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arXiv:2010.05621 · Physical review. B./Physical review. B

Inhomogeneity induced shortcut to adiabaticity in Ising chains with long-range interactions

Aritra Sinha, Debasis Sadhukhan, Marek M. Rams, Jacek Dziarmaga

Published Dec 10, 2020• 10 citations
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Abstract

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 $z$ 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 $v$ 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.

DOI
10.1103/physrevb.102.214203

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Driving a homogeneous system across a quantum phase transition in a quench time \ensuremathτQ{\ensuremath{\tau}}_{Q}\ensuremathτQ​ generates excitations on wavelengths longer than the Kibble-Zurek (KZ) length \ensuremathξ\ifmmode^\elseˆ\fi\ensuremath∝\ensuremathτQ\ensuremathν/(1+z\ensuremathν)\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\xi}}\ensuremath{\propto}{\ensuremath{\tau}}_{Q}^{\ensuremath{\nu}/(1+z\ensuremath{\nu})}\ensuremathξ\ifmmode^\elseˆ\fi​\ensuremath∝\ensuremathτQ\ensuremathν/(1+z\ensuremathν)​ within the KZ time window t\ifmmode^\elseˆ\fi\ensuremath∝\ensuremathτQz\ensuremathν/(1+z\ensuremathν)\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{t}\ensuremath{\propto}{\ensuremath{\tau}}_{Q}^{z\ensuremath{\nu}/(1+z\ensuremath{\nu})}t\ifmmode^\elseˆ\fi​\ensuremath∝\ensuremathτQz\ensuremathν/(1+z\ensuremathν)​, where zzz and \ensuremathν\ensuremath{\nu}\ensuremathν 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 \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ from the critical point in the parameter space of the Hamiltonian: \ensuremathϵ(t,x)=t\ensuremath−x/v\ensuremathτQ\ensuremath≡\ensuremathθ(vt\ensuremath−x).\ensuremath{\epsilon}(t,x)=\frac{t\ensuremath{-}x/v}{{\ensuremath{\tau}}_{Q}}\ensuremath{\equiv}\ensuremath{\theta}(vt\ensuremath{-}x).\ensuremathϵ(t,x)=\ensuremathτQ​t\ensuremath−x/v​\ensuremath≡\ensuremathθ(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 v\ifmmode^\elseˆ\fi\ensuremath∝\ensuremathξ\ifmmode^\elseˆ\fi/t\ifmmode^\elseˆ\fi\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{v}\ensuremath{\propto}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\xi}}/\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{t}v\ifmmode^\elseˆ\fi\ensuremath∝\ensuremathξ\ifmmode^\elseˆ\fi​/t\ifmmode^\elseˆ\fi​. 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 v\ifmmode~\else˜\fi\ensuremath∝\ensuremathθ(z\ensuremath−1)\ensuremathν/(1+\ensuremathν)\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{v}\ensuremath{\propto}{\ensuremath{\theta}}^{(z\ensuremath{-}1)\ensuremath{\nu}/(1+\ensuremath{\nu})}v\ifmmode~\else˜\fi\ensuremath∝\ensuremathθ(z\ensuremath−1)\ensuremathν/(1+\ensuremathν), where \ensuremathθ\ensuremath{\theta}\ensuremathθ 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.
We argue that, in a second-order quantum phase transition driven by an inhomogeneous quench, the density of quasi-particle excitations is suppressed when velocity at which a critical point propagates across a system falls below a threshold velocity equal to the Kibble–Zurek correlation length times the energy gap at freeze-out divided by ℏ. This general prediction is supported by an analytic solution in the quantum Ising chain. Our results suggest, in particular, that adiabatic quantum computers can be made more adiabatic when operated in an 'inhomogeneous' way.
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 vvv 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=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 $\ensuremath{\alpha}$ 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<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.
In the nonadiabatic dynamics across a quantum phase transition, the Kibble-Zurek mechanism predicts that the formation of topological defects is suppressed as a universal power law with the quench time. In inhomogeneous systems, the critical point is reached locally and causality reduces the effective system size for defect formation to regions where the velocity of the critical front is slower than the sound velocity, favoring adiabatic dynamics. The reduced density of excitations exhibits a much steeper dependence on the quench rate and is also described by a universal power law that we demonstrated in a quantum Ising chain.
We consider an inhomogeneous quantum phase transition across a multicritical point of the XY quantum spin chain. This is an example of a Lifshitz transition with a dynamical exponent z =2. Just like in the case z =1 considered by Dziarmaga and Rams (2010 New J. Phys. 12 055007), when a critical front propagates much faster than the maximal group velocity of quasiparticles v q , then the transition is effectively homogeneous: the density of excitations obeys a generalized Kibble–Zurek mechanism and scales with the sixth root of the transition rate. However, unlike for the case z =1, the inhomogeneous transition becomes adiabatic not below v q but at a lower threshold velocity , proportional to the inhomogeneity of the transition, where the excitations are suppressed exponentially. Interestingly, the adiabatic threshold is nonzero despite the vanishing minimal group velocity of low-energy quasiparticles. In the adiabatic regime below , the inhomogeneous transition can be used for efficient adiabatic quantum state preparation in a quantum simulator: the time required for the critical front to sweep across a chain of N spins adiabatically is merely linear in N , while the corresponding time for a homogeneous transition across the multicritical point scales with the sixth power of N . What is more, excitations after the adiabatic inhomogeneous transition, if any, are brushed away by the critical front to the end of the spin chain.
A system gradually driven through a symmetry-breaking phase transition is subject to the Kibble-Zurek mechanism (KZM). As a consequence of the critical slowing down, its state cannot follow local equilibrium, and its evolution becomes nonadiabatic near the critical point. In the simplest approximation, that stage can be regarded as an ``impulse'' where the state of the system remains unchanged. It leads to the correct KZM scaling laws. However, such a ``freeze-out'' might suggest that the coherence length of the nascent order parameter remains unchanged as the critical region is traversed. By contrast, the original causality-based discussion emphasized the role of the sonic horizon: Domains of the broken symmetry phase can expand with a velocity limited by the speed of the relevant sound. This effect was demonstrated in the quantum Ising chain where the dynamical exponent z=1z=1z=1 and quasiparticles excited by the transition have a fixed speed of sound. To elucidate the role of the sonic horizon, in this paper we study two systems with z>1z>1z>1 where the speed of sound is no longer fixed, and the fastest excited quasiparticles set the size of the sonic horizon. Their effective speed decays with increasing transition time. In the extreme case, the dynamical exponent zzz can diverge such as in the Griffiths region of the random Ising chain where the localization of excited quasiparticles freezes the growth of the correlation range when the critical region is traversed. Of particular interest is an example with z<1z<1z<1, the long-range extended Ising chain, where there is no upper limit to the velocity of excited quasiparticles with small momenta. Initially, the power-law tail of the correlation function grows adiabatically, but in the nonadiabatic stage it lags behind the adiabatic evolution.