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arXiv:0904.0115 · New Journal of Physics

Dynamics of an inhomogeneous quantum phase transition

Jacek Dziarmaga, Marek M. Rams

Published May 28, 2010• 66 citations
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Abstract

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.

DOI
10.1088/1367-2630/12/5/055007

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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.
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.
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.
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 τQ−βδ/(1+zν)\tau_Q^{-\beta\delta/(1+z\nu)}τQ−βδ/(1+zν)​, where β,δ,z,ν\beta,\delta,z,\nuβ,δ,z,ν are the critical exponents and τQ\tau_QτQ​ 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.
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 a formulation for investigating quench dynamics across quantum phase transitions in the presence of decoherence. We formulate decoherent dynamics induced by continuous quantum non-demolition measurements of the instantaneous Hamiltonian. We generalize the well-studied universal Kibble-Zurek behavior for linear temporal drive across the critical point. We identify a strong decoherence regime wherein the decoherence time is shorter than the standard correlation time, which varies as the inverse gap above the groundstate. In this regime, we find that the freeze-out time tˉ∼τ2νz/(1+2νz)\bar{t}\simτ^{{2νz}/({1+2νz})}tˉ∼τ2νz/(1+2νz) for when the system falls out of equilibrium and the associated freeze-out length ξˉ∼τν/(1+2νz)\barξ\simτ^{ν/({1+2νz})}ξˉ​∼τν/(1+2νz) show power-law scaling with respect to the quench rate 1/τ1/τ1/τ, where the exponents depend on the correlation length exponent ννν and the dynamical exponent zzz associated with the transition. The universal exponents differ from those of standard Kibble-Zurek scaling. We explicitly demonstrate this scaling behavior in the instance of a topological transition in a Chern insulator system. We show that the freeze-out time scale can be probed from the relaxation of the Hall conductivity. Furthermore, on introducing disorder to break translational invariance, we demonstrate how quenching results in regions of imbalanced excitation density characterized by an emergent length scale which also shows universal scaling. We perform numerical simulations to confirm our analytical predictions and corroborate the scaling arguments that we postulate as universal to a host of systems.