Tunable many-body burst in isolated quantum systems

Thermalization in isolated quantum many-body systems can be nonmonotonic, with its process dependent on an initial state. We propose a numerical method to construct a low-entangled initial state that creates a “burst” – a transient deviation of an observable from its thermal equilibrium value – at a designated time. We apply this method to demonstrate that a burst of magnetization can be realized for a nonintegrable mixed-field Ising chain on a timescale comparable to the onset of quantum scrambling. Contrary to the typical spreading of information in this regime, the created burst is accompanied by a slow or even negative entanglement growth. Analytically, we show that a burst becomes probabilistically rare after a long time. Our results suggest that a nonequilibrium state is maintained for an appropriately chosen initial state until scrambling becomes dominant. These predictions can be tested with programmable quantum simulators.

💡 Research Summary

The paper investigates a non‑monotonic thermalization phenomenon—called a “burst”—in isolated quantum many‑body systems, where an observable temporarily deviates from its thermal equilibrium value before eventually relaxing. The authors develop a numerical framework to construct low‑entangled initial states that generate a burst at a pre‑chosen time τ, and they apply this framework to a non‑integrable mixed‑field Ising chain.

Two complementary methods are introduced. Method 1 simply evolves the ground state of the observable backwards in time to τ, then truncates the resulting state to a matrix‑product state (MPS) with a fixed bond dimension χ. While computationally cheap, this approach offers little control over the energy distribution of the initial state. Method 2 formulates a cost function
(H_{\text{DMRG}} = O(\tau) + \lambda_{L}\bigl(H-\langle H\rangle_{\beta}\bigr)^{2})
where (O(\tau)=e^{iH\tau} O e^{-iH\tau}) is the Heisenberg‑evolved observable, λL penalises energy fluctuations, and β sets a target temperature (often β = 0, i.e., infinite temperature). By running a DMRG optimisation with a maximal bond dimension χ, the algorithm yields an MPS that simultaneously minimises the expectation value of the observable at time τ and keeps the energy close to the desired value. The resulting state is then evolved forward in real time to monitor the burst.

The physical model is the mixed‑field Ising Hamiltonian
(H = \sum_{i=1}^{L-1} J_{z} S^{z}{i} S^{z}{i+1} + \sum_{i=1}^{L}\bigl(h_{x} S^{x}{i}+h{z} S^{z}{i}\bigr))
with parameters ((J{z},2h_{x},2h_{z})=(1.0,0.9045,0.8090)), which is known to be quantum‑chaotic. The authors focus on the average magnetizations (M_{y}) and (M_{z}) as observables. Because of time‑reversal symmetry, (\langle M_{y}\rangle_{\text{eq}}=0) for any temperature, making any non‑zero value a clear signature of a burst.

Numerical simulations with χ = 10 (far smaller than the χ ≈ 10⁶ required to represent a typical volume‑law state) reveal a pronounced peak in both (M_{y}(t)) and (M_{z}(t)) at the designated time τ ≈ 20. Remarkably, the bipartite entanglement entropy (S_{A}(t)) (with A being half the chain) grows only very slowly, and in some intervals even decreases, contrary to the linear growth expected in generic scrambling dynamics. This indicates that, up to the burst, local information remains relatively protected despite the onset of ballistic scrambling.

A systematic finite‑size analysis shows that, for a fixed χ, the burst amplitude remains large as the system size L increases, especially for short τ (τ ≲ 20). In the thermodynamic limit the burst survives, confirming that low‑entanglement initial states can generate substantial non‑equilibrium signatures even in large chaotic systems. However, as τ grows, the burst amplitude gradually diminishes, suggesting that the complexity of the time‑evolved observable’s eigenstates eventually exceeds the expressive power of a low‑χ MPS.

To understand the long‑time behavior, the authors turn to a local random quantum circuit model that approximates time‑dependent 2‑local Hamiltonian dynamics. In each step a random nearest‑neighbor unitary drawn from the Haar measure is applied. Using concentration‑of‑measure results for approximate unitary k‑designs, they derive an upper bound on the probability that any MPS with bond dimension ≤ χ yields a burst larger than a chosen threshold ΔOₐ. The bound scales roughly as
(\Pr