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arXiv:1005.3763 · New Journal of Physics

Adiabatic dynamics of an inhomogeneous quantum phase transition: the case of az>1 dynamical exponent

Jacek Dziarmaga, Marek M. Rams

Published Oct 5, 2010• 31 citations
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Abstract

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.

DOI
10.1088/1367-2630/12/10/103002

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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.
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.
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.
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 explore the robustness of universal dynamical scaling behavior in a quantum system near criticality with respect to initialization in a large class of states with finite energy. By focusing on a homogeneous XY\mathit{XY}XY quantum spin chain in a transverse field, we characterize the nonequilibrium response under adiabatic and sudden quench processes originating from a pure as well as a mixed excited initial state, and involving either a regular quantum critical or a multicritical point. We find that the critical exponents of the ground-state quantum phase transition can be encoded in the dynamical scaling exponents despite the finite energy of the initial state. In particular, we identify conditions on the initial distribution of quasiparticle excitation that ensure that Kibble-Zurek scaling persists. The emergence of effective thermal equilibrium behavior following a sudden quench toward criticality is also investigated, with focus on the long-time expectation value of the quasiparticle number operator. Despite the integrability of the XY\mathit{XY}XY model, this observable is found to behave thermally in quenches to a regular quantum critical point, provided that the system is initially prepared at sufficiently high temperature. However, a similar thermalization behavior fails to occur in quenches toward a multicritical point. We argue that the observed lack of thermalization originates in this case in the asymmetry of the impulse region that is also responsible for anomalous multicritical dynamical scaling.
The Kibble-Zurek mechanism predicts the formation of topological defects and other excitations that quantify how much a quantum system driven across a quantum critical point fails to be adiabatic. We point out that, thanks to the divergent linear susceptibility at the critical point, even a tiny symmetry breaking bias can restore the adiabaticity. The minimal required bias scales like ${\ensuremath{\tau}}_{Q}^{\ensuremath{-}\ensuremath{\beta}\ensuremath{\delta}/(1+z\ensuremath{\nu})}$, where $\ensuremath{\beta}$, $\ensuremath{\delta}$, zzz, $\ensuremath{\nu}$ are the critical exponents and ${\ensuremath{\tau}}_{Q}$ is a quench time. We test this prediction by DMRG simulations of the quantum Ising chain. It is directly applicable to the recent emulation of quantum phase transition dynamics in the Ising chain with ultracold Rydberg atoms.
We present here our study of the adiabatic quantum dynamics of a random Ising chain across its quantum critical point. The model investigated is an Ising chain in a transverse field with disorder present both in the exchange coupling and in the transverse field. The transverse field term is proportional to a function $\ensuremath{\Gamma}(t)$ which, as in the Kibble-Zurek mechanism, is linearly reduced to zero in time with a rate ${\ensuremath{\tau}}^{\ensuremath{-}1}$, $\ensuremath{\Gamma}(t)=\ensuremath{-}t∕\ensuremath{\tau}$, starting at $t=\ensuremath{-}\ensuremath{\infty}$ from the quantum disordered phase $(\ensuremath{\Gamma}=\ensuremath{\infty})$ and ending at t=0t=0t=0 in the classical ferromagnetic phase $(\ensuremath{\Gamma}=0)$. We first analyze the distribution of the gaps, occurring at the critical point ${\ensuremath{\Gamma}}_{c}=1$, which are relevant for breaking the adiabaticity of the dynamics. We then present extensive numerical simulations for the residual energy Eres{E}_{\mathrm{res}}Eres​ and density of defects ${\ensuremath{\rho}}_{k}$ at the end of the annealing, as a function of the annealing inverse rate $\ensuremath{\tau}$. Both the average ${E}_{\mathrm{res}}(\ensuremath{\tau})$ and ${\ensuremath{\rho}}_{k}(\ensuremath{\tau})$ are found to behave logarithmically for large $\ensuremath{\tau}$, but with different exponents, $[{E}_{\mathrm{res}}(\ensuremath{\tau})∕L{]}_{\mathrm{av}}\ensuremath{\sim}1∕{\mathrm{ln}}^{\ensuremath{\zeta}}(\ensuremath{\tau})$ with $\ensuremath{\zeta}\ensuremath{\approx}3.4$, and $[{\ensuremath{\rho}}_{k}(\ensuremath{\tau}){]}_{\mathrm{av}}\ensuremath{\sim}1∕{\mathrm{ln}}^{2}(\ensuremath{\tau})$. We propose a mechanism for $1∕{\mathrm{ln}}^{2}\ensuremath{\tau}$ behavior of $[{\ensuremath{\rho}}_{k}{]}_{\mathrm{av}}$ based on the Landau-Zener tunneling theory and on a Fisher's-type real-space renormalization group analysis of the relevant gaps. The model proposed shows therefore a paradigmatic example of how an adiabatic quantum computation can become very slow when disorder is at play, even in the absence of any source of frustration.