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:1103.1212 · Physical Review A

Density of defects and the scaling law of the entanglement entropy in quantum phase transition of one-dimensional spin systems induced by a quench

Banasri Basu, Pratul Bandyopadhyay, Priyadarshi Majumdar

Published Mar 21, 2011• 10 citations
arXivHTMLPDF

Abstract

We have studied quantum phase transition induced by a quench in different one-dimensional spin systems. Our analysis is based on the dynamical mechanism which envisages nonadiabaticity in the vicinity of the critical point. This causes spin fluctuation which leads to the random fluctuation of the Berry phase factor acquired by a spin state when the ground state of the system evolves in a closed path. The two-point correlation of this phase factor is associated with the probability of the formation of defects. In this framework, we have estimated the density of defects produced in several one-dimensional spin chains. At the critical region, the entanglement entropy of a block of $L$ spins with the rest of the system is also estimated which is found to increase logarithmically with $L$. The dependence on the quench time puts a constraint on the block size $L$. It is also pointed out that the Lipkin-Meshkov-Glick model in point-splitting regularized form appears as a combination of the $\mathit{XXX}$ model and Ising model with magnetic field in the negative $z$ axis. This unveils the underlying conformal symmetry at criticality which is lost in the sharp point limit. Our analysis shows that the density of defects as well as the scaling behavior of the entanglement entropy follows a universal behavior in all these systems.

DOI
10.1103/physreva.83.032312

Related papers

Scaling of entanglement entropy in a quantum phase transition in the transverse Ising model induced by a quench

Priyadarshi Majumdar, Pratul Bandyopadhyay

It is known that at the critical point of a zero-temperature quantum phase transition in a one-dimensional spin system the entanglement entropy of a block of $L$ spins with the rest of the system scales logarithmically with $L$ with a prefactor determined by the central charge of the relevant conformal field theory. When we introduce critical slowing down incorporating the Kibble-Zurek mechanism of defect formation induced by a quench, the implicit nonadiabatic transition disturbs the scaling behavior. We have shown that in this case the entanglement entropy also obeys a scaling law such that it increases logarithmically with $L$ but the prefactor depends on the quench time. This puts a constraint on the block size $L$ so that we cannot arbitrarily choose it. Thus, the entanglement entropy obeys the scaling law only in a restrictive sense due to the formation of defects.

Physical Review A•Jan 19, 2010•8 citations

Entropy of entanglement and correlations induced by a quench: Dynamics of a quantum phase transition in the quantum Ising model

Łukasz Cincio, Jacek Dziarmaga, Marek M. Rams +1 more

Quantum Ising model in one dimension is an exactly solvable example of a quantum phase transition. We investigate its behavior during a quench caused by a gradual turning off of the transverse bias field. The system is then driven at a fixed rate characterized by the quench time ${\ensuremath{\tau}}_{Q}$ across the critical point from a paramagnetic to ferromagnetic phase. In agreement with Kibble-Zurek mechanism (which recognizes that evolution is approximately adiabatic far away, but becomes approximately impulse sufficiently near the critical point), quantum state of the system after the transition exhibits a characteristic correlation length $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\xi}}$ proportional to the square root of the quench time ${\ensuremath{\tau}}_{Q}$: $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\xi}}=\sqrt{{\ensuremath{\tau}}_{Q}}$. The inverse of this correlation length is known to determine average density of defects (e.g., kinks) after the transition. In this paper, we show that this same $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\xi}}$ controls the entropy of entanglement, e.g., entropy of a block of $L$ spins that are entangled with the rest of the system after the transition from the paramagnetic ground state induced by the quench. For large $L$, this entropy saturates at $\frac{1}{6}\phantom{\rule{0.2em}{0ex}}{\mathrm{log}}_{2}\phantom{\rule{0.2em}{0ex}}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\xi}}$, as might have been expected from the Kibble-Zurek mechanism. Close to the critical point, the entropy saturates when the block size $L\ensuremath{\approx}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\xi}}$, but---in the subsequent evolution in the ferromagnetic phase---a somewhat larger length scale $l=\sqrt{{\ensuremath{\tau}}_{Q}}\phantom{\rule{0.2em}{0ex}}\mathrm{ln}\phantom{\rule{0.2em}{0ex}}{\ensuremath{\tau}}_{Q}$ develops as a result of a dephasing process that can be regarded as a quantum analog of phase ordering, and the entropy saturates when $L\ensuremath{\approx}l$. We also study the spin-spin correlation using both analytic methods and real time simulations with the Vidal algorithm. We find that at an instant when quench is crossing the critical point, ferromagnetic correlations decay exponentially with the dynamical correlation length $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\xi}}$, but (as for entropy of entanglement) in the following evolution length scale $l$ gradually develops. The correlation function becomes oscillatory at distances less than this scale. However, both the wavelength and the correlation length of these oscillations are still determined by $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\xi}}$. We also derive probability distribution for the number of kinks in a finite spin chain after the transition.

arXiv:cond-mat/0701768•Physical Review A•May 17, 2007•182 citations

Magnetic-field dependence of the entanglement entropy of one-dimensional spin systems in quantum phase transitions induced by a quench

Banasri Basu, Pratul Bandyopadhyay, Priyadarshi Majumdar

We study the magnetic-field dependence of the entanglement entropy in quantum phase transition induced by a quench of the $XX$, $XXX,$ and Lipkin-Meshkov-Glick (LMG) models. The entropy for a block of $L$ spins with the rest follows a logarithmic scaling law where the block size $L$ is restricted due to the dependence of the prefactor on the quench time. Within this restricted region the entropy undergoes a renormalization group (RG) flow. From the RG flow equation we have analytically determined the magnetic field dependence of the entropy. The anisotropy parameter dependence of the entropy for the $XY$ and the LMG models has also been studied in this framework. The results are found to be in excellent agreement with that obtained by other authors from numerical studies without any quench.

arXiv:1207.2267•Physical Review A•Aug 1, 2012•16 citations

Evolution of entanglement towards criticality in the XY model in a quantum phase transition induced by a quench

Banasri Basu, Pratul Bandyopadhyay, Priyadarshi Majumdar

We have studied here the evolution of entanglement towards criticality in the $\mathit{XY}$ model when quantum phase transition (QPT) is induced by a quench. To this end, we have utilized the relation of the Berry phase acquired by a spin state when it traverses a closed path at criticality with the concurrence measuring the entanglement for a pair of nearest-neighbor spins. It has been argued that when the spin state is rotated towards the generation of the closed path, the segments of noncyclic paths with open ends can be mapped onto cyclic paths enclosing flux lines associated with $q$-deformed monopoles. This essentially gives rise to the sequence of Berry phases attained by the quasispin states belonging to the quantum group ${\mathrm{SU}}_{q}(2)$ when rotated in a closed path. As the standard Berry phase at criticality is obtained when the time-dependent deformation parameter $q$ attains the value 1, this formulation helps us to analyze the evolution of entanglement towards criticality when QPT is induced by a quench. From this analysis we also formulate a scaling law for the entanglement entropy for a block of $L$ spins with the rest at off-criticality. As in QPT off-critical points correspond to thermal fluctuations, this formulation represents the evolution of entanglement towards criticality with temperature as $T\ensuremath{\rightarrow}0$.

Physical Review A•Aug 19, 2015•2 citations

Universal aspects in the behavior of the entanglement spectrum in one dimension: Scaling transition at the factorization point and ordered entangled structures

S. M. Giampaolo, Simone Montangero, Fabio Dell’Anno +2 more

We investigate the scaling of the entanglement spectrum and of the R\'enyi block entropies and determine its universal aspects in the ground state of critical and noncritical one-dimensional quantum spin models. In all cases, the scaling exhibits an oscillatory behavior that terminates at the factorization point and whose frequency is universal. Parity effects in the scaling of the R\'enyi entropies for gapless models at zero field are thus shown to be a particular case of such universal behavior. Likewise, the absence of oscillations for the Ising chain in transverse field is due to the vanishing value of the factorizing field for this particular model. In general, the transition occurring at the factorizing field between two different scaling regimes of the entanglement spectrum corresponds to a quantum transition to the formation of finite-range, ordered structures of quasidimers, quasitrimers, and quasipolymers. This entanglement-driven transition is superimposed to and independent of the long-range magnetic order in the broken symmetry phase. Therefore, it conforms to recent generalizations that identify and classify the quantum phases of matter according to the structure of ground-state entanglement patterns. We characterize this form of quantum order by a global order parameter of entanglement defined as the integral, over blocks of all lengths, of the R\'enyi entropy of infinite order. Equivalently, it can be defined as the integral of the bipartite single-copy or geometric entanglement. The global entanglement order parameter remains always finite at fields below the factorization point and vanishes identically above it.

arXiv:1307.7718•Physical Review B•Sep 30, 2013•40 citations

The geometric phase and the dynamics of quantum phase transition induced by a linear quench

Banasri Basu, P Bandyopadhyay

We have analysed here the role of geometric phase in the dynamical mechanism of quantum phase transition in the transverse Ising model. We have investigated the system when it is driven at a fixed rate characterized by a quench time τ q across the critical point from a paramagnetic to ferromagnetic phase. Our argument is based on the fact that the spin fluctuation occurring during the critical slowing down causes a random fluctuation in the ground-state geometric phase at the critical regime. The correlation function of the random geometric phase determines the excitation probability of the quasiparticles, which are excited during the transition from the inital paramagnetic to the ferromagnetic phase. This helps us to evaluate the number density of the kinks formed during the transition, which is found to scale as . In addition, we have also estimated the spin–spin correlation at criticality.

arXiv:1006.5814•Journal of Physics A Mathematical and Theoretical•Aug 12, 2010•3 citations
We have studied quantum phase transition induced by a quench in different one-dimensional spin systems. Our analysis is based on the dynamical mechanism which envisages nonadiabaticity in the vicinity of the critical point. This causes spin fluctuation which leads to the random fluctuation of the Berry phase factor acquired by a spin state when the ground state of the system evolves in a closed path. The two-point correlation of this phase factor is associated with the probability of the formation of defects. In this framework, we have estimated the density of defects produced in several one-dimensional spin chains. At the critical region, the entanglement entropy of a block of LLL spins with the rest of the system is also estimated which is found to increase logarithmically with LLL. The dependence on the quench time puts a constraint on the block size LLL. It is also pointed out that the Lipkin-Meshkov-Glick model in point-splitting regularized form appears as a combination of the XXX\mathit{XXX}XXX model and Ising model with magnetic field in the negative zzz axis. This unveils the underlying conformal symmetry at criticality which is lost in the sharp point limit. Our analysis shows that the density of defects as well as the scaling behavior of the entanglement entropy follows a universal behavior in all these systems.
It is known that at the critical point of a zero-temperature quantum phase transition in a one-dimensional spin system the entanglement entropy of a block of LLL spins with the rest of the system scales logarithmically with LLL with a prefactor determined by the central charge of the relevant conformal field theory. When we introduce critical slowing down incorporating the Kibble-Zurek mechanism of defect formation induced by a quench, the implicit nonadiabatic transition disturbs the scaling behavior. We have shown that in this case the entanglement entropy also obeys a scaling law such that it increases logarithmically with LLL but the prefactor depends on the quench time. This puts a constraint on the block size LLL so that we cannot arbitrarily choose it. Thus, the entanglement entropy obeys the scaling law only in a restrictive sense due to the formation of defects.
Quantum Ising model in one dimension is an exactly solvable example of a quantum phase transition. We investigate its behavior during a quench caused by a gradual turning off of the transverse bias field. The system is then driven at a fixed rate characterized by the quench time ${\ensuremath{\tau}}_{Q}$ across the critical point from a paramagnetic to ferromagnetic phase. In agreement with Kibble-Zurek mechanism (which recognizes that evolution is approximately adiabatic far away, but becomes approximately impulse sufficiently near the critical point), quantum state of the system after the transition exhibits a characteristic correlation length $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\xi}}$ proportional to the square root of the quench time ${\ensuremath{\tau}}_{Q}$: $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\xi}}=\sqrt{{\ensuremath{\tau}}_{Q}}$. The inverse of this correlation length is known to determine average density of defects (e.g., kinks) after the transition. In this paper, we show that this same $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\xi}}$ controls the entropy of entanglement, e.g., entropy of a block of LLL spins that are entangled with the rest of the system after the transition from the paramagnetic ground state induced by the quench. For large LLL, this entropy saturates at $\frac{1}{6}\phantom{\rule{0.2em}{0ex}}{\mathrm{log}}_{2}\phantom{\rule{0.2em}{0ex}}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\xi}}$, as might have been expected from the Kibble-Zurek mechanism. Close to the critical point, the entropy saturates when the block size $L\ensuremath{\approx}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\xi}}$, but---in the subsequent evolution in the ferromagnetic phase---a somewhat larger length scale $l=\sqrt{{\ensuremath{\tau}}_{Q}}\phantom{\rule{0.2em}{0ex}}\mathrm{ln}\phantom{\rule{0.2em}{0ex}}{\ensuremath{\tau}}_{Q}$ develops as a result of a dephasing process that can be regarded as a quantum analog of phase ordering, and the entropy saturates when $L\ensuremath{\approx}l$. We also study the spin-spin correlation using both analytic methods and real time simulations with the Vidal algorithm. We find that at an instant when quench is crossing the critical point, ferromagnetic correlations decay exponentially with the dynamical correlation length $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\xi}}$, but (as for entropy of entanglement) in the following evolution length scale lll gradually develops. The correlation function becomes oscillatory at distances less than this scale. However, both the wavelength and the correlation length of these oscillations are still determined by $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\xi}}$. We also derive probability distribution for the number of kinks in a finite spin chain after the transition.
We study the magnetic-field dependence of the entanglement entropy in quantum phase transition induced by a quench of the XXXXXX, XXX,XXX,XXX, and Lipkin-Meshkov-Glick (LMG) models. The entropy for a block of LLL spins with the rest follows a logarithmic scaling law where the block size LLL is restricted due to the dependence of the prefactor on the quench time. Within this restricted region the entropy undergoes a renormalization group (RG) flow. From the RG flow equation we have analytically determined the magnetic field dependence of the entropy. The anisotropy parameter dependence of the entropy for the XYXYXY and the LMG models has also been studied in this framework. The results are found to be in excellent agreement with that obtained by other authors from numerical studies without any quench.
We have studied here the evolution of entanglement towards criticality in the XY\mathit{XY}XY model when quantum phase transition (QPT) is induced by a quench. To this end, we have utilized the relation of the Berry phase acquired by a spin state when it traverses a closed path at criticality with the concurrence measuring the entanglement for a pair of nearest-neighbor spins. It has been argued that when the spin state is rotated towards the generation of the closed path, the segments of noncyclic paths with open ends can be mapped onto cyclic paths enclosing flux lines associated with qqq-deformed monopoles. This essentially gives rise to the sequence of Berry phases attained by the quasispin states belonging to the quantum group SUq(2){\mathrm{SU}}_{q}(2)SUq​(2) when rotated in a closed path. As the standard Berry phase at criticality is obtained when the time-dependent deformation parameter qqq attains the value 1, this formulation helps us to analyze the evolution of entanglement towards criticality when QPT is induced by a quench. From this analysis we also formulate a scaling law for the entanglement entropy for a block of LLL spins with the rest at off-criticality. As in QPT off-critical points correspond to thermal fluctuations, this formulation represents the evolution of entanglement towards criticality with temperature as $T\ensuremath{\rightarrow}0$.
We investigate the scaling of the entanglement spectrum and of the R\'enyi block entropies and determine its universal aspects in the ground state of critical and noncritical one-dimensional quantum spin models. In all cases, the scaling exhibits an oscillatory behavior that terminates at the factorization point and whose frequency is universal. Parity effects in the scaling of the R\'enyi entropies for gapless models at zero field are thus shown to be a particular case of such universal behavior. Likewise, the absence of oscillations for the Ising chain in transverse field is due to the vanishing value of the factorizing field for this particular model. In general, the transition occurring at the factorizing field between two different scaling regimes of the entanglement spectrum corresponds to a quantum transition to the formation of finite-range, ordered structures of quasidimers, quasitrimers, and quasipolymers. This entanglement-driven transition is superimposed to and independent of the long-range magnetic order in the broken symmetry phase. Therefore, it conforms to recent generalizations that identify and classify the quantum phases of matter according to the structure of ground-state entanglement patterns. We characterize this form of quantum order by a global order parameter of entanglement defined as the integral, over blocks of all lengths, of the R\'enyi entropy of infinite order. Equivalently, it can be defined as the integral of the bipartite single-copy or geometric entanglement. The global entanglement order parameter remains always finite at fields below the factorization point and vanishes identically above it.
We have analysed here the role of geometric phase in the dynamical mechanism of quantum phase transition in the transverse Ising model. We have investigated the system when it is driven at a fixed rate characterized by a quench time τ q across the critical point from a paramagnetic to ferromagnetic phase. Our argument is based on the fact that the spin fluctuation occurring during the critical slowing down causes a random fluctuation in the ground-state geometric phase at the critical regime. The correlation function of the random geometric phase determines the excitation probability of the quasiparticles, which are excited during the transition from the inital paramagnetic to the ferromagnetic phase. This helps us to evaluate the number density of the kinks formed during the transition, which is found to scale as . In addition, we have also estimated the spin–spin correlation at criticality.