The Lieb-Robinson correlation function for long disordered transverse-field Ising chains

TL;DR

The paper develops and applies the operator Pauli walk (OPW) method to compute the Lieb–Robinson (LR) correlation function $C_k(t)$ in long disordered transverse-field Ising chains, enabling direct calculation for chains with hundreds to thousands of qubits. By mapping the TFIM dynamics to a compact operator space, the authors derive a closed-form expression for $C_k(t)$ that scales with the chain length as $2N_q$, avoiding exponential growth. They demonstrate that disorder in the Ising couplings leads to localization of quantum correlations, with light cones becoming vertical at late times and a disorder-dependent localization length, quantified by $k_{ ext{thresh}}$. In both paramagnetic and ferromagnetic phases, the results show stronger confinement with larger disorder, and comparisons to the Hamza–Sims–Stolz bound confirm the localization behavior while highlighting that the observed decay can be faster than the bound. Overall, the work provides a scalable, direct tool for characterizing information propagation in large, disordered quantum spin chains and clarifies the impact of disorder on LR propagation beyond bound analyses.

Abstract

The transverse-field Ising model is useful for studying interacting qubit arrays. The Lieb--Robinson correlation function can be used to characterize the propagation of quantum information in Ising chains. Considerable work has been done to establish bounds on this correlation function in various circumstances. To actually calculate the value of the correlation function directly typically requires a state space which grows exponentially with system size, and so is intractable for all but relatively small systems. We employ a recently-developed method that enables direct calculation of the value of the Lieb--Robinson correlation function and which scales linearly with system size. This enables the computation for systems with many hundreds of qubits, revealing the propagation of quantum information down the chain. We extend this technique to the problem of Ising chains with randomly disordered coupling strengths. Increasing disorder causes localization of the quantum correlations and halts propagation of quantum information.

Figures (15)

• Figure 1: Snapshots of the probability for a tight-binding chain of 600 sites. $P(k,t)$, the probability of finding the particle at site $k$, is shown at different times in (a)-(c). The probability $P_R(k,t)$ of finding the particle to the right of site $k$ is shown in (d)-(f).
• Figure 2: Snapshots of the probability for the tight-binding chain of 600 sites with disorder. The site probability $P(k,t)$ is shown in (a)-(c). The probability of finding the particle to the right of site $k$, $P_R(k,t)$ is shown in (d)-(f). The disorder in the hopping matrix element is characterized by $\Delta\gamma/\gamma_0=1$.
• Figure 3: Light-cone plots for 1D tight-binding chain of 300 sites. (a) The color indicates the site probability $P(x,t)$ for the case of no disorder, $\Delta\gamma=0$. The color indicates the disorder-averaged probability $\overline{P}(x,t)$ for (b) weak disorder with $\Delta\gamma/\gamma=1$, and (c) strong disorder with $\Delta\gamma/\gamma=2$. The red dashed line in each plot corresponds to the velocity of Eq. (\ref{['eq:Vtb']}).
• Figure 4: Configuration averaged probability $\overline{P}_R$ for two long times. The average is over 2000 sample configurations and $N=600$. The disorder is characterized by $\Delta\gamma/\gamma_0=0.5$ (black), 1 (blue), 1.5 (green) and 2 (red).
• Figure 5: Two phases of the TFIM chain. (a) Energies of the ground state and first excited state as a function of coupling $J'$. For $J'<1$, the system is in the paramagnetic state; for $J'>1$ it is in the ferromagnetic state. (b) Front velocity as a function of the coupling $J'$ given by Eq. (\ref{['eq:vFront']}). (c) Probability distributions for disordered couplings as given by Eq. (\ref{['eq:UniformPDF_inh']})