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arXiv:2610.14591

Exploring Dynamical Quantum Phase Transitions via the Dynamical Phase Accumulated at an Equilibrium Critical Point

Han-Chuan Kou, Peng Li, Zhi-Han Zhang, Zhen-Yu Zheng

quant-phSubmitted Oct 10, 2026• Announced Oct 13, 2026• 0 citations
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Abstract

We investigate dynamical quantum phase transitions (DQPTs) generated by a two-step linear quench with a finite waiting time at the equilibrium critical point. The waiting stage leads to the accumulation of a dynamical phase that depends on the waiting time and directly reflects the critical spectral structure. In the transverse Ising chain, this accumulated phase modifies the critical structure of DQPTs from a single-critical-mode (SCM) regime to a multiple-critical-mode (MCM) regime. We analytically and numerically determine the boundary between the two regimes. Along this boundary, the waiting time approaches a constant in the sudden-quench limit, whereas in the slow-quench limit it follows Kibble--Zurek (KZ) scaling with the quench time. In the SCM regime, the rate function exhibits a conventional linear singularity, $r(t)-r(t^{*})\propto |t-t^{*}|^α$, with the critical exponent $α=1$ near the critical time. At the SCM--MCM boundary, this exponent changes to $α=1/2$, leading to a divergent first derivative of the rate function. Additional discrete parameter lines with $α=1/2$ also appear within the MCM regime in the parameter plane spanned by the waiting and quench times. Our findings establish a direct connection between the accumulated dynamical phase and the critical structure of DQPTs, further supported by an extended Su--Schrieffer--Heeger (SSH) model.

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We investigate dynamical quantum phase transitions (DQPTs) in the extended Agassi model following sudden quenches of same level pairing ($Σ$), monopole-monopole ($χ$) and cross level pairing ($Λ$) 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 $χ$-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 $β\simeq 1/2$, while the $Σ$- and $Λ$-driven transitions are first order. We further characterize the associated dynamical singularities through a local scaling exponent $α$, with $α_χ\simeq 2/3$, $α_Λ\simeq 1/3$, and $α_Σ\simeq 3/2$ 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.

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We investigate dynamical quantum phase transitions (DQPTs) generated by a two-step linear quench with a finite waiting time at the equilibrium critical point. The waiting stage leads to the accumulation of a dynamical phase that depends on the waiting time and directly reflects the critical spectral structure. In the transverse Ising chain, this accumulated phase modifies the critical structure of DQPTs from a single-critical-mode (SCM) regime to a multiple-critical-mode (MCM) regime. We analytically and numerically determine the boundary between the two regimes. Along this boundary, the waiting time approaches a constant in the sudden-quench limit, whereas in the slow-quench limit it follows Kibble--Zurek (KZ) scaling with the quench time. In the SCM regime, the rate function exhibits a conventional linear singularity, r(t)−r(t∗)∝∣t−t∗∣αr(t)-r(t^{*})\propto |t-t^{*}|^αr(t)−r(t∗)∝∣t−t∗∣α, with the critical exponent α=1α=1α=1 near the critical time. At the SCM--MCM boundary, this exponent changes to α=1/2α=1/2α=1/2, leading to a divergent first derivative of the rate function. Additional discrete parameter lines with α=1/2α=1/2α=1/2 also appear within the MCM regime in the parameter plane spanned by the waiting and quench times. Our findings establish a direct connection between the accumulated dynamical phase and the critical structure of DQPTs, further supported by an extended Su--Schrieffer--Heeger (SSH) model.
We study the dynamical quantum phase transitions (DQPTs) manifested in the subsequent unitary dynamics of an extended Ising model with an additional three spin interactions following a sudden quench. Revisiting the equilibrium phase diagram of the model, where different quantum phases are characterized by different winding numbers, we show that in some situations the winding number may not change across a gap closing point in the energy spectrum. Although, usually there exists a one-to-one correspondence between the change in winding number and the number of critical time scales associated with DQPTs, we show that the extended nature of interactions may lead to unusual situations. Importantly, we show that in the limit of the cluster Ising model, three critical modes associated with DQPTs become degenerate, thereby leading to a single critical time scale for a given sector of Fisher zeros.
Previous investigations of dynamical quantum phase transition (DQPT) have predominantly focused on sudden quench protocols. In this work, we systematically explore the fate of DQPT from the sudden to the slow quench regime. We establish that DQPT remains robust under slow quenching when the protocol crosses an equilibrium quantum phase transition. Conversely, we show that accidental DQPTs,which occur when the pre and post quench Hamiltonians reside in the same equilibrium phase, can be filtered out in the slow quench limit. This demonstrates that slow quenches can reveal the intrinsic connection between DQPT and equilibrium quantum phase transitions, thereby enabling DQPT to serve as a dynamical probe for identifying equilibrium quantum critical points. We illustrate these findings with examples from the XY chain, the Aubry-André model, and the trimer Su-Schrieffer-Heeger model.
We investigate the nonequilibrium dynamics induced by a finite-time linear quench in the XYXYXY chain. Initially, we examine the dynamical quantum phase transition (DQPT-LE), characterized by the nonanalytic behavior of the Loschmidt amplitude. We find distinct behaviors of DQPT-LEs during and following the ramp. Following the ramp, the ramp crossing the critical point hc{h}_{c}hc​ is the sufficient condition for the occurrence of DQPT-LE, but it is not during the ramp. Through adiabatic-impulse-adiabatic (AIA) approximation analysis, we establish that adequate distancing from the critical point is crucial for DQPT-LE manifestation during the ramp, elucidating the absence of DQPT-LE as the ramp gets faster. Additionally, we explore another type of dynamical phase transition (DQPT-OP), describing the long-term relaxation behavior of the order parameter. Our finding indicates that the asymptotic behavior of the time-dependent part induced by the linear quench is equivalent to that following a sudden quench, i.e., the time-dependent part exhibits power-law decays of $\ensuremath{\sim}{t}^{\ensuremath{-}3/2}$ and $\ensuremath{\sim}{t}^{\ensuremath{-}1/2}$ for the ramp to the commensurate and incommensurate phases, respectively. Moreover, we also delve into the steady part, which showcases nonanalytic singularities at the critical point.
Abstract Dynamical phase transition in quantum many body systems is usually studied by taking it in the ground state and then quenching a parameter to a new value. We investigate here the dynamics when one performs the time evolution of a generic state and observe that the rate function related to the Loschmidt echo shows non-analytic behavior of two types, one related, and the other unrelated, to the appearance of a quantum phase transition. Specifically, we consider a quantum Ising chain in an initial configuration which is a generic superposition of the eigenstates, and follow its dynamics under the transverse Ising Hamiltonian with a constant field. Depending on the the configuration of the initial state, some singularities appear in the rate function which do not correspond to the equilibrium phase transition of the Hamiltonian. However another class of singularity is found having a connection with the quantum critical point. Some features of the singularities of the rate function have been derived analytically and it is observed that the occupancy of quasiparticle eigenstates plays the key role in triggering the non-analyticities.
The nonequilibrium dynamics of a periodically driven extended XY model, in the presence of linear time dependent magnetic field, is investigated using the notion of dynamical quantum phase transitions (DQPTs). Along the similar lines to the equilibrium phase transition, the main purpose of this work is to search fundamental concepts such as scaling and universality at the ramped quench DQPTs. We have shown that the critical points of the model, where the gap closing occurs, can be moved by tuning the driven frequency and consequently the presence of or absence of DQPTs can be flexibly controlled by adjusting the driven frequency. We have uncovered that, for a ramp across the single quantum critical point, the critical mode at which DQPTs occur is classified into three regions: the Kibble-Zurek (KZ) region, where the critical mode scales linearly with the square root of the sweep velocity, the pre-saturated (PS) region, and the saturated (S) region where the critical mode makes a plateau versus the sweep velocity. While for a ramp that crosses two critical points, the critical modes disclose just the KZ and PS regions. On the basis of numerical simulations, we find that the dynamical free energy scales linearly with time, as approaches to DQPT time, with the exponentν=1±0.01for all sweep velocities and driven frequencies.
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.