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:0802.3986 · Physical Review B

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

Shreyoshi Mondal, Diptiman Sen, K. Sengupta

Published Jul 1, 2008• 96 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.78.045101

Related papers

Exact Results for Quench Dynamics and Defect Production in a Two-Dimensional Model

K. Sengupta, Diptiman Sen, Shreyoshi Mondal

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.

arXiv:0710.1712•Physical Review Letters•Feb 21, 2008•191 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 of the Kitaev model: Anisotropic critical point and effect of disorder

T. Hikichi, Sei Suzuki, K. Sengupta

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.

arXiv:1009.0323•Physical Review B•Nov 18, 2010•24 citations

Theory of defect production in nonlinear quench across a quantum critical point

Shreyoshi Mondal, K. Sengupta, Diptiman Sen

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 $n$ 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.

arXiv:0808.1175•Physical Review B•Jan 29, 2009•64 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

Reverse quenching in a one-dimensional Kitaev model

Uma Divakaran, Amit Dutta

We present an exact result for the nonadiabatic transition probability and hence the defect density in the final state of a one-dimensional Kitaev model following a slow quench of the parameter ${J}_{\ensuremath{-}}$, which estimates the anisotropy between the interactions, as ${J}_{\ensuremath{-}}(t)\ensuremath{\sim}\ensuremath{-}|t/\ensuremath{\tau}|$. Here, time $t$ goes from $\ensuremath{-}\ensuremath{\infty}$ to $+\ensuremath{\infty}$ and $\ensuremath{\tau}$ defines the rate of change in the Hamiltonian. In other words, the spin chain initially prepared in its ground state is driven by changing ${J}_{\ensuremath{-}}$ linearly in time up to the quantum critical point, which in the model considered here occurs at $t=0$, reversed and then gradually decreased to its initial value at the same rate. We have thoroughly compared the reverse quenching with its counterpart forward quenching, i.e., ${J}_{\ensuremath{-}}\ensuremath{\sim}t/\ensuremath{\tau}$. Our exact calculation shows that the probability of excitations is zero for the wave vector at which the instantaneous energy gap is zero at the critical point ${J}_{\ensuremath{-}}=0$ as opposed to the maximum value of unity in the forward quenching. It is also shown that the defect density in the final state following a reverse quenching, we propose here, is nearly half of the defects generated in the forward quenching. We argue that the defects produced when the system reaches the quantum critical point get redistributed in the wave-vector space at the final time in case of reverse quenching whereas it keeps on increasing until the final time in the forward quenching. We study the entropy density and also the time evolution of the diagonal entropy density in the case of the reverse quenching and compare it with the forward case.

arXiv:0901.3260•Physical Review B•Jun 8, 2009•23 citations
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∼1/\ensuremathτn\ensuremath{\sim}1/\sqrt{\ensuremath{\tau}}n\ensuremath∼1/\ensuremathτ​, where 1/\ensuremathτ1/\ensuremath{\tau}1/\ensuremathτ 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−md\ensuremath{-}md\ensuremath−m dimensional gapless (critical) surface characterized by correlation length exponent \ensuremathν\ensuremath{\nu}\ensuremathν and dynamical critical exponent zzz, the defect density n\ensuremath∼1/\ensuremathτm\ensuremathν/(z\ensuremathν+1)n\ensuremath{\sim}1/{\ensuremath{\tau}}^{m\ensuremath{\nu}/(z\ensuremath{\nu}+1)}n\ensuremath∼1/\ensuremathτm\ensuremathν/(z\ensuremathν+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 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.
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.
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 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 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 present an exact result for the nonadiabatic transition probability and hence the defect density in the final state of a one-dimensional Kitaev model following a slow quench of the parameter ${J}_{\ensuremath{-}}$, which estimates the anisotropy between the interactions, as ${J}_{\ensuremath{-}}(t)\ensuremath{\sim}\ensuremath{-}|t/\ensuremath{\tau}|$. Here, time ttt goes from $\ensuremath{-}\ensuremath{\infty}$ to $+\ensuremath{\infty}$ and $\ensuremath{\tau}$ defines the rate of change in the Hamiltonian. In other words, the spin chain initially prepared in its ground state is driven by changing ${J}_{\ensuremath{-}}$ linearly in time up to the quantum critical point, which in the model considered here occurs at t=0t=0t=0, reversed and then gradually decreased to its initial value at the same rate. We have thoroughly compared the reverse quenching with its counterpart forward quenching, i.e., ${J}_{\ensuremath{-}}\ensuremath{\sim}t/\ensuremath{\tau}$. Our exact calculation shows that the probability of excitations is zero for the wave vector at which the instantaneous energy gap is zero at the critical point ${J}_{\ensuremath{-}}=0$ as opposed to the maximum value of unity in the forward quenching. It is also shown that the defect density in the final state following a reverse quenching, we propose here, is nearly half of the defects generated in the forward quenching. We argue that the defects produced when the system reaches the quantum critical point get redistributed in the wave-vector space at the final time in case of reverse quenching whereas it keeps on increasing until the final time in the forward quenching. We study the entropy density and also the time evolution of the diagonal entropy density in the case of the reverse quenching and compare it with the forward case.