--- title: Excited-State Quantum Phase Transition url: https://www.emergentmind.com/topics/excited-state-quantum-phase-transition type: topic --- # Excited-State Quantum Phase Transition 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 [2011.01662]. 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 [1405.3514]. 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 [2103.10762, 2201.03976]. ## 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 $\alpha < \alpha_c$ [1705.04547, 1604.04289]. The underlying classical double-well potential yields a separatrix at $E_c$ [1604.04289]. - **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) [1610.08350, 1607.00283]. 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) [2011.02823, 2211.15142]. - **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 [1506.06765, 1604.06851, 2202.11413]. ## 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 [1506.06765, 1604.04289, 2202.11413]. - **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 [1705.04547, 2101.04985]. - **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) [1705.04547]. Density-of-states peaks and PR minima decrease and narrow as $N$ grows (power-law scaling observed) [1506.06765, 1604.04289]. - **Out-of-time-ordered correlators (OTOC)**: OTOCs show distinct regimes: steady plateau for $EE_c$. The long-time average of the OTOC serves as an effective order parameter for the ESQPT [1812.01920]. - **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 [2209.02510]. ## 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