We determine dynamical response functions in the scaling limit of the quantum Ising chain on the half-line in the presence of a boundary magnetic field. Using a spectral representation in terms of infinite volume form factors and a boundary state, we derive an expansion for the correlator that is found to be rapidly convergent as long as , where ξ is the correlation length. At sufficiently late times we observe oscillatory behaviour of the correlations arbitrarily far away from the boundary. We investigate the effects of the boundary bound state that is present for a range of boundary magnetic fields.We determine dynamical response functions in the scaling limit of the quantum Ising chain on the half-line in the presence of a boundary magnetic field. Using a spectral representation in terms of infinite volume form factors and a boundary state, we derive an expansion for the correlator that is found to be rapidly convergent as long as , where ξ is the correlation length. At sufficiently late times we observe oscillatory behaviour of the correlations arbitrarily far away from the boundary. We investigate the effects of the boundary bound state that is present for a range of boundary magnetic fields.

arXiv:0709.1809Journal of Statistical Mechanics Theory and ExperimentNov 12, 200713 citations

Dynamics of 2D Ising Model in linearly varying magnetic field

Na Xu, Cheng‐Wei Liu, Anatoli Polkovnikov +1 more

Bulletin of the American Physical SocietyMar 4, 20140 citations

Response of Ising systems to oscillating and pulsed fields: Hysteresis, ac, and pulse susceptibility

Muktish Acharyya, Bikas K. Chakrabarti

We have studied, using Monte Carlo (MC) simulation for ferromagnetic Ising systems in one to four dimensions and solving numerically the mean-field (MF) equation of motion, the nature of the response magnetization m(t) of an Ising system in the presence of a periodically varying external field [h(t)= ${\mathit{h}}_{0}$ cos(\ensuremath{\omega}t)]. From these studies, we determine the m-h loop or hysteresis loop area A(=\ensuremath{\oint}mdh) and the dynamic order parameter Q(=\ensuremath{\oint}mdt) and investigate their variations with the frequency (\ensuremath{\omega}) and amplitude ( ${\mathit{h}}_{0}$ ) of the applied external magnetic field and the temperature (T) of the system. The variations in A are fitted to a scaling form, assumed to be valid over a wide range of parameter (\ensuremath{\omega}, ${\mathit{h}}_{0}$ ,T) values, and the best-fit exponents are obtained in all three dimensions (D=2,3,4). The scaling function is Lorentzian in the MF case and is exponentially decaying, with an initial power law, for the MC cases. The dynamic phase boundary (in the ${\mathit{h}}_{0}$ -T plane) is found to be frequency dependent and the transition (from Q\ensuremath{\ne}0 for low T and ${\mathit{h}}_{0}$ to Q=0 for high T and ${\mathit{h}}_{0}$ ) across the boundary crosses over from a discontinuous to a continuous one at a tricritical point. These boundaries are determined in various cases.We find that the response can be generally expressed as m(t)=P(\ensuremath{\omega}(t-${\mathrm{\ensuremath{\tau}}}_{\mathrm{eff}}$)) where P denotes a periodic function with the same frequency \ensuremath{\omega} of the perturbing field and ${\mathrm{\ensuremath{\tau}}}_{\mathit{e}\mathit{f}\mathit{f}}$( ${\mathit{h}}_{0}$ ,\ensuremath{\omega},T) denotes the effective delay. We established that this effective delay ${\mathrm{\ensuremath{\tau}}}_{\mathit{e}\mathit{f}\mathit{f}}$ of the response is the crucial term and it practically determines all the above observations for A, Q, etc. Investigating the nature of the in-phase (\ensuremath{\chi}\ensuremath{'}) and the out-of-phase (\ensuremath{\chi}\ensuremath{'}\ensuremath{'}) susceptibility, defined as \ensuremath{\chi}\ensuremath{'}=( ${\mathit{m}}_{0}$ / ${\mathit{h}}_{0}$ )cos(\ensuremath{\varphi}) and \ensuremath{\chi}\ensuremath{'}\ensuremath{'}=( ${\mathit{m}}_{0}$ / ${\mathit{h}}_{0}$ )sin(\ensuremath{\varphi}); \ensuremath{\varphi}=\ensuremath{\omega}${\mathrm{\ensuremath{\tau}}}_{\mathit{e}\mathit{f}\mathit{f}}$ [and ${\mathit{m}}_{0}$ is the amplitude of m(t)], we find that the loop area A is directly given by \ensuremath{\chi}\ensuremath{'}\ensuremath{'} and also the temperature variation of \ensuremath{\chi}\ensuremath{'}\ensuremath{'}, at fixed \ensuremath{\omega} and ${\mathit{h}}_{0}$ , gives a prominent peak at the dynamic transition point. We have also studied the behavior of the response magnetization by the application of a short-duration (compared with the relaxation time) pulsed magnetic field.Here, we observed (both in the MC and MF cases) that the width ratio (of the half-width and the width of the response magnetization and of the pulsed field, respectively) and the susceptibility (the ratio of excess magnetization over its equilibrium value and the height of the pulsed field) both show sharp peaks at the order-disorder (ferromagnetic-paramagnetic) transition point. We have also studied similar response behavior of Ising systems in the presence of time-varying longitudinal and transverse fields, solving numerically the mean-field equation of motion. We have again studied here the nature of the dynamic phase transition and the behavior of the ac susceptibility (both longitudinal and transverse) across the dynamic phase boundary. For a short-duration pulse of the transverse field, the width ratio and the pulse susceptibility are again seen to diverge at the order-disorder transition point.We have studied, using Monte Carlo (MC) simulation for ferromagnetic Ising systems in one to four dimensions and solving numerically the mean-field (MF) equation of motion, the nature of the response magnetization m(t) of an Ising system in the presence of a periodically varying external field [h(t)=${\mathit{h}}_{0}$cos(\ensuremath{\omega}t)]. From these studies, we determine the m-h loop or hysteresis loop area A(=\ensuremath{\oint}mdh) and the dynamic order parameter Q(=\ensuremath{\oint}mdt) and investigate their variations with the frequency (\ensuremath{\omega}) and amplitude (${\mathit{h}}_{0}$) of the applied external magnetic field and the temperature (T) of the system. The variations in A are fitted to a scaling form, assumed to be valid over a wide range of parameter (\ensuremath{\omega},${\mathit{h}}_{0}$,T) values, and the best-fit exponents are obtained in all three dimensions (D=2,3,4). The scaling function is Lorentzian in the MF case and is exponentially decaying, with an initial power law, for the MC cases. The dynamic phase boundary (in the ${\mathit{h}}_{0}$-T plane) is found to be frequency dependent and the transition (from Q\ensuremath{\ne}0 for low T and ${\mathit{h}}_{0}$ to Q=0 for high T and ${\mathit{h}}_{0}$) across the boundary crosses over from a discontinuous to a continuous one at a tricritical point. These boundaries are determined in various cases.We find that the response can be generally expressed as m(t)=P(\ensuremath{\omega}(t-${\mathrm{\ensuremath{\tau}}}_{\mathrm{eff}}$)) where P denotes a periodic function with the same frequency \ensuremath{\omega} of the perturbing field and ${\mathrm{\ensuremath{\tau}}}_{\mathit{e}\mathit{f}\mathit{f}}$(${\mathit{h}}_{0}$,\ensuremath{\omega},T) denotes the effective delay. We established that this effective delay ${\mathrm{\ensuremath{\tau}}}_{\mathit{e}\mathit{f}\mathit{f}}$ of the response is the crucial term and it practically determines all the above observations for A, Q, etc. Investigating the nature of the in-phase (\ensuremath{\chi}\ensuremath{'}) and the out-of-phase (\ensuremath{\chi}\ensuremath{'}\ensuremath{'}) susceptibility, defined as \ensuremath{\chi}\ensuremath{'}=(${\mathit{m}}_{0}$/${\mathit{h}}_{0}$)cos(\ensuremath{\varphi}) and \ensuremath{\chi}\ensuremath{'}\ensuremath{'}=(${\mathit{m}}_{0}$/${\mathit{h}}_{0}$)sin(\ensuremath{\varphi}); \ensuremath{\varphi}=\ensuremath{\omega}${\mathrm{\ensuremath{\tau}}}_{\mathit{e}\mathit{f}\mathit{f}}$ [and ${\mathit{m}}_{0}$ is the amplitude of m(t)], we find that the loop area A is directly given by \ensuremath{\chi}\ensuremath{'}\ensuremath{'} and also the temperature variation of \ensuremath{\chi}\ensuremath{'}\ensuremath{'}, at fixed \ensuremath{\omega} and ${\mathit{h}}_{0}$, gives a prominent peak at the dynamic transition point. We have also studied the behavior of the response magnetization by the application of a short-duration (compared with the relaxation time) pulsed magnetic field.Here, we observed (both in the MC and MF cases) that the width ratio (of the half-width and the width of the response magnetization and of the pulsed field, respectively) and the susceptibility (the ratio of excess magnetization over its equilibrium value and the height of the pulsed field) both show sharp peaks at the order-disorder (ferromagnetic-paramagnetic) transition point. We have also studied similar response behavior of Ising systems in the presence of time-varying longitudinal and transverse fields, solving numerically the mean-field equation of motion. We have again studied here the nature of the dynamic phase transition and the behavior of the ac susceptibility (both longitudinal and transverse) across the dynamic phase boundary. For a short-duration pulse of the transverse field, the width ratio and the pulse susceptibility are again seen to diverge at the order-disorder transition point.

Physical review. B, Condensed matterSep 1, 1995232 citations

Magnetic responses in the presence of a partial magnetic field on the rectangular Ising lattice

Shu-Chiuan Chang, Masuo Suzuki

Physica A Statistical Mechanics and its ApplicationsJun 16, 20043 citations

Dynamic response of an Ising system to a pulsed field

Muktish Acharyya, Jayanta K. Bhattacharjee, Bikas K. Chakrabarti

The dynamical response to a pulsed magnetic field has been studied here both using Monte Carlo simulation and by solving numerically the mean-field dynamical equation of motion for the Ising model. The ratio ${\mathrm{R}}_{\mathrm{p}}$ of the response magnetization half-width to the width of the external field pulse has been observed to diverge and pulse susceptibility ${\mathrm{\ensuremath{\chi}}}_{\mathrm{p}}$ (ratio of the response magnetization peak height and the pulse height) gives a peak near the order-disorder transition temperature ${\mathrm{T}}_{\mathrm{c}}$ (for the unperturbed system). The Monte Carlo results for the Ising system on a square lattice show that ${\mathrm{R}}_{\mathrm{p}}$ diverges at ${\mathrm{T}}_{\mathrm{c}}$ , with the exponent \ensuremath{\nu}z\ensuremath{\cong}2.0, while ${\mathrm{\ensuremath{\chi}}}_{\mathrm{p}}$ shows a peak at ${\mathrm{T}}_{\mathrm{c}}^{\mathrm{e}}$ , which is a function of the field pulse width \ensuremath{\delta}t. A finite-size (in time) scaling analysis shows that ${\mathrm{T}}_{\mathrm{c}}^{\mathrm{e}}$ = ${\mathrm{T}}_{\mathrm{c}}$ +C(\ensuremath{\delta}t${)}^{\mathrm{\ensuremath{-}}1\mathrm{/}\mathrm{x}}$, with x=\ensuremath{\nu}z\ensuremath{\cong}2.0. The mean-field results show that both the divergence of R and the peak in ${\mathrm{\ensuremath{\chi}}}_{\mathrm{p}}$ occur at the mean-field transition temperature, while the peak height in ${\mathrm{\ensuremath{\chi}}}_{\mathrm{p}}$\ensuremath{\sim}(\ensuremath{\delta}t ${)}^{\mathrm{y}}$ , y\ensuremath{\cong}1 for small values of \ensuremath{\delta}t. These results also compare well with an approximate analytical solution of the mean-field equation of motion.The dynamical response to a pulsed magnetic field has been studied here both using Monte Carlo simulation and by solving numerically the mean-field dynamical equation of motion for the Ising model. The ratio ${\mathrm{R}}_{\mathrm{p}}$ of the response magnetization half-width to the width of the external field pulse has been observed to diverge and pulse susceptibility ${\mathrm{\ensuremath{\chi}}}_{\mathrm{p}}$ (ratio of the response magnetization peak height and the pulse height) gives a peak near the order-disorder transition temperature ${\mathrm{T}}_{\mathrm{c}}$ (for the unperturbed system). The Monte Carlo results for the Ising system on a square lattice show that ${\mathrm{R}}_{\mathrm{p}}$ diverges at ${\mathrm{T}}_{\mathrm{c}}$, with the exponent \ensuremath{\nu}z\ensuremath{\cong}2.0, while ${\mathrm{\ensuremath{\chi}}}_{\mathrm{p}}$ shows a peak at ${\mathrm{T}}_{\mathrm{c}}^{\mathrm{e}}$, which is a function of the field pulse width \ensuremath{\delta}t. A finite-size (in time) scaling analysis shows that ${\mathrm{T}}_{\mathrm{c}}^{\mathrm{e}}$=${\mathrm{T}}_{\mathrm{c}}$+C(\ensuremath{\delta}t${)}^{\mathrm{\ensuremath{-}}1\mathrm{/}\mathrm{x}}$, with x=\ensuremath{\nu}z\ensuremath{\cong}2.0. The mean-field results show that both the divergence of R and the peak in ${\mathrm{\ensuremath{\chi}}}_{\mathrm{p}}$ occur at the mean-field transition temperature, while the peak height in ${\mathrm{\ensuremath{\chi}}}_{\mathrm{p}}$\ensuremath{\sim}(\ensuremath{\delta}t${)}^{\mathrm{y}}$, y\ensuremath{\cong}1 for small values of \ensuremath{\delta}t. These results also compare well with an approximate analytical solution of the mean-field equation of motion.

arXiv:cond-mat/9611182Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topicsMar 1, 199718 citations

{\mathit{h}}_{0}

cos(\ensuremath{\omega}t)]. From these studies, we determine the m-h loop or hysteresis loop area A(=\ensuremath{\oint}mdh) and the dynamic order parameter Q(=\ensuremath{\oint}mdt) and investigate their variations with the frequency (\ensuremath{\omega}) and amplitude (

{\mathit{h}}_{0}

) of the applied external magnetic field and the temperature (T) of the system. The variations in A are fitted to a scaling form, assumed to be valid over a wide range of parameter (\ensuremath{\omega},

{\mathit{h}}_{0}

,T) values, and the best-fit exponents are obtained in all three dimensions (D=2,3,4). The scaling function is Lorentzian in the MF case and is exponentially decaying, with an initial power law, for the MC cases. The dynamic phase boundary (in the

{\mathit{h}}_{0}

-T plane) is found to be frequency dependent and the transition (from Q\ensuremath{\ne}0 for low T and

{\mathit{h}}_{0}

to Q=0 for high T and

{\mathit{h}}_{0}

) across the boundary crosses over from a discontinuous to a continuous one at a tricritical point. These boundaries are determined in various cases.We find that the response can be generally expressed as m(t)=P(\ensuremath{\omega}(t-${\mathrm{\ensuremath{\tau}}}_{\mathrm{eff}}$)) where P denotes a periodic function with the same frequency \ensuremath{\omega} of the perturbing field and ${\mathrm{\ensuremath{\tau}}}_{\mathit{e}\mathit{f}\mathit{f}}$(

{\mathit{h}}_{0}

,\ensuremath{\omega},T) denotes the effective delay. We established that this effective delay ${\mathrm{\ensuremath{\tau}}}_{\mathit{e}\mathit{f}\mathit{f}}$ of the response is the crucial term and it practically determines all the above observations for A, Q, etc. Investigating the nature of the in-phase (\ensuremath{\chi}\ensuremath{'}) and the out-of-phase (\ensuremath{\chi}\ensuremath{'}\ensuremath{'}) susceptibility, defined as \ensuremath{\chi}\ensuremath{'}=(

{\mathit{m}}_{0}

{\mathit{h}}_{0}

)cos(\ensuremath{\varphi}) and \ensuremath{\chi}\ensuremath{'}\ensuremath{'}=(

{\mathit{m}}_{0}

{\mathit{h}}_{0}

)sin(\ensuremath{\varphi}); \ensuremath{\varphi}=\ensuremath{\omega}${\mathrm{\ensuremath{\tau}}}_{\mathit{e}\mathit{f}\mathit{f}}$ [and

{\mathit{m}}_{0}

is the amplitude of m(t)], we find that the loop area A is directly given by \ensuremath{\chi}\ensuremath{'}\ensuremath{'} and also the temperature variation of \ensuremath{\chi}\ensuremath{'}\ensuremath{'}, at fixed \ensuremath{\omega} and

{\mathit{h}}_{0}

, gives a prominent peak at the dynamic transition point. We have also studied the behavior of the response magnetization by the application of a short-duration (compared with the relaxation time) pulsed magnetic field.Here, we observed (both in the MC and MF cases) that the width ratio (of the half-width and the width of the response magnetization and of the pulsed field, respectively) and the susceptibility (the ratio of excess magnetization over its equilibrium value and the height of the pulsed field) both show sharp peaks at the order-disorder (ferromagnetic-paramagnetic) transition point. We have also studied similar response behavior of Ising systems in the presence of time-varying longitudinal and transverse fields, solving numerically the mean-field equation of motion. We have again studied here the nature of the dynamic phase transition and the behavior of the ac susceptibility (both longitudinal and transverse) across the dynamic phase boundary. For a short-duration pulse of the transverse field, the width ratio and the pulse susceptibility are again seen to diverge at the order-disorder transition point.

{\mathrm{R}}_{\mathrm{p}}

of the response magnetization half-width to the width of the external field pulse has been observed to diverge and pulse susceptibility ${\mathrm{\ensuremath{\chi}}}_{\mathrm{p}}$ (ratio of the response magnetization peak height and the pulse height) gives a peak near the order-disorder transition temperature

{\mathrm{T}}_{\mathrm{c}}

(for the unperturbed system). The Monte Carlo results for the Ising system on a square lattice show that

{\mathrm{R}}_{\mathrm{p}}

diverges at

{\mathrm{T}}_{\mathrm{c}}

, with the exponent \ensuremath{\nu}z\ensuremath{\cong}2.0, while ${\mathrm{\ensuremath{\chi}}}_{\mathrm{p}}$ shows a peak at

{\mathrm{T}}_{\mathrm{c}}^{\mathrm{e}}

, which is a function of the field pulse width \ensuremath{\delta}t. A finite-size (in time) scaling analysis shows that

{\mathrm{T}}_{\mathrm{c}}^{\mathrm{e}}

{\mathrm{T}}_{\mathrm{c}}

+C(\ensuremath{\delta}t${)}^{\mathrm{\ensuremath{-}}1\mathrm{/}\mathrm{x}}$, with x=\ensuremath{\nu}z\ensuremath{\cong}2.0. The mean-field results show that both the divergence of R and the peak in ${\mathrm{\ensuremath{\chi}}}_{\mathrm{p}}$ occur at the mean-field transition temperature, while the peak height in ${\mathrm{\ensuremath{\chi}}}_{\mathrm{p}}$\ensuremath{\sim}(\ensuremath{\delta}t

{)}^{\mathrm{y}}

, y\ensuremath{\cong}1 for small values of \ensuremath{\delta}t. These results also compare well with an approximate analytical solution of the mean-field equation of motion.