Interaction quantum quenches in the one-dimensional Fermi-Hubbard model with spin imbalance

Using the time-dependent density matrix renormalization-group method and exact diagonalization, we study the nonequilibrium dynamics of the one-dimensional Fermi-Hubbard model following a quantum quench or a ramp of the on-site interaction strength. We are particularly interested in the nonequilibrium evolution of Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) correlations, which, at finite spin polarizations and for attractive interactions, are the dominant two-body correlations in the ground state. For quenches from the noninteracting to the attractive regime, we investigate the dynamical emergence of FFLO correlations and their signatures in the pair quasi–momentum distribution function. We observe that the postquench double occupancy exhibits a maximum as the interaction strength becomes of the order of the bandwidth. Finally, we study quenches and ramps from attractive to repulsive interactions, which imprint FFLO correlations onto repulsively bound pairs. We show that a quite short ramp time is sufficient to wipe out the characteristic FFLO features in the postquench pair momentum distribution functions.

Figure 1

(a) System-size dependence [lattice sizes $L=12$ (ED), $L=24,36$ (DMRG)] of the time evolution of the double occupancy $d\left(t\right)/N_{\downarrow }$ after the interaction quench from $U_i=0$ to $U_f=-8J$ at $n=1/3,p=1/2$. (b) Fit (dashed black line) of a single, exponentially damped oscillation $-acos(\omega t+\varphi )exp(-t/{\tau }_d)+\overline{d}$ to the dynamics of $d\left(t\right)$ [for $t\lesssim 3/J$: ${\tau }_d=(1.05\pm 0.02)/J,\omega =(6.15\pm 0.02)J,\overline{d}$ from $t>3/J$]. Note that we observe a short relaxation time, ${\tau }_d\sim 1/J$, for this set of parameters. (c) $d\left(t\right)$ at low densities. In this regime, the relaxation follows a power law (dotted black line), $-acos(\omega t+\varphi )/t^{\alpha }+\overline{d}$ [for $0.1/J\lesssim t\lesssim 5/J$: $\alpha =0.53\pm 0.01,\omega =(5.097\pm 0.005)J$] and cannot be described by the fit of an exponentially damped oscillation (dashed black line).

Figure 2

Time average $\overline{d}$ of the double occupancy $d\left(t\right)$ versus the postquench interaction strength $U_f$ after the quench from $U_i=0$ for different fillings, $n=2/3$ (diamonds) and $n=1/3$ (asterisks), for $L=36$ (DMRG). Black X's indicate the corresponding results for $L=12$ (ED). Dashed lines show the ground-state values for $U=U_f$. Error bars indicate the standard deviation from the mean value (not visible for $L=36$). The dotted line shows the result for the low-density limit, Eq. (9).

Figure 3

Time average $\overline{d}$ of the double occupancy after the quench from $U_i=0$ to $U_f=-8J$ as a function of the polarization $p$. Thin and thick lines correspond to fillings $n=1/3$ and $n=2/3$, respectively. Dotted lines show the prequench value of $\overline{d}$ from Eq. (8); dashed lines indicate the postquench ground-state value with $U=U_f$. All data for $L=36$ (DMRG); the time average was computed for $t>3/J$.

Figure 4

Relaxation time ${\tau }_d$ of the double occupancy in the quench from $U_i=0$ to $U_f<0$ versus filling $n$ for different polarizations $p$ and $U_f=-8J$ ($L=36$; DMRG). Error bars indicate the asymptotic standard error from the least-squares fit of Eq. (11) to the data.

Figure 5

(a) Pair momentum distribution $n_k^{\mathrm{pair}}\left(t\right)$ after the interaction quench from $U_i=0$ to $U_f=-8J$ at $n=1/3,p=1/2,L=36$ (DMRG); the horizontal line marks the position of the FFLO momentum $q$ set by the density and polarization. (b) $n_k^{\mathrm{pair}}\left(t\right)$ at the FFLO momentum $q$. (c) $n_{k=0}^{\mathrm{pair}}\left(t\right)$ at zero momentum; solid lines for $L=36$ (DMRG) and dashed lines for $L=12$ (ED). Dotted horizontal lines in (b) and (c) indicate postquench ground-state values of the respective variables.

Figure 6

(a) Lowest natural orbital $|{\varphi }_0{\left(i\right)|}^2$ in the ground state of the noninteracting system ($U_i=0$; diamonds), and in the ground state of the attractively interacting system ($U_f=-8J$; squares), and time average (for $t>3/J$) after the initial quench dynamics (circles). (b) Time-averaged spatial decay (circles) of pair-pair correlations $|{\rho }_{ij}^{\mathrm{pair}}|$ after a quench from $U_i=0$ to $U_f=-8J$ (circles; time averages calculated for $t>3/J$), in the ground state of the noninteracting system ($U=0$; diamonds), and in the ground state of the attractively interacting system ($U=-8J$; squares). Dotted line: Fit of $f\left(\right|i-j\left|\right)=a\frac{|cos(q|i-j|+\varphi \left)\right|}{{|i-j|}^{\alpha }}$ to the $U=-8J$ ground state for $|i-j|\ge 3$, with $q$ taken from the position of the maximum in the pair-momentum distribution and fitting parameters $a,\alpha$, and $\varphi$. All data are for $n=1/3,p=1/2$, and $L=36$ (DMRG).

Figure 7

(a) Time evolution of the lowest natural orbital $|{\varphi }_0{\left(i\right)|}^2$ [for better color contrast, we plot $|{\varphi }_0\left(i\right)|$] and (b) time evolution of the FFLO visibility $V$ [see Eq. (7)]. The time at which the natural orbital $|{\varphi }_0{\left(i\right)|}^2$ develops an oscillatory pattern roughly coincides with the time at which the visibility goes through 0. For the prequench system with $U_i=0$ the lowest natural orbital is twofold degenerate, where one of them has a deviation from the constant background near the left, and the other near the right, boundary. Numerically, we select one of the degenerate orbitals, which explains the spatial anisotropy of $|{\varphi }_0{\left(i\right)|}^2$ for $t\lesssim {\tau }_{\mathrm{FFLO}}$. All data for $n=2/3,p=1/2$, and $L=36$ (DMRG).

Figure 8

Time scale ${\tau }_{\mathrm{FFLO}}$ for the formation of FFLO correlations in the quench from $U_i=0$ to $U_f<0$, versus (a) filling $n$, (b) polarization $p$, and (c) postquench interaction strength $U_f$, all for $L=36$ (DMRG). The time scale is extracted as the first 0 of the visibility $V\left(t\right)$.

Figure 9

(a) Visibility $V$ of the FFLO peak in $n_k^{\mathrm{pair}}$ in (a) the ground state at interaction strength $U$ and (b) the long-time average (calculated for $t>3/J$) of $V$ versus the postquench interaction strength $U_f$ after the quench from $U_i=0$, all for $L=36$ (DMRG) and $L=12$ (ED). Error bars indicate the standard deviation from the time average (not visible for $L=36$).

Figure 10

Time average of the pair-momentum distribution $n_k^{\mathrm{pair}}$ between $t=t_{\mathrm{ramp}}$ and $t_{\max }=4/J$ after linear ramps from the noninteracting system $\left(U_i=0\right)$ to attractive interactions $\left(U_f=-8J\right)$, for different ramp times $t_{\mathrm{ramp}}$ and polarizations (symbols with dotted lines): (a) $p=0$; (b) $p=1/4$; (c) $p=1/2$; (d) $p=3/4$. We also include the data for the ground state of the noninteracting system (diamonds) and the ground state of the attractively interacting system (squares). Dotted vertical lines show the maximum position at $k=q$. All results are for $n=2/3$ and $L=36$ (DMRG).

Figure 11

Spectra of the scattering continuum and the bound states for different on-site interaction strengths $U$ from a two-particle calculation $\left(N_{\uparrow }=N_{\downarrow }=1\right)$ for pairs with total momentum $K$ in the Fermi-Hubbard model. The shading of the scattering continuum is proportional to the density of states (DOS); darker gray indicates a higher DOS (note that the DOS diverges at the edges of the continuum region). The bandwidth of the scattering continuum is $8J$. Bound states are discrete and symmetric for $U\to -U$. There is no energy overlap of repulsively bound states with the continuum for $U>4J$ (dotted black line).

Figure 12

(a) Relative fraction of excess double occupancy $f_d\left(t\right)$ [see Eq. (15)] after the interaction quench from $U_i<0$ to repulsive $U_f=\left|U_i\right|>0$, normalized to the difference between post- and prequench ground-state values. (b) Time-averaged double occupancy $\overline{d}$ (circles; for $t>4/J$), postquench ground-state (squares), and initial state (diamonds) versus postquench interaction strength $U_f/J$. All data are for $n=2/3,p=1/2$, and $L=36$ (DMRG).

Figure 13

Time average of the pair momentum distribution function $n_k^{\mathrm{pair}}$, calculated for $t>3/J$, for the quench from attractive interactions $U_i<0$ to repulsive interactions $U_f=-U_i>0$, for different $U_f$—(a) $U_f/J=32$, (b) $U_f/J=8$, (c) $U_f/J=4$, and (d) $U_f/J=2$ (circles); for the ground state of the attractive system (diamonds); and for the ground state of the repulsive system at $U_f$ (squares). All data for $n=2/3,p=1/2$, and $L=36$ (DMRG).

Figure 14

Time average of the pair-momentum distribution $n_k^{\mathrm{pair}}$ (calculated for $t>4/J$) after ramps from attractive interactions $\left(U_i<0\right)$ to repulsive interactions $U_f=\left|U_i\right|>0$, for different ramp times $t_{\mathrm{ramp}}$ (symbols with dashed lines). We also include results for the instantaneous quench (circles), the ground state of the attractive system at $U_i$ (diamonds), and the ground state of the repulsive system (squares) for comparison. All results are for $n=2/3,p=1/2$, and $L=36$ (DMRG).

Figure 15

Time average of $f_o$ [see Eq. (15)] for visibility (diamonds) and double occupancy (circles), calculated for $t>4/J$, versus ramp time $t_{\mathrm{ramp}}$ after the quench and ramps from attractive $U_i<0$ to repulsive interactions $U_f=-U_i>0$. All data for $n=2/3,p=1/2$, and $L=36$ (DMRG).