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arXiv:2208.03060 · PRX Quantum

Probing Critical Behavior of Long-Range Transverse-Field Ising Model through Quantum Kibble-Zurek Mechanism

B.-W. Li, Yukai Wu, Q.-X. Mei, R. Yao, Wentao Lian, M.-L. Cai, Y. Wang, Ben Qi, Lin Yao, Longfei He, Z.-C. Zhou, Luming Duan

Published Jan 5, 2023• 39 citations
arXivHTMLPDF

Abstract

The trapped-ion quantum simulator has demonstrated qualitative properties of different physical models for up to tens of ions.In particular, a linear ion chain naturally hosts long-range Ising interactions under the laser driving, which has been used for various phenomena, such as quantum phase transition, localization, thermalization, and information propagation.For near-term practical usage, a central task is to find more quantitative applications of the noisy quantum simulators that are robust to small errors in the parameters.Here we report the quantum simulation of a long-range transverse-field Ising model using up to 61 ions and probe the critical behavior of its quantum phase transition through the Kibble-Zurek mechanism.By calibrating and verifying the coupling coefficients, we realize the same model for increasing ion numbers, so as to extract a critical exponent free of the finite-size effect.For ferromagnetic interaction, our experimental result agrees with the previous numerical prediction.As for the antiferromagnetic case, signals are too weak to fit a critical exponent due to the frustration in the interaction, but still consistent with the theory.

DOI
10.1103/prxquantum.4.010302

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The trapped-ion quantum simulator has demonstrated qualitative properties of different physical models for up to tens of ions.In particular, a linear ion chain naturally hosts long-range Ising interactions under the laser driving, which has been used for various phenomena, such as quantum phase transition, localization, thermalization, and information propagation.For near-term practical usage, a central task is to find more quantitative applications of the noisy quantum simulators that are robust to small errors in the parameters.Here we report the quantum simulation of a long-range transverse-field Ising model using up to 61 ions and probe the critical behavior of its quantum phase transition through the Kibble-Zurek mechanism.By calibrating and verifying the coupling coefficients, we realize the same model for increasing ion numbers, so as to extract a critical exponent free of the finite-size effect.For ferromagnetic interaction, our experimental result agrees with the previous numerical prediction.As for the antiferromagnetic case, signals are too weak to fit a critical exponent due to the frustration in the interaction, but still consistent with the theory.
We analyze the quantum phase transitions taking place in a one-dimensional transverse-field Ising model with long-range couplings that decay algebraically with distance. We are interested in the Kibble-Zurek universal scaling laws emerging in nonequilibrium dynamics and in the potential for the unambiguous observation of such behavior in a realistic experimental setup based on trapped ions. To this end, we determine the phase diagram of the model and the critical exponents characterizing its quantum phase transitions by means of density-matrix renormalization group calculations and finite-size scaling theory, which allows us to obtain good estimates for different range of ferromagnetic and antiferromagnetic interactions. Beyond critical equilibrium properties, we tackle a nonequilibrium scenario in which quantum Kibble-Zurek scaling laws may be retrieved. Here, it is found that the predicted nonequilibrium universal behavior, i.e., the scaling laws as a function of the quench rate and critical exponents, can be observed in systems comprising an experimentally feasible number of spins. Finally, a scheme is introduced to simulate the algebraically decaying couplings accurately by means of a digital quantum simulation with trapped ions. Our results suggest that quantum Kibble-Zurek physics can be explored and observed in state-of-the-art experiments with trapped ions realizing long-range Ising models.
We investigate an extension of the quantum Ising (QI) model in one spatial dimension including long-range interactions in its statics and dynamics with possible applications from heteronuclear polar molecules in optical lattices to trapped ions described by two-state spin systems. We introduce the statics of the system via both numerical techniques with finite size and infinite size matrix product states (MPSs) and a theoretical approaches using a truncated Jordan–Wigner transformation for the ferromagnetic and antiferromagnetic case and show that finite size effects have a crucial role shifting the quantum critical point of the external field by fifteen percent between thirty-two and around five-hundred spins. We numerically study the Kibble–Zurek hypothesis in the long-range QI model with MPSs. A linear quench of the external field through the quantum critical point yields a power-law scaling of the defect density as a function of the total quench time. For example, the increase of the defect density is slower for longer-range models and the critical exponent changes by twenty-five percent. Our study emphasizes the importance of such long-range interactions in statics and dynamics that could point to similar phenomena in a different setup of dynamical systems or for other models.
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.
The quantum Kibble-Zurek mechanism (QKZM) predicts universal dynamical behavior near the quantum phase transitions (QPTs). It is now well understood for the one-dimensional quantum matter. Higher-dimensional systems, however, remain a challenge, complicated by the fundamentally different character of the associated QPTs and their underlying conformal field theories. In this work, we take the first steps toward theoretical exploration of the QKZM in two dimensions for interacting quantum matter. We study the dynamical crossing of the QPT in the paradigmatic Ising model by a joint effort of modern state-of-the-art numerical methods, including artificial neural networks and tensor networks. As a central result, we quantify universal QKZM behavior close to the QPT. We also note that, upon traversing further into the ferromagnetic regime, deviations from the QKZM prediction appear. We explain the observed behavior by proposing an extended QKZM taking into account spectral information as well as phase ordering. Our work provides a testing platform for higher-dimensional quantum simulators.
The quantum-critical properties of the transverse-field Ising model with algebraically decaying interactions are investigated by means of stochastic series expansion quantum Monte Carlo, on both the one-dimensional linear chain and the two-dimensional square lattice. We extract the critical exponents $\ensuremath{\nu}$ and $\ensuremath{\beta}$ as a function of the decay exponent of the long-range interactions. For ferromagnetic Ising interactions, we resolve the limiting regimes known from field theory, ranging from the nearest-neighbor Ising to the long-range Gaussian universality classes, as well as the intermediate regime with continuously varying critical exponents. In the long-range Gaussian regime, we treat the effect of dangerous irrelevant variables on finite-size scaling forms. For antiferromagnetic and therefore competing Ising interactions, the stochastic series expansion algorithm displays growing autocorrelation times leading to a reduced performance. Nevertheless, our results are consistent with the nearest-neighbor Ising universality for all investigated interaction ranges both on the linear chain and the square lattice.
The Kibble-Zurek mechanism is the paradigm to account for the nonadiabatic dynamics of a system across a continuous phase transition. Its study in the quantum regime is hindered by the requisite of ground state cooling. We report the experimental quantum simulation of critical dynamics in the transverse-field Ising model by a set of Landau-Zener crossings in pseudo-momentum space, that can be probed with high accuracy using a single trapped ion. We test the Kibble-Zurek mechanism in the quantum regime in the momentum space and find the measured scaling of excitations is in accordance with the theoretical prediction.