Tunable transport in the mass-imbalanced Fermi-Hubbard model

The late-time dynamics of quantum many-body systems is organized in distinct dynamical universality classes, characterized by their conservation laws and thus by their emergent hydrodynamic transport. Here, we study transport in the one-dimensional Hubbard model with different masses of the two fermionic species. To this end, we develop a quantum Boltzmann approach valid in the limit of weak interactions. We explore the crossover from ballistic to diffusive transport, whose timescale strongly depends on the mass ratio of the two species. For timescales accessible with matrix product operators, we find excellent agreement between these numerically exact results and the quantum Boltzmann equation, even for intermediate interactions. We investigate two scenarios which have been recently studied with ultracold-atom experiments. First, in the presence of a tilt, the quantum Boltzmann equation predicts that transport is significantly slowed down and becomes subdiffusive, consistent with previous studies. Second, we study transport probed by displacing a harmonic confinement potential and find good quantitative agreement with recent experimental data [N. D. Oppong et al., arXiv:2011.12411]. Our results demonstrate that the quantum Boltzmann equation is a useful tool to study complex nonequilibrium states in inhomogeneous potentials, as often probed with synthetic quantum systems.

Figure 1

Dynamics in the mass-imbalanced Fermi-Hubbard model. (a) Illustration of the one-dimensional mass-imbalanced Fermi-Hubbard model, with the onsite Hubbard interaction $U$ and distinct hopping amplitudes $t_{\uparrow }$ (light) and $t_{\downarrow }$ (heavy). (b) Expansion after trap release of an initial double well at inverse temperature $\beta =0.1/t_{\uparrow }$ and for fixed mass ratio $t_{\uparrow }/t_{\downarrow }=0.1$. The kinetic theory (solid lines) is in good agreement with numerically exact calculations based on matrix product operators (dashed black line) for different interaction strengths (legend).

Figure 2

Short-time evolution of correlations in the linear-response regime. The density-density correlation function of the light species $C_{\uparrow \uparrow }(x,t)={\langle {\widehat{n}}_{x\uparrow }\left(t\right){\widehat{n}}_{0\uparrow }\rangle }_c$ is evaluated on a thermal background with infinite temperature at half-filling and zero magnetization. (a) At weak interaction $U/t_{\uparrow }=0.1$ and small hopping imbalance $t_{\downarrow }/t_{\uparrow }=0.5$ ballistic propagation lasts for long time. (b) For stronger interaction $U/t_{\uparrow }=1$, correlations quickly become Gaussian, compatible with diffusive transport. (c) If the mass imbalance is large $t_{\downarrow }/t_{\uparrow }=0.05$, transport is impeded by the slow species and profiles are strongly peaked and non-Gaussian on shown timescales. Data obtained from the QBE are compared with numerically exact matrix product operator simulations (dashed lines).

Figure 3

Limitations of the Boltzmann theory at strong interactions. We show the decay of an initially peaked correlation profile at time (a) $t=10/t_{\uparrow }$ and (b) $t=40/t_{\uparrow }$ for two interaction strengths $U/t_{\uparrow }=1$ (blue) and $U/t_{\uparrow }=4$ (red), where $t_{\downarrow }/t_{\uparrow }=0.2$. Solid lines are obtained by solving the linearized QBE and dashed lines correspond to exact matrix product operator simulations.

Figure 4

Slow decay of the autocorrelation function. (a) Autocorrelation function for different mass ratios $t_{\uparrow }/t_{\downarrow }$ evaluated up to late times. The correlations are computed at infinite temperature for $U/t_{\uparrow }=1$ with an initial Gaussian profile of $w=2$. (b) Flow of the scaling exponent of the autocorrelation function $z_{\sigma }\left(t\right)=-{[dlog{C}_{\sigma }/dlogt]}^{-1}$ for different values of $t_{\uparrow }/t_{\downarrow }$ at fixed interactions $U/t_{\uparrow }=1$ (upper panels) and different $U$ at fixed mass imbalance $t_{\uparrow }/t_{\uparrow }=0.1$ (lower panels). The left and right columns correspond to the $\uparrow$ and $\downarrow$ species, respectively.

Figure 5

Correlation profiles. (a) Rescaled correlation profiles $\sqrt{t}{C}_{\uparrow \uparrow }(x/\sqrt{t},t)$ for several times between $t{t}_{\uparrow }={10}^2\cdots {10}^3$, where a collapse indicates diffusive scaling. (b) Spatial variance $d{\mathrm{\Sigma }}^2/dt=2D\left(t\right)$ for light (left column) and heavy (right column) particles; the late-time saturation value corresponds to twice the diffusion constant. (c) A finite excess kurtosis ${\tilde{\kappa }}^{\left(4\right)}={\kappa }^{\left(4\right)}/{\mathrm{\Sigma }}^2$ quantifies the non-Gaussianity of the correlation profiles.

Figure 6

Scaling of the spectral gap of the linearized collision operator. The gap determines the onset of diffusion at $t_{\mathrm{hydro}}\sim 1/\mathrm{\Delta }$, indicating the scale when all nonconserved charges are decayed. (a) Inverse gap as a function of the mass ratio at infinite temperature. The left (right) inset illustrates the quadratic divergence of $t_{\mathrm{hydro}}$ in vicinity of $t_{\downarrow }/t_{\uparrow }=0$ ($t_{\downarrow }/t_{\uparrow }=1)$. (b) Temperature dependence of the inverse gap $1/\mathrm{\Delta }$ for different mass ratios. Note that kinks occurring in both panels correspond to crossings of the lowest eigenvalues of $\mathrm{\Gamma }$.

Figure 7

Tunable diffusion constants. (a) Diffusion constants $D_{\alpha \alpha }$ as a function of the mass ratio $t_{\downarrow }/t_{\uparrow }$ for $U/t_{\uparrow }=1$ at infinite temperature. (b) Temperature dependence of the diffusion constants. In the left panel the infinite temperature limit (solid) is compared to $\beta t_{\uparrow }=1,2$ (dotted, dashed). In the right panel, the temperature dependence is plotted for $t_{\downarrow }/t_{\uparrow }=0.5$.

Figure 8

Subdiffusive transport in a linear potential. (a) Decaying hydrodynamic modes ${\gamma }_n\left(k\right)$ obtained for $t_{\downarrow }/t_{\uparrow }=0.5$ at infinite temperature. The black dashed line corresponds to the analytic asymptotics, Eq. (8), for the subdiffusive mode. Solid lines are extracted from linearizing the hydrodynamic equations, and symbols are obtained from solving the inhomogeneous QBE for various density wave initial conditions. (b) Left panel: Decay of a density wave $n_{\uparrow }\left(x\right)\sim cos\left(kx\right)$ for $t_{\downarrow }/t_{\uparrow }=0.5, U/t_{\uparrow }=1.0$ at infinite temperature in a linear potential $V_{\mathrm{ext}}\left(x\right)=-Fx$ at different times $t=0$ (blue), $t=100/t_{\uparrow }$ (gray), and $t=500/t_{\uparrow }$ (pink). Right panel: The amplitude $A\left(t\right)$ for a wide range of wave vectors $k$. The markers in (a) are obtained from the decay constants for a range of different tilts $F/t_{\uparrow }\in [0.01,0.1]$ and wave vectors $k\in [\pi /100,\pi /10]$.

Figure 9

Comparison of the inhomogeneous QBE with experimental results of Ref. [30]. Recent experiments studied the relaxation dynamics of a heavy-light mixture of ytterbium atoms prepared in an optical lattice with harmonic confinement [30]. After a slow translation of the trap minimum by $\approx 20$ lattice sites, the dynamics of the light species is monitored for a time $t$. Experimental data are compared with the inhomogeneous QBE. In the upper row normalized density profiles at different points in time are shown, while the lower panel shows $\delta n_{\uparrow }$, Eq. (9), which quantifies the residual dynamics. We show data for interaction strength $U/t_{\uparrow }=-2.0$ and mass ratio $t_{\downarrow }/t_{\uparrow }=0.3$.

Figure 10

Diagrammatic notation for multipoint correlation functions. (a) The interaction tensor $I_{s_1,s_2,s_3,s_4}$ defined in Eq. (A2). (b) The two-point $W_{\sigma \tau }$, four-point $\langle C_{\left\{s_n\right\}}^{\left(4\right)}\left(k_n\right)\rangle$, and six-point $\langle C_{\left\{s_n\right\}}^{\left(6\right)}\left(k_n\right)\rangle$ correlation functions. Leg indices are abbreviated by ${\xi }_n=(s_n,k_n)$ and the legs' positions are relevant (note the different conventions for $I$ and $C$). For the correlators, incoming legs connect to creation and outgoing legs to annihilation operators. (c) The diagram corresponding to the right-hand side of Eq. (A5), without the Hermitian conjugate term. Contracting two legs amounts to integrating the momentum over $k\in \mathbb{B}$ and summing the spin index over $s\in \{\uparrow ,\downarrow \}$. For each interaction tensor, a global factor of $U$ must be included.

Figure 11

Interaction terms in the equation of motion for the four-point correlator. These diagrams correspond to the functional $\mathcal{F}[\langle C^{\left(4\right)}|,\rangle \langle C^{\left(6\right)}\left|\right]\rangle$ in Eq. (A6), which we want to approximate in the kinetic limit to second order in $U$.

Figure 12

Connected part of the interaction contributions to the four-point correlator in Gaussian approximation. Here, the shown diagrams represent the connected part of the functional $\mathcal{F}{\left[W\right]}_{\mathrm{c}}$. We obtained the diagrams in two steps: First, we approximate higher-order correlators by their Gaussian part, truncating the series expansion to second order in $U$; second, we identify all connected diagrams appearing in this approximation.

Figure 13

Collision operator in kinetic scaling limit. In Gaussian approximation, the equation for the collision operator is integrated by going to the kinetic limit, where we defined the modified interaction vertex $I^{+}={\mathrm{\Delta }}^{+}\times I$.