Universal Kardar-Parisi-Zhang Dynamics in Integrable Quantum Systems

Although the Bethe ansatz solution of the spin-$1/2$ Heisenberg model dates back nearly a century, the anomalous nature of its high-temperature transport dynamics has only recently been uncovered. Indeed, numerical and experimental observations have demonstrated that spin transport in this paradigmatic model falls into the Kardar-Parisi-Zhang (KPZ) universality class. This has inspired the significantly stronger conjecture that KPZ dynamics, in fact, occur in all integrable spin chains with non-Abelian symmetry. Here, we provide extensive numerical evidence affirming this conjecture. Moreover, we observe that KPZ transport is even more generic, arising in both supersymmetric and periodically driven models. Motivated by recent advances in the realization of $\mathrm{SU}(N)$-symmetric spin models in alkaline-earth-based optical lattice experiments, we propose and analyze a protocol to directly investigate the KPZ scaling function in such systems.

Figure 1

(a) Schematic depicting a one-dimensional chain of alkaline-earth atoms (each with $N$ levels) trapped in an optical lattice and interacting via nearest-neighbor superexchange. The equilibration of an initial domain-wall-like imbalance encodes the underlying KPZ dynamics. (b) Domain-wall dynamics as a function of time for an SU(3)-symmetric, integrable spin chain. (c) The polarization profiles at different times collapse upon rescaling with $t^{-1/z}$. The dynamical exponent, $z=3/2$, indicates superdiffusion and is consistent with KPZ transport.

Figure 2

(a) Conjectured landscape of KPZ transport in integrable, non-Abelian-symmetric models (blue dot). The non-Abelian symmetry can be broken in two distinct ways, either by adding a finite charge density to the initial state (orange line) or by perturbing the underlying Hamiltonian (purple line). (b) The total polarization transferred across the domain wall, $\mathcal{P}(t)$, directly determines the dynamical exponent. For the integrable SU(3) model, $z=3/2$; when either the integrability or the symmetry is broken in the Hamiltonian, $z=2$ [49]; when the initial state has nonzero charge density, $z=1$. Note that the curve for the integrability breaking case (green) is shifted down for clarity. (c) Depicts the charge transport velocity $v$ as a function of charge density $\delta$ for both the SU(3) model and the SU(2) model (inset) [53]. (d) The diffusion coefficient $D$ diverges as the Izergin-Korepin and XXZ (inset) integrable models approach the SU(3) and SU(2) (inset) symmetric points. The DMT bond dimension $\chi$ is chosen to be $\{64,128,256\}$ and $\{64,128,256,512\}$ for the SU(3) and SU(2) cases, respectively. Green crosses in the inset mark previous numerical results obtained from tDMRG simulations with bond dimension $\chi \sim 2000$ [54].

Figure 3

(a)–(d) The KPZ scaling function emerges from a wide variety of integrable dynamics: static, non-Abelian-symmetric models, their Floquet counterparts, and supersymmetric models. (a)[(d)] At late times, the rescaled polarization profiles of the $\mathrm{SU}(3)[\mathrm{SU}(2|1)]$ model differ from both the Gaussian and Lévy-flight expectations, but exhibit excellent agreement with the KPZ scaling function. Insets of (a)[(d)]: relative difference with respect to the KPZ scaling function. We note that the agreement extends to longer length scales as time is increased. (b)[(c)] Late-time, rescaled polarization profiles of static [Floquet] integrable models with different non-Abelian symmetries. For all symmetries explored, the dynamics exhibit excellent agreement with the KPZ scaling function. Insets of (b)[(c)]: zoom-in of the polarization profiles. The system sizes in the numerical simulation are chosen as: $L=600$ for all static models, $L=1200$ for Floquet SU(3) and SO(3) models, and $L=800$ for other Floquet models. (e) For all models considered, the ratio between the polarization gradient and the current is inhomogeneous, in stark contrast with the expectation for any linear transport equation. The observed curvature is instead in agreement with KPZ transport. (f) In integrable supersymmetric models, the total charge transferred across the domain wall (upper panel) and the extracted dynamical exponent $z$ (lower panel) are consistent with superdiffusion. (g) Polarization gradients in an integrable $\mathrm{SU}(2|1)$ model with varying hole density. At the same evolution time, systems with a smaller hole density are closer to the KPZ expectation.