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arXiv:0803.2081 · Physical Review Letters

Defect Production in Nonlinear Quench across a Quantum Critical Point

Diptiman Sen, K. Sengupta, Shreyoshi Mondal

Published Jul 3, 2008• 215 citations
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Abstract

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.

DOI
10.1103/physrevlett.101.016806

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We show that the defect density nnn, for a slow nonlinear power-law quench with a rate \ensuremathτ\ensuremath−1{\ensuremath{\tau}}^{\ensuremath{-}1}\ensuremathτ\ensuremath−1 and an exponent \ensuremathα>0\ensuremath{\alpha}>0\ensuremathα>0, which takes the system through a critical point characterized by correlation length and dynamical critical exponents \ensuremathν\ensuremath{\nu}\ensuremathν and zzz, scales as n\ensuremath∼\ensuremathτ\ensuremath−\ensuremathα\ensuremathνd/(\ensuremathαz\ensuremathν+1)n\ensuremath{\sim}{\ensuremath{\tau}}^{\ensuremath{-}\ensuremath{\alpha}\ensuremath{\nu}d/(\ensuremath{\alpha}z\ensuremath{\nu}+1)}n\ensuremath∼\ensuremathτ\ensuremath−\ensuremathα\ensuremathνd/(\ensuremathαz\ensuremathν+1) [n\ensuremath∼(\ensuremathαg(\ensuremathα\ensuremath−1)/\ensuremathα/\ensuremathτ)\ensuremathνd/(z\ensuremathν+1)n\ensuremath{\sim}(\ensuremath{\alpha}{g}^{(\ensuremath{\alpha}\ensuremath{-}1)/\ensuremath{\alpha}}/\ensuremath{\tau}{)}^{\ensuremath{\nu}d/(z\ensuremath{\nu}+1)}n\ensuremath∼(\ensuremathαg(\ensuremathα\ensuremath−1)/\ensuremathα/\ensuremathτ)\ensuremathνd/(z\ensuremathν+1)] if the quench takes the system across the critical point at time t=0t=0t=0 [t=t0\ensuremath≠0t={t}_{0}\ensuremath{\ne}0t=t0​\ensuremath=0], where ggg is a nonuniversal constant and ddd 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 (\ensuremathα=1\ensuremath{\alpha}=1\ensuremathα=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 defect production in a quantum system subjected to a nonlinear power-law quench which takes it either through a quantum critical or multicritical point or along a quantum critical line. We elaborate on our earlier work [D. Sen, K. Sengupta, and S. Mondal, Phys. Rev. Lett. 101, 016806 (2008)] and present a detailed analysis of the scaling of the defect density nnn with the quench rate $\ensuremath{\tau}$ and exponent $\ensuremath{\alpha}$ for each of the above-mentioned cases. We also compute the correlation functions for defects generated in nonlinear quenches through a quantum critical point and discuss the dependence of the amplitudes of such correlation functions on the exponent $\ensuremath{\alpha}$. We discuss several experimental systems where these theoretical predictions can be tested.
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 nnn and residual energy QQQ for a slow linear quench. We provide a general expression for the scaling of n(Q)n(Q)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 ddd dimensions, for which the energy gap ${\ensuremath{\Delta}}_{\stackrel{P\vec}{k}}\ensuremath{\sim}{k}_{i}^{z}$ for mmm momentum components (i=1,…,m)(i=1,\dots{},m)(i=1,…,m) and $\ensuremath{\sim}{k}_{i}^{{z}^{\ensuremath{'}}}$ for the rest $d\ensuremath{-}m$ components (i=m+1,…,d)(i=m+1,\dots{},d)(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 nnn and QQQ 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{D}_{n}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 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 ddd-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 zzz, 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 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 show that for a d-dimensional model in which a quench with a rate tau(-1) takes the system across a (d-m)-dimensional critical surface, the defect density scales as n approximately 1/tau(mnu/(znu+1)), where nu and z are the correlation length and dynamical critical exponents characterizing the critical surface. We explicitly demonstrate that the Kitaev model provides an example of such a scaling with d = 2 and m = nu = z = 1. We also provide the first example of an exact calculation of some multispin correlation functions for a two-dimensional model that can be used to determine the correlation between the defects. We suggest possible experiments to test our theory.
When a quantum system exhibiting a second order phase transition is quenched across the critical point in large but finite time, the dynamics are not adiabatic in the critical region and the Kibble-Zurek (KZ) mechanism provides a framework to determine local observables such as the mean defect density. However, to find higher-point functions, one has to go beyond the KZ paradigm asshown in recent works on one-dimensional transverse field Ising model (TFIM) following a linear quench. It has been found that (i) besides the KZ scale, the quench dynamics depend on another length scale that arises due to the finite phase difference between the low energy modes, and (ii) contrary to the expectations based on the KZ mechanism, in general, the correlation functions do not decay exponentially with distance. Motivated by these results for the linear quench, we are interested in understanding if these properties are universal, and consider the 1D TFIM when the transverse field varies algebraically in the vicinity of the critical field. We focus on the equal-time,longitudinal kink-kink correlation function at the end of the quench from the paramagnetic to the ferromagnetic phase, and find that (i) the correlator depends only on the KZ length for superlinear quenches, otherwise an additional dephasing length is required to describe it, and (ii) the dephased correlator decays as a compressed exponential with an exponent that changes continuously with the quench exponent. Our results are obtained using an adiabatic perturbation theory, analytical arguments and exact numerical integration of the relevant equations.