Magnetic excitations, nonclassicality, and quantum wake spin dynamics in the Hubbard chain

Recent work has demonstrated that quantum Fisher information (QFI), a witness of multipartite entanglement, and magnetic Van Hove correlations $G(r,t)$, a probe of local real-space real-time spin dynamics, can be successfully extracted from inelastic neutron scattering on spin systems through accurate measurements of the dynamical spin structure factor $S(k,\omega )$. Here we apply theoretically these ideas to the half-filled Hubbard chain with nearest-neighbor hopping, away from the strong-coupling limit. This model has nontrivial redistribution of spectral weight in $S(k,\omega )$ going from the noninteracting limit ($U=0$) to strong coupling ($U\to \infty$), where it reduces to the Heisenberg quantum spin chain. We use the density matrix renormalization group to find $S(k,\omega )$, from which QFI is then calculated. We find that QFI grows with $U$. With realistic energy resolution it becomes capable of witnessing bipartite entanglement above $U=2.5$ (in units of the hopping), where it also changes slope. This point is also proximate to slope changes of the bandwidth $W\left(U\right)$ and the half-chain von Neumann entanglement entropy. We compute $G(r,t)$ by Fourier transforming $S(k,\omega )$. The results indicate a crossover in the short-time short-distance dynamics at low $U$ characterized by ferromagnetic light-cone wavefronts, to a Heisenberg-type behavior at large $U$ featuring antiferromagnetic light cones and spatially period-doubled antiferromagnetism. We find this crossover has largely been completed by $U=3$. Our results thus provide evidence that, in several aspects, the strong-coupling limit of the Hubbard chain is reached qualitatively already at a relatively modest interaction strength. We discuss experimental candidates for observing the $G(r,t)$ dynamics found at low $U$.

Figure 1

$S(k,\omega )$ for the Hubbard chain as function of $u=U/\tilde{t}$ and the Heisenberg chain for chains of length $L=128$ with broadening parameter $\eta =0.05\tilde{t}$. As $u$ increases, the bandwidth shrinks and spectral weight moves from the top of the dispersion towards its bottom edge. The resulting spectrum approaches that of the Heisenberg chain shown in (o).

Figure 2

$S(k,\omega )$ for the Hubbard chain as function of $u=U/\tilde{t}$ and the Heisenberg chain for chains of length $L=128$ with broadening parameter $\eta =0.05\tilde{t}$. Note that the panels have been plotted with different $u$-dependent energy scale factors (see Table 1) in order to keep the apparent bandwidth constant. The redistribution of spectral weight from the top of the scattering continuum to the bottom is apparent already at relatively low values of $u$.

Figure 3

(a) The normalized QFI (nQFI) as function of $u=U/\tilde{t}$, calculated from $S(k=\pi ,\omega )$ where elastic contributions have been removed and with $\eta =0.05\tilde{t}$ for several system sizes. For $\mathrm{nQFI}>1$ (i.e., outside the shaded region), at least bipartite entanglement is witnessed by QFI. For this energy resolution, we find $u=2.5$ to be the lowest interaction strength at which bipartite entanglement may be witnessed. (b) The nQFI evaluated for $\eta \propto 1/L$, i.e., with size-dependent energy resolution. A suppression of nQFI is found for smaller systems, primarily at higher $u$ values. (c) The first derivative of the $L=128$ nQFI, calculated using both a standard forward finite difference (circles) and the Fornberg finite-difference method [81] (diamonds). Together these curves indicate a broad peak around $u=2.5$.

Figure 4

Real-time real-space correlations in the noninteracting (left column) and strong-coupling limits (right column). The top row panels show the real part, and the bottom row panels show the imaginary part. Ferromagnetic (antiferromagnetic) correlations $\langle S_r^z\left(t\right)S_0^z\rangle$ are indicated in red (blue).

Figure 5

Real-time real-space correlations at low to intermediate $u=U/\tilde{t}$. Left (right) column shows the real (imaginary part) of $\langle S_r^z\left(t\right)S_0^z\rangle$.

Figure 6

Real-time real-space correlations at intermediate to high $u=U/\tilde{t}$. Left (right) column shows the real (imaginary part) of $\langle S_r^z\left(t\right)S_0^z\rangle$.

Figure 7

(a) Averaged onsite spin-spin correlation $\langle S_j^z{S}_j^z\rangle$ (filled symbols, left $y$ axis) and single-site entropy $\mathcal{E}$ (black line, open symbols, right $y$ axis). The single-site entropy indicates a reduction in the number of local basis states with $u$, which causes the onsite spin-spin correlation to grow as the weights of empty and doubly occupied states decrease. (b) The first derivative of $\langle S_j^z{S}_j^z\rangle$ peaks near $u=2$ for all studied sizes, while the peak of $d\mathcal{E}/du$ occurs at a higher value $u=3.5$, in agreement with Eq. (25). (c) The second derivative of $\langle S_j^z{S}_j^z\rangle$ changes sign near $u=2$ and has peaks near $u=1.4$ and 3.5 for $L=64,128$ ($u=1.3$ and 3.6 for $L=32$, which is sampled more densely). The second derivative of $\mathcal{E}$ peaks at $u=1.75$ and 5.5, and changes sign near $u=3.5$. All derivatives were calculated using the Fornberg finite-difference method [81].

Figure 8

(a) The half-chain von Neumann entanglement entropy decreases as $u=U/\tilde{t}$ is increased, tending to the value for the Heisenberg chain as $u\to \infty$. There is a change in curvature of the half-chain entropy at low $u$ below $u=2$. To see this we calculate the first and second derivatives in (b) and (c) using the Fornberg finite-difference method [81]. The first derivative peaks near $u=1.9$ for $L=32$, and near 1.5 for $L=64$ and 128. The second derivative changes sign near $u=1.8$ for $L=32$, 1.5 for $L=64$, and 1.25 for $L=128$. The second peak in the second derivative occurs at $u=2.6$ for $L=32$, 2 for $L=64$, and 1.75 for $L=128$. Although the shown data are very limited in scope, we find that naive finite-size scaling in $1/L$ is consistent with finite values for all three quantities. The finite-size scaling prediction is that the zero of the second derivative occurs at $u\approx 0.93$ and the second peak occurs at $u\approx 1.53$ in the thermodynamic limit. The peak in the first derivative is predicted to occur at $u=1.63$.

Figure 9

Reduction of central charge as function of $u$. (a) Plotted data series represent $\mathrm{\Delta }{S}_{\mathrm{vN}}=S_{\mathrm{vN}}\left(2L\right)-S_{\mathrm{vN}}\left(L\right)$. The horizontal lines are drawn at $ln\left(2\right)/3$ and $ln\left(2\right)/6$. (b) Finite-size scaling of $S_{\mathrm{vN}}$ at weak and strong coupling. The lines are linear least-squares fits of the $L\ge 64$ entropies, with functional forms $0.518190+0.342946ln\left(L\right)$ and $0.134746+0.171053ln\left(L\right)$, respectively. The fits yield central charges $c_{u=0}^{\mathrm{fit}}\approx 2.058$ and $c_{\mathrm{Heis}}^{\mathrm{fit}}\approx 1.026$ and $c_{\mathrm{Heis}}^{\mathrm{fit}}/c_{u=0}^{\mathrm{fit}}\approx 2.005$, consistent with the expected $c_{u=0}=2$ and $c_{\mathrm{Heis}}=1$.

Figure 10

(a) Bandwidth $W\left(u\right)$ with $u=U/\tilde{t}$ according to the strong-coupling prediction ($u\to \infty$) and the Bethe ansatz. The difference becomes asymptotically small as $u\to \infty$. (b) The first derivative of $W\left(u\right)$ peaks near $u=2.25$, while (c) the second derivative changes sign near $u=2.25$. This value is marked by the vertical line in (b) and (c).