Efficient Simulation of One-Dimensional Quantum Many-Body Systems

We present a numerical method to simulate the time evolution, according to a generic Hamiltonian made of local interactions, of quantum spin chains and systems alike. The efficiency of the scheme depends on the amount of entanglement involved in the simulated evolution. Numerical analysis indicates that this method can be used, for instance, to efficiently compute time-dependent properties of low-energy dynamics in sufficiently regular but otherwise arbitrary one-dimensional quantum many-body systems. As by-products, we describe two alternatives to the density matrix renormalization group method.

Figure 1

Propagation of a spin wave in the ferromagnetic chain (24) with $n=30$ spins, $B=J=1$, $T=25$, and $\delta =0.005$. The eigenvalues $p_{\alpha }=|{\lambda }_{\alpha }^{\left[15\right]}{|}^2$, $\alpha =1,\dots ,17$, of the reduced density matrix ${\rho }_A$ for half a chain [see Eq. (7)] are plotted as a function of time (thin, irregular, rapidly oscillating lines). Notice the exponential decay of $p_{\alpha }$ in $\alpha$ for any fixed value of $t/J$, a seemingly common feature of low-energy dynamics in sufficiently regular, but otherwise arbitrary local models in one dimension. The time scale is such that the spin wave, originating at the left end of the open chain [see state $\mathrm{|}{\mathrm{\Psi }}_{\mathrm{0}}\mathrm{\rangle }$ after Eq. (24)] travels along the open chain 7 times, bouncing backward each time it reaches one of the extremes of the chain. The figure shows the error fidelity $\epsilon \left(t\right)$ corresponding to keeping all 17 terms in (7) in an order $p=2$ Trotter expansion. Notice that, as predicted by Eq. (22), $\epsilon \left(t\right)$ grows quadratically in the simulated time. Errors ${\epsilon }^{\prime }\left(t\right)$ and ${\epsilon }^{\prime \prime }\left(t\right)$ correspond to keeping ${\chi }_{\epsilon }=12$ and ${\chi }_{\epsilon }=8$ terms in Eq. (8). Truncation errors are of the order of the neglected eigenvalues $p_{\alpha }$.