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:0708.0278 · Physical Review B

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

Victor Mukherjee, Uma Divakaran, Amit Dutta, Diptiman Sen

Published Nov 19, 2007• 157 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.76.174303

Related papers

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

Victor Mukherjee, Amit Dutta, Diptiman Sen

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.

arXiv:0803.1351•Physical Review B•Jun 20, 2008•54 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

Anti-Kibble-Zurek behavior of a noisy transverse-field XY chain and its quantum simulation with two-level systems

Zhipeng Gao, Dan-Wei Zhang, Yang Yu +1 more

We study the dynamics of a transverse-field $\mathrm{XY}$ chain driven across quantum critical points by noisy control fields. We characterize the defect density as a function of the quench time and the noise strength and demonstrate that the defect productions for three quench protocols with different scaling exponents exhibit the anti-Kibble-Zurek behavior, whereby slower driving results in more defects. The protocols are quenching through the boundary line between paramagnetic and ferromagnetic phases, quenching across the isolated multicritical point, and quenching along the gapless line, respectively. We also show that the optimal quench time to minimize defects scales as a universal power law of the noise strength in all three cases. Furthermore, by using quantum simulation of the quench dynamics in the spin system with well-designed Landau-Zener crossings in pseudomomentum space, we propose an experimentally feasible scheme to test the predicted anti-Kibble-Zurek behavior of this noisy transverse-field $\mathrm{XY}$ chain with two-level systems under controllable fluctuations.

arXiv:1704.05248•Physical review. B./Physical review. B•Jun 19, 2017•30 citations

Quantum phase transitions in transverse field spin models: from statistical physics to quantum information

Amit Dutta, G. Aeppli, Bikas K. Chakrabarti +3 more

We review quantum phase transitions of spin systems in transverse magnetic fields taking the examples of the spin-1/2 Ising and XY models in a transverse field. Beginning with an overview of quantum phase transitions, we introduce a number of model Hamiltonians. We provide exact solutions in one spatial dimension connecting them to conformal field theoretical studies. We also discuss Kitaev models and some other exactly solvable spin systems. Studies of quantum phase transitions in the presence of quenched randomness and with frustrating interactions are presented in detail. We discuss novel phenomena like Griffiths-McCoy singularities. We then turn to more recent topics like information theoretic measures of the quantum phase transitions in these models such as concurrence, entanglement entropy, quantum discord and quantum fidelity. We then focus on non-equilibrium dynamics of a variety of transverse field systems across quantum critical points and lines. After mentioning rapid quenching studies, we dwell on slow dynamics and discuss the Kibble-Zurek scaling for the defect density following a quench across critical points and its modifications for quenching across critical lines, gapless regions and multicritical points. Topics like the role of different quenching schemes, local quenching, quenching of models with random interactions and quenching of a spin chain coupled to a heat bath are touched upon. The connection between non-equilibrium dynamics and quantum information theoretic measures is presented at some length. We indicate the connection between Kibble-Zurek scaling and adiabatic evolution of a state as well as the application of adiabatic dynamics as a tool of a quantum optimization technique known as quantum annealing. The final section is dedicated to a detailed discussion on recent experimental studies of transverse Ising-like systems.

arXiv:1012.0653•Dec 3, 2010•171 citations

Scaling regimes in slow quenches within a gapped phase

Lakshita Jindal, Kavita Jain

We consider the finite-time quench dynamics in the quantum transverse field Ising model which exhibits a second order phase transition from a paramagnetic to a ferromagnetic phase, as the transverse magnetic field is decreased. These dynamics have been thoroughly investigated in previous studies when the critical point is crossed during the quench; here, we quench the system from deep in the paramagnetic phase to just above the critical field so that the system remains in the gapped phase throughout the quench duration. On linearly quenching the infinitely large system, we find that the behavior of mean longitudinal defect density and mean transverse magnetization at the end of the quench falls into three distinct scaling regimes as the quench time is increased. For sufficiently small quench times, these observables remain roughly constant, but for larger quench times, a crossover occurs from the Kibble-Zurek scaling law to the quadratic quench rate law when the Kibble-Zurek time is of the order of relaxation time at the final quench field. These results are shown analytically using power series and uniform asymptotic expansions of the exact solution of the model, and also compared with an adiabatic perturbation theory in the third regime. We find that the above mentioned scaling regimes hold for quenches within the ferromagnetic phase also, and provide a general scaling argument for crossover from the Kibble-Zurek regime to an adiabatic regime for slow quenches within a gapped phase.

arXiv:2504.04925•Physical review. B./Physical review. B•Jun 26, 2025•1 citation

The effect of the three-spin interaction and the next nearest neighbor interaction on the quenching dynamics of a transverse Ising model

Uma Divakaran, Amit Dutta

We study the zero-temperature quenching dynamics of various extensions of the transverse Ising model (TIM) for when the transverse field is linearly quenched from to (or zero) at a finite and uniform rate. The rate of quenching is dictated by a characteristic scale given by τ. The density of kinks produced in these extended models while crossing the quantum critical points during the quenching process is calculated using a many-body generalization of the Landau–Zener transition theory. The density of kinks in the final state is found to decay as τ −1/2 . In the first model considered here, the transverse Ising Hamiltonian includes an additional ferromagnetic three-spin interaction term of strength J 3 . For J 3 <0.5, the kink density is found to increase monotonically with J 3 whereas it decreases with J 3 for J 3 >0.5. The point with J 3 = 0.5 and the transverse field h = −0.5 is multicritical, where the density shows a slower decay given by τ −1/6 . We also study the effect of ferromagnetic or antiferromagnetic next nearest neighbor (NNN) interactions on the dynamics of the TIM under the same quenching scheme. In a mean field approximation, the transverse Ising Hamiltonians with NNN interactions are identical to the three-spin Hamiltonian. The NNN interactions non-trivially modify the dynamical behavior; for example an antiferromagnetic NNN interaction results in a larger number of kinks in the final state in comparison to the case when the NNN interaction is ferromagnetic.

arXiv:0801.2621•Journal of Statistical Mechanics Theory and Experiment•Nov 9, 2007•18 citations
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∕\ensuremathτ1∕\sqrt{\ensuremath{\tau}}1∕\ensuremathτ​, where \ensuremathτ\ensuremath{\tau}\ensuremathτ 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∕\ensuremathτ1∕61∕{\ensuremath{\tau}}^{1∕6}1∕\ensuremathτ1∕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 quenching dynamics of a one-dimensional spin-1/2 XYXYXY 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 ln 2\text{ln}\text{ }2ln 2, respectively, for a large number of repeated quenching.
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 study the dynamics of a transverse-field XY\mathrm{XY}XY chain driven across quantum critical points by noisy control fields. We characterize the defect density as a function of the quench time and the noise strength and demonstrate that the defect productions for three quench protocols with different scaling exponents exhibit the anti-Kibble-Zurek behavior, whereby slower driving results in more defects. The protocols are quenching through the boundary line between paramagnetic and ferromagnetic phases, quenching across the isolated multicritical point, and quenching along the gapless line, respectively. We also show that the optimal quench time to minimize defects scales as a universal power law of the noise strength in all three cases. Furthermore, by using quantum simulation of the quench dynamics in the spin system with well-designed Landau-Zener crossings in pseudomomentum space, we propose an experimentally feasible scheme to test the predicted anti-Kibble-Zurek behavior of this noisy transverse-field XY\mathrm{XY}XY chain with two-level systems under controllable fluctuations.
We review quantum phase transitions of spin systems in transverse magnetic fields taking the examples of the spin-1/2 Ising and XY models in a transverse field. Beginning with an overview of quantum phase transitions, we introduce a number of model Hamiltonians. We provide exact solutions in one spatial dimension connecting them to conformal field theoretical studies. We also discuss Kitaev models and some other exactly solvable spin systems. Studies of quantum phase transitions in the presence of quenched randomness and with frustrating interactions are presented in detail. We discuss novel phenomena like Griffiths-McCoy singularities. We then turn to more recent topics like information theoretic measures of the quantum phase transitions in these models such as concurrence, entanglement entropy, quantum discord and quantum fidelity. We then focus on non-equilibrium dynamics of a variety of transverse field systems across quantum critical points and lines. After mentioning rapid quenching studies, we dwell on slow dynamics and discuss the Kibble-Zurek scaling for the defect density following a quench across critical points and its modifications for quenching across critical lines, gapless regions and multicritical points. Topics like the role of different quenching schemes, local quenching, quenching of models with random interactions and quenching of a spin chain coupled to a heat bath are touched upon. The connection between non-equilibrium dynamics and quantum information theoretic measures is presented at some length. We indicate the connection between Kibble-Zurek scaling and adiabatic evolution of a state as well as the application of adiabatic dynamics as a tool of a quantum optimization technique known as quantum annealing. The final section is dedicated to a detailed discussion on recent experimental studies of transverse Ising-like systems.
We consider the finite-time quench dynamics in the quantum transverse field Ising model which exhibits a second order phase transition from a paramagnetic to a ferromagnetic phase, as the transverse magnetic field is decreased. These dynamics have been thoroughly investigated in previous studies when the critical point is crossed during the quench; here, we quench the system from deep in the paramagnetic phase to just above the critical field so that the system remains in the gapped phase throughout the quench duration. On linearly quenching the infinitely large system, we find that the behavior of mean longitudinal defect density and mean transverse magnetization at the end of the quench falls into three distinct scaling regimes as the quench time is increased. For sufficiently small quench times, these observables remain roughly constant, but for larger quench times, a crossover occurs from the Kibble-Zurek scaling law to the quadratic quench rate law when the Kibble-Zurek time is of the order of relaxation time at the final quench field. These results are shown analytically using power series and uniform asymptotic expansions of the exact solution of the model, and also compared with an adiabatic perturbation theory in the third regime. We find that the above mentioned scaling regimes hold for quenches within the ferromagnetic phase also, and provide a general scaling argument for crossover from the Kibble-Zurek regime to an adiabatic regime for slow quenches within a gapped phase.
We study the zero-temperature quenching dynamics of various extensions of the transverse Ising model (TIM) for when the transverse field is linearly quenched from to (or zero) at a finite and uniform rate. The rate of quenching is dictated by a characteristic scale given by τ. The density of kinks produced in these extended models while crossing the quantum critical points during the quenching process is calculated using a many-body generalization of the Landau–Zener transition theory. The density of kinks in the final state is found to decay as τ −1/2 . In the first model considered here, the transverse Ising Hamiltonian includes an additional ferromagnetic three-spin interaction term of strength J 3 . For J 3 <0.5, the kink density is found to increase monotonically with J 3 whereas it decreases with J 3 for J 3 >0.5. The point with J 3 = 0.5 and the transverse field h = −0.5 is multicritical, where the density shows a slower decay given by τ −1/6 . We also study the effect of ferromagnetic or antiferromagnetic next nearest neighbor (NNN) interactions on the dynamics of the TIM under the same quenching scheme. In a mean field approximation, the transverse Ising Hamiltonians with NNN interactions are identical to the three-spin Hamiltonian. The NNN interactions non-trivially modify the dynamical behavior; for example an antiferromagnetic NNN interaction results in a larger number of kinks in the final state in comparison to the case when the NNN interaction is ferromagnetic.