Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1009.0323 · Physical Review B

Slow quench dynamics of the Kitaev model: Anisotropic critical point and effect of disorder

T. Hikichi, Sei Suzuki, K. Sengupta

Published Nov 18, 2010• 24 citations
arXivHTMLPDF

Abstract

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 $(i=1,\dots{},m)$ and $\ensuremath{\sim}{k}_{i}^{{z}^{\ensuremath{'}}}$ for the rest $d\ensuremath{-}m$ components $(i=m+1,\dots{},d)$ 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 ${D}_{n}$ 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.

DOI
10.1103/physrevb.82.174305

Related papers

Quench dynamics and defect production in the Kitaev and extended Kitaev models

Shreyoshi Mondal, Diptiman Sen, K. Sengupta

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.

arXiv:0802.3986•Physical Review B•Jul 1, 2008•96 citations

Non-adiabatic quench dynamics near anisotropic quantum critical point

T Hikichi, Sei Suzuki, K. Sengupta

We study non-equilibrium dynamics induced by a quench of a parameter near a quantum phase transition in the Kitaev model. On the basis of an exact solution for the time-dependent Schrödinger equation, we show that the density of excitations and residual energy after a quench scale with a quench rate τ −1 as τ −3/4 and τ −5/4 respectively, when a parameter is quenched from a gapped phase toward the quantum critical point. These unusual scaling laws originate from an anisotropic nature of the quantum critical point in the Kitaev model. Gerenalizing these results, we provide novel scaling laws for a generic anisotropic quantum critical point in d -dimensional systems.

Journal of Physics Conference Series•May 1, 2011•0 citations

Sudden quenching in the Kitaev honeycomb model: Study of defect and heat generation

Aavishkar A. Patel, Amit Dutta

We study the behavior of the defect and heat densities under sudden quenching near the quantum critical points in the two-dimensional Kitaev honeycomb model both in the thermodynamic and nonthermodynamic limits. We consider quenches starting from a quantum critical point into the gapped as well as the gapless phases. We choose points on the lines of anisotropic quantum critical points as well as different points of intersection of these lines as the initial points from where the quenching starts. We find that the defect and heat densities display the expected power-law scalings along with logarithmic corrections to scaling (or cusp singularities) in certain cases. In the vicinity of some of the intersection points, the scaling behaviors change, indicating an effective dimensional reduction; the scaling behavior near these points depends on the number of critical lines crossed in the process of quenching. All the analytical predictions are also verified by numerical integration.

arXiv:1209.0072•Physical Review B•Nov 28, 2012•12 citations

Defect Production in Nonlinear Quench across a Quantum Critical Point

Diptiman Sen, K. Sengupta, Shreyoshi Mondal

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.

arXiv:0803.2081•Physical Review Letters•Jul 3, 2008•215 citations

Defect production and quench dynamics in the three-dimensional Kitaev model

Subhajit Sarkar, Dibyendu Rana, Saptarshi Mandal

We study the quench dynamics of the three-dimensional Kitaev (spin) model under a linear drive using both exact numerical calculations and analytical independent crossing approximation. Unlike the two-dimensional Kitaev model, the three-dimensional Kitaev model reduces to a multilevel Landau-Zener problem for each momentum. We show that for the slow quench, the defect density is proportional to the quench rate $1/\ensuremath{\tau}$. We find that the zeros of the relevant coupling between the levels determine the nonadiabatic condition for the production of defects. The contour on which the energy spectrum becomes gapless does not play an active role. The asymptotic behavior of the defect density crucially depends on the way the system reaches the nonadiabatic regime during the quenching process. We analytically show that defect correlation varies as ${\ensuremath{\tau}}^{\ensuremath{-}1}{e}^{\ensuremath{-}A/\ensuremath{\tau}}$ where $A$ is a constant independent of $\ensuremath{\tau}$. For the slow quench, the qualitative dependence of the entropy (produced during the quenching process) on the quench time is the same as that of the defect correlation, indicating a close connection between the defect correlation and the entropy content of the final state. Possible experimental realization of such quench dynamics is also described briefly.

arXiv:1812.09923•Physical review. B./Physical review. B•Oct 21, 2020•7 citations

Slow quench dynamics in classical systems: kinetic Ising model and zero-range process

Priyanka, Sayani Chatterjee, Kavita Jain

Abstract While a large number of studies have focused on the nonequilibrium dynamics of a system when it is quenched instantaneously from a disordered phase to an ordered phase, such dynamics have been relatively less explored when the quench occurs at a finite rate. Here, we study the slow quench dynamics in two paradigmatic models of classical statistical mechanics, a one-dimensional kinetic Ising model and a mean-field zero-range process, when the system is annealed slowly to the critical point. Starting from the time evolution equations for the spin–spin correlation function in the Ising model and the mass distribution in the zero-range process, we derive the Kibble–Zurek scaling laws. We then test a recent proposal that critical coarsening, which is ignored in the Kibble–Zurek argument, plays a role in the nonequilibrium dynamics close to the critical point. We find that the defect density in the Ising model and a scaled mass distribution in the zero-range process decay linearly to their respective values at the critical point with the time remaining until the end of the quench provided the final quench point is approached sufficiently fast, and sublinearly otherwise. As the linear scaling for the approach to the critical point also holds when a system following an instantaneous quench is allowed to coarsen for a finite time interval, we conclude that critical coarsening captures the scaling behavior in the vicinity of the critical point if the annealing is not too slow.

arXiv:2011.05251•Journal of Statistical Mechanics Theory and Experiment•Mar 1, 2021•9 citations
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 \ensuremathτ\ensuremath−1{\ensuremath{\tau}}^{\ensuremath{-}1}\ensuremathτ\ensuremath−1 and exponent \ensuremathα\ensuremath{\alpha}\ensuremathα, 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∼kiz\ensuremath′\ensuremath{\sim}{k}_{i}^{{z}^{\ensuremath{'}}}\ensuremath∼kiz\ensuremath′​ for the rest d\ensuremath−md\ensuremath{-}md\ensuremath−m components (i=m+1,…,d)(i=m+1,\dots{},d)(i=m+1,…,d) with z\ensuremath≤z\ensuremath′z\ensuremath{\le}{z}^{\ensuremath{'}}z\ensuremath≤z\ensuremath′: n\ensuremath∼\ensuremathτ\ensuremath−[m+(d\ensuremath−m)z/z\ensuremath′]\ensuremathν\ensuremathα/(z\ensuremathν\ensuremathα+1)(Q\ensuremath∼\ensuremathτ\ensuremath−[(m+z)+(d\ensuremath−m)z/z\ensuremath′]\ensuremathν\ensuremathα/(z\ensuremathν\ensuremathα+1))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)})n\ensuremath∼\ensuremathτ\ensuremath−[m+(d\ensuremath−m)z/z\ensuremath′]\ensuremathν\ensuremathα/(z\ensuremathν\ensuremathα+1)(Q\ensuremath∼\ensuremathτ\ensuremath−[(m+z)+(d\ensuremath−m)z/z\ensuremath′]\ensuremathν\ensuremathα/(z\ensuremathν\ensuremathα+1)). These general expressions reproduce both the corresponding results for the Kitaev model as a special case for d=z\ensuremath′=2d={z}^{\ensuremath{'}}=2d=z\ensuremath′=2 and m=z=\ensuremathν=1m=z=\ensuremath{\nu}=1m=z=\ensuremathν=1 and the well-known scaling laws of nnn and QQQ for isotropic critical points for z=z\ensuremath′z={z}^{\ensuremath{'}}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∼\ensuremathτ\ensuremath−1/2n\ensuremath{\sim}{\ensuremath{\tau}}^{\ensuremath{-}1/2}n\ensuremath∼\ensuremathτ\ensuremath−1/2 and Q\ensuremath∼\ensuremathτ\ensuremath−1Q\ensuremath{\sim}{\ensuremath{\tau}}^{\ensuremath{-}1}Q\ensuremath∼\ensuremathτ\ensuremath−1 (Q\ensuremath∼\ensuremathτ\ensuremath−1/2)(Q\ensuremath{\sim}{\ensuremath{\tau}}^{\ensuremath{-}1/2})(Q\ensuremath∼\ensuremathτ\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 non-equilibrium dynamics induced by a quench of a parameter near a quantum phase transition in the Kitaev model. On the basis of an exact solution for the time-dependent Schrödinger equation, we show that the density of excitations and residual energy after a quench scale with a quench rate τ −1 as τ −3/4 and τ −5/4 respectively, when a parameter is quenched from a gapped phase toward the quantum critical point. These unusual scaling laws originate from an anisotropic nature of the quantum critical point in the Kitaev model. Gerenalizing these results, we provide novel scaling laws for a generic anisotropic quantum critical point in d -dimensional systems.
We study the behavior of the defect and heat densities under sudden quenching near the quantum critical points in the two-dimensional Kitaev honeycomb model both in the thermodynamic and nonthermodynamic limits. We consider quenches starting from a quantum critical point into the gapped as well as the gapless phases. We choose points on the lines of anisotropic quantum critical points as well as different points of intersection of these lines as the initial points from where the quenching starts. We find that the defect and heat densities display the expected power-law scalings along with logarithmic corrections to scaling (or cusp singularities) in certain cases. In the vicinity of some of the intersection points, the scaling behaviors change, indicating an effective dimensional reduction; the scaling behavior near these points depends on the number of critical lines crossed in the process of quenching. All the analytical predictions are also verified by numerical integration.
We show that the defect density nnn, 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 zzz, 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=0t=0t=0 [$t={t}_{0}\ensuremath{\ne}0$], 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 ($\ensuremath{\alpha}=1$) 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 quench dynamics of the three-dimensional Kitaev (spin) model under a linear drive using both exact numerical calculations and analytical independent crossing approximation. Unlike the two-dimensional Kitaev model, the three-dimensional Kitaev model reduces to a multilevel Landau-Zener problem for each momentum. We show that for the slow quench, the defect density is proportional to the quench rate $1/\ensuremath{\tau}$. We find that the zeros of the relevant coupling between the levels determine the nonadiabatic condition for the production of defects. The contour on which the energy spectrum becomes gapless does not play an active role. The asymptotic behavior of the defect density crucially depends on the way the system reaches the nonadiabatic regime during the quenching process. We analytically show that defect correlation varies as ${\ensuremath{\tau}}^{\ensuremath{-}1}{e}^{\ensuremath{-}A/\ensuremath{\tau}}$ where AAA is a constant independent of $\ensuremath{\tau}$. For the slow quench, the qualitative dependence of the entropy (produced during the quenching process) on the quench time is the same as that of the defect correlation, indicating a close connection between the defect correlation and the entropy content of the final state. Possible experimental realization of such quench dynamics is also described briefly.
Abstract While a large number of studies have focused on the nonequilibrium dynamics of a system when it is quenched instantaneously from a disordered phase to an ordered phase, such dynamics have been relatively less explored when the quench occurs at a finite rate. Here, we study the slow quench dynamics in two paradigmatic models of classical statistical mechanics, a one-dimensional kinetic Ising model and a mean-field zero-range process, when the system is annealed slowly to the critical point. Starting from the time evolution equations for the spin–spin correlation function in the Ising model and the mass distribution in the zero-range process, we derive the Kibble–Zurek scaling laws. We then test a recent proposal that critical coarsening, which is ignored in the Kibble–Zurek argument, plays a role in the nonequilibrium dynamics close to the critical point. We find that the defect density in the Ising model and a scaled mass distribution in the zero-range process decay linearly to their respective values at the critical point with the time remaining until the end of the quench provided the final quench point is approached sufficiently fast, and sublinearly otherwise. As the linear scaling for the approach to the critical point also holds when a system following an instantaneous quench is allowed to coarsen for a finite time interval, we conclude that critical coarsening captures the scaling behavior in the vicinity of the critical point if the annealing is not too slow.