Excited-State Quantum Phase Transition

• Excited-state quantum phase transitions are nonanalytic spectral singularities occurring at finite excitation energies that mark drastic changes in eigenstate structure and phase-space topology.
• They are identified by logarithmic divergences in the density of states and sharp shifts in observables like participation ratios and Loschmidt echoes.
• These phenomena arise in various models, including Lipkin–Meshkov–Glick, Dicke, and Bose-Einstein condensates, providing insights for quantum control and coherence protection.

An excited-state quantum phase transition (ESQPT) is a nonanalytic phenomenon occurring at finite excitation energy in the spectrum of a quantum many-body system. Whereas ground-state quantum phase transitions (QPTs) are triggered by variation of a control parameter at zero temperature and involve nonanalyticities only in ground-state properties, ESQPTs are driven by changes in excitation energy (or, equivalently, by scanning a control parameter at fixed excited level) and manifest as singularities—typically logarithmic divergences or discontinuities—in the density of states, level spacing statistics, or expectation values of observables. These spectral singularities reflect underlying stationary points (saddles or maxima) in the classical energy surface and are associated with drastic reorganization of eigenstate structure, bifurcation in phase-space topology, and sharp dynamical anomalies.

1. Definition and Structural Origin of ESQPTs

An ESQPT is rigorously defined by the appearance of a nonanalyticity in the density of states $\rho(E)$ as a function of energy at a critical value $E_c$, typically characterized by

$\rho(E) \sim -\frac{1}{\pi}\log|E-E_c|$

or analogous power-law/cusp behavior, depending on the system's degrees of freedom (Cejnar et al., 2020).

Classical analysis via coherent-state or mean-field theory identifies stationary points (minima, maxima, saddles) of the Hamiltonian $H_{\rm cl}(\vec{x})$ as the locus for ESQPTs; the separatrix formed at energy $E_c$ partitions phase space into structurally different regions, marking the transition between excited-state phases (Engelhardt et al., 2014).

In models with discrete symmetry (e.g., $\mathbb{Z}_2$ parity), the ESQPT is concomitant with a change in eigenstate symmetry: below $E_c$, the spectrum is split into symmetry sectors (often with degenerate doublets), while above $E_c$ this degeneracy is lifted and symmetry is restored (Corps et al., 2021, Corps et al., 2022).

2. Model Hamiltonians and Universal Critical Energies

ESQPTs are observed across various collective and bosonic models:

• Lipkin–Meshkov–Glick (LMG) model:

$H(\alpha) = -\frac{1}{N} S_x^2 + \alpha(S_z + N/2)$

With control parameter $\alpha$, the critical energy is typically $E_c=0$ for $\alpha < \alpha_c$ (Wang et al., 2017, Santos et al., 2016). The underlying classical double-well potential yields a separatrix at $E_c$ (Santos et al., 2016).

• Dicke and Rabi models:

Both exhibit ESQPTs in the superradiant phase, displaying logarithmic singularities in $\rho(E)$ at $E_c=-N/2$ (Dicke) and $E_c = -\Omega/2$ (Rabi) (Pérez-Fernández et al., 2016, Puebla et al., 2016). The ESQPT coincides with a topological change in the classical energy surface.

• Spinor Bose-Einstein condensates:

The mean-field Hamiltonian produces a saddle at $n_0=0$ or $n_0=1$, leading to an ESQPT at $E_c=-N|c||\xi|$ (with $c$ the spin-mixing interaction and $\xi$ a rescaled quadratic Zeeman shift) (Feldmann et al., 2020, Niu et al., 2022).

• Two-level pairing and vibron-like models:

For Hamiltonians interpolating between $U(n)$ and $SO(n+1)$ dynamical symmetries:

$$E_{\rm ESQPT}(\xi) = \frac{[1-5\xi]^2}{16\xi} \quad (\xi > \xi_c)$$

marks the ESQPT separatrix energy (Santos et al., 2015, Pérez-Bernal et al., 2016, Gamito et al., 2022).

3. Spectral and Dynamical Signatures

The hallmark of an ESQPT is a singularity in $\rho(E)$ and related quantities:

• Level density and participation ratio:

At $E=E_c$, $\rho(E)$ spikes sharply, and the participation ratio (PR) in a chosen basis exhibits a pronounced dip, denoting eigenstate localization (Santos et al., 2015, Santos et al., 2016, Gamito et al., 2022).

• Work statistics and Loschmidt echo:

In sudden quench protocols, the mean and variance of quantum work $P(W)$ develop nonanalytic minima at $E_c$, while the survival probability $L(t)$ transitions from regular collapse–revival below $E_c$ to highly irregular, Gaussian statistics at criticality (Wang et al., 2017, Mzaouali et al., 2021).

• Finite-size scaling:

The critical signatures sharpen with increasing $N$. For example, derivatives of $\langle W\rangle/N$ with respect to a control parameter scale as $\ln N$, allowing extraction of critical exponents (e.g., $\nu_e\approx 1$ in LMG) (Wang et al., 2017). Density-of-states peaks and PR minima decrease and narrow as $N$ grows (power-law scaling observed) (Santos et al., 2015, Santos et al., 2016).

• Out-of-time-ordered correlators (OTOC):

OTOCs show distinct regimes: steady plateau for $E<E_c$, irregular flutter at $E\simeq E_c$, and decay to zero for $E>E_c$. The long-time average of the OTOC serves as an effective order parameter for the ESQPT (Wang et al., 2018).

• Multiple quantum coherence spectrum:

The zero-quantum mode $I_0(t)$ and coherence width $\Sigma(t)$ undergo abrupt changes at the ESQPT, revealing phase-space bifurcation (Wang et al., 2022).

4. Phase-Structure and Constants of Motion

The ESQPT divides the spectrum into distinct excited-state phases:

Region	Symmetry/Constant of Motion	Eigenstate Structure
$E < E_c$	$\hat{\mathcal C}$ conserved; $\pm 1$ eigenvalues	Localized, symmetry sectors (e.g., doublets)
$E > E_c$	$\hat{\mathcal C}$ not conserved	Delocalized, symmetry restored

The operator $\hat{\mathcal C}$, constructed as the sign of a properly chosen quantum observable, is conserved only in $E<E_c$ and splits the spectrum into two quantum phases. Observables and equilibrium expectation values depend on both energy and $\langle\hat{\mathcal C}\rangle$ in the symmetry-broken phase, analogous to a generalized Gibbs ensemble (Corps et al., 2021, Corps et al., 2022).

In parity-broken systems, the ESQPT can induce three distinct phases: single-well, double-well with level crossings, and connected (delocalized) (Corps et al., 2022).

5. Impact on Dynamics and Quantum Information

ESQPTs induce critical slowing down: initial states or wave packets with energies near $E_c$ exhibit dramatically suppressed evolution, manifesting as slow decay of survival probability and extremely long relaxation times (Pérez-Bernal et al., 2016, Santos et al., 2015).

Coupling to a system at ESQPT criticality engenders vanishing decoherence rates for embedded qubits, causing maximal quantum speed limit times (QSL), i.e., quantum dynamics is frozen longer when the environment is at $E_c$ (Wang et al., 2018). This phenomenon provides both a diagnostic tool for ESQPTs and a potentially exploitable regime for coherence protection, but impedes fast quantum control.

Experimentally, these signatures can be detected via:

• Measurement of Loschmidt echo statistics, work distribution, and OTOC plateaux in collective spin platforms (cold atoms, ion traps) (Wang et al., 2017, Wang et al., 2018, Niu et al., 2022).
• Interferometric measurement protocols in spinor BECs, revealing jumps in population expectation as energy is tuned across $E_c$ (Feldmann et al., 2020).
• Quantum coherence spectrum analysis (MQC) in NMR and cavity-QED setups (Wang et al., 2022).

6. Extensions: Anharmonicity, Chaos, and Lattice Generalizations

The inclusion of higher-order (anharmonic) terms in models can produce additional ESQPTs linked to boundary-induced separatrices in phase space (Gamito et al., 2022). In the Kerr parametric oscillator and related driven systems, ESQPT singularities are associated with classical separatrices governing cat state formation and decoherence protection; chaos introduced by strong nonlinearities or drive destroys the ESQPT signature and melts protected states (García-Mata et al., 1 Aug 2024).

Generalizations to lattice models (van Hove singularities) and scattering systems (resonant tunneling/delay-time singularities) recast ESQPTs as universal spectral phenomena tied to stationary points in the dispersion or scattering phase shift structure (Cejnar et al., 2020).

7. Non-Hermitian Perspective and Exceptional Points

Non-Hermitian extensions of Hermitian models reveal that the avoided level crossings signaling ESQPTs in finite $N$ are shadows of exceptional points (EPs) in the complexified control parameter plane. As $N\rightarrow\infty$, EPs coalesce onto the real axis, yielding true nonanalyticities characteristic of ESQPTs. Padé extrapolation of EP trajectories as a function of $1/N$ provides precise determination of the thermodynamic ESQPT critical point (Šindelka et al., 2016).

8. Summary and Physical Implications

ESQPTs represent a universal framework for understanding singularities in excitation spectra, eigenstate localization, dynamical slowing, symmetry breaking, and phase-space topology transitions in quantum many-body systems. They unify phenomena across spin systems, light–matter interacting models, Bose-Einstein condensates, and parametric oscillators, with direct consequences for spectral engineering, quantum control, coherence protection, and the design of quantum technologies.

The detection and exploitation of ESQPTs requires: analysis of level density peaks, participation ratio minima, nonanalytic dynamics in quench protocols, and symmetry-resolved observable tracking; and care in accounting for finite-size effects, composite system subtleties, and the role of chaos in melting ESQPT-induced protection (Wang et al., 2017, Cejnar et al., 2020, Santos et al., 2016, Niu et al., 2022, Wang et al., 2022, Wang et al., 2018, Šindelka et al., 2016, García-Mata et al., 1 Aug 2024).