Dynamical Quantum Phase Transitions

• DQPTs are non-equilibrium phenomena in quantum many-body systems characterized by nonanalyticities in the Loschmidt echo rate function.
• They are diagnosed by singularities and Fisher zero crossings in the Loschmidt amplitude, revealing critical real-time dynamics.
• Experimental realizations in ultracold atoms, trapped ions, and photonic simulators validate the role of symmetry and topology in driving DQPTs.

Dynamical quantum phase transitions (DQPTs) are non-equilibrium phenomena in quantum many-body systems characterized by temporally localized nonanalytic behavior in the real-time evolution of the system, analogously to equilibrium phase transitions exhibiting singularities in free energy upon parameter tuning. DQPTs are typically diagnosed by singularities in the Loschmidt echo rate function, signaling points of dynamical orthogonality between the time-evolved state and its initial condition. They are observed in a variety of systems, including integrable and non-integrable spin chains, free-fermion models, topological insulators, periodically driven systems, and open quantum systems, and can be underpinned by symmetry, topology, or more general dynamical mechanisms.

1. Definitions, Diagnostics, and Theoretical Framework

The central quantity in DQPT analysis is the Loschmidt amplitude,

$L(t) = \langle\psi_0|e^{-i H_f t}|\psi_0\rangle,$

where $|\psi_0\rangle$ is typically the ground state of an initial Hamiltonian $H_0$, and $H_f$ is the post-quench Hamiltonian. The rate function, also called the "dynamical free-energy density," is defined as

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

in the thermodynamic limit $N \to \infty$ (Heyl, 2017, Zvyagin, 2017). DQPTs manifest at critical times $t_c$ as nonanalyticities (cusps, kinks, or discontinuities in higher derivatives) of $g(t)$ (Zvyagin, 2017, Schmitt et al., 2015).

Many models allow factorization over momentum, leading to a mode-resolved analysis:

$$L(t) = \prod_k L_k(t), \qquad L_k(t) = \langle \psi_k^0 | e^{-i H_f(k) t} | \psi_k^0 \rangle,$$

where $\psi_k^0$ denotes the pre-quench Bloch eigenstate at momentum $k$.

A DQPT is associated with the crossing of Fisher zeros—zeros of $L(z)$ as a function of complexified time $z$—through the real axis $z = i t$, in analogy to Lee-Yang or Fisher zeros in equilibrium statistical mechanics (Heyl, 2017, Zvyagin, 2017).

2. Classification and Mechanisms

2.1 Symmetry-Breaking and Manifold DQPTs

For models with discrete symmetry-broken ground state manifolds, the dynamical return probability projects onto the manifold, and the rate function develops cusps when dominance among overlaps shifts between components of the initial manifold:

$$g(t) = \min_\alpha \{-N^{-1}\ln |\langle \psi^\alpha_0 | \psi(t) \rangle|^2\}$$

(Damme et al., 2022, Damme et al., 2022). This frequently yields periodic DQPTs synchronized with zeros of a corresponding Landau order parameter in two-level systems, but need not do so in systems with higher degeneracies (Damme et al., 2022, Damme et al., 2022).

2.2 Topological DQPTs

When $H_0$ and $H_f$ differ in their topological invariants (e.g., winding or Chern numbers), there must exist momenta $k^*$ with orthogonal pre- and post-quench Bloch vectors, enforcing zeros of $L_k(t)$ and guaranteeing DQPTs at times $t_c^{(m)} = (2m+1)\pi/(2\epsilon_{k^*})$ (Heyl, 2017, Mendl et al., 2019, Okugawa et al., 2021).

Changes in topological invariants under a quench enforce the existence of one or more DQPTs, with associated quantized jumps in appropriate dynamical topological order parameters such as the winding of the Pancharatnam geometric phase across the Brillouin zone (Mendl et al., 2019, Okugawa et al., 2021).

2.3 Role of Disorder and Non-Topological DQPTs

DQPTs can occur without underlying equilibrium phase transitions, topological changes, or local order parameters. In disordered systems, new critical times may arise triggered by Anderson-orthogonality-type catastrophes, invisible to local or topological observables (Kuliashov et al., 2022, Gurarie, 2018). The presence of disorder leads to new universality classes of DQPTs with distinct scaling and singularity structure, for example, a logarithmic divergence in $g'(t)$ at the critical point in the random field Ising chain (Gurarie, 2018).

3. Analytical Approaches and Key Models

3.1 Real-Space Renormalization and Complex Dynamics

An explicit connection between the renormalization group (RG) and DQPTs is realized by mapping the RG transformation onto an iterated map in the complex plane. For the 1D transverse field Ising chain,

$$y' = R(y) = \tfrac{1}{2}(y + y^{-1}), \qquad y = e^{i J t},$$

and the Julia set $J(R)$—the complex fractal separating basins of RG flow—determines the locus of Fisher zeros that accumulate in the thermodynamic limit (Kaur et al., 18 Sep 2025). The intersections of the temporal trajectory $y=e^{iJt}$ with $J(R)$ fix the critical times of DQPTs (Kaur et al., 18 Sep 2025).

3.2 Free Fermion and Integrable Models

In 1D and 2D BCS-type systems, exact evaluation of Loschmidt amplitudes reveals Fisher zeros forming lines (1D) or areas (2D) in complex time. In 1D, DQPTs appear as discontinuities in $g'(t)$, while in 2D, they emerge in higher derivatives due to the coalescence of Fisher zeros over finite $t$-intervals (Schmitt et al., 2015).

3.3 Floquet DQPTs and Periodic Driving

In driven systems, DQPTs can occur solely due to the micromotion within a period of the time-dependent Hamiltonian (Floquet DQPTs), even in the absence of any quench. These are marked by periodic singularities and topological invariants such as winding numbers of the geometric phase. The occurrence and multiplicity of Floquet DQPTs are controlled by the underlying Floquet band topology (1901.10365, Zhou et al., 2020).

3.4 Mixed States and Open Quantum Systems

DQPTs have been generalized to mixed states using the Uhlmann or interferometric phase and to open systems via quantum trajectories and Lindblad dynamics. In open systems, DQPTs are robust under pure loss or pure gain, but are generically smoothed out by many-body backflow when both processes coexist (Fu et al., 21 Jul 2025, Zhang et al., 3 Sep 2025).

4. Topological Aspects and Dynamical Order Parameters

DQPTs in topological systems are accompanied by quantized jumps in dynamical topological order parameters (DTOPs), such as:

$$\nu_D(t) = \frac{1}{2\pi} \int_{BZ} dk\, \partial_k\, \varphi_k^G(t),$$

where $\varphi_k^G(t)$ is the Pancharatnam geometric phase (Heyl, 2017, Okugawa et al., 2021). Topologically protected DQPTs are enforced whenever equilibrium topological indices change under the quench. For crystalline and mirror-symmetric systems, mirror-symmetry-enforced DQPTs and corresponding mirror-protected DTOPs have been constructed (Okugawa et al., 2021).

Non-Hermitian DQPTs extend this framework to systems described by effective non-Hermitian Hamiltonians or biorthogonal quantum mechanics. The universal condition for a DQPT is the orthogonality of two vectors in a geometric construction on the relevant phase space, with topological jumps predicted under appropriate symmetry constraints (Fu et al., 21 Jul 2025).

5. Experimental Realizations and Observable Signatures

DQPTs have been observed in controlled quantum systems including trapped ions, nitrogen-vacancy centers in diamond, ultracold atomic gases, and photonic simulators. Detection has been achieved by monitoring the Loschmidt echo through full-state tomography, Ramsey interferometry, or measurement of projectors onto the initial state or small subsystems (1901.10365, Halimeh et al., 2020, Heyl, 2017). Local, real-space (Halimeh et al., 2020) and momentum-space (Halimeh et al., 2020) DQPT measures have been proposed for systems where global fidelity measurements are impractical.

Signatures include nonanalytic peak sharpening in effective rate functions, quantized jumps in dynamical topological invariants, and the creation of phase vortices in geometric phases (1901.10365, Heyl, 2017). Experiments have mapped both symmetry-breaking and topologically-protected DQPTs and established scaling behavior consistent with theoretical predictions.

6. Universal Properties, Scaling, and Open Problems

DQPTs exhibit universality classes characterized by the scaling of $g(t)$ or its derivatives near $t_c$. For instance, in the 1D Ising chain, $g(t)\sim |t-t_c|$ (linear cusp), while the random field variant shows a logarithmic singularity, $g(t)\sim \epsilon\ln(1/|\epsilon|)$, at criticality (Gurarie, 2018, Kaur et al., 18 Sep 2025). The underlying RG eigenvalue governs the critical exponent, and the Julia set construction provides fractal boundaries for Fisher-zero accumulation (Kaur et al., 18 Sep 2025).

Outstanding issues include the fate of DQPTs in genuinely interacting, non-integrable, or higher-dimensional systems, the existence of new universality classes, the effect of disorder and symmetry-breaking perturbations, and the macroscopic classification of DQPTs beyond analogy with equilibrium transitions. The role of entanglement, as disentangled from classical precession, has been investigated using matrix product state and transfer matrix analyses, distinguishing precession-driven and entanglement-driven DQPTs (Nicola et al., 2020).

7. Tables: Key Diagnostic Quantities and Models

Quantity	Definition / Criterion	DQPT Signature
Loschmidt amplitude $L(t)$	$\langle\psi_0|e^{-iH_ft}|\psi_0\rangle$	Zero at $t_c$
Rate function $g(t)$	$-N^{-1} \ln |L(t)|^2$	Cusp/kink at $t_c$
Fisher zeros $z_j$	$L(z_j) = 0$ (complex $z$)	$z_j$ crosses $z=it$ axis
Dynamical topological OP	Winding of geometric phase or phase vortices	Quantized jump at $t_c$

Model	DQPT Mechanism	Topological Link
1D TFIM, XY chain	Orthogonality in Bloch vector	Winding/jump in $k$
Kitaev honeycomb (2D)	Area of Fisher zeros	Chern number difference
Floquet/topological drives	Driving/periodicity, micromotion	Floquet winding number
Disordered Ising	Anderson-orthogonality, nonlocality	Absent
Open quantum/Lindblad	Many-body backflow, loss/gain	Robust if single channel

DQPTs constitute a broad and unifying framework for understanding real-time critical phenomena in quantum many-body dynamics, encompassing symmetry, topology, disorder, entanglement, and open-system effects (Heyl, 2017, Kaur et al., 18 Sep 2025, Mendl et al., 2019, Kuliashov et al., 2022, Fu et al., 21 Jul 2025, Zhang et al., 3 Sep 2025).