Quench dynamics of the quantum XXZ chain with staggered interactions: Exact results and simulations on digital quantum computers

TL;DR

The paper analyzes quenches in a fully dimerized XXZ chain with alternating couplings, deriving exact Bell-basis time evolution and closed-form expressions for half-chain entanglement entropies and Loschmidt echoes in finite systems. It reveals persistent oscillations and finite-size Loschmidt zeros that depend on system size and anisotropy, with clear periodicity when $J\Delta= p/q$ is rational. Beyond theory, it validates the framework with IBM-Q experiments: Hadamard-test reconstruction of Bell-state amplitudes for small systems and randomized measurement/classical-shadow protocols for larger systems, achieving strong agreement with exact results. The work lays groundwork for scalable quantum-dynamics tomography in non-relaxing flat-band spin chains and highlights practical pathways for near-term quantum hardware to probe non-equilibrium many-body phenomena.

Abstract

We investigate quench dynamics in the quantum $S=1/2$ XXZ antiferromagnetic chain with staggered and anisotropic interactions in the flat-band limit. Our quench protocol interchanges the odd- and even-bond strengths of a fully dimerized chain, enabling us to derive exact time-dependent states for arbitrary even system sizes by working in the Bell basis. We obtain closed-form, size-independent expressions for the von Neumann and second-order Rényi entanglement entropies. We further calculate exact Loschmidt echoes and the corresponding return rate functions across various anisotropies and system sizes, and identify Loschmidt zeros in finite chains. Our analysis reveals the precise conditions on the anisotropy parameter that govern the periodicity of the dynamical observables. In addition to the analytic study, we perform two types of numerical experiments on IBM-Q quantum devices. First, we use the Hadamard test to estimate the Bell-basis expansion coefficients and reconstruct the dynamical states, achieving accurate entanglement entropies and the Loschmidt echo for small systems. Second, we implement Trotter-error-free time-evolution circuits combined with randomized Pauli measurements. Post-processing via statistical correlations and classical shadows yields reliable estimates of the second-order Rényi entanglement entropy and the Loschmidt echo, showing satisfactory agreement with exact results.

Figures (16)

• Figure 1: Overlap diagrams for a positive (a), negative (b) and zero (c) coefficient $a_k=\langle {\Phi_k} | {\Psi_0}\rangle$ in a four-site chain with PBC. Gray and white circles represent spin-up state $\left| {0}\right\rangle$ and spin-down state $\left| {1}\right\rangle$, respectively. Black and red arches represent valence bonds (Bell dimers) $\psi_i$ in the states $\left| {\Psi_0}\right\rangle$ and $\left| {\Phi_k}\right\rangle$, with solid (dotted) arches indicating positive (negative) contributions determined by the spin orientations. In each panel, the right diagram is obtained from the left by a global spin inversion. In (a), two negative dimers remain under spin inversion, resulting in a positive coefficient $a_k = 2\cdot 2^{-2}$; in (b), one negative dimer remains, corresponding to a negative coefficient $a_k = -2\cdot 2^{-2}$; in (c) the number of negative dimers change from one to two after spin inversion, representing a case with zero coefficient $a_k=0$.
• Figure 2: Sketch of the bipartition for chains with PBC and $n=2, 3, 4, 5$. Gray lines denote the singlet dimers in the initial state $\left| {\Psi_0}\right\rangle$, and red lines represent the couplings of the postquench Hamiltonian. The dotted line in each figure marks a symmetric bipartition of the chain. (a) and (b): For even $n$, the bipartition cuts two couplings of the post-quench Hamiltonian. (c) and (d): For odd $n$, the bipartition cuts one postquench coupling and one dimer in the initial state.
• Figure 3: Time dependence of the half-chain von Neumann entanglement entropy for the XXZ chain with even $n\ge 4$ at various values of the anisotropy parameter $\Delta$ with $J=1$. All curves are plots of the analytical results using Eq. \ref{['eq:S_even']}.
• Figure 4: Time dependence of the exact Loschmidt echo for the XXZ chain of length $N=2n,\,n=2, 4, 6$ and $8$ with $J=1$ and various values of the anisotropy parameter: (a) $\Delta = 0$, corresponding to the XX chain; (b) $\Delta =1/2$; (c) $\Delta=1$, the XXX chain; (d) $\Delta=7/4$.
• Figure 5: Time dependence of the return rate function calculated from the Loschmidt echo in Fig. \ref{['fig:echo_even']} for the XXZ chain with $J=1$ and the anisotropy parameter at (a) $\Delta = 0$, corresponding to the XX chain; (b) $\Delta =1/2$; (c) $\Delta=1$, the XXX chain; (d) $\Delta=7/4$.