Disorder-induced enhancement and critical scaling of spontaneous magnetization in random-field quantum spin systems

We investigate the effect of a unidirectional quenched random field on the anisotropic quantum spin-1/2 $XY$ model, which magnetizes spontaneously in the absence of the random field. We adopt a mean-field approach for this analysis. In general, the models considered have Ising symmetry, and as such they exhibit ferromagnetic order in two and three dimensions in the presence of not too large disorder. Even in the special case when the model without disorder has $\text{U}\left(1\right)$ symmetry, a small $\text{U}\left(1\right)$-symmetry-breaking random field induces ferromagnetic long-range order in two dimensions. The mean-field approach, consequently, provides a rather good qualitative and even quantitative description when applied not too close to the criticality. We show that spontaneous magnetization persists even in the presence of the random field, but the magnitude of magnetization gets suppressed due to disorder, and the system magnetizes in the directions parallel and transverse to the random field. Our results are obtained via analytical calculations within a perturbative framework and by numerical simulations. Interestingly, we show that it is possible to enhance a component of magnetization in the presence of the disorder field provided that we apply an additional constant field in the $XY$ plane. Moreover, we derive generalized expressions for the critical temperature and the scalings of the magnetization near the critical point for the $XY$ spin system with arbitrary fixed quantum spin angular momentum.

Figure 1

The system magnetizes in directions parallel and transverse to the disorder field. Zero contour lines of the $F_x^{\epsilon ,2}\left(m\right)$ and $F_y^{\epsilon ,2}\left(m\right)$ in Eqs. (10) (solid red line) and (11) (dotted blue line) for $\gamma =0.3,\epsilon /J=0.1$, and $J\beta =2$, respectively, as functions of $m_x$ and $m_y$. All quantities are dimensionless.

Figure 2

Numerical and analytical results exhibit persistence of spontaneous magnetization in specific directions even after insertion of disorder. Numerical results for the magnetization as a function of $J\beta$, in the directions (a) transverse and (b) parallel to the disordered field. Red circles correspond to the roots of Eqs. (6) and (7) with $\epsilon /J=0.1$ and $\gamma =0.1$. Insets: The blue solid lines correspond to the analytic solutions derived for small $m$ given in Eqs. (18) and (20) for the same set of parameters. The red circles are the numerical results. We find that the numerical and analytical results agree in the small-$m$ regime. All quantities are dimensionless.

Figure 3

Magnetization as a function of $J\beta$ for different choices of the anisotropy constant, $\gamma$, in the directions (a) transverse and (b) parallel to the random field. Circles, squares, and crosses correspond to the magnetization of the system with $\epsilon /J=0.1$ and $\gamma =0.2$, 0.4, and 0.6, respectively. Insets show the inverse critical temperatures as functions of $\gamma$ for $\epsilon /J=0.1$. All quantities are dimensionless.

Figure 4

Disorder-induced enhancement. Effect of the random field in the presence of a constant field. The $X$ component of magnetization, $m_x$, as a function of $J\beta$ for (a) $\gamma =0.05$ and (b) $\gamma =0.1$. Red pluses show $m_x$ for the case when the $XY$ model is subjected to a constant field $\vec{h}$ with $h/J=0.3$ and $\theta =\pi /3$. The blue solid and green dashed lines represent $m_x$ when the system is treated with a random field of strength $\epsilon /J=0.1$ and $0.2$, respectively, along with the constant field [the corresponding Hamiltonian is given in Eq. (23)]. The insets show blowups of the same for a smaller range of $J\beta$. The enhancement of $m_x$ in the presence of the disorder field uncovers a “random field induced order.” Comparing panels (a) and (b), we observe disorder-induced enhancement as the anisotropy parameter is cranked down for fixed $h/J$ and $\theta$. $\theta$ is in radians. All other quantities are dimensionless.

Figure 5

The span of disorder-induced enhancement in parameter space. Plots of the functions (a) $P(\theta ,j)$, (b) $Q(\theta ,j)$, (c) $R(\theta ,j)$, and (d) $S(\theta ,j)$ with respect to $\theta$ and $j=J/h$. Note that there are ranges of the $(\theta ,j)$, for which the function $P(\theta ,j)$ is positive, signaling disorder-induced enhancement in the system described by the Hamiltonian $H_{h,\epsilon }$. However, this is not true for $Q(\theta ,j)$ and $R(\theta ,j)$, which are negative for the entire range of $\theta$ and $j$. $\theta$ is in radians. The fact that $S(\theta ,j)$ is also negative in the entire range implies that in the presence of disorder, the magnetization vector moves away from the direction of the applied random field. All other quantities are dimensionless. Here, $\gamma =0.1$.

Figure 6

${\delta }_{\beta }$ as function of $1/s$ for the transverse (red circles) and parallel (blue squares) magnetizations for $\epsilon /J=0.05$. The lines serve as guides to the eye. All quantities are dimensionless.