Nonlinear susceptibility of a quantum spin glass under uniform transverse and random longitudinal magnetic fields

The interplay between quantum fluctuations and disorder is investigated in a quantum spin-glass model, in the presence of a uniform transverse field $\mathrm{\Gamma }$, as well as of a longitudinal random field $h_i$, which follows a Gaussian distribution characterized by a width proportional to $\mathrm{\Delta }$. The interactions are infinite-ranged, and the model is studied through the replica formalism, within a one-step replica-symmetry-breaking procedure; in addition, the dependence of the Almeida-Thouless eigenvalue ${\lambda }_{\mathrm{AT}}$ (replicon) on the applied fields is analyzed. This study is motivated by experimental investigations on the ${\mathrm{LiHo}}_x{\mathrm{Y}}_{1-x}{\mathrm{F}}_4$ compound, where the application of a transverse magnetic field yields rather intriguing effects, particularly related to the behavior of the nonlinear magnetic susceptibility ${\chi }_3$, which have led to a considerable experimental and theoretical debate. We have analyzed two physically distinct situations, namely, $\mathrm{\Delta }$ and $\mathrm{\Gamma }$ considered as independent, as well as these two quantities related, as proposed recently by some authors. In both cases, a spin-glass phase transition is found at a temperature $T_f$, with such phase being characterized by a nontrivial ergodicity breaking; moreover, $T_f$ decreases by increasing $\mathrm{\Gamma }$ towards a quantum critical point at zero temperature. The situation where $\mathrm{\Delta }$ and $\mathrm{\Gamma }$ are related $[\mathrm{\Delta }\equiv \mathrm{\Delta }(\mathrm{\Gamma }\left)\right]$ appears to reproduce better the experimental observations on the ${\mathrm{LiHo}}_x{\mathrm{Y}}_{1-x}{\mathrm{F}}_4$ compound, with the theoretical results coinciding qualitatively with measurements of the nonlinear susceptibility ${\chi }_3$. In this later case, by increasing $\mathrm{\Gamma }$ gradually, ${\chi }_3$ becomes progressively rounded, presenting a maximum at a temperature $T^{*}$ $\left(T^{*}>T_f\right)$, with both the amplitude of the maximum and the value of $T^{*}$ decreasing gradually. Moreover, we also show that the random field is the main responsible for the smearing of the nonlinear susceptibility, acting significantly inside the paramagnetic phase, leading to two regimes delimited by the temperature $T^{*}$, one for $T_f<T<T^{*}$, and another one for $T>T^{*}$. It is argued that the conventional paramagnetic state corresponds to $T>T^{*}$, whereas the temperature region $T_f<T<T^{*}$ may be characterized by a rather unusual dynamics, possibly including Griffiths singularities.

Figure 1

The one-step RSB parameter $\delta \equiv q_1-q_0$ is presented vs the dimensionless temperature $T/J$, for $\mathrm{\Delta }=0$ and typical values of $\mathrm{\Gamma }/J$ (a), as well as for $\mathrm{\Gamma }=0$ and typical values $\mathrm{\Delta }/J$ (b). The parameters $q_1,q_0$, and $p$ are also exhibited vs temperature in the respective insets; one notices that the quadrupolar parameter $p$ becomes relevant only in the cases $\mathrm{\Gamma }>0$, for which it decreases by lowering the temperature. Due to the usual numerical difficulties, the low-temperature results [typically $(T/J)<0.05\left)\right]$ correspond to smooth extrapolations from higher-temperature data.

Figure 2

Plots of the dimensionless nonlinear susceptibility [computed from Eq. (3)] are exhibited for $\mathrm{\Delta }=0$: (a) $J^3{\chi }_3$ vs $T/J$ for two different values of $\mathrm{\Gamma }/J$; (b) $J^3{\chi }_3$ vs $\mathrm{\Gamma }/J$ for two different temperatures. In all cases, one notices sharp divergences of ${\chi }_3$, signaling evident phase transitions. The corresponding critical exponents are estimated through log-log plots (shown in the respective insets), where in each case, the fitting proposal is represented by a dashed-dotted line (see text).

Figure 3

The behavior of the dimensionless nonlinear susceptibility (in two typical cases exhibited in Fig. 2) is presented for increasing values of $\mathrm{\Delta }/J$: (a) $J^3{\chi }_3$ vs $T/J$, for $(\mathrm{\Gamma }/J)=0.0$; (b) $J^3{\chi }_3$ vs $\mathrm{\Gamma }/J$, for $(T/J)=1.0$. In each case, one notices a rounded peak for $(\mathrm{\Delta }/J)>0$, with its maximum value located at a temperature $T^{*}$ (a), or at a transverse field ${\mathrm{\Gamma }}^{*}$ (b), such that its height decreases for increasing values of $\mathrm{\Delta }/J$. The log-log plots in the respective insets show that the divergences of Eq. (7), leading to the exponent $\gamma$ [inset of (a)], or in Eq. (6), leading to the exponent ${\delta }^{\prime }$ [inset of (b)], are fulfilled only for $(\mathrm{\Delta }/J)=0$.

Figure 4

(a) The dimensionless nonlinear susceptibility is represented vs $\mathrm{\Gamma }/J$, for typical fixed temperatures and a nonzero width for the random fields $\left[\right(\mathrm{\Delta }/J)=0.25]$. The divergences that occur for $\mathrm{\Delta }=0$ at ${\mathrm{\Gamma }}_f\left(T\right)$ [following Eq. (6)], signalled by arrows in some cases, get smoothened due to the random fields, so that their corresponding maxima [located at ${\mathrm{\Gamma }}^{*}\left(T\right)]$ are shifted towards higher values of the transverse field, i.e., ${\mathrm{\Gamma }}^{*}\left(T\right)>{\mathrm{\Gamma }}_f\left(T\right)$. (b) The behavior of the dimensionless nonlinear susceptibility is shown versus $\mathrm{\Gamma }/J$, for typical fixed temperatures, by considering a particular relation involving $\mathrm{\Delta }$ and $\mathrm{\Gamma }$ $[(\mathrm{\Delta }/J)=0.02{(\mathrm{\Gamma }/J)}^2]$; the inset represents an amplification of the region for higher values of $\mathrm{\Gamma }/J$. In all cases, the maxima [located at ${\mathrm{\Gamma }}^{*}\left(T\right)]$ appear shifted with respect to the onset of RSB [located at ${\mathrm{\Gamma }}_f\left(T\right)$, signalled by arrows in some cases] towards higher values of the transverse field, i.e., ${\mathrm{\Gamma }}^{*}\left(T\right)>{\mathrm{\Gamma }}_f\left(T\right)$.

Figure 5

The softening of the nonlinear susceptibility is illustrated by means of the denominator of $q_2$, i.e., $q_2\propto b^{-1},b=1-2{\left(\beta J\right)}^2{I}_0\left(\mathrm{\Gamma }\right)$ [cf. Eq. (24)], which appears in the expression of ${\chi }_3$ calculated in Appendix, within the RS approximation. The Almeida-Thouless eigenvalue ${\lambda }_{\mathrm{AT}}$, associated with the onset of RSB and defining the SG critical temperature $T_f$ is also shown, for comparison. (a) Results for $(\mathrm{\Gamma }/J)=1.0$ are exhibited for two typical values of $\mathrm{\Delta }/J$, namely, $(\mathrm{\Delta }/J)=0.0$ and $(\mathrm{\Delta }/J)=0.1$, in the case where $\mathrm{\Gamma }$ and $\mathrm{\Delta }$ are independent; similar results are presented in the inset for $(\mathrm{\Gamma }/J)=0$. The full lines represent the cases $(\mathrm{\Delta }/J)=0.0$, showing that ${\lambda }_{\mathrm{AT}}$ and the denominator $b$ coincide, becoming zero at the temperature $T_f$. The cases $(\mathrm{\Delta }/J)=0.1$ show that $b$ is always positive, presenting a smooth minimum around a temperature $T^{*}$, leading to the rounding of ${\chi }_3$, whereas ${\lambda }_{\mathrm{AT}}$ becomes zero at a lower temperature $T_f$. (b) Results for $(\mathrm{\Gamma }/J)=1.0$ and $(\mathrm{\Delta }/J)=0.1$ are shown, by comparing the case where these two quantities are considered as independent (dashed lines), with the one where they follow the relation proposed in in in Fig. 4 $[(\mathrm{\Delta }/J)=0.02{(\mathrm{\Gamma }/J)}^2]$ (dotted lines). In all cases, the arrows locate the freezing temperature $T_f$.

Figure 6

Phase diagrams $T/J$ vs $\mathrm{\Gamma }/J$ showing the frontiers separating the SG and paramagnetic phases (full lines), which represent the behavior of the temperature $T_f$ for increasing values of $\mathrm{\Gamma }$, located by the onset of RSB, i.e., ${\lambda }_{\mathrm{AT}}=0$. The dashed lines correspond to the temperature $T^{*}$, associated with the maximum of the nonlinear susceptibility ${\chi }_3$, and herein interpreted as a crossover between two distinct regions of the paramagnetic phase (PM1 and PM2). (a) Phase diagram for $(\mathrm{\Delta }/J)=0.25$, in the case where $\mathrm{\Gamma }$ and $\mathrm{\Delta }$ are independent. (b) Phase diagram for which $\mathrm{\Gamma }$ and $\mathrm{\Delta }$ follow the relation proposed in Fig. 4 $[(\mathrm{\Delta }/J)=0.02{(\mathrm{\Gamma }/J)}^2]$, whose parabolic behavior is presented in the inset. Due to the usual numerical difficulties, the low-temperature results [typically $(T/J)<0.05\left)\right]$ correspond to smooth extrapolations from higher-temperature data.