We study quench dynamics and defect production in the Kitaev and the extended Kitaev models. For the Kitaev model in one dimension, we show that in the limit of slow quench rate, the defect density $n\ensuremath{\sim}1/\sqrt{\ensuremath{\tau}}$, where $1/\ensuremath{\tau}$ is the quench rate. We also compute the defect correlation function by providing an exact calculation of all independent nonzero spin correlation functions of the model. In two dimensions, where the quench dynamics takes the system across a critical line, we elaborate on the results of earlier work [K. Sengupta, D. Sen, and S. Mondal, Phys. Rev. Lett. 100, 077204 (2008)] to discuss the unconventional scaling of the defect density with the quench rate. In this context, we outline a general proof that for a
d-dimensional quantum model, where the quench takes the system through a $d\ensuremath{-}m$ dimensional gapless (critical) surface characterized by correlation length exponent $\ensuremath{\nu}$ and dynamical critical exponent
z, the defect density $n\ensuremath{\sim}1/{\ensuremath{\tau}}^{m\ensuremath{\nu}/(z\ensuremath{\nu}+1)}$. We also discuss the variation of the shape and spatial extent of the defect correlation function with both the rate of quench and the model parameters and compute the entropy generated during such a quenching process. Finally, we study the defect scaling law, entropy generation and defect correlation function of the two-dimensional extended Kitaev model.
We show that the defect density
n, for a slow nonlinear power-law quench with a rate ${\ensuremath{\tau}}^{\ensuremath{-}1}$ and an exponent $\ensuremath{\alpha}>0$, which takes the system through a critical point characterized by correlation length and dynamical critical exponents $\ensuremath{\nu}$ and
z, scales as $n\ensuremath{\sim}{\ensuremath{\tau}}^{\ensuremath{-}\ensuremath{\alpha}\ensuremath{\nu}d/(\ensuremath{\alpha}z\ensuremath{\nu}+1)}$ [$n\ensuremath{\sim}(\ensuremath{\alpha}{g}^{(\ensuremath{\alpha}\ensuremath{-}1)/\ensuremath{\alpha}}/\ensuremath{\tau}{)}^{\ensuremath{\nu}d/(z\ensuremath{\nu}+1)}$] if the quench takes the system across the critical point at time
t=0 [$t={t}_{0}\ensuremath{\ne}0$], where
g is a nonuniversal constant and
d is the system dimension. These scaling laws constitute the first theoretical results for defect production in nonlinear quenches across quantum critical points and reproduce their well-known counterpart for a linear quench ($\ensuremath{\alpha}=1$) as a special case. We supplement our results with numerical studies of well-known models and suggest experiments to test our theory.
We study the generation of defects when a quantum spin system is quenched through a multicritical point by changing a parameter of the Hamiltonian as t /τ, where τ is the characteristic timescale of quenching. We argue that when a quantum system is quenched across a multicritical point, the density of defects ( n ) in the final state is not necessarily given by the Kibble–Zurek scaling form n ∼1/τ d ν/( z ν+1) , where d is the spatial dimension, and ν and z are respectively the correlation length and dynamical exponent associated with the quantum critical point. We propose a generalized scaling form of the defect density given by n ∼1/τ d /(2 z 2 ) , where the exponent z 2 determines the behavior of the off-diagonal term of the 2 × 2 Landau–Zener matrix at the multicritical point. This scaling is valid not only at a multicritical point but also at an ordinary critical point.
We study adiabatic quantum quenches across a quantum multicritical point (MCP) using a quenching scheme that enables the system to hit the MCP along different paths. We show that the power-law scaling of the defect density with the rate of driving depends non-trivially on the path, i.e. , the exponent varies continuously with the parameter α that defines the path, up to a critical value α=α c ; on the other hand for α⩾α c , the scaling exponent saturates to a constant value. We show that dynamically generated and path (α)- dependent effective critical exponents associated with the quasicritical points lying close to the MCP (on the ferromagnetic side), where the energy-gap is minimum, lead to this continuously varying exponent. The scaling relations are established using the integrable transverse XY spin chain and generalized to a MCP associated with a d -dimensional quantum many-body systems (not reducible to two-level systems) using adiabatic perturbation theory. We also calculate the effective path-dependent dimensional shift d 0 (α) (or the shift in center of the impulse region) that appears in the scaling relation for special paths lying entirely in the paramagnetic phase. The numerically obtained results are in good agreement with analytical predictions.
We study the nonequilibrium slow dynamics for the Kitaev model both in the presence and the absence of disorder. For the case without disorder, we demonstrate, via an exact solution, that the model provides an example of a system with an anisotropic critical point and exhibits unusual scaling of defect density
n and residual energy
Q for a slow linear quench. We provide a general expression for the scaling of
n(Q) generated during a slow power-law dynamics, characterized by a rate ${\ensuremath{\tau}}^{\ensuremath{-}1}$ and exponent $\ensuremath{\alpha}$, from a gapped phase to an anisotropic quantum critical point in
d dimensions, for which the energy gap ${\ensuremath{\Delta}}_{\stackrel{P\vec}{k}}\ensuremath{\sim}{k}_{i}^{z}$ for
m momentum components
(i=1,…,m) and $\ensuremath{\sim}{k}_{i}^{{z}^{\ensuremath{'}}}$ for the rest $d\ensuremath{-}m$ components
(i=m+1,…,d) with $z\ensuremath{\le}{z}^{\ensuremath{'}}$: $n\ensuremath{\sim}{\ensuremath{\tau}}^{\ensuremath{-}[m+(d\ensuremath{-}m)z/{z}^{\ensuremath{'}}]\ensuremath{\nu}\ensuremath{\alpha}/(z\ensuremath{\nu}\ensuremath{\alpha}+1)}(Q\ensuremath{\sim}{\ensuremath{\tau}}^{\ensuremath{-}[(m+z)+(d\ensuremath{-}m)z/{z}^{\ensuremath{'}}]\ensuremath{\nu}\ensuremath{\alpha}/(z\ensuremath{\nu}\ensuremath{\alpha}+1)})$. These general expressions reproduce both the corresponding results for the Kitaev model as a special case for $d={z}^{\ensuremath{'}}=2$ and $m=z=\ensuremath{\nu}=1$ and the well-known scaling laws of
n and
Q for isotropic critical points for $z={z}^{\ensuremath{'}}$. We also present an exact computation of all nonzero, independent, multispin correlation functions of the Kitaev model for such a quench and discuss their spatial dependence. For the disordered Kitaev model, where the disorder is introduced via random choice of the link variables
Dn in the model's fermionic representation, we find that $n\ensuremath{\sim}{\ensuremath{\tau}}^{\ensuremath{-}1/2}$ and $Q\ensuremath{\sim}{\ensuremath{\tau}}^{\ensuremath{-}1}$ $(Q\ensuremath{\sim}{\ensuremath{\tau}}^{\ensuremath{-}1/2})$ for a slow linear quench ending in the gapless (gapped) phase. We provide a qualitative explanation of such scaling.
We study the quenching dynamics of a many-body system in one dimension described by a Hamiltonian that has spatial periodicity. Specifically, we consider a spin-
21 chain with equal
xx and
yy couplings and subject to a periodically varying magnetic field in the $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{z}$ direction or, equivalently, a tight-binding model of spinless fermions with a periodic local chemical potential, having period
2q, where
q is a positive integer. For a linear quench of the strength of the magnetic field (or chemical potential) at a rate $1/\ensuremath{\tau}$ across a quantum critical point, we find that the density of defects thereby produced scales as $1/{\ensuremath{\tau}}^{q/(q+1)}$, deviating from the $1/\sqrt{\ensuremath{\tau}}$ scaling that is ubiquitous in a range of systems. We analyze this behavior by mapping the low-energy physics of the system to a set of fermionic two-level systems labeled by the lattice momentum
k undergoing a nonlinear quench as well as by performing numerical simulations. We also show that if the magnetic field is a superposition of different periods, the power law depends only on the smallest period for very large values of $\ensuremath{\tau}$, although it may exhibit a crossover at intermediate values of $\ensuremath{\tau}$. Finally, for the case where a
zz coupling is also present in the spin chain, or equivalently, where interactions are present in the fermionic system, we argue that the power associated with the scaling law depends on a combination of
q and the interaction strength.