Symmetry oscillations sensitivity to SU(2)-symmetry breaking in quantum mixtures

In one-dimensional bosonic quantum mixtures with SU(2)-symmetry breaking Hamiltonian, the dynamical evolution explores different particle exchange symmetry sectors. For the case of infinitely strong intra-species repulsion, the hallmark of such symmetry oscillations are time modulations of the momentum distribution [Phys. Rev. Lett. 133, 183402 (2024)], an observable routinely accessed in experiments with ultracold atoms. In this work we show that this phenomenon is robust in strongly interacting quantum mixtures with arbitrary inter-species to intra-species interaction strength ratio. Taking as initial state the ground state of the SU(2) symmetric Hamiltonian and time-evolving with the symmetry breaking Hamiltonian, we analyze how the amplitude and frequency of symmetry oscillations, and thus of the momentum distribution oscillations, depend on the strength of the symmetry-breaking perturbation. We find that the set of symmetry sectors which are coupled during the time evolution is dictated by the spin-flip symmetry of the initial state and show that the population of the initial state may vanish periodically, even in the thermodynamic limit, thus revealing the robustness and universality of the symmetry oscillations.

💡 Research Summary

The authors investigate the dynamics of one‑dimensional bosonic mixtures when the Hamiltonian deviates from SU(2) symmetry by introducing a tunable interaction imbalance λ = g↑↓/g − 1, where g is the intra‑species contact strength and g↑↓ the inter‑species one. In the limit of infinitely strong intra‑species repulsion (hard‑core bosons), the many‑body wavefunction can be factorized via a generalized Bose‑Fermi mapping into a spatial part (identical to non‑interacting spinless fermions) and a spin part. The spin dynamics are governed by an effective spin‑chain Hamiltonian ˆHλ = ˆHSU + λ ˆVSB, where ˆHSU is the SU(2)‑symmetric XXX Heisenberg chain and ˆVSB is an axial anisotropy term that breaks the symmetry, turning the model into an XXZ chain. The anisotropy parameter Δ = 1 + 2λ determines whether the chain is in an XY‑like, paramagnetic, ferromagnetic, or antiferromagnetic regime.

The paper focuses on the situation where the initial state is the ground state of the SU(2)‑symmetric Hamiltonian, i.e. the fully symmetric Young‑tableau state |χSU0⟩ with maximal eigenvalue γ0 of the two‑cycle class‑sum operator Γ(2) = ∑i<j Pij. Because the full Hamiltonian ˆHλ is invariant under a global spin‑flip operator U = ∏iσx,i, the eigenstates of ˆHλ can be classified by the parity of the number q of boxes in the second row of the Young tableau. Consequently, only symmetry sectors with the same parity of q as the initial state can be coupled during the time evolution. This selection rule dramatically reduces the number of accessible sectors: for a balanced mixture (N↑ = N↓) only the fully symmetric sector and the sector with two boxes moved to the second row (e.g. (N‑2, 2)) become populated.

Exact diagonalization for N = 4 and N = 6 particles is performed across a wide range of λ. In the weak‑symmetry‑breaking regime (|λ| ≪ 1) a second‑order perturbative expansion accurately reproduces the dynamics: the populations Wγ(t) of the coupled symmetry sectors oscillate with a frequency proportional to |λ| and an amplitude that grows linearly with |λ|. As |λ| increases, the amplitude saturates and at λ = 1 the system exhibits a full population swap between the two coupled sectors, i.e. a π‑pulse‑like behavior. In the strong‑breaking regime (|λ| ≫ 1) the initial sector is completely depleted, yet the dynamics remain periodic, indicating a universal mechanism independent of microscopic details.

A central observable is the momentum distribution n(k,t), which can be expressed as a sum over symmetry‑resolved contributions because Γ(2) commutes with the momentum‑density operator n̂k. At large momenta the distribution follows a k⁻⁴ tail, with a coefficient C(t) directly proportional to the expectation value of the SU(2)‑symmetric part of the spin Hamiltonian. Therefore, the temporal modulation of C(t) provides a direct experimental signature of symmetry oscillations. The authors show that C(t) mirrors the oscillations of the symmetry‑sector populations, confirming that the momentum‑distribution dynamics are a faithful probe of the underlying spin‑symmetry dynamics.

Overall, the work demonstrates that symmetry‑oscillation phenomena are robust and universal across the entire range of interaction imbalance λ. The key ingredients are (i) preparation of a state with well‑defined SU(2) symmetry, (ii) the presence of a global spin‑flip symmetry of the Hamiltonian, and (iii) the mapping to an effective XXZ spin chain. The results suggest that current ultracold‑atom platforms, which can tune inter‑ and intra‑species interactions and measure momentum distributions with high resolution, are ideally suited to observe these symmetry oscillations, even when SU(2) symmetry is only approximately realized.