# IndoCulture: Exploring Geographically-Influenced Cultural
Commonsense Reasoning Across Eleven Indonesian Provinces

Fajri Koto1 Rahmad Mahendra2,3 Nurul Aisyah4 Timothy Baldwin1,5
1Department of Natural Language Processing, MBZUAI
2Universitas Indonesia 3Royal Melbourne Institute of Technology
4Quantic School of Business and Technology 5The University of Melbourne
<EMAIL_ADDRESS><EMAIL_ADDRESS>

###### Abstract

Although commonsense reasoning is greatly shaped by cultural and geographical
factors, previous studies on language models have predominantly centered on
English cultures, potentially resulting in an Anglocentric bias. In this
paper, we introduce IndoCulture, aimed at understanding the influence of
geographical factors on language model reasoning ability, with a specific
emphasis on the diverse cultures found within eleven Indonesian provinces. In
contrast to prior works that relied on template Yin et al. (2022) and online
scrapping Fung et al. (2024), we create IndoCulture by asking local people to
manually develop the context and plausible options based on predefined topics.
Evaluations of 23 language models reveal several insights: (1) even the best
open-source model struggles with an accuracy of 53.2%, (2) models often
provide more accurate predictions for specific provinces, such as Bali and
West Java, and (3) the inclusion of location contexts enhances performance,
especially in larger models like GPT-4, emphasizing the significance of
geographical context in commonsense reasoning.111IndoCulture can be accessed
at https://huggingface.co/datasets/indolem/IndoCulture

##  1 Introduction

Figure 1: IndoCulture covers eleven provinces spanning from eastern to western
Indonesia. The highlighted regions in the map represent the provinces examined
in IndoCulture. We offer examples from Aceh, South Borneo, and Papua, with
correct answers indicated in bold. English translations are for illustrative
purposes.

Reasoning abilities of multilingual language models are frequently evaluated
using English texts, potentially amplifying an Anglocentric bias toward
English culture and leading to less inclusive models Thomas (1983); Ponti et
al. (2020). Cultures, however, vary significantly from one location to another
and profoundly shape the way a speaker of a language reasons Hershcovich et
al. (2022). Recent evaluations of models reasoning ability OpenAI (2023);
Touvron et al. (2023); Jiang et al. (2023); Sengupta et al. (2023); Liu et al.
(2023b) conducted on the English datasets, such as PIQA Bisk et al. (2020a),
HellaSwag Zellers et al. (2019), Social IQA Sap et al. (2019), often overlook
geographical aspects, thereby risking cultural bias.

Culture is a multifaceted concept encompassing the way of life Giddens and
Sutton (2021), including our thoughts and actions Macionis John . It includes
tangible elements like food art, and clothing, as well as intangible aspects
such as ideas, values, attitudes, and norms. Culture is shaped by geographical
location and ethnicity, influencing the commonsense reasoning of people within
a region. For example, in Indonesia, it is culturally acceptable to eat rice
with your hands but it is considered unusual to use a fork. Similarly, at
traditional Indonesian weddings, it is common to sit on the floor while
eating, whereas this practice is less common in Australia.

This work focuses on understanding the influence of geographical contexts in
cultural commonsense reasoning, with the main focus on Indonesian culture.
Indonesia is a highly multicultural country Putra et al. (2019), home to over
1,300 recognized ethnic groups and more than 700 languages Zarbaliyev (2017);
Aji et al. (2022). As the the largest archipelagic country in the world,
Indonesia has a population exceeding 270 million spread across 38 provinces,
stretching from Aceh province in the west to Papua province in the east. Few
prior studies on commonsense reasoning in Indonesian contexts Mahendra et al.
(2021); Wibowo et al. (2023); Putri et al. (2024) overlooked the geographical
nuance and the rich diversity of Indonesian cultures.

This paper introduces IndoCulture, a novel dataset to evaluate cultural
reasoning in eleven Indonesian provinces, manually developed by local people
in each province based on predefined topics. In prior works, cultural
reasoning has primarily relied on datasets constructed through templates Yin
et al. (2022), and online scraping Nguyen et al. (2023a); Fung et al. (2024).
While these studies offer valuable insights, they may be susceptible to
training data contamination when utilized to assess large language models
(LLMs). For instance, Fung et al. (2024) reported a zero-shot accuracy of 92%
when using ChatGPT to evaluate low-resource data.

Figure 1 illustrates the eleven provinces examined in IndoCulture including
Aceh, North Sumatra, West Sumatra, West Java, Central Java, East Java, Bali,
South Borneo, East Nusa Tenggara (NTT), South Sulawesi, and Papua.222Although
Papua consists of six provinces, for the purpose of this study, we treat it as
a single entity (referred to as Papua) due to the relatively recent
establishment of most of these provinces. These provinces span from the
eastern to the western parts of Indonesia, each representing a major island in
the country, with the addition of Bali and NTT. Figure 1 also shows three
examples in IndoCulture for three provinces: Aceh, South Borneo, and Papua.
The first two examples focus on cultural artifacts: the traditional wedding
dress from Aceh and the Bagandut dance from South Borneo. The second example
also examines cultural symbols. The third example focuses on cultural beliefs
and norms regarding pregnancy in Papua.

Can large language models effectively reason based on the diverse cultures of
Indonesia? To capture the rich diversity of Indonesian cultures, we
meticulously predefined 12 fine-grained topics as guidelines for data
construction. Figure 2 displays the topic distribution in IndoCulture, with
the majority focusing on food, weddings, art, pregnancy and children, and
family relationships. Additionally, we also pose the question: Is there an
influence of geographical location on commonsense reasoning by language
models? We address these questions through comprehensive experiments across
different language models, incorporating several levels of location
granularity as additional context in the prompt.

Our contributions can be summarized as follows:

  * •

We present IndoCulture, a high-quality cultural reasoning dataset in the
Indonesian language, covering eleven provinces of Indonesia and twelve fine-
grained cultural topics. Our dataset has 2,429 instances, and was meticulously
developed by local people with rigorous quality controls in place.

  * •

We assess 15 open-source multilingual models, 6 open-source Indonesian-centric
models, and 2 closed-source models. Although local individuals can answer all
questions correctly (i.e., 100% accuracy), we observed that all open-source
models struggle to comprehend Indonesian cultures, with the highest
performance reaching only 53.2%.

  * •

We conduct a thorough analysis over various dimensions: (1) models performance
for each province and topics, (2) the influence of different granularities of
location contexts (i.e., none, province, country), (3) model performance over
English translation, (4) analysis over model explanation behind its answer.

##  2 Related Work

#### Commonsense Reasoning in English

Many studies have focused on commonsense reasoning in English, often
overlooking considerations of cultures and geographical locations. Early works
include the Winograd Schema Challenge Levesque et al. (2012) and WinoGrande
Sakaguchi et al. (2021) for pronoun coreference resolution tasks. Other
research areas include reasoning based on cause-effect relationships Roemmele
et al. (2011), physical activities Bisk et al. (2020b), social interaction Sap
et al. (2019), cloze story completion Mostafazadeh et al. (2016), sentence
completion Zellers et al. (2019), numerical Lin et al. (2020), and temporal
Qin et al. (2021). Additionally, pretrained language models are employed in
other works to extract structured commonsense knowledge by providing seed
words Davison et al. (2019), and using code language model Madaan et al.
(2022).

#### Cultural Commonsense Reasoning with Geographical Contexts

Previous studies have explored commonsense reasoning with geographical
contexts. Shwartz (2022) investigated time perception (e.g., morning and night
hours) across different locations, while Yin et al. (2022) delved into
cultural knowledge of language models across five countries by masking and
template strategies. Other works have focused on automatically extracting
cultural knowledge from various sources, including Wikipedia Fung et al.
(2024), conversations Fung et al. (2023), and Common Crawl Nguyen et al.
(2023a), incorporating location contexts with the assistance of large language
models. Related, Ziems et al. (2023) created a knowledge bank for situational
norms, utilizing English-speaking Mechanical Turk annotators and incorporating
country taxonomy. Unlike these works, IndoCulture specifically concentrates on
cultural reasoning within Indonesian provinces, developed and validated
manually by local people (experts). Compared to the automatic method and
English-speaking crowdsource workers for data construction, IndoCulture
arguably contains less noise, and is free from training data contamination of
large language models (LLMs).

#### Commonsense Reasoning with Indonesian contexts

Commonsense reasoning in Indonesian language models has been studied using
translated English–Indonesian datasets, such as XCOPA Ponti et al. (2020) and
XStoryCloze Lin et al. (2021). However, these datasets potentially introduce a
cultural bias toward English culture. IndoCloze Koto et al. (2022) is the
first commonsense reasoning dataset in Indonesian, developed by native
Indonesian workers following cloze story completion framework Mostafazadeh et
al. (2016). IndoMMLU Koto et al. (2023), a benchmark for assessing knowledge
in the Indonesian context, includes questions pertaining to Indonesian
culture. However, its primary focus lies on testing proficiency in the local
language rather than reasoning abilities. In a contemporary study, Wibowo et
al. (2023) followed the COPA framework Roemmele et al. (2011) to build a
dataset with contexts limited to Jakarta. In other work, Putri et al. (2024)
studied the capability of LLMs in generating questions with cultural norms,
for both general Indonesian and specific Sundanese contexts. In comparison,
IndoCulture offers broader coverage across eleven provinces, thereby fostering
greater inclusivity for local communities in Indonesia.

##  3 IndoCulture

As illustrated in Figure 1, IndoCulture is a sentence completion task in
Indonesian language featuring a one-sentence premise, three plausible options,
and one correct option to evaluate reasoning ability and cultural knowledge
across eleven Indonesian provinces. In total, there are 2,429 instances, with
the distribution across provinces detailed in Table 1.

###  3.1 Data Construction

IndoCulture is constructed manually by human and verified through a two-step
process.

#### Worker Recruitment

Culture often arises from the shared experiences, traditions, and beliefs of a
specific group over time, often closely intertwined with native populations.
Keeping this in mind, we engaged individuals from various provinces across
Indonesia to assist in preparing data for the IndoCulture benchmark. During
the recruitment, we presented a few examples of the intended IndoCulture data
and requested each candidate to generate similar instances tailored to the
context of their respective provinces. From a pool of 58 applicants, we
carefully selected 22 expert workers representing 11 provinces (with 2 workers
selected per province). These recruited expert workers are local residents and
have resided in their respective provinces for a minimum of 10 years, thereby
possessing a profound understanding of local customs and culture. The age
range of our workforce spans from 21 to 35 years old, with educational
backgrounds distributed as follows: 3 high school graduates, 14 bachelor’s
degree holders, 4 master’s degree holders, and 1 PhD holder. During data
construction, each expert worker will fulfill the dual roles of instance
writer and quality controller.333Each worker is compensated above the monthly
minimum wage in Indonesia.

#### Province Selection

The provinces covered in this study represent diversity of Indonesian
cultures. Those 11 provinces (in Figure 1) spread across 6 islands and large
archipelagos in Indonesia which are inhabited by various tribes who speak
different regional languages and adhere to different religions.

#### Topic Taxonomy

IndoCulture consists of 12 topics and 66 fine-grained subtopics. Below is a
list of topics along with their detailed subtopics. The numbers following the
topic indicate the total instances required to be written by one worker (with
a total of 150 per worker).

  1. 1.

Food (22): breakfast (2); lunch (3); dinner (2); snacks (2); food souvenir
(3); traditional foods and beverages (5); eating habit (1); cutlery (1);
cooking ware (1), fruits (2).

  2. 2.

Wedding (20): traditions before marriage (3); traditions when getting married
(3); traditions after marriage (3); men’s wedding clothes (2); women’s wedding
clothes (2); invited guests (2); wedding location (1); foods at a wedding (2);
gifts brought to weddings (2).

  3. 3.

Family relationship (13): relationships within the main family (3);
relationships in the extended family (3); relations with society/neighbors
(5); clan/descendant system (2).

  4. 4.

Pregnancy and kids (16): traditions during pregnancy (4); traditions after
birth (2); how to care for a newborn baby (2); how to care for toddlers (2);
how to care for children (2); how to care for teenagers (2); parents and
children interactions as adults (2).

  5. 5.

Death (10): when death occurs (2); the process of caring for a corpse (2);
traditions after the body is buried (2); the clothes of the mourners (2);
inheritance matters (2).

  6. 6.

Religious holiday (12): traditions before religious holidays (2); traditions
leading up to religious holidays (4); traditions during religious holidays
(5); traditions after religious holidays (1).

  7. 7.

Agriculture (6): what to plant (2); traditions when planting (2); harvest (2).

  8. 8.

Fisheries and trade (7): traditions of taking care of livestock/fish (5);
buying and selling traditions (2)

  9. 9.

Art (16): musical instruments (3); folk songs (3); traditional dances (3); use
of art at certain events (5); poetry or similar literature (2)

  10. 10.

Traditional games (5): game types; (3), location played (2).

  11. 11.

Daily activities (10): morning activities (1); afternoon activities (1);
evening activities (1); leisure activities (3); house, household and
transportation (4).

  12. 12.

Socio-religious aspects of life (13): regular religious activities (2);
mystical things (2); traditional ceremonies (1); lifestyle (3); self care (1);
traditional medicine (3); traditional saying (1).

#### Instance Writing

For each instance, workers craft two culturally relevant sentences that align
with the predefined subtopic. The first sentence serves as the premise
context, while the last sentence acts as the correct answer. Subsequently, the
annotator generates two additional plausible sentences as distractors by
modifying cultural objects or activities from the correct sentence. Workers
are given a period of two months to complete the task.

#### Two Stages of Quality Control

In stage 1, we implement quality control by pairing two annotators from the
same province. Each annotator is tasked with answering a set of questions
prepared by the other annotator, and vice versa. During this phase, the
annotator is presented with a premise sentence and three shuffled options.
They are allowed to search for the answer from any source if they are unsure.
Instances that are incorrectly answered by the second annotator are discarded,
as we hypothesize that these instances may contain incorrect answers or
possess a level of ambiguity. Additionally, annotators are required to
identify whether the instance is province-specific (binary annotation:
True/False), meaning it is uniquely relevant in their province and not in
others.

In stage 2 of quality control, the first two authors of this paper perform
post-editing on the data that has passed the first stage of quality control.
We first focus on correcting the linguistic aspects of the text, which
includes checking for spelling typos. Although the data is written in
Indonesian, some annotators may use dialects or be influenced by the structure
or style of regional languages. In these cases, we correct the text to adhere
to correct Indonesian grammar.

To maintain the quality of IndoCulture, we rigorously filter instances that
display (1) poor writing, as enhancing their quality substantially is
difficult; (2) obvious answer options, which allow for easy guessing of the
correct choice without understanding the cultural contexts; and (3) ambiguous
contexts, where all options are equally valid as the correct answer. For
example, in a topic about breakfast, the three options might include one
traditional food alongside two other very commonly consumed foods in
Indonesia.

Furthermore, we manually verify the province-specific annotation for each
instance using the Google search engine. We annotate whether the instance
pertains to national-level culture. If the example is specific to a province,
it will be annotated as uncommon in national culture, and vice versa.

###  3.2 Data Statistics

After the annotation process, we initially collected 3,162
instances.444Although we requested each annotator to produce 150 instances,
not all were able to complete their allotted tasks within the given timeframe.
Unfortunately, we were unable to find additional candidates from the same
local province to address the data deficiencies Winata et al. (2023). In stage
1 of quality control, the initial pool of 3,162 instances was reduced to 2,801
instances. Our thorough stage 2 of quality control further reduced the sample
to 2,429 final high-quality samples. The data distribution of IndoCulture per
province is presented in Table 1. Approximately three-quarters of IndoCulture
instances contain province-specific content, with an average length of around
35 words. IndoCulture covers multiple topics, as illustrated in Figure 2.

Figure 2: Topic distribution in IndoCulture.

Province | # | province | $\mu$(word) | $\mu$(char)
---|---|---|---|---
specific (%)
Aceh | 246 | 70.7 | 28.0 | 175.9
North Sumatra | 234 | 83.8 | 36.8 | 246.0
West Sumatra | 299 | 74.6 | 39.6 | 261.4
West Java | 231 | 58.0 | 37.5 | 244.8
Central Java | 171 | 66.7 | 39.3 | 260.5
East Java | 233 | 69.5 | 46.0 | 310.4
Bali | 241 | 76.3 | 33.3 | 216.1
NTT | 103 | 72.8 | 31.8 | 203.6
South Borneo | 233 | 83.7 | 33.3 | 226.0
South Sulawesi | 185 | 90.3 | 33.6 | 227.8
Papua | 253 | 88.1 | 37.3 | 245.0
All | 2429 | 76.0 | NA | NA

Table 1: Overall statistics of IndoCulture by province.

##  4 Experiments

###  4.1 Set-Up

We focus on evaluating 23 language models in zero-shot settings: (1) 15 open-
source multilingual language models including BLOOMZ Muennighoff et al.
(2022), mT0 Muennighoff et al. (2022), LlaMA2 Touvron et al. (2023),
Bactrian-X Li et al. (2023a) in different sizes, (2) 2 South East Asian
language models including SeaLLM Nguyen et al. (2023b), and SeaLion Singapore
(2023), (3) 4 Indonesian-centric language models, including IndoBART
Cahyawijaya et al. (2021), IndoGPT Cahyawijaya et al. (2021), Merak Ichsan
(2023), Komodo Owen et al. (2024), (3) 2 closed-source models, including
ChatGPT: gpt-3.5-turbo Ouyang et al. (2022) and GPT–4: gpt-4-0613 OpenAI
(2023). Please refer to Appendix A for further details.

Initially, we evaluate the effectiveness of sentence completion and multiple-
choice question strategies in predicting the correct options. In both
scenarios, we conduct benchmarks across three distinct location contexts.
Formally, given a premise $s$, three candidate options $c_{1}$, $c_{2}$,
$c_{3}$, and location $l\in\\{\texttt{none},\texttt{Indonesia},p_{j}\\}$, for
sentence completion, we select the correct option based on:

$\displaystyle\underset{c}{\mathrm{argmax}}\log P(\text{concat}(s,c)|l)$

Here, $\text{concat}(s,c)$ denotes the concatenation of premise $s$ and
candidate option $c$, separated by a space. In the case of multiple-choice
questions, we devise a template for the prompt question and determine the
answer by selecting the option with the highest probability among letters A,
B, and C. Figure 3 displays the prompt templates in Indonesian and English for
both sentence completion and multiple-choice question.

Figure 3: Templates for sentence completion and multiple-choice questions
prompts.

###  4.2 Results

Model (#parameter) | Completion |  | MCQ
---|---|---|---
$l=\text{None}$ | $l=\text{Ind}$ | $l=\text{Prov}$ |  | $l=\text{None}$ | $l=\text{Ind}$ | $l=\text{Prov}$
Human | – | – | 100.0 |  | – | – | 100.0
Random | 33.3 | 33.3 | 33.3 |  | 33.3 | 33.3 | 33.3
BLOOMZ (560M) | 37.5 | 35.3 | 35.3 |  | 32.1 | 32.0 | 32.1
BLOOMZ (1.1B) | 36.2 | 36.9 | 37.2 |  | 32.0 | 32.0 | 32.0
BLOOMZ (3B) | 38.5 | 40.5 | 41.2 |  | 45.9 | 47.6 | 48.6
BLOOMZ (7B) | 40.9 | 43.8 | 44.4 |  | 48.8 | 49.8 | 49.7
mT0${}_{\text{small}}$ (300M) | 28.3 | 28.2 | 28.3 |  | 33.7 | 32.7 | 32.7
mT0${}_{\text{base}}$ (580M) | 28.6 | 28.3 | 28.7 |  | 35.6 | 35.3 | 35.6
mT0${}_{\text{large}}$ (1.2B) | 29.5 | 29.5 | 30.1 |  | 35.7 | 35.8 | 35.9
mT0${}_{\text{xl}}$ (3.7B) | 32.0 | 30.9 | 31.2 |  | 49.2 | 50.0 | 50.3
mT0${}_{\text{xxl}}$ (13B) | 33.0 | 33.3 | 34.4 |  | 52.9 | 51.5 | 52.3
LLaMA2 (7B) | 36.7 | 37.0 | 37.2 |  | 40.0 | 39.0 | 37.7
LLaMA2-chat (7B) | 36.9 | 36.9 | 37.4 |  | 40.5 | 41.1 | 40.3
LLaMA2 (13B) | 39.4 | 39.7 | 39.9 |  | 46.9 | 46.9 | 46.6
LLaMA2-chat (13B) | 38.2 | 38.4 | 38.9 |  | 47.2 | 49.2 | 49.2
Bactrian-X${}_{\text{LLaMA}}$ (7B) | 33.2 | 33.8 | 33.8 |  | 38.1 | 38.5 | 38.7
Bactrian-X${}_{\text{LLaMA}}$ (13B) | 33.0 | 35.1 | 34.9 |  | 38.7 | 38.5 | 38.7
IndoBART (132M) | 41.9 | 41.1 | 41.7 |  | 32.1 | 32.0 | 32.2
IndoGPT (117M) | 42.6 | 42.1 | 42.7 |  | 33.7 | 34.1 | 35.0
Merak (7B) | 40.2 | 40.6 | 42.6 |  | 51.5 | 52.5 | 52.5
SeaLLM (7B) | 39.1 | 39.7 | 41.1 |  | 51.7 | 52.4 | 52.5
SEA-LION (7B) | 38.8 | 38.7 | 39.6 |  | 33.4 | 32.8 | 33.0
Komodo (7B) | 45.1 | 45.4 | 46.1 |  | 37.6 | 35.1 | 36.1
GPT–3.5 (NA) | – | – | – |  | 58.4 | 60.2 | 61.7
GPT–4 (NA) | – | – | – |  | 68.8 | 71.5 | 75.8

Table 2: Zero-shot accuracy across various models and settings. “MCQ” refers
to the multiple-choice question method, and $l$ denotes the location as
additional context. The bold numbers highlight the highest score within each
model group.

#### Overall observation

The results presented in Table 2 display the performance across various models
and settings. The overall observation is that all open-source models struggle
with understanding Indonesian culture, contrasting sharply with the 100%
accuracy achieved by human (i.e., native people). Among open-source models,
Merak and mT0${}_{\texttt{xxl}}$ achieves the highest accuracy at 52%, while
closed-source models, such as GPT–3.5 and GPT–4.0, achieve performances of
61.7% and 75.8%, respectively. These findings underscore the challenging
nature of the IndoCulture dataset.

#### Multiple-choice question method is generally better.

Our findings suggest that the multiple-choice question method tends to
outperform the sentence completion method, with exceptions noted for BLOOMZ
(560M, 1.1B), IndoBART, IndoGPT, and Komodo. Interestingly, within the
sentence completion method, the small Indonesian-centric language model,
IndoGPT, surpasses the performance of almost all large multilingual models and
remains competitive with BLOOMZ (7B). However, IndoGPT’s performance
experiences a significant decline in the multiple-choice question method, with
a notable margin of nearly 8 to 10 points. While Komodo (7B) achieves the
highest accuracy with the sentence completion method, its performance also
decreases by 10-12 points in the multiple-choice method. This discrepancy
could potentially be attributed to differences in the nature of language model
training and instruction-tuning.

#### Impact of location context on model performance

Our investigation reveals that incorporating various levels of location
granularity has a noticeable effect on zero-shot performances, especially
models with larger parameter sizes. Detailed location contexts notably enhance
the accuracy of BLOOMZ (7B), LlaMA2 (13B), Merak (7B), SeaLLM (7B), Komodo
(7B), GPT–3.5, and GPT–4. For instance, in GPT–4, the accuracy gap between
$l=\texttt{none}$ and $l=\texttt{Indonesia}$ is 2.7, and this gap further
increases by 7 points when $l=\texttt{Prov}$ is assigned.

###  4.3 Analysis

Given the superior performance of models with large parameter sizes using the
multiple-choice question method and location $l=\texttt{Prov}$, we employ
these settings for analysis.

Province | Merak | GPT–3.5 | GPT–4
---|---|---|---
$\neg$PS | PS |  $\neg$PS | PS |  $\neg$PS | PS
Aceh | 59.7 | 53.4 | 79.2 | 65.5 | 93.1 | 73.6
North Sumatra | 58.3 | 46.9 | 63.9 | 57.2 | 75.0 | 73.2
West Sumatra | 50.0 | 41.3 | 63.2 | 48.4 | 85.5 | 63.7
West Java | 65.3 | 58.8 | 84.2 | 71.8 | 92.6 | 81.7
Cental Java | 58.8 | 47.1 | 67.6 | 52.9 | 82.4 | 72.9
East Java | 57.4 | 37.6 | 72.2 | 46.5 | 87.0 | 63.4
Bali | 78.9 | 65.6 | 87.7 | 81.7 | 93.0 | 89.4
NTT | 64.3 | 52.0 | 57.1 | 49.3 | 85.7 | 68.0
South Borneo | 65.7 | 46.4 | 68.6 | 53.6 | 77.1 | 67.4
South Sulawesi | 53.3 | 47.9 | 66.7 | 55.8 | 80.0 | 70.6
Papua | 76.7 | 55.2 | 76.7 | 62.8 | 90.0 | 71.3

Table 3: Top-3 model accuracy by province. "PS" indicates instances containing
province-specific contexts, while "$\neg$PS" indicates otherwise. The green
and red cells indicate the top three and bottom three scores, respectively.

#### Results by province

Table 3 highlights that the top 3 performing LLMs exhibit a nuanced
understanding of cultures within Indonesian provinces, particularly excelling
in the cultures of West Java and Bali compared to other provinces. GPT–4, for
instance, achieves the best accuracy of 93%, while Merak and GPT–3.5 also
demonstrate strong performance in Bali and West Java, with accuracies ranging
between 55% and 88%. In other provinces like West Sumatra and South Borneo,
the models typically exhibit poorer performance. Specifically, for GPT-3.5,
the performance gap compared to Bali ranges from 19 to 33%. This highlights
the presence of cultural biases and a lack of inclusivity in model reasoning
abilities, likely stemming from the distribution of training data. The
proximity of West Java to Jakarta (Indonesia’s capital) and Bali’s global
status as a tourism destination may contribute to the abundance of textual
data on these two cultures.

We also note a consistent disparity between non-province and province-specific
contexts across all models, with models generally finding non-province
contexts easier to comprehend. On average, this gap ranges from 12 to 13
points for the three models, highlighting the challenge posed by province-
specific content and emphasizing the significant influence of location context
on the reasoning ability of LLMs.

#### Results by topic

Table 4 shows the accuracy of the top 3 performing models across topics.
Similar to Table 3, the models perform better in province-specific context for
all topics, with death customs being an exception. For province-specific
contexts, GPT–4 excels in themes of wedding, death, and arts while for non-
specific contexts, GPT–4 achieves accuracy more than 90.0 for family
relationship, religious holidays, and farm and fishery.

Province | Merak | GPT–3.5 | GPT–4
---|---|---|---
$\neg$PS | PS |  $\neg$PS | PS |  $\neg$PS | PS
Food | 56.9 | 51.0 | 65.9 | 62.1 | 79.3 | 63.5
Wedding | 66.7 | 64.3 | 77.6 | 56.0 | 92.8 | 78.6
Family relationship | 65.5 | 54.5 | 72.5 | 53.9 | 93.1 | 72.7
Pregnancy and kids | 67.9 | 42.9 | 69.2 | 50.6 | 82.1 | 72.7
Death | 38.5 | 55.2 | 60.9 | 62.5 | 76.9 | 82.5
Religious holidays | 68.5 | 48.8 | 79.3 | 68.8 | 94.4 | 64.6
Agriculture | 70.5 | 52.3 | 71.4 | 62.1 | 88.5 | 76.1
Farm and fishery | 77.8 | 51.0 | 77.8 | 51.9 | 94.4 | 75.8
Art | 58.7 | 50.2 | 85.2 | 61.1 | 78.3 | 77.3
Traditional game | 60.0 | 50.2 | 80.3 | 59.8 | 88.0 | 66.9
Daily activities | 55.1 | 39.6 | 66.0 | 64.8 | 87.8 | 63.0
Socio-religious | 57.1 | 44.3 | 79.6 | 61.1 | 85.7 | 64.6

Table 4: Top-3 model accuracy by topic. "PS" indicates instances containing
province-specific contexts, while "$\neg$PS" indicates otherwise. The green
and red cells indicate the top three and bottom three scores, respectively.

#### Does language affect model performance?

We automatically translated IndoCulture to English using the Google Translate
API555Accessed in March 2024. and used the English prompt in Figure 3 to
evaluate the models. All model performances in English text dropped except for
LLaMA-2 and Merak. This could be attributed to two reasons. First, LLaMA-2 is
an English-centric model, and Merak is pretrained from LLaMA-2. Second, the
performance drop in other models could be caused by translation errors. We
further investigated this with 100 random samples and found that 81 samples
had acceptable translations. We observed translation errors such as pronoun
mismatches, inaccurate proverb translations, and inaccurate translations of
local terms, such as “pupuik” translated as “fertilizer”.666pupuik is a
traditional musical instrument in West Sumatra. It is worth noting that the
word pupuik closely resembles the word pupuk in Indonesian, which means
fertilizer. To better understand the cultural gap in language models, we
followed the approach of Liu et al. (2023a) to manually correct the
translations and reevaluate the models. We found that GPT-4’s performance over
the 100 random samples was 77.0 for the original Indonesian text, 68.0 for the
English machine-translated text, and 72.0 for the English translation fixed by
humans.

Figure 4: The accuracy of Indonesian and English translations across BLOOMZ
(7B), mT0${}_{\text{xxl}}$ (13B), LLaMA2-chat (13B), Merak (7B), GPT-3.5, and
GPT-4.

#### Results by fine-grained culture elements

We tasked two expert workers with annotating 200 random samples from
IndoCulture based on six cultural elements, as derived from Axtell and
Fornwald (1998); Williams
(2014).777https://www.languageeducatorsassemble.com/elements-of-
culture888https://pressbooks.howardcc.edu/soci101/chapter/3-2-the-elements-of-
culture/ While these elements may not encompass every cultural aspect, we
contend that they cover the most prominent or pivotal elements, including: (1)
symbols (material or non-material objects representing meaning), (2) artifacts
(material or non-material objects produced by society), (3) values and beliefs
(principles, ideas, and concepts assumed to be ideal and correct in society),
(4) norms (rules guiding values and beliefs), (5) language, (6) rituals
(established procedures and ceremonies), and (7) other, for examples that do
not fit into any of the defined elements. This annotation is a multi-label
task, and the average Kappa score across the cultural elements is 0.56, with
each ranging from 0.4 to 0.75. These scores indicate moderate to substantial
agreement.

Table 5 displays the distribution of each cultural element in our dataset,
along with the performance breakdown across Merak, GPT-3.5, and GPT-4. Among
the 200 random samples, we observe that 42.5% of our data contains artifacts,
37.5% norms, and 30% rituals. Only 4% of the data pertains to symbols, while
7.5% belongs to the other category. Merak exhibits lower performance in norms,
with a 24% decrease compared to values and beliefs. Conversely, GPT-3.5 and
GPT-4 perform better in norms, with the performance gap in values and beliefs
being only $-4$ points. Additionally, language poses a challenge in Merak,
with only 38% accuracy, while GPT-3.5 and GPT-4 achieve 60% and 72% accuracy,
respectively.

Cultural element (%) |  Merak |  GPT–3.5 |  GPT-4
---|---|---|---
Symbols (4) |  50.0 |  58.9 |  70.8
Artefacts (42.5) |  55.3 |  59.1 |  67.8
Values and Beliefs (10.5) |  61.9 |  57.5 |  69.8
Norms (37.5) |  38.7 |  63.2 |  73.6
Language (19.5) |  38.5 |  60.2 |  72.0
Ritual (30) |  53.3 |  60.0 |  70.7
Other (7.5) |  66.7 |  58.9 |  69.2

Table 5: Accuracy comparison of Merak, GPT–3.5, and GPT–4 across 200 random
samples, categorized by cultural elements. The numerical value following each
cultural element indicates its proportion within the samples.

#### Can the model provide a reasonable explanation to support the answer?

Figure 5: Performance comparison between Merak (7B) and GPT–4 based on text
generation output. “Answer (T)” indicates that the generated answer is true,
while ”Exp(F)” denotes that the answer explanation is false.

We conduct a manual investigation of the text generation output for Merak (7B)
and GPT–4 across 200 random samples. This involves manually examining the
generated answer along with its explanation. To obtain explanation, we modify
the Indonesian prompt in Figure 3 by adding the string Jelaskan jawabanmu!
("Explain your answer!"). Our annotation process is binary, categorizing
explanations as either True or False. We label an explanation as False if it
is absent, contains hallucinations, or provides inaccurate information.999We
use Google search engine to verify the correctness of the explanation.

As anticipated, there is a significant drop in accuracy for Merak (7B) from
52.5% (as shown in Table 2) to 29.5%. This discrepancy underscores the
limitations of relying solely on token probability to assess the true
capability of a language model. Interestingly, only 4.5% of the samples are
answered correctly with the appropriate explanation by Merak, despite it being
the top performer among the open-source models. GPT–4 achieves an accuracy of
69.5%, which is 5 points lower than indicated in Table 2, possibly due to a
smaller sample size and biases in regular expressions. However, GPT–4 provides
better explanations, with 51.5% of the samples correctly answered along with
the appropriate explanation.

##  5 Discussion

The recent study Wang et al. (2024) has demonstrated that evaluating language
models using token probability in multiple-choice question types does not
align well with the generated text. This discrepancy is reported to be more
pronounced in models fine-tuned on conversational or safety data. In response
to this issue, we have conducted a manual evaluation on 200 random samples (in
Section 4.3) and found that the performance in generated texts, especially for
Merak, the best open-source language model, is deteriorating. However, this
issue is less apparent in GPT-4, as the initial evaluation is based on regular
expressions. Conducting manual evaluation on all data and models is expensive,
and we plan to address this issue in future works. This work primarily focuses
on introducing a novel dataset meticulously constructed for evaluating
cultural commonsense reasoning within the Indonesian context, including
preliminary evaluation results based on the standard methods used in previous
studies OpenAI (2023); Touvron et al. (2023); Koto et al. (2023); Li et al.
(2023b); Koto et al. (2024).

##  6 Conclusion

IndoCulture is a cultural commonsense reasoning dataset encompassing the
diversity of Indonesian cultures, spanning from Aceh province in the west to
Papua province in the east. Through collaboration with local individuals
across eleven provinces and rigorous quality control measures, we introduce
IndoCulture for the purpose of evaluating language models. Our findings reveal
that large language models, whether Indonesian-centric or multilingual,
demonstrate a limited understanding of Indonesian culture. Notably,
incorporating location as additional context significantly enhances model
performance, particularly in GPT–3.5 and GPT–4.

## Limitations

IndoCulture is specifically designed to explore the influence of geographical
location on cultural commonsense reasoning, with a focus on the present time.
It does not consider temporal aspects. Our dataset was created in the year
2023, recognizing that cultures may evolve over time, as discussed by Mesoudi
(2016).

Furthermore, as demonstrated in Section 4.3, a significant portion of
IndoCulture comprises cultural elements such as artifacts, norms, and rituals.
Symbols, values and beliefs, and language represent smaller proportions,
ranging from 4% to 20%. We encourage future research to further explore these
cultural elements and to expand the geographical coverage beyond the eleven
provinces studied in this paper.

## Ethical Considerations

IndoCulture is licensed under a Creative Commons Attribution-NonCommercial-
ShareAlike 4.0 International
License.101010https://creativecommons.org/licenses/by-nc-sa/4.0/ Our data is
intended for academic research and non-commercial purposes. Workers are
compensated above the minimum monthly salary in Indonesia and are fully aware
that the data will be released to the public. It is important to note that no
private or sensitive information of the workers is included in IndoCulture.

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##  Appendix A Model Artifacts

Models (#parameters) | Source
---|---
BLOOMZ (560M) | bigscience/bloomz-560m
BLOOMZ (1.1B) | bigscience/bloomz-1b1
BLOOMZ (1.7B) | bigscience/bloomz-1b7
BLOOMZ (3B) | bigscience/bloomz-3b
BLOOMZ (7.1B) | bigscience/bloomz-7b1
mT0${}_{\text{small}}$ (300M) | bigscience/mt0-small
mT0${}_{\text{base}}$ (580M) | bigscience/mt0-base
mT0${}_{\text{large}}$ (1.2B) | bigscience/mt0-large
mT0${}_{\text{xl}}$ (3.7B) | bigscience/mt0-xl
mT0${}_{\text{xxl}}$ (13B) | bigscience/mt0-xxl
LLamA2 (7B) | meta-llama/Llama-2-7b
LLamA2-chat (7B) | meta-llama/Llama-2-7b-chat
LLamA2 (13B) | meta-llama/Llama-2-13b
LLamA2-chat (13B) | meta-llama/Llama-2-13b-chat
Bactrian-X${}_{\texttt{LLaMa}}$ (7B) | MBZUAI/bactrian-x-llama-7b-merged
Bactrian-X${}_{\texttt{LLaMa}}$ (13B) | MBZUAI/bactrian-x-llama-13b-merged
IndoBART (132M) | indobenchmark/indobart-v2
IndoGPT (117M) | indobenchmark/indogpt
Merak (7B) | Ichsan2895/Merak-7B-v5-PROTOTYPE1
SeaLLM (7B) | SeaLLMs/SeaLLM-7B-v2
SEA-LION (7B) | aisingapore/sea-lion-7b
Komodo (7B) | Yellow-AI-NLP/komodo-7b-base

Table 6: With the exception of GPT-3.5 and GPT-4, all the models used in this
study were sourced from Huggingface Wolf et al. (2020).


11institutetext: Department of Theoretical Physics and Astrophysics, Faculty
of Science,
P. J. Šafárik University, Park Angelinum 9, 040 01 Košice, Slovak Republic
22institutetext: Laboratoire de Physique Théorique et Modélisation, CY Cergy
Paris Université,
Avenue A. Chauvin, Pontoise, 95302 Cergy-Pontoise, France

# Pseudo-transition between antiferromagnetic and charge orders in a minimal
spin-pseudospin model of one-dimensional cuprates

Jozef Strečka 11<EMAIL_ADDRESS>Katarína Karl’ová 22
<EMAIL_ADDRESS>
(Received: date / Revised version: date)

###### Abstract

This study rigorously investigates the phenomenon of a pseudo-transition in a
minimal spin-pseudospin model, serving as a simplified model of one-
dimensional cuprate chains [CuO]∞, by making use of the transfer-matrix
method. The studied spin-pseudospin model of one-dimensional cuprate chains
[CuO]∞ reveals a peculiar pseudo-transition between a charged-ordered phase
and an antiferromagnetic phase, which is accompanied with anomalous behavior
of magnetic and thermodynamic quantities. While the entropy and short-range
correlations undergo near the pseudo-transition steep continuous changes
reminiscent of a discontinuous phase transition, the specific heat displays a
sizable sharp peak akin to a continuous phase transition.

###### pacs:

05.50.+qLattice theory and statistics and 75.10.-bGeneral theory and models of
magnetic ordering and 75.10.PqSpin chain models and 75.30.KzMagnetic phase
boundaries and 75.40.CxStatic properties

##  1 Introduction

One-dimensional lattice-statistical spin models traditionally attract a great
deal of attention due to their conceptual simplicity and exact solvability
mat93 . Previous exact solutions for one-dimensional spin models with short-
range interactions and non-singular potential have consistently ruled out the
existence of finite-temperature phase transitions in accordance with the non-
existence theorems rigorously proved by van Hove hov50 , Cuesta and Sánchez
cue04 . However, recent studies revealed that a variety of enigmatic one-
dimensional lattice-statistical models may display pseudo-transitions at
nonzero temperatures, which closely resemble genuine thermal phase
transitions. Notable examples of one-dimensional lattice-statistical models
displaying thermal pseudo-transitions include the coupled spin-electron
double-tetrahedral chain gal15 , spin-1/2 Ising-Heisenberg ladder roj16  and
tube str16 , spin-1/2 Ising-XYZ diamond chain car18 ; car19 , spin-1/2 and
mixed spin-(1/2, 1) Ising-Heisenberg double-tetrahedral chains ono20 ; roj20 ,
spin-1/2 Ising diamond and tetrahedral chain str20 ; nov20 , mixed
spin-(1/2,1) Ising hexagonal nanowire pim22 , spin-1/2 Ising ladder with
trimer rungs hut21 ; yin24 , as well as, one-dimensional q-state Potts models
pan21 ; pan23 . Thermal pseudo-transitions of all aforementioned one-
dimensional lattice-statistical models differ from genuine phase transitions
in several important respects: first-order derivatives of a free energy
display in vicinity of a pseudo-transition a steep but continuous change
instead of a true discontinuity and second-order derivatives of a free energy
exhibit in vicinity of a pseudo-transition a sizable finite peak instead of a
power-law divergence sou18 ; oro20 . In spite of this fact, sharp (albeit
finite) peaks in correlation length, specific heat and magnetic susceptibility
closely mimic peculiar power-law divergences near pseudo-transitions,
characterized by a set of universal quasi-critical exponents roj19 ; kro21 .
Moreover, anomalous behavior of quantities observed around pseudo-transitions
is accompanied by significant finite-size effects roj21 , whereby the pseudo-
transitions can be approached arbitrarily closely yin24 .

In this paper, we introduce and exactly solve a minimal spin-pseudospin model
for [CuO] centers of the polymeric cuprate chain [CuO]∞, which will extend the
realm of one-dimensional lattice-statistical models displaying pseudo-
transition. The simplified spin-pseudospin model of the one-dimensional
cuprate chain [CuO]∞ will take into consideration three valence states of
Cu1+, Cu2+ and Cu3+ centers corresponding to the many-electron valence states
of [CuO]-, [CuO]0 and [CuO]+ centers as three components of the pseudospin
triplet mos11 ; mos13 ; mos19 . The spin-pseudospin model of one-dimensional
cuprate chain [CuO]∞ will take into account a mutual electrostatic correlation
between non-magnetic valence states of [CuO]- and [CuO]+ centers (nominally
Cu1+ and Cu3+ centers), as well as, an antiferromagnetic superexchange
interaction between the spin-1/2 Cu2+ centers mediated by O2- anions within
the electroneutral [CuO]0 centers. The approach based on a minimal spin-
pseudospin model has already demonstrated its utility for two-dimensional
cuprates composed of CuO2 planes, which exhibit rather complex phase diagrams
due to competition between insulating antiferromagnetic, charged ordered and
superconducting phases mos19 ; pan16 ; pan17 ; pan19 ; pan20 ; pan22 .
Although the present spin-pseudospin model was primarily developed for the
purpose of investigating a potential pseudo-transition between
antiferromagnetic and charge ordered phases, it may also shed light on
essential features of one-dimensional cuprates Sr2CuO3 ami95 ; mai97 ; sch12 ;
sch18 , Ca2CuO3 yam95  or SrCuO2 nag97 ; bou18  incorporating cuprate chains
[CuO]∞ as their basic structural motif.

The paper is structured as follows. The spin-pseudospin model will be
introduced in Section 2 along with a concise overview of the method used for
its exact treatment. The discussion of the most interesting results is
presented in Section 3 and finally, some concluding remarks summarizing the
key findings and implications for future studies are given in Section 4.

##  2 Model and method

Figure 1: An exemplary configuration of the considered spin-pseudospin model
of the one-dimensional cuprate chain [CuO]∞. Large green spheres represent
Cu2+ centers (i.e., the electroneutral bare centers [CuO]0) characterized by
the pseudospin value $S=0$, whereby up- and down-pointing arrows specify a
magnetic polarization characterized by the spin values $\sigma=1$ and $-1$,
respectively. Large red and blue spheres represent Cu3+ and Cu1+ centers
(i.e., positively charged hole centers [CuO]+ and negatively charged electron
centers [CuO]-) characterized by the pseudospin values $S=1$ and $-1$,
respectively. Small violet spheres correspond to O2- anions.

Let us consider the minimal spin-pseudospin model of one-dimensional cuprate
chain [CuO]∞, which takes into consideration three valence states of Cu1+,
Cu2+ and Cu3+ centers described by the pseudospin triplet with the projections
$S=-1,0,+1$, respectively. Within the polymeric cuprate chain [CuO]∞, the
three nominal valence states of Cu1+, Cu2+ and Cu3+ centers surrounded by O2-
anions give rise to three distinct valence states of the [CuO] center: a
negatively charged electron center [CuO]-, an electroneutral bare center
[CuO]0, and a positively charged hole center [CuO]+. While two charged
pseudospin states [CuO]- and [CuO]+ involve nonmagnetic Cu1+ and Cu3+ centers,
respectively, the electroneutral pseudospin state [CuO]0 encompasses a
magnetic spin-1/2 Cu2+ center characterized by the Ising spin $\sigma=\pm 1$.
The non-magnetic charged states of [CuO]- and [CuO]+ centers (nominally Cu1+
and Cu3+) are subject to electrostatic forces, which are absent in the
electroneutral bare centers [CuO]0 subjected solely to short-range magnetic
forces. An exemplary configuration of the spin-pseudospin model of the one-
dimensional cuprate chain [CuO]∞ is illustrated in Fig. 1. Bearing all this in
mind, the minimal spin-pseudospin model of the one-dimensional cuprate chain
[CuO]∞ can be described by the following effective Hamiltonian:

$\displaystyle{\cal
H}\\!=\\!J\\!\sum_{i=1}^{N}(1\\!-\\!S_{i}^{2})\sigma_{i}(1\\!-\\!S_{i+1}^{2})\sigma_{i+1}\\!+\\!\Delta\\!\sum_{i=1}^{N}\\!S_{i}^{2}\\!+\\!V\\!\sum_{i=1}^{N}\\!S_{i}S_{i+1},$
(1)

which is expressed in terms of the pseudospin triplet $S_{i}=-1,0,1$
corresponding to the three valence states of Cu1+, Cu2+, Cu3+ centers and the
Ising spin $\sigma_{i}=\pm 1$ ascribed to the magnetic Cu2+ center only (i.e.
$S_{i}=0$). The first term of the Hamiltonian (1) accounts for the
antiferromagnetic exchange interaction $J>0$ between nearest-neighbor Cu2+
centers, whereby $(1\\!-\\!S_{i}^{2})$ acts as the respective projection
operator for the magnetic pseudospin state $S_{i}=0$. The second term of the
Hamiltonian (1) represents the on-site energy cost $\Delta>0$ associated with
the creation of either a negatively charged electron center [CuO]- or a
positively charged hole center [CuO]+ corresponding to the nominal valence
states Cu1+ and Cu3+, respectively. The third term of the Hamiltonian (1)
takes into account the inter-site electrostatic energy between two nearest-
neighbor charged centers, which lowers (raises) the overall electrostatic
energy provided that two neighboring centers are oppositely (equally) charged.
It should be noted that we impose electroneutrality constraints on the cuprate
chain [CuO]∞, which means that the nominal valence states Cu1+ and Cu3+
leading to negatively [CuO]- and positively [CuO]+ charged centers may only
arise in pairs as a result of one-electron charge transfer: Cu2+ \+ Cu2+ $\to$
Cu1+ \+ Cu3+.

The total Hamiltonian (1) of the spin-pseudospin model can be written as a sum
over bond Hamiltonians:

$\displaystyle{\cal H}=\sum_{i=1}^{N}{\cal H}_{i},$ (2)

where the $i$-th bond Hamiltonian can be cast into the following symmetrized
form:

$\displaystyle{\cal
H}_{i}\\!=\\!J(1\\!-\\!S_{i}^{2})\sigma_{i}(1\\!-\\!S_{i+1}^{2})\sigma_{i+1}\\!+\\!\frac{\Delta}{2}(S_{i}^{2}\\!+\\!S_{i+1}^{2})\\!+\\!VS_{i}S_{i+1}.$
(3)

The partition function of the spin-pseudospin model can be factorized into the
following product:

$\displaystyle{\cal Z}\\!$ $\displaystyle=$
$\displaystyle\\!\sum_{\\{S\\}}\sum_{\\{\sigma\\}}\prod_{i=1}^{N}\frac{1}{2^{S_{i}^{2}}}{\rm
e}^{-\beta{\cal
H}_{i}}=\sum_{\\{S\\}}\sum_{\\{\sigma\\}}\prod_{i=1}^{N}\frac{1}{\sqrt{2^{S_{i}^{2}+S_{i+1}^{2}}}}{\rm
e}^{-\beta{\cal H}_{i}}$ (4) $\displaystyle=$
$\displaystyle\\!\sum_{\\{S\\}}\sum_{\\{\sigma\\}}\prod_{i=1}^{N}{\rm
T}(S_{i},\sigma_{i};S_{i+1},\sigma_{i+1}),$

where $\beta=1/(k_{\rm B}T)$, $k_{\rm B}$ is Boltzmann’s constant, $T$ is
absolute temperature and the summations $\sum_{\\{S\\}}$ and
$\sum_{\\{\sigma\\}}$ are carried out over all pseudospin and spin variables,
respectively. The factor $\frac{1}{2^{S_{i}^{2}}}$ and its symmetrized
counterpart $\frac{1}{\sqrt{2^{S_{i}^{2}\\!+\\!S_{i+1}^{2}}}}$ are introduced
as correction factors for two charged states $S_{i}=\pm 1$ to avoid their
double counting in the partition function when performing the summation over
the Ising spins $\sum_{\\{\sigma\\}}$. The Boltzmann factor ${\rm
T}(S_{i},\sigma_{i};S_{i+1},\sigma_{i+1})=\frac{1}{\sqrt{2^{S_{i}^{2}\\!+\\!S_{i+1}^{2}}}}{\rm
e}^{-\beta{\cal H}_{i}}$ depending on the pseudospin and spin variables from
two adjacent sites can be alternatively viewed as the transfer matrix:

$\displaystyle{\rm T}\\!\\!\\!\\!$
$\displaystyle\\!\\!\\!\\!(S_{i},\sigma_{i};S_{i+1},\sigma_{i+1})\\!=\\!\frac{1}{\sqrt{2^{S_{i}^{2}\\!+\\!S_{i+1}^{2}}}}\exp\left[-\frac{\beta\Delta}{2}(S_{i}^{2}+S_{i+1}^{2})\right]$
(5) $\displaystyle\times$ $\displaystyle\\!\exp\left[-\beta
VS_{i}S_{i+1}\\!-\\!\beta
J(1\\!-\\!S_{i}^{2})\sigma_{i}(1\\!-\\!S_{i+1}^{2})\sigma_{i+1}\right].$

The pseudospin and spin variables $S_{i}$ and $\sigma_{i}$ from $i$-th site
define rows of the transfer matrix, while the pseudospin and spin variables
$S_{i+1}$ and $\sigma_{i+1}$ from $(i+1)$-st site define columns of the
transfer matrix, which has the following explicit form:

$\displaystyle{\rm T}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccccc}{\rm T}(1,1;1,1)&{\rm
T}(1,1;1,-1)&{\rm T}(1,1;0,1)&{\rm T}(1,1;0,-1)&{\rm T}(1,1;-1,1)&{\rm
T}(1,1;-1,-1)\\\ {\rm T}(1,-1;1,1)&{\rm T}(1,-1;1,-1)&{\rm T}(1,-1;0,1)&{\rm
T}(1,-1;0,-1)&{\rm T}(1,-1;-1,1)&{\rm T}(1,-1;-1,-1)\\\ {\rm T}(0,1;1,1)&{\rm
T}(0,1;1,-1)&{\rm T}(0,1;0,1)&{\rm T}(0,1;0,-1)&{\rm T}(0,1;-1,1)&{\rm
T}(0,1;-1,-1)\\\ {\rm T}(0,-1;1,1)&{\rm T}(0,-1;1,-1)&{\rm T}(0,-1;0,1)&{\rm
T}(0,-1;0,-1)&{\rm T}(0,-1;-1,1)&{\rm T}(0,-1;-1,-1)\\\ {\rm T}(-1,1;1,1)&{\rm
T}(-1,1;1,-1)&{\rm T}(-1,1;0,1)&{\rm T}(-1,1;0,-1)&{\rm T}(-1,1;-1,1)&{\rm
T}(-1,1;-1,-1)\\\ {\rm T}(-1,-1;1,1)&{\rm T}(-1,-1;1,-1)&{\rm
T}(-1,-1;0,1)&{\rm T}(-1,-1;0,-1)&{\rm T}(-1,-1;-1,1)&{\rm T}(-1,-1;-1,-1)\\\
\end{array}\right)$ (12) $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccccc}\frac{1}{2}{\rm
e}^{-\beta(\Delta+V)}&\frac{1}{2}{\rm
e}^{-\beta(\Delta+V)}&\frac{1}{\sqrt{2}}{\rm
e}^{-\frac{\beta\Delta}{2}}&\frac{1}{\sqrt{2}}{\rm
e}^{-\frac{\beta\Delta}{2}}&\frac{1}{2}{\rm
e}^{-\beta(\Delta-V)}&\frac{1}{2}{\rm e}^{-\beta(\Delta-V)}\\\ \frac{1}{2}{\rm
e}^{-\beta(\Delta+V)}&\frac{1}{2}{\rm
e}^{-\beta(\Delta+V)}&\frac{1}{\sqrt{2}}{\rm
e}^{-\frac{\beta\Delta}{2}}&\frac{1}{\sqrt{2}}{\rm
e}^{-\frac{\beta\Delta}{2}}&\frac{1}{2}{\rm
e}^{-\beta(\Delta-V)}&\frac{1}{2}{\rm e}^{-\beta(\Delta-V)}\\\
\frac{1}{\sqrt{2}}{\rm e}^{-\frac{\beta\Delta}{2}}&\frac{1}{\sqrt{2}}{\rm
e}^{-\frac{\beta\Delta}{2}}&{\rm e}^{-\beta J}&{\rm e}^{\beta
J}&\frac{1}{\sqrt{2}}{\rm e}^{-\frac{\beta\Delta}{2}}&\frac{1}{\sqrt{2}}{\rm
e}^{-\frac{\beta\Delta}{2}}\\\ \frac{1}{\sqrt{2}}{\rm
e}^{-\frac{\beta\Delta}{2}}&\frac{1}{\sqrt{2}}{\rm
e}^{-\frac{\beta\Delta}{2}}&{\rm e}^{\beta J}&{\rm e}^{-\beta
J}&\frac{1}{\sqrt{2}}{\rm e}^{-\frac{\beta\Delta}{2}}&\frac{1}{\sqrt{2}}{\rm
e}^{-\frac{\beta\Delta}{2}}\\\ \frac{1}{2}{\rm
e}^{-\beta(\Delta-V)}&\frac{1}{2}{\rm
e}^{-\beta(\Delta-V)}&\frac{1}{\sqrt{2}}{\rm
e}^{-\frac{\beta\Delta}{2}}&\frac{1}{\sqrt{2}}{\rm
e}^{-\frac{\beta\Delta}{2}}&\frac{1}{2}{\rm
e}^{-\beta(\Delta+V)}&\frac{1}{2}{\rm e}^{-\beta(\Delta+V)}\\\ \frac{1}{2}{\rm
e}^{-\beta(\Delta-V)}&\frac{1}{2}{\rm
e}^{-\beta(\Delta-V)}&\frac{1}{\sqrt{2}}{\rm
e}^{-\frac{\beta\Delta}{2}}&\frac{1}{\sqrt{2}}{\rm
e}^{-\frac{\beta\Delta}{2}}&\frac{1}{2}{\rm
e}^{-\beta(\Delta+V)}&\frac{1}{2}{\rm e}^{-\beta(\Delta+V)}\\\
\end{array}\right).$ (19)

After straightforward analytical diagonalization of the transfer matrix (19),
one obtains all six eigenvalues:

$\displaystyle\lambda_{1,2}$ $\displaystyle=$ $\displaystyle 0,$
$\displaystyle\lambda_{3}$ $\displaystyle=$ $\displaystyle-2\sinh(\beta J),$
$\displaystyle\lambda_{4}$ $\displaystyle=$
$\displaystyle-2\exp(-\beta\Delta)\sinh(\beta V),$ $\displaystyle\lambda_{5}$
$\displaystyle=$ $\displaystyle\cosh(\beta
J)\\!+\\!\exp(-\beta\Delta)\cosh(\beta V)\\!-\\!\\!\sqrt{D},$
$\displaystyle\lambda_{6}$ $\displaystyle=$ $\displaystyle\cosh(\beta
J)\\!+\\!\exp(-\beta\Delta)\cosh(\beta V)\\!+\\!\\!\sqrt{D},$ (20)

where $D=\left[\cosh(\beta J)-{\rm e}^{-\beta\Delta}\cosh(\beta
V)\right]^{2}+4{\rm e}^{-\beta\Delta}$. In the spirit of the transfer-matrix
method, the subsequent summation over the pseudospin and spin variables in Eq.
(4) is equivalent to a matrix multiplication. Utilizing a trace invariance,
the partition function of the spin-pseudospin model can be consequently
calculated from the transfer-matrix eigenvalues (20):

$\displaystyle{\cal Z}=\sum_{\sigma_{1}=\pm 1}\sum_{S_{1}=\pm
1,0}\\!\\!\\!\\!{\rm T}(S_{1},\sigma_{1};S_{1},\sigma_{1})=\mbox{Tr}\,{\rm
T}^{N}=\sum_{j=1}^{6}\lambda_{j}^{N}\\!.$ (21)

It can be readily proved that the largest transfer-matrix eigenvalue is
$\lambda_{max}=\lambda_{6}$, and hence, the free energy of the spin-pseudospin
model in the thermodynamic limit $N\to\infty$ can be expressed solely in terms
of the largest transfer-matrix eigenvalue:

$\displaystyle F=-k_{\rm B}T\lim_{N\to\infty}\ln{\cal Z}=-Nk_{\rm
B}T\ln\lambda_{6}.$ (22)

The exact result (22) derived for the free energy allows straightforward
derivation of local observables such as the mean value of the pseudospin
$\langle S_{i}\rangle$, the mean value of the square of the pseudospin
$\langle S_{i}^{2}\rangle$, the pair correlation function between nearest-
neighbor pseudospins $\langle S_{i}S_{i+1}\rangle$, and the pair correlation
function between nearest-neighbor spins
$\langle\sigma_{i}\sigma_{i+1}\rangle$. In addition, standard thermodynamic
relations can be utilized to derive exact results for other quantities such as
the magnetic entropy and specific heat from the free energy (22).

##  3 Results and discussion

Let us proceed to a discussion of the most intriguing findings obtained for
the minimal pseudospin-spin model of the one-dimensional cuprate chain [CuO]∞.
It turns out that the spin-pseudospin model exhibits only two ground states
exemplified by configurations illustrated in Fig. 2. The first ground state
comprises solely the electroneutral [CuO]0 centers indicating a regular
alternation of Cu2+ and O2- ions within the entire one-dimensional cuprate
chain [CuO]∞. Moreover, the spins of nearest-neighbor Cu2+ centers display
antiferromagnetic alignment characterizing this ground state as the
antiferromagnetic (AF) phase. The AF phase evidently manifests spontaneously
broken symmetry and consists of two-fold degenerate eigenvectors with the
energy $E_{\rm AF}=-NJ$:

$\displaystyle|{\rm
AF}\rangle=\left\\{\begin{array}[]{l}\displaystyle\prod_{i=1}^{N/2}|S=0,\sigma=+1\rangle_{2i-1}|S=0,\sigma=-1\rangle_{2i}\\\
\displaystyle\prod_{i=1}^{N/2}|S=0,\sigma=-1\rangle_{2i-1}|S=0,\sigma=+1\rangle_{2i}\end{array}\right..$
(25)

In the second ground state, one contrarily finds a regular alternation of
negatively charged electron centers [CuO]- with positively charged hole
centers [CuO]+, which means that nonmagnetic Cu1+ and Cu3+ centers (separated
by O2- anions) regularly alternate within the one-dimensional cuprate chain
[CuO]∞. Consequently, the second ground state can be regarded as the charged
ordered (CO) phase, which also exhibits spontaneously broken symmetry and
consists of two-fold degenerate eigenvectors with the energy $E_{\rm
CO}=-NV+N\Delta$:

$\displaystyle|{\rm
CO}\rangle=\left\\{\begin{array}[]{l}\displaystyle\prod_{i=1}^{N/2}|S=+1\rangle_{2i-1}|S=-1\rangle_{2i}\\\
\displaystyle\prod_{i=1}^{N/2}|S=-1\rangle_{2i-1}|S=+1\rangle_{2i}\end{array}\right..$
(28)

It is noteworthy that the AF and CO phases coexist whenever the attractive
inter-site interaction $V$ between the negatively charged electron centers
[CuO]- (nominally Cu1+ centers) and the positively charged hole centers [CuO]+
(nominally Cu3+ centers) is balanced by the antiferromagnetic superexchange
interaction $J$ between the spin-1/2 Cu2+ ions inherent to the electroneutral
[CuO]0 centers and the on-site energy cost associated with the creation of the
charged centers. The corresponding line of first-order phase transitions
$V=\Delta+J$ between the AF and CO phases is depicted in the ground-state
phase diagram in Fig. 3. The most intriguing behavior of the pseudospin-spin
model could be thus anticipated if the interaction parameters drive the one-
dimensional cuprate chain [CuO]∞ close to a first-order phase transition
between the AF and CO phases.

Figure 2: A schematic illustration of two possible ground states of the spin-
pseudospin model of the one-dimensional cuprate chain [CuO]∞: (a) the
antiferromagnetic (AF) phase; (b) the charged ordered (CO) phase. The notation
for the three nominal valence states of Cu1+, Cu2+, and Cu3+ centers is the
same as in Fig. 1.

Figure 3: The ground-state phase diagram of the spin-pseudospin model of the
one-dimensional cuprate chain [CuO]∞ in the $\Delta/J-V/J$ plane. The
antiferromagnetic (AF) and the charge ordered (CO) phases are separated by the
line of first-order phase transitions $V/J=1+\Delta/J$.

The temperature variations of two local observables ($\langle S_{i}\rangle$,
$\langle S_{i}^{2}\rangle$) and two pair correlation functions ($\langle
S_{i}S_{i+1}\rangle$, $\langle\sigma_{i}\sigma_{i+1}\rangle$) are depicted in
Fig. 4 for the spin-pseudospin model of the one-dimensional cuprate chain
[CuO]∞ with specific values of the interaction parameters $V/J=11.1$ and
$\Delta/J=10$. Notably, the mean value of the pseudospin remains zero $\langle
S_{i}\rangle=0$ irrespective of temperature, which convincingly evidences an
equal proliferation of both charged centers [CuO]- and [CuO]+ (nominally Cu1+
and Cu3+ centers), and hence, satisfying the electroneutrality condition.
Besides, the mean square value of pseudospin tends towards its maximum value
$\langle S_{i}^{2}\rangle=1$ in the zero-temperature limit in accordance with
the CO ground state. This mean value remains nearly constant up to the
temperature $k_{\rm B}T/J\approx 1$, at which it abruptly decreases almost to
zero indicating the absence of charged centers. The abrupt breakdown of the
mean square value of pseudospin $\langle S_{i}^{2}\rangle$ from unity to zero
signifies the pseudo-transition from the CO phase towards the AF phase. A
rather steep increase of the pair correlation function between nearest-
neighbor pseudospins from the perfectly anticorrelated value $\langle
S_{i}S_{i+1}\rangle=-1$ almost towards zero provides further evidence of the
pseudo-transition from the CO phase towards the AF phase. The absence of a
pair correlation function between nearest-neighbor spins
$\langle\sigma_{i}\sigma_{i+1}\rangle=0$ at absolute zero temperature also
confirms the CO ground state. However, a steep decline towards the value
$\langle\sigma_{i}\sigma_{i+1}\rangle\approx-0.75$ observed around the
temperature $k_{\rm B}T/J\approx 1$ indicates the pseudo-transition towards
the AF phase, where a perfectly anticorrelated value
$\langle\sigma_{i}\sigma_{i+1}\rangle=-1$ would be expected at absolute zero
temperature.

Figure 4: Temperature variations of the mean value of pseudospin $\langle
S_{i}\rangle$, the mean square value of pseudospin $\langle S_{i}^{2}\rangle$,
the pair correlation function between nearest-neighbor pseudospins $\langle
S_{i}S_{i+1}\rangle$, and the pair correlation function between nearest-
neighbor spins $\langle\sigma_{i}\sigma_{i+1}\rangle$ of the spin-pseudospin
model of the one-dimensional cuprate chain [CuO]∞ with fixed values of the
interaction parameters $V/J=11.1$ and $\Delta/J=10$.

Next, let us bring insight into how the pseudo-transition between the CO and
AF phases manifests itself in the temperature dependencies of the magnetic
entropy and specific heat. To this end, the temperature dependencies of the
entropy of the spin-pseudospin model of the one-dimensional cuprate chain
[CuO]∞ are depicted in Fig. 5 for a fixed value of the on-site interaction
$\Delta/J=10$ and a few selected values of the inter-site interaction $V/J$.
Generally, the entropy maintains zero value at sufficiently low temperatures,
then it exhibits a sudden rise within a relatively narrow range of moderate
temperatures before it finally shows a more gradual increase at higher
temperatures. It is evident from Fig. 5 that the sudden rise in entropy
becomes sharper as the inter-site interaction approaches the ground-state
phase boundary between the CO and AF phases. Although the steep increase in
entropy entropy resembles a discontinuous thermal phase transition, the
observed increase in entropy still remains continuous.

To complete the overall picture, temperature variations of the specific heat
of the spin-pseudospin model of the one-dimensional cuprate chain [CuO]∞ are
displayed in Fig. 6 for the same fixed value of the on-site interaction
$\Delta/J=10$ and a few different values of the inter-site interaction $V/J$
selected slightly above the ground-state phase boundary between the CO and AF
phases ($V/J=11$ for $\Delta/J=10$). It is obvious from Fig. 6 that the
specific heat exhibits an extremely high peak at relatively low temperatures,
which is thus quite reminiscent of a continuous thermal phase transition.
However, it should be pointed out that the relevant peak of specific heat,
although it may show increase over several orders of magnitude, always remains
finite (see the insert of Fig. 6). If the inter-site interaction is chosen
further away from the ground-state phase boundary between the CO and AF
phases, the specific-heat peak shifts towards higher temperatures, reduces its
height and becomes broader.

Figure 5: Temperature dependencies of the entropy of the spin-pseudospin model
of the one-dimensional cuprate chain [CuO]∞ for the fixed value of the on-site
interaction $\Delta/J=10$ and a few different values of the inter-site
interaction $V/J$.

Figure 6: Temperature dependencies of the specific heat of the spin-pseudospin
model of one-dimensional cuprate chain [CuO]∞ for the fixed value the on-site
interaction $\Delta/J=10$ and four selected values of the inter-site
interaction $V/J=11.05$, $11.1$, $11.2$, and $11.4$ specified by the
respective labels. The inset shows in a semi-logarithmic scale detail from a
temperature range, where a sizable but still finite maximum of the specific
heat is observed.

Furthermore, it might be also interesting to explore the asymptotic behavior
of the specific heat near the pseudo-critical temperature. To this end, we
plotted in Fig. 7 in a log-log scale temperature dependence of the specific
heat for the spin-pseudospin model of the one-dimensional cuprate chain [CuO]∞
with fixed values of the on-site interaction $\Delta/J=10$ and the inter-site
interaction $V/J=11.05$. Fig. 7 illustrates that the specific heat closely
follows a power-law dependence with the respective quasi-critical exponent
$\alpha^{\prime}=3$ slightly below the pseudo-critical temperature even though
it eventually bends and converges to a finite value as the pseudo-critical
temperature is approached. On the other hand, the relevant power-law
dependence of the specific heat cannot be clearly discerned in a temperature
range slightly above the pseudo-critical temperature where the specific heat
rapidly drops to an almost constant nonzero value (see the insert in Fig. 6).
Nonetheless, it can be inferred that the spin-pseudospin model of the one-
dimensional cuprate chain [CuO]∞ falls within the universality class of one-
dimensional models exhibiting a pseudo-transition at finite temperatures roj19
.

Figure 7: Temperature dependence of the specific heat of the spin-pseudospin
model of the one-dimensional cuprate chain [CuO]∞ close to the
pseudotransition in a log-log scale for fixed values of the parameters
$\Delta/J=10$ and $V/J=11.05$. A blue (red) line displays the temperature
dependence of the specific heat slightly below (above) of the pseudotransition
temperature. A linear fit of the specific heat $\log(C/Nk_{\rm
B})=-5.92-3\log(k_{\rm B}|T-T_{p}|/J)$ shown by a black dotted line verifies a
power-law dependence below the pseudotransition temperature.

One can exploit the abrupt change in entropy and the extremely sharp peak of
the specific heat, which accompany the pseudo-transition between the CO and AF
phases, to construct the pseudo-phase diagrams of the spin-pseudospin model of
the one-dimensional cuprate chain [CuO]∞. The density plot of the entropy
shown in Fig. 8 in the inter-site interaction versus temperature ($V/J-k_{\rm
B}T/J$) plane directly implies that the AF phase gains much higher entropy
upon increasing temperature compared to the CO phase. This finding stems from
the fact that all excited charged configurations emerging above the CO ground
state still have to satisfy the electroneutrality constraint with an equal
number of positively and negatively charged centers [CuO]- and [CuO]+
(nominally Cu1+ and Cu3+ centers), whereas all spin configurations emerging
above the AF ground state naturally satisfy the electroneutrality constraint
as they are composed of electroneutral centers [CuO]0 (nominally Cu2+). Owing
to this fact, the line of pseudo-phase transitions between the AF and CO
phases bends towards the CO phase due to entropic favoring of the AF phase.

The density plot of the specific heat presented in Fig. 9 in the inter-site
interaction versus temperature ($V/J-k_{\rm B}T/J$) provides another clear
pseudo-phase diagram of the spin-pseudospin model of the one-dimensional
cuprate chain [CuO]∞. It is evident from Fig. 9 that the pseudo-phase boundary
between the CO and AF phases is almost perfectly vertical up to moderate
temperatures $k_{\rm B}T/J\approx 0.4$ before it begins to bend towards the CO
phase due to the entropically-driven favoring of the AF phase. It should also
be stressed that the pseudo-phase transition between the CO and AF phases
gradually melts upon increasing temperature. It actually turns out that the
vigorous peak of specific heat observed in a very narrow temperature range
persists up to moderate temperatures $k_{\rm B}T/J\approx 1.0$, while at
higher temperatures, the relevant peak of specific heat is gradually
suppressed and spreads over a wider temperature range.

Figure 8: A density plot of the entropy of the spin-pseudospin model of the
one-dimensional cuprate chain [CuO]∞ in the $V/J-k_{\rm B}T/J$ plane for a
fixed value of the parameter $\Delta/J=10$. The specific value $V/J=11$
corresponds to the ground-state phase boundary between the AF and CO phases.

Figure 9: A density plot of the specific heat of the spin-pseudospin model of
the one-dimensional cuprate chain [CuO]∞ in the $V/J-k_{\rm B}T/J$ plane for a
fixed value of the parameter $\Delta/J=10$. The specific value $V/J=11$
corresponds to the ground-state phase boundary between the AF and CO phases.

##  4 Conclusion

In the present article, we have addressed the minimal spin-pseudospin model of
the one-dimensional cuprate chain [CuO]∞, which takes into consideration the
negatively charged electron center [CuO]-, the electroneutral center [CuO]0,
and positively charged hole center [CuO]+ corresponding to three nominal
valence states of Cu1+, Cu2+, and Cu3+ ions, respectively. The transfer-matrix
solution of the model under investigation afforded exact results for the free
energy, entropy, specific heat, as well as, local order parameters and pair
correlation functions. It has been demonstrated that the investigated spin-
pseudospin model has two available ground states, which can be characterized
as the AF and CO phases. Although the AF and CO phases do not exhibit true
spontaneous long-range order, the low-temperature behavior of local
observables and pair correlation functions clearly verifies characteristic
short-range arrangement of the AF and CO phases.

The most intriguing discovery highlighted in this study concerns with the
manifestation of a pseudo-transition between the CO and AF phases. This
phenomenon emerges when the spin-pseudospin model of the one-dimensional
cuprate chain [CuO]∞ approaches the CO ground state near the ground-state
phase boundary with the AF phase. Under these conditions, the entropy exhibits
a sudden thermally-induced increase reminiscent of a discontinuous phase
transition, while the specific heat displays a prominent sharp peak
reminiscent of a continuous phase transition. Despite the absence of a true
jump in entropy jump and divergence in specific heat, all aforedescribed
features serve as clear signatures of the pseudo-transition. For future
investigations, it would be valuable to relax the electroneutrality condition
to explore the effects of electron or hole doping on the pseudo-transition
phenomenon. In addition, it would be intriguing to investigate whether similar
pseudo-transition phenomena can be found in more complex low-dimensional
cuprates such ladders or plane square lattices.

###### Acknowledgements.

This work was financially supported by the grant of the Slovak Research and
Development Agency provided under the contract Nos. APVV-16-0186 and
APVV-20-0150 and by the grant of The Ministry of Education, Science, Research,
and Sport of the Slovak Republic provided under the contract No. VEGA
1/0695/23. This project has received funding from the European Union’s Horizon
2020 research and innovation programme under the Marie Skłodowska-Curie grant
agreement No. 945380.

## Author contributions

Both authors contributed equally to the paper.

## Conflict of interest

The authors have no conflicts of interest to disclose.

## Data availability statements

This manuscript has no associated data or the data will not be deposited
[Authors’ comment: This is a theoretical study with no experimental data. All
plots directly follow from exact analytical formulas included in the
manuscript.].

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