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arXiv:2609.23513

Critical Touching of Temporal Entanglement Transitions

Ting-Long Wang, Shi-Xin Zhang, Shuai Yin, Yi-Fan Jiang

cond-mat.stat-mechquant-phSubmitted Sep 20, 2026• Announced Sep 22, 2026• 0 citations
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Abstract

Equilibrium phase transitions are conventionally categorized into first-order and continuous phase transitions. Far from equilibrium, many new transitions emerge during the real-time evolution of quantum systems. One such transition is the temporal entanglement transition (TET) characterized by the nonanalyticity of the entanglement spectrum. So far, all TETs occur through a linear crossing of the leading Schmidt levels in different symmetry sectors, resembling a first-order transition in equilibrium. A natural question is whether a continuous TET, featured by entanglement spectrum touching, is possible. In a periodically driven transverse $J_1$-$J_2$ Ising chain, we show that two such TETs can merge into a critical touching, where the leading levels meet tangentially without exchanging, realizing the temporal analog of a continuous phase transition. Near the critical frequency, the temporal separation of the two TETs vanishes continuously and its derivative with respect to frequency diverges, a nonanalytic signature that is absent in a single first-order TET. This finite-frequency touching arises from the interplay between a weak symmetry-preserving perturbation of the product initial state and Floquet corrections. Further extending the frequency scan reveals a second critical touching at a higher frequency, and the two critical frequencies enclose a finite window with no TET. These features can be understood from a second-order Floquet Hamiltonian and persist across a broad range of coupling ratios, establishing the critical touching as a distinct form of TET.

Comments
4.5 pages, 3 figures + supplemental material

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Equilibrium phase transitions are conventionally categorized into first-order and continuous phase transitions. Far from equilibrium, many new transitions emerge during the real-time evolution of quantum systems. One such transition is the temporal entanglement transition (TET) characterized by the nonanalyticity of the entanglement spectrum. So far, all TETs occur through a linear crossing of the leading Schmidt levels in different symmetry sectors, resembling a first-order transition in equilibrium. A natural question is whether a continuous TET, featured by entanglement spectrum touching, is possible. In a periodically driven transverse J1J_1J1​-J2J_2J2​ Ising chain, we show that two such TETs can merge into a critical touching, where the leading levels meet tangentially without exchanging, realizing the temporal analog of a continuous phase transition. Near the critical frequency, the temporal separation of the two TETs vanishes continuously and its derivative with respect to frequency diverges, a nonanalytic signature that is absent in a single first-order TET. This finite-frequency touching arises from the interplay between a weak symmetry-preserving perturbation of the product initial state and Floquet corrections. Further extending the frequency scan reveals a second critical touching at a higher frequency, and the two critical frequencies enclose a finite window with no TET. These features can be understood from a second-order Floquet Hamiltonian and persist across a broad range of coupling ratios, establishing the critical touching as a distinct form of TET.
Periodically driven quantum systems can host nonequilibrium phenomena without static analogs, including in their entanglement dynamics. Here, we discover temporal entanglement transitions (TETs) in a Floquet spin chain, which correspond to a quantum phase transition in the spectrum of the entanglement Hamiltonian and are signaled by dynamical spontaneous symmetry breaking. We identify the symmetry principles underlying these transitions: they appear when the driven Hamiltonian preserves global symmetry (here, Z_{2}), the initial state respects this symmetry, and the reduced density matrix carries weight in both subsystem-parity sectors, with TETs occurring precisely when the sector weights become equal (given the previous two conditions are also satisfied). Intriguingly, we find these transitions across a broad range of driving frequencies (from adiabatic to high-frequency regime) and independent of drive details, where they manifest as periodic, sharp entanglement spectrum reorganizations marked by the Schmidt-gap closure, a vanishing entanglement echo, and symmetry-quantum-number flips, while remaining invisible to conventional local observables. At high frequencies, the entanglement Hamiltonian acquires an intrinsic timescale decoupled from the drive period, rendering the transitions genuine steady-state features. Finite-size scaling reveals universal critical behavior with correlation-length exponent ν=1, matching equilibrium Ising universality despite its emergence from purely dynamical mechanisms decoupled from static criticality. Our Letter establishes TETs as novel features in Floquet quantum matter.
We study entanglement transitions in a periodically driven Ising chain in the presence of an imaginary transverse field $\ensuremath{\gamma}$ as a function of drive frequency ${\ensuremath{\omega}}_{D}$. In the high drive amplitude and frequency regime, we find a critical value $\ensuremath{\gamma}={\ensuremath{\gamma}}_{c}$ below which the steady-state half-chain entanglement entropy, SL/2{S}_{L/2}SL/2​, scales with chain length LLL as ${S}_{L/2}\ensuremath{\sim}lnL$; in contrast, for $\ensuremath{\gamma}>{\ensuremath{\gamma}}_{c}$, it becomes independent of LLL. In the small $\ensuremath{\gamma}$ limit, we compute the coefficient, $\ensuremath{\alpha}$, of the lnLlnLlnL term analytically using a Floquet perturbation theory and trace its origin to the presence of Fisher-Hartwig jump singularities in the correlation function of the driven chain. We also study the frequency dependence of ${\ensuremath{\gamma}}_{c}$ and show that ${\ensuremath{\gamma}}_{c}\ensuremath{\rightarrow}0$ at special drive frequencies; at these frequencies, which we analytically compute, SL/2{S}_{L/2}SL/2​ remain independent of LLL for all $\ensuremath{\gamma}$. This behavior can be traced to an approximate emergent symmetry of the Floquet Hamiltonian at these drive frequencies which we identify. Finally, we discus the behavior of the driven system at low and intermediate drive frequencies. Our analysis shows the presence of volume law behavior of the entanglement in this regime ${S}_{\ensuremath{\ell}}\ensuremath{\sim}\ensuremath{\ell}$ for small subsystem length $\ensuremath{\ell}\ensuremath{\le}{\ensuremath{\ell}}^{*}({\ensuremath{\omega}}_{D})$. We identify ${\ensuremath{\ell}}^{*}({\ensuremath{\omega}}_{D})$ and tie its existence to the effective long-range nature of the Floquet Hamiltonian of the driven chain for small subsystem size. We discuss the applicability of our results to other integrable non-Hermitian models.
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.
Dynamical phase transitions (DPTs) occur after quenching some global parameters in quantum systems, and are signalled by the nonanalytical time evolution of the dynamical free energy, which is calculated from the Loschmidt overlap between the initial and time evolved states. In a recent Letter [M. Heyl et al., Phys. Rev. Lett. 110, 135704 (2013)], it was suggested that DPTs are closely related to equilibrium phase transitions (EPTs) for the transverse field Ising model. By studying a minimal model, the XY chain in a transverse magnetic field, we show analytically that this connection does not hold generally. We present examples where DPT occurs without crossing any equilibrium critical lines by the quench, and a nontrivial example with no DPT but crossing a critical line by the quench. Although the nonanalyticities of the dynamical free energy on the real time axis do not indicate the presence or absence of an EPT, the structure of Fisher lines for complex times reveals a qualitative difference.
By considering a solvable driven-dissipative quantum model, we demonstrate that continuous phase transitions in dissipative systems may occur without an accompanying symmetry breaking. As such, the underlying mechanism for this type of transition is qualitatively different from that of continuous equilibrium phase transitions. In our model, the transition is solely a result of the interplay between Hamiltonian and dissipative dynamics and manifests as a nonanalyticity in the steady state ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{\mathrm{ss}}$ in the thermodynamic limit. Based on knowledge from critical classical models we suggest that this behavior derives from a rounding of a first-order phase transition into a continuous one due to large environment-induced fluctuations. Despite being conceptually different from the traditional continuous transitions, we show that expectations of local observables can still be characterized by a set of critical exponents.
Understanding phase transitions in quantum matters constitutes a significant part of present day condensed matter physics. Quantum phase transitions concern ground state properties of many-body systems, and hence their signatures are expected to be pronounced in low-energy states. Here we report signature of a quantum critical point manifested in strongly out-of-equilibrium states with finite energy density with respect to the ground state and extensive (subsystem) entanglement entropy, generated by an external pulse. These non-equilibrium states are evidently completely disordered (e.g., paramagnetic in case of a magnetic ordering transition). The pulse is applied by switching a coupling of the Hamiltonian from an initial value (λI) to a final value (λF) for sufficiently long time and back again. The signature appears as non-analyticities (kinks) in the energy absorbed by the system from the pulse as a function of λF at critical-points (i.e., at values of λF corresponding to static critical-points of the system). As one excites higher and higher eigenstates of the final Hamiltonian H(λF) by increasing the pulse height (|λF - λI|), the non-analyticity grows stronger monotonically with it. This implies adding contributions from higher eigenstates help magnifying the non-analyticity, indicating strong imprint of the critical-point on them. Our findings are grounded on exact analytical results derived for Ising and XY chains in transverse field.