Physical review. B, Condensed matter

Critical scaling of frequency-dependent magnetic susceptibility in YBa2 Cu3 O7 single crystals near the vortex-glass transition

D. S. Reed, N.-C. Yeh, Wei‐Teh Jiang, U. Kriplani, D. A. Beam, Frederic Holtzberg

Published Feb 1, 1994• 33 citations

Abstract

A demonstration of critical scaling of the magnetic susceptibility in a ${\mathrm{YBa}}_{2}$${\mathrm{Cu}}_{3}$${\mathrm{O}}_{7}$ single crystal is performed for 350 frequencies (f) from 1 kHz to 2 MHz. Near the vortex-glass melting temperature ${\mathit{T}}_{\mathit{M}}$(H) and in the linear-response limit, the susceptibility \ensuremath{\chi}\ensuremath{\equiv}\ensuremath{\Vert}\ensuremath{\chi}\ensuremath{\Vert}${\mathit{e}}_{\mathrm{\ensuremath{\chi}}}^{\mathit{i}(\mathrm{\ensuremath{\pi}}\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\varphi}}}$) follows the scaling relations \ensuremath{\Vert}\ensuremath{\chi}\ensuremath{\Vert}\ensuremath{\sim}\ensuremath{\Vert}1-(T/${\mathit{T}}_{\mathit{M}}$)${\mathrm{\ensuremath{\Vert}}}^{\ensuremath{\nu}/2}$ for f\ensuremath{\rightarrow}0, \ensuremath{\Vert}\ensuremath{\chi}\ensuremath{\Vert}\ensuremath{\sim}${\mathit{f}}^{1/(2\mathit{z})}$ and ${\mathrm{\ensuremath{\varphi}}}_{\mathrm{\ensuremath{\chi}}}$=\ensuremath{\pi}/4z for T\ensuremath{\rightarrow}${\mathit{T}}_{\mathit{M}}$. The static and dynamic exponents (\ensuremath{\nu}\ensuremath{\approxeq}2/3 and z\ensuremath{\approxeq}3) as well as ${\mathit{T}}_{\mathit{M}}$(H) are found to be in excellent agreement with our previous dc and ac transport measurements, lending strong support for a second-order phase transition.

DOI: 10.1103/physrevb.49.4384

Related papers

Bose-Glass Phase in Twinned YBa2Cu3O7−δ

S. A. Grigera, E. Morré, E. J. Osquiguil +3 more

arXiv:cond-mat/9806376Physical Review LettersSep 14, 1998100 citations

Experimental verification of activated critical dynamics in the d=3 random-field Ising model

A. E. Nash, A. R. King, Vincent Jaccarino

Physical review. B, Condensed matterJan 1, 199118 citations

Dynamical scaling in Ising and vector spin glasses

Helmut G. Katzgraber, Ian A. Campbell

arXiv:cond-mat/0504082Physical Review BJul 25, 200533 citations

Critical behavior of the three-dimensional random-field Ising model: Two-exponent scaling and discontinuous transition

Heiko Rieger

arXiv:cond-mat/9503041Physical review. B, Condensed matterSep 1, 1995138 citations

High-frequency magnetic field on crystal field diluted S = 1 Ising system: magnetic relaxation near continuous phase transition points

Gül Gülpınar, Rıza Erdem

Canadian Journal of PhysicsApr 11, 20184 citations

ac Susceptibility Study of the d=3 Random-Field Critical Dynamics

A. R. King, John A. Mydosh, Vincent Jaccarino

Physical Review LettersJun 9, 198662 citations

Physical review. B, Condensed matter

Critical scaling of frequency-dependent magnetic susceptibility in YBa2 Cu3 O7 single crystals near the vortex-glass transition

D. S. Reed, N.-C. Yeh, Wei‐Teh Jiang, U. Kriplani, D. A. Beam, Frederic Holtzberg

Published Feb 1, 1994• 33 citations

Publisher

Abstract

A demonstration of critical scaling of the magnetic susceptibility in a

{\mathrm{YBa}}_{2}$${\mathrm{Cu}}_{3}$${\mathrm{O}}_{7}

single crystal is performed for 350 frequencies (f) from 1 kHz to 2 MHz. Near the vortex-glass melting temperature

{\mathit{T}}_{\mathit{M}}

(H) and in the linear-response limit, the susceptibility \ensuremath{\chi}\ensuremath{\equiv}\ensuremath{\Vert}\ensuremath{\chi}\ensuremath{\Vert}

{\mathit{e}}_{\mathrm{\ensuremath{\chi}}}^{\mathit{i}(\mathrm{\ensuremath{\pi}}\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\varphi}}}

) follows the scaling relations \ensuremath{\Vert}\ensuremath{\chi}\ensuremath{\Vert}\ensuremath{\sim}\ensuremath{\Vert}1-(T/

{\mathit{T}}_{\mathit{M}}

)

{\mathrm{\ensuremath{\Vert}}}^{\ensuremath{\nu}/2}

for f\ensuremath{\rightarrow}0, \ensuremath{\Vert}\ensuremath{\chi}\ensuremath{\Vert}\ensuremath{\sim}

{\mathit{f}}^{1/(2\mathit{z})}

and

{\mathrm{\ensuremath{\varphi}}}_{\mathrm{\ensuremath{\chi}}}

=\ensuremath{\pi}/4z for T\ensuremath{\rightarrow}

{\mathit{T}}_{\mathit{M}}

. The static and dynamic exponents (\ensuremath{\nu}\ensuremath{\approxeq}2/3 and z\ensuremath{\approxeq}3) as well as

{\mathit{T}}_{\mathit{M}}

(H) are found to be in excellent agreement with our previous dc and ac transport measurements, lending strong support for a second-order phase transition.

DOI: 10.1103/physrevb.49.4384

Related papers

Bose-Glass Phase in Twinned YBa2Cu3O7−δ

S. A. Grigera, E. Morré, E. J. Osquiguil +3 more

Using an extensive scaling analysis of the transport properties in twinned ${\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{7\ensuremath{-}\ensuremath{\delta}}$ crystals we have experimentally found the predicted change in the universality class of the Bose-glass to liquid transition when the magnetic field is applied at small angles away from the direction of the correlated defects. The new dynamical critical exponent is ${s}^{\ensuremath{'}}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.1\ifmmode\pm\else\textpm\fi{}0.3$.Using an extensive scaling analysis of the transport properties in twinned ${\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{7\ensuremath{-}\ensuremath{\delta}}$ crystals we have experimentally found the predicted change in the universality class of the Bose-glass to liquid transition when the magnetic field is applied at small angles away from the direction of the correlated defects. The new dynamical critical exponent is ${s}^{\ensuremath{'}}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.1\ifmmode\pm\else\textpm\fi{}0.3$.

arXiv:cond-mat/9806376Physical Review LettersSep 14, 1998100 citations

Experimental verification of activated critical dynamics in the d=3 random-field Ising model

A. E. Nash, A. R. King, Vincent Jaccarino

ac susceptibility studies of the d=3 random-field Ising-model (RFIM) system ${\mathrm{Fe}}_{0.46}$ ${\mathrm{Zn}}_{0.54}$ ${\mathrm{F}}_{2}$ in the 5\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}3}$\ensuremath{\leqslant}\ensuremath{\omega}/2\ensuremath{\pi}\ensuremath{\leqslant} ${10}^{5}$ Hz frequency range show the \ensuremath{\omega} scaling of the peak value [${\mathrm{\ensuremath{\chi}}}_{\mathrm{c}}^{\ensuremath{'}}$(\ensuremath{\omega}) ${]}_{\mathrm{p}}$ and the characteristic rounding temperature ${\mathrm{t}}^{\mathrm{*}}$ (\ensuremath{\omega}) are consistent with activated, but not conventional, dynamic scaling. We find [${\mathrm{\ensuremath{\chi}}}_{\mathrm{c}}^{\ensuremath{'}}$(\ensuremath{\chi}) ${]}_{\mathrm{p}}$ \ensuremath{\sim}log|log(\ensuremath{\omega}/${\mathrm{\ensuremath{\omega}}}_{0}$)| and ${\mathrm{t}}^{\mathrm{*}}$ (\ensuremath{\omega})\ensuremath{\sim}|log(\ensuremath{\omega}/${\mathrm{\ensuremath{\omega}}}_{0}$${\mathrm{|}}^{\mathrm{\ensuremath{-}}1/\mathrm{y}}$ with y=1.05\ifmmode\pm\else\textpm\fi{}0.16 and ${\mathrm{\ensuremath{\omega}}}_{0}$\ensuremath{\simeq} ${10}^{8}$ ${\mathrm{s}}^{\mathrm{\ensuremath{-}}1}$ in agreement with the random-exchange Ising-model (REIM) dynamic frequency at the REIM-to-RFIM crossover. The relation between these results and the Villain-Fisher theories is discussed.ac susceptibility studies of the d=3 random-field Ising-model (RFIM) system ${\mathrm{Fe}}_{0.46}$${\mathrm{Zn}}_{0.54}$${\mathrm{F}}_{2}$ in the 5\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}3}$\ensuremath{\leqslant}\ensuremath{\omega}/2\ensuremath{\pi}\ensuremath{\leqslant}${10}^{5}$ Hz frequency range show the \ensuremath{\omega} scaling of the peak value [${\mathrm{\ensuremath{\chi}}}_{\mathrm{c}}^{\ensuremath{'}}$(\ensuremath{\omega})${]}_{\mathrm{p}}$ and the characteristic rounding temperature ${\mathrm{t}}^{\mathrm{*}}$(\ensuremath{\omega}) are consistent with activated, but not conventional, dynamic scaling. We find [${\mathrm{\ensuremath{\chi}}}_{\mathrm{c}}^{\ensuremath{'}}$(\ensuremath{\chi})${]}_{\mathrm{p}}$\ensuremath{\sim}log|log(\ensuremath{\omega}/${\mathrm{\ensuremath{\omega}}}_{0}$)| and ${\mathrm{t}}^{\mathrm{*}}$(\ensuremath{\omega})\ensuremath{\sim}|log(\ensuremath{\omega}/${\mathrm{\ensuremath{\omega}}}_{0}$${\mathrm{|}}^{\mathrm{\ensuremath{-}}1/\mathrm{y}}$ with y=1.05\ifmmode\pm\else\textpm\fi{}0.16 and ${\mathrm{\ensuremath{\omega}}}_{0}$\ensuremath{\simeq}${10}^{8}$ ${\mathrm{s}}^{\mathrm{\ensuremath{-}}1}$ in agreement with the random-exchange Ising-model (REIM) dynamic frequency at the REIM-to-RFIM crossover. The relation between these results and the Villain-Fisher theories is discussed.

Physical review. B, Condensed matterJan 1, 199118 citations

Dynamical scaling in Ising and vector spin glasses

Helmut G. Katzgraber, Ian A. Campbell

We have studied numerically the dynamics of spin glasses with Ising and $XY$ symmetry (gauge glass) in space dimensions 2, 3, and 4. The nonequilibrium spin-glass susceptibility ${\ensuremath{\chi}}_{\mathrm{ne}}({t}_{\mathrm{w}},T)$ and the nonequilibrium energy per spin, ${e}_{\mathrm{ne}}({t}_{\mathrm{w}},T)$ , of samples of large size ${L}_{\mathrm{b}}$ are measured as a function of anneal time ${t}_{\mathrm{w}}$ after a quench to temperatures $T$ . The two observables are compared to the equilibrium spin-glass susceptibility ${\ensuremath{\chi}}_{\mathrm{eq}}(L,T)$ and the equilibrium energy ${e}_{\mathrm{eq}}(L,T)$ , respectively, measured as functions of temperature $T$ and system size $L$ for a range of system sizes. For any time and temperature a nonequilibrium time-dependent length scale ${L}^{*}({t}_{\mathrm{w}},T)$ can be defined by writing ${\ensuremath{\chi}}_{\mathrm{ne}}({t}_{\mathrm{w}},T)={\ensuremath{\chi}}_{\mathrm{eq}}({L}^{*},T)$ (or the equivalent expression for the energy). Our analysis shows that for all systems studied, an ``effective dynamical critical exponent'' parametrization ${L}^{*}({t}_{\mathrm{w}},T)=A(T){t}^{1∕z(T)}$ fits the data well at each temperature within the whole temperature range studied, which extends from well above the critical temperature ${T}_{\mathrm{c}}$ to near $T=0$ for dimension 2 or to well below ${T}_{\mathrm{c}}$ for the other space dimensions studied. In addition, the data suggest that the dynamical exponent $z$ varies smoothly when crossing the transition temperature.We have studied numerically the dynamics of spin glasses with Ising and $XY$ symmetry (gauge glass) in space dimensions 2, 3, and 4. The nonequilibrium spin-glass susceptibility ${\ensuremath{\chi}}_{\mathrm{ne}}({t}_{\mathrm{w}},T)$ and the nonequilibrium energy per spin, ${e}_{\mathrm{ne}}({t}_{\mathrm{w}},T)$, of samples of large size ${L}_{\mathrm{b}}$ are measured as a function of anneal time ${t}_{\mathrm{w}}$ after a quench to temperatures $T$. The two observables are compared to the equilibrium spin-glass susceptibility ${\ensuremath{\chi}}_{\mathrm{eq}}(L,T)$ and the equilibrium energy ${e}_{\mathrm{eq}}(L,T)$, respectively, measured as functions of temperature $T$ and system size $L$ for a range of system sizes. For any time and temperature a nonequilibrium time-dependent length scale ${L}^{*}({t}_{\mathrm{w}},T)$ can be defined by writing ${\ensuremath{\chi}}_{\mathrm{ne}}({t}_{\mathrm{w}},T)={\ensuremath{\chi}}_{\mathrm{eq}}({L}^{*},T)$ (or the equivalent expression for the energy). Our analysis shows that for all systems studied, an ``effective dynamical critical exponent'' parametrization ${L}^{*}({t}_{\mathrm{w}},T)=A(T){t}^{1∕z(T)}$ fits the data well at each temperature within the whole temperature range studied, which extends from well above the critical temperature ${T}_{\mathrm{c}}$ to near $T=0$ for dimension 2 or to well below ${T}_{\mathrm{c}}$ for the other space dimensions studied. In addition, the data suggest that the dynamical exponent $z$ varies smoothly when crossing the transition temperature.

arXiv:cond-mat/0504082Physical Review BJul 25, 200533 citations

Critical behavior of the three-dimensional random-field Ising model: Two-exponent scaling and discontinuous transition

Heiko Rieger

In extensive Monte Carlo simulations the phase transition of the random-field Ising model in three dimensions is investigated. The values of the critical exponents are determined via finite-size scaling. For a Gaussian distribution of the random fields it is found that the correlation length \ensuremath{\xi} diverges with an exponent \ensuremath{\nu}=1.1\ifmmode\pm\else\textpm\fi{}0.2 at the critical temperature and that \ensuremath{\chi}\ensuremath{\sim}${\ensuremath{\xi}}^{2\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\eta}}}$ with \ensuremath{\eta}=0.50\ifmmode\pm\else\textpm\fi{}0.05 for the connected susceptibility and ${\mathrm{\ensuremath{\chi}}}_{\mathrm{dis}}$\ensuremath{\sim}${\ensuremath{\xi}}^{4\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\eta}}\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}$ with \ensuremath{\eta}\ifmmode\bar\else\textasciimacron\fi{}=1.03\ifmmode\pm\else\textpm\fi{}0.05 for the disconnected susceptibility. Together with the amplitude ratio A=${\mathrm{lim}}_{\mathit{T}\ensuremath{\rightarrow}\mathit{T}\mathit{c}}$${\mathrm{\ensuremath{\chi}}}_{\mathrm{dis}}$/${\mathrm{\ensuremath{\chi}}}^{2}$( ${\mathit{h}}_{\mathit{r}}$ /T ${)}^{2}$ being close to one this gives further support for a two-exponent scaling scenario implying \ensuremath{\eta}\ifmmode\bar\else\textasciimacron\fi{}=2\ensuremath{\eta}. The magnetization behaves discontinuously at the transition, i.e., \ensuremath{\beta}=0. However, no divergence for the specific heat and in particular no latent heat is found. Also the probability distribution of the magnetization does not show a multipeak structure that would be characteristic for the phase-coexistence at first-order phase-transition points.In extensive Monte Carlo simulations the phase transition of the random-field Ising model in three dimensions is investigated. The values of the critical exponents are determined via finite-size scaling. For a Gaussian distribution of the random fields it is found that the correlation length \ensuremath{\xi} diverges with an exponent \ensuremath{\nu}=1.1\ifmmode\pm\else\textpm\fi{}0.2 at the critical temperature and that \ensuremath{\chi}\ensuremath{\sim}${\ensuremath{\xi}}^{2\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\eta}}}$ with \ensuremath{\eta}=0.50\ifmmode\pm\else\textpm\fi{}0.05 for the connected susceptibility and ${\mathrm{\ensuremath{\chi}}}_{\mathrm{dis}}$\ensuremath{\sim}${\ensuremath{\xi}}^{4\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\eta}}\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}$ with \ensuremath{\eta}\ifmmode\bar\else\textasciimacron\fi{}=1.03\ifmmode\pm\else\textpm\fi{}0.05 for the disconnected susceptibility. Together with the amplitude ratio A=${\mathrm{lim}}_{\mathit{T}\ensuremath{\rightarrow}\mathit{T}\mathit{c}}$${\mathrm{\ensuremath{\chi}}}_{\mathrm{dis}}$/${\mathrm{\ensuremath{\chi}}}^{2}$(${\mathit{h}}_{\mathit{r}}$/T${)}^{2}$ being close to one this gives further support for a two-exponent scaling scenario implying \ensuremath{\eta}\ifmmode\bar\else\textasciimacron\fi{}=2\ensuremath{\eta}. The magnetization behaves discontinuously at the transition, i.e., \ensuremath{\beta}=0. However, no divergence for the specific heat and in particular no latent heat is found. Also the probability distribution of the magnetization does not show a multipeak structure that would be characteristic for the phase-coexistence at first-order phase-transition points.

arXiv:cond-mat/9503041Physical review. B, Condensed matterSep 1, 1995138 citations

High-frequency magnetic field on crystal field diluted S = 1 Ising system: magnetic relaxation near continuous phase transition points

Gül Gülpınar, Rıza Erdem

Magnetization relaxation and the steady state response of the S = 1 Ising model with random crystal field to a time varying magnetic field with a frequency ω is modelled and studied here by a method that combines the statistical equilibrium theory with the theory of irreversible thermodynamics. The method offers information on the relaxation time (τ) of the system as well as the temperature (θ) and ω dependencies of the complex (AC or dynamical) susceptibility (i.e., χ(ω) = χ′(ω) − iχ″(ω)). The so-called low- and high-frequency regions are separated by τ because τ−1 → 0 as θ approaches the critical temperatures (θc). One can choose to keep the frequency ω fixed and observe the low-frequency behaviors followed by the high-frequency behaviors when θ → θc. It is shown that χ(ω) exhibits different behaviors in low- and high-frequency regimes that are separated by the quantity ωτ: χ′(ω) converges to static susceptibility and χ″(ω) → 0 for ωτ ≪ 1. However, in the high-frequency region where ωτ ≫ 1, χ′(ω) vanishes and χ″(ω) displays a peak at the critical temperature (θc). Besides the above, the logarithm of the susceptibility components versus log(ω) is also plotted. From these plots, one plateau (a step-like) region and a shifted peak with rising temperature is observed for the real and imaginary parts, respectively.Magnetization relaxation and the steady state response of the S = 1 Ising model with random crystal field to a time varying magnetic field with a frequency ω is modelled and studied here by a method that combines the statistical equilibrium theory with the theory of irreversible thermodynamics. The method offers information on the relaxation time (τ) of the system as well as the temperature (θ) and ω dependencies of the complex (AC or dynamical) susceptibility (i.e., χ(ω) = χ′(ω) − iχ″(ω)). The so-called low- and high-frequency regions are separated by τ because τ−1 → 0 as θ approaches the critical temperatures (θc). One can choose to keep the frequency ω fixed and observe the low-frequency behaviors followed by the high-frequency behaviors when θ → θc. It is shown that χ(ω) exhibits different behaviors in low- and high-frequency regimes that are separated by the quantity ωτ: χ′(ω) converges to static susceptibility and χ″(ω) → 0 for ωτ ≪ 1. However, in the high-frequency region where ωτ ≫ 1, χ′(ω) vanishes and χ″(ω) displays a peak at the critical temperature (θc). Besides the above, the logarithm of the susceptibility components versus log(ω) is also plotted. From these plots, one plateau (a step-like) region and a shifted peak with rising temperature is observed for the real and imaginary parts, respectively.

Canadian Journal of PhysicsApr 11, 20184 citations

ac Susceptibility Study of the d=3 Random-Field Critical Dynamics

A. R. King, John A. Mydosh, Vincent Jaccarino

The dynamic critical behavior in a $d=3$ random-field Ising-model system is studied with ac susceptibility $\ensuremath{\chi}(\ensuremath{\omega})$ techniques on ${\mathrm{Fe}}_{0.46}$ ${\mathrm{Zn}}_{0.54}$ ${\mathrm{F}}_{2}$ . Extraordinary critical slowing down is seen in the $\ensuremath{\omega}$ dependence of ${\ensuremath{\chi}}^{\ensuremath{'}}(\ensuremath{\omega})$ at very low frequencies. In the range 2 Hz to 8.5 kHz, the peak value ${[{\ensuremath{\chi}}_{c}^{\ensuremath{'}}(\ensuremath{\omega})]}_{p}$ decreases and the width ${t}^{*}(\ensuremath{\omega})$ increases with increasing $\ensuremath{\omega}$. A conventional dynamicscaling interpretation with ${t}^{*}(\ensuremath{\omega})\ensuremath{\sim}{\ensuremath{\omega}}^{\frac{1}{z\ensuremath{\nu}}}$ gives ${[{\ensuremath{\chi}}_{c}^{\ensuremath{'}}(\ensuremath{\omega})]}_{p}\ensuremath{\sim}|\mathrm{ln}\ensuremath{\omega}|$ and yields $z\ensuremath{\nu}\ensuremath{\simeq}14$. An equally good fit is obtained with ${t}^{*}(\ensuremath{\omega})\ensuremath{\sim}{|\mathrm{ln}\ensuremath{\omega}|}^{\ensuremath{-}1}$ and ${[{\ensuremath{\chi}}_{c}^{\ensuremath{'}}(\ensuremath{\omega})]}_{p}\ensuremath{\sim}\mathrm{ln}|\mathrm{ln}\ensuremath{\omega}|$, consistent with an activated dynamic scaling.The dynamic critical behavior in a $d=3$ random-field Ising-model system is studied with ac susceptibility $\ensuremath{\chi}(\ensuremath{\omega})$ techniques on ${\mathrm{Fe}}_{0.46}$${\mathrm{Zn}}_{0.54}$${\mathrm{F}}_{2}$. Extraordinary critical slowing down is seen in the $\ensuremath{\omega}$ dependence of ${\ensuremath{\chi}}^{\ensuremath{'}}(\ensuremath{\omega})$ at very low frequencies. In the range 2 Hz to 8.5 kHz, the peak value ${[{\ensuremath{\chi}}_{c}^{\ensuremath{'}}(\ensuremath{\omega})]}_{p}$ decreases and the width ${t}^{*}(\ensuremath{\omega})$ increases with increasing $\ensuremath{\omega}$. A conventional dynamicscaling interpretation with ${t}^{*}(\ensuremath{\omega})\ensuremath{\sim}{\ensuremath{\omega}}^{\frac{1}{z\ensuremath{\nu}}}$ gives ${[{\ensuremath{\chi}}_{c}^{\ensuremath{'}}(\ensuremath{\omega})]}_{p}\ensuremath{\sim}|\mathrm{ln}\ensuremath{\omega}|$ and yields $z\ensuremath{\nu}\ensuremath{\simeq}14$. An equally good fit is obtained with ${t}^{*}(\ensuremath{\omega})\ensuremath{\sim}{|\mathrm{ln}\ensuremath{\omega}|}^{\ensuremath{-}1}$ and ${[{\ensuremath{\chi}}_{c}^{\ensuremath{'}}(\ensuremath{\omega})]}_{p}\ensuremath{\sim}\mathrm{ln}|\mathrm{ln}\ensuremath{\omega}|$, consistent with an activated dynamic scaling.

Physical Review LettersJun 9, 198662 citations

A demonstration of critical scaling of the magnetic susceptibility in a

{\mathrm{YBa}}_{2}$${\mathrm{Cu}}_{3}$${\mathrm{O}}_{7}

single crystal is performed for 350 frequencies (f) from 1 kHz to 2 MHz. Near the vortex-glass melting temperature

{\mathit{T}}_{\mathit{M}}

(H) and in the linear-response limit, the susceptibility \ensuremath{\chi}\ensuremath{\equiv}\ensuremath{\Vert}\ensuremath{\chi}\ensuremath{\Vert}

{\mathit{e}}_{\mathrm{\ensuremath{\chi}}}^{\mathit{i}(\mathrm{\ensuremath{\pi}}\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\varphi}}}

) follows the scaling relations \ensuremath{\Vert}\ensuremath{\chi}\ensuremath{\Vert}\ensuremath{\sim}\ensuremath{\Vert}1-(T/

{\mathit{T}}_{\mathit{M}}

)

{\mathrm{\ensuremath{\Vert}}}^{\ensuremath{\nu}/2}

for f\ensuremath{\rightarrow}0, \ensuremath{\Vert}\ensuremath{\chi}\ensuremath{\Vert}\ensuremath{\sim}

{\mathit{f}}^{1/(2\mathit{z})}

and

{\mathrm{\ensuremath{\varphi}}}_{\mathrm{\ensuremath{\chi}}}

=\ensuremath{\pi}/4z for T\ensuremath{\rightarrow}

{\mathit{T}}_{\mathit{M}}

. The static and dynamic exponents (\ensuremath{\nu}\ensuremath{\approxeq}2/3 and z\ensuremath{\approxeq}3) as well as

{\mathit{T}}_{\mathit{M}}

(H) are found to be in excellent agreement with our previous dc and ac transport measurements, lending strong support for a second-order phase transition.

ac susceptibility studies of the d=3 random-field Ising-model (RFIM) system

{\mathrm{Fe}}_{0.46}

{\mathrm{Zn}}_{0.54}

{\mathrm{F}}_{2}

in the 5\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}3}$\ensuremath{\leqslant}\ensuremath{\omega}/2\ensuremath{\pi}\ensuremath{\leqslant}

{10}^{5}

Hz frequency range show the \ensuremath{\omega} scaling of the peak value [${\mathrm{\ensuremath{\chi}}}_{\mathrm{c}}^{\ensuremath{'}}$(\ensuremath{\omega})

{]}_{\mathrm{p}}

and the characteristic rounding temperature

{\mathrm{t}}^{\mathrm{*}}

(\ensuremath{\omega}) are consistent with activated, but not conventional, dynamic scaling. We find [${\mathrm{\ensuremath{\chi}}}_{\mathrm{c}}^{\ensuremath{'}}$(\ensuremath{\chi})

{]}_{\mathrm{p}}

\ensuremath{\sim}log|log(\ensuremath{\omega}/${\mathrm{\ensuremath{\omega}}}_{0}$)| and

{\mathrm{t}}^{\mathrm{*}}

(\ensuremath{\omega})\ensuremath{\sim}|log(\ensuremath{\omega}/${\mathrm{\ensuremath{\omega}}}_{0}$${\mathrm{|}}^{\mathrm{\ensuremath{-}}1/\mathrm{y}}$ with y=1.05\ifmmode\pm\else\textpm\fi{}0.16 and ${\mathrm{\ensuremath{\omega}}}_{0}$\ensuremath{\simeq}

{10}^{8}

${\mathrm{s}}^{\mathrm{\ensuremath{-}}1}$ in agreement with the random-exchange Ising-model (REIM) dynamic frequency at the REIM-to-RFIM crossover. The relation between these results and the Villain-Fisher theories is discussed.

We have studied numerically the dynamics of spin glasses with Ising and

XY

symmetry (gauge glass) in space dimensions 2, 3, and 4. The nonequilibrium spin-glass susceptibility ${\ensuremath{\chi}}_{\mathrm{ne}}({t}_{\mathrm{w}},T)$ and the nonequilibrium energy per spin,

{e}_{\mathrm{ne}}({t}_{\mathrm{w}},T)

, of samples of large size

{L}_{\mathrm{b}}

are measured as a function of anneal time

{t}_{\mathrm{w}}

after a quench to temperatures

T

. The two observables are compared to the equilibrium spin-glass susceptibility ${\ensuremath{\chi}}_{\mathrm{eq}}(L,T)$ and the equilibrium energy

{e}_{\mathrm{eq}}(L,T)

, respectively, measured as functions of temperature

T

and system size

L

for a range of system sizes. For any time and temperature a nonequilibrium time-dependent length scale

{L}^{*}({t}_{\mathrm{w}},T)

can be defined by writing ${\ensuremath{\chi}}_{\mathrm{ne}}({t}_{\mathrm{w}},T)={\ensuremath{\chi}}_{\mathrm{eq}}({L}^{*},T)$ (or the equivalent expression for the energy). Our analysis shows that for all systems studied, an ``effective dynamical critical exponent'' parametrization

{L}^{*}({t}_{\mathrm{w}},T)=A(T){t}^{1∕z(T)}

fits the data well at each temperature within the whole temperature range studied, which extends from well above the critical temperature

{T}_{\mathrm{c}}

to near

T=0

for dimension 2 or to well below

{T}_{\mathrm{c}}

for the other space dimensions studied. In addition, the data suggest that the dynamical exponent

z

varies smoothly when crossing the transition temperature.

{\mathit{h}}_{\mathit{r}}

{)}^{2}

being close to one this gives further support for a two-exponent scaling scenario implying \ensuremath{\eta}\ifmmode\bar\else\textasciimacron\fi{}=2\ensuremath{\eta}. The magnetization behaves discontinuously at the transition, i.e., \ensuremath{\beta}=0. However, no divergence for the specific heat and in particular no latent heat is found. Also the probability distribution of the magnetization does not show a multipeak structure that would be characteristic for the phase-coexistence at first-order phase-transition points.

The dynamic critical behavior in a

d=3

random-field Ising-model system is studied with ac susceptibility $\ensuremath{\chi}(\ensuremath{\omega})$ techniques on

{\mathrm{Fe}}_{0.46}

{\mathrm{Zn}}_{0.54}

{\mathrm{F}}_{2}

. Extraordinary critical slowing down is seen in the $\ensuremath{\omega}$ dependence of ${\ensuremath{\chi}}^{\ensuremath{'}}(\ensuremath{\omega})$ at very low frequencies. In the range 2 Hz to 8.5 kHz, the peak value ${[{\ensuremath{\chi}}_{c}^{\ensuremath{'}}(\ensuremath{\omega})]}_{p}$ decreases and the width ${t}^{*}(\ensuremath{\omega})$ increases with increasing $\ensuremath{\omega}$. A conventional dynamicscaling interpretation with ${t}^{*}(\ensuremath{\omega})\ensuremath{\sim}{\ensuremath{\omega}}^{\frac{1}{z\ensuremath{\nu}}}$ gives ${[{\ensuremath{\chi}}_{c}^{\ensuremath{'}}(\ensuremath{\omega})]}_{p}\ensuremath{\sim}|\mathrm{ln}\ensuremath{\omega}|$ and yields $z\ensuremath{\nu}\ensuremath{\simeq}14$. An equally good fit is obtained with ${t}^{*}(\ensuremath{\omega})\ensuremath{\sim}{|\mathrm{ln}\ensuremath{\omega}|}^{\ensuremath{-}1}$ and ${[{\ensuremath{\chi}}_{c}^{\ensuremath{'}}(\ensuremath{\omega})]}_{p}\ensuremath{\sim}\mathrm{ln}|\mathrm{ln}\ensuremath{\omega}|$, consistent with an activated dynamic scaling.