₂ spin liquid state. The tensor that gives the TN of the partition function is also (2 × 2 × 2 × 2), whose non-zero elements are

$$\displaystyle \begin{aligned} \begin{array}{rcl} T_{s_1,\cdots, s_N}&=& \left\{ \begin{array}{lll} 1, \ \ s_1+\cdots + s_N=even, \\ 0, \ \ \text{otherwise}. \end{array} \right. \end{array} \end{aligned} $$

(3.7)

We shall remark that there are more than one ways to define the TN of the partition function of a classical system. For example, when there only exist nearest-neighbor couplings, one can define a matrix $M_{ss'} = e^{-\beta H_{ss'}}$ on each bond and put on each site a super-digonal tensor I (or called copy tensor) defined as

$$\displaystyle \begin{aligned} I_{s_1,\cdots, s_N}= \begin{cases} 1, \ \ s_1 = \cdots = s_N;\\ 0, \ \ \text{otherwise}. \end{cases} \end{aligned} $$

(3.8)

Then the TN of the partition function is the contraction of copies of M and I, and possesses exactly the same geometry of the original lattice (instead of the dual one).

3.1.2 Quantum Observables

With a TN state, the computations of quantum observables as $\langle \psi | \hat {O} | \psi \rangle $ and 〈ψ|ψ〉 are the contraction of a scalar TN, where $\hat {O}$ can be any operator. For a 1D MPS , this can be easily calculated, since one only needs to deal with a 1D TN stripe. For 2D PEPS , such calculations become contractions of 2D TNs. Taking 〈ψ|ψ〉 as an example, the TN of such an inner product is the contraction of the copies of the local tensor (Fig. 3.1c) defined as

$$\displaystyle \begin{aligned} \begin{array}{rcl} T_{a_1a_2a_3a_4} = \sum_{s} P_{s,a^{\prime\prime}_1a^{\prime\prime}_2a^{\prime\prime}_3a^{\prime\prime}_4}^{\ast} P_{s,a_1^{\prime}a_2^{\prime}a_3^{\prime}a_4^{\prime}}, \end{array} \end{aligned} $$

(3.9)

with P the tensor of the PEPS and $a_i = (a^{\prime }_i, a^{\prime \prime }_i)$. There are no open indexes left and the TN gives the scalar 〈ψ|ψ〉. The TN for computing the observable $\langle \hat {O} \rangle $ is similar. The only difference is that we should substitute some small number of $T_{a_1a_2a_3a_4}$ in original TN of 〈ψ|ψ〉 with “impurities” at the sites where the operators locate. Taking one-body operator as an example, the “impurity” tensor on this site can be defined as

$$\displaystyle \begin{aligned} \begin{array}{rcl} \widetilde{T}_{a_1a_2a_3a_4}^{[i]} = \sum_{s,s'} P_{s,a^{\prime\prime}_1a^{\prime\prime}_2a^{\prime\prime}_3a^{\prime\prime}_4}^{\ast} \hat{O}_{s,s'}^{[i]} P_{s',a_1^{\prime}a_2^{\prime}a_3^{\prime}a_4^{\prime}}. \end{array} \end{aligned} $$

(3.10)

In such a case, the single-site observables can be represented by the TN contraction of

$$\displaystyle \begin{aligned} \begin{array}{rcl} \frac{\langle \psi| \hat{O}^{[i]}|\psi \rangle}{\langle \psi| \psi \rangle} = \frac{\text{tTr }\ \widetilde{T}^{[i]} \prod_{n\neq i} T}{\text{tTr }\ \prod_{n=1}^{N} T}. \end{array} \end{aligned} $$

(3.11)

For some non-local observables, e.g., the correlation function, the contraction of $\langle \psi | \hat {O}^{[i]} \hat {O}^{[j]} |\psi \rangle $ is nothing but adding another “impurity” by

$$\displaystyle \begin{aligned} \begin{array}{rcl} \langle \psi| \hat{O}^{[i]} \hat{O}^{[j]} |\psi \rangle = \text{tTr }\ \widetilde{T}^{[i]}\widetilde{T}^{[j]} \prod_{n\neq i,j}^{N} T. \end{array} \end{aligned} $$

(3.12)

3.1.3 Ground-State and Finite-Temperature Simulations

Ground-state simulations of 1D quantum models with short-range interactions can also be efficiently transferred to 2D TN contractions. When minimizing the energy

$$\displaystyle \begin{aligned} \begin{array}{rcl} E=\frac{\langle \psi | \hat{H} | \psi \rangle} {\langle \psi | \psi \rangle}, \end{array} \end{aligned} $$

(3.13)

where we write |ψ〉 as an MPS . Generally speaking, there are two ways to solve the minimization problem: (1) simply treat all the tensor elements as variational parameters; (2) simulate the imaginary-time evolution

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} |\psi_{gs}\rangle = \lim_{\beta \rightarrow \infty} \frac{e^{-\beta \hat{H}}|\psi\rangle}{\parallel{e^{-\beta \hat{H}}|\psi\rangle}\parallel}. \end{array} \end{aligned} $$

(3.14)

The first way can be realized by, e.g., Monte Carlo methods where one could randomly change or choose the value of each tensor element to locate the minimal of energy. One can also use the Newton method and solve the partial-derivative equations ∂E∕∂x _n = 0 with x _n standing for an arbitrary variational parameter. Anyway, it is inevitable to calculate E (i.e., $\langle \psi | \hat {H} | \psi \rangle $ and 〈ψ|ψ〉) for most cases, which is to contraction the corresponding TNs as explained above.

We shall stress that without TN , the dimension of the ground state (i.e., the number of variational parameters) increases exponentially with the system size, which makes the ground-state simulations impossible for large systems.

The second way of computing the ground state with imaginary-time evolution is more or less like an “annealing” process. One starts from an arbitrarily chosen initial state and acts the imaginary-time evolution operator on it. The “temperature” is lowered a little for each step, until the state reaches a fixed point. Mathematically speaking, by using Trotter-Suzuki decomposition, such an evolution is written in a TN defined on (D + 1)-dimensional lattice, with D the dimension of the real space of the model.

Here, we take a 1D chain as an example. We assume that the Hamiltonian only contains at most nearest-neighbor couplings, which reads

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} \hat{H}=\sum_{n} \hat{h}_{n,n+1}, \end{array} \end{aligned} $$

(3.15)

with $\hat {h}_{n,n+1}$ containing the on-site and two-body interactions of the n-th and n + 1-th sites. It is useful to divide $\hat {H}$ into two groups, $\hat {H}=\hat {H}^e+\hat {H}^{o}$ as

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} \begin{aligned} \hat{H}^e \equiv\sum_{even\ n}\hat{h}_{n,n+1}, \ \ \ \hat{H}^{o} \equiv\sum_{odd\ n}\hat{h}_{n,n+1}. \end{aligned} \end{array} \end{aligned} $$

(3.16)

By doing so, each two terms in $\hat {H}^e$ or $\hat {H}^{o}$ commutes with each other. Then the evolution operator $\hat {U}(\tau )$ for infinitesimal imaginary time τ → 0 can be written as

$$\displaystyle \begin{aligned} \begin{array}{rcl} \hat{U}(\tau) = e^{-\tau \hat{H}} = e^{-\tau \hat{H}^e} e^{-\tau \hat{H}^o} + O\left(\tau^2\right) \left[\hat{H}^e, \hat{H}^o\right]. {} \end{array} \end{aligned} $$

(3.17)

If τ is small enough, the high-order terms are negligible, and the evolution operator becomes

$$\displaystyle \begin{aligned} \begin{array}{rcl} \hat{U}(\tau) \simeq \prod_{n} \hat{U}(\tau)_{n,n+1}, {} \end{array} \end{aligned} $$

(3.18)

with the two-site evolution operator $\hat {U}(\tau )_{n,n+1} = e^{-\tau \hat {H}_{n,n+1}}$.

The above procedure is known as the first-order Trotter-Suzuki decomposition [3,4,5]. Note that higher-order decomposition can also be adopted. For example, one may use the second-order Trotter-Suzuki decomposition that is written as

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} e^{-\tau \hat{H}} \simeq e^{-\frac{\tau}{2} \hat{H}^e} e^{-\tau \hat{H}^o} e^{-\frac{\tau}{2} \hat{H}^e}. \end{array} \end{aligned} $$

(3.19)

With Eq. (3.18), the time evolution can be transferred to a TN, where the local tensor is actually the coefficients of $\hat {U}(\tau )_{n,n+1}$, satisfying

$$\displaystyle \begin{aligned} \begin{array}{rcl} T_{s_n s_{n+1} s^{\prime}_n s^{\prime}_{n+1}} = \langle s^{\prime}_n s^{\prime}_{n+1} | \hat{U}(\tau)_{n,n+1} | s_n s_{n+1}\rangle. {} \end{array} \end{aligned} $$

(3.20)

Such a TN is defined in a plain of two dimensions that corresponds to the spatial and (real or imaginary) time, respectively. The initial state is located at the bottom of the TN (β = 0) and its evolution is to do the TN contraction which can be efficiently solved by TN algorithms (presented later).

In addition, one can readily see that the evolution of a 2D state leads to the contraction of a 3D TN. Such a TN scheme provides a straightforward picture to understand the equivalence between a (d + 1)-dimensional classical and a d-dimensional quantum theory. Similarly, the finite-temperature simulations of a quantum system can be transferred to TN contractions with Trotter-Suzuki decomposition. For the density operator $\hat {\rho }(\beta ) = e^{-\beta \hat {H}}$, the TN is formed by the same tensor given by Eq. (3.20).

3.2 Tensor Renormalization Group

In 2007, Levin and Nave proposed TRG approach [1] to contract the TN of 2D classical lattice models. In 2008, Gu et al. further developed TRG to handle 2D quantum topological phases [6]. TRG can be considered as a coarse-graining contraction algorithm. To introduce the TRG algorithm, let us consider a square TN formed by infinite number of copies of a fourth-order tensor $T_{a_1a_2a_3a_4}$ (see the left side of Fig. 3.2).

Contraction and Truncation

The idea of TRG is to iteratively “coarse-grain” the TN without changing the bond dimensions, the geometry of the network, and the translational invariance. Such a process is realized by two local operations in each iteration. Let us denote the tensor in the t-th iteration as T ^(t) (we take T ⁽⁰⁾ = T). For obtaining T ^(t+1), the first step is to decompose T ^(t) by SVD in two different ways (Fig. 3.2) as

$$\displaystyle \begin{aligned} \begin{array}{rcl} T^{(t)}_{a_1a_2a_3a_4} = \sum_{b} U_{a_1a_2b}V_{a_3a_4b}, \end{array} \end{aligned} $$

(3.21)

$$\displaystyle \begin{aligned} \begin{array}{rcl} T^{(t)}_{a_1a_2a_3a_4} = \sum_{b} X_{a_4a_1b}Y_{a_2a_3b}. {} \end{array} \end{aligned} $$

(3.22)

Note that the singular value spectrum can be handled by multiplying it with the tensor(s), and the dimension of the new index satisfies dim(b) = χ ² with χ the dimension of each bond of T ^(t).

The purpose of the first step is to deform the TN, so that in the second step, a new tensor T ^(t+1) can be obtained by contracting the four tensors that form a square (Fig. 3.2) as

$$\displaystyle \begin{aligned} \begin{array}{rcl} T^{(t+1)}_{b_1b_2b_3b_4} \leftarrow \sum_{a_1a_2a_3a_4} V_{a_1a_2b_1} Y_{a_2a_3b_2} U_{a_3a_4b_3} X_{a_4a_1b_4}. {} \end{array} \end{aligned} $$

(3.23)

We use an arrow instead of the equal sign, because one may need to divide the tensor by a proper number to keep the value of the elements from being divergent. The arrows will be used in the same way below.

These two steps define the contraction strategy of TRG. By the first step, the number of tensors in the TN (i.e., the size of the TN ) increases from N to 2N, and by the second step, it decreases from 2N to N∕2. Thus, after t times of each iterations, the number of tensors decreases to the $\frac {1}{2^t}$ of its original number. For this reason, TRG is an exponential contraction algorithm.

Error and Environment

The dimension of the tensor at the t-th iteration becomes $\chi ^{2^t}$, if no truncations are implemented. This means that truncations of the bond dimensions are necessary. In its original proposal, the dimension is truncated by only keeping the singular vectors of the χ-largest singular values in Eq. (3.22). Then the new tensor T ^(t+1) obtained by Eq. (3.23) has exactly the same dimension as T ^(t).

Each truncation will absolutely introduce some error, which is called the truncation error. Consistent with Eq. (2.7), the truncation error is quantified by the discarded singular values λ as

$$\displaystyle \begin{aligned} \begin{array}{rcl} \varepsilon = \frac{\sqrt{\sum_{b=\chi}^{\chi^2-1} \lambda_{b}^2}}{\sqrt{\sum_{b=0}^{\chi^2-1} \lambda_{b}^2}}. {} \end{array} \end{aligned} $$

(3.24)

According to the linear algebra, ε in fact gives the error of the SVD given in Eq. (3.22), meaning that such a truncation minimizes the error of reducing the rank of T ^(t), which reads

$$\displaystyle \begin{aligned} \begin{array}{rcl} \varepsilon = |T^{(t)}_{a_1a_2a_3a_4} - \sum_{b=0}^{\chi-1} U_{a_1a_2b} V_{a_3a_4b}|. {} \end{array} \end{aligned} $$

(3.25)

One may repeat the contraction-and-truncation process until T ^(t) converges. It usually only takes ∼10 steps, after which one in fact contract a TN of 2^t tensors to a single tensor.

The truncation is optimized according to the SVD of T ^(t). Thus, T ^(t) is called the environment. In general, the tensor(s) that determines the truncations is called the environment. It is a key factor to the accuracy and efficiency of the algorithm. For those that use local environments, like TRG, the efficiency is relatively high since the truncations are easy to compute. But, the accuracy is bounded since the truncations are only optimized according to some local information (like in TRG the local partitioning T ^(t)).

One may choose other tensors or even the whole TN as the environment. In 2009, Xie et al. proposed the second renormalization group (SRG) algorithm [7]. The idea is in each truncation step of TRG , they define the global environment that is a fourth-order tensor $\mathscr {E}_{a_1^{\tilde {n}} a_2^{\tilde {n}} a_3^{\tilde {n}} a_4^{\tilde {n}}} = \sum _{\{a\}} \prod _{n \neq \tilde {n}} T^{(n, t)}_{a_1^na_2^na_3^na_4^n}$ with T ^{(n, t)} the n-th tensor in the t-th step and $\tilde {n}$ the tensor to be truncated. $\mathscr {E}$ is the contraction of the whole TN after getting rid of $T^{(\tilde {n}, t)}$, and is computed by TRG. Then the truncation is obtained not by the SVD of $T^{(\tilde {n}, t)}$, but by the SVD of $\mathscr {E}$. The word “second” in the name of the algorithm comes from the fact that in each step of the original TRG, they use a second TRG to calculate the environment. SRG is obviously more consuming, but bears much higher accuracy than TRG. The balance between accuracy and efficiency, which can be controlled by the choice of environment, is one main factor to consider while developing or choosing the TN algorithms.

3.3 Corner Transfer Matrix Renormalization Group

In the 1960s, the corner transfer matrix (CTM) idea was developed originally by Baxter in Refs. [8, 9] and a book [10]. Such ideas and methods have been applied to various models, for example, the chiral Potts model [11,12,13], the 8-vertex model [2, 14, 15], and to the 3D Ising model [16]. Combining CTM with DMRG, Nishino and Okunishi proposed the CTMRG [17] in 1996 and applied it to several models [17,18,19,20,21,22,23,24,25,26,27]. In 2009, Orús and Vidal further developed CTMRG to deal with TNs [28]. What they proposed to do is to put eight variational tensors to be optimized in the algorithm, which are four corner transfer matrices C ^[1], C ^[2], C ^[3], C ^[4] and four row (column) tensors R ^[1], R ^[2], R ^[3], R ^[4], on the boundary, and then to contract the tensors in the TN to these variational tensors in a specific order shown in Fig. 3.3. The TN contraction is considered to be solved with the variational tensors when they converge in this contraction process. Compared with the boundary-state methods in the last subsection, the tensors in CTMRG define the states on both the boundaries and corners.

Contraction

In each iteration step of CTMRG , one choses two corner matrices on the same side and the row tensor between them, e.g., C ^[1], C ^[2], and R ^[2]. The update of these tensors (Fig. 3.4) follows

$$\displaystyle \begin{aligned} \begin{array}{rcl} \tilde{C}^{[1]}_{\tilde{b}_2b^{\prime}_1} &\displaystyle \leftarrow&\displaystyle \sum_{b_1} C^{[1]}_{b_1b_2}R^{[1]}_{b_1a_1b^{\prime}_1}, \end{array} \end{aligned} $$

(3.26)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \tilde{R}^{[2]}_{\tilde{b}_2 a_4 \tilde{b_3}} &\displaystyle \leftarrow&\displaystyle \sum_{a_2}R^{[2]}_{b_2a_2b_3}T_{a_1a_2a_3a_4}, {} \end{array} \end{aligned} $$

(3.27)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \tilde{C}^{[2]}_{\tilde{b}_3b^{\prime}_4} &\displaystyle \leftarrow&\displaystyle \sum_{b_4} C^{[2]}_{b_3b_4}R^{[3]}_{b_4a_3b^{\prime}_4}, \end{array} \end{aligned} $$

(3.28)

where $\tilde {b}_2=(b_2,a_1)$ and $\tilde {b}_3=(b_3,a_1)$.

After the contraction given above, it can be considered that one column of the TN (as well as the corresponding row tensors R ^[1] and R ^[3]) is contracted. Then one chooses other corner matrices and row tensors (such as $\tilde {C}^{[1]}$, C ^[4], and R ^[1]) and implement similar contractions. By iteratively doing so, the TN is contracted in the way shown in Fig. 3.3.

Note that for a finite TN , the initial corner matrices and row tensors should be taken as the tensors locating on the boundary of the TN. For an infinite TN, they can be initialized randomly, and the contraction should be iterated until the preset convergence is reached.

CTMRG can be regarded as a polynomial contraction scheme. One can see that the number of tensors that are contracted at each step is determined by the length of the boundary of the TN at each iteration time. When contracting a 2D TN defined on a (L × L) square lattice as an example, the length of each side is L − 2t at the t-th step. The boundary length of the TN (i.e., the number of tensors contracted at the t-th step) bears a linear relation with t as 4(L − 2t) − 4. For a 3D TN such as cubic TN, the boundary length scales as 6(L − 2t)² − 12(L − 2t) + 8, thus the CTMRG for a 3D TN (if exists) gives a polynomial contraction.

Truncation

One can see that after the contraction in each iteration step, the bond dimensions of the variational tensors increase. Truncations are then in need to prevent the excessive growth of the bond dimensions. In Ref. [28], the truncation is obtained by inserting a pair of isometries V and V ^† in the enlarged bonds. A reasonable (but not the only choice) of V for translational invariant TN is to consider the eigenvalue decomposition on the sum of corner transfer matrices as

$$\displaystyle \begin{aligned} \begin{array}{rcl} \sum_{b}\tilde{C}^{[1]\dagger}_{\tilde{b}b}{\tilde{C}^{[1]}}_{\tilde{b}'b} + \sum_{b}\tilde{C}^{[2]\dagger}_{\tilde{b}b}{\tilde{C}^{[1]}}_{\tilde{b}'b} \simeq \sum_{b=0}^{\chi-1} V_{\tilde{b}b} \varLambda_{b} V^{*}_{\tilde{b}'b}. {} \end{array} \end{aligned} $$

(3.29)

Only the χ largest eigenvalues are preserved. Therefore, V is a matrix of the dimension Dχ × χ, where D is the bond dimension of T and χ is the dimension cut-off. We then truncate $\tilde {C}^{[1]}$, $\tilde {R}^{[2]}$, and $\tilde {C}^{[2]}$ using V as

$$\displaystyle \begin{aligned} \begin{array}{rcl} C^{[1]}_{b^{\prime}_1b_2} &\displaystyle =&\displaystyle \sum_{\tilde{b}_2}\tilde{C}^{[1]}_{\tilde{b}_2b^{\prime}_1}V^{*}_{\tilde{b}_2b_2}, \end{array} \end{aligned} $$

(3.30)

$$\displaystyle \begin{aligned} \begin{array}{rcl} R^{[2]}_{b_2a_4b_3} &\displaystyle =&\displaystyle \sum_{\tilde{b}_2,\tilde{b}_3}\tilde{R}^{[2]}_{\tilde{b}_2a_4\tilde{b}_3} V_{\tilde{b}_2b_2} V^{*}_{\tilde{b}_3b_3},{} \end{array} \end{aligned} $$

(3.31)

$$\displaystyle \begin{aligned} \begin{array}{rcl} C^{[2]}_{b_3b^{\prime}_4} &\displaystyle =&\displaystyle \sum_{\tilde{b}_3}\tilde{C}^{[2]}_{\tilde{b}_3b^{\prime}_4} V_{\tilde{b}_3b_3}. \end{array} \end{aligned} $$

(3.32)