From a magnetic material to a spin chain
A spin-1/2 has two possible outcomes when we measure its magnetization along a chosen direction. In our simulation, each spin is represented by a qubit. X, Y and Z are Pauli operators describing different spin components. The Hamiltonian specifies how the spins interact and how their state changes over time.
We use an open chain: each spin couples to its nearest neighbour, but the final spin does not couple back to the first.
H=H_{\mathrm{XXZ}}+H_f,
\qquad
H_{\mathrm{XXZ}}=\sum_{i=0}^{N-2}\left[J_{xy}(X_iX_{i+1}+Y_iY_{i+1})+J_zZ_iZ_{i+1}\right].
\]
The XX and YY terms allow neighbouring spins to change together: they can exchange a local spin excitation. The ZZ term assigns different energies to parallel and antiparallel configurations. The ratio of Jz to Jxy controls the exchange anisotropy. In our test, both are 1: the isotropic exchange point within the XXZ family, supplemented by a spatially varying field.
H_f=\sum_{i=0}^{N-1}h_iZ_i,\qquad
h_i=0.5\cos(2\pi\beta i),\qquad
\beta=\frac{\sqrt{5}-1}{2}.
\]
Because beta is irrational, this field is quasiperiodic: it has structure but does not repeat with a fixed integer lattice period. This lets us study how spatial variation and interactions together govern the decay of an initial pattern. It is a simplified model of many-body dynamics, not a calculation of the properties of a particular industrial material.
The setup and source implementation come from Imbalance_XXZ_hz_1D by Vishal Khatkar, Akash Yadav, Md. Osama Ali, Ritajit Majumdar, Rudranil Basu and Sambuddha Sanyal. Our local implementation preserves their code as an unchanged submodule pinned to commit 8416e815da5d7a5df147a276b9f2434fef9626a8.
Imbalance: how much of the initial pattern remains?
We start with a Neel pattern: alternating 0 and 1. This is easy to prepare by applying X gates to odd-indexed qubits. We then let the chain evolve. Rather than measuring the full wavefunction, we measure a weighted average of the local magnetizations:
I(t)=\frac{1}{N}\sum_{i=0}^{N-1}(-1)^i\langle Z_i(t)\rangle.
\]
For the ideal initial pattern, I(0)=1. As the alternating order disappears, the imbalance approaches zero. A change of sign indicates that the opposite sublattice orientation predominates at that time. A value near zero does not, by itself, establish complete thermalization: one observable does not describe the entire state.
We study all 31 time points from t=0 to t=5, spaced by 1/6. Time is expressed in model units with hbar=Jxy=1. These are not five seconds of physical evolution. Runtime in seconds is a separate quantity.