Classical Simulation of Infinite-Size Quantum Lattice Systems in One Spatial Dimension

Invariance under translation is exploited to efficiently simulate one-dimensional quantum lattice systems in the limit of an infinite lattice. Both the computation of the ground state and the simulation of time evolution are considered.

Figure 1

Diagrammatic representation of several decompositions of $|\Psi ⟩$ in terms of a network of tensors, see 8 for more details. Each filled circle represents a tensor $\Gamma$ and each edge represents an index, such as $\alpha$ or $i$. Hanging from an open side of an edge there is an (omitted) orthonormal basis, such as $|{\Phi }_{\alpha }^{[◃0]}⟩$ or $|i^{[1]}⟩$. An open diamond on top of an edge represents a set of weights $\lambda$ (Schmidt coefficients) for the corresponding index. (i) Schmidt decomposition, Eq. (3), according to semi-infinite sublattices $[◃0]$ and $[1▹]$. The bases $|{\Phi }_{\alpha }^{[◃0]}⟩$ and $|{\Phi }_{\alpha }^{[1▹]}⟩$ are linked through index $\alpha$ with weights ${\lambda }_{\alpha }^{[0]}$. (ii) Expansion of $|\Psi ⟩$ in terms of the Schmidt bases for the semi-infinite sublattices $[◃0]$ and $[2▹]$ and an orthonormal basis for site 1; see Eq. (8). (iii) Expansion of $|\Psi ⟩$ in terms of orthonormal sets of vectors for sublattices $[◃0]$ and $[3▹]$ and sites 1 and 2.

Figure 2

Two diagrammatical representations (see also Fig. 1 and Ref. 8) for state $U^{AB}|\Psi ⟩$: (i) tensor network representation containing both an MPS for $|\Psi ⟩$ and two-site gates $U$ acting on each pair of sites $\{2r,2r+1\}$, $\forall r\in \mathbb{Z}$; (ii) new MPS for $U^{AB}|\Psi ⟩$. Notice that both structures are invariant under shifts by two lattice sites and are completely specified by a small number of tensors, in spite of the fact that they represent a state of an infinite 1D lattice.

Figure 3

In order to update the MPS after gate $U$ has been applied, see Fig. 2, we first contract the tensor network (i) into a single tensor ${\Theta }_{\alpha ij\gamma }$ (ii). We then compute the singular value decomposition of $\Theta$ according to the index bipartition $[\alpha i]:[j\gamma ]$, namely, $\Theta =\sum _{\beta }{X}_{[\alpha i]\beta }{\tilde{\lambda }}_{\beta }^A{Y}_{\beta [j\gamma ]}$ as in (iii). We introduce ${\lambda }^B$ back into the network (iv) and form tensors ${\tilde{\Gamma }}^A$ and ${\tilde{\Gamma }}^{[B]}$ in (iv) by attaching to $X$ and $Y$ the inverse of the Schmidt coefficients ${\lambda }^B$. All such matrix manipulations are achieved with $O(d^2{\chi }^2)$ space and $O(d^3{\chi }^3)$ and need to be performed only once in order to update the MPS for the whole infinite chain.

Figure 4

Left: spectrum $p_{\alpha }=({\lambda }_{\alpha }{)}^2$ of the density matrix for half of the infinite chain, see Eq. (4), for different values of the magnetic field $h_z$. For $h_z\gg 1$ the eigenvalues $p_{\alpha }$ decay very fast with $\alpha$. As $h_z$ approaches the critical value $h_z^{*}=1$, the decay is less pronounced and the computation of the spectrum becomes more demanding. Right: however, for $h_z=1.001$ we still obtain the 50, 100, and 250 first eigenvalues with remarkable accuracy, the error on $p_{\alpha }$ being typically several orders of magnitude smaller than its value.

Figure 5

Two-point correlator $C_2(r)$ of Eq. (13). Upper part: for $h_z=1.001$, when the distance $r$ between spins is up to several hundred sites, $C_2(r)$ seems to decay as a power law (as in the critical case), but for $r$ of the order of 1000, the exponential character of the decay becomes manifest. Lower part: even though iTEBD is unable to properly reproduce the spectrum $p_{\alpha }$ of half an infinite chain at criticality, $h_z=1$, (see Fig. 4), we still obtain a very accurate approximation to the correlator $C_2(r)$ when $r$ is of the order of several thousands of spins.

Figure 6

Simulation of the evolution in time of $m_z(t)$ in Eq. (14). The infinite chain is initially in the ground state of $H$ with $h_z=10$. At $t=0$ the magnetic field is changed to $h_z=3$. Upper part: the magnetization, originally at a value $m_z(0)\approx 0.5$, oscillates in time until it stabilizes around a lower value $m_z(\infty )\approx 0.48$. By considering increasingly large values of $\chi$ we are able to simulate the evolution of $m_z(t)$ for long times with remarkable accuracy. Lower part: scaling of errors with time for different values of $\chi$.