We study non-equilibrium dynamics induced by a quench of a parameter near a quantum phase transition in the Kitaev model. On the basis of an exact solution for the time-dependent Schrödinger equation, we show that the density of excitations and residual energy after a quench scale with a quench rate τ −1 as τ −3/4 and τ −5/4 respectively, when a parameter is quenched from a gapped phase toward the quantum critical point. These unusual scaling laws originate from an anisotropic nature of the quantum critical point in the Kitaev model. Gerenalizing these results, we provide novel scaling laws for a generic anisotropic quantum critical point in d -dimensional systems.We study non-equilibrium dynamics induced by a quench of a parameter near a quantum phase transition in the Kitaev model. On the basis of an exact solution for the time-dependent Schrödinger equation, we show that the density of excitations and residual energy after a quench scale with a quench rate τ −1 as τ −3/4 and τ −5/4 respectively, when a parameter is quenched from a gapped phase toward the quantum critical point. These unusual scaling laws originate from an anisotropic nature of the quantum critical point in the Kitaev model. Gerenalizing these results, we provide novel scaling laws for a generic anisotropic quantum critical point in d -dimensional systems.

Journal of Physics Conference SeriesMay 1, 20110 citations

Critical behavior of the quantum spin- anisotropic Heisenberg model

J. Ricardo de Sousa

Physica A Statistical Mechanics and its ApplicationsOct 1, 19985 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.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:1009.0323Physical Review BNov 18, 201024 citations

Phase transitions for the anisotropic Heisenberg model

Harvey S. Leff, Michael Flicker, Kenneth Millard

Physics Letters AMay 1, 19713 citations

Universal dynamics of the superradiant phase transition in the anisotropic quantum Rabi model

Xunda Jiang, Bo Lü, Chengyin Han +5 more

We investigate the universal nonequilibrium dynamics of the superradiant phase transition in the anisotropic quantum Rabi model. By introducing position and momentum operators, we find the generators and the displaced operators to diagonalize the anisotropic Hamiltonian, and then obtain the ground states and their excitation gaps. According to the Kibble-Zurek mechanism, we find that the excitation gap near the critical point vanishes as $\ensuremath{\omega}\ensuremath{\sim}{|\ensuremath{\epsilon}|}^{\ensuremath{\nu}z}$ with $\ensuremath{\nu}z=1/2$, and the length scale diverges according to $\ensuremath{\zeta}\ensuremath{\sim}{\ensuremath{\epsilon}}^{\ensuremath{-}\ensuremath{\nu}}$, which determines the universality class $(z=2,\ensuremath{\nu}=1/4)$ of the anisotropic quantum Rabi model. Moreover, by simulating the real-time dynamics across the critical point, we numerically extract the critical exponents from the phase transition delay ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{b}}_{d}\ensuremath{\sim}{\ensuremath{\tau}}_{Q}^{\ensuremath{-}1/(1+\ensuremath{\nu}z)}$ and the diverging length scale $\mathrm{\ensuremath{\Delta}}x$, $\mathrm{\ensuremath{\Delta}}p\ensuremath{\sim}{\ensuremath{\tau}}_{Q}^{\ensuremath{\nu}/(1+\ensuremath{\nu}z)}$ (where ${\ensuremath{\tau}}_{Q}$ is the quench time). The numerical critical exponents are well consistent with the analytical ones. Our study provides a dynamic way to explore universal critical behaviors in the anisotropic quantum Rabi model.We investigate the universal nonequilibrium dynamics of the superradiant phase transition in the anisotropic quantum Rabi model. By introducing position and momentum operators, we find the generators and the displaced operators to diagonalize the anisotropic Hamiltonian, and then obtain the ground states and their excitation gaps. According to the Kibble-Zurek mechanism, we find that the excitation gap near the critical point vanishes as $\ensuremath{\omega}\ensuremath{\sim}{|\ensuremath{\epsilon}|}^{\ensuremath{\nu}z}$ with $\ensuremath{\nu}z=1/2$, and the length scale diverges according to $\ensuremath{\zeta}\ensuremath{\sim}{\ensuremath{\epsilon}}^{\ensuremath{-}\ensuremath{\nu}}$, which determines the universality class $(z=2,\ensuremath{\nu}=1/4)$ of the anisotropic quantum Rabi model. Moreover, by simulating the real-time dynamics across the critical point, we numerically extract the critical exponents from the phase transition delay ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{b}}_{d}\ensuremath{\sim}{\ensuremath{\tau}}_{Q}^{\ensuremath{-}1/(1+\ensuremath{\nu}z)}$ and the diverging length scale $\mathrm{\ensuremath{\Delta}}x$, $\mathrm{\ensuremath{\Delta}}p\ensuremath{\sim}{\ensuremath{\tau}}_{Q}^{\ensuremath{\nu}/(1+\ensuremath{\nu}z)}$ (where ${\ensuremath{\tau}}_{Q}$ is the quench time). The numerical critical exponents are well consistent with the analytical ones. Our study provides a dynamic way to explore universal critical behaviors in the anisotropic quantum Rabi model.

arXiv:2009.11047Physical Review AOct 13, 20219 citations

Relaxation in an anisotropic Heisenberg model

Ivana Dimitrova, Niklas Jepsen, Jesse Amato-Grill +4 more

Bulletin of the American Physical SocietyJun 4, 20200 citations

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.