Hybrid infinite time-evolving block decimation algorithm for long-range multidimensional quantum many-body systems

In recent years, the infinite time-evolving block decimation (iTEBD) method has been demonstrated to be one of the most efficient and powerful numerical schemes for time evolution in one-dimensional quantum many-body systems. However, a major shortcoming of the method, along with other state-of-the-art algorithms for many-body dynamics, has been their restriction to one spatial dimension. We present an algorithm based on a hybrid extension of iTEBD where finite blocks of a chain are first locally time evolved before an iTEBD-like method combines these processes globally. This in turn permits simulating the dynamics of many-body systems in spatial dimensions $d\ge 1$ where the thermodynamic limit is achieved along one spatial dimension and where long-range interactions can also be included. Our work paves the way for simulating the dynamics of many-body phenomena that occur exclusively in higher dimensions and whose numerical treatments have hitherto been limited to exact diagonalization of small systems, which fundamentally limits a proper investigation of dynamical criticality. We expect the algorithm presented here to be of significant importance to validating and guiding investigations in state-of-the-art ion-trap and ultracold-atom experiments.

Figure 1

Tensor network diagram of the time evolution of a state by one time step using the h-iTEBD algorithm for a unit cell of four sites. Each square represents a matrix. The horizontal legs are the indices of a matrix and vertical legs represent the local state ($\langle \sigma |$ if going downwards and $|\sigma \rangle$ if going upwards). The connected legs represent a contraction of the matrices with matrix products (horizontal) and taking the overlap of the two local states (vertical).

Figure 2

Fidelity scaling of the different local subspace dimensions for the h-iTEBD algorithm along with second-order iTEBD, for the quasiexact reference state calculated in fourth-order iTEBD. The h-iTEBD algorithm with local time evolution using third-order and seventh-order Krylov dimensions is compared with the quasiexact state, and we see that with seventh-order Krylov h-iTEBD, we already reach its limiting behavior, which is equivalent to second-order iTEBD. The early time scaling of the fidelity gives the expected scaling of $t^3$ and $t^2$ for h-iTEBD-K3 and h-iTEBD-K7, respectively.

Figure 3

A two-dimensional square lattice of size $N$ sites $\times \infty$ sites and its mapping onto a one-dimensional chain with a unit cell of $N$ sites. The sites are numbered sequentially from top to bottom and then back to the top of the next vertical slice.

Figure 4

Comparison of the Loschmidt-echo return rate calculated in h-iTEBD to that in TDVP for quenches in (a) the XXZ model from a Néel state, and (b) the ANNNI model from a fully $z$-up polarized state (see text for details). The results show excellent agreement.

Figure 5

Top panel shows the time evolution of the longitudinal magnetization in the ferromagnetic nearest-neighbor triangular lattice after a quench of the fully ordered state ($h_i=0$) with the Hamiltonian (26) at various final values $h_f$ of the transverse-field strength. The lower panel shows the return rate exhibiting anomalous cusps for a quench to $h_f=2.1J$, which is below the dynamical critical point $h_c^d\approx 2.25J$.

Figure 6

Value of the longitudinal magnetization per site ($m_z={\sum }_i\langle {\widehat{\sigma }}_i^z\rangle /N$) with respect to the transverse field strength $h$ of a triangular lattice of size $N=6$ sites $\times \infty$ sites. The equilibrium quantum phase transition is at $h\approx 4.53J$, vastly different from its 1D counterpart due to much higher connectivity on the triangular lattice.