The quantum Rabi model is a standard effective Hamiltonian in studies of light-matter interaction, capturing the simplest nontrivial setting in which a qubit couples to a single harmonic oscillator. Within the broader Rabi family, we focus on two special cases: the Jaynes-Cummings (JC) model, which carries an explicit $U(1)$ symmetry, and the asymmetric quantum Rabi model (AQRM), which possesses a parameter-dependent"hidden''symmetry that appears only at integer bias, $\varepsilon/\omega\in\mathbb{Z}$, and is not manifest in the Hamiltonian. We use the time-averaged entangling power as an operator-level diagnostic of these symmetry structures. Since the oscillator Hilbert space is infinite-dimensional, we compare two finite input ensembles: a Haar average after Fock-space truncation and a coherent-state average at fixed mean occupation $\bar{n}$. Both diagnostics show peaks at the integer-bias points of the AQRM, where the hidden symmetry resides. In contrast, the manifest $U(1)$ symmetry at the JC point instead gives a weak dip. Thus, the time-averaged entangling power responds to both the hidden symmetry and the $U(1)$ symmetry in the Rabi family, with the sign of the response indicating how the symmetry reorganizes the spectral expansion. These results demonstrate that the entangling power can serve as an operator diagnostic to reveal the presence and properties of hidden and manifest symmetries in light-matter systems.
Partial dynamical symmetry (PDS) is an algebraic structure in which a prescribed symmetry is neither exact nor completely broken: a subset of eigenstates keeps good quantum numbers and remains solvable while the rest of the spectrum mixes. PDS is currently identified from spectroscopic data, band-head energies, level systematics, and $B(E2)$ ratios. We ask whether it also has a purely structural signature in the eigenstates, and find that it does, though not in the magnitude of entanglement. The natural diagnostic is the variance of a symmetry Casimir, the label variance $\Var\,\C[G]$, which we show coincides with a block-coherence entropy and a block impurity: all three vanish exactly when a state carries a single irreducible-representation label. Resolved state by state, this quantity is zero on the solvable subset and of order $N^2$ on the mixed states, at stable symmetry points and at Leviatan's first- and second-order critical points, where it takes two distinct forms set by the order of the transition. The magnitude of bipartite entanglement, by contrast, does not separate solvable from mixed states and drifts even where the labels are exact. We anchor the analysis in $^{168}$Er, connect the block purity to the ``purity/coherence''language of the quasi-dynamical-symmetry literature, and show the label variance is uncorrelated with multipartite entanglement and with magic. Finally we encode the model on a qubit register and prepare its solvable and mixed eigenstates variationally, as a step toward evaluating the diagnostic on a quantum device.
Entanglement entropy (EE) is commonly studied using real-space bipartitions. We show that, in quantum impurity models, an energy-space bipartition, equivalent to the momentum-space bipartition of the bath, can display universal behavior. Motivated by poor man's scaling, we logarithmically discretize the bath and partition it into high- and low-energy sectors. For models with Fermi-liquid fixed points, including the Anderson model and fully screened or underscreened Kondo models, the low-energy EE flows to constants independent of model parameters. These constants are integer multiples of $\ln 2$ plus corrections that depend only on the logarithmic discretization parameter $\Lambda$. We show that scale invariance of the fixed-point wavefunction in energy space maps to effective translation invariance along a one-dimensional chain, allowing the fixed points to be classified by one-dimensional topological band theory. With low-energy chiral symmetry, each $\ln 2$ contribution originates from a topological edge mode. We also study transitions between distinct Fermi-liquid fixed points using the local-singlet--Kondo-singlet transition in a two-orbital Anderson model driven by an inter-orbital antiferromagnetic coupling. The local-singlet phase has an effectively decoupled impurity and nearly vanishing EE, whereas the Kondo-singlet phase has finite EE larger than $\ln 2$ per spin and orbital. When chiral symmetry holds at low energies, this distinction corresponds to a topological transition of the effective bath chain. At the non-Fermi-liquid critical point, the EE develops an unstable plateau. Its $\Lambda$ dependence resembles that of the overscreened two-channel Kondo model, supporting universality within the same non-Fermi-liquid universality class.
We investigate a spin-$1/2$ anisotropic Heisenberg model on a lattice consisting of two identical bipartite sublattices. A family of exact eigenstates generated by the restricted spectrum generating algebra (RSGA) constitutes quantum many-body scar states, characterized by subextensive entanglement entropy and supporting. These scar states are magnon-pair condensates exhibiting off-diagonal long-range order (ODLRO). At the resonance point of the inter-sublattice interaction, the model exactly maps onto a mixed spin-$1$ and spin-$0$ XY model on a bipartite lattice, which decomposes into independent sub-Hamiltonians labeled by all possible spin configurations. Each spin-$0$ particle is dynamically isolated from its neighbors and acts as a kinetic constraint, giving rise to emergent Hilbert space fragmentation (HSF). Our work establishes an exactly solvable platform in which quantum many-body scars, magnon-pair condensation exhibiting off-diagonal long-range order, and Hilbert space fragmentation naturally coexist.
We present an exact analytical solution of a generic two-mode quantum Rabi model with non-identical bosonic modes. By means of a Bogoliubov transformation, the Hamiltonian is reduced to invariant subspaces labeled by a conserved quantity, and a transcendental G-function is derived, whose zeros determine the regular energy spectrum. We show that the spectra in different subspaces are related by a simple energy shift, revealing the crucial role of mode asymmetry in shaping the spectral structure. In the symmetric limit, the model exhibits an enhanced degeneracy, including fourfold degenerate energy levels arising from the merging of invariant subspaces. The structure of exceptional solutions is analyzed in detail. We identify finite-dimensional Juddian solutions associated with level crossings, as well as special nondegenerate solutions that give rise to isolated states. The degeneracy of these solutions is shown to be controlled by the underlying symmetry of the model. Furthermore, we investigate spectral collapse in the generic two-mode quantum Rabi model. Unlike in the two-photon quantum Rabi model, the collapse energy depends explicitly on the subspace, leading to a richer collapse scenario. We demonstrate that spectral collapse originates from the vanishing spacing between adjacent poles of the G-function, and that the collapse point supports an infinite sequence of bound states. These states exhibit exponential accumulation toward the collapse energy, revealing a hierarchical structure of collapse-induced bound states.