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:1112.6092 · Physical Review B

Quenching across quantum critical points in periodic systems: Dependence of scaling laws on periodicity

Manisha Thakurathi, Wade DeGottardi, Diptiman Sen, Smitha Vishveshwara

Published Apr 12, 2012• 7 citations
arXivHTMLPDF

Abstract

We study the quenching dynamics of a many-body system in one dimension described by a Hamiltonian that has spatial periodicity. Specifically, we consider a spin-$\frac{1}{2}$ chain with equal $xx$ and $yy$ couplings and subject to a periodically varying magnetic field in the $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{z}$ direction or, equivalently, a tight-binding model of spinless fermions with a periodic local chemical potential, having period $2q$, where $q$ is a positive integer. For a linear quench of the strength of the magnetic field (or chemical potential) at a rate $1/\ensuremath{\tau}$ across a quantum critical point, we find that the density of defects thereby produced scales as $1/{\ensuremath{\tau}}^{q/(q+1)}$, deviating from the $1/\sqrt{\ensuremath{\tau}}$ scaling that is ubiquitous in a range of systems. We analyze this behavior by mapping the low-energy physics of the system to a set of fermionic two-level systems labeled by the lattice momentum $k$ undergoing a nonlinear quench as well as by performing numerical simulations. We also show that if the magnetic field is a superposition of different periods, the power law depends only on the smallest period for very large values of $\ensuremath{\tau}$, although it may exhibit a crossover at intermediate values of $\ensuremath{\tau}$. Finally, for the case where a $zz$ coupling is also present in the spin chain, or equivalently, where interactions are present in the fermionic system, we argue that the power associated with the scaling law depends on a combination of $q$ and the interaction strength.

DOI
10.1103/physrevb.85.165425

Related papers

Quenching across quantum critical points: Role of topological patterns

Diptiman Sen, Smitha Vishveshwara

We introduce a one-dimensional version of the Kitaev model consisting of spins on a two-legged ladder and characterized by Z(2) invariants on the plaquettes of the ladder. We map the model to a fermionic system and identify the topological sectors associated with different Z2 patterns in terms of fermion occupation numbers. Within these different sectors, we investigate the effect of a linear quench across a quantum critical point. We study the dominant behavior of the system by employing a Landau-Zener-type analysis of the effective Hamiltonian in the low-energy subspace for which the effective quenching can sometimes be non-linear. We show that the quenching leads to a residual energy which scales as a power of the quenching rate, and that the power depends on the topological sectors and their symmetry properties in a non-trivial way. This behavior is consistent with the general theory of quantum quenching, but with the correlation length exponent nu being different in different sectors. Copyright (C) EPLA, 2010

•Sep 1, 2010•22 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

Quenching across quantum critical points: Role of topological patterns

Diptiman Sen, Smitha Vishveshwara

We introduce a one-dimensional version of the Kitaev model consisting of spins on a two-legged ladder and characterized by Z 2 invariants on the plaquettes of the ladder. We map the model to a fermionic system and identify the topological sectors associated with different Z 2 patterns in terms of fermion occupation numbers. Within these different sectors, we investigate the effect of a linear quench across a quantum critical point. We study the dominant behavior of the system by employing a Landau-Zener–type analysis of the effective Hamiltonian in the low-energy subspace for which the effective quenching can sometimes be non-linear. We show that the quenching leads to a residual energy which scales as a power of the quenching rate, and that the power depends on the topological sectors and their symmetry properties in a non-trivial way. This behavior is consistent with the general theory of quantum quenching, but with the correlation length exponent ν being different in different sectors.

arXiv:1003.0058•Europhysics Letters (EPL)•Sep 1, 2010•21 citations

Adiabatic multicritical quantum quenches: Continuously varying exponents depending on the direction of quenching

Victor Mukherjee, Amit Dutta

We study adiabatic quantum quenches across a quantum multicritical point (MCP) using a quenching scheme that enables the system to hit the MCP along different paths. We show that the power-law scaling of the defect density with the rate of driving depends non-trivially on the path, i.e. , the exponent varies continuously with the parameter α that defines the path, up to a critical value α=α c ; on the other hand for α⩾α c , the scaling exponent saturates to a constant value. We show that dynamically generated and path (α)- dependent effective critical exponents associated with the quasicritical points lying close to the MCP (on the ferromagnetic side), where the energy-gap is minimum, lead to this continuously varying exponent. The scaling relations are established using the integrable transverse XY spin chain and generalized to a MCP associated with a d -dimensional quantum many-body systems (not reducible to two-level systems) using adiabatic perturbation theory. We also calculate the effective path-dependent dimensional shift d 0 (α) (or the shift in center of the impulse region) that appears in the scaling relation for special paths lying entirely in the paramagnetic phase. The numerically obtained results are in good agreement with analytical predictions.

arXiv:1006.3343•Europhysics Letters (EPL)•Nov 1, 2010•39 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

Quantum critical scaling of fidelity in BCS-like model

Mariusz Adamski, Janusz Jȩdrzejewski, Taras Krokhmalskii

We study scaling of the ground-state fidelity in neighborhoods of quantum critical points in a model of interacting spinfull fermions - a BCS-like model. Due to the exact diagonalizability of the model, in one and higher dimensions, scaling of the ground-state fidelity can be analyzed numerically with great accuracy, not only for small systems but also for macroscopic ones, together with the crossover region between them. Additionally, in one-dimensional case we have been able to derive a number of analytical formulae for fidelity and show that they accurately fit our numerical results; these results are reported in the article. Besides regular critical points and their neighborhoods, where well-known scaling laws are obeyed, there is the multi-critical point and critical points in its proximity where anomalous scaling behavior is found. We consider also scaling of fidelity in neighborhoods of critical points where fidelity oscillates strongly as the system size or the chemical potential is varied. Our results for a one-dimensional version of a BCS-like model are compared with those obtained by Rams and Damski in similar studies of a quantum spin chain - an anisotropic XY model in transverse magnetic field.

arXiv:1304.2555•Journal of Statistical Mechanics Theory and Experiment•Jul 4, 2013•4 citations
We study the quenching dynamics of a many-body system in one dimension described by a Hamiltonian that has spatial periodicity. Specifically, we consider a spin-12\frac{1}{2}21​ chain with equal xxxxxx and yyyyyy couplings and subject to a periodically varying magnetic field in the z\ifmmode^\elseˆ\fi\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{z}z\ifmmode^\elseˆ\fi direction or, equivalently, a tight-binding model of spinless fermions with a periodic local chemical potential, having period 2q2q2q, where qqq is a positive integer. For a linear quench of the strength of the magnetic field (or chemical potential) at a rate 1/\ensuremathτ1/\ensuremath{\tau}1/\ensuremathτ across a quantum critical point, we find that the density of defects thereby produced scales as 1/\ensuremathτq/(q+1)1/{\ensuremath{\tau}}^{q/(q+1)}1/\ensuremathτq/(q+1), deviating from the 1/\ensuremathτ1/\sqrt{\ensuremath{\tau}}1/\ensuremathτ​ scaling that is ubiquitous in a range of systems. We analyze this behavior by mapping the low-energy physics of the system to a set of fermionic two-level systems labeled by the lattice momentum kkk undergoing a nonlinear quench as well as by performing numerical simulations. We also show that if the magnetic field is a superposition of different periods, the power law depends only on the smallest period for very large values of \ensuremathτ\ensuremath{\tau}\ensuremathτ, although it may exhibit a crossover at intermediate values of \ensuremathτ\ensuremath{\tau}\ensuremathτ. Finally, for the case where a zzzzzz coupling is also present in the spin chain, or equivalently, where interactions are present in the fermionic system, we argue that the power associated with the scaling law depends on a combination of qqq and the interaction strength.
We introduce a one-dimensional version of the Kitaev model consisting of spins on a two-legged ladder and characterized by Z(2) invariants on the plaquettes of the ladder. We map the model to a fermionic system and identify the topological sectors associated with different Z2 patterns in terms of fermion occupation numbers. Within these different sectors, we investigate the effect of a linear quench across a quantum critical point. We study the dominant behavior of the system by employing a Landau-Zener-type analysis of the effective Hamiltonian in the low-energy subspace for which the effective quenching can sometimes be non-linear. We show that the quenching leads to a residual energy which scales as a power of the quenching rate, and that the power depends on the topological sectors and their symmetry properties in a non-trivial way. This behavior is consistent with the general theory of quantum quenching, but with the correlation length exponent nu being different in different sectors. Copyright (C) EPLA, 2010
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 introduce a one-dimensional version of the Kitaev model consisting of spins on a two-legged ladder and characterized by Z 2 invariants on the plaquettes of the ladder. We map the model to a fermionic system and identify the topological sectors associated with different Z 2 patterns in terms of fermion occupation numbers. Within these different sectors, we investigate the effect of a linear quench across a quantum critical point. We study the dominant behavior of the system by employing a Landau-Zener–type analysis of the effective Hamiltonian in the low-energy subspace for which the effective quenching can sometimes be non-linear. We show that the quenching leads to a residual energy which scales as a power of the quenching rate, and that the power depends on the topological sectors and their symmetry properties in a non-trivial way. This behavior is consistent with the general theory of quantum quenching, but with the correlation length exponent ν being different in different sectors.
We study adiabatic quantum quenches across a quantum multicritical point (MCP) using a quenching scheme that enables the system to hit the MCP along different paths. We show that the power-law scaling of the defect density with the rate of driving depends non-trivially on the path, i.e. , the exponent varies continuously with the parameter α that defines the path, up to a critical value α=α c ; on the other hand for α⩾α c , the scaling exponent saturates to a constant value. We show that dynamically generated and path (α)- dependent effective critical exponents associated with the quasicritical points lying close to the MCP (on the ferromagnetic side), where the energy-gap is minimum, lead to this continuously varying exponent. The scaling relations are established using the integrable transverse XY spin chain and generalized to a MCP associated with a d -dimensional quantum many-body systems (not reducible to two-level systems) using adiabatic perturbation theory. We also calculate the effective path-dependent dimensional shift d 0 (α) (or the shift in center of the impulse region) that appears in the scaling relation for special paths lying entirely in the paramagnetic phase. The numerically obtained results are in good agreement with analytical predictions.
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 scaling of the ground-state fidelity in neighborhoods of quantum critical points in a model of interacting spinfull fermions - a BCS-like model. Due to the exact diagonalizability of the model, in one and higher dimensions, scaling of the ground-state fidelity can be analyzed numerically with great accuracy, not only for small systems but also for macroscopic ones, together with the crossover region between them. Additionally, in one-dimensional case we have been able to derive a number of analytical formulae for fidelity and show that they accurately fit our numerical results; these results are reported in the article. Besides regular critical points and their neighborhoods, where well-known scaling laws are obeyed, there is the multi-critical point and critical points in its proximity where anomalous scaling behavior is found. We consider also scaling of fidelity in neighborhoods of critical points where fidelity oscillates strongly as the system size or the chemical potential is varied. Our results for a one-dimensional version of a BCS-like model are compared with those obtained by Rams and Damski in similar studies of a quantum spin chain - an anisotropic XY model in transverse magnetic field.