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arXiv:2609.24636

Phase space anatomy of dynamical quantum phase transitions

Zakaria Mzaouali

quant-phSubmitted Sep 21, 2026• Announced Sep 22, 2026• 0 citations
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Abstract

Dynamical quantum phase transitions (DQPT) are commonly defined either by the behavior of a late-time order parameter or by nonanalyticities in a quantum state's return rate. We show that discrete phase space fundamentally separates these two notions by the scale of information they require. An order parameter transition is determined by a local reduced state, requiring no quasiprobability negativity. In contrast, general global returns can contain information absent from every proper reduced state. For stabilizer returns, we derive an exact decomposition of the rate into distinct costs from loss of support and destructive quantum interference. Dynamics of a qutrit Potts chain show that selective cancellation can reverse the ranking of competing block returns and delay their exchange. An exact control also exhibits a thermodynamic return cusp with zero negativity at the crossing. The existence of a return singularity and the role of interference in selecting its branches are therefore distinct physical questions.

Comments
17 pages, 6 figures

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Dynamical quantum phase transitions (DQPT) are commonly defined either by the behavior of a late-time order parameter or by nonanalyticities in a quantum state's return rate. We show that discrete phase space fundamentally separates these two notions by the scale of information they require. An order parameter transition is determined by a local reduced state, requiring no quasiprobability negativity. In contrast, general global returns can contain information absent from every proper reduced state. For stabilizer returns, we derive an exact decomposition of the rate into distinct costs from loss of support and destructive quantum interference. Dynamics of a qutrit Potts chain show that selective cancellation can reverse the ranking of competing block returns and delay their exchange. An exact control also exhibits a thermodynamic return cusp with zero negativity at the crossing. The existence of a return singularity and the role of interference in selecting its branches are therefore distinct physical questions.
The analogy between an equilibrium partition function and the return probability in many-body unitary dynamics has led to the concept of dynamical quantum phase transition (DQPT). DQPTs are defined by nonanalyticities in the return amplitude and are present in many models. In some cases, DQPTs can be related to equilibrium concepts, such as order parameters, yet their universal description is an open question. In this Letter, we provide first steps toward a classification of DQPTs by using a matrix product state description of unitary dynamics in the thermodynamic limit. This allows us to distinguish the two limiting cases of "precession" and "entanglement" DQPTs, which are illustrated using an analytical description in the quantum Ising model. While precession DQPTs are characterized by a large entanglement gap and are semiclassical in their nature, entanglement DQPTs occur near avoided crossings in the entanglement spectrum and can be distinguished by a complex pattern of nonlocal correlations. We demonstrate the existence of precession and entanglement DQPTs beyond Ising models, discuss observables that can distinguish them, and relate their interplay to complex DQPT phenomenology.
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Dynamical quantum phase transitions (DQPTs) are an exciting paradigm of out-of-equilibrium criticality in many-body systems manifested in nonanalytic behavior in the return rate to the initial state following a sudden quench. While previous work has tried to distinguish between distinct types of DQPTs, such as regular and anomalous, or manifold and branch, a comprehensive understanding of why each type appears in a given scenario is still lacking. In this work, we propose a unified framework addressing this gap in terms of the energy structure of different product state configurations. In particular, while manifold DQPTs are governed by resonances within the initial state manifold, branch DQPTs are governed by resonances with a transitional manifold of states dynamically connected to the initial manifold by low-order processes. We show that the (ir)regularity of branch DQPTs is related to the multiplicity of this transitional manifold, and we also observe exotic periods of extended degeneracy in the return rate (beyond the conventional level crossing of a DQPT) which are also conditioned on the structure of this transitional manifold. We demonstrate this by studying quenches of two different configurations in the 1 + 1D Z_2 LGT to various parameter regimes. Our findings provide a dynamical mechanism underlying branch DQPTs and frames DQPTs as probes of resonant connectivity in constrained Hilbert spaces, paving the way to a more complete understanding of the multifaceted nature of dynamical criticality.
A sudden quantum quench of a Bloch band from one topological phase toward another has been shown to exhibit an intimate connection with the notion of a dynamical quantum phase transition (DQPT), where the returning probability of the quenched state to the initial state-i.e., the Loschmidt echo-vanishes at critical times {t^{*}}. Analytical results to date are limited to two-band models, leaving the exact relation between topology and DQPT unclear. In this Letter, we show that, for a general multiband system, a robust DQPT relies on the existence of nodes (i.e., zeros) in the wave function overlap between the initial band and the postquench energy eigenstates. These nodes are topologically protected if the two participating wave functions have distinctive topological indices. We demonstrate these ideas in detail for both one and two spatial dimensions using a three-band generalized Hofstadter model. We also discuss possible experimental observations.
We investigate dynamical quantum phase transitions (DQPTs) in the extended Agassi model following sudden quenches of same level pairing (ΣΣΣ), monopole-monopole (χχχ) and cross level pairing (ΛΛΛ) strengths. Using exact diagonalization, we characterize the non-equilibrium dynamics through the Loschmidt echo and intensive return-rate function. Quenches across equilibrium phase boundaries produce cusp-like structures in the return rate that shows non-analyticities with increasing system size. Equilibrium finite size to large scaling identifies the χχχ-driven transition as a second order phase transition similar to mean-field, Dicke, and Lipkin-Meshkov-Glick (LMG) models critical behaviour with the exponent of the order parameter β≃1/2β\simeq 1/2β≃1/2, while the ΣΣΣ- and ΛΛΛ-driven transitions are first order. We further characterize the associated dynamical singularities through a local scaling exponent ααα, with αχ≃2/3α_χ\simeq 2/3αχ​≃2/3, αΛ≃1/3α_Λ\simeq 1/3αΛ​≃1/3, and αΣ≃3/2α_Σ\simeq 3/2αΣ​≃3/2 of all three parameters. The distinct scaling behaviours across the parameter channels show that dynamical criticality is not determined solely by the equilibrium order of the transition, revealing multiple dynamical scaling regimes. These non-analyticities are associated with the Fisher zeros of the Loschmidt amplitude, which cross the real time axis at critical times and the time derivatives of the return-rate function to characterize the order of DQPTs within the extended Agassi model.