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Dynamical Phase Boundary in Long-Range Quantum Ising Chains

igor merlini

Published Apr 3, 2026• 0 citations
Publisher

Abstract

We systematically characterize a previously unidentified dynamical phase boundary alpha*(h/J) in the one-dimensional transverse-field Ising model with power-law decaying interactions ~1/r^alpha. Three main results are established: (i) the dominant oscillation frequency of the mean magnetization satisfies omega = Delta/2pi universally across all interaction ranges alpha tested; (ii) the finite-size scaling of the energy gap changes qualitatively at alpha*(h/J), separating exponential (long-range) from power-law (short-range effective) scaling; (iii) the boundary follows alpha*(h/J) = (c*J/h)^3 + b with c ~ e/2 and b ~ 5/3, and is characterized by the effective crossover condition Jeff(alpha*) ~ (h/J)/sqrt(3) + 1/2, suggesting the crossover is controlled by a balance between collective interaction strength and transverse field energy scale. Results are obtained via exact diagonalization (QuTiP, N=4-12) and symbolic regression (PySR).

DOI
10.5281/zenodo.19410024

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We systematically characterize a previously unidentified dynamical phase boundary alpha*(h/J) in the one-dimensional transverse-field Ising model with power-law decaying interactions ~1/r^alpha. Three main results are established: (i) the dominant oscillation frequency of the mean magnetization satisfies omega = Delta/2pi universally across all interaction ranges alpha tested; (ii) the finite-size scaling of the energy gap changes qualitatively at alpha*(h/J), separating exponential (long-range) from power-law (short-range effective) scaling; (iii) the boundary follows alpha*(h/J) = (cJ/h)^3 + b with c ~ e/2 and b ~ 5/3, and is characterized by the effective crossover condition Jeff(alpha) ~ (h/J)/sqrt(3) + 1/2, suggesting the crossover is controlled by a balance between collective interaction strength and transverse field energy scale. Results are obtained via exact diagonalization (QuTiP, N=4-12) and symbolic regression (PySR).
Abstract We present a detailed study of the finite-size one-dimensional quantum XY chain in a transverse field in the presence of boundary fields coupled with the order-parameter spin operator. We consider fields located at the chain boundaries that have the same strength and that are oppositely aligned. We derive exact expressions for the gap Δ as a function of the model parameters for large values of the chain length L. These results allow us to characterize the nature of the ordered phases of the model. We find a magnetic (M) phase ( Δ ∼ e − a L ), a magnetic-incommensurate (MI) phase ( Δ ∼ e − a L f MI ( L ) ), a kink (K) phase ( Δ ∼ L − 2 ) and a kink-incommensurate (KI) phase ( Δ ∼ L − 2 f KI ( L ) ); f MI ( L ) and f KI ( L ) are bounded oscillating functions of L. We also analyze the behavior along the phase boundaries. In particular, we characterize the universal crossover behavior across the K-KI phase boundary. On this boundary, the dynamic critical exponent is z = 4, i.e. Δ ∼ L − 4 for large values of L.
We study the low-energy properties of the long-range random transverse-field Ising chain with ferromagnetic interactions decaying as a power of the distance. Using variants of the strong-disorder renormalization group method, the critical behavior is found to be controlled by a strong-disorder fixed point with a finite dynamical exponent . Approaching the critical point, the correlation length diverges exponentially. In the critical point, the magnetization shows an -independent logarithmic finite-size scaling and the entanglement entropy satisfies the area law. These observations are argued to hold for other systems with long-range interactions, even in higher dimensions.
We study the phase diagram and critical properties of quantum Ising chains with long-range ferromagnetic interactions decaying in a power-law fashion with exponent α \\alpha , in regimes of direct interest for current trapped ion experiments. Using large-scale path integral Monte Carlo simulations, we investigate both the ground-state and the nonzero-temperature regimes. We identify the phase boundary of the ferromagnetic phase and obtain accurate estimates for the ferromagnetic-paramagnetic transition temperatures. We further determine the critical exponents of the respective transitions. Our results are in agreement with existing predictions for interaction exponents α > 1 \\alpha > 1 up to small deviations in some critical exponents. We also address the elusive regime α < 1 \\alpha < 1 , where we find that the universality class of both the ground-state and nonzero-temperature transition is consistent with the mean-field limit at α = 0 \\alpha = 0 . Our work not only contributes to the understanding of the equilibrium properties of long-range interacting quantum Ising models, but can also be important for addressing fundamental dynamical aspects, such as issues concerning the open question of thermalization in such models.
We theoretically study the dynamics of a transverse-field Ising chain with power-law decaying interactions characterized by an exponent α, which can be experimentally realized in ion traps. We focus on two classes of emergent dynamical critical phenomena following a quantum quench from a ferromagnetic initial state: The first one manifests in the time-averaged order parameter, which vanishes at a critical transverse field. We argue that such a transition occurs only for long-range interactions α≤2. The second class corresponds to the emergence of time-periodic singularities in the return probability to the ground-state manifold which is obtained for all values of α and agrees with the order parameter transition for α≤2. We characterize how the two classes of nonequilibrium criticality correspond to each other and give a physical interpretation based on the symmetry of the time-evolved quantum states.
We study the relaxation of the local ferromagnetic order in the transverse field quantum Ising chain with power-law decaying interactions \sim 1/r^\alpha ∼ 1 / r α . We prepare the system in the GHZ state and study the time evolution of the probability distribution function (PDF) of the order parameter within a block of \ell ℓ when quenching the transverse field. The model is known to support long range order at finite temperature for \alpha\leq 2.0 α ≤ 2.0 . In this regime, quasi-localized topological magnetic defects are expected to strongly affect the equilibration of the full probability distribution. We highlight different dynamical regimes where gaussification mechanism may be slowed down by confinement and eventually breaks. We further study the PDF dynamics induced by changing the effective dimensionality of the system; we mimic this by quenching the range of the interactions. As a matter of fact, the behavior of the system crucially depends on the value of \alpha α governing the unitary evolution.
We analyze the quantum phases, correlation functions and edge modes for a class of spin-1/2 and fermionic models related to the one-dimensional Ising chain in the presence of a transverse field. These models are the Ising chain with anti-ferromagnetic long-range interactions that decay with distance r as 1 / r α , as well as a related class of fermionic Hamiltonians that generalize the Kitaev chain, where both the hopping and pairing terms are long-range and their relative strength can be varied. For these models, we provide the phase diagram for all exponents α , based on an analysis of the entanglement entropy, the decay of correlation functions, and the edge modes in the case of open chains. We demonstrate that violations of the area law can occur for α ≲ 1 , while connected correlation functions can decay with a hybrid exponential and power-law behavior, with a power that is α -dependent. Interestingly, for the fermionic models we provide an exact analytical derivation for the decay of the correlation functions at every α . Along the critical lines, for all models breaking of conformal symmetry is argued at low enough α . For the fermionic models we show that the edge modes, massless for α ≳ 1 , can acquire a mass for α < 1 . The mass of these modes can be tuned by varying the relative strength of the kinetic and pairing terms in the Hamiltonian. Interestingly, for the Ising chain a similar edge localization appears for the first and second excited states on the paramagnetic side of the phase diagram, where edge modes are not expected. We argue that, at least for the fermionic chains, these massive states correspond to the appearance of new phases, notably approached via quantum phase transitions without mass gap closure. Finally, we discuss the possibility to detect some of these effects in experiments with cold trapped ions.