Spin crossovers and superdiffusion in the one-dimensional Hubbard model

We use tools from integrability and generalized hydrodynamics to study finite-temperature dynamics in the one-dimensional Hubbard model. First, we examine charge, spin, and energy transport away from half-filling and zero magnetization, focusing on the strong coupling regime where we identify a rich interplay of temperature and energy scales, with crossovers between distinct dynamical regimes. We identify an intermediate-temperature regime analogous to the spin-incoherent Luttinger liquid, where spin degrees of freedom are hot but charge degrees of freedom are at low temperature. We demonstrate that the spin Drude weight exhibits sharp features at the crossover between this regime and the low-temperature Luttinger liquid regime, which are absent in the charge and energy response, and rationalize this behavior in terms of the properties of Bethe ansatz quasiparticles. We then turn to the dynamics along special lines in the phase diagram corresponding to half-filling and/or zero magnetization where on general grounds we anticipate that the transport is subballistic but superdiffusive. We provide analytical and numerical evidence for Kardar-Parisi-Zhang (KPZ) dynamical scaling (with length and time scales related via $x\sim t^{2/3}$) along both lines and at the $\mathrm{SO}\left(4\right)$-symmetric point where they intersect. Our results suggest that both spin-coherence crossovers and KPZ scaling may be accessed in near-term experiments with optical lattice Hubbard emulators.

Figure 1

Regimes of transport for the Hubbard model. At strong coupling $U\gg t,$ we can distinguish four temperature regimes delineated by sharp crossovers (indicated by the solid lines) in dynamics. In descending order of temperature $T$ these are (i) the ‘high temperature Hubbard’ regime, where $T$ is the biggest energy scale; (ii) the high-temperature $t-J$ regime, where we can effectively ignore double-occupancies since $U\gg t$, but $T$ still exceeds both the charge scale $t$ (i.e., the holon bandwidth) and the effective spin-exchange scale $T\gg J\sim t^2/U$; (iii) the “spin-incoherent” regime, where the charge fluctuations of the $t-J$ model are cold ($T\ll t$) but the spins remain hot ($T\gg J$); and finally, the low-temperature regime where the system is described as spin-charge separated Luttinger liquid of coherent charge and spin degrees of freedom, where $T$ is the lowest energy scale. At weak coupling, regimes (i) and (iv) are broadly similar and we expect a crossover at $T\sim t$. However, the weak-coupling crossovers for $t\sim J$ and $t\sim U$ are less significant and hence we do not discuss them further in this work.

Figure 2

Spin Drude weight $D_m$ at the thermal crossover between spin-incoherent and spin-coherent regimes [(iii) and (iv) respectively in Fig. 1]. We fix $\langle \widehat{Q}\rangle /L=0.3$ and $h/t=0.04$. (a) $D_m$ as a function of $U$ for various $\beta$. Dashed line indicates the asymptotic value $D_m^{\infty }$ for $U\to \infty$ in regime (iii). (b) The same data as in (a). We observe that for $\beta h\lesssim 1, D_m/D_m^{\infty }$ departs from 1 when $J\sim T$. Instead, if $\beta h\gtrsim 1, D_m/D_m^{\infty }$ departs from 1 when $J\sim h$ (inset). (c) We highlight that the crossover in the Drude weight is a consequence of a change of the dressed magnetization of the $y$ particles at the Fermi points.

Figure 3

Dynamical spin-spin correlation function in (a) the spin-coherent regime ($U=10$) and (b) the spin-incoherent regime ($U=160$). In each panel, the main figure displays the rich structure due to slow magnon modes, while the insets show the fast-moving leading front due to $y$ particles. The peaks corresponding to $1|w$ and $2|w$ magnons are marked in the figure. In both figures, $h=0.04, \beta =35.9$, and $\langle \widehat{Q}\rangle /L=0.3$.

Figure 4

The crossover between the spin-incoherent regime (iii) and spin-coherent regime (iv) is also visible in subleading corrections to the charge and energy Dude weights. The plots show the magnitude of the relative correction to the Drude weight compared to the leading order expressions in $t/U$ ($D_q^{\infty }$ and $D_{\tilde{e}}^{\infty }$) given by Eq. (11). Apart from a tail at large $T$, which is due to the crossover to regime (ii), we see that the corrections indeed scale like $t/U$ and depend on the ratio $T/J$, signaling that their change is really a consequence of the (iii)-(iv) crossover. Numerical parameters: $h=0$ and $\langle n\rangle =0.3$.

Figure 5

TEBD data for the Hubbard model at $h=\mu =0$ and $U/t=1,2,4$ and for various maximum-bond-dimensions ${\chi }_{\text{max}}=256,512,\text{and}1024$. From the return probability (top) and from the profile width (bottom), we have fit the dynamical exponent $z$ (insets) as described in the main text.

Figure 6

Signatures of superdiffusion in small Hubbard chains directly accessible in current quantum microscope experiments. TEBD data for the Hubbard model at $U/t=4$ for finite chains of size $L=12,16,\text{and}20$. The return probability saturates due to the finite size effects, cutting off the $t^{-2/3}$ scaling.

Figure 7

Superdiffusion at finite temperature. TEBD data for the Hubbard model at $U/t=4$ for $\beta t=0.1,0.2,\text{and}0.5$.

Figure 8

TEBD data for the Hubbard model at $h=0.1,\mu =0$, and $U/t=4$. Ballistically moving peaks coexist with a superdiffusive central peak that lasts for a timescale $\sim h^{-3}$.

Figure 9

(a) Collapsed profile of $\langle S^z(x{}_j,t)S^z(0,0)\rangle$ for $U/t=1$ assuming $z=3/2$. We can observe that the tails of the correlators did not converge yet up to the maximum time we were able to reach in our simulations. (b) Same as (a), but for $U/t=4$. The drift of the profile with time is less pronounced than in (a), making it harder to understand if the profile is converged in $t$. If convergence has been reached, this would indicate that the profile are described by $f_{\text{KPZ}}$, in contrast with our previous analysis. (c) and (d) report the same data of (a) and (b) respectively, but in a linear scale.

Figure 10

Collapsed profile of $\langle S^z(x{}_j,t)S^z(0,0)\rangle$ for the Heisenberg XXX chain assuming $z=3/2$, computed using TEBD with bond dimension $\chi =512$. Significant deviations in the tails from $f_{\text{KPZ}}$ persist to accessible timescales.