Infinite time-evolving block decimation algorithm beyond unitary evolution

The infinite time-evolving block decimation algorithm [G. Vidal, Phys. Rev. Lett. 98, 070201 (2007)] allows to simulate unitary evolution and to compute the ground state of one-dimensional (1D) quantum lattice systems in the thermodynamic limit. Here we extend the algorithm to tackle a much broader class of problems, namely, the simulation of arbitrary one-dimensional evolution operators that can be expressed as a (translationally invariant) tensor network. Relatedly, we also address the problem of finding the dominant eigenvalue and eigenvector of a one-dimensional transfer matrix that can be expressed in the same way. New applications include the simulation, in the thermodynamic limit, of open (i.e., master equation) dynamics and thermal states in 1D quantum systems, as well as calculations with partition functions in two-dimensional (2D) classical systems, on which we elaborate. The present extension of the algorithm also plays a prominent role in the infinite projected entangled-pair states approach to infinite 2D quantum lattice systems.

Figure 1

(Color online) (i) Diagrammatic representation of the tensors $\Gamma$ and $\lambda$ that form an iMPS for a pure state $|\Psi ⟩$ of a translationally invariant chain. (ii) The iMPS is actually an infinite one-dimensional tensor network consisting of alternating copies of the tensors $\Gamma$ and $\lambda$. In the diagram, a leg shared by two tensors corresponds to an index over which there is an implicit sum. (iii) Diagrammatic representation of the conditions in Eqs. (4, 5), which are fulfilled if and only if the iMPS $\left\{\Gamma ,\lambda \right\}$ is in its canonical form. (iv) Tensor network corresponding to the expectation value $⟨\Psi |O^{\left[r\right]}|\Psi ⟩$ of an operator $O^{\left[r\right]}$, acting on site $r$, when $|\Psi ⟩$ is represented by an iMPS $\left\{\Gamma ,\lambda \right\}$ in its canonical form. As a result of the orthonormality of the Schmidt vectors in Eq. (1), only a very small number of tensors need to be considered. (v) Tensor network for the computation of the two-point correlator $⟨\Psi |O^{\left[r\right]}{O}^{\left[s\right]}|\Psi ⟩$ when $|\Psi ⟩$ is represented by an iMPS $\left\{\Gamma ,\lambda \right\}$ in the canonical form.

Figure 2

(Color online) (i)–(iii) The three steps involved in the computation of the canonical form $\left\{{\Gamma }^{\prime },{\lambda }^{\prime }\right\}$ for an iMPS $\left\{\Gamma ,\lambda \right\}$ for $|\Psi ⟩$, as explained in the text. In particular, ${\lambda }^{\prime }$ is obtained in step (ii) as the singular values of $Y^T\lambda X$, whereas ${\Gamma }^{\prime }$ is defined in step (iii). (iv) Both $\left\{\Gamma ,\lambda \right\}$ and $\left\{{\Gamma }^{\prime },{\lambda }^{\prime }\right\}$ represent the same state $|\Psi ⟩$, as can be verified by direct substitution.

Figure 3

(Color online) Proof that $\left\{{\Gamma }^{\prime }{\lambda }^{\prime }\right\}$, the iMPS obtained from $\left\{\Gamma ,\lambda \right\}$ by reorthonormalizing its bond indices following Fig. 2 is indeed in the canonical form. The diagram only shows the condition of Eq. (4) for $R^{\prime }$. The proof of Eq. (5) for $L^{\prime }$ is analogous.

Figure 4

(Color online) Construction of the canonical form $\left\{{\Gamma }^{A^{\prime }},{\lambda }^{A^{\prime }},{\Gamma }^{B^{\prime }},{\lambda }^{B^{\prime }}\right\}$ from an iMPS $\left\{{\Gamma }^A,{\lambda }^A,{\Gamma }^B,{\lambda }^B\right\}$ that is invariant under translations by $n=2$ sites: (i) Coarse-grained iMPS $\left\{\Gamma ,\lambda \right\}$. (ii) Canonical form $\left\{{\Gamma }^{\prime },{\lambda }^{\prime }\right\}$ for the coarse-grained imps $\left\{\Gamma ,\lambda \right\}$. (iii) ${\lambda }^B$ corresponds to ${\lambda }^{\prime }$. (iv) ${\lambda }^{A^{\prime }}$ is obtained through a singular value decomposition, from where also (v) ${\Gamma }^{A^{\prime }}$ and ${\Gamma }^{B^{\prime }}$ are obtained after minor manipulations.

Figure 5

(Color online) Four types of gates $G$ acting on an iMPS: (i) infinite matrix product operator (iMPO); (ii) product of two-site operators; (iii) product of two-to-one-site operators; (iv) tensor product of one-to-two-sites operators. Detailed explanations on how to update the iMPS for cases (ii)–(iv) can be found in Appendix. Notice that gates (iii) and (iv) change the number of sites in the lattice. These gates appear, e.g., when coarse-graining (or fine-graining) the chain.

Figure 6

(Color online) Sequence of transformations (i)–(iii) that produce a truncated iMPS $\left\{{\Gamma }^{\prime }{\lambda }^{\prime }\right\}$ for $|{\Psi }^{\prime }⟩=G|\Psi ⟩$ from an iMPS $\left\{\Gamma ,\lambda \right\}$ for $|\Psi ⟩$, as explained in the text. Notice in particular that, as a result of the truncation step, the initial and final iMPSs have the same rank $\chi$.

Figure 7

(Color online) The partition function $Z\left(\beta \right)$ of a 2D classical system can be written as the contraction of a 2D tensor network. In the case of an infinite square lattice with isotropic and homogeneous interactions, this tensor network consists of infinitely many copies of the tensor $a$ in Eq. (14).

Figure 8

(Color online) Computation of the partition function through Eq. (18). (i) $|{\Psi }_U⟩$ is the up dominant eigenvector of the transfer matrix $T$ with dominant eigenvalue $\theta$. (ii) Similarly, $|{\Psi }_D⟩$ is a down dominant eigenvector of $T$ with dominant eigenvalue $\theta$. (iii) $|{\phi }_R⟩$ is the right dominant eigenvector of matrix $W$ with dominant eigenvalue $\omega$. (iv) $|{\phi }_L⟩$ is the left dominant eigenvector of matrix $W$ with dominant eigenvalue $\omega$.

Figure 9

(Color online) The expectation value $⟨f\left(s^{\left[\vec{r}\right]}\right)⟩$ from Eq. (20) is the ratio of the contraction of two infinite 2D tensor networks. By introducing the dominant eigenvectors of the one-dimensional transfer matrix $T$, we can rewrite $⟨f\left(s^{\left[\vec{r}\right]}\right)⟩$ as the ratio of the trace of two infinite 1D tensor networks. Finally, by introducing the dominant eigenvectors of matrix $W$, we obtain a ratio of two simple tensor networks.

Figure 10

(Color online) Following steps analogous as those of Fig. 9 for the expectation value $⟨f\left(s^{\left[\vec{r}\right]}\right)⟩$, the two-point correlator $⟨f\left(s^{\left[\vec{r}\right]}\right)g\left(s^{\left[{\vec{r}}^{\prime }\right]}\right)⟩$ can also be reduced to the ratio of two simple tensor networks.

Figure 11

(Color online) Magnetization per lattice site for the infinite-size 2D classical Ising model at different temperatures $\beta$. The exact solution $m={\mathbf{\left(}1-\left\{{\left[\text{sinh}\left(2\beta \right)\right]}^{-4}\right\}\mathbf{\right)}}^{1/8}$ has been included. The numerical results have been obtained by approximating the dominant eigenvectors of the one-dimensional transfer matrix $T$ with an iMPS of rank $\chi =40$. The inset shows the relative error.

Figure 12

(Color online) Two-point correlators for the infinite-size 2D classical Ising model at critical temperature, ${\beta }_c=\frac{1}{2}\text{log}\left(1+\sqrt{2}\right)$, along a row of the square lattice. The exact solution, which scales as ${⟨s^{\left[\vec{r}\right]}{s}^{\left[{\vec{r}}^{\prime }\right]}⟩}_{{\beta }_c}\approx c{\left|\vec{r}-{\vec{r}}^{\prime }\right|}^{-1/4}$, has been included. The numerical results have been obtained by approximating the dominant eigenvectors of the one-dimensional transfer matrix $T$ with an iMPS of rank $\chi =40$, $\chi =60$, and $\chi =80$.

Figure 13

(Color online) Steps in the updating of the iMPS after the action of a MPO.

Figure 14

(Color online) Steps in the updating of the iMPS after the action of the tensor product of two-site operators.

Figure 15

(Color online) Steps in the updating of the iMPS after the action of the tensor product of two-to-one-site gates.

Figure 16

(Color online) Steps in the updating of the iMPS after the action of the tensor product of one-to-two-sites gates.