# Text-guided HuBERT: Self-Supervised Speech Pre-training via Generative
Adversarial Networks

Duo Ma, Xianghu Yue∗, Junyi Ao, Xiaoxue Gao, Haizhou Li Duo Ma, Junyi Ao, and
Haizhou Li, Shenzhen Research Institute of Big Data, School of Data Science,
The Chinese University of Hong Kong, Shenzhen 518172, China (email:
<EMAIL_ADDRESS><EMAIL_ADDRESS>haizhouli@cuhk.edu.cn).Xianghu
Yue, Xiaoxue Gao, Department of Electrical and Computer and Engineering,
National University of Singapore, Singapore 119077 (e-mail:
<EMAIL_ADDRESS>xiaoxue.gao@u.nus.edu).This work is supported by Huawei
Noah’s Ark Lab, and the National Natural Science Foundation of China (Grant
No. 62271432). ∗ Corresponding author.

###### Abstract

Human language can be expressed in either written or spoken form, i.e. text or
speech. Humans can acquire knowledge from text to improve speaking and
listening. However, the quest for speech pre-trained models to leverage
unpaired text has just started. In this paper, we investigate a new way to
pre-train such a joint speech-text model to learn enhanced speech
representations and benefit various speech-related downstream tasks.
Specifically, we propose a novel pre-training method, text-guided HuBERT, or
T-HuBERT, which performs self-supervised learning over speech to derive
phoneme-like discrete representations. And these phoneme-like pseudo-label
sequences are firstly derived from speech via the generative adversarial
networks (GAN) to be statistically similar to those from additional unpaired
textual data. In this way, we build a bridge between unpaired speech and text
in a unsupervised manner. Extensive experiments demonstrate the significant
superiority of our proposed method over various strong baselines, which
achieves up to 15.3% relative Word Error Rate (WER) reduction on the
LibriSpeech dataset.

###### Index Terms:

self-supervised learning, pre-training, speech representation

##  I Introduction

Speech pre-trained models [1, 2, 3] have recently advanced the state-of-the-
art in speech-related tasks, such as speech recognition [3, 1, 2], speaker
verification [4, 5] and speech enhancement [6, 7]. Speech and text are two
important mediums of human communication, and they can be directly converted
into each other through speech synthesis and recognition systems. We human
beings can acquire knowledge from additional text to improve speaking and
listening. However, the quest for speech pre-trained models to leverage
unpaired text has just started. SpeechLM [8], MAESTRO [9], and SLAM [10],
utilize paired data to bridge the gap between speech and text and learn
enhanced speech representations. token2vec [11] offers another approach by
converting speech waveforms into discrete code sequences and text into phoneme
sequences, thus leveraging unpaired textual data to enhance speech pre-
training. Despite the conceptual similarity between speech codes and text
phonemes, token2vec does not explicitly establish a direct correspondence
between the two modalities.

In this work, we propose a novel technique to leverage additional unpaired
text for speech pre-training. By enhancing the phonetic information of speech
representations, we finally benefit various downstream speech-related task. To
achieve this, we build a bridge between unpaired speech and text in an
unsupervised manner. Specifically, we derive phoneme sequences from speech
data via the generative adversarial networks (GAN) to be statistically similar
to those from the text data. These derived phoneme-like pseudo labels are then
together with the original targets, following the principles of HuBERT [2],
serve as hierarchical targets for speech pre-training to learn high-level
semantic speech representations.

The contributions of T-HuBERT can be summarized as follows: 1) We first
leverage unpaired text information to obtain phoneme-like modelling units from
speech via unsupervised training, thus eliminating the need of paired data as
a bridge for joint pre-training of speech and text [8, 12, 10, 13]. 2)
Compared with the hidden unit of HuBERT, the derived phoneme-like modelling
units could effectively guide the pre-trained model to improve its speech
representation. 3) Based on previous empirical evidence in [14], we show that
the phoneme-like modelling units and the hidden units can jointly guide the
pre-trained model to achieve better speech representation.

Figure 1:  Illustration of our T-HuBERT model. In the left part, we use HuBERT
to discretize the speech to get the pseudo code labels, and the middle part is
the main structure of our model. In the right part, we use the GAN network to
discretize the speech to get the phoneme-like pseudo-label sequences.

##  II Methodology

Given unpaired speech and text data, how does the speech pre-trained model
leverage the additional unpaired text to benefit the downstream speech-related
tasks? As speech and text are two different forms of a language, they are
inherently similar in terms of phonetic, phonotactic, and lexical
distribution. The question is how we draw an equivalence between the self-
organized phonetic units, i.e. HuBERT codes, and their phonetic equivalents
derived from text.

The success of HuBERT shows that a poor teacher can teach a good student, so
it is logical that if a better teacher is found, a better student speech pre-
training model can be delivered. To this end, inspired by wav2vec-U 2.0 [15],
we introduce an unsupervised approach to obtain the discrete representations
of frame-level speech waveform via the adversarial training between speech and
text. In this way, we build a bridge between unpaired speech and text without
an explicit supervision signal. Subsequently, these obtained phoneme-like
sequences serves as additional self-supervised targets on the intermediate
layer during pre-training, thus enhancing the phonetic information of the
learned speech representations. The overall framework of our proposed method,
T-HuBERT, is depicted in Figure 1.

###  II-A HuBERT-based Tokenizer

We follow the main settings of Hidden Unit BERT (HuBERT), a classic self-
supervised speech pre-training framework. In HuBERT, there are two iterative
re-clustering and re-training stages. In the first stage, the discrete targets
are obtained by clustering on the MFCC features of the training data. In the
second stage, a new generation of training targets are derived by clustering
on the learned representations from the first stage model. Following the
common practice of self-supervised learning (SSL), such as wav2vec 2.0 and
HuBERT, we apply these k-means codes in the second stage as the targets of the
top layer of the T-HuBERT model.

###  II-B GAN-based Tokenizer

To leverage additional text data and improve the phonetic information of the
pre-training, we utilize the GANs [16] as a bridge between unpaired speech and
text, to discretize the speech waveform into phoneme-like pseudo-label
sequences in a unsupervised manner. Specifically, as shown in the right part
of Figure 1, the generator consumes speech, while the input of the
discriminator is the real phoneme sequences of unpaired text. During
adversarial training, the discriminator is trained to distinguish the output
of the generator and real phoneme sequences. The goal of the generator is to
output phoneme sequences that cannot be distinguished by the discriminator.

We adopt the same unsupervised training strategy as wav2vec-U 2.0 [15].
Specifically, we train the network, on a traditional GAN loss,
$\mathcal{L}_{gan}$, a gradient penalty term, $\mathcal{L}_{gp}$, for better
convergence, a smoothness penalty term, $\mathcal{L}_{sp}$, to encourage
consecutive speech frames to output the same phonemes, a phoneme diversity
term, $\mathcal{L}_{pd}$, to diverse phoneme usage, and a self-supervised loss
$\mathcal{L}_{ss}$, to encourage the generated phonemes to match the input
speech. The final GAN training objective is formulated following:

$\mathcal{L}=\min_{\mathcal{G}}\max_{\mathcal{C}}[\mathcal{L}_{gan}+\lambda\mathcal{L}_{gp}+\gamma\mathcal{L}_{sp}+\eta\mathcal{L}_{pd}+\delta\mathcal{L}_{ss}]$
(1)

After adversarial training, we use the generator to discretize the speech
waveform into frame-level phoneme-like pseudo sequences, which can be
considered to be obtained under the supervision of additional unpaired text.

###  II-C Multi-Unit and Multi-Layer Supervision Objective

Our model architecture design mainly follows HuBERT, which contains a
convolutional feature encoder and a Transformer encoder. The convolutional
encoder is composed of seven temporal convolution blocks, which have 512
channels with strides $(5,2,2,2,2,2,2)$ and kernel widths $(10,3,3,3,3,2,2)$,
resulting in a 20ms frame rate for audio sampled at 16kHz. Before these
features are feed to the transformer encoder, we randomly mask them in time
dimension with a probability of 0.8. For the transformer encoder, we use a
bucket relative position embedding[17], which generates different learned
embeddings based on the offset between the ”key” and the ”query” in the
vanilla self-attention.

To leverage the HuBERT-based and GAN-based targets, we further propose a
multi-unit and multi-layer supervision objective, which could effectively
guide the pre-trained model to learn rich phonetic information, thus
benefiting downstream tasks. Specifically, during pre-training, we further
select one intermediate layer $k$ as the supervised layer of GAN-based
targets, besides the traditional HuBERT-based targets for the top layers, and
calculate the masked prediction loss, which can summarized as follows:

$\mathcal{L}=-\sum_{l\in\\{k,L\\}}\sum_{t\in\mathcal{M}}{log\
p^{l}(c^{l}_{t}|{\mathbf{o}^{l}_{t})}}$ (2)

where $L$ is the number of encoder layer, $\mathcal{M}$ denotes the set of
masked timesteps, $c^{l}_{t}$ denotes the corresponding codes at timestep $t$,
and $\mathbf{o}^{l}_{t}$ denotes the output hidden states of layer $l$ at
timestep $t$. And the distribution over the codes is parameterized with:

${p^{l}(c^{l}_{t}|{o^{l}_{t})}=\frac{exp(sim(\mathbf{W}^{l}{o^{l}_{t}},\mathbf{e}^{l}(c^{l}_{t}))/\tau)}{\sum_{c^{\prime}\in
C}{exp(sim(\mathbf{W}^{l}{o^{l}_{t}},\mathbf{e}^{l}(c^{\prime}))/\tau})}}$ (3)

where $\mathbf{W}$ is the project matrix, and $\mathbf{e}$ is the embedding
for code $c$.

TABLE I: WER results on LibriSpeech test sets when fine-tuned on the Libri-Light low-resource labeled data setups of 1 hour, 10 hours and the train-clean-100 splits of LibriSpeech. Model | No LM | LM
---|---|---
clean | other | clean | other
1-hour subset |  |  |  |
DeCoAR 2.0 [18] | - | - | 13.8 | 29.1
DiscreteBERT [19] | - | - | 9.0 | 17.6
wav2vec 2.0 [1] | 24.5 | 29.7 | 5.5 | 11.3
HuBERT [2] | 20.9 | 27.5 | 6.1 | 11.3
WavLM [3] | 24.5 | 29.2 | 5.7 | 10.8
token2vec [11] | 20.0 | 25.8 | - | -
T-HuBERT | 16.4 | 22.2 | 6.4 | 11.9
10-hour subset |  |  |  |
DeCoAR 2.0 [18] | - | - | 5.4 | 13.3
DiscreteBERT [19] | - | - | 5.9 | 14.1
wav2vec 2.0 [1] | 11.1 | 17.6 | 4.3 | 9.5
HuBERT [2] | 10.1 | 16.8 | 4.3 | 9.4
WavLM [3] | 9.8 | 16.0 | 4.3 | 9.2
token2vec [11] | 9.0 | 14.9 | - | -
T-HuBERT | 7.8 | 13.7 | 4.1 | 9.1
100-hour subset |  |  |  |
DeCoAR 2.0 [18] | - | - | 5.0 | 12.1
DiscreteBERT [19] | - | - | 4.5 | 12.1
wav2vec 2.0 [1] | 6.1 | 13.3 | 3.4 | 8.0
HuBERT [2] | 6.3 | 13.2 | 3.4 | 8.1
WavLM [3] | 5.7 | 12.0 | 3.4 | 7.7
token2vec [11] | 5.1 | 11.8 | - | -
PBERT [20] | 4.2 | 9.5 | 3.1 | 7.2
SpeechLM-P [8] | 3.4 | 8.1 | 2.7 | 6.2
T-HuBERT | 4.8 | 10.0 | 3.1 | 7.0

##  III Experiments

###  III-A Datasets

For unsupervised speech pre-training, we use the publicly available 960-hour
LibriSpeech [21] dataset. For unpaired text data, we use the LibriSpeech LM
corpus 111https://www.openslr.org/resources/11/librispeech-lm-norm.txt.gz,
containing about 40M English sentences. For the downstream ASR task, we use
the train-clean-100 split of LibriSpeech, the 1-hour and 10-hour split data of
Libri-Light [22] as the low-resource scenarios. We further evaluate on the
SUPERB benchmark [23], which is designed to provide a standard and
comprehensive testbed for pre-trained speech models on various speech tasks.

###  III-B Experimental Setup

All pre-training experiments are conducted using fairseq toolkit [24]. For
GAN-based tokenizer, we mainly follow the training process and hyper-
parameters in [15]. Specifically, we first use the unsupervised voice activity
detection [25] to remove the silences from speech, and then feed these speech
into the pre-trained speech model, i.e., wav2vec 2.0, to extract high-level
speech representations. For unpaired text, we convert the word sequences into
phoneme sequences using a off-the-shelf phonemizer [26], which contains a
total of 41 monophones, including a silence symbol. The generator consists of
a single-layer CNN with batch normalization and the discriminator employs a
three-layer CNN that takes either the output of the generator or a one-hot
representation of the real phoneme sequence as input.

All pre-training models are trained on 8 GPUs for a total of 400k steps. We
randomly sample starting positions with a probability of 0.08 and apply a mask
to the subsequent 10 time steps. We employ the Adam optimizer [27] with a
weight decay of 0.01 and $\beta=(0.9,0.98)$. The learning rate follows a
linear increase for the first 32k steps and subsequently decays linearly to 0.
The peaking learning rate is 5e-4. Following [2, 1], we use the Connectionist
Temporal Classification (CTC) [28] loss for downstream ASR fine-tuning. The
target vocabulary includes 26 English characters, a space token, and an
apostrophe. We train the model with Adam optimizer and a tri-stage rate
schedule where the learning rate is warmed up for the first 10% of the
updates, held constant for the next 40%, and then linearly decayed for the
remainder. During the inference stage, we use the flashlight [29]
222https://github.com/flashlight/text beam search decoder with beam size 1500
for a 4-gram language model (LM)
333https://www.openslr.org/resources/11/4gram.arpa.gz.

TABLE II: Universal speech representation evaluation on the SUPERB benchmark. Method | #Params | Corpus | Speaker | Content | Semantics | ParaL
---|---|---|---|---|---|---
SID Acc$\uparrow$ |  ASV EER$\downarrow$ |  SD DER$\downarrow$ |  PR PER$\downarrow$ |  ASR WER$\downarrow$ |  KS Acc$\uparrow$ |  QbE MTWV$\uparrow$ |  IC Acc$\uparrow$ |  SF F1$\uparrow$ CER$\downarrow$ |  ER Acc$\uparrow$
Fbank | 0 | - | 8.5E-4 | 9.56 | 10.05 | 82.01 | 23.18 | 8.63 | 0.0058 | 9.10 | 69.64 52.94 | 35.39
PASE+ | 7.83M | LS50hr | 37.99 | 11.61 | 8.68 | 58.87 | 25.11 | 82.54 | 0.0072 | 29.82 | 62.17 60.17 | 57.86
TERA | 21.33M | LS960hr | 57.57 | 15.89 | 9.96 | 49.17 | 18.17 | 89.48 | 0.0013 | 58.42 | 67.50 54.17 | 56.27
DeCoAR 2.0 | 89.84M | LS960hr | 74.42 | 7.16 | 6.59 | 14.93 | 13.02 | 94.48 | 0.0406 | 90.80 | 83.28 34.73 | 62.47
wav2vec | 32.54M | LS960hr | 56.56 | 7.99 | 9.90 | 31.58 | 15.86 | 95.59 | 0.0485 | 84.92 | 76.37 43.71 | 59.79
vq-wav2vec | 34.15M | LS960hr | 38.80 | 10.38 | 9.93 | 33.48 | 17.71 | 93.38 | 0.0410 | 85.68 | 77.68 41.54 | 58.24
wav2vec 2.0 | 95.04M | LS960hr | 75.18 | 6.02 | 6.08 | 5.74 | 6.43 | 96.23 | 0.0233 | 92.35 | 88.30 24.77 | 63.43
HuBERT | 94.68M | LS960hr | 81.42 | 5.11 | 5.88 | 5.41 | 6.42 | 96.30 | 0.0736 | 98.34 | 88.53 25.20 | 64.92
SpeechLM-P | 94.70M | LS960hr | 75.24 | 5.97 | 7.34 | 3.10 | 4.98 | 94.09 | 0.0410 | 97.68 | 87.67 25.90 | 61.84
T-HuBERT | 95.05M | LS960hr | 61.64 | 5.37 | 6.43 | 3.87 | 5.12 | 96.27 | 0.0799 | 98.81 | 89.27 22.72 | 65.28

##  IV Results and Analysis

###  IV-A Main Results

To demonstrate the effectiveness of our proposed T-HuBERT model, we first
evaluate it on the standard LibriSpeech test sets. We mainly compare T-HuBERT
with three kinds of models, including 1) speech-only pre-training models
(e.g., wav2vec 2.0, HuBERT and WavLM), 2) speech-text pre-training models that
use paired text data (e.g., PBERT and SpeechLM), 3) speech-text pre-training
model that uses unpaired text data (e.g., token2vec). Table I shows the WER
results when the model is fine-tuned on 1 hour, 10 hours, and 100 hours of
labeled data, respectively. Firstly, our T-HuBERT achieves state-of-the art
performance with significant improvement over all methods on low-resource
scenarios (e.g., 1-hour and 10-hour subsets). Secondly, on the train-clean-100
split, T-HuBERT obtains 4.8/10.0% WER, which outperforms all pre-trained
models that only use speech (e.g., 5.7/12.0% of WavLM) and unpaired speech and
text data (e.g., 5.1/11.8% of token2vec). This indicates that our T-HuBERT can
better leverage additional text data in a unsupervised manner than token2vec.
However, our T-HuBERT (3.1/7.0%) falls behind SpeechLM (2.7/6.2%) on the
train-clean-100 subset, which is due to that SpeechLM uses 100 hours of paired
data for pre-training, while T-HuBERT doesn’t involve any paired data during
the whole training process.

TABLE III: Performance comparison when the derived GAN-based codes are used as the targets of the last layer. The model is fine-tuned on the train-clean-100 subset. Model | No LM | LM
---|---|---
clean | other | clean | other
wav2vec 2.0 | 6.1 | 13.3 | 3.4 | 8.0
HuBERT | 6.3 | 13.2 | 3.4 | 8.1
token2vec | 5.1 | 11.8 | - | -
T-HuBERT (only phoneme) | 5.0 | 11.2 | 3.1 | 7.4
T-HuBERT | 4.8 | 10.0 | 3.1 | 7.0

###  IV-B Ablation Study

####  IV-B1 The effectiveness of the GAN-based tokenizer

To verify whether the derived phoneme-like pseudo codes through GAN networks
are good teachers for speech pre-training, we replace the targets of original
HuBERT with our GAN-based pseudo codes and pre-train it using 960h
LibriSpeech. Table III shows the results when the model is fine-tuned on the
train-clean-100 subset. Obviously, when we only use the codes derived from GAN
for pre-training, our T-HuBERT (5.0/11.2%) outperforms HuBERT (6.3/13.2%) with
a large margin, and yields comparable results with token2vec (5.1/11.8%). It
suggests the effectiveness of our proposed GAN-based tokenizer that can
effectively leverage additional unpaired text in a unsupervised way to provide
better targets for speech pre-training.

####  IV-B2 The choice of supervision layer for the GAN-based pseudo codes

Here we study the effect of layer choices for the GAN-based pseudo codes. The
HuBERT-based codes are always put at the 12-th layer of the model, and we only
vary the position of the GAN-based codes. Due to limited resources, we conduct
experiments using smaller models with an embedding size of 384, six attention
heads, and feed-forward embedding size of 1536 in each transformer block. As
shown in Table IV, the model that uses 7-th layer for the GAN-based codes
yields best performance. When putting the GAN-based codes at higher layer of
the model, the performance starts to drop.

####  IV-B3 The ratio between unpaired speech and text

In order to investigate the impact of the amount of the unpaired text, we
gradually reduce the unpaired text used during the adversarial training of GAN
and keep the speech data fixed. From Table V, it is obvious that the
performance drops dramatically as we reduce the amount of text, which suggests
that we need large amount of additional unpaired text to provide better
statistics of the phoneme distribution.

TABLE IV: Results comparisons with GAN-based pseudo codes on different intermediate layers. Phoneme layer | Target layers | test-clean | test-other
---|---|---|---
None | 12 | 8.4 | 18.8
6 | 6,12 | 6.6 | 14.3
7 | 7,12 | 6.4 | 14.3
8 | 8,12 | 6.5 | 14.5
9 | 9,12 | 6.9 | 14.9
TABLE V: WER comparison when varying the ratio between speech and text during GAN training. The model is fine-tuned on 1-hour label data of Libri-light. Ratio (Speech/Text) | test-other
---|---
1:110 | 22.2
1:50 | 29.7
1:1 | 31.6
1:0 (speech-only) | 27.5

###  IV-C Universal Speech Representation Evaluation

We further evaluate our T-HuBERT on SUPERB[23] benchmark, which provides a
standard and comprehensive test-bed for evaluating the generalisability of
speech pre-trained models on various tasks from different aspects of speech,
including content, speaker, semantics, and paralinguistics (ParaL). Table II
compares T-HuBERT with other pre-training models on SUPERB, we observe that
T-HuBERT achieves better performance on semantic and paralinguistic related
tasks, while falls behind other models on speaker related tasks. This is might
be due to that the derived phoneme-like pseudo codes via GAN force the model
to learn more phonetic information and discard the speaker characteristics
during pre-training.

##  V Conclusion

This paper proposes a novel pre-training method, T-HuBERT, by leveraging
additional unpaired text to improve its learned speech representations. We
build a bridge between unpaired speech and text via the generative adversarial
networks in a unsupervised way and encourage the pre-trained model to capture
more phonetic information. Extensive experiments and analysis show that our
model outperforms various competitive baselines on LibriSpeech and SUPERB
benchmark. For future work, we will further improve the capabilities of pre-
trained speech models with the help of large language models that contain rich
semantic information.

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Also at ]Dokuz Eylül University, Graduate School of Natural and Applied
Sciences, Turkey

# Effective field theory analysis of $3D$ random field Ising model on
isometric lattices

Ümit Akıncı  Yusuf Yüksel  [  Department of Physics, Dokuz Eylül University,
TR-35160 Izmir, Turkey  Hamza Polat<EMAIL_ADDRESS>Department of
Physics, Dokuz Eylül University, TR-35160 Izmir, Turkey

###### Abstract

Ising model with quenched random magnetic fields is examined for single
Gaussian, bimodal and double Gaussian random field distributions by
introducing an effective field approximation that takes into account the
correlations between different spins that emerge when expanding the
identities. Random field distribution shape dependencies of the phase diagrams
and magnetization curves are investigated for simple cubic, body centered and
face centered cubic lattices. The conditions for the occurrence of reentrant
behavior and tricritical points on the system are also discussed in detail.

PACS numbers

75.10.Hk, 75.30.Kz, 75.50.Lk

††preprint: APS/123-QED

###### Contents

  1. I Introduction
  2. II Formulation
  3. III Results and Discussion
    1. III.1 Phase diagrams of single Gaussian distribution
    2. III.2 Phase diagrams of bimodal distribution
    3. III.3 Phase diagrams of double Gaussian distribution
  4. IV Conclusions

##  I Introduction

Ising model ising ; onsager  which was originally introduced as a model
describing the phase transition properties of ferromagnetic materials has been
widely examined in statistical mechanics and condensed matter physics. In the
course of time, basic conclusions of this simple model has been improved by
introducing new concepts such as disorder effects on the critical behavior of
the systems in question. Ising model in a quenched random field (RFIM) which
has been studied over three decades is an example of this situation. The model
which is actually based on the local fields acting on the lattice sites which
are taken to be random according to a given probability distribution was
introduced for the first time by Larkin larkin  for superconductors and later
generalized by Imry and Ma imry_ma . Lower critical dimension $d_{c}$ of RFIM
has been remained as an unsolved mystery for many years and various
theoretical methods have been introduced. For example, domain wall argument of
Imry and Ma imry_ma  suggests that a transition should exist in three and
higher dimensions for finite temperature and randomness which means that
$d_{c}=2$ grinstein ; fernandez ; imbrie ; bricmont . On the contrary,
dimensional reduction arguments parisi  conclude that the system should not
have a phase transition at finite temperature in three dimensions or fewer, so
$d_{c}=3$ binder ; mukamel2 ; mukamel ; niemi . On the other hand, Frontera
and Vives frontera  showed that a two dimensional ferromagnetic RFIM with
Gaussian random field distribution exhibits order at zero temperature.

At the same time, a great many of theoretical and experimental works have paid
attention on the RFIM and quite noteworthy results have been obtained. For
instance, it has been shown that diluted antiferromagnets such as
$\mathrm{Fe}_{x}\mathrm{Zn}_{1-x}\mathrm{F}_{2}$belanger ; king ,
$\mathrm{Rb}_{2}\mathrm{Co}_{x}\mathrm{Mg}_{1-x}\mathrm{F}_{4}$ferreira ;
yoshizawa  and $\mathrm{Co}_{x}\mathrm{Zn}_{1-x}\mathrm{F}_{2}$yoshizawa  in a
uniform magnetic field just correspond to a ferromagnet in a random uniaxial
magnetic field fishman ; cardy . Following studies have been devoted to
investigate the phase diagrams of these systems in depth and in the mean field
level it was found that different random field distributions lead to different
phase diagrams for infinite dimensional models. For example, using a Gaussian
probability distribution Schneider and Pytteschneider_pytte  have shown that
phase diagrams of the model exhibit only second order phase transition
properties. On the other hand, Aharony aharony  and Matthis matthis  have
introduced bimodal and trimodal distributions, respectively and they have
reported the observation of tricritical behavior. In a recent series of
papers, phase transition properties of infinite dimensional RFIM with
symmetric double crokidakis  and triple salmon  Gaussian random fields have
also been studied by means of a replica method and a rich variety of phase
diagrams have been presented. The situation has also been handled on three
dimensional lattices with nearest neighbor interactions by a variety of
theoretical works such as effective field theory (EFT) borges ; liang2 ;
sarmento ; sebastianes ; kaneyoshi , Monte Carlo (MC) simulations landau ;
machta ; fytas1 ; fytas2 , pair approximation (PA) albayrak  and series
expansion (SE) method gofman . By using EFT, Borges and Silva borges  studied
the system for square $(q=4)$ and simple cubic $(q=6)$ lattices and they
observed a tricritical point only for $(q\geq 6)$. Similarly, Sarmento and
Kaneyoshi sarmento  investigated the phase diagrams of RFIM by means of EFT
with correlations for a bimodal field distribution and they concluded that
reentrant behavior of second order is possible for the system with $(q\geq
6)$. Recently, Fytas et al. fytas1  applied Monte Carlo simulations on a
simple cubic lattice. They found that the transition remains continuous for a
bimodal field distribution, while Hadjiagapiou hadjiagapiou  observed
reentrant behavior and confirmed the existence of tricritical point for an
asymmetric bimodal probability distribution within the mean-field
approximation based on Landau expansion.

Conventional EFT approximations include spin-spin correlations resulting from
the usage of the Van der Waerden identities and provide results that are
superior to those obtained within the traditional MFT. However, these
conventional EFT approximations are not sufficient enough to improve the
results, due to the usage of a decoupling approximation (DA) that neglects the
correlations between different spins that emerge when expanding the
identities. Therefore, taking these correlations into consideration will
improve the results of conventional EFT approximations. In order to overcome
this point, recently we proposed an approximation that takes into account the
correlations between different spins in the cluster of considered lattice
polat ; canpolat ; yuksel_1 ; yuksel_2 ; yuksel_3 . Namely, an advantage of
the approximation method proposed by these studies is that no decoupling
procedure is used for the higher order correlation functions. On the other
hand, as far as we know, EFT studies in the literature dealing with RFIM are
based only on discrete probability distributions (bimodal or trimodal). Hence,
in this work we intent to study the phase diagrams of the RFIM with single
Gaussian, bimodal and double Gaussian random field distributions on isometric
lattices.

Organization of the paper is as follows: In section II we briefly present the
formulations. The results and discussions are presented in section III, and
finally section IV contains our conclusions.

##  II Formulation

In this section, we give the formulation of present study for simple cubic
lattice with $q=6$. A detailed explanation of the method for body centered and
face centered cubic lattices can be found in Appendix. As a simple cubic
lattice, we consider a three-dimensional lattice which has $N$ identical spins
arranged. We define a cluster on the lattice which consists of a central spin
labeled $S_{0}$, and $q$ perimeter spins being the nearest neighbors of the
central spin. The cluster consists of $(q+1)$ spins being independent from the
spin operator $\hat{S}$. The nearest neighbor spins are in an effective field
produced by the outer spins, which can be determined by the condition that the
thermal average of the central spin is equal to that of its nearest neighbor
spins. The Hamiltonian describing our model is

$H=-J\sum_{<i,j>}S_{i}^{z}S_{j}^{z}-\sum_{i}h_{i}S_{i}^{z},$ (1)

where the first term is a summation over the nearest neighbor spins with
$S_{i}^{z}=\pm 1$ and the second term represents the Zeeman interactions on
the lattice. Random magnetic fields are distributed according to a given
probability distribution function. Present study deals with three kinds of
field distribution. Namely, a normal distribution which is defined as

$P(h_{i})=\left(\frac{1}{2\pi\sigma^{2}}\right)^{1/2}\exp\left[-\frac{h_{i}^{2}}{2\sigma^{2}}\right],$
(2)

with a width $\sigma$ and zero mean, a bimodal discrete distribution

$P(h_{i})=\frac{1}{2}\left[\delta(h_{i}-h_{0})+\delta(h_{i}+h_{0})\right],$
(3)

where half of the lattice sites subject to a magnetic field $h_{0}$ and the
remaining lattice sites have a field $-h_{0}$, and a double peaked Gaussian
distribution

$P(h_{i})=\frac{1}{2}\left(\frac{1}{2\pi\sigma^{2}}\right)^{1/2}\left\\{\exp\left[-\frac{(h_{i}-h_{0})^{2}}{2\sigma^{2}}\right]+\exp\left[-\frac{(h_{i}+h_{0})^{2}}{2\sigma^{2}}\right]\right\\}.$
(4)

In a double peaked distribution defined in equation (4), random fields $\pm
h_{0}$ are distributed with equal probability and the form of the distribution
depends on the $h_{0}$ and $\sigma$ parameters, where $\sigma$ is the width of
the distribution.

According to Callen identity callen  for the spin-1/2 Ising ferromagnetic
system, the thermal average of the spin variables at the site $i$ is given by

$\left\langle\\{f_{i}\\}S_{i}^{z}\right\rangle=\left\langle\\{f_{i}\\}\tanh\left[\beta\left(J\sum_{j}S_{j}+h_{i}\right)\right]\right\rangle,$
(5)

where $j$ expresses the nearest neighbor sites of the central spin and
$\\{f_{i}\\}$ can be any function of the Ising variables as long as it is not
a function of the site. From equation (5) with $f_{i}=1$, the thermal and
random-configurational averages of a central spin can be represented on a
simple cubic lattice by introducing the differential operator technique
honmura_kaneyoshi ; kaneyoshi_1

$m_{0}=\left\langle\left\langle
S_{0}^{z}\right\rangle\right\rangle_{r}=\left\langle\left\langle\prod_{j=1}^{q=6}\left[\cosh(J\nabla)+S_{j}^{z}\sinh(J\nabla)\right]\right\rangle\right\rangle_{r}F(x)|_{x=0},$
(6)

where $\nabla$ is a differential operator, $q$ is the coordination number of
the lattice and the inner $\langle...\rangle$ and the outer
$\langle...\rangle_{r}$ brackets represent the thermal and configurational
averages, respectively. The function $F(x)$ in equation (6) is defined by

$F(x)=\int dh_{i}P(h_{i})\tanh[\beta(x+h_{i})],$ (7)

and it has been calculated by numerical integration and by using the
distribution functions defined in equations (2), (3) and (4). By expanding the
right hand side of equation (6) we get the longitudinal spin correlation as

$\displaystyle\langle\langle S_{0}\rangle\rangle_{r}$ $\displaystyle=$
$\displaystyle k_{0}+6k_{1}\langle\langle
S_{1}\rangle\rangle_{r}+15k_{2}\langle\langle
S_{1}S_{2}\rangle\rangle_{r}+20k_{3}\langle\langle
S_{1}S_{2}S_{3}\rangle\rangle_{r}+15k_{4}\langle\langle
S_{1}S_{2}S_{3}S_{4}\rangle\rangle_{r}$ (8)
$\displaystyle+6k_{5}\langle\langle
S_{1}S_{2}S_{3}S_{4}S_{5}\rangle\rangle_{r}+k_{6}\langle\langle
S_{1}S_{2}S_{3}S_{4}S_{5}S_{6}\rangle\rangle_{r}.$

The coefficients in equation (8) are defined as follows

$\displaystyle k_{0}$ $\displaystyle=$
$\displaystyle\cosh^{6}(J\nabla)F(x)|_{x=0},$ $\displaystyle k_{1}$
$\displaystyle=$ $\displaystyle\cosh^{5}(J\nabla)\sinh(J\nabla)F(x)|_{x=0},$
$\displaystyle k_{2}$ $\displaystyle=$
$\displaystyle\cosh^{4}(J\nabla)\sinh^{2}(J\nabla)F(x)|_{x=0},$ $\displaystyle
k_{3}$ $\displaystyle=$
$\displaystyle\cosh^{3}(J\nabla)\sinh^{3}(J\nabla)F(x)|_{x=0},$ $\displaystyle
k_{4}$ $\displaystyle=$
$\displaystyle\cosh^{2}(J\nabla)\sinh^{4}(J\nabla)F(x)|_{x=0},$ $\displaystyle
k_{5}$ $\displaystyle=$
$\displaystyle\cosh(J\nabla)\sinh^{5}(J\nabla)F(x)|_{x=0},$ $\displaystyle
k_{6}$ $\displaystyle=$ $\displaystyle\sinh^{6}(J\nabla)F(x)|_{x=0}.$ (9)

Next, the average value of the perimeter spin in the system can be written as
follows and it is found as

$\displaystyle m_{1}=\langle\langle S_{1}\rangle\rangle_{r}$ $\displaystyle=$
$\displaystyle\langle\langle\cosh(J\nabla)+S_{0}\sinh(J\nabla)\rangle\rangle_{r}F(x+\gamma),$
(10) $\displaystyle=$ $\displaystyle a_{1}+a_{2}\langle\langle
S_{0}\rangle\rangle_{r}.$

For the sake of simplicity, the superscript $z$ is omitted from the left and
right hand sides of equations (8) and (10). The coefficients in equation (10)
are defined as

$\displaystyle a_{1}$ $\displaystyle=$
$\displaystyle\cosh(J\nabla)F(x+\gamma)|_{x=0},$ $\displaystyle a_{2}$
$\displaystyle=$ $\displaystyle\sinh(J\nabla)F(x+\gamma)|_{x=0}.$ (11)

In equation (II), $\gamma=(q-1)A$ is the effective field produced by the
$(q-1)$ spins outside of the system and $A$ is an unknown parameter to be
determined self-consistently. Equations (8) and (10) are the fundamental
correlation functions of the system. When the right-hand side of equation (6)
is expanded, the multispin correlation functions appear. The simplest
approximation, and one of the most frequently adopted is to decouple these
correlations according to

$\left\langle\left\langle
S_{i}S_{j}...S_{l}\right\rangle\right\rangle_{r}\cong\left\langle\left\langle
S_{i}\right\rangle\right\rangle_{r}\left\langle\left\langle
S_{j}\right\rangle\right\rangle_{r}...\left\langle\left\langle
S_{l}\right\rangle\right\rangle_{r},$ (12)

for $i\neq j\neq...\neq l$ tamura_kaneyoshi . The main difference of the
method used in this study from the other approximations in the literature
emerges in comparison with any decoupling approximation (DA) when expanding
the right-hand side of equation (6). In other words, one advantage of the
approximation method used in this study is that such a kind of decoupling
procedure is not used for the higher order correlation functions. For spin-1/2
Ising system in a random field, taking equations (8) and (10) as a basis we
derive a set of linear equations of the spin correlation functions in the
system. At this point, we assume that $(i)$ the correlations depend only on
the distance between the spins and $(ii)$ the average values of a central spin
and its nearest-neighbor spin (it is labeled as the perimeter spin) are equal
to each other with the fact that, in the matrix representations of spin
operator $\hat{S}$, the spin-1/2 system has the property $(\hat{S})^{2}=1$ .
Thus, the number of linear equations obtained for a simple cubic lattice
$(q=6)$ reduces to twelve and the complete set is as follows

$\displaystyle\left\langle\left\langle S_{1}\right\rangle\right\rangle_{r}$
$\displaystyle=$ $\displaystyle a_{1}+a_{2}\langle\langle
S_{0}\rangle\rangle_{r},$ $\displaystyle\left\langle\left\langle
S_{1}S_{2}\right\rangle\right\rangle_{r}$ $\displaystyle=$ $\displaystyle
a_{1}\langle\langle S_{1}\rangle\rangle_{r}+a_{2}\langle\langle
S_{0}S_{1}\rangle\rangle_{r},$ $\displaystyle\left\langle\left\langle
S_{1}S_{2}S_{3}\right\rangle\right\rangle_{r}$ $\displaystyle=$ $\displaystyle
a_{1}\langle\langle S_{1}S_{2}\rangle\rangle_{r}+a_{2}\langle\langle
S_{0}S_{1}S_{2}\rangle\rangle_{r},$ $\displaystyle\left\langle\left\langle
S_{1}S_{2}S_{3}S_{4}\right\rangle\right\rangle_{r}$ $\displaystyle=$
$\displaystyle a_{1}\langle\langle
S_{1}S_{2}S_{3}\rangle\rangle_{r}+a_{2}\langle\langle
S_{0}S_{1}S_{2}S_{3}\rangle\rangle_{r},$
$\displaystyle\left\langle\left\langle
S_{1}S_{2}S_{3}S_{4}S_{5}\right\rangle\right\rangle_{r}$ $\displaystyle=$
$\displaystyle a_{1}\langle\langle
S_{1}S_{2}S_{3}S_{4}\rangle\rangle_{r}+a_{2}\langle\langle
S_{0}S_{1}S_{2}S_{3}S_{4}\rangle\rangle_{r},$
$\displaystyle\left\langle\left\langle
S_{1}S_{2}S_{3}S_{4}S_{5}S_{6}\right\rangle\right\rangle_{r}$ $\displaystyle=$
$\displaystyle a_{1}\langle\langle
S_{1}S_{2}S_{3}S_{4}S_{5}\rangle\rangle_{r}+a_{2}\langle\langle
S_{0}S_{1}S_{2}S_{3}S_{4}S_{5}\rangle\rangle_{r},$
$\displaystyle\langle\langle S_{0}\rangle\rangle_{r}$ $\displaystyle=$
$\displaystyle k_{0}+6k_{1}\langle\langle
S_{1}\rangle\rangle_{r}+15k_{2}\langle\langle
S_{1}S_{2}\rangle\rangle_{r}+20k_{3}\langle\langle
S_{1}S_{2}S_{3}\rangle\rangle_{r}+15k_{4}\langle\langle
S_{1}S_{2}S_{3}S_{4}\rangle\rangle_{r}$ $\displaystyle+6k_{5}\langle\langle
S_{1}S_{2}S_{3}S_{4}S_{5}\rangle\rangle_{r}+k_{6}\langle\langle
S_{1}S_{2}S_{3}S_{4}S_{5}S_{6}\rangle\rangle_{r},$
$\displaystyle\left\langle\left\langle
S_{0}S_{1}\right\rangle\right\rangle_{r}$ $\displaystyle=$ $\displaystyle
6k_{1}+(k_{0}+15k_{2})\langle\langle
S_{1}\rangle\rangle_{r}+20k_{3}\langle\langle
S_{1}S_{2}\rangle\rangle_{r}+15k_{4}\langle\langle
S_{1}S_{2}S_{3}\rangle\rangle_{r}$ $\displaystyle+6k_{5}\langle\langle
S_{1}S_{2}S_{3}S_{4}\rangle\rangle_{r}+k_{6}\langle\langle
S_{1}S_{2}S_{3}S_{4}S_{5}\rangle\rangle_{r},$
$\displaystyle\left\langle\left\langle
S_{0}S_{1}S_{2}\right\rangle\right\rangle_{r}$ $\displaystyle=$
$\displaystyle(6k_{1}+20k_{3})\langle\langle
S_{1}\rangle\rangle_{r}+(k_{0}+15k_{2}+15k_{4})\langle\langle
S_{1}S_{2}\rangle\rangle_{r}$ $\displaystyle+6k_{5}\langle\langle
S_{1}S_{2}S_{3}\rangle\rangle_{r}+k_{6}\langle\langle
S_{1}S_{2}S_{3}S_{4}\rangle\rangle_{r},$
$\displaystyle\left\langle\left\langle
S_{0}S_{1}S_{2}S_{3}\right\rangle\right\rangle_{r}$ $\displaystyle=$
$\displaystyle(6k_{1}+20k_{3}+6k_{5})\langle\langle
S_{1}S_{2}\rangle\rangle_{r}+(k_{0}+15k_{2}+15k_{4}+k_{6})\langle\langle
S_{1}S_{2}S_{3}\rangle\rangle_{r},$ $\displaystyle\left\langle\left\langle
S_{0}S_{1}S_{2}S_{3}S_{4}\right\rangle\right\rangle_{r}$ $\displaystyle=$
$\displaystyle(6k_{1}+20k_{3}+6k_{5})\langle\langle
S_{1}S_{2}S_{3}\rangle\rangle_{r}+(k_{0}+15k_{2}+15k_{4}+k_{6})\langle\langle
S_{1}S_{2}S_{3}S_{4}\rangle\rangle_{r},$
$\displaystyle\left\langle\left\langle
S_{0}S_{1}S_{2}S_{3}S_{4}S_{5}\right\rangle\right\rangle_{r}$ $\displaystyle=$
$\displaystyle(6k_{1}+20k_{3}+6k_{5})\langle\langle
S_{1}S_{2}S_{3}S_{4}\rangle\rangle_{r}+(k_{0}+15k_{2}+15k_{4}+k_{6})\langle\langle
S_{1}S_{2}S_{3}S_{4}S_{5}\rangle\rangle_{r}$ (13)

If equation (II) is written in the form of a $12\times 12$ matrix and solved
in terms of the variables $x_{i}[(i=1,2,...,12)(e.g.,x_{1}=\langle\langle
S_{1}\rangle\rangle_{r},x_{2}=\langle\langle
S_{1}S_{2}\rangle\rangle_{r},x_{3}=\langle\langle
S_{1}S_{2}S_{3}\rangle\rangle_{r},...)]$ of the linear equations, all of the
spin correlation functions can be easily determined as functions of the
temperature and Hamiltonian/random field parameters. Since the thermal and
configurational average of the central spin is equal to that of its nearest-
neighbor spins within the present method, the unknown parameter $A$ can be
numerically determined by the relation

$\langle\langle S_{0}\rangle\rangle_{r}=\langle\langle
S_{1}\rangle\rangle_{r}\qquad{\rm{or}}\qquad x_{7}=x_{1}.$ (14)

By solving equation (14) numerically at a given fixed set of
Hamiltonian/random field parameters we obtain the parameter $A$. Then we use
the numerical values of $A$ to obtain the spin correlation functions which can
be found from equation (II). Note that $A=0$ is always the root of equation
(14) corresponding to the disordered state of the system. The nonzero root of
$A$ in equation (14) corresponds to the long range ordered state of the
system. Once the spin correlation functions have been evaluated then we can
give the numerical results for the thermal and magnetic properties of the
system. Since the effective field $\gamma$ is very small in the vicinity of
$k_{B}T_{c}/J$, we can obtain the critical temperature for the fixed set of
Hamiltonian/random field parameters by solving equation (14) in the limit of
$\gamma\rightarrow 0$ then we can construct the whole phase diagrams of the
system. Depending on the Hamiltonian/random field parameters, there may be two
solutions (i.e. two critical temperature values satisfy the equation (14))
corresponding to the first/second and second order phase transition points,
respectively. We determine the type of the transition by looking at the
temperature dependence of magnetization for selected values of system
parameters.

##  III Results and Discussion

In this section, we discuss how the type of random field distribution effects
the phase diagrams of the system. Also, in order to clarify the type of the
transitions in the system, we give the temperature dependence of the order
parameter.

###  III.1 Phase diagrams of single Gaussian distribution

The form of single Gaussian distribution which is defined in equation (2) is
governed by only one parameter $\sigma$ which is the width of the
distribution. In Fig. 1, we show the phase diagram of the system for simple
cubic, body centered cubic and face centered cubic lattices in
$(k_{B}T_{c}/J-\sigma)$ plane. We can clearly see that as $\sigma$ increases
then the width of the distribution function gets wider and randomness effect
of magnetic field distribution on the system becomes significantly important.
Therefore, increasing $\sigma$ value causes a decline in the critical
temperature $k_{B}T_{c}/J$ of the system. We note that the critical
temperature of the system reaches to zero at $\sigma=3.8501$, $5.450$ and
$8.601$ for simple cubic $(q=6)$, body centered $(q=8)$ and face centered
cubic $(q=12)$ lattices, respectively. Besides, we have not observed any
reentrant/tricritical behavior for single Gaussian distribution, or in other
words, the system undergoes only a second order phase transition.
$k_{B}T_{c}/J$ value in the absence of any randomness i.e. when $\sigma=0$ is
obtained as $k_{B}T_{c}/J=4.5274$, $6.5157$ and $10.4986$ for $q=6,8,12$,
respectively. These values can be compared with the other works in the
literature. In Table 1, we see that present work improves the results of the
other works. Hence, it is obvious that the present method is superior to the
other EFT methods in the literature. The reason is due to the fact that, in
contrast to the previously published works mentioned above, there is no
uncontrolled decoupling procedure used for the higher order correlation
functions within the present approximation.

Table 1: Critical temperature $k_{B}T_{c}/J$ at $h_{0}/J=0$ and $\sigma=0$
obtained by several methods and the present work for $q=6,8,12$.
---
Lattice | EBPAdu  | CEFTkaneyoshi_2  |  PAbalcerzak  | EFTkaneyoshi_3  |  BAkikuchi  | EFRGsousa  | MFRGreinerhr  | MClandau2  |  SEfisher  | Present Work
SC | 4.8108 | 4.9326 | 4.9328 |  5.0732 |  4.6097 | 4.85 |  4.93 | 4.51 | 4.5103 |  4.5274
BCC |  - | 6.9521 |  - |  - |  - | 6.88 |  6.95 | 6.36 | 6.3508 |  6.5157
FCC |  - | 10.9696 |  - |  - |  - |  - |  - |  - | 9.7944 |  10.4986

Magnetization surfaces and their projections on $(k_{B}T/J-\sigma)$ plane
corresponding to the phase diagrams shown in Fig. 1 are depicted in Fig. 2
with $q=6,8$ and $12$. We see that as the temperature increases starting from
zero, magnetization of the system decreases continuously and it falls rapidly
to zero at the critical temperature for all $\sigma$ values. Moreover,
critical temperature of the system decreases and saturation value of
magnetization curves remains constant for a while then reduces as $\sigma$
value increases. This is a reasonable result since, if $\sigma$ value
increases, then the randomness effects increasingly play an important role on
the system and random fields have a tendency to destruct the long range
ferromagnetic order on the system, and hence magnetization weakens. These
observations are common properties of both three lattices.

Figure 1: Phase diagrams of simple cubic $(q=6)$, body centered cubic $(q=8)$
and face centered cubic $(q=12)$ lattices in $(k_{B}T_{c}/J-\sigma)$ plane
corresponding to single Gaussian distribution of random fields. The numbers on
each curve denote the coordination numbers.
Figure 2: Variation of magnetization with $k_{B}T/J$ and $\sigma$
corresponding to the phase diagrams in Fig. 1 for simple cubic, body centered
and face centered cubic lattices. (a)-(c) 3D contour plot surfaces, (d)-(f)
projections on $(k_{B}T/J-\sigma)$ plane.

###  III.2 Phase diagrams of bimodal distribution

Next, in order to investigate the effect of the bimodal random fields defined
in equation (3) on the phase diagrams of the system, we show the phase
diagrams in $(k_{B}T_{c}/J-h_{0}/J)$ plane and corresponding magnetization
profiles with coordination numbers $q=6,8,12$ in Figs. 3 and 4. In these
figures solid and dashed lines correspond to second and first order phase
transition points, respectively and hollow circles in Fig. 3 denote
tricritical points. As seen in Fig. 3, increasing values of $h_{0}/J$ causes
the critical temperature to decrease for a while, then reentrant behavior of
first order occurs at a specific range of $h_{0}/J$. According to our
calculations, reentrant phenomena and the first order phase transitions can be
observed within the range of $2.0<h_{0}/J<3.0$ for $q=6$, $3.565<h_{0}/J<3.95$
for $q=8$ and $4.622<h_{0}/J<5.81$ for $q=12$. If $h_{0}/J$ value is greater
than the upper limits of these field ranges, the system exhibits no phase
transition. The tricritical temperatures $k_{B}T_{t}/J$ which are shown as
hollow circles in Fig. 2 are found as $k_{B}T_{t}/J=1.5687,2.4751$ and
$4.3769$ for $q=6,8,12$, respectively.

Figure 3: Phase diagrams of the system $(q=6,8,12)$ in
$(k_{B}T_{c}/J-h_{0}/J)$ plane, corresponding to bimodal random field
distribution. Solid and dashed lines correspond to second and first order
phase transitions, respectively. Hollow circles denote the tricritical points.
Figure 4: Temperature dependence of magnetization corresponding to Fig. 3 with
$\sigma=0$ and $h_{0}/J=1.0,2.5$ for simple cubic lattice (left panel),
$h_{0}/J=1.0,3.8$ for body centered cubic lattice (middle panel) and
$h_{0}/J=1.0,5.5$ for face centered cubic lattice (right panel). Solid and
dashed lines correspond to second and first order phase transitions,
respectively.

In Fig. 4, we show two typical magnetization profiles of the system. Namely,
the system always undergoes a second order phase transition for $h_{0}/J=1.0$.
On the other hand, two successive phase transitions (i.e. a first order
transition is followed by a second order phase transition) occur at the values
of $h_{0}/J=2.5,3.8$ and $5.5$ for $q=6,8$ and $12$, respectively, which puts
forward the existence of a first order reentrant phenomena on the system. We
observe that the increasing $h_{0}/J$ values do not effect the saturation
values of magnetization curves.

###  III.3 Phase diagrams of double Gaussian distribution

Figure 5: Phase diagrams of the system for a double Gaussian random field
distribution with $q=6,8,12$ in $(k_{B}T_{c}/J-h_{0}/J)$ and
$(k_{B}T_{c}/J-\sigma)$ planes.

Double Gaussian distribution in equation (4) with nearest neighbor
interactions have not yet been examined in the literature. Therefore, it would
be interesting to investigate the phase diagrams of the system with random
fields corresponding to equation (4). Now the shape of the random fields is
governed by two parameters $h_{0}/J$ and $\sigma$. As shown in preceding
results, increasing $\sigma$ value tends to reduce the saturation value of the
order parameter and destructs the second order phase transitions by decreasing
the critical temperature of the system without exposing any reentrant
phenomena for $h_{0}/J=0$. Besides, as $h_{0}/J$ value increases then the
second order phase transition temperature decreases again and the system may
exhibit a reentrant behavior for $\sigma=0$, while saturation value of
magnetization curves remains unchanged. Hence, the presence of both $h_{0}/J$
and $\sigma$ on the system should produce a cooperative effect on the phase
diagrams of the system. Fig. 5 shows the phase diagrams of the system in
$(k_{B}T_{c}/J-h_{0}/J)$ and $(k_{B}T_{c}/J-\sigma)$ planes for $q=6,8$ and
$12$. As we can see on the left panels in Fig. 5, the system exhibits
tricritical points and reentrant phenomena for narrow width of the random
field distribution and as the width $\sigma$ of the distribution gets wider
then the reentrant phenomena and tricritical behavior disappear. In other
words, both the reentrant behavior and tricritical points disappear as
$\sigma$ parameter becomes significantly dominant on the system. Our results
predict that tricritical points depress to zero at $\sigma=1.421,2.238$ and
$3.985$ for $q=6,8$ and $12$, respectively. For distribution widths greater
than these values, all transitions are of second order and as we increase
$\sigma$ value further, then ferromagnetic region gets narrower. Similarly, on
the right panels in Fig. 5, we investigate the phase diagrams of the system in
$(k_{B}T_{c}/J-\sigma)$ plane with selected values of $h_{0}/J$. We note that
for the values of $h_{0}/J\leq 2.0$ $(q=6)$, $h_{0}/J\leq 3.565$ $(q=8)$ and
$h_{0}/J\leq 4.622$ $(q=12)$, system always undergoes a second order phase
transition between paramagnetic and ferromagnetic phases at a critical
temperature which decreases with increasing values of $h_{0}/J$ as in Fig. 1,
where $h_{0}/J=0$. Moreover, for the values of $h_{0}/J$ greater than these
threshold values the system exhibits a reentrant behavior of first order and
the transition lines exhibit a bulge which gets smaller with increasing values
of $h_{0}/J$ which again means that ferromagnetic phase region gets narrower.
Besides, for $h_{0}/J>2.9952$ $(q=6)$, $h_{0}/J>3.9441$ $(q=8)$ and
$h_{0}/J>5.8085$ $(q=12)$ tricritical points appear on the system. In Fig. 6,
we show magnetization curves corresponding to the phase diagrams shown in Fig.
5 for simple cubic lattice. Fig. 6a shows the temperature dependence of
magnetization curves for $q=6$ with $h_{0}/J=0.5$ (left panel) and
$h_{0}/J=2.5$ (right panel) with selected values of $\sigma$. As we can see in
Fig. 6a, as $\sigma$ increases then the critical temperature of the system
decreases and first order phase transitions disappear (see the right panel in
Fig. 6a). Moreover, rate of decrease of the saturation value of magnetization
curves increases as $h_{0}/J$ increases. On the other hand, in Fig. 6b,
magnetization versus temperature curves have been plotted with $\sigma=0.5$
(left panel) and $\sigma=2.5$ (right panel) with some selected values of
$h_{0}/J$. In this figure, it is clear that saturation values of magnetization
curves remain unchanged for $\sigma=0.5$ and tend to decrease rapidly to zero
with increasing values of $h_{0}/J$ when $\sigma=2.5$. In addition, as
$h_{0}/J$ increases when the value of $\sigma$ is fixed then the critical
temperature of the system decreases and ferromagnetic phase region of the
system gets narrower. These observations show that there is a collective
effect originating from the presence of both $h_{0}/J$ and $\sigma$ parameters
on the phase diagrams and magnetization curves of the system.

Figure 6: Magnetization curves for simple cubic lattice corresponding to the
phase diagrams in Fig. 5 for a double Gaussian distribution with some selected
values of $h_{0}/J$ and $\sigma$.

Finally, Fig. 7 represents the variation of the tricritical point
$(k_{B}T_{t}/J,h_{t}/J)$ with $\sigma$ for $q=6,8$ and $12$. As seen from Fig.
7, $k_{B}T_{t}/J$ value decreases monotonically as $\sigma$ increases and
reaches to zero at a critical value $\sigma_{t}$. According to our
calculations, the critical distribution width $\sigma_{t}$ value can be given
as $\sigma_{t}=1.421,2.238,3.985$ for $q=6,8$ and $12$, respectively. This
implies that $\sigma_{t}$ value depends on the coordination number of the
lattice. Besides, $h_{t}/J$ curves exhibit relatively small increment with
increasing $\sigma$ value.

Figure 7: Variations of tricritical temperature $k_{B}T_{t}/J$ and tricritical
field $h_{t}/J$ as function of distribution width $(\sigma)$ for (a) simple
cubic, (b) body centered cubic and (c) face centered cubic lattices.

##  IV Conclusions

In this work, we have studied the phase diagrams of a spin-$1/2$ Ising model
in a random magnetic field on simple cubic, body centered cubic and face
centered cubic lattices. RFIM with nearest neighbor interactions with double
Gaussian distribution as well as its single Gaussian counterpart has not been
examined within the EFT approximation before. Hence, we have introduced an
effective field approximation that takes into account the correlations between
different spins in the cluster of considered lattice and examined the phase
diagrams as well as magnetization curves of the system for different types of
random field distributions, namely single Gaussian, bimodal and double
Gaussian distributions. For a single Gaussian distribution, we have found that
the system always undergoes a second order phase transition between
paramagnetic and ferromagnetic phases. In the absence of any randomness (i.e.
$h_{0}/J=0,\sigma=0$) critical temperature values corresponding to the
coordination numbers $q=6,8$ and $12$ are found to be in an excellent
agreement with those of the other theoretical methods such as MC and SE. For
bimodal and double Gaussian distributions, we have given the proper phase
diagrams, especially the first-order transition lines that include reentrant
phase transition regions. Our numerical analysis clearly indicate that such
field distributions lead to a tricritical behavior. Moreover, we have
discussed a collective effect which arise from the presence of both $h_{0}/J$
and $\sigma$ parameters together and we have observed that saturation values
of magnetization curves are strongly related to this collective effect.

As a result, we can conclude that all of the points mentioned above show that
our method is superior to the conventional EFT methods based on decoupling
approximation and therefore we think that the calculation results are more
accurate. Therefore, we hope that the results obtained in this work may be
beneficial from both theoretical and experimental points of view.

## Acknowledgements

One of the authors (YY) would like to thank the Scientific and Technological
Research Council of Turkey (TÜBİTAK) for partial financial support. This work
has been completed at the Dokuz Eylül University, Graduate School of Natural
and Applied Sciences and is the subject of the forthcoming Ph.D. thesis of Y.
Yüksel. The partial financial support from SRF (Scientific Research Fund) of
Dokuz Eylül University (2009.KB.FEN.077) is also acknowledged.

## Appendix

The number of distinct correlations for the spin-$1/2$ system with $q$ nearest
neighbors is $2q-1$. With the magnetization of the central spin
$\langle\langle S_{0}\rangle\rangle_{r}$ and its nearest neighbor spin
$\langle\langle S_{1}\rangle\rangle_{r}$, there has to be $2q+1$ unknowns
which forms a system of linear equations. $q$ of the correlations which
include central spin are derived from the central spin magnetization
$\langle\langle S_{0}\rangle\rangle_{r}$, and rest of the $q-1$ correlations
which include only perimeter spin are derived from perimeter spin
magnetization expression $\langle\langle S_{1}\rangle\rangle_{r}$. Let us
label these correlations with $x_{i},i=1,2,\ldots,2q+1$, such that first $q$
of them include only perimeter spins and last $q+1$ of them include central
spin. We can represent all of the spin correlations and centeral/perimeter
spin magnetization with $x_{i}$ as $x_{1}=\langle\langle
S_{1}\rangle\rangle_{r}$, $x_{2}=\langle\langle
S_{1}S_{2}\rangle\rangle_{r}$,…$x_{q}=\langle\langle S_{1}S_{2}\ldots
S_{q}\rangle\rangle_{r}$, $x_{q+1}=\langle\langle S_{0}\rangle\rangle_{r}$,
$x_{q+2}=\langle\langle S_{0}S_{1}\rangle\rangle_{r},\ldots$ and
$x_{2q+1}=\langle\langle S_{0}S_{1}\ldots S_{q}\rangle\rangle_{r}$. The
complete set of correlation functions for $q=8,12$ can be obtained as the same
procedure given between equations (6)-(II) in section II.

The complete set of correlation functions for body centered cubic lattice
$(q=8)$:

$\displaystyle x_{1}$ $\displaystyle=$ $\displaystyle a_{1}+a_{2}x_{9},$
$\displaystyle x_{2}$ $\displaystyle=$ $\displaystyle a_{1}x_{1}+a_{2}x_{10},$
$\displaystyle x_{3}$ $\displaystyle=$ $\displaystyle a_{1}x_{2}+a_{2}x_{11},$
$\displaystyle x_{4}$ $\displaystyle=$ $\displaystyle a_{1}x_{3}+a_{2}x_{12},$
$\displaystyle x_{5}$ $\displaystyle=$ $\displaystyle a_{1}x_{4}+a_{2}x_{13},$
$\displaystyle x_{6}$ $\displaystyle=$ $\displaystyle a_{1}x_{5}+a_{2}x_{14},$
$\displaystyle x_{7}$ $\displaystyle=$ $\displaystyle a_{1}x_{6}+a_{2}x_{15},$
$\displaystyle x_{8}$ $\displaystyle=$ $\displaystyle a_{1}x_{7}+a_{2}x_{16},$
$\displaystyle x_{9}$ $\displaystyle=$ $\displaystyle
k_{0}+8k_{1}x_{1}+28k_{2}x_{2}+56k_{3}x_{3}+70k_{4}x_{4}+56k_{5}x_{5}+28k_{6}x_{6}+8k_{7}x_{7}+k_{8}x_{8},$
$\displaystyle x_{10}$ $\displaystyle=$ $\displaystyle
8k_{1}+(28k_{2}+k_{0})x_{1}+56k_{3}x_{2}+70k_{4}x_{3}+56k_{5}x_{4}+28k_{6}x_{5}+8k_{7}x_{6}+k_{8}x_{7},$
$\displaystyle x_{11}$ $\displaystyle=$
$\displaystyle(56k_{3}+8k_{1})x_{1}+(70k_{4}+28k_{2}+k_{0})x_{2}+56k_{5}x_{3}+28k_{6}x_{4}+8k_{7}x_{5}+k_{8}x_{6},$
$\displaystyle x_{12}$ $\displaystyle=$
$\displaystyle(56k_{3}+56k_{5}+8k_{1})x_{2}+(70k_{4}+28k_{2}+k_{0}+28k_{6})x_{3}+8k_{7}x_{4}+k_{8}x_{5},$
$\displaystyle x_{13}$ $\displaystyle=$
$\displaystyle(56k_{3}+56k_{5}+8k_{7}+8k_{1})x_{3}+(70k_{4}+k_{8}+28k_{2}+k_{0}+28k_{6})x_{4},$
$\displaystyle x_{14}$ $\displaystyle=$
$\displaystyle(56k_{3}+56k_{5}+8k_{7}+8k_{1})x_{4}+(70k_{4}+k_{8}+28k_{2}+k_{0}+28k_{6})x_{5},$
$\displaystyle x_{15}$ $\displaystyle=$
$\displaystyle(56k_{3}+56k_{5}+8k_{7}+8k_{1})x_{5}+(70k_{4}+k_{8}+28k_{2}+k_{0}+28k_{6})x_{6},$
$\displaystyle x_{16}$ $\displaystyle=$
$\displaystyle(56k_{3}+56k_{5}+8k_{7}+8k_{1})x_{6}+(70k_{4}+k_{8}+28k_{2}+k_{0}-28k_{6})x_{7},$
$\displaystyle x_{17}$ $\displaystyle=$
$\displaystyle(56k_{3}+56k_{5}+8k_{7}+8k_{1})x_{7}+(70k_{4}+k_{8}+28k_{2}+k_{0}+28k_{6})x_{8}.$

Similarly, correlation functions for face centered cubic lattice $(q=12)$:

$\displaystyle x_{1}$ $\displaystyle=$ $\displaystyle a_{1}+a_{2}x_{13},$
$\displaystyle x_{2}$ $\displaystyle=$ $\displaystyle a_{1}x_{1}+a_{2}x_{14},$
$\displaystyle x_{3}$ $\displaystyle=$ $\displaystyle a_{1}x_{2}+a_{2}x_{15},$
$\displaystyle x_{4}$ $\displaystyle=$ $\displaystyle a_{1}x_{3}+a_{2}x_{16},$
$\displaystyle x_{5}$ $\displaystyle=$ $\displaystyle a_{1}x_{4}+a_{2}x_{17},$
$\displaystyle x_{6}$ $\displaystyle=$ $\displaystyle a_{1}x_{5}+a_{2}x_{18},$
$\displaystyle x_{7}$ $\displaystyle=$ $\displaystyle a_{1}x_{6}+a_{2}x_{19},$
$\displaystyle x_{8}$ $\displaystyle=$ $\displaystyle a_{1}x_{7}+a_{2}x_{20},$
$\displaystyle x_{9}$ $\displaystyle=$ $\displaystyle a_{1}x_{8}+a_{2}x_{21},$
$\displaystyle x_{10}$ $\displaystyle=$ $\displaystyle
a_{1}x_{9}+a_{2}x_{22},$ $\displaystyle x_{11}$ $\displaystyle=$
$\displaystyle a_{1}x_{10}+a_{2}x_{23},$ $\displaystyle x_{12}$
$\displaystyle=$ $\displaystyle a_{1}x_{11}+a_{2}x_{24},$ $\displaystyle
x_{13}$ $\displaystyle=$ $\displaystyle
k_{0}+12k_{1}x_{1}+66k_{2}x_{2}+220k_{3}x_{3}+495k_{4}x_{4}+792k_{5}x_{5}+924k_{6}x_{6}$
$\displaystyle+792k_{7}x_{7}+495k_{8}x_{8}+220k_{9}x_{9}+66k_{10}x_{10}+12k_{11}x_{11}+k_{12}x_{12},$
$\displaystyle x_{14}$ $\displaystyle=$ $\displaystyle
12k_{1}+(k_{0}+66k_{2})x_{1}+220k_{3}x_{2}+495k_{4}x_{3}+792k_{5}x_{4}+924k_{6}x_{5},$
$\displaystyle+792k_{7}x_{6}+495k_{8}x_{7}+220k_{9}x_{8}+66k_{10}x_{9}+12k_{11}x_{10}+k_{12}x_{11},$
$\displaystyle x_{15}$ $\displaystyle=$
$\displaystyle(12k_{1}+220k_{3})x_{1}+(k_{0}+66k_{2}+495k_{4})x_{2}+792k_{5}x_{3}+924k_{6}x_{4},$
$\displaystyle+792k_{7}x_{5}+495k_{8}x_{6}+220k_{9}x_{7}+66k_{10}x_{8}+12k_{11}x_{9}+k_{12}x_{10},$
$\displaystyle x_{16}$ $\displaystyle=$
$\displaystyle(12k_{1}+220k_{3}+792k_{5})x_{2}+(k_{0}+66k_{2}+495k_{4}+924k_{6})x_{3},$
$\displaystyle+792k_{7}x_{4}+495k_{8}x_{5}+220k_{9}x_{6}+66k_{10}x_{7}+12k_{11}x_{8}+k_{12}x_{9},$
$\displaystyle x_{17}$ $\displaystyle=$
$\displaystyle(12k_{1}+220k_{3}+792k_{5}+792k_{7})x_{3}+(k_{0}+66k_{2}+495k_{4}+495k_{8}+924k_{6})x_{4},$
$\displaystyle+220k_{9}x_{5}+66k_{10}x_{6}+12k_{11}x_{7}+k_{12}x_{8},$
$\displaystyle x_{18}$ $\displaystyle=$
$\displaystyle(12k_{1}+220k_{3}+220k_{9}+792k_{5}+792k_{7})x_{4},$
$\displaystyle+(k_{0}+66k_{2}+495k_{4}+495k_{8}+66k_{10}+924k_{6})x_{5}+12k_{11}x_{6}+k_{12}x_{7},$
$\displaystyle x_{19}$ $\displaystyle=$
$\displaystyle(12k_{1}+220k_{3}+220k_{9}+792k_{5}+12k_{11}+792k_{7})x_{5},$
$\displaystyle+(k_{0}+66k_{2}+k_{12}+495k_{4}+495k_{8}+66k_{10}+924k_{6})x_{6},$
$\displaystyle x_{20}$ $\displaystyle=$
$\displaystyle(12k_{1}+220k_{3}+220k_{9}+792k_{5}+12k_{11}+792k_{7})x_{6},$
$\displaystyle+(k_{0}+66k_{2}+k_{12}+495k_{4}+495k_{8}+66k_{10}+924k_{6})x_{7},$
$\displaystyle x_{21}$ $\displaystyle=$
$\displaystyle(12k_{1}+220k_{3}+220k_{9}+792k_{5}+12k_{11}+792k_{7})x_{7},$
$\displaystyle+(k_{0}+66k_{2}+k_{12}+495k_{4}+495k_{8}+66k_{10}+924k_{6})x_{8},$
$\displaystyle x_{22}$ $\displaystyle=$
$\displaystyle(12k_{1}+220k_{3}+220k_{9}+792k_{5}+12k_{11}+792k_{7})x_{8},$
$\displaystyle+(k_{0}+66k_{2}+k_{12}+495k_{4}+495k_{8}+66k_{10}+924k_{6})x_{9},$
$\displaystyle x_{23}$ $\displaystyle=$
$\displaystyle(12k_{1}+220k_{3}+220k_{9}+792k_{5}+12k_{11}+792k_{7})x_{9},$
$\displaystyle+(k_{0}+66k_{2}+k_{12}+495k_{4}+495k_{8}+66k_{10}+924k_{6})x_{10},$
$\displaystyle x_{24}$ $\displaystyle=$
$\displaystyle(12k_{1}+220k_{3}+220k_{9}+792k_{5}+12k_{11}+792k_{7})x_{10},$
$\displaystyle+(k_{0}+66k_{2}+k_{12}+495k_{4}+495k_{8}+66k_{10}+924k_{6})x_{11},$
$\displaystyle x_{25}$ $\displaystyle=$
$\displaystyle(12k_{1}+220k_{3}+220k_{9}+792k_{5}+12k_{11}+792k_{7})x_{11},$
$\displaystyle+(k_{0}+66k_{2}+k_{12}+495k_{4}+495k_{8}+66k_{10}+924k_{6})x_{12}.$

where the coefficients are given by

$\displaystyle a_{0}=\cosh(J\nabla)F(x+\gamma)|_{x=0},\ \ \ \ \ \ \ \ \ \ $
$\displaystyle a_{1}=\sinh(J\nabla)F(x+\gamma)|_{x=0},$

with $\gamma=(q-1)A$,

$k_{i}=\cosh^{q-i}(J\nabla)\sinh^{i}(J\nabla)F(x)|_{x=0},\ \ i=0,\ldots,q$
(15)

where $q$ is the coordination number. Since the first $2q$ equations are
independent of the correlation labeled $x_{2q+1}$ it is not necessary to
include this correlation function in the set of linear equations. Therefore it
is adequate to take $2q$ linear equations for calculations.

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