Entanglement growth and correlation spreading with variable-range interactions in spin and fermionic tunneling models

We investigate the dynamics following a global parameter quench for two one-dimensional models with variable-range power-law interactions: a long-range transverse Ising model, which has recently been realized in chains of trapped ions, and a long-range lattice model for spinless fermions with long-range tunneling. For the transverse Ising model, the spreading of correlations and growth of entanglement are computed using numerical matrix product state techniques, and are compared with exact solutions for the fermionic tunneling model. We identify transitions between regimes with and without an apparent linear light cone for correlations, which correspond closely between the two models. For long-range interactions (in terms of separation distance $r$, decaying slower than $1/r$), we find that despite the lack of a light cone, correlations grow slowly as a power law at short times, and that—depending on the structure of the initial state—the growth of entanglement can also be sublinear. These results are understood through analytical calculations, and should be measurable in experiments with trapped ions.

Figure 1

Illustration of the long-range models. (a) The long-range transverse Ising (LRTI) model, which has been realized experimentally in chains of trapped ions. The spins interact over a distance with amplitude $J_{lj}$, in the presence of a transverse field term $B$. (b) The long-range fermionic hopping (LRFH) model. Spinless fermions tunnel between distant sites in a 1D lattice with amplitude $J_{lj}$, and pairing on neighboring sites is induced with strength $\mathrm{\Delta }$.

Figure 2

Correlation spreading after global quenches in long-range spin and fermionic models, respectively. (a)–(c) ${log}_{10}\left|{\tilde{C}}_d\left(t\right)\right|$ for the LRTI model after the global quench $B/J=\infty \to 1$ is applied. These results are obtained using TDVP approach with MPOs for chains of $M=100$ spins (converged with MPS bond dimension $D=256$). (d)–(f) ${log}_{10}\left|C_d\left(t\right)\right|$ for the LRFH model after the global quench $\mathrm{\Delta }/J=10\to 1$ is applied. These results are obtained using exact numerical computations for $M={10}^4$ sites. (a) and (d) $\alpha =3$, short-range interactions with a strongly suppressed leakage of correlations outside of the light cone $d/t>v_{\mathrm{gr}}^{max}$ are observed for both models. Markers indicate contour lines at levels ${log}_{10}\left|{\tilde{C}}_d\left(t\right)\right|=[-4,-3\frac{1}{2},-3]$ for the LRTI model and ${log}_{10}\left|C_d\left(t\right)\right|=[-6,-5,-4,-3]$ for the LRFH model. (b) and (e) $\alpha =3/2$, intermediate-range interactions, the light cone is not sharply defined, but a light-cone effect is observed. Markers indicate averaged contour lines at levels ${log}_{10}\left|{\tilde{C}}_d\left(t\right)\right|=[-3\frac{1}{4},-3,-2\frac{3}{4}]$ for the LRTI model and ${log}_{10}\left|C_d\left(t\right)\right|=[-6,-5,-4,-3]$ for LRFH. (c) and (f) $\alpha =1/2$, no light cone, instant spread of correlations through the entire system. The suppression of correlations at large distances in (f) is discussed in Sec. 3c.

Figure 3

Determining the light cone in the LRFH model. We plot the two-site correlation function ${log}_{10}\left|C_d\left(t\right)\right|$ after global quenches $\mathrm{\Delta }/J=10\to 1$ in the LRFH model with $M={10}^4$ sites at different times $tJ$: (a) short-range interactions $\alpha =3$ and (b) midrange interactions $\alpha =3/2$. Dashed lines indicate different threshold levels $\delta =[-6,-5,-4,-3]$ that the correlation function reaches, see the markers in Figs. 2 and 2. Analogous markers for the LRTI model are in Figs. 2 and 2. In each case here, on the right-hand side, we reproduce the same plots on a double logarithmic scale, showing algebraically decay of correlations outside the light cone as $d\to \infty$ in both cases.

Figure 4

(a) The dispersion relation $\epsilon \left(k\right)$ and (b) density of states in velocity $D\left(k\right)$ for the LRFH model for $\mathrm{\Delta }=J$ and various interaction ranges. In the case of short-range interactions $\left(\alpha >2\right)$ one can see the smooth behavior of $\epsilon \left(k\right)$ leading to a finite maximum group velocity $v_{\mathrm{gr}}^{max}={\epsilon }^{\prime }\left(k^{*}\right)$. The corresponding density of states in velocity $D\left(k^{*}\right)$ diverges, which justifies the strong light-cone effect. In the case $\alpha <2,v_{\mathrm{gr}}^{max}\propto k^{\alpha -2}$ for $k\to 0$, but density of states in velocity is suppressed, $D\left(k\right)\propto k^{3-\alpha }$. As a result there is no domination of infinite velocity excitations and the light cone broadens. In the case $\alpha <1$, the quasiexcitation spectrum become unbounded as well.

Figure 5

Correlation spreading in the LRTI model after the global quench $B/J=\infty \to 2$ is applied in the system of $M=100$ spins. (a)–(c) ${log}_{10}\left|{\tilde{C}}_d\left(t\right)\right|$ calculated numerically via MPS/MPO methods [as in Figs. 2–2(c)], and (d)–(f) ${log}_{10}\left|{\widehat{C}}_d\left(t\right)\right|$ calculated analytically via the Holstein-Primakoff approximation. (a) and (d) $\alpha =3$, short-range interactions case. (b) and (e) $\alpha =3/2$, intermediate-range interactions. (c) and (f) $\alpha =1/2$, long-range interactions.

Figure 6

Bipartite entanglement growth after the global quench in LRTI model with open boundary conditions. (a) Half-chain entanglement entropy as a function of time for $M=20$ spins beginning in the fully polarized state $\left|{\psi }_0\right.〉=\left|\downarrow \downarrow \downarrow \cdots \right.〉$ along the axis of the magnetic field $B=J$ (from exact diagonalization). (b) The same as (a), but for $\alpha =1/2$, and starting from a selection of initial states (two fully polarized states in the opposite directions, Néel ordered state, and a product state with spins down in the left half of the chain and up in the right half). (c) Linear and (d) sublinear growth of entanglement entropy (shown on a logarithmic scale) for the LRTI model and selection of initial states, now for $M=50$ spins, computed with MPS methods (converged with MPS bond dimension $D=256$).

Figure 7

The entanglement entropy in LRFH model with $\alpha =0.8$, computed using exact numerical techniques for various system sizes $M$ and subsystem sizes $\mu$ after the global quench of $\mathrm{\Delta }/J=4\to 1/5$. (a) Entanglement dynamics for the equal partition case with $\mu =M/2$. The circular markers show the limit described by (23) for large subsystems in the bulk (i.e., $\mu ,M\to \infty$ and $\mu \ll M$). (b) Entanglement dynamics for a selection of system and subsystem sizes, which is shown to converge to the prediction (23) with increasing $\mu$ for $\mu \ll M=640$. The circular markers again indicate (23) for large subsystems in the bulk.

Figure 8

The entanglement entropy in LRFH model computed using exact numerical techniques with a selection of interaction range exponents $\alpha$, showing results for $\mathrm{\Delta }=J/5$, and two different initial states two cases: (a) and (b) Beginning from the ground state for $\mathrm{\Delta }=4J$, and (c) and (d) beginning from the ground state of a critical Ising model, as detailed in the text. (a) and (c) The half-chain entanglement entropy $\mu =M/2=40$, and (b) and (d) the subsystem entanglement entropy for $\mu =M/8=40$. Markers denote the bulk prediction from (23). (a) For $\alpha \ge 2$ there is a time window of clean linearity. At short times the leading behavior seems to be linear, but the subleading terms (in $tJ/\mu$) are also affected by the divergence of the dispersion relation. (b) The agreement with the bulk prediction is excellent, but at sufficiently large $tJ/\mu$ the discrepancy due to the finiteness of $M$ is clearly visible, especially for the largest values of $\alpha$. (c) There are evident differences with respect to (a) due to the fact that quasiparticles with momentum close to zero turn out to give a finite contribution to the entropy. While for $\alpha \ge 2$ the entropy seems to grow linearly, for small $\alpha$ the leading contribution at short time growth logarithmically in time. (d) The agreement with prediction (23) is still good, although the leading correction seems to be more complicated than a simple constant.