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:0803.1351 · Physical Review B

Defect generation in a spin- 12 transverse XY chain under repeated quenching of the transverse field

Victor Mukherjee, Amit Dutta, Diptiman Sen

Published Jun 20, 2008• 54 citations
arXivHTMLPDF

Abstract

We study the quenching dynamics of a one-dimensional spin-1/2 $XY$ model in a transverse field when the transverse field $h(=t/\ensuremath{\tau})$ is quenched repeatedly between $\ensuremath{-}\ensuremath{\infty}$ and $+\ensuremath{\infty}$. A single passage from $h\ensuremath{\rightarrow}\ensuremath{-}\ensuremath{\infty}$ to $h\ensuremath{\rightarrow}+\ensuremath{\infty}$ or the other way around is referred to as a half period of quenching. For an even number of half periods, the transverse field is brought back to the initial value of $\ensuremath{-}\ensuremath{\infty}$; in the case of an odd number of half periods, the dynamics is stopped at $h\ensuremath{\rightarrow}+\ensuremath{\infty}$. The density of defects produced due to the nonadiabatic transitions is calculated by mapping the many-particle system to an equivalent Landau-Zener problem and is generally found to vary as $1/\sqrt{\ensuremath{\tau}}$ for large $\ensuremath{\tau}$; however, the magnitude is found to depend on the number of half periods of quenching. For two successive half periods, the defect density is found to decrease in comparison to a single half period, suggesting the existence of a corrective mechanism in the reverse path. A similar behavior of the density of defects and the local entropy is observed for repeated quenching. The defect density decays as $1/\sqrt{\ensuremath{\tau}}$ for large $\ensuremath{\tau}$ for any number of half periods, and shows an increase in kink density for small $\ensuremath{\tau}$ for an even number; the entropy shows qualitatively the same behavior for any number of half periods. The probability of nonadiabatic transitions and the local entropy saturate to 1/2 and $\text{ln}\text{ }2$, respectively, for a large number of repeated quenching.

DOI
10.1103/physrevb.77.214427

Related papers

Quenching dynamics of a quantum XY spin- 12 chain in a transverse field

Victor Mukherjee, Uma Divakaran, Amit Dutta +1 more

We study the quantum dynamics of a one-dimensional spin-$\frac{1}{2}$ anisotropic $XY$ model in a transverse field when the transverse field or the anisotropic interaction is quenched at a slow but uniform rate. The two quenching schemes are called transverse and anisotropic quenching, respectively. Our emphasis in this paper is on the anisotropic quenching scheme, and we compare the results with those of the other scheme. In the process of anisotropic quenching, the system crosses all the quantum critical lines of the phase diagram where the relaxation time diverges. The evolution is nonadiabatic in the time interval when the parameters are close to their critical values, and is adiabatic otherwise. The density of defects produced due to nonadiabatic transitions is calculated by mapping the many-particle system to an equivalent Landau-Zener problem and is generally found to vary as $1∕\sqrt{\ensuremath{\tau}}$, where $\ensuremath{\tau}$ is the characteristic time scale of quenching, a scenario that supports the Kibble-Zurek mechanism. Interestingly, in the case of anisotropic quenching, there exists an additional nonadiabatic transition, in comparison to the transverse quenching case, with the corresponding probability peaking at an incommensurate value of the wave vector. In the special case in which the system passes through a multicritical point, the defect density is found to vary as $1∕{\ensuremath{\tau}}^{1∕6}$. The von Neumann entropy of the final state is shown to maximize at a quenching rate around which the ordering of the final state changes from antiferromagnetic to ferromagnetic.

arXiv:0708.0278•Physical Review B•Nov 19, 2007•157 citations

Effects of interference in the dynamics of a spin- 1/2 transverseXYchain driven periodically through quantum critical points

Victor Mukherjee, Amit Dutta

We study the effects of interference on the quenching dynamics of a one-dimensional spin 1/2 XY model in the presence of a transverse field ( h ( t )) which varies sinusoidally with time as h = h 0 cosω t , with | t |≤ t f = π/ω. We have explicitly shown that the finite values of t f make the dynamics inherently dependent on the phases of probability amplitudes, which had been hitherto unseen in all cases of linear quenching with large initial and final times. In contrast, we also consider the situation where the magnetic field consists of an oscillatory as well as a linearly varying component, i.e., h ( t ) = h 0 cosω t + t /τ, where the interference effects lose importance in the limit of large τ. Our purpose is to estimate the defect density and the local entropy density in the final state if the system is initially prepared in its ground state. For a single crossing through the quantum critical point with h = h 0 cosω t , the density of defects in the final state is calculated by mapping the dynamics to an equivalent Landau–Zener problem by linearizing near the crossing point, and is found to vary as in the limit of small ω. On the other hand, the local entropy density is found to attain a maximum as a function of ω near a characteristic scale ω 0 . Extending to the situation of multiple crossings, we show that the role of finite initial and final times of quenching are manifested non-trivially in the interference effects of certain resonance modes which solely contribute to the production of defects. Kink density as well as the diagonal entropy density show oscillatory dependence on the number of full cycles of oscillation. Finally, the inclusion of a linear term in the transverse field on top of the oscillatory component results in a kink density which decreases continuously with τ while it increases monotonically with ω. The entropy density also shows monotonic change with the parameters, increasing with τ and decreasing with ω, in sharp contrast to the situations studied earlier. We also propose appropriate scaling relations for the defect density in the above situations and compare the results with the numerical results obtained by integrating the Schrödinger equations.

arXiv:0901.1933•Journal of Statistical Mechanics Theory and Experiment•May 6, 2009•53 citations

Quenching along a gapless line: A different exponent for defect density

Uma Divakaran, Amit Dutta, Diptiman Sen

We use a quenching scheme to study the dynamics of a one-dimensional anisotropic $XY$ spin-1/2 chain in the presence of a transverse field which alternates between the values $h+\ensuremath{\delta}$ and $h\ensuremath{-}\ensuremath{\delta}$ from site to site. In this quenching scheme, the parameter denoting the anisotropy of interaction $(\ensuremath{\gamma})$ is linearly quenched from $\ensuremath{-}\ensuremath{\infty}$ to $+\ensuremath{\infty}$ as $\ensuremath{\gamma}=t/\ensuremath{\tau}$, keeping the total strength of interaction $J$ fixed. The system traverses through a gapless phase when $\ensuremath{\gamma}$ is quenched along the critical surface ${h}^{2}={\ensuremath{\delta}}^{2}+{J}^{2}$ in the parameter space spanned by $h$, $\ensuremath{\delta}$, and $\ensuremath{\gamma}$. By mapping to an equivalent two-level Landau-Zener problem, we show that the defect density in the final state scales as $1/{\ensuremath{\tau}}^{1/3}$, a behavior that has not been observed in previous studies of quenching through a gapless phase. We also generalize the model incorporating additional alternations in the anisotropy or in the strength of the interaction and derive an identical result under a similar quenching. Based on the above results, we propose a general scaling of the defect density with the quenching rate $\ensuremath{\tau}$ for quenching along a gapless critical line.

arXiv:0805.3328•Physical Review B•Oct 17, 2008•92 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

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

Quench dynamics and defects formation in the Ising chain in a transverse magnetic field

Alexander I. Nesterov, Mónica F. Ramírez

We study analytically and numerically quench dynamics and defects formation in the quantum Ising model in the presence of a time-dependent transverse magnetic field. We generalize the Landau-Ziner formula to the case of non-adiabatic evolution of the quantum system. For a quasi-static magnetic field, with a slow dependence on time, our outcomes are similar to the results predicted by the Landau-Zener formula. However, a quench dynamics under a shock-wave load is more complicated. The final state of the system depends on the amplitude and pulse velocity, resulting in the mixture of ground and excited states and significant density of defects.

arXiv:1812.11887•Dec 31, 2018•0 citations
We study the quenching dynamics of a one-dimensional spin-1/2 XYXYXY model in a transverse field when the transverse field h(=t/\ensuremathτ)h(=t/\ensuremath{\tau})h(=t/\ensuremathτ) is quenched repeatedly between \ensuremath−\ensuremath∞\ensuremath{-}\ensuremath{\infty}\ensuremath−\ensuremath∞ and +\ensuremath∞+\ensuremath{\infty}+\ensuremath∞. A single passage from h\ensuremath→\ensuremath−\ensuremath∞h\ensuremath{\rightarrow}\ensuremath{-}\ensuremath{\infty}h\ensuremath→\ensuremath−\ensuremath∞ to h\ensuremath→+\ensuremath∞h\ensuremath{\rightarrow}+\ensuremath{\infty}h\ensuremath→+\ensuremath∞ or the other way around is referred to as a half period of quenching. For an even number of half periods, the transverse field is brought back to the initial value of \ensuremath−\ensuremath∞\ensuremath{-}\ensuremath{\infty}\ensuremath−\ensuremath∞; in the case of an odd number of half periods, the dynamics is stopped at h\ensuremath→+\ensuremath∞h\ensuremath{\rightarrow}+\ensuremath{\infty}h\ensuremath→+\ensuremath∞. The density of defects produced due to the nonadiabatic transitions is calculated by mapping the many-particle system to an equivalent Landau-Zener problem and is generally found to vary as 1/\ensuremathτ1/\sqrt{\ensuremath{\tau}}1/\ensuremathτ​ for large \ensuremathτ\ensuremath{\tau}\ensuremathτ; however, the magnitude is found to depend on the number of half periods of quenching. For two successive half periods, the defect density is found to decrease in comparison to a single half period, suggesting the existence of a corrective mechanism in the reverse path. A similar behavior of the density of defects and the local entropy is observed for repeated quenching. The defect density decays as 1/\ensuremathτ1/\sqrt{\ensuremath{\tau}}1/\ensuremathτ​ for large \ensuremathτ\ensuremath{\tau}\ensuremathτ for any number of half periods, and shows an increase in kink density for small \ensuremathτ\ensuremath{\tau}\ensuremathτ for an even number; the entropy shows qualitatively the same behavior for any number of half periods. The probability of nonadiabatic transitions and the local entropy saturate to 1/2 and ln 2\text{ln}\text{ }2ln 2, respectively, for a large number of repeated quenching.
We study the quantum dynamics of a one-dimensional spin-12\frac{1}{2}21​ anisotropic XYXYXY model in a transverse field when the transverse field or the anisotropic interaction is quenched at a slow but uniform rate. The two quenching schemes are called transverse and anisotropic quenching, respectively. Our emphasis in this paper is on the anisotropic quenching scheme, and we compare the results with those of the other scheme. In the process of anisotropic quenching, the system crosses all the quantum critical lines of the phase diagram where the relaxation time diverges. The evolution is nonadiabatic in the time interval when the parameters are close to their critical values, and is adiabatic otherwise. The density of defects produced due to nonadiabatic transitions is calculated by mapping the many-particle system to an equivalent Landau-Zener problem and is generally found to vary as $1∕\sqrt{\ensuremath{\tau}}$, where $\ensuremath{\tau}$ is the characteristic time scale of quenching, a scenario that supports the Kibble-Zurek mechanism. Interestingly, in the case of anisotropic quenching, there exists an additional nonadiabatic transition, in comparison to the transverse quenching case, with the corresponding probability peaking at an incommensurate value of the wave vector. In the special case in which the system passes through a multicritical point, the defect density is found to vary as $1∕{\ensuremath{\tau}}^{1∕6}$. The von Neumann entropy of the final state is shown to maximize at a quenching rate around which the ordering of the final state changes from antiferromagnetic to ferromagnetic.
We study the effects of interference on the quenching dynamics of a one-dimensional spin 1/2 XY model in the presence of a transverse field ( h ( t )) which varies sinusoidally with time as h = h 0 cosω t , with | t |≤ t f = π/ω. We have explicitly shown that the finite values of t f make the dynamics inherently dependent on the phases of probability amplitudes, which had been hitherto unseen in all cases of linear quenching with large initial and final times. In contrast, we also consider the situation where the magnetic field consists of an oscillatory as well as a linearly varying component, i.e., h ( t ) = h 0 cosω t + t /τ, where the interference effects lose importance in the limit of large τ. Our purpose is to estimate the defect density and the local entropy density in the final state if the system is initially prepared in its ground state. For a single crossing through the quantum critical point with h = h 0 cosω t , the density of defects in the final state is calculated by mapping the dynamics to an equivalent Landau–Zener problem by linearizing near the crossing point, and is found to vary as in the limit of small ω. On the other hand, the local entropy density is found to attain a maximum as a function of ω near a characteristic scale ω 0 . Extending to the situation of multiple crossings, we show that the role of finite initial and final times of quenching are manifested non-trivially in the interference effects of certain resonance modes which solely contribute to the production of defects. Kink density as well as the diagonal entropy density show oscillatory dependence on the number of full cycles of oscillation. Finally, the inclusion of a linear term in the transverse field on top of the oscillatory component results in a kink density which decreases continuously with τ while it increases monotonically with ω. The entropy density also shows monotonic change with the parameters, increasing with τ and decreasing with ω, in sharp contrast to the situations studied earlier. We also propose appropriate scaling relations for the defect density in the above situations and compare the results with the numerical results obtained by integrating the Schrödinger equations.
We use a quenching scheme to study the dynamics of a one-dimensional anisotropic XYXYXY spin-1/2 chain in the presence of a transverse field which alternates between the values $h+\ensuremath{\delta}$ and $h\ensuremath{-}\ensuremath{\delta}$ from site to site. In this quenching scheme, the parameter denoting the anisotropy of interaction $(\ensuremath{\gamma})$ is linearly quenched from $\ensuremath{-}\ensuremath{\infty}$ to $+\ensuremath{\infty}$ as $\ensuremath{\gamma}=t/\ensuremath{\tau}$, keeping the total strength of interaction JJJ fixed. The system traverses through a gapless phase when $\ensuremath{\gamma}$ is quenched along the critical surface ${h}^{2}={\ensuremath{\delta}}^{2}+{J}^{2}$ in the parameter space spanned by hhh, $\ensuremath{\delta}$, and $\ensuremath{\gamma}$. By mapping to an equivalent two-level Landau-Zener problem, we show that the defect density in the final state scales as $1/{\ensuremath{\tau}}^{1/3}$, a behavior that has not been observed in previous studies of quenching through a gapless phase. We also generalize the model incorporating additional alternations in the anisotropy or in the strength of the interaction and derive an identical result under a similar quenching. Based on the above results, we propose a general scaling of the defect density with the quenching rate $\ensuremath{\tau}$ for quenching along a gapless critical line.
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.
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 analytically and numerically quench dynamics and defects formation in the quantum Ising model in the presence of a time-dependent transverse magnetic field. We generalize the Landau-Ziner formula to the case of non-adiabatic evolution of the quantum system. For a quasi-static magnetic field, with a slow dependence on time, our outcomes are similar to the results predicted by the Landau-Zener formula. However, a quench dynamics under a shock-wave load is more complicated. The final state of the system depends on the amplitude and pulse velocity, resulting in the mixture of ground and excited states and significant density of defects.