Emergent Asymptotic Excited Stationary States

• Emergent asymptotic highly excited stationary states are eigenstates in quantum many-body systems, defined by robust nonthermal correlations and fixed-point dynamics.
• They arise through integrability, quantum quenches, and non-Hermitian extensions, yielding observable energy scaling, density profiles, and coherence phenomena.
• These states provide actionable insights into ergodicity breakdown, anomalous entanglement, and phase transitions in both closed and dissipative quantum systems.

Emergent asymptotic highly excited stationary states are eigenstates of quantum many-body Hamiltonians that manifest robust stationary properties in the far-from-equilibrium or high-energy-density regime. These states arise in diverse contexts—including quenched integrable chains, nonuniform Bose condensates, open photonic arrays, Landau level electron droplets, and non-Hermitian extensions of nonlinear electronic-structure methods. Unlike ground or low-lying excited states, these highly excited stationary solutions can exhibit nonthermal correlations, anomalous entanglement, and ergodic or nonergodic behavior, and often correspond to fixed points or limit cycles under time evolution, sometimes protected by integrability, symmetry, or topological selection rules.

1. Structural Characterization and Origins

Highly excited stationary states typically reside near the center of the many-body spectrum, where the microcanonical entropy is maximal and local observables may equilibrate to time-independent values even after strong perturbation or quantum quench. For integrable systems, a sudden quench from an initial excited eigenstate generates time evolution governed by a diagonal or generalized Gibbs ensemble (GGE), with stationary values for local observables and entropies computable from overlaps and integrals of motion (Kormos et al., 2014). In open quantum systems with dissipation, the approach to stationary behavior can define quasi-steady fluid phases, distinguished by the relaxation of collective observables against persistent microscopic fluctuations (Mondal et al., 2023). In the Gross-Pitaevskii framework for 1D interacting Bose gases, a discrete ladder of stationary highly excited condensate states emerges, labeled by Bethe-ansatz quantum numbers, each supporting nontrivial density and coherence structure (Tomchenko, 2023). In nonlinear quantum chemistry, complex-analytic extension of interaction parameters uncovers a topological hierarchy of stationary solutions corresponding to highly excited determinants, accessible by a continuous path from the ground state (Burton et al., 2018).

2. Ensemble Descriptions and Entropic Structure

The description of stationary states in integrable models after a quench leverages two complementary ensembles. The diagonal ensemble (DE),

$$\rho_D = \sum_j |c_j|^2 |j\rangle \langle j|,\quad c_j = \langle j|\Psi_0\rangle,$$

retains full information about initial-state overlaps and thus encodes correlations and nonlocal information. The generalized Gibbs ensemble (GGE),

$\rho_{\text{GGE}} = \exp(-\sum_k \lambda_k n_k)/Z,$

is constructed from the (quasi)local integrals of motion, fixing occupation numbers $n_k$ from the initial state. The entropic comparison reveals $S_{\text{GGE}} = 2 S_{\text{diag}}$ in the thermodynamic limit for Ising chain quenches from highly excited states, reflecting the loss of pairing correlations in the GGE (Kormos et al., 2014). Distinct averaging protocols for microstates yield “annealed” and “quenched” thermodynamic entropies, influencing the subleading behavior in large systems.

3. Specific Examples: Doubly Coherent States and Bubble Eigenstates

In the 1D repulsive Bose gas with zero boundaries, the stationary highly excited states are given by time-independent solutions to the GP or GP$_N$ equation,

$$i\hbar \partial_t \Psi = -\frac{\hbar^2}{2m} \partial_x^2\Psi + 2c |\Psi|^2\Psi,$$

with the $j$th excited condensate constructed via an “elementary $j$–series” basis. These solutions exhibit “doubly coherent” character: all $N$ atoms occupy the same orbital mode $\Phi_j(x)$, and simultaneously all $N$ Bogoliubov quasiparticles align in a single phonon mode of quasimomentum $\hbar\pi j/L$ (Tomchenko, 2023). The energy scales quadratically with $j$ as $E_j \sim N (\hbar \pi j/L)^2$, and the density profile manifests $j$ identical domains. Experimentally accessible via Bragg/Raman pulse or adiabatic phonon pumping, such states evidence unique quantum coherence properties.

In strongly interacting LLL electron droplets, exceptional “bubble” states arise as highly excited eigenstates with tight spatial clustering: the density exhibits ring-shaped concentration with local filling $\nu(r) \approx 1$ inside each bubble, and the pair correlation $g(r, r_0)$ resolves discrete electron localization in ordered polygons (Dai et al., 12 Jul 2024). These states are robust under $1/r$ and $1/r^3$ interactions and exemplify nonthermal eigenstates scattered throughout the spectrum, serving as composite objects protected by selection rules and symmetry.

4. Nonlocal Integrals, Integrability, and Emergence under Dynamics

In both integrable and disordered lattice models, highly excited stationary states can be constructed via emergent local integrals of motion (LIOMs, or “l-bits”). In the MBL regime, each eigenstate is parametrized by a set of quasi-local $\tau^z_i$ operators, rendering the Hamiltonian diagonal and forbidding internal thermalization (Parameswaran et al., 2016). A sharp dichotomy exists: thermal eigenstates obey the eigenstate thermalization hypothesis (ETH), show volume-law entanglement and rapid transport, while MBL eigenstates are characterized by area-law entanglement and infinite slow dynamics. Phase transitions between these regimes manifest universal scaling (e.g., $\xi \sim |W-W_c|^{-\nu}$) and anomalous criticality at infinite temperature.

Within the emergent Hamiltonian paradigm, quantum quenches from imbalance generate time-dependent local Hamiltonians $H_e(t)$ whose zero-energy eigenstate exactly describes the time-evolving state, often at highly excited energy density (Vidmar et al., 2015). Entanglement scales logarithmically in subsystem size, despite the energetic extremity, mimicking ground-state properties with critical (non-volume-law) growth.

5. Open Quantum Systems, Ergodicity, and Scarring

Driven-dissipative systems, such as Tavis–Cummings cavity arrays, support the dynamical emergence of quasi-steady highly excited stationary states under chaotic quench dynamics when superradiant fixed points become dynamically unstable (Mondal et al., 2023). Here, ergodic and non-ergodic dynamical observables bifurcate:

• Class I (ergodic): collective photon number and magnetization, stationary regardless of initial condition
• Class II (non-ergodic): phase variables, persistent memory of initial state

Asymptotic distributions of ergodic variables concentrate around unstable fixed points (collective scarring), and the approach is marked by positive Lyapunov exponents and light-cone decorrelator spreading. This collective ergodicity breakdown parallels the many-body scar phenomenon in closed systems.

6. Complex-Analytic and Non-Hermitian Approaches to Excited State Structure

Stationary highly excited solutions can be systematically generated by analytic continuation in the coupling constant of nonlinear quantum chemistry methods. Non-Hermitian extensions of the Hartree–Fock Hamiltonian $H(\lambda)$, with $\lambda \in \mathbb{C}$, reveal that Coulson–Fischer points act as branch points yielding exceptional symmetry-broken solutions (Burton et al., 2018). By traversing complex $\lambda$ contours, one morphs ground-state determinants into excited determinants, uncovering an infinite hierarchy of asymptotic highly excited stationary states in the physical limit. These branches carry geometric connections, and Berry phases arise upon encircling non-Hermitian degeneracies. A plausible implication is that similar topological landscapes exist in coupled-cluster and DFT methods, awaiting explicit construction.

7. Scaling Behavior, Asymptotics, and Experimental Observability

Energy and density scaling in asymptotic stationary states is governed by the underlying single-particle or composite structure. In doubly coherent condensates, energy scales as $E_j \sim j^2$ and wavefunction localization becomes sharper with $j \to \infty$ (Tomchenko, 2023). For electron bubbles, radius and intra-bubble binding scale with system size and interaction power law, preserving state identity in the thermodynamic limit (Dai et al., 12 Jul 2024). In open systems, lifetimes of quasi-steady states scale extensively with system size in the semiclassical limit (Mondal et al., 2023). Experimental detection schemes focus on spectroscopic identification of density domains, vibrational resonances, and collective excitation plateaux.

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References:

• Stationary entropies after a quench from excited states in the Ising chain (Kormos et al., 2014)
• Complex Adiabatic Connection: a Hidden Non-Hermitian Path from Ground to Excited States (Burton et al., 2018)
• Electron bubbles in highly excited states of the lowest Landau level (Dai et al., 12 Jul 2024)
• Eigenstate phase transitions and the emergence of universal dynamics in highly excited states (Parameswaran et al., 2016)
• Emergent eigenstate solution to quantum dynamics far from equilibrium (Vidmar et al., 2015)
• Emergence of a quasi-ergodic steady state in a dissipative Tavis-Cummings array (Mondal et al., 2023)
• Nonuniform Bose-Einstein condensate. II. Doubly coherent states (Tomchenko, 2023)