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arXiv:2610.30647

String Order in Quench Dynamics as a Probe of Quantum Phase Transitions

H. H. Zhang, Z. Song

Published Oct 22, 2026• 0 citations
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Abstract

We study the anisotropic XY chain in a transverse field by introducing a global string order parameter and examining its equilibrium behavior and its quench dynamics from a fully polarized initial state. The corresponding string operator is defined as the product of local projected spin operators over the entire chain. The nonequilibrium order parameter is the long-time average of the logarithm of the string-operator expectation value per site after the quench. For the ground state, its field derivative is singular on the Ising critical lines; after the quench, the steady-state response also records the field boundaries and the anisotropy transition. We derive analytically the nonanalytic behavior of the order parameter at the quantum critical lines. Our results reveal a hidden nonlocal order underlying this paradigmatic model of conventional quantum phase transitions. They also connect measurable spin populations with equilibrium and dynamical topological structure.

DOI
10.48550/arxiv.2610.30647

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We study dynamical properties of the finite-size fully connected Ising model with a transverse field at zero temperature. In a quench dynamics, we study the time period and the first critical time, which play important roles in the dynamical phase transitions, based on a dynamical order parameter and the Loschmidt rate, respectively. When all the spins are initially polarized in the direction of their mutual interaction, we show that both the time period and critical time diverge logarithmically with the system size at the dynamical critical point. When all the spins are initially in the direction of transverse field, both the time period and critical time exhibit logarithmic or power-law divergences depending on the final field strength. In the case of convergence, we provide estimates for the finite-size scaling and converged value. We also investigate the equilibrium phase transition, presenting approximate ground and first excited states away from the criticality, and compare their energy gap and bipartite and multipartite entanglements with the exact eigenstates.
Topological order is defined by topological invariants, rather than symmetries and local order parameters. Nonetheless some topological phases can be characterized by string order parameters and entanglement. In this article we study how string order parameters and entanglement spectra behave out-of-equilibrium following quenches in one dimensional topological models with Z\mathbb{Z}Z invariants. Previously it has been suggested that string order parameters could serve as an experimental probe of dynamical quantum phase transitions. Despite the existence of clear zeroes in the order parameters at critical times, we show that in general there is no exact quantitative or qualitative connection with the critical times of dynamical quantum phase transitions. Another possible connection is of dynamical string order parameter zeroes and dynamical crossings at the center of entanglement spectra. Here we see that there can sometimes be a connection, but it is not typical. Again there is no general quantitative or qualitative connection. Each dynamical form of criticality behaves independently, though we do see that critical times tend to be of the same order of magnitude and give an argument for why this is the case. We also find that a string order parameter which labels one topological phase can undergo non-trivial dynamics even following a quench between \emph{other} topological phases. We elucidate where connections can be made, and where they result from a consideration of insufficiently general models. These results cast doubt on the idea of genuine dynamical phases following quenches in such models.
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We study the time evolution of spin squeezing in the one-dimensional spin-1/2 XY model subject to a sudden quantum quench of a transverse magnetic field. The initial state is selected from the ground state phase diagram of the model, consisting of ferro- and paramagnetic phases separated by a critical value of the transverse field. Our analysis, based on exact results for the model, reveals that by a proper choice of protocol, a quantum quench from an unsqueezed state can create spin-squeezed nonequilibrium states. Moreover, we identify a nonanalyticity in the long-time average of the spin-squeezing parameter when quenching to the equilibrium quantum critical point. This suggests that the ferro- and paramagnetic phases also define distinct phases for how the transverse field redistributes quantum fluctuations among the spin components away from equilibrium.
We investigate a transverse-field Ising chain with dimerized anisotropic Gamma interactions and demonstrate that, both in the absence and presence of Ising interactions, the competition between Gamma anisotropy and the transverse field gives rise to vector chiral order and dynamical quantum phase transitions. The phase diagram comprises positive and negative vector chiral phases and a paramagnetic phase, characterized by the vector spin chirality and the vector chiral order parameter. In the absence of Ising interactions, we identify a gapless critical line that exhibits no finite-size drift, along which the spin-spin correlations and subsystem entanglement entropy display behavior distinct from that of Ising criticality. Furthermore, we find that the critical times of dynamical quantum phase transitions form periodic or aperiodic sequences, depending on the quench path. Our findings highlight how dimerized off-diagonal exchange enriches both equilibrium ordering and dynamical critical behavior in a quantum Ising chain.