Large deviation theory provides the framework to study the probability of rare fluctuations of time-averaged observables, opening new avenues of research in nonequilibrium physics. Some of the most appealing results within this context are dynamical phase transitions (DPTs), which might occur at the level of trajectories in order to maximize the probability of sustaining a rare event. While macroscopic fluctuation theory has underpinned much recent progress on the understanding of symmetry-breaking DPTs in driven diffusive systems, their microscopic characterization is still challenging. In this work we shed light on the general spectral mechanism giving rise to continuous DPTs not only for driven diffusive systems, but for any jump process in which a discrete Z_{n} symmetry is broken. By means of a symmetry-aided spectral analysis of the Doob-transformed dynamics, we provide the conditions whereby symmetry-breaking DPTs might emerge and how the different dynamical phases arise from the specific structure of the degenerate eigenvectors. In particular, we show explicitly how all symmetry-breaking features are encoded in the subleading eigenvectors of the degenerate subspace. Moreover, by partitioning configuration space into equivalence classes according to a proper order parameter, we achieve a substantial dimensional reduction which allows for the quantitative characterization of the spectral fingerprints of DPTs. We illustrate our predictions in several paradigmatic many-body systems, including (1) the one-dimensional boundary-driven weakly asymmetric exclusion process (WASEP), which exhibits a particle-hole symmetry-breaking DPT for current fluctuations, (2) the three- and four-state Potts model for spin dynamics, which displays discrete rotational symmetry-breaking DPTs for energy fluctuations, and (3) the closed WASEP which presents a continuous symmetry-breaking DPT into a time-crystal phase characterized by a rotating condensate.
Large deviation theory provides the framework to study the probability of rare fluctuations of time-averaged observables, opening new avenues of research in nonequilibrium physics. Some of the most appealing results within this context are dynamical phase transitions (DPTs), which might occur at the level of trajectories in order to maximize the probability of sustaining a rare event. While macroscopic fluctuation theory has underpinned much recent progress on the understanding of symmetry-breaking DPTs in driven diffusive systems, their microscopic characterization is still challenging. In this work we shed light on the general spectral mechanism giving rise to continuous DPTs not only for driven diffusive systems, but for any jump process in which a discrete Z_{n} symmetry is broken. By means of a symmetry-aided spectral analysis of the Doob-transformed dynamics, we provide the conditions whereby symmetry-breaking DPTs might emerge and how the different dynamical phases arise from the specific structure of the degenerate eigenvectors. In particular, we show explicitly how all symmetry-breaking features are encoded in the subleading eigenvectors of the degenerate subspace. Moreover, by partitioning configuration space into equivalence classes according to a proper order parameter, we achieve a substantial dimensional reduction which allows for the quantitative characterization of the spectral fingerprints of DPTs. We illustrate our predictions in several paradigmatic many-body systems, including (1) the one-dimensional boundary-driven weakly asymmetric exclusion process (WASEP), which exhibits a particle-hole symmetry-breaking DPT for current fluctuations, (2) the three- and four-state Potts model for spin dynamics, which displays discrete rotational symmetry-breaking DPTs for energy fluctuations, and (3) the closed WASEP which presents a continuous symmetry-breaking DPT into a time-crystal phase characterized by a rotating condensate.
In this PhD thesis, I investigate the properties of symmetry-breaking dynamical phase transitions that manifest in the fluctuations of time-integrated observables within classical systems. In particular, I analyze how these phase transitions impose stringent constraints on the structure of the eigenvectors of the system dynamical generator of the dynamics. Additionally, I identify a dynamical phase transition to a time-crystal phase in a model of driven-diffusive lattice gas. The study of this transition then allows the identification of the "packing-field" mechanism responsible for its emergence. This mechanism is then exploited to propose new transport models displaying time-crystal behavior.
We present a theory for the two kinds of dynamical quantum phase transitions, termed DPT-I and DPT-II, based on a minimal set of symmetry assumptions. In the special case of collective systems with infinite-range interactions, both are triggered by excited-state quantum phase transitions. For quenches below the critical energy, the existence of an additional conserved charge, identifying the corresponding phase, allows for a nonzero value of the dynamical order parameter characterizing DPTs-I, and precludes the main mechanism giving rise to nonanalyticities in the return probability, trademark of DPTs-II. We propose a statistical ensemble describing the long-time averages of order parameters in DPTs-I, and provide a theoretical proof for the incompatibility of the main mechanism for DPTs-II with the presence of this additional conserved charge. Our results are numerically illustrated in the fully connected transverse-field Ising model, which exhibits both kinds of dynamical phase transitions. Finally, we discuss the applicability of our theory to systems with finite-range interactions, where the phenomenology of excited-state quantum phase transitions is absent. We illustrate our findings by means of numerical calculations with experimentally relevant initial states.
Quantum phases and their transitions play a pivotal role to characterize macroscopic matter. Recent studies have revealed that critical transitions reminiscent of equilibrium phase transitions can occur even out of equilibrium, as a singularity of dynamical counterpart of free energy. In this paper, we report a novel mechanism of dynamical phase transitions caused by spontaneous hidden antiunitary-symmetry breaking in periodically driven unitary dynamics. We characterize dynamical phases in such systems with nonunitary many-body operator obtained by switching space and time and find that its symmetry constrains the nature of phases and their transitions. In particular, at some parameters of stroboscopic Ising models, dynamical free energy density exhibits a critical transition with novel singularity absent in equilibrium. The criticality and its universal exponent originate from a many-body exceptional point of the nonunitary operator associated with spontaneous antiunitary-symmetry breaking. We also show that the generalized correlation function has the divergent correlation length at transition and exhibits oscillatory long-range order after symmetry breaking. Our results open up a quest for hitherto unknown phases with their hidden connection to nonunitary physics in macroscopic nonequilibrium systems, which can be tested in state-of-the-art quantum simulators.
The breaking of the continuous time-translation symmetry manifests, in Markovian open quantum systems, through the emergence of nonstationary dynamical phases. Systems that display nonequilibrium transitions into these phases are referred to as time crystals, and they can be realized, for example, in many-body systems governed by collective dissipation and long-ranged interactions. Here, we provide a complete analytical characterization of a boundary time-crystal phase transition. This involves exact expressions for the order parameter and for the dynamics of quantum fluctuations, which, in the time-crystalline phase, remains asymptotically non-Markovian as a consequence of the time-translation symmetry breaking. We demonstrate that boundary time crystals are intrinsically critical phases, where fluctuations exhibit a power-law divergence with time. Our results show that a dissipative time-crystal phase is far more than merely a classical nonlinear and nonstationary (limit cycle) dynamics of a macroscopic order parameter. It is rather a genuine many-body phase where the properties of correlations distinctly differ from those of stationary ones.
A parity-time (PT) transition is a spectral transition characteristic of non-Hermitian generators; it typically occurs at an exceptional point, where multiple eigenvectors coalesce. The concept of a PT transition has been extended to Markovian open quantum systems, which are described by the GKSL equation. Interestingly, the PT transition in many-body Markovian open quantum systems, the so-called \textit{Lindbladian PT (L-PT) phase transition}, is closely related to two classes of exotic nonequilibrium many-body phenomena: \textit{continuous-time crystals} and \textit{non-reciprocal phase transitions}. In this review, we describe the recent advances in the study of L-PT phase transitions. First, we define PT symmetry in three distinct contexts: non-Hermitian systems, nonlinear dynamical systems, and Markovian open quantum systems, highlighting the interconnections between these frameworks. Second, we develop mean-field theories of L-PT phase transitions for collective-spin systems and for bipartite bosonic systems with particle-number conservation. Within these classes of models, we show that L-PT symmetry can induce a breaking of continuous time-translation symmetry down to a discrete one, leading to persistent periodic dynamics. We further demonstrate that the L-PT phase transition point is typically \textit{a critical exceptional point}, where multiple collective excitation modes with zero excitation spectrum coalesce. These findings establish an explicit connection to continuous-time crystals and non-reciprocal phase transitions. Third, going beyond the mean-field theory, we analyze statistical and quantum properties, such as purity and quantum entanglement indicators of time-independent steady states for several specific models with the L-PT symmetry. Finally, we discuss future research directions for L-PT phase transitions.
We study dynamical phase transitions (DPTs) in quantum many-body systems with infinite-range interaction, and present a theory connecting the two kinds of known DPTs (sometimes referred to as DPTs-I and DPTs-II) with the concept of excited-state quantum phase transition (ESQPT), traditionally found in collective models. We show that DPTs-I appear as a manifestation of symmetry restoration after a quench from the broken-symmetry phase, the limits between these two phases being demarcated precisely by an ESQPT. We describe the order parameters of DPTs-I with a generalization of the standard microcanonical ensemble incorporating the information of two additional conserved charges identifying the corresponding phase. We also show that DPTs-I are linked to a mechanism of information erasure brought about by the ESQPT, and quantify this information loss with the statistical ensemble that we propose. Finally, we show analytically the main mechanism for DPTs-II is forbidden in these systems for quenches leading a broken-symmetry initial state to the same broken-symmetry phase, on one side of the ESQPT, and we provide a formulation of DPTs-II depending on the side of the ESQPT where the quench ends. We analyze the connections between various indicators of DPTs-II. Our results are numerically illustrated in the infinite-range transverse-field Ising model and are applicable to a large class of collective quantum systems satisfying a set of conditions.