Dynamical Phase Transitions in Many-Body Systems

• Dynamical phase transitions are defined by time-local singularities in system observables, marking abrupt changes in many-body dynamics.
• They are classified into Type I (long-time averages) and Type II (Loschmidt echo based) transitions, each with distinct signatures.
• Analytical methods such as partition function zeros and large-deviation theory enable experimental detection across quantum quenches, open systems, and classical dynamics.

Dynamical phase transitions (DPTs) are temporally localized singularities in the evolution of many-body systems, marked by non-analyticities in trajectory-level statistical or dynamical observables. Unlike equilibrium phase transitions, which are defined by non-analyticities of thermodynamic quantities as a function of control parameters like temperature or field, DPTs manifest as critical behavior in real time, or in space–time ensembles, across quantum, classical, and open or dissipative systems. The DPT concept unifies critical behavior in quantum quenches, nonequilibrium statistical ensembles, large-deviation trajectories, and emergent feedback-driven instabilities in complex systems.

1. Fundamental Definitions and Classification

DPTs are identified by non-analytic points in a dynamical “free energy.” In closed quantum systems, the central object is the Loschmidt amplitude

$$G(t) = \langle \psi_0 | e^{-iH t} | \psi_0 \rangle,$$

where $|\psi_0\rangle$ is the initial state, $H$ is the post-quench Hamiltonian. The corresponding intensive rate is

$g(t) = -\frac{1}{N}\ln|G(t)|^2,$

with non-analyticities in $g(t)$ (or its derivatives) signaling DPTs (Vajna et al., 2014, Sedlmayr, 2019).

In open systems, or for time-ordered or trajectory observables, the framework generalizes to large deviation theory: the scaled cumulant generating function (SCGF), or “dynamical free energy,”

$\psi(s) = \lim_{T\to\infty} \frac{1}{T}\ln Z(s,T)$

is defined from the moment-generating function $Z(s,T)$ of a time-extensive observable (e.g., quantum jumps, dynamical activity). Non-analyticities of $\psi(s)$ as a function of the conjugate field $s$ define DPTs in trajectory space, with distinct dynamical phases associated to different behaviors of trajectory ensembles (Lesanovsky et al., 2012, Ye et al., 2022).

A major classification distinguishes:

• Type I (DPT-I): Singularities in long-time averages of macroscopic observables (dynamical order-parameter transitions).
• Type II (DPT-II): Non-analyticities in the Loschmidt echo rate function or dynamical free energy (return-probability–based transitions). In quantum systems with symmetry, these may be separated by the presence of emergent conserved quantities or excited-state quantum phase transitions (Corps et al., 2022, Puebla, 2020).

2. Mechanisms and Mathematical Structures

The universal mathematical structure underlying DPTs is rooted in partition function zeros, echoing the Yang-Lee mechanism of equilibrium transitions:

• In Loschmidt or dynamical partition functions, zeros in the complex time or field plane accumulate and pinch the real axis as system size or observation time diverges, leading to real-time non-analyticities in rate functions [$Z(s,t)$; (Ye et al., 2022, Vajna et al., 2014, Hickey et al., 2013)].
• In Markovian trajectory ensembles, the SCGF $\psi(s)$ is the dominant eigenvalue of a tilted generator (quantum or classical), with transitions characterized by level crossings or gap closings in this spectrum (Lesanovsky et al., 2012, Hurtado-Gutiérrez et al., 2023).

For diffusive systems described by macroscopic fluctuation theory (MFT), DPTs emerge as singularities in the large-deviation rate function $I(J)$ for time-averaged currents, often associated with phase-space competition between multiple optimal density profiles ("additivity principle") and transitions between symmetry-related trajectory sectors (Shpielberg, 2017, Agranov et al., 2022).

3. Physical Realizations and Phenomenology

3.1 Isolated Quantum Systems and Quantum Quenches

After a sudden quench, non-analyticities in the Loschmidt echo rate are observed at critical times given by the crossing of Fisher zeros, which depend sensitively on the overlap between pre- and post-quench eigenstates. Topological systems display DPTs protected or indexed by changes in the winding number or Chern number, with the number or character of non-analyticities directly linked to topological invariants (Vajna et al., 2014, Sedlmayr, 2019, 1901.10365).

3.2 Open Quantum Systems and Quantum Trajectory DPTs

In Lindbladian (open) quantum systems, dynamical phases are classified via singularities of the trajectory-space SCGF, which can be first or second order as a function of control fields, and may or may not coincide with stationarity or thermodynamic order. Glassy and kinetically constrained models demonstrate that trajectories can be dynamically critical even when stationary states are trivial (Lesanovsky et al., 2012, Canovi et al., 2014).

3.3 Classical and Markovian Dynamics

Classical non-equilibrium settings, including boundary-driven diffusive systems, exclusion processes, and interacting networks, exhibit DPTs as singularities in current fluctuation LDFs, order–disorder transitions of density-wave modulation, or finite-time blowup in nonlinear dynamical variables. These transitions are universal, manifest across domains such as active matter, social and financial networks, and ecological cascades (Liu et al., 2024, Agranov et al., 2022, Shpielberg, 2017).

3.4 Glasses and Trajectory-Space First-Order Transitions

Glassy dynamics are characterized by trajectory-space first-order DPTs, separating active (liquid-like) and inactive (glassy) phases, observable as thermodynamic-like coexistence in dynamical activity statistics. This dynamical phase framework reconciles kinetic-facilitation and configurational-entropy-based thermodynamic theories, with first-order lines terminating in dynamical lower critical points near the Kauzmann temperature (Royall et al., 2020).

4. Topological and Symmetry-Breaking DPTs

Topology and symmetry play a central role in the structure and classification of DPTs:

• In 1D and 2D two-band fermionic systems, the difference in winding numbers or Chern numbers between pre- and post-quench Hamiltonians dictates the minimal number and nature of DPTs; these are robust to perturbations that do not close the gap (Vajna et al., 2014, Sedlmayr, 2019, 1901.10365).
• Symmetry breaking in trajectory ensembles appears as spectral degeneracies of the tilted or Doob-transformed generator, with the emergent structure of degenerate eigenvectors encoding Z$_n$ (Potts, particle–hole) or U(1) (time-crystal) symmetry-broken dynamical phases (Hurtado-Gutiérrez et al., 2023).

Topological invariants may also be time-dependent, as in Floquet DQPTs, or realized in squeezed trajectory states with diverging geometric phases (Berry/Chern), manifesting critical geometry in dynamical space (1901.10365, Hickey et al., 2013).

5. Orders, Tricriticality, and Universality Classes

DPTs exhibit both first-order (rate/discontinuity) and second-order (derivative/cusp) transitions in dynamical free energy. Macroscopic fluctuation theory has uncovered Landau theory scenarios with tricritical points, where first- and second-order DPT lines meet, controlled by analyticity-breaking in the observable bias field and the sign of quartic terms in the order-parameter expansion (Agranov et al., 2022). In disordered quantum systems (e.g., random-field Ising), DPTs can feature anomalous universality classes, such as logarithmically divergent slopes at DPT points (Gurarie, 2018).

Table: Types of dynamical phase transitions in representative models.

Model/Class	DPT Mechanism	Order Parameter / Signature
Quantum quench (Ising, SSH)	Loschmidt echo Fisher zeros	$g(t)$ nonanalytic at $t_c$, linked to $\Delta w,\Delta Q$
Open quantum glass	Tilted generator eigenvalue	SCGF $\psi(s)$ nonanalytic, activity rate jumps
SSEP / driven diffusive	MFT, optimal path competition	LDF $I(J)$ singularity, density profile change
Markovian jump process, WASEP	Symmetry-degenerate spectrum	Reduced order parameter becomes multimodal
Classical networks	Nonlinear feedback & Riccati ODE	Mean degree blows up as $(t_c-t)^{-1}$
Glass-former trajectory	Large-deviation, $s$ or $\mu$-ensemble	First-order transition in time-averaged activity/structure

6. Experimental Realizations and Detection

Direct experimental signatures of DPTs include time-dependent collapse of Loschmidt echoes (observed in trapped-ion and cold atom systems), cusp singularities in return probabilities, singularities in trajectory-based entropy measures, and geometric phase jumps detected via state tomography. In time-series experiments, robust estimators based on Rényi-entropy derivatives (KS entropy, bit-variance) serve as early-warning or detection tools for DPTs in observed data streams, accurately predicting bifurcation points in chaotic and noisy dynamical systems (Sándor et al., 2024).

7. Outlook and Connections

Dynamical phase transitions unify ideas across quantum dynamics, statistical mechanics, and complex systems: trajectory ensemble theory, partition function zeros (Yang-Lee), and large-deviation approaches provide a statistical-mechanical framework for dynamics in space–time. The recognition of DPTs as finite-time singularities offers new paradigms for the criticality of emergent phenomena in networks, classical and quantum many-body systems, and opens avenues to classify, predict, and control critical transitions purely from dynamical observables (Ye et al., 2022, Liu et al., 2024).

Despite substantial advances, major open questions persist: universality classes for DPTs in disordered and many-body localized systems, the precise structure and robustness of trajectory symmetry breaking in non-Markovian environments, and the development of universally applicable order parameters for arbitrary observable classes. Emerging experimental and data-driven diagnostics, including direct geometric and spectral analyses, are poised to play a decisive role in advancing the theory and application of dynamical phase transitions across physics and complex systems.