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arXiv:2505.02661 · Journal of Physics Condensed Matter

Role of noise on defect formation and correlations in a long-range Ising model under adiabatic driving

Santanu Dhara, Suhas Gangadharaiah

Published Apr 8, 2026• 0 citations
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Abstract

Abstract We study an exactly solvable long-range (LR) transverse-field Ising model with a power-law decaying interaction characterized by the decay exponent α . For 1 < α < 2 , the model exhibits two quantum critical points with distinct critical exponents. In the thermodynamic limit, the system is adiabatically driven across these critical points in the presence of noise, from the paramagnetic phase with all spins down to one with all spins up. In the noiseless case, within the LR regime, the steady-state properties are primarily influenced by the modes near the k c = π region, while the low-energy modes around k c = 0 remain nearly adiabatic and contribute negligibly. The defect density follows Kibble–Zurek (KZ) scaling law and scales as n ∝ τ Q − 1 / 2 , implying KZ scaling exponent independent of decay exponent. However, the magnitude of the defect density increases with increasing the range (i.e. decreasing the value of α ). On the other hand in the presence of noise, the low-energy modes around k c = 0 become highly susceptible to excitation, making them the dominant source of excitations, with their contribution gradually diminishing as the interaction range increases. This differs from the short-range model, where previous studies have shown that under noisy condition modes around k = π / 2 play a significant role. The contrasting influence of LR interactions under noisy and noiseless driving, is also observed in spin–spin correlation function and defect distribution, highlighting how LR interactions fundamentally reshape the interplay between adiabatic driving and noise.

DOI
10.1088/1361-648x/ae5d0f

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Abstract We study an exactly solvable long-range (LR) transverse-field Ising model with a power-law decaying interaction characterized by the decay exponent α . For 1 < α < 2 , the model exhibits two quantum critical points with distinct critical exponents. In the thermodynamic limit, the system is adiabatically driven across these critical points in the presence of noise, from the paramagnetic phase with all spins down to one with all spins up. In the noiseless case, within the LR regime, the steady-state properties are primarily influenced by the modes near the k c = π region, while the low-energy modes around k c = 0 remain nearly adiabatic and contribute negligibly. The defect density follows Kibble–Zurek (KZ) scaling law and scales as n ∝ τ Q − 1 / 2 , implying KZ scaling exponent independent of decay exponent. However, the magnitude of the defect density increases with increasing the range (i.e. decreasing the value of α ). On the other hand in the presence of noise, the low-energy modes around k c = 0 become highly susceptible to excitation, making them the dominant source of excitations, with their contribution gradually diminishing as the interaction range increases. This differs from the short-range model, where previous studies have shown that under noisy condition modes around k = π / 2 play a significant role. The contrasting influence of LR interactions under noisy and noiseless driving, is also observed in spin–spin correlation function and defect distribution, highlighting how LR interactions fundamentally reshape the interplay between adiabatic driving and noise.
We study the impact of noise on the dynamics of entanglement in the transverse-field Ising chain, with the field quenched linearly across one or both of the quantum critical points of the model. Taking concurrence as a measure of entanglement, we find that a quench generates entanglement between nearest- and next-nearest-neighbor spins, with noise reducing the amount of entanglement. Focusing on the next-nearest-neighbor concurrence, known to exhibit Kibble-Zurek scaling with the square root of the quench rate in the noiseless case, we find a different result when noise is present: The concurrence now scales logarithmically with the quench rate, with a noise-dependent amplitude. This is also different from the “anti-Kibble-Zurek” scaling of defect density with quench rate when noise is present, suggesting that noisy entanglement generation is largely independent of the rate of defect formation. Intriguingly, the critical timescale beyond which no entanglement is produced by a noisy quench scales as a power law with the strength of noise, with the same exponent as that which governs the optimal quench time for which defect formation is at a minimum in a standard quantum annealing scheme.
The Kibble-Zurek mechanism describes defect production due to non-adiabatic passage through a critical point. Here we study its variant from ramping the environment temperature to a critical point. We find that the defect density scales as [Formula: see text] or [Formula: see text] for thermal or quantum critical points, respectively, in terms of the usual critical exponents and [Formula: see text] the speed of the drive. Both scalings describe reduced defect density compared to conventional Kibble-Zurek mechanism, which stems from the enhanced relaxation due to bath-system interaction. Ramping to the quantum critical point is investigated by studying the Lindblad equation for the transverse field Ising chain in the presence of thermalizing bath, with couplings to environment obeying detailed balance, confirming the predicted scaling. The von-Neumann or the system-bath entanglement entropy follows the same scaling. Our results are generalized to a large class of dissipative systems with power-law energy dependent bath spectral densities as well.
We study the effect of noise on the dynamics of the transverse-field Ising model quenched across a quantum critical point. To quantify two-spin correlations, we employ the quantum Fisher information (QFI) and the Wigner-Yanase skew information (WYSI) as measures of quantum coherence. In the noiseless case, in contrast to the dynamics of entanglement in anisotropic XY chains, both QFI and WYSI increase monotonically with the ramp quench time, approaching their adiabatic limits without exhibiting any Kibble-Zurek-type scaling with quench duration. In contrast, when noise is added to the quench protocol, the coherence dynamics change qualitatively: QFI and WYSI both decay exponentially with the timescale of a ramp quench, with an exponent determined by the noise intensity. Furthermore, the maximum ramp time, at which either of these measures reaches its maximum, scales linearly with the noise variance, featuring the same exponent that determines the optimal annealing time for minimizing defect production in noisy quantum annealing.
We study defect generation in a quantum XY-spin chain arising due to the linear drive of the many-body Hamiltonian in the presence of a time-dependent fast Gaussian noise. The main objective of this work is to quantify analytically the effects of noise on the defect density production. In the absence of noise, it is well known that in the slow sweep regime, the defect density follows the Kibble-Zurek (KZ) scaling behavior with respect to the sweep speed. We consider time-dependent fast Gaussian noise in the anisotropy of the spin-coupling term $[{\ensuremath{\gamma}}_{0}=({J}_{1}\ensuremath{-}{J}_{2})/({J}_{1}+{J}_{2})]$ and show via analytical calculations that the defect density exhibits anti-Kibble-Zurek (AKZ) scaling behavior in the slow sweep regime. In the limit of large chain length and long time, we calculate the entropy and magnetization density of the final decohered state and show that their scaling behavior is consistent with the AKZ picture in the slow sweep regime. We have also numerically calculated the sublattice spin correlators for finite separation by evaluating the Toeplitz determinants and find results consistent with the KZ picture in the absence of noise, while in the presence of noise and slow sweep speeds the correlators exhibit the AKZ behavior. Furthermore, by considering the large nnn-separation asymptotes of the Toeplitz determinants, we further quantify the effect of the noise on the spin-spin correlators in the final decohered state. We show that while the correlation length of the sublattice correlator scales according to the AKZ behavior, we obtain different scaling for the magnetization correlators.
We analyze mechanisms for universal out-of-equilibrium dynamics near criticality by exploring the effect of randomized quantum resetting (QR) under a finite-time quench across a quantum phase transition. Using the transverse-field Ising chain as a generic model and exploiting its exact solution, QR is found to cause a crossover of the scaling of the topological defect density with the time scale τττ of the quench, from Kibble-Zurek to anti-Kibble-Zurek scaling as τττ increases. This reflects a competition between non-adiabatic quench-driven excitations and QR, giving rise to local minima of the defect densities at optimal annealing times. These times and the corresponding local minima are shown to scale as universal power laws with the rate of QR. Additional results for the scaling of the mean excess energy suggest that a system driven across a quantum critical point exhibits the same scaling behavior under a linear quench with QR as with uncorrelated noise.
We consider the ddd-dimensional transverse-field Ising model with power-law interactions $J/{r}^{d+\ensuremath{\sigma}}$ in the presence of a noisy longitudinal field with zero average. We study the longitudinal-magnetization dynamics of an initial paramagnetic state after a sudden switch-on of both the interactions and the noisy field. While the system eventually relaxes to an infinite-temperature state with vanishing magnetization correlations, we find that two-time correlation functions show aging at intermediate times. Moreover, for times shorter than the inverse noise strength $\ensuremath{\kappa}$ and distances longer than $a{(J/\ensuremath{\kappa})}^{2/\ensuremath{\sigma}}$ with aaa being the lattice spacing, we find a critical scaling regime of correlation and response functions consistent with the model A dynamical universality class with an initial-slip exponent $\ensuremath{\theta}=1$ and dynamical critical exponent $z=\ensuremath{\sigma}/2$. We obtain our results analytically by deriving an effective action for the magnetization field including the noise in a nonperturbative way. The above scaling regime is governed by a nonequilibrium fixed point dominated by the noise fluctuations.