Random-field-induced disordering mechanism in a disordered ferromagnet: Between the Imry-Ma and the standard disordering mechanism

TL;DR

This paper identifies a new disordering mechanism in disordered ferromagnets driven by finite-size glassy domains at moderate random fields, expanding understanding beyond traditional Imry-Ma and standard mechanisms.

Contribution

It demonstrates that a third disordering mechanism exists in ferromagnets with competing interactions, applicable to a broad class of disordered magnetic systems.

Findings

Disordered ferromagnets can be disordered by finite-size glassy domains at moderate random fields.

The new mechanism is supported by large-scale Monte Carlo simulations and zero-temperature numerical methods.

The transition may be first order based on finite-size scaling analysis.

Abstract

Random fields disorder Ising ferromagnets by aligning single spins in the direction of the random field in three space dimensions, or by flipping large ferromagnetic domains at dimensions two and below. While the former requires random fields of typical magnitude similar to the interaction strength, the latter Imry-Ma mechanism only requires infinitesimal random fields. Recently, it has been shown that for dilute anisotropic dipolar systems a third mechanism exists, where the ferromagnetic phase is disordered by finite-size glassy domains at a random field of finite magnitude that is considerably smaller than the typical interaction strength. Using large-scale Monte Carlo simulations and zero-temperature numerical approaches, we show that this mechanism applies to disordered ferromagnets with competing short-range ferromagnetic and antiferromagnetic interactions, suggesting its generality in ferromagnetic systems with competing interactions and an underlying spin-glass phase. A finite-size-scaling analysis of the magnetization distribution suggests that the transition might be first order.

Tables

Table 1. Table 1: Parameters of the simulations for J ~ = J / J 0 = 0.5 ~ 𝐽 𝐽 subscript 𝐽 0 0.5 \tilde{J}=J/J_{0}=0.5 where J 0 = 1 subscript 𝐽 0 1 J_{0}=1 . N sa subscript 𝑁 sa N_{\rm sa} is the number of samples, N sw subscript 𝑁 sw N_{\rm sw} is the total number of Monte Carlo sweeps used for equilibration, T min subscript 𝑇 min T_{\rm min} is the lowest temperature simulated, T max subscript 𝑇 max T_{\rm max} is the highest temperature simulated, and N T subscript 𝑁 𝑇 N_{T} is the number of temperatures used in the parallel tempering method for each system size L 𝐿 L .

$\tilde{J}$	$H_r$	$L$	$N_{\mathrm{sa}}$	$N_{\mathrm{sw}}$	$T_{\min }$	$T_{\max }$	$N_T$
$0.50$	0.00	$16,20,24$	$512$	$1024$	$4.200$	$5.000$	$100$
$0.50$	0.00	$28,32$	$512$	$2048$	$4.200$	$5.000$	$100$
$0.50$	0.00	$40$	$512$	$4096$	$4.200$	$5.000$	$100$
$0.50$	0.00	$48$	$512$	$8192$	$4.200$	$5.000$	$40$
$0.50$	0.20	$16,20,24$	$512$	$2048$	$4.100$	$5.000$	$30$
$0.50$	0.20	$28,32,36,40$	$512$	$2048$	$4.100$	$5.000$	$50$
$0.50$	0.30	$16,20,24$	$1024$	$2048$	$4.100$	$5.000$	$30$
$0.50$	0.30	$28,32$	$512$	$2048$	$4.100$	$5.000$	$50$
$0.50$	0.30	$40$	$512$	$4096$	$4.100$	$5.000$	$50$
$0.50$	0.50	$16$	$1024$	$4096$	$4.100$	$5.000$	$30$
$0.50$	0.50	$20$	$1024$	$8192$	$4.100$	$5.000$	$30$
$0.50$	0.50	$24$	$1024$	$16384$	$4.100$	$5.000$	$30$
$0.50$	0.50	$28,32,40$	$1024$	$16384$	$4.100$	$5.000$	$50$
$0.50$	0.70	$16$	$1024$	$2048$	$3.950$	$4.850$	$30$
$0.50$	0.70	$20$	$1024$	$8196$	$3.950$	$4.850$	$30$
$0.50$	0.70	$24$	$1024$	$16384$	$3.950$	$4.850$	$30$
$0.50$	0.70	$28,32,40$	$1024$	$16384$	$3.950$	$4.850$	$50$
$0.50$	0.90	$12$	$1024$	$1024$	$3.700$	$4.800$	$30$
$0.50$	0.90	$16$	$1024$	$4096$	$3.700$	$4.800$	$30$
$0.50$	0.90	$20$	$1024$	$8192$	$3.700$	$4.800$	$30$
$0.50$	0.90	$24$	$1024$	$16384$	$3.700$	$4.800$	$30$
$0.50$	0.90	$28,32$	$1024$	$65536$	$3.700$	$4.800$	$40$
$0.50$	1.30	$12$	$1024$	$512$	$2.200$	$5.000$	$40$
$0.50$	1.30	$16$	$1900$	$32768$	$2.200$	$5.000$	$50$
$0.50$	1.30	$20$	$1424$	$262144$	$2.200$	$5.000$	$50$
$0.50$	1.30	$24$	$1424$	$1048576$	$2.200$	$5.000$	$50$
$0.50$	1.60	$8$	$2300$	$8192$	$1.500$	$5.000$	$50$
$0.50$	1.60	$10$	$2300$	$65536$	$1.500$	$5.000$	$50$
$0.50$	1.60	$12$	$2073$	$4194304$	$1.500$	$5.000$	$50$
$0.50$	1.60	$14$	$2116$	$4194304$	$1.500$	$5.000$	$50$
$0.50$	1.80	$6$	$1274$	$32768$	$1.400$	$5.000$	$50$
$0.50$	1.80	$8$	$2230$	$65536$	$1.400$	$5.000$	$50$
$0.50$	1.80	$10$	$2392$	$2097152$	$1.400$	$5.000$	$50$
$0.50$	1.80	$12$	$2752$	$16777216$	$1.400$	$5.000$	$50$
$0.50$	2.00	$4,6$	$3000$	$16384$	$0.600$	$4.500$	$50$
$0.50$	2.00	$8$	$4500$	$1048576$	$0.600$	$4.500$	$50$
$0.50$	2.00	$10$	$1200$	$33554432$	$0.600$	$4.500$	$50$

Table 2. Table 2: Parameters of the simulations for J ~ = J / J 0 = 1.50 ~ 𝐽 𝐽 subscript 𝐽 0 1.50 \tilde{J}=J/J_{0}=1.50 , where J 0 = 1 subscript 𝐽 0 1 J_{0}=1 . N sa subscript 𝑁 sa N_{\rm sa} is the number of samples, N sw subscript 𝑁 sw N_{\rm sw} is the total number of Monte Carlo sweeps used for equilibration, T min subscript 𝑇 min T_{\rm min} is the lowest temperature simulated, T max subscript 𝑇 max T_{\rm max} is the highest temperature simulated, and N T subscript 𝑁 𝑇 N_{T} is the number of temperatures used in the parallel tempering method for each system size L 𝐿 L .

$\tilde{J}$	$H_r$	$L$	$N_{\mathrm{sa}}$	$N_{\mathrm{sw}}$	$T_{\min }$	$T_{\max }$	$N_T$
$1.50$	0.00	$16,20$	$1024$	$16384$	$3.30$	$3.80$	$100$
$1.50$	0.00	$24,28$	$1024$	$65536$	$3.30$	$3.80$	$100$
$1.50$	0.00	$32,40$	$1024$	$262144$	$3.30$	$3.80$	$30$
$1.50$	0.02	$16$	$1024$	$65536$	$3.22$	$3.983$	$18$
$1.50$	0.02	$20,24$	$1024$	$65536$	$3.22$	$3.983$	$30$
$1.50$	0.02	$28$	$1024$	$131072$	$3.22$	$3.983$	$30$
$1.50$	0.02	$32,40$	$1024$	$131072$	$3.22$	$3.983$	$40$
$1.50$	0.04	$16$	$1024$	$65536$	$3.22$	$3.983$	$18$
$1.50$	0.04	$20$	$1024$	$65536$	$3.22$	$3.983$	$40$
$1.50$	0.04	$24,28,32,40$	$1024$	$131072$	$3.22$	$3.983$	$40$
$1.50$	0.06	$16$	$1024$	$32768$	$3.22$	$3.983$	$18$
$1.50$	0.06	$20,24$	$1024$	$65536$	$3.22$	$3.983$	$40$
$1.50$	0.06	$28,32$	$1024$	$262144$	$3.22$	$3.983$	$40$
$1.50$	0.06	$40$	$1200$	$524288$	$3.22$	$3.983$	$40$
$1.50$	0.08	$16,20$	$1024$	$16384$	$3.25$	$4.00$	$20$
$1.50$	0.08	$24$	$1024$	$65536$	$3.22$	$4.00$	$18$
$1.50$	0.08	$28$	$1024$	$262144$	$3.22$	$4.00$	$40$
$1.50$	0.08	$32,40$	$1024$	$524288$	$3.22$	$4.00$	$40$
$1.50$	0.10	$16$	$2048$	$16384$	$3.20$	$4.00$	$16$
$1.50$	0.10	$20$	$2048$	$32768$	$3.20$	$4.00$	$16$
$1.50$	0.10	$24$	$1024$	$65536$	$3.20$	$4.00$	$16$
$1.50$	0.10	$28$	$1024$	$262144$	$3.20$	$4.00$	$40$
$1.50$	0.10	$32,40$	$1024$	$524288$	$3.20$	$4.00$	$40$
$1.50$	0.20	$12$	$2048$	$16384$	$3.00$	$4.00$	$16$
$1.50$	0.20	$16$	$1024$	$32768$	$3.00$	$4.00$	$16$
$1.50$	0.20	$20,24$	$1024$	$131072$	$3.00$	$4.00$	$16$
$1.50$	0.20	$28,32$	$1024$	$524288$	$3.00$	$4.00$	$30$
$1.50$	0.30	$12,14$	$2048$	$32768$	$2.78$	$4.00$	$20$
$1.50$	0.30	$16$	$1024$	$65536$	$2.78$	$4.00$	$20$
$1.50$	0.30	$20$	$2043$	$131072$	$2.78$	$4.00$	$20$
$1.50$	0.30	$24$	$3285$	$1048576$	$2.78$	$4.00$	$30$
$1.50$	0.40	$8,10,12$	$2048$	$32768$	$2.00$	$4.50$	$60$
$1.50$	0.40	$14,16$	$2048$	$65536$	$2.00$	$4.50$	$60$
$1.50$	0.40	$20$	$1024$	$1048576$	$2.00$	$4.50$	$60$
$1.50$	0.50	$4$	$4096$	$32768$	$1.00$	$5.00$	$50$
$1.50$	0.50	$6$	$3637$	$65536$	$1.00$	$5.00$	$50$
$1.50$	0.50	$8$	$2048$	$131072$	$1.00$	$5.00$	$50$
$1.50$	0.50	$10$	$2048$	$262144$	$1.00$	$5.00$	$50$
$1.50$	0.50	$12$	$1441$	$1048576$	$1.00$	$5.00$	$50$
$1.50$	0.50	$14$	$2645$	$2097152$	$1.00$	$5.00$	$50$
$1.50$	0.60	$4,6$	$2048$	$16384$	$1.00$	$5.00$	$50$
$1.50$	0.60	$8$	$1387$	$65536$	$1.00$	$5.00$	$50$
$1.50$	0.60	$10$	$2048$	$262144$	$1.00$	$5.00$	$50$
$1.50$	0.60	$12$	$5275$	$524288$	$1.00$	$5.00$	$50$
$1.50$	0.60	$14$	$1199$	$4194304$	$1.00$	$5.00$	$50$
$1.50$	0.60	$16$	$1024$	$33554432$	$1.00$	$5.00$	$50$

Table 3. Table 3: Parameters of the simulations for J ~ = J / J 0 = 1.60 ~ 𝐽 𝐽 subscript 𝐽 0 1.60 \tilde{J}=J/J_{0}=1.60 , where J 0 = 1 subscript 𝐽 0 1 J_{0}=1 . N sa subscript 𝑁 sa N_{\rm sa} is the number of samples, N sw subscript 𝑁 sw N_{\rm sw} is the total number of Monte Carlo sweeps used for equilibration, T min subscript 𝑇 min T_{\rm min} is the lowest temperature simulated, T max subscript 𝑇 max T_{\rm max} is the highest temperature simulated, and N T subscript 𝑁 𝑇 N_{T} is the number of temperatures used in the parallel tempering method for each system size L 𝐿 L .

$\tilde{J}$	$H_r$	$L$	$N_{\mathrm{sa}}$	$N_{\mathrm{sw}}$	$T_{\min }$	$T_{\max }$	$N_T$
$1.60$	0.00	$12$	$1024$	$8192$	$2.75$	$5.00$	$100$
$1.60$	0.00	$16$	$1024$	$32768$	$2.75$	$5.00$	$100$
$1.60$	0.00	$20$	$1024$	$65536$	$2.75$	$5.00$	$100$
$1.60$	0.00	$24$	$1024$	$262144$	$2.75$	$5.00$	$30$
$1.60$	0.00	$28$	$1024$	$524288$	$2.75$	$5.00$	$30$
$1.60$	0.00	$32$	$1024$	$1048576$	$2.75$	$5.00$	$30$
$1.60$	0.10	$12$	$1024$	$32768$	$1.75$	$5.00$	$37$
$1.60$	0.10	$16,20$	$1024$	$131072$	$1.75$	$5.00$	$37$
$1.60$	0.10	$24$	$1024$	$262144$	$2.75$	$5.00$	$30$
$1.60$	0.10	$28$	$1024$	$524288$	$2.75$	$5.00$	$30$
$1.60$	0.10	$32$	$1024$	$1048576$	$2.75$	$5.00$	$30$
$1.60$	0.20	$8$	$2048$	$8192$	$1.75$	$5.00$	$37$
$1.60$	0.20	$10$	$2840$	$16384$	$1.75$	$5.00$	$37$
$1.60$	0.20	$12$	$1500$	$65536$	$1.75$	$5.00$	$37$
$1.60$	0.20	$14$	$2048$	$1048576$	$1.75$	$5.00$	$40$
$1.60$	0.20	$16$	$2048$	$4194304$	$1.75$	$5.00$	$40$
$1.60$	0.30	$4$	$3024$	$8192$	$0.5$	$4.00$	$71$
$1.60$	0.30	$6$	$3024$	$32768$	$0.5$	$4.00$	$71$
$1.60$	0.30	$8$	$4096$	$131072$	$0.5$	$4.00$	$71$
$1.60$	0.30	$10$	$2440$	$1048576$	$0.5$	$4.00$	$71$
$1.60$	0.30	$12$	$1782$	$8388608$	$0.5$	$4.00$	$71$
$1.60$	0.40	$4$	$4000$	$8192$	$0.50$	$4.00$	$71$
$1.60$	0.40	$6$	$2048$	$32768$	$0.50$	$4.00$	$71$
$1.60$	0.40	$8$	$5800$	$131072$	$0.50$	$4.00$	$71$
$1.60$	0.40	$10$	$3500$	$524288$	$0.50$	$4.00$	$71$
$1.60$	0.40	$12$	$2100$	$4194304$	$0.50$	$4.00$	$71$

Table 4. Table 4: Parameters of the zero-temperature simulations with J ~ = J / J 0 ~ 𝐽 𝐽 subscript 𝐽 0 \tilde{J}=J/J_{0} , where J 0 = 1 subscript 𝐽 0 1 J_{0}=1 . H r subscript 𝐻 𝑟 H_{r} is the random field strength, N sa subscript 𝑁 sa N_{\rm sa} is the number of samples, N eo subscript 𝑁 eo N_{\rm eo} is the total number of simulation steps for each system size L 𝐿 L .

$\tilde{J}$	$H_r$	$L$	$N_{\mathrm{sa}}$	$N_{\mathrm{eo}}$
$0.00$	$1.40$, $1.50$, $1.55$, $1.60$, $1.70$, $1.80$	$4$	$1024$	$10000000$
$0.00$	$1.40$, $1.50$, $1.55$, $1.60$, $1.65$, $1.70$, $1.75$, $1.80$	$6$	$1024$	$25000000$
$0.00$	$1.40$, $1.50$, $1.55$, $1.60$, $1.65$, $1.68$, $1.70$, $1.75$	$8$	$1024$	$35000000$
$0.00$	$1.50$, $1.55$, $1.60$, $1.70$	$10$	$1024$	$268435456$
$0.15$	$1.50$, $1.60$, $1.70$	$4$	$1024$	$15000000$
$0.15$	$1.50$, $1.55$, $1.60$, $1.65$, $1.70$	$6$	$1024$	$25000000$
$0.15$	$1.50$, $1.55$, $1.60$, $1.65$, $1.70$	$8$	$1024$	$35000000$
$0.25$	$1.50$, $1.55$, $1.60$, $1.70$	$4$	$1024$	$25000000$
$0.25$	$1.40$, $1.50$, $1.55$, $1.60$, $1.65$, $1.70$	$6$	$1024$	$25000000$
$0.25$	$1.40$, $1.50$, $1.55$, $1.60$, $1.65$	$8$	$1024$	$35000000$
$0.50$	$1.40$, $1.50$, $1.55$, $1.60$, $1.65$	$4$	$1024$	$25000000$
$0.50$	$1.40$, $1.45$, $1.50$, $1.55$, $1.60$	$6$	$1024$	$35000000$
$0.50$	$1.40$, $1.45$, $1.50$, $1.55$, $1.60$	$8$	$1024$	$35000000$

Equations

$$\mathcal{H}$$

$$g=\frac{1}{2}\left(3-\frac{{\left[m^4\right]}_{\mathrm{av}}}{{\left[m^2\right]}_{\mathrm{av}}^2}\right),$$

$${\xi }_L^{\mathrm{sg},\mathrm{fm}}=\frac{1}{2\sin \left(k_{\min }/2\right)}\sqrt{\frac{{\chi }_{\mathrm{sg},\mathrm{fm}}(0)}{{\chi }_{\mathrm{sg},\mathrm{fm}}({\mathbf{k}}_{\min })}-1},$$

$$\displaystyle\chi_{\rm sg}\left({\bf k}\right)=\frac{1}{N}\Big{[}\langle\sum_{i,j}q_{i}q_{j}e^{\text{i}{\bf k}{\bf r}_{ij}}\rangle\Big{]}_{\rm av}\,,$$

$$\displaystyle\chi_{\rm fm}\left({\bf k}\right)=\frac{1}{N}\Big{[}\langle\sum_{i,j}S_{i}S_{j}e^{\text{i}{\bf k}{\bf r}_{ij}}\rangle\Big{]}_{\rm av}\,.$$

$$H_r\left(J^{-1}\right)\propto {\left(J^{-1}-J_c^{-1}\right)}^{\frac{(3/2-\theta )}{(3-\theta )}}.$$

Full text

I Introduction

Spin glasses, where frustration and disorder are introduced through random competing ferromagnetic and antiferromagnetic interactions Edwards and Anderson (1975), and random-field ferromagnets, where disorder is introduced through an effective longitudinal field Imry and Ma (1975), are two archetypal models for the study of disordered magnetic systems Binder and Young (1986); Young (1998). While usually studied independently from each other, random interactions and random fields are generic in many nonmagnetic systems Schechter and Stamp (2009), and dominate thermodynamic and dynamic properties in e.g. orientational glassesVugmeister and Glinchuk (1990) and relaxor ferroelectricsWestphal et al. (1992); Sherrington (2013). In magnetic systems, random interactions are abundant, yet the presence of an effective longitudinal random field is nontrivial. Applied magnetic fields cannot be locally randomized and nonmagnetic disorder cannot produce an effective random magnetic field, because it violates time-reversal symmetry.

Anisotropic dipolar magnets, and specifically the ${{\rm LiHo_{x}Y_{1-x}F_{4}}}$ compound, are an intriguing exception. The interplay of an applied field in the direction transverse to the easy axis of the magnetic holmium ions and the off-diagonal elements of the dipolar interaction gives rise to an effective longitudinal field Schechter and Laflorencie (2006); Tabei et al. (2006); Schechter (2008). This field is locally random, transforming spatial disorder coming from the dilution of the Ho ions by the nonmagnetic yttrium ions into a disorder in the effective longitudinal field. Furthermore, as a function of holmium concentration, ${{\rm LiHo_{x}Y_{1-x}F_{4}}}$ has both a ferromagnetic phase at $x\gtrsim 0.3$ and a spin-glass phase at $0<x\lesssim 0.3$ including the extreme dilute limit Tam and Gingras (2009); Andresen et al. (2013, 2014). The ${{\rm LiHo_{x}Y_{1-x}F_{4}}}$ system is therefore ideal for the study of the interplay of competing interactions and effective longitudinal random fields, and the effect of this interplay on the thermodynamic and dynamical properties of the system.

Recently, we have shown Andresen et al. (2013) that disordered anisotropic dipolar magnets in their ferromagnetic phase, are driven into a quasi-spin-glass phase upon the introduction of a finite effective random magnetic field which is considerably smaller than the typical nearest-neighbor interaction. This occurs also in three space dimensions, where it displays a novel disordering mechanism, intermediate between the standard disordering of a ferromagnet and the Imry-Ma Imry and Ma (1975) mechanism at dimensions two and below. The disordering field is neither infinitesimal, nor of the order of the interaction. The smallness of the random field at the transition is dictated by the proximity of the system to the zero-field transition between the ferromagnet and the spin-glass phase. Moreover, the disordered phase near the transition consists of neither single spins pointing in the direction of the random field, nor of large ferromagnetic domains, but of finite-size glassy domains, reminiscent of the competing spin-glass phase. We denote this phase “quasi-spin-glass” (QSG) in the regime of low temperatures, where it is frozen, and “paramagnetic QSG” (pQSG) in the high temperature regime. The size of the glassy domains is a function of the magnitude of the disordering field and temperature Schechter and Laflorencie (2006); Andresen et al. (2013); com .

Here we consider a more general model of an Ising magnet where the interactions are short ranged, taken from a Gaussian distribution with a mean ferromagnetic value $J_{0}$ and a standard deviation $J$, and with longitudinal random fields taken from a Gaussian distribution of mean zero and standard deviation $H_{r}$. Using jaded extremal optimization Middleton (2004) and large-scale parallel tempering Monte Carlo simulations Hukushima and Nemoto (1996) we analyze the thermodynamic properties of the system at zero and finite temperature. For zero random field we establish the phase diagram of the system, consisting of low-temperature ferromagnetic (FM) and spin-glass (SG) phases and a high-temperature paramagnetic (PM) phase. The zero-temperature transition between the FM and SG phases occurs at a ratio of $J/J_{0}\approx 1.65$. We then analyze the disordering of the FM phase at $J/J_{0}<1.65$ with increasing temperature and random field. For zero temperature we find that the disordering of the FM phase occurs at finite random field, which is much smaller than the typical interaction, $0<H_{r}\ll J_{0}$, the value of $H_{r}$ depending on the proximity of the system at zero field to the SG phase. At finite temperature we find that the critical temperature of the FM-to-pQSG transition increases linearly as a function of decreasing random field, down to a rather small value of $H_{r}$. Our results here are in agreement with our previous results for the dipolar-interacting ${{\rm LiHo_{x}Y_{1-x}F_{4}}}$ system Andresen et al. (2013), as well as with experimental data for this material Silevitch et al. (2007). This suggests that the disordering mechanism by finite-size glassy domains is a generic feature of disordered ferromagnets with competing interactions in the presence of random fields. By analyzing the distribution of magnetization values near the random-field-driven transition, we find evidence for a first-order transition between the FM and pQSG phases in the regime where the FM phase is in proximity to the SG phase. No evidence for a first-order transition is found in the regime where the interactions are strongly ferromagnetic dominated, i.e., $J/J_{0}\ll 1$.

The paper is structured as follows. In Sec. II we introduce the model. In Sec. III we introduce the methods used for zero-temperature and for finite-temperature calculations. Our results are presented in Sec. IV, and discussion and conclusions are given in Sec. V. An Appendix lists all parameters of the different simulations.

II Model

The model we simulate is given by the Hamiltonian

[TABLE]

where the sum is over nearest neighbors, the spin couplings $J_{ij}$ are chosen from a Gaussian distribution with standard deviation $J$ and mean $J_{0}$, and the $S_{i}$ are Ising spins on the vertices of a hypercube of dimension $d=3$ and linear size $L$. Throughout this work we fix the mean of the distribution $J_{0}=1$ and use the standard deviation of the Gaussian distribution $J$ as a means of tuning the interaction disorder in the model. The second term describes the coupling of the Ising spins to a site-dependent random field $h_{i}$. The random fields are taken from a Gaussian distribution with zero mean and standard deviation $H_{r}$.

Different parameters of the model lead to different phase diagrams. If $J$ is small compared to $|{J_{0}}|$ and $H_{r}=0$, as the temperature is reduced, the system undergoes a phase transition from a PM phase to a FM phase for $J_{0}>0$. For $J_{0}<0$ the system undergoes a phase transition from a PM phase to an antiferromagnetic (AFM) phase. By increasing the value of the parameter $J$ it is possible to introduce disorder and frustration into the system. These two ingredients are known to be essential for the emergence of a SG phase. The ratio $J_{0}/J$ quantifies the amount of interaction disorder in the system. Below a critical ratio $J_{0}/J_{c}$, the system has a PM phase at high temperatures and a SG phase at low temperatures (keeping $H_{r}=0$). Extreme cases occur when $J=0$, $J_{0}\neq 0$ ($J_{0}/J\to\infty$) where the model reduces to the well-known Ising model if $H_{r}=0$, and to the random-field Ising model Imry and Ma (1975) if $H_{r}>0$; and when $J\neq 0$, $J_{0}=0$ ($J_{0}/J=0$) where the model reduces to the Edwards-Anderson (EA) spin-glass model Edwards and Anderson (1975). In Fig. 1 we plot the phase diagram of the system as a function of temperature and interaction disorder $J$ (keeping $H_{r}=0$) presenting a phase transition at zero temperature between a FM phase at small disorder and a SG phase at large disorder, as well as a PM phase at high temperatures.

The random-field term in Eq. (1) introduces a third axis to Fig. 1. For both limits of zero $J$ and finite $J_{0}$ (pure ferromagnet) and zero $J_{0}$ and finite $J$ (pure EA spin glass) the effect of finite random field has been thoroughly studied Young and Nauenberg (1985); Rieger (1993, 1995); Ogielski and Huse (1986); Young (1998); Sethna et al. (1993); Perkovic et al. (1995); Bertotti (2006); Thouless et al. (1980); Fisher and Huse (1986, 1987); Katzgraber and Young (2005). For the latter limit two different pictures to describe finite-dimensional spin glasses have been proposed: the replica symmetry breaking (RSB) picture based on the solution of the Sherrington-Kirkpatrick model Parisi (1979, 1983); Rammal et al. (1986); Mézard et al. (1987); Parisi (2008), which predicts the existence of a spin-glass phase at finite fields and the droplet picture McMillan (1984); Fisher and Huse (1986, 1987, 1988), which contrary to the RSB predicts the instability of the spin-glass phase for any infinitesimal small field. Which of these pictures correctly describes the three-dimensional spin glass is still an open question Young and Katzgraber (2004); Katzgraber and Young (2005); Jörg et al. (2008); Katzgraber et al. (2009); Fernández (2010); Yucesoy et al. (2012); Billoire et al. (2014); Morais et al. (2016). In this paper, our theoretical considerations are based on the droplet picture, but we do not undermine the possibility to derive an analogous prediction within the RSB picture. Specifically, within the droplet picture of spin glasses, it is argued Fisher and Huse (1986) that the spin-glass phase is unstable to infinitesimal random fields, as finite-size glassy domains are created, destroying long-range glass order. Here we study the effect of the random field in the range, where the system is a ferromagnet, albeit with competing interactions ($0<J/J_{0}<1.65$). At $T=0$ we obtain the phase diagram as a function of $J$ and $H_{r}$. At finite temperature we obtain the phase diagram for different values of $J$ in the ferromagnetic regime, in proximity to the spin-glass phase and deep inside the ferromagnetic regime. We find the dependence of the critical temperature $T_{c}$ on random-field strength and study the nature of the phase transition.

III Methods

III.1 Zero-temperature simulations

For the zero-temperature simulations we use the jaded extremal optimization heuristic Boettcher and Percus (2001); Middleton (2004). We set the algorithm parameter $\tau=1.6$, $1.8$, $1.9$ with an aging parameter $\Gamma=0.0001$ for at least $2^{24}$ simulation steps. Each disorder realization undergoes at least $512$ independent runs. We monitor how many times the lowest energy configuration is found. When the success rate is more than $\sim 5\%$, we assume to have found the ground-state configuration. Ground states are found with high confidence for $L\leqslant 10$ for $H_{r}=0$ and $L\leqslant 8$ for $H_{r}\neq 0$.

The FM to SG (PM) phase boundary is identified through the Binder ratio Binder (1981)

[TABLE]

where $m=1/N\sum_{i}S_{i}$ is the magnetization of the system, $N=L^{3}$ is the number of spins and $\left[\cdots\right]_{\rm av}$ represents a disorder average. The Binder ratio $g$ is a dimensionless observable that scales as $g\sim\tilde{G}\left[L^{1/\nu}(J-J_{c})\right]$. The argument vanishes if $J=J_{c}$ for all linear system sizes $L$. Therefore, the crossing of the curves for different system sizes gives an estimate of the transition value $J_{c}$ up to finite-size effects. Simulation parameters are shown in table 4.

The critical disorder $J_{c}$ and the standard deviation for each random field $H_{r}$ are estimated using a Levenberg-Marquardt minimization combined with a bootstrap analysis, where we assume that the universal function $g$ is well approximated by a third-order polynomial. We show in Figure 2 how the estimated $J_{c}$ agrees with the crossing of the Binder ratio curves when $H=0.00$. The dashed lines are fit functions to the data used only for visualization purposes and the vertical line is the average of the intersections of all samples, in this case $J_{c}=1.63(1)$.

III.2 Finite-temperature simulations

The simulations at finite temperature are done using a combination of single-spin-flip Monte Carlo with parallel tempering Monte Carlo Hukushima and Nemoto (1996). To determine the finite-temperature transitions for a fixed random field strength $H_{r}$ and disorder $J$ we measure the ferromagnetic and spin-glass two-point correlation length Ballesteros et al. (2000)

[TABLE]

where ${\bf k}_{\rm min}=(2\pi/L,0,0)$, $\chi_{\rm sg}\left({\bf k}\right)$ is the spin-glass wave-dependent susceptibility

[TABLE]

$q_{i}=S_{i}^{\alpha}S_{i}^{\beta}$ is the two-replica spin overlap, and $\chi_{\rm fm}\left({\bf k}\right)$ is the ferromagnetic wave-dependent susceptibility

[TABLE]

Here, $\langle\cdots\rangle$ denotes a thermal average and $\left[\cdots\right]_{\rm av}$ a disordered average.

Near the transition the dimensionless ratio of the two-point correlation functions $\xi_{L}^{\rm sg,fm}$ and the linear system size $L$ scales as $\xi_{L}^{\rm sg,fm}/L\sim\tilde{X}\left[L^{1/\nu}\left(T-T_{c}\right)\right]$. At the critical temperature the argument vanishes and the dimensionless quantity becomes size independent (up to scaling corrections), hence we expect lines of different system sizes to cross at $T_{c}$. If, however, the lines do not cross, no transition takes place at the studied temperature range. Some example cases are shown in Fig. 3. In Fig. 3(a) the FM two-point correlation length for $J=1.60$ ($J_{0}/J=0.625$) and $H_{r}=0.00$ is depicted. Clearly, the curves cross at a putative point which indicates a transition at a temperature $T_{c}$. In Fig. 3(b) we plot the same quantity, but for a higher disorder value $J=1.80$ ($J_{0}/J=0.5\bar{5}$). In this case, clearly the curves do not intersect at any studied temperature, showing a lack of a ferromagnetic phase transition in this temperature range. Moreover, it suggests a possible lack of ferromagnetic transition at any $T>0$. In Fig. 3(c) we show the spin-glass two-point correlation length for the same disorder value $J=1.80$ as in Fig. 3(b). The curves do cross in the simulated temperature range, signaling a PM to SG phase transition.

For small random field strengths and interactions distribution widths the equilibration time is relatively short, making it possible to simulate large system sizes. With increasing random fields and interaction distribution widths equilibration times become longer. To determine the critical temperature we approximate the scaling function by a third-order polynomial and perform a fit with six free parameters. To estimate the error bars we use a bootstrap analysis as described by Katzgraber et al. Katzgraber et al. (2006). Simulation parameters for the finite-temperature simulations are shown in Tables 1, 2, and 3.

IV Results

The phase diagram of the model presented in Sec. II is shown in Fig. 1. It has a high temperature PM phase and low temperature FM and SG phases at small and strong disorder $J$, respectively. The boundary between the FM and the SG phase is at $J\approx 1.65$ ($J_{0}/J\approx 0.606$) with a weak reentrance at $T=0$ [$J_{c}=1.63(1)$]. The obtained phase diagram qualitatively agrees with previous resultsSherrington and Southern (1975); Sherrington and Kirkpatrick (1975); Southern and Young (1977) for the model in Eq. (1), and for the closely related diluted bimodal Ising SG modelHasenbusch et al. (2007). In comparison to the real space rescaling method by Southern and Young Southern and Young (1977), we find larger values for $T_{c}$ and for $J_{c}$, in accordance with the typical behavior of real-space rescaling method Southern and Young (1977). Note the linear dependence of the PM to SG transition on the mean interaction value. The same is true for the PM to FM phase transition in the regime $J\lesssim 1.2$ ($J_{0}/J\gtrsim 0.83\bar{3}$).

We now study the phase diagram with the inclusion of finite random fields and consider first the case where $T=0$, see Fig. 4. The FM phase at low disorder $J$ and low random-field strength $H_{r}$ is disordered at large $J$ into a QSG phase, consisting of finite size glassy domains, but no long range glass order except at $H_{r}=0$. The data point at $H_{r}=0$ corresponds to the FM to SG transition point at $T=0$ plotted also in the $T/J$ vs $J_{0}/J$ phase diagram shown in Fig. 1. At small $J$ and large $H_{r}$ the FM phase disorders into a PM phase. The data point at $J=0$ corresponds to the critical random field value $H_{r}\approx 2.16$ of the random-field Ising model. For low disorder $J$ we expect the critical random field to be of the order of the mean interaction $J_{0}$, because in the PM phase single spins order along their local effective field.

The situation for $J\lesssim J_{c}$ is, however, different. At $H_{r}=0$ the FM ground state and the lowest-energy SG state differ in energy, the difference being linear in $J-J_{c}$ Andresen et al. (2013). At finite $H_{r}$, the low-energy SG state changes profoundly: long-range order is destroyed, finite-size glassy domains appear, and the energy of the resulting QSG state is reduced accordingly Schechter and Laflorencie (2006); Andresen et al. (2013). This leads to a reduction of the energy of the QSG state below that of the ferromagnetic state, and thus to a zero-temperature phase transition at finite $H_{r}$, which is much smaller than $J_{0}$. The predicted functional form for the boundary between the FM and the PM phases at $T=0$ is given by Andresen et al. (2013)

[TABLE]

Here, $\theta\approx 0.19$ is the stiffness exponent Hartmann (1999). Indeed, our numerical results are in agreement with this functional form, as can be seen by fitting the data for $H_{r}<1$, fixing the power to its theoretically obtained value Andresen et al. (2013) $\beta=(3/2-\theta)/(3-\theta)=0.466$, and using the above estimated critical disorder $J_{c}\approx 1.63$ for the fit. This leaves the proportionality prefactor $\alpha$ as the single free fitting parameter. The inset of Fig. 4 shows the fitting result. We note that our data are limited to $H_{r}\leq 0.5$. It was not possible to determine the crossing point of the Binder ratio curves for different system sizes for random fields with $H_{r}>0.5$ because of the proximity of the crossing point to $g=1$.

At finite temperature and for $J\lesssim J_{c}$, similar considerations to the ones mentioned above manifest themselves in the dependence of $T_{c}$ on the random field. The underlying glassy state at finite temperature consists of paramagnetic SG domains of typical size $\xi$. We denote this phase as “pQSG”. From moderate $H_{r}$, which does not affect the typical domain size, the reduction of the energy per spin of the pQSG state is $\propto H_{r}/\xi^{3/2}$. As a result, only for $H_{r}>h^{*}\propto(J_{c}-J)\xi^{3/2}$ the disordering of the FM phase is by the pQSG phase. In this regime theory predicts a linear dependence of $T_{c}$ on $H_{r}$ Andresen et al. (2013). At $H_{r}<h^{*}$ the disordering is to the standard PM phase, with the known weak dependence of $T_{c}$ on $H_{r}$. All the above considerations do not apply to $J\ll J_{c}$, where the FM is far from the SG phase, and the disordering to the PM phase is standard for all strengths of random field.

In Fig. 5 we present the $T$ vs $H_{r}$ phase diagrams for disorders $J=0.50$, $J=1.50$, and $J=1.60$. In Fig. 5(a) $J\ll J_{c}$. The disordering is into the standard PM phase, with quadratic dependence $T(H_{r})-T_{c}(0)\propto H_{r}^{2}$ for small $H_{r}$ and a small reentrance at low temperatures. For $J=1.50$ [panel (b)] the system is close to the SG phase. Indeed, for $H_{r}<h^{*}\approx 0.1$ the dependence of $T_{c}$ on $H_{r}$ is weak, as is expected for disordering to the standard PM phase. For $H_{r}>h^{*}$ we find the dependence of $T_{c}$ on $H_{r}$ to be linear, in agreement with disordering to the pQSG phase. We note that at $0.5<H_{r}<0.6$ there is a sharp decrease of $T_{c}$ to zero within the finite-temperature simulations, and we find a value of $H_{r}=0.39$ for the zero temperature transition (inferred from the fitting function in the inset of Fig. 4). This is in agreement with the disordering field by the QSG phase being much smaller than the interaction strength also at low temperatures. We note that as the QSG phase is frozen, susceptibility measurements are expected to depict only the crossover between the QSG and the PM phase at large random fields. This is in agreement with Ref. Silevitch et al. (2007), where a sharp feature in the susceptibility is observed at higher temperatures, depicting the transition to the pQSG phase, but a smooth crossover is observed for the lower temperatures. The observed peak value at the crossover occurs at larger $H_{r}$ with diminishing peak value as temperature is reduced Silevitch et al. (2007), in agreement with the scenario of the QSG having smaller glassy domains as a function of increasing $H_{r}$ Schechter and Laflorencie (2006); Andresen et al. (2013). In panel (c) of Fig. 5 we present the results for $J=1.60$, closer to the zero-field FM-to-SG transition. Results are similar to the case of $J=1.50$, with a weaker linear dependence, and with a smaller random field, $0.3<H_{r}<0.4$, which disorders the FM at all temperatures.

Let us now consider the order of the FM to PM/pQSG transition. The order of the phase transition of the RFIM is a long standing question. Whereas analytic arguments support a second-order transition controlled by a zero-temperature fixed point Bray and Moore (1985); Villain (1985); Fisher and Huse (1986), some numerical results support a first-order transition Young and Nauenberg (1985); Rieger (1993) while others support a second-order transition Ogielski and Huse (1986); Ogielski (1986); Rieger (1995); Ahrens and Hartmann (2011); Ahrens et al. (2013); Fytas and Martin-Mayor (2013); Theodorakis and Fytas (2014). For bimodal disorder distributions a mean-field solution suggests a first-order transition for large enough values of the random field Aharony (1978a). Recent numerical work supports universality in the RFIM Fytas and Martin-Mayor (2013), suggesting that the nature of the phase transition is independent of the random-field distribution. Moreover, high-accuracy estimates for the magnetic exponent ratio $\beta/\nu$ Theodorakis and Fytas (2014) found the value to be very small, but clearly finite. In the present work, for small exchange disorder ($J=0.50$) we do not find any signature of a first-order phase transition. This suggests that the recent phase transition behavior found for the RFIM Fytas and Martin-Mayor (2013); Theodorakis and Fytas (2014) also applies for models with small exchange disorder.

We now consider the order of the transition in the presence of both random fields and strong random interactions. Intriguingly, our results suggest, for strong interaction disorder, a continuous transition for $H_{r}<h^{*}$ where the FM disorders into a “standard” PM phase, and a first-order transition for $H_{r}>h^{*}$ where the disordering is into the pQSG phase. In the latter, we find a discontinuous jump in the magnetization across the transition, but from our microcanonical simulations, based on a recently proposed method Martin-Mayor (2007); Ferniández et al. (2008); Fernández et al. (2012), we did not detect any latent heat, which is either zero, or too small to be detected in the current accessible system sizes, similar to Refs. Rieger (1993, 1995).

Specifically, we study the distribution of the magnetization above and below $T_{c}$ for $J=1.50$ at different field strengths $H_{r}$, as shown in Fig. 6. A bimodal distribution close to $T_{c}$ is a sign of a first-order phase transition, whereas a normal distribution with mean $\mu_{0}=\left[\langle|m|\rangle\right]_{\rm av}$ would be the expected distribution for a continuous phase transition. For small $H_{r}=0.04<h^{*}$ [panel (a) of Fig.  6] we note that the single-Gaussian fit and the double-Gaussian fit agree, suggesting that the magnetization distribution is likely normal close to $T_{c}$ and therefore the phase transition is continuous. However, we find that for large $H_{r}=0.40>h^{*}$ the double-Gaussian fit does differ considerably from the single-Gaussian fit, moreover, the double Gaussian fit reproduces better the magnetization histogram below and above $T_{c}$. This suggests that the magnetization jumps at $T_{c}$, and thus the existence of a first-order phase transition.

To strengthen the claim of a bimodally-distributed magnetization for large random fields close to $T_{c}$, we perform a bootstrap analysis of the double-Gaussian fit for $J=1.50$ and $H_{r}=0.30$, $0.40$, and $0.50$, and show the normalized difference between the mean of the two modes as a function of the temperature $T$ (see Fig. 7). We observe that the maximum values of the curves are close to $T_{c}$. The fact that the relative difference between the modes is maximal close to the critical temperature gives further evidence for a first-order phase transition. These distributions show large finite-size corrections and a clean determination of the critical temperature is difficult.

V Discussion

Disordered nonmagnetic ferroic systems naturally show randomness in both their inter-particle interactions and in the effective field presented as a bias between the two degenerate single-particle states. Recently, it was shown that this interplay of randomness of interactions and fields is present also in the ferromagnetic phase of the ${{\rm LiHo_{x}Y_{1-x}F_{4}}}$ compound. This has allowed for new experiments Silevitch et al. (2007, 2010) investigating this interplay of randomness, as well as for new insights into older experiments, e.g., demonstrating different final states obtained by in-field and zero-field annealing of the disordered ferromagnetic ${{\rm LiHo_{x}Y_{1-x}F_{4}}}$ system Brooke et al. (1999).

The effective random field in the ${{\rm LiHo_{x}Y_{1-x}F_{4}}}$ system is a consequence of the interplay between the off-diagonal terms of the dipolar interactions and the applied transverse field. The applied field also induces transitions between the spin up ${\mid\uparrow\rangle}$ and spin down ${\mid\downarrow\rangle}$ states in Ho, thus giving rise to an effective transverse field term in the Ising Hamiltonian. However, at low field, the effective random field dominates because it is linear in the applied transverse field Schechter and Laflorencie (2006); Schechter (2008), whereas the effective transverse field is negligible at small fields Schechter and Stamp (2005, 2008).

In a previous study Andresen et al. (2013) we analyzed the ${{\rm LiHo_{x}Y_{1-x}F_{4}}}$ system in the regime where the system is ferromagnetic, albeit with disorder in both the interaction and the effective longitudinal field. We have suggested a novel disordering mechanism where at a finite random field that is much smaller than the typical interaction, finite-size glassy domains disorder the FM phase into a PM phase. This mechanism explains various experimental features, such as the linear dependence of the critical temperature with increased random field, and the diminishing and rounding of the susceptibility peak with decreasing temperature Silevitch et al. (2007).

The scaling theory within which the novel disordering mechanism was obtained Andresen et al. (2013) is not particular to dipolar systems. In the present work we have shown that the same mechanism applies to a generic short-range random-field Ising model with competing interactions having a ferromagnetic mean. Our results thus support the generality of the disordering mechanism to random-field ferroic systems with competing interactions, and it would be of interest to check their applicability, e.g., to relaxor ferroelectricsWestphal et al. (1992); Akbarzadeh et al. (2012); Sherrington (2013, 2015).

We find an excellent agreement, quantitative and qualitative, between our numerical results and the predictions of the scaling theory based on the picture of the disordering of a FM with competing interactions by a QSG (pQSG) phase at low (high) temperature. It would be of much interest to further corroborate our results here with a direct microscopic analysis of the domain structure in the disordered phase.

We further analyze here the nature of the FM to PM transition and show evidence suggesting that once the disordering is induced via the above-mentioned mechanism, where the FM phase is disordered by finite size glassy domains (i.e. to the pQSG phase), then the transition is first order. This differs from the RFIM with no competing interaction. In the latter, it is believed that the thermodynamic phase transition is second orderImry and Ma (1975), albeit experimentally hard to be observed because of slow relaxation. It would be of much interest to further study this transition between the FM and pQSG phases, and the dynamics of the system near the transition, as hysteresis and slow relaxation are expected. With regard to the latter, nonequilibrium dynamics of the Ising model were recently studied in the presence of random bondsManssen and Hartmann (2015) or in the presence of random fieldsvon Ohr et al. (2017). For both systems it was shown that the equilibrium disorder-driven transition shows up when measuring the nonequilibrium aging properties. It would be of interest to study whether the quasi-SG driven transition discussed here has distinct characteristics in the non equilibrium aging properties of the system.

Acknowledgements.

We would like to thank Amnon Aharony, Michael Moore and David Sherrington for useful discussions and valuable input. J.C.A. acknowleges support by the Göran Gustafsson Foundation. M.S. acknowledges support from the Israel Science Foundation (Grant No. 821/14). H.G.K. acknowledges support from the National Science Foundation (Grant No. DMR-1151387) and would like to thank Quantum Kellerbier for inspiration. The research of H.G.K. is based upon work supported in part by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), via MIT Lincoln Laboratory Air Force Contract No. FA8721-05-C-0002. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of ODNI, IARPA, or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purpose notwithstanding any copyright annotation thereon. We thank Texas A&M University for access to their Ada and Curie clusters, the Swedish National Infrastructure for Computing for access to Beskow and Triolith clusters, the KTH Department of Theoretical Physics for access to the Octopus cluster, Mikael Twenstöm for access to the Termina cluster, and ETH Zurich for access to the Brutus cluster.

VI Appendix

The simulation parameters for Fig. 5 are shown in Tab. 1, Tab. 2 and Tab. 3 for $\tilde{J}=0.50$, $\tilde{J}=1.50$, and $\tilde{J}=1.60$, respectively. The simulation parameters of the zero-temperature simulations are shown in Tab. 4.