Superdiffusion resilience in Heisenberg Chains with 2D interactions on a quantum processor

TL;DR

This work investigates how integrability-breaking 2D couplings affect superdiffusive spin transport in a Heisenberg-type system by introducing a 2D Floquet model on a heavy-hex lattice that reduces to the 1D XXX chain as interlayer coupling vanishes. The authors compute infinite-temperature edge-spin autocorrelations $C^{zz}(t)$ and analyze the scaling exponent transitions from the known $-2/3$ superdiffusive value toward diffusive ($-1/2$) or ballistic ($-1$) regimes, depending on the 2D interaction type. They find that SU(2)-symmetry-preserving 2D interactions, specifically $(1,1,1)$, are most resilient against breakdown, while other types show varying resilience tied to cross-chain transmission and symmetry constraints, supported by both noiseless simulations and IBM hardware experiments. The results illuminate how interchain couplings can sustain or degrade anomalous transport in 2D lattices, with direct implications for real quasi-1D materials and for scaling quantum simulations of non-equilibrium quantum many-body dynamics on current hardware.

Abstract

Observing superdiffusive scaling in the spin transport of the integrable 1D Heisenberg model is one of the key discoveries in non-equilibrium quantum many-body physics. Despite this remarkable theoretical development and the subsequent experimental observation of the phenomena in KCuF$_3$, real materials are often imperfect and contain integrability breaking interactions. Understanding the effect of such terms on the superdiffusion is crucial in identifying connections to such materials. Current quantum hardware has already ascertained its utility in studying such non-equilibrium phenomena by simulating the superdiffusion of the 1D Heisenberg model. In this work, we perform a quantum simulation of the superdiffusion breakdown by generalizing the superdiffusive Floquet-type 1D Heisenberg model to a general 2D model. We comprehensively study the effect of different 2D interactions on the superdiffusion breakdown by tuning up their strength from zero, corresponding to the 1D Heisenberg chain, to finite nonzero values. We observe that certain 2D interactions are more resilient against superdiffusion breakdown than others and that the $SU(2)$ preserving 2D interaction has the highest resilience among all the 2D interactions we study. Importantly, this observed resilience has direct implications for sustaining superdiffusive spin transport in two-dimensional lattices. We reason out the relative resilience against the superdiffusion breakdown through an analysis of the scattering coefficients off the 2D interaction in otherwise 1D chains. The relative resilience of different interaction types against superdiffusion breakdown was also captured in quantum hardware with remarkable accuracy, further establishing the current quantum hardware's applicability in simulating interesting non-equilibrium quantum many-body phenomena.

Figures (9)

• Figure 1: Summary:(a) We study the extension of the 1D Floquet Heisenberg Hamiltonian, represented by black bonds, known to exhibit superdiffusive scaling in discrete time evolution Discrete_1D_1KPZ_floquet, to a 2D Floquet model native to the heavy-hex lattice, represented by the vertical brown bonds. Discrete time evolution is performed by sequentially applying three types of bonds (colored red, green, and blue) at a kicking period $\tau$, with interaction strengths tuned relative to the 1D Heisenberg interaction and expressed as $\vec{J_\perp} = \vec{\lambda} \times J_\perp$ using a representative set of vectors (b)$\vec{\lambda}$ defined in Eq. \ref{['interaction_type']}. (c) We measure the infinite temperature auto spin-spin correlation function of the edge probe qubit marked by the square box using a quantum algorithm Richter_Pal to study (d) the breakdown and resilience of superdiffusion under different 2D interaction types.
• Figure 2: Different types of superdiffusion breakdown Using noiseless simulations of a 28-qubit system based on the model in Fig. \ref{['model_description']}, we show that superdiffusion ($\propto t^{-2/3}$) can break down toward either the diffusive ($\propto t^{-1/2}$) or ballistic ($\propto t^{-1}$) regimes, depending on the interaction type. The figure presents running averages of scaling exponents $\left\langle \frac{d(\ln(C^{zz}(t)))}{d(\ln(N))} \right\rangle$, compared against the 1D reference model ($\vec{\lambda} = (0,0,0)$), with insets showing the corresponding $C^{zz}(t)$. Notably, $\vec{\lambda} = (1,1,1)$ remains most resilient against breakdown. Diffusive breakdown is observed for $\vec{\lambda} = (0,0,1)$ and $(1,1,0)$, with the latter showing greater resilience, while ballistic breakdown occurs for $\vec{\lambda} = (1,0,0)$ and $(1,0,1)$, with the latter being more resilient. All simulations were performed with a fixed interaction strength ratio of $\frac{J_{\perp}}{J} = 1.0$, by probing the 2D rung either closer (a) or farther (b) from the activated probe qubit; probing the farther qubit delays, but does not alter, the nature of the breakdown.
• Figure 3: (a) Scattering coefficients. Computed from 45-qubit simulations of two coupled 22-qubit 1D chains joined by a 2D interaction rung. The plots show $|T_{\text{same}}|$, $|T_{\text{cross}}|$, and $|R|$ as functions of time steps $N$ for three interaction cases: uncoupled 1D, $\vec{\lambda} = (1,1,0)$, and $\vec{\lambda} = (0,0,1)$ with a fixed interaction strength ratio $J_{\perp}/J = 4$. Only $T_{\text{cross}}$ transmission is observed for $\vec{\lambda} = (1,1,0)$, indicating reduced reflection and enhanced transport resilience. (b) Directional dependence of superdiffusion breakdown. Heatmaps show how breakdown varies with the correlation measurement direction $\hat{n} = n_x \hat{x} + n_z \hat{z}$ for $\vec{\lambda} = (0,0,1)$ and $\vec{\lambda} = (1,1,0)$, with fixed interaction strength ratio $J_{\perp}/J = 1$. $\vec{\lambda} = (0,0,1)$ transitions to ballistic or diffusive behavior depending on $\hat{n}$, while $\vec{\lambda} = (1,1,0)$ remains resilient across all directions. Dashed lines serve as guides to the eye.
• Figure 4: Comparison of superdiffusion breakdown in noiseless simulations (a) and IBM Heron QPU experiments (b) for various 2D interaction types: The plots display running averages of scaling exponents with propagated standard errors derived from correlation measurements. Early-time agreement validates the simulation protocol, while intermediate-time deviations demonstrate the QPU’s capability to resolve the relative resilience of interaction types against superdiffusion breakdown, despite noise introduced by deep circuits involved in the protocol.
• Figure 5: Trotter circuit for the time evolution of $J_h \times \vec{\lambda} = J_h (\lambda_x,\lambda_y,\lambda_z)$ for time $t$: The circuit is equal to $e^{-i J_h (\frac{\lambda_x}{4} \sigma_x \sigma_x + \frac{\lambda_y}{4} \sigma_y \sigma_y +\frac{\lambda_z}{4} \sigma_z \sigma_z ) t }$ up to a global phase.