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arXiv:2212.07499 · Physical review. B./Physical review. B

Spatiotemporal quenches in long-range Hamiltonians

Simon Bernier, Kartiek Agarwal

Published Jul 28, 2023• 1 citation
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Abstract

Spatiotemporal quenches are efficient at preparing ground states of critical Hamiltonians that have emergent low-energy descriptions with Lorentz invariance [Agarwal, Bhatt, and Sondhi, Phys. Rev. Lett. 120, 210604 (2018)]. The critical transverse field Ising model with nearest-neighbor interactions, for instance, maps to free fermions with a relativistic low energy dispersion. However, spin models realized in artificial quantum simulators based on neutral Rydberg atoms, or trapped ions, generically exhibit long-range power-law decay of interactions with $J(r)\ensuremath{\sim}1/{r}^{\ensuremath{\alpha}}$ for a wide range of $\ensuremath{\alpha}$. In this paper, we study the fate of spatiotemporal quenches in these models with a fixed velocity $v$ for the propagation of the quench front, using the numerical time-dependent variational principle. For $\ensuremath{\alpha}\ensuremath{\gtrsim}3$, where the critical theory is suggested to have a dynamical critical exponent $z=1$, our simulations show that optimal cooling is achieved when the front velocity $v$ approaches $c$, the effective speed of excitations in the critical model. The energy density is inhomogeneously distributed in space, with prominent hot regions populated by excitations copropagating with the quench front, and cold regions populated by counterpropagating excitations. Lowering $\ensuremath{\alpha}$ largely blurs the boundaries between these regions. For $\ensuremath{\alpha}<3$, we find that the Doppler cooling effect disappears, as expected from renormalization group results for the critical model, which suggest a dispersion $\ensuremath{\omega}\ensuremath{\sim}{q}^{z}$ with $z<1$. Instead, we show that excitations are controlled by two relevant length scales whose ratio is related to that of the front velocity to a threshold velocity that ultimately determines the adiabaticity of the quench.

DOI
10.1103/physrevb.108.024310

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Spatiotemporal quenches are efficient at preparing ground states of critical Hamiltonians that have emergent low-energy descriptions with Lorentz invariance [Agarwal, Bhatt, and Sondhi, Phys. Rev. Lett. 120, 210604 (2018)]. The critical transverse field Ising model with nearest-neighbor interactions, for instance, maps to free fermions with a relativistic low energy dispersion. However, spin models realized in artificial quantum simulators based on neutral Rydberg atoms, or trapped ions, generically exhibit long-range power-law decay of interactions with J(r)\ensuremath∼1/r\ensuremathαJ(r)\ensuremath{\sim}1/{r}^{\ensuremath{\alpha}}J(r)\ensuremath∼1/r\ensuremathα for a wide range of \ensuremathα\ensuremath{\alpha}\ensuremathα. In this paper, we study the fate of spatiotemporal quenches in these models with a fixed velocity vvv for the propagation of the quench front, using the numerical time-dependent variational principle. For \ensuremathα\ensuremath≳3\ensuremath{\alpha}\ensuremath{\gtrsim}3\ensuremathα\ensuremath≳3, where the critical theory is suggested to have a dynamical critical exponent z=1z=1z=1, our simulations show that optimal cooling is achieved when the front velocity vvv approaches ccc, the effective speed of excitations in the critical model. The energy density is inhomogeneously distributed in space, with prominent hot regions populated by excitations copropagating with the quench front, and cold regions populated by counterpropagating excitations. Lowering \ensuremathα\ensuremath{\alpha}\ensuremathα largely blurs the boundaries between these regions. For \ensuremathα<3\ensuremath{\alpha}<3\ensuremathα<3, we find that the Doppler cooling effect disappears, as expected from renormalization group results for the critical model, which suggest a dispersion \ensuremathω\ensuremath∼qz\ensuremath{\omega}\ensuremath{\sim}{q}^{z}\ensuremathω\ensuremath∼qz with z<1z<1z<1. Instead, we show that excitations are controlled by two relevant length scales whose ratio is related to that of the front velocity to a threshold velocity that ultimately determines the adiabaticity of the quench.
Quantum simulators have the potential to shed light on the study of quantum many-body systems and materials, offering unique insights into various quantum phenomena. Although adiabatic evolution has been conventionally employed for state preparation, it faces challenges when the system evolves too quickly or the coherence time is limited. In such cases, shortcuts to adiabaticity, such as spatiotemporal quenches, provide a promising alternative. This paper numerically investigates the application of spatiotemporal quenches in the two-dimensional transverse field Ising model with ferromagnetic interactions, focusing on the emergence of the ground state and its correlation properties at criticality when the gap vanishes. We demonstrate the effectiveness of these quenches in rapidly preparing ground states in critical systems. Our simulations reveal the existence of an optimal quench front velocity at the emergent speed of light, leading to minimal excitation energy density and correlation lengths of the order of finite system sizes we can simulate. These findings emphasize the potential of spatiotemporal quenches for efficient ground state preparation in quantum systems, with implications for the exploration of strongly correlated phases and programmable quantum computing.
Quantum simulators have the potential to shed light on the study of quantum many-body systems and materials, offering unique insights into various quantum phenomena. While adiabatic evolution has been conventionally employed for state preparation, it faces challenges when the system evolves too quickly or the coherence time is limited. In such cases, shortcuts to adiabaticity, such as spatiotemporal quenches, provide a promising alternative. This paper numerically investigates the application of spatiotemporal quenches in the two-dimensional transverse field Ising model with ferromagnetic interactions, focusing on the emergence of the ground state and its correlation properties at criticality when the gap vanishes. We demonstrate the effectiveness of these quenches in rapidly preparing ground states in critical systems. Our simulations reveal the existence of an optimal quench front velocity at the emergent speed of light, leading to minimal excitation energy density and correlation lengths of the order of finite system sizes we can simulate. These findings emphasize the potential of spatiotemporal quenches for efficient ground state preparation in quantum systems, with implications for the exploration of strongly correlated phases and programmable quantum computing.
Driving a homogeneous system across a quantum phase transition in a quench time ${\ensuremath{\tau}}_{Q}$ generates excitations on wavelengths longer than the Kibble-Zurek (KZ) length $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\xi}}\ensuremath{\propto}{\ensuremath{\tau}}_{Q}^{\ensuremath{\nu}/(1+z\ensuremath{\nu})}$ within the KZ time window $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{t}\ensuremath{\propto}{\ensuremath{\tau}}_{Q}^{z\ensuremath{\nu}/(1+z\ensuremath{\nu})}$, where zzz and $\ensuremath{\nu}$ are the critical exponents. Quenches designed with local time-dependent inhomogeneity can introduce a gap in the spectrum. They can be parametrized by a time- and space-dependent distance $\ensuremath{\epsilon}$ from the critical point in the parameter space of the Hamiltonian: $\ensuremath{\epsilon}(t,x)=\frac{t\ensuremath{-}x/v}{{\ensuremath{\tau}}_{Q}}\ensuremath{\equiv}\ensuremath{\theta}(vt\ensuremath{-}x).$ For a variety of setups with short-range interactions, they have been shown to suppress excitations if the spatial velocity vvv of the inhomogeneous front is below the characteristic KZ velocity $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{v}\ensuremath{\propto}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\xi}}/\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{t}$. Ising-like models with long-range interactions can have no sonic horizon, spreading information instantaneously across the system. Usually, this should imply that inhomogeneous transitions will render the dynamics adiabatic regardless of the velocity of the front. However, we show that we get an adiabatic transition with no defects only when the inhomogeneous front moves slower than the characteristic crossover velocity $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{v}\ensuremath{\propto}{\ensuremath{\theta}}^{(z\ensuremath{-}1)\ensuremath{\nu}/(1+\ensuremath{\nu})}$, where $\ensuremath{\theta}$ is the slope of the inhomogeneous front at the critical point. The existence of this crossover velocity and adiabaticity of the model results from the energy gap in the quasiparticle spectrum that is opened by the inhomogeneity. This effect can be employed for efficient adiabatic quantum state preparation in systems with long-range interactions.
Abstract Quantum many body systems with long range interactions are known to display many fascinating phenomena experimentally observable in trapped ions, Rydberg atoms and polar molecules. Among these are dynamical phase transitions which occur after an abrupt quench in spin chains with interactions decaying as and whose critical dynamics depend crucially on the power : for systems with the transition is sharp while for it fans out in a chaotic crossover region. In this paper we explore the fate of critical dynamics in Ising chains with long-range interactions when the transverse field is ramped up with a finite speed. While for abrupt quenches we observe a chaotic region that widens as is increased, the width of the crossover region diminishes as the time of the ramp increases, suggesting that chaos will disappear altogether and be replaced by a sharp transition in the adiabatic limit.
We propose a spatiotemporal quench protocol that allows for the fast preparation of ground states of gapless models with Lorentz invariance. Assuming the system initially resides in the ground state of a corresponding massive model, we show that a superluminally moving "front" that locally quenches the mass, leaves behind it (in space) a state arbitrarily close to the ground state of the gapless model. Importantly, our protocol takes time O(L) to produce the ground state of a system of size ∼L^{d} (d spatial dimensions), while a fully adiabatic protocol requires time ∼O(L^{2}) to produce a state with exponential accuracy in L. The physics of the dynamical problem can be understood in terms of relativistic rarefaction of excitations generated by the mass front. We provide proof of concept by solving the proposed quench exactly for a system of free bosons in arbitrary dimensions, and for free fermions in d=1. We discuss the role of interactions and UV effects on the free-theory idealization, before numerically illustrating the usefulness of the approach via simulations on the quantum Heisenberg spin chain.
Long-range interacting spin systems are ubiquitous in physics and exhibit a variety of ground-state disorder-to-order phase transitions. We consider a prototype of infinite-range interacting models known as the Lipkin-Meshkov-Glick model describing the collective interaction of N spins and investigate the dynamical properties of fluctuations and correlations after a sudden quench of the Hamiltonian. Specifically, we focus on critical quenches, where the initial state and/or the postquench Hamiltonian are critical. Depending on the type of quench, we identify three distinct behaviors where both the short-time dynamics and the stationary state at long times are effectively thermal, quantum, and genuinely nonequilibrium, characterized by distinct universality classes and static and dynamical critical exponents. These behaviors can be identified by an infrared effective temperature that is finite, zero, and infinite (the latter scaling with the system size as N^{1/3}), respectively. The quench dynamics is studied through a combination of exact numerics and analytical calculations utilizing the nonequilibrium Keldysh field theory. Our results are amenable to realization in experiments with trapped-ion experiments where long-range interactions naturally arise.