# Assessing the influence of attractor-verb distance on grammatical agreement
in humans and language models

Christos-Nikolaos Zacharopoulos
Cognitive Neuroimaging Unit,
NeuroSpin center, France
Sensome SAS,
Massy,
France
<EMAIL_ADDRESS>Théo Desbordes
Meta AI Research;
Cognitive Neuroimaging Unit,
NeuroSpin center, France;
<EMAIL_ADDRESS>
Mathias Sablé-Meyer
Cognitive Neuroimaging Unit,
NeuroSpin center, France;
Collège de France,
Université PSL
<EMAIL_ADDRESS>
###### Abstract

Subject-verb agreement in the presence of an attractor noun located between
the main noun and the verb elicits complex behavior: judgments of
grammaticality are modulated by the grammatical features of the attractor. For
example, in the sentence “The girl near the boys likes climbing”, the
attractor (_boys_) disagrees in grammatical number with the verb (_likes_),
creating a locally implausible transition probability. Here, we parametrically
modulate the distance between the attractor and the verb while keeping the
length of the sentence equal. We evaluate the performance of both humans and
two artificial neural network models: both make more mistakes when the
attractor is closer to the verb, but neural networks get close to the chance
level while humans are mostly able to overcome the attractor interference.
Additionally, we report a linear effect of attractor distance on reaction
times. We hypothesize that a possible reason for the proximity effect is the
calculation of transition probabilities between adjacent words. Nevertheless,
classical models of attraction such as the cue-based model might suffice to
explain this phenomenon, thus paving the way for new research. Data and
analyses available at https://osf.io/d4g6k

##  1 Introduction

On the surface, language appears to be produced and understood linearly, as
humans read or hear words one after the other. Yet, formal linguistic theories
postulate the existence of an underlying structure that governs language
processing Chomsky (1957); Rizzi (2004); Dehaene et al. (2015); Vigliocco and
Nicol (1998); Hauser et al. (2002). This hypothesis is supported by both
behavioral Fossum and Levy (2012); Shi et al. (2020); Coopmans et al. (2021)
and neural Brennan et al. ; Nelson et al. (2017); Pallier et al. (2011) data.
According to a competing view, unstructured probabilistic models capture
behavior without explicitly relying on structures Frank and Bod (2011). The
discrepancy between the two views has led to a decade-long debate on linear
versus structural effects in language Haskell and MacDonald (2005); Ding et
al. (2015); Willer Gold et al. (2017); Arana et al. (2021).

Figure 1: Experimental design Our design aims to separate two possible
mechanisms involved in language processing, by altering the distance of the
embedded noun (attractor) from the verb. We compare the impact of noun-verb
agreement (_Violation_) and noun-noun dependency (_Congruency_). The
_Transition_ effect symbolizes the consequence of the transition probability
between the second noun and the verb. As a baseline, we also include a
condition with a similar number of words, but without a noun in between. The
summary of the experimental conditions is shown in Table 1

Analyzing subject-verb agreement errors in the presence of distracting
elements is a standard way of separating linear and structural accounts of
language Molinaro et al. (2011). Attraction effects in number agreement have
been studied extensively in humans Bock and Miller (1991); Franck et al.
(2002); Hammerly et al. (2019) and Neural Language Models (NLMs) Linzen et al.
(2016); Jumelet et al. (2019); Lakretz et al. (2021).

Condition Sentence Violation Congruency Proximal Attractor The writer who
knows the happy young journalist climbs. ✗ ✓ The writer who knows the happy
young journalists climbs. ✗ ✗ The writer who knows the happy young journalist
climb. ✓ ✓ The writer who knows the happy young journalists climb. ✓ ✗ Distal
Attractor The writer who knows the journalist the most climbs. ✗ ✓ The writer
who knows the journalists the most climbs. ✗ ✗ The writer who knows the
journalist the most climb. ✓ ✓ The writer who knows the journalists the most
climb. ✓ ✗ Baseline The writer who walks fast but rather clumsily climbs. ✗ -
The writer who walks fast but rather clumsily climb. ✓ - Filler (Number) The
writer who knows the journalist climbs. ✗ - The writer who know the journalist
climbs. ✓ - Filler (POS) The writer whom the journalist knows climbs. ✗ - The
writer whom knows the journalist climbs. ✓ -

Table 1: Conditions. The design contrasts the attribution of two distinct
effects to the overall error rate in humans and NLMs. To prevent the subjects
from developing strategies for the effective resolution of the violation-
detection task, we included two different kinds of filler trials that
contained violations at the beginning of the sentence.

In the present work, we introduce a parametric manipulation of the distance
between the distracting nouns and the verb (Figure 1). We include a baseline
condition where the subject-verb distance is matched but no word that carries
number marking is introduced in-between. We thereby provide a minimal triplet
of experimental conditions (Table 1) that can disentangle structural from
linear mechanisms by contrasting operations directly ascribed to each
mechanism. We posit that there are structural operations at play to explain
the fact that participants are always able to detect violations, and at the
same time there are linear operations at play which lead to the attraction
effect of the intervening noun. We additionally hypothesize that the linear
effect might be modulated by transition probabilities, i.e., the prior
probability that a given word follows another Dehaene et al. (2015); Friston
et al. (2021). Finally, despite reaching on par with or even supra-human
performance on many tasks Brown et al. (2020); Minaee et al. (2021), NLMs are
known to be sensitive to superficial statistical properties of their training
data McCoy et al. (2021). Thus, we compare two NLMs with the behavior of human
participants.

##  2 Experimental Evidence

###  2.1 Method

We tested the grammatical judgments of the participants in a forced-choice,
online violation detection task where the words were presented one at a time
on the screen (RSVP), and the participants had to press a button to indicate
whether a given sentence was grammatically correct or not.

#### Participants

Fifty-four native speakers of English took part in our experiment, which was
advertised on social media and mailing lists. The procedure and the consent
were approved by the local ethical committee (Université Paris-Saclay, CER-
Paris-Saclay-2019-063). We used filler trials (Table 1) to avoid potential
confounding strategies from the participants, such as actively ignoring the
middle of the sentences Pearlmutter et al. (1999). Any participant whose
answer to fillers was not significantly different from chance (binomial test,
null hypothesis $p_{0}=.5$) was rejected. We also rejected participants whose
success rate on the main task was below $70\%$ Overall, we rejected 20
participants, and reported analyses from 34 participants (all analyses
reported are consistent with corresponding analyses performed with the full
dataset, reported in appendix B. For example, one can compare Table 2 and
Table 5, and Figure 2 and Figure 5).

We also tested two transformer models Wolf et al. (2020): a replication of the
GPT-3 language model Brown et al. (2020) made available by
EleutherAI111https://huggingface.co/EleutherAI/gpt-neo-1.3B Black et al.
(2021) and a Text-To-Text Transformer
(T5)222https://huggingface.co/vennify/t5-base-grammar-correction Raffel et al.
(2019) fine-tuned on a grammatical error correction benchmark Napoles et al.
(2017). To evaluate the GPT-3 model, we input it with the sentence up to
(excluding) the target verb and compare the probabilities associated with the
grammatical and ungrammatical tokens (e.g., “climb” vs “climbs” for sentences
in Table 1). Thus, for this model, we get a comparative performance per
condition but cannot evaluate performances between grammatical and
ungrammatical sentences. On the other hand, we compare humans directly with
the grammaticality judgment of T5.

#### Experimental Procedure

Participants were given a description of a violation-detection task (see
Appendix A for the exact wording) including which button to press and three
examples of grammatical and ungrammatical sentences. We presented sentences to
participants, with a fixation cross between words, in white on black
background, using a presentation time of 200ms and an SOA of 366ms.
Participants could answer at any point by pressing the keyboard to indicate
their judgment of grammaticality (left and right arrow keys randomized across
subjects). Participants received auditory and visual feedback with each trial:
green/red fixation and upward/downward tune (correct/incorect). At the end of
the experiment, participants answered questions about the experiment
(difficulty, weirdness, strategies), we provided them with an overall score
and invited them to share the experiment on social media.

#### Stimuli

We generated sentences from a fixed lexicon balanced for low-level features,
yielding many sentences. Then we filtered them based on the perplexity of
GPT-3, keeping only sentences that had an overall perplexity between the
median and median $+2std$ — in order to keep sentences of low and consistent
perplexity. We sampled $5$ sentences for each condition (baseline, distal
congruent, distal incongruent, proximal congruent, proximal incongruent),
subject number (singular, plural), and grammaticality, as well as 32 filler
sentences. No two sentences shared 5 words or more. The same sentences were
presented to participants in randomized order.

###  2.2 Results

We call _incongruent_ the trials in which the numbers of $N_{1}$ and $N_{2}$
disagree (Figure 1) — which is independent from _grammaticality_. Figure 2
shows the main effect of the presence of an attractor, and how its distance to
the target (distal or proximal) modulates the performances of humans and NLMs.

In all cases, the baseline elicits fewer errors (ER) and faster reaction times
(RT) compared to the attractor conditions. Errors occur more often in
ungrammatical sentences than in grammatical ones, irrespective of the
condition: this phenomenon is called grammatical illusions Phillips et al.
(2011) and indicates that participants accept ungrammatical sentences more
often than they reject grammatical ones. We also replicate grammatical
asymmetry Wagers et al. (2009): incongruent trials lead to higher ER in the
ungrammatical sentences.

To investigate the attractor effects, we analyzed the main factors of our
design. The main effect of Violation refers to the dependency that controls
the grammatical configuration of the sentence (Figure 1) and we use this
effect as a proxy into structural processing Rizzi (2004). The Congruency
effect refers to a dependency realized between two non-structurally related
words and is used as a proxy for linear processing.

Figure 3 shows the main factors and their interaction. We report the
corresponding ANOVA for the human participants in Table 2. We replicate the
markedness phenomenon Bock and Miller (1991); Wagers et al. (2009): attraction
effects surface only with plural attractors (see Table 4 and Figure 4 for
analyses with singular attractors). Results shown in Figure 3 and Table 4
correspond to sentences with plural attractors.

Figure 2: Performances of humans and NLMs: colors indicate grammaticality, error bars indicate (all figures) SEM over participants (humans) or sentences (NLMs). Effect | $F_{1,33}$ |  $p$-value | $\eta_{G}^{2}$
---|---|---|---
Response Time; Distal attractor
congruency | $7.04$ | $.012$ | $.011$
violation | $11.64$ | $.002$ | $.031$
interaction | $0.01$ | $.915$ | $<.001$
Response Time; Proximal attractor
congruency | $26.90$ | $<.001$ | $.046$
violation | $11.81$ | $.002$ | $.027$
interaction | $0.10$ | $.752$ | $<.001$
Error Rate; Distal attractor
congruency | $2.29$ | $.140$ | $.013$
violation | $18.01$ | $<.001$ | $.135$
interaction | $0.04$ | $.846$ | $<.001$
Error Rate; Proximal attractor
congruency | $15.79$ | $<.001$ | $.085$
violation | $5.33$ | $.027$ | $.040$
interaction | $0.57$ | $.454$ | $.003$
Table 2: One-way between-subjects ANOVAs were conducted to compare the effect
of congruency, violation, and their interaction on RT and ER for both proximal
and distal attractors. Highlighted rows indicate significance at the $p<.05$
level. The last column provides $\eta_{G}^{2}$ values, an estimator of the
variance explained by the ANOVA, similar to $r^{2}$ for linear models. Figure
3: Effect of grammaticality, congruency, and distance of the attractor on our
dependent variables.

In the human data, the main effect of Violation is significant across all
conditions and dependent variables. Nevertheless, for the ER, the
$\eta_{G}^{2}$ value is larger by an order of magnitude in the distal
attractor condition: this illustrates that participants make more mistakes in
judging the grammaticality of sentences in which the attractor is far from the
verb, especially when the sentence is ungrammatical. The distance of the
attractor does not affect the magnitude of the effect in the case of RTs.

To elucidate this facilitatory effect, we focus on Congruency: its effect is
only significant in the proximal condition, which illustrates that
participants make more errors in incongruent sentences compared to the
congruent ones (figure 3).

NLMs displayed super-human performances in evaluating grammatical sentences.
T5 is sensitive to the mere presence of an attractor, whose distance strongly
modulates performance. GPT-3 is not sensitive to the mere presence of an
attractor, but elicits a significant distance effect. Both models, but not
humans, have near-chance performances in the case of a proximal attractor.
These results might have a two-fold interpretation. The presence of
grammatical illusions in both humans and models might be informative on the
role of training in linguistic performance: in real sentences, grammatical
sentences vastly outnumber ungrammatical ones. We might therefore be
describing a similar training bias between humans and NLMs. There is a common
response profile between the NLMs, which demonstrates the sensitivity of
transformer models to statistical regularities. This allows us to zoom in on
T5, which is more directly comparable to humans, and investigate how the
Congruency factor modulates the ER.

The reduction in the ER in the human data was traced to a facilitatory effect
of congruency: congruent trials led to fewer errors in the proximal compared
to the distal attractor. On the contrary, in T5, the incongruent trials
yielded chance-level performances, but there was no distance effect in the
congruent trials. This indicates that congruent sentences help humans in
performing the task, but not NLMs. Additionally, incongruent sentences have a
detrimental effect in the models, but no such effect is observed in humans.
These observations point to a common sensitivity to the attractor-verb
distance in both systems, but a fundamental difference as to the outcome of
this sensitivity.

##  3 Discussion

In this study, we revisited the classical attraction effect in subject-verb
number agreement in humans and neural networks and sought to assess the
influence of the attractor-verb distance.

Our results draw a picture of a shared distance sensitivity between humans and
NLMs, but also a fundamental difference in the weight of this sensitivity. We
observed that the artificial system operates on the basis of word-level
statistics, and is thus driven to chance-level performance in the presence of
a deviant bigram (Figure 3-bottom right-Incongruent trials). On the contrary,
the incongruency of the sentence leads to comparable error rates irrespective
of the attractor distance in humans (Figure 3-central column). The significant
effect of congruency observed in the human data is due to a facilitatory
effect of congruency in judging grammaticality, and not an inhibitory effect
of incongruency, unlike with the neural networks. This effect is mostly
evident in the agrammatical sentences and can be decomposed as follows.

Consider the following two sentences:

  * $(a)$

[Congruent]
The writers who like the happy young
editors cries.

  * $(b)$

[Incongruent]
The writers who like the happy young
editor cries.

Participants and $T5$ made fewer errors in $(a)$ compared to $(b)$.
Participants also made fewer errors in $(a)$ when the attractor was further
away. We thus observe a facilitatory effect. Notice that here we have the
realization of the locally implausible bigram “editors cries”. One possible
interpretation might be that the participants were lured into judging this
sentence as invalid, based on the presence of this bigram. A facilitatory
effect is hence observed because the sentence was indeed agrammatical.

The error rate for the incongruent cases like $(b)$ remained unchanged for the
humans, with respect to the distal attractor (see Figure 3-central column).
Importantly, though, in this case, where a plausible bigram is found (“editor
cries”), the NLM reached chance-level performance. Given the local agreement
of the attractor with the verb, the NLM was lured into judging the sentence as
grammatical, when the sentence was wrong, whereas humans were able to mostly
overcome the attraction effect.

Thus, our results suggest that operations at the n-gram level might play a key
role in explaining the observed phenomena, in both humans and NLMs.

Nevertheless, alternative explanations can be described. In many behavioral
studies, significant effects of congruency have been reported in non-
intervening attractor structures such as Object Relative Clauses (e.g.: The
player that the [teacher encourage/s] climbs) Wagers et al. (2009). The n-gram
mechanism cannot explain attraction phenomena in this setup, something that
the dominant model of attraction (cue-retrieval model, Wagers et al. (2009))
can do. In this model, a memory mechanism is enabled upon a cue, that
retrieves the number of the subject from the memory system and the errors can
be attributed to retrieval interference. Under this interpretation, the memory
representation attractors fade with distance and therefore the distal
attractor does not compete for retrieval. In contrast, when the attractor is
in the vicinity of the verb, similarity-based retrieval interference can
occur, and thus, attraction effects can only be realized in that condition
McElree et al. (2003).

However, it is important to note that none of the dominant models of
grammatical agreement (cue-based retrieval model & feature percolation) are
complete, as both are conclusive in ungrammatical conditions, but not always
for the grammatical ones. Thus, a need of model revision seems necessary. In
this study, we tried to point to the calculation of transition probabilities
as a candidate factor for model development.

###  3.1 Limitations

There are a number of limitations that narrows the scope of our results.
First, it is difficult to draw general conclusions from a single experiment on
grammatical number agreement. Second, although all conditions are balanced,
the stimuli we used are not devoid of semantic content which might induce some
biases. These results will need to be confirmed in future experiments using
different tasks and stimuli, for example using morphosyntactically marked
stimuli but devoid of semantics, so-called “jabberwocky” sentences (Hahne and
Jescheniak, 2001; Desbordes et al., 2023). This subject represents a
significant area for our ongoing research and exploration.

Third, we compared human participants to language models under the assumption
that NLMs express sensitivity to probabilistic relationships at the word
level, and thus a comparison under the same conditions might shed light on the
processing of language in the human brain. We are fully aware that this
comparison is indirect, and that an LSTM architecture might have been more
appropriate for such a comparison. Nevertheless, the literature has pointed to
differences between grammatical and ungrammatical conditions, effects that we
replicated in this study, and we therefore sought for a model that would allow
us for a direct comparison for each condition.

###  3.2 Conclusion

Our results show that, in humans and NLMs, language processing is affected by
the attractor-verb distance. We additionally hypothesize that this is due to
the calculation of transition probabilities at the word level, which can
either run contrary or reinforce the overall structure-based processing.

Humans are less affected by this local interference, suggesting that language
models process language in ways that are still fundamentally different from
humans, even though they superficially coincide, e.g. in grammatical cases in
our data.

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##  Appendix A Questionnaire, Instructions, and Stimuli

## Instructions

The exact instructions given to the participants are provided below. They
consisted of three separate pages, participants could go back and forth
between pages freely.

Remember that the key/response binding was randomized across subjects, so page
3 provided below only applied to half of our participants, where the other
half had a corresponding, flipped association of key and responses.

Page 1 • This experiment is about sentence processing • You will read
sentences on the screen, with the words presented one after the other, at the
center of the screen • Some of these sentences will contain mistakes • Your
task is to find these mistakes

Page 2 Here are a few examples to show what we mean by correct and incorrect.
Remember that the sentence will not be presented as a whole, but rather one
word after another. Incorrect examples: • The boy drink water while listening
to music • The farmer near the two pilot detests boxing • The athletes that
dislike the happy proud banker sings Correct examples: • The boy drinks water
while listening to music • The farmer near the two pilots detests boxing • The
athletes that dislike the happy proud banker sing Some sentences might be a
bit weird, like in example 3, but you should always be able to perform the
task if you remain focused.

Page 3 You have to look at the cross at the center of the screen, which is
always present when there is no word to read. Make sure the luminosity of your
screen is high enough for you to read. Then you will read sentences one word
after the other and you have to do the following: • As soon as you think a
given sentence is INCORRECT, please press the -> right arrow key on your
keyboard • When the sentence ends, if you think it is CORRECT, please press
the <\- left arrow key on your keyboard • You have to answer every time, even
when you’re not sure or you feel you don’t know. Only after you answer, the
following sentence will start. Answer the best you can! After each answer you
will receive feedback: the central cross will turn green if you answered
correctly, and red otherwise. If you can, please turn your computer audio on:
that way, you will receive feedback with sounds for each trial. This is the
last instruction page. You can go back to the other pages, but when you move
forward the experiment ask you to go fullscreen. Then the experiment will
start with 5 training examples so that you understand the task.

## Material

All stimuli are provided below, first the grammatical ones (1-50), then the
ungrammatical ones (51-100):

  1. 1.

The lawyers who avoid the kind gentle judge lie.

  2. 2.

The athlete who hates the farmers the least sings.

  3. 3.

The judges that fear the proud charming man sing.

  4. 4.

The builder who dislikes the proud gentle farmer cheats.

  5. 5.

The plumbers that run happily although rather quickly lie.

  6. 6.

The painter who loves the young lazy farmers cheats.

  7. 7.

The waiter who avoids the judge the least cooks.

  8. 8.

The waiters that love the chefs the least pray.

  9. 9.

The tailor who avoids the farmer the least prays.

  10. 10.

The waiters that walk happily albeit pretty quickly pray.

  11. 11.

The judges that avoid the tailors the most swim.

  12. 12.

The women who love the happy clever chefs sing.

  13. 13.

The lawyer that runs carefully yet fairly quickly swims.

  14. 14.

The athlete that loves the vet the most lies.

  15. 15.

The waiters who walk carefully yet pretty quickly lie.

  16. 16.

The teacher who fears the lawyers the most cheats.

  17. 17.

The teachers who fear the plumber the most climb.

  18. 18.

The waiters who avoid the clumsy clever plumber cheat.

  19. 19.

The waiter who dislikes the proud nice woman swims.

  20. 20.

The chefs who avoid the waiters the least climb.

  21. 21.

The painters who fear the funny nice women pray.

  22. 22.

The vets that like the farmer the most smoke.

  23. 23.

The doctor that runs happily albeit rather carefully cheats.

  24. 24.

The teachers that dislike the tailors the least cheat.

  25. 25.

The lawyers that love the waiter the least climb.

  26. 26.

The plumber who fears the lawyer the most climbs.

  27. 27.

The painters that love the careless proud judges smoke.

  28. 28.

The farmers that run happily though fairly quickly sing.

  29. 29.

The waiters that walk carefully although rather quickly cheat.

  30. 30.

The man who laughs happily though pretty quickly lies.

  31. 31.

The plumber that rides happily although pretty quickly swims.

  32. 32.

The actor that dislikes the lawyer the most prays.

  33. 33.

The chef who dislikes the authors the least cheats.

  34. 34.

The vet that likes the proud helpful painters cheats.

  35. 35.

The farmer who fears the clever lazy tailors cheats.

  36. 36.

The painter who dislikes the nice careless teacher cheats.

  37. 37.

The painters that dislike the careless helpful tailors climb.

  38. 38.

The waiters who avoid the men the most sing.

  39. 39.

The actors that hate the painter the most cook.

  40. 40.

The teacher who likes the bakers the most sings.

  41. 41.

The actor who dislikes the chefs the most swims.

  42. 42.

The lawyers that fear the funny kind athlete pray.

  43. 43.

The bakers that fear the man the least pray.

  44. 44.

The actors who fear the nice proud men cook.

  45. 45.

The man who laughs carefully yet rather quickly smokes.

  46. 46.

The athlete who hates the proud funny woman prays.

  47. 47.

The tailor that loves the clever clumsy bakers cheats.

  48. 48.

The man who avoids the clumsy helpful chef sings.

  49. 49.

The vets who dislike the young nice man cheat.

  50. 50.

The man who fears the lazy nice authors lies.

  51. 51.

The baker who likes the clever happy plumber swims.

  52. 52.

The baker that hates the lazy gentle man cooks.

  53. 53.

The waiter who likes the actors the most smokes.

  54. 54.

The tailors who like the nice cool men swim.

  55. 55.

The women who fear the lazy clumsy plumber sing.

  56. 56.

The baker who dislikes the judge the least cooks.

  57. 57.

The woman that fears the baker the most cheats.

  58. 58.

The plumber who talks happily yet rather quickly swims.

  59. 59.

The lawyer that likes the farmers the most swims.

  60. 60.

The bakers that love the careless clever chef pray.

  61. 61.

The man who walks carefully although fairly quickly cheats.

  62. 62.

The man that runs carefully though rather quickly lies.

  63. 63.

The bakers who love the actor the least cheat.

  64. 64.

The tailor that dislikes the cool lazy baker prays.

  65. 65.

The plumbers who love the kind charming doctor cheat.

  66. 66.

The pilots that hate the waiter the most cook.

  67. 67.

The baker who dislikes the kind helpful plumbers lies.

  68. 68.

The farmer who fears the doctors the least swims.

  69. 69.

The doctor who hates the actor the least cheats.

  70. 70.

The waiter who hates the cool clumsy man swims.

  71. 71.

The farmer who likes the builders the least prays.

  72. 72.

The waiter who dislikes the painter the least prays.

  73. 73.

The woman who avoids the gentle lazy waiters prays.

  74. 74.

The author that hates the waiters the least smokes.

  75. 75.

The doctor that hates the careless young teacher lies.

  76. 76.

The chef that dislikes the proud clumsy tailors sings.

  77. 77.

The athletes who love the judges the least sing.

  78. 78.

The bakers that love the lovely helpful women cook.

  79. 79.

The tailors that drive happily although fairly quickly climb.

  80. 80.

The tailors who run carefully yet fairly happily cheat.

  81. 81.

The chefs who love the proud cool bakers cook.

  82. 82.

The bakers who fear the woman the most climb.

  83. 83.

The teacher who dislikes the helpful charming builders cheats.

  84. 84.

The chefs who run carefully albeit rather quickly cook.

  85. 85.

The painter who loves the helpful friendly judges prays.

  86. 86.

The waiters that run carefully though pretty quickly cook.

  87. 87.

The teachers that avoid the waiters the most pray.

  88. 88.

The waiters who drive happily yet fairly quickly swim.

  89. 89.

The painter that avoids the waiter the most prays.

  90. 90.

The tailors that love the charming young authors smoke.

  91. 91.

The lawyers that dislike the proud young farmer cheat.

  92. 92.

The vets that love the woman the least cook.

  93. 93.

The doctors that like the bakers the least pray.

  94. 94.

The builder that drives happily though rather quickly cheats.

  95. 95.

The plumbers who dislike the careless clever bakers sing.

  96. 96.

The athletes who hate the bakers the most swim.

  97. 97.

The doctors that like the chef the most cook.

  98. 98.

The bakers who fear the friendly nice man swim.

  99. 99.

The men who dislike the pilots the least pray.

  100. 100.

The plumber that laughs carefully yet pretty quickly prays.

##  Appendix B Additional Results

Table 3: ANOVAs for two dependent variables in humans (response time and error rates) and for error rates in T5, testing whether there was a significant effect of violation in three possible conditions: Baseline, (no attractor), and Distal and Proximal attractors. condition | Statistic |  $p$-value | $\eta_{G}^{2}$
---|---|---|---
Response Time (humans)
Baseline | $F_{1,33}=1.81$ | $.187$ | $.023$
Distal | $F_{1,33}=27.86$ | $<.001$ | $.274$
Proximal | $F_{1,33}=13.22$ | $<.001$ | $.139$
Error Rate (humans)
Baseline | $F_{1,33}=10.91$ | $.002$ | $.040$
Distal | $F_{1,33}=22.76$ | $<.001$ | $.032$
Proximal | $F_{1,33}=28.60$ | $<.001$ | $.047$
Error Rate (T5)
Baseline | $F_{1,110}=6.29$ | $.014$ | $.054$
Distal | $F_{1,233}=71.59$ | $<.001$ | $.235$
Proximal | $F_{1,216}=147.35$ | $<.001$ | $.406$
Table 4: Corresponding table to Table 1 with the filter operation when the attractor is singular. Effect | $F_{1,33}$ |  $p$-value | $\eta_{G}^{2}$
---|---|---|---
Response Time; Distal attractor
congruency | $1.12$ | $.297$ | $.001$
violation | $14.02$ | $<.001$ | $.021$
interaction | $0.15$ | $.704$ | $<.001$
Response Time; Proximal attractor
congruency | $0.00$ | $.961$ | $<.001$
violation | $13.66$ | $<.001$ | $.045$
interaction | $0.32$ | $.578$ | $<.001$
Error Rate; Distal attractor
congruency | $0.03$ | $.867$ | $<.001$
violation | $14.52$ | $<.001$ | $.091$
interaction | $0.88$ | $.356$ | $.003$
Error Rate; Proximal attractor
congruency | $0.00$ | $>.999$ | $<.001$
violation | $8.17$ | $.007$ | $.053$
interaction | $0.38$ | $.539$ | $.002$
Figure 4: Corresponding figure to 3 filtering for singular attractors: observe the overall collapse of the effect of congruency. Table 5: Table 2 with full dataset of N=54 participants included Effect | $F_{1,53}$ |  $p$-value | $\eta_{G}^{2}$
---|---|---|---
Response Time; Distal attractor
congruency | $0.14$ | $.714$ | $<.001$
violation | $11.77$ | $.001$ | $.024$
interaction | $0.08$ | $.773$ | $<.001$
Response Time; Proximal attractor
congruency | $0.16$ | $.691$ | $<.001$
violation | $17.98$ | $<.001$ | $.038$
interaction | $0.97$ | $.329$ | $.002$
Error Rate; Distal attractor
congruency | $0.01$ | $.918$ | $<.001$
violation | $20.70$ | $<.001$ | $.062$
interaction | $0.09$ | $.768$ | $<.001$
Error Rate; Proximal attractor
congruency | $0.75$ | $.391$ | $.002$
violation | $18.21$ | $<.001$ | $.064$
interaction | $0.27$ | $.607$ | $<.001$
Figure 5: Performances of humans and NLMs, using all participants: colors
indicate grammaticality, error bars indicate (all figures) SEM over
participants (humans) or sentences (NLMs). Results are comparable to the ones
after rejection of low-performing participants


# Bias induced circular current in a loop nanojunction with AAH modulation:
Role of hopping dimerization

Moumita Mondal<EMAIL_ADDRESS>Physics and Applied Mathematics
Unit, Indian Statistical Institute, 203 Barrackpore Trunk Road, Kolkata-700
108, India  Santanu K. Maiti<EMAIL_ADDRESS>Physics and Applied
Mathematics Unit, Indian Statistical Institute, 203 Barrackpore Trunk Road,
Kolkata-700 108, India

###### Abstract

In this work, we put forward, for the first time, the interplay between
correlated disorder and hopping dimerization on bias driven circular current
in a loop conductor that is clamped between two electrodes. The correlated
disorder is introduced in site energies of the ring in the form of Aubry-
André-Harper (AAH) model. Simulating the quantum system within a tight-binding
framework all the results are worked out based on the standard wave-guide
theory. Unlike transport current, commonly referred to drain current, circular
current in the loop conductor can get enhanced with increasing disorder
strength. This enhancement becomes much effective when hopping dimerization is
included which is taken following the Su-Schrieffer-Heeger (SSH) model. The
characteristic features of bias driven circular current are studied under
different input conditions and we find the results are robust for wide range
of physical parameters. Our analysis may provide a new insight in analyzing
transport behavior in different disordered lattices in presence of additional
restrictions in hopping integrals.

##  I Introduction

The generation of circular current due to voltage bias in a nanojunction
possessing single and/or multiple loops is relatively a new phenomenon cr1 ;
cr2 ; cr3 ; cr4  compared to the well established non-decaying circular
current in an isolated loop in presence of magnetic flux p1 ; p2 ; p3 ; p4 ;
p5 ; p6 ; p7 ; p8 ; p9 ; p10 ; p11 ; p12 . For the latter case, the magnetic
flux is responsible to break the symmetry among clockwise and counter-
clockwise propagating electronic waves which yields a net circular charge
current that even persists when the flux is removed. This phenomenon is
referred to as flux-driven circular current in an isolated loop conductor (not
coupled to external baths) and has been extensively studied in the literature
over the past several years p1 ; p2 ; p3 ; p4 ; p5 ; p6 ; p7 ; p8 ; p9 ; p10 ;
p11 ; p12 .

When we talk about current in a nanojunction, we essentially consider
transport current i.e., the current measured in drain electrode. But along
with transport current, a circular current can be established cr1 ; cr2 ; cr3
; cr4  in a loop conductor that is directly clamped to electronic baths under
suitable conditions. Here the current is generated due to voltage bias and no
magnetic flux is required unlike isolated quantum loop. The study of this
phenomenon, called as bias driven circular current, leads to several important
physical pictures as it directly involves current distribution in different
segments of a conductor. A very little progress has been made so far by a few
groups including us considering some molecular junctions as well as different
tailored geometries cr1 ; cr2 ; cr3 ; cr4 . These works have not investigated
the effect of disorder in a deeper level, though for junction current the role
of disorder is well known. Thus, the specific role of disorder on bias driven
circular current is highly required, and it is one of our primary motivations
behind the present study. In condensed matter systems different kinds of
disorder are taken into account. Among them, correlated disorder in the form
of Aubry-André-Harper models aa1 ; aa2 ; aa3 ; aa4 ; aa5 ; aa6 ; aa7 ; aa8 ;
aa9  nowadays draws significant attention due to its unique and diverse
characteristic features, compared to usual random disordered lattices ran1 ;
ran2 ; ran3 ; ran4 . The response of circular current on disorder is quite
different. Unlike junction current, the circular current can get enhanced with
increasing the disorder strength. This phenomenon becomes more effective when
a restriction is imposed in hopping integrals. Instead of identical hopping in
all bonds, if we take dimerized hopping scenario, following the Su-Schrieffer-
Heeger (SSH) model ss1 ; ss2 ; ss3 ; ss4 ; ss5 ; ss6 ; ss7 , then much higher
circular current can be achieved.

Figure 1: (Color online). Schematic diagram of a ring nanojunction. A circular
current $I_{c}$ is established in the ring when a bias is applied among the
electrodes, source (S) and drain (D).

The combined effect of AAH correlation and hopping dimerization brings several
interesting features which we want to investigate in the present work. Such an
effect has not been discussed so far to the best of our concern and might be
helpful in analyzing transport phenomena in nanoscale regime.

We take a nanojunction where a single ring is coupled to two electronic baths
(see Fig. 1). A tight-binding (TB) framework is given to describe the
nanojunction, and, all the results are worked out based on the standard wave-
guide theory. Different aspects are taken into account which include the ring-
electrode junction configuration, Fermi energy, system size, AAH phase and
related issues. The key findings are (i) observation of higher circular
current with disorder, (ii) more enhancement can occur with the inclusion of
hopping dimerization, and (iii) all the results are valid for broad range of
physical parameters.

The rest part of our work is organized as follows. Section II includes the
description of model ring nanojunction along with the required theoretical
framework to compute the results. All the essential results are presented and
critically analyzed in Sec. III. Finally, Sec. IV contains the summary of the
work.

##  II Model and Theoretical framework

###  II.1 Ring nanojunction and TB Hamiltonian

Let us begin with the junction setup, shown in Fig. 1, where a $N$-site nano
ring is coupled to source (S) and drain (D) electrodes. $I_{\mbox{\small tr}}$
is the transport current which is divided into two arms, upper and lower, of
the ring and the arm currents are $I_{1}$ and $I_{2}$, respectively. The
circular current is defined as cr1 ; cr2 ; cr3

$I_{c}=\frac{I_{1}L_{1}+I_{2}L_{2}}{L}$ (1)

where $L=L_{1}+L_{2}$ being the circumference of the ring. It is easy to
follow that when the two arms are identical (status and lengthwise) then
$I_{1}=I_{2}$ (positive sign is used for clockwise motion) and $L_{1}=L_{2}$.
Under that condition $I_{c}$ will be exactly zero. Thus, in order to have a
finite $I_{c}$, we need to break the symmetry between the two arms.

To describe the nanojunction, we employ a TB framework. The setup contains
different parts: source, ring, drain and the coupling of the ring between two
electrodes. For all these parts the TB Hamiltonian can be written in a general
form as cr1 ; cr3

$H=\sum_{\alpha}\left[\sum_{n}\epsilon_{\alpha,n}c^{\dagger}_{\alpha,n}c_{\alpha,n}+\sum_{n,m}t_{\alpha,n,m}c^{\dagger}_{\alpha,n}c_{\alpha,m}\right]$
(2)

where $\alpha=S,D,R$ respectively for source (S), drain (D), and ring (R).
$c^{\dagger}_{\alpha,n}$, $c_{\alpha,n}$ are the usual fermionic operators,
$\epsilon_{\alpha,n}$ is the on-site energy and $t_{\alpha,n,m}$ is the
nearest-neighbor hopping (NNH) strength. The index $m$ is used to describe the
neighboring sites of the site $n$. Now, we explicitly describe the physical
parameters $\epsilon_{\alpha,n}$ and $t_{\alpha,n}$ that are associated with
different parts of the junction. For the ring, $\epsilon_{\alpha,n}$ is
denoted by $\epsilon_{R,n}$ and the site energies are chosen in the AAH form
governed by the relation aa1 ; aa2 ; aa3 ; aa4

$\epsilon_{R,n}=W\cos\left(2\pi bn+\phi_{\nu}\right).$ (3)

Here $W$ measures the AAH modulation strength or simply called as the
correlated disorder strength, $\phi_{\nu}$ is the phase factor that can be
tuned externally with a suitable arrangement and $b$ is an irrational number
which makes the site energies non-uniform but correlated (non-random). For
rational $b$ (viz, $b=1/Q$, $Q$ being an integer) and if the total number of
lattice sites $N$ of the ring is divisible by the factor $Q$ then the ring
becomes an ordered one with $Q$ periodicity. For a perfect ring, the
phenomenon of bias induced circular current is quite known, and thus, in the
present work we concentrate on the disordered one. For $W=0$, the ring is
always perfect irrespective of $b$ and this is a trivial case. The other
factor $t_{R,n,m}$ in the ring is associated with electron hopping. In our
case, we choose two different hopping strengths that repeat one after the
other, following the SSH model. These two hopping strengths are denoted as
$t_{1}$ and $t_{2}$. Both $t_{1}<t_{2}$ and $t_{1}>t_{2}$ conditions are
imposed in our calculations for the dimerized ring, and, for $t_{1}=t_{2}$ we
have the ring with conventional uniform hopping. The side attached electrodes
are taken as one-dimensional perfect and reflection-less. The electrodes are
parameterized by the on-site energy $\epsilon_{0}$ and NNH strength $t_{0}$.
In our junction setup the source electrode is always coupled to site $1$ of
the ring, whereas drain electrode is attached to any arbitrary point $q$
(which is variable). The coupling strengths of these electrodes S and D, to
the ring are denoted by the parameters $\tau_{s}$ and $\tau_{d}$ respectively.

###  II.2 Theoretical Framework

To determine circular current inside the ring we have to find bond currents
and for that we need to calculate current densities in individual bonds. For
any particular bond, the current density is defined as cr1 ; cr3 ; wg1

$J_{n,n+1}(E)=\left(\frac{2e}{\hbar}\right)\mbox{Im}\left[t_{R,n}A^{*}_{R,n}A_{R,n+1}\right]$
(4)

where $t_{R,n}=t_{1}$ or $t_{2}$ depending on the bond of the ring,
$A_{R,n}$’s are the wave amplitudes at different lattices sites, $e$ and
$\hbar$ ($\hbar=h/2\pi$) are the fundamental constants. Solving $(N+2)$
coupled equations involving wave amplitudes of different lattice sites, the
coefficients $A_{R,n}$’s are computed. Among which $N$ equations are
associated with $N$ lattice sites of the ring, and $2$ other equations are
related to the sites of the source and drain with which the ring is coupled.
This is a standard prescription and has been actively used in the literature.
The general form of $(N+2)$ equations is expressed as wg1

$\left(E-\epsilon_{\alpha,n}\right)A_{\alpha,n}=\sum_{m}t_{\alpha,m}A_{m}$ (5)

where the index $m$ runs over all the nearest-neighbor sites of the site n.

Integrating the bond current density $J_{n,n+1}(E)$ over a suitable energy
window, the bond current is calculated. It is expressed as cr1 ; cr3 ; wg1

$I_{n,n+1}=\int_{E_{F}-eV/2}^{E_{F}+eV/2}J_{n,n+1}(E)\,dE.$ (6)

Determining the currents in all the bonds of the upper and lower arms, the net
circular current in the ring is evaluated. It is written as cr1 ; cr3

$I_{c}=\frac{1}{N}\sum_{n}I_{n,n+1}.$ (7)

##  III Numerical results and discussion

This section contains all the essential results of our investigation, which
include current density and circular currents under different input conditions
of the junction setup. Before starting our discussion let us mention the
parameter values that are kept unchanged throughout the work. For the side-
attached electrodes we choose $\epsilon_{0}=0$. The ring-to-electrode coupling
strengths $\tau_{s}$ and $\tau_{d}$ are fixed to $1\,$eV. Unless specified, we
choose $N=10$, $W=0.5$, $\phi_{\nu}=0$, $E_{F}=0$, $t_{0}=2$ and we compute
the results for a symmetric junction configuration setting the temperature to
zero. The other physical parameters that are different compared to the stated
above, are mentioned in the appropriate parts. All the energies are measured
in units of electron-volt (eV). Depending on the condition imposed on $t_{1}$
and $t_{2}$, we have two different kinds of ring systems.

Figure 2: (Color online). Total circular current density $J_{c}$ as a function
of energy $E$ for the (a) AAH ring with $t_{1}=t_{2}=1$ and (b) AAH SSH ring
with $t_{1}=1.5$ and $t_{2}=1$. The source and drain are connected to sites
$1$ and $6$ of the ring, respectively.

In one case we consider a ring with only correlated disorder which is referred
to as AAH ring, and, for the other case both AAH modulation and hopping
dimerization are present which we may call, for simplification, as AAH SSH
ring.

Now we present and analyze the results one by one. Let us begin with Fig. 2
where the variation of total current density $J_{c}$ with incoming electron
energy $E$ is shown for the (a) AAH and (b) AAH SSH ring junctions. The total
current density is obtained by adding the current densities associated with
all the individual bonds of the ring. For both the ring nanojunctions, peaks
and dips are observed at some discrete energies and in some energies they are
very sharp, while for other energy regions $J_{c}$ becomes zero. These peaks
and dips appear at the discrete energy eigenvalues of the ring which is
clamped between the electrodes. Unlike the current density determined at the
drain electrode, which provides only one sign, the current density measured
inside the ring can have positive and negative signs, as electrons can move
both in the clockwise and counterclockwise directions.

Figure 3: (Color online). Variation of circular current $I_{c}$ with applied
bias voltage $V$ for three different conditions of the hopping integrals
$t_{1}$ and $t_{2}$, where the black, magenta, and pink curves correspond to
$t_{1}=t_{2}$ ($t_{1}=t_{2}=1$), $t_{1}>t_{2}$ ($t_{1}=1.5$, $t_{2}=1$), and
$t_{1}<t_{2}$ ($t_{1}=1$, $t_{2}=1.5$), respectively.

Quite an irregular shape of the density envelop across a peak or dip arises
due to the disorder in the ring. For a perfect ring ($W=0$), uniform peaks and
dips are obtained (not shown here in the figure). Comparing the results given
in Figs. 2(a) and (b) it is found that much higher peaks/dips are obtained
once we include hopping dimerization in the AAH ring. Naturally, higher
circular current is expected for the AAH SSH ring compared to the AAH one. It
can be visualized from the forthcoming results. This enhancement of peak/dip
heights is entirely due to the inclusion of specific restriction in hopping
integrals.

Figure 3 displays the variation of circular current $I_{c}$ as a function of
bias voltage $V$ under three different conditions of the hopping strengths
$t_{1}$ and $t_{2}$, those are represented by three colored curves. For the
dimerized case we take both the situations viz, $t_{1}<t_{2}$ and
$t_{1}>t_{2}$. The nature of $I_{c}$ is directly connected with $J_{c}$, as
integrating the current density profile over a suitable energy window (see Eq.
6), $I_{c}$ is obtained. A finite $I_{c}$ appears when the peaks and dips of
$J_{c}$ asymmetric over the selected energy window. If they are symmetric then
the net current becomes zero due to their mutual cancellation. For the AAH
ring with identical hopping ($t_{1}=t_{2}$), the contributions from peaks and
dips are quite comparable and thus small circular current is obtained
throughout the chosen bias window (black curve of Fig. 3). But once the
dimerized hopping is imposed, the circular current becomes reasonably large
compared to the only AAH ring, and it is clearly reflected by comparing the
curves given in Fig. 3. This enhancement of current is solely due to the
modification of the current density profile in presence of the dimerized
hopping. In addition, we find that depending on the condition i.e., whether
$t_{1}>t_{2}$ or $t_{1}<t_{2}$, there is a sign reversal of current. The
enhancement of $I_{c}$ by introducing hopping asymmetry ($t_{1}\neq t_{2}$)
can be referred to as “delocalization” phenomenon which is of course a new
observation in the context of bias driven circular current and might be useful
in different other directions.

Following the above fact that $I_{c}$ gets significantly modified with the
hopping factors,

Figure 4: (Color online). Dependence of maximum circular current $I_{cmax}$ on
the ratio of $t_{1}$ and $t_{2}$ for three distinct values of disorder
strength $W$. Here $t_{2}$ is fixed at $1.5$ and $t_{1}$ is varied from the
$0$ to $3$. To have all the energy levels of the ring inside the allowed
energy window of the electrodes here we set $t_{0}=3$. The rest other physical
parameters and the junction configuration remain unchanged as used in Fig. 2.

Figure 5: (Color online). Density plot. Simultaneous variation of maximum
circular current $I_{cmax}$ as functions of Fermi energy $E_{F}$ and AAH
modulation strength $W$ for the AAH SSH ring. Here we take $t_{1}=1.5$ and
$t_{2}=1$.The color bars denote $I_{cmax}$ in units of mA.

it is now necessary to check how the current is altered if we continuously
vary anyone of the two hopping strengths in a broad range keeping the other
constant. The results are shown in Fig. 4 for three typical values of AAH
modulation strength $W$. Here we essentially focus on the current magnitude,
and hence, we plot the absolute maximum of $I_{c}$ (referred to as $I_{cmax}$)
computing $I_{c}$ in a large bias window ($0$ to $3\,$V) and taking the
maximum from that. The effect of $\alpha(=t_{1}/t_{2})$ is really appreciable.
The circular current can significantly be modified by altering the hopping
strength and the underlying mechanism solely relies on the modification of
energy eigenspectrum of the ring and thus the current density profile. In each
$W$, the circular current becomes too low for the ring with with only AAH
modulation (viz, $\alpha=1$) compared to the cases when $t_{1}\neq t_{2}$, and
this effect becomes more prominent for lower values of $W$. It gives clear
indication that hopping dimerization can lead to a delocalization in a
disordered system.

All the above results are worked out for a particular value of Fermi energy
$E_{F}$

Figure 6: (Color online). Dependence of $I_{cmax}$ on the AAH phase factor
$\phi_{\nu}$ both for the AAH ($t_{1}=t_{2}=1$) and AAH SSH ($t_{1}=1.5$ and
$t_{2}=1$) rings.

Figure 7: (Color online). Effect of ring-electrode junction configuration on
$I_{c}$. Variation of $I_{c}$ as a function of bias voltage for some distinct
ring-electrode junction configurations in the AAH SSH ring with $t_{1}=1.5$
and $t_{2}=1$.

and for some typical disorder strengths. Now to inspect the characteristics of
$I_{c}$ more clearly, in Fig. 5, we present a density plot where the variation
of $I_{cmax}$ is shown by simultaneously changing $E_{F}$ and $W$ in a wide
range. The results are computed for the symmetric junction configuration. From
the density plot it is clearly seen that there are multiple Fermi energies for
which the current get enhanced with increasing $W$. The other important
finding is that for a broad range of $E_{F}$ we can have the possibility to
get large $I_{c}$ which suggests that our results are not valid only for a
specific choice of $E_{F}$.

Figure 6 displays another interesting behavior. The dependence of $I_{cmax}$
with the AAH phase factor $\phi_{\nu}$ is shown for the two rings, AAH and AAH
SSH. In both the cases $I_{cmax}$ varies with $\phi_{\nu}$, and the effect
becomes more prominent for the AAH SSH ring. The change of $\phi_{\nu}$ alters
the site energies of the ring which directly modifies the energy eigenvalues
and the current density profile, resulting in a variation in $I_{c}$. The
appearance of higher circular current in presence of the hopping dimerization
is clearly visible.

It is well known that for a ring nanojunction, the ring-electrode junction
configuration plays an important role

Figure 8: (Color online). Variation of $I_{cmax}$ as a function of ring size
$N$ for the lengthwise symmetric junction configuration. The ring size is
varied from $4$ to $20$ with the interval $\Delta N=2$. The hopping parameters
are: $t_{1}=1.5$ and $t_{2}=1$.

on transport phenomena, and thus, its effect on circular current needs to be
discussed. In Fig. 7 we present the results of $I_{c}$ as a function of bias
voltage for five distinct ring-electrode junction configurations. The current
is highly sensitive to junction configuration and it is due to the effect of
quantum interference of electronic waves passing through different arms of the
nano ring. The overall current magnitude becomes maximum for the lengthwise
symmetric junction (where drain is connected to site number $6$ of the ring of
our chosen $10$-site ring). For other junction configurations the currents are
also appreciable.

For the sake of completion of our analysis, finally, we discuss the effect of
ring size on circular current. Figure 8 displays the variation of $I_{cmax}$
as a function of ring size $N$ by varying it from $4$ to $20$ with the
interval $\Delta N=2$. In each case we choose lengthwise symmetric junction.
First of all, a finite oscillation is observed with $N$ which is the generic
property of the low-dimensional quantum systems. The fluctuation is directly
connected to the quantum interference of electronic waves. Though for all the
chosen ring sizes, a finite current is obtained, the overall current envelop
shows a decreasing nature with the ring size. This is expected as we are
discussing a quantum mechanical phenomenon. For large enough ring size the
fluctuation will be suppressed significantly and the net current will be
vanishingly small which we confirm through our detailed numerical analysis.
Thus, smaller ring sizes are highly recommended to have finite bias-driven
circular current.

##  IV Summary

To summarize, the present work deals with the interplay between quasi-periodic
site energies and hopping dimerization on bias-driven circular current in a
ring nanojunction. The site energies are taken in a cosine form following the
AAH model and the dimerized hopping integrals are included in the form of SSH
model. Employing a tight-binding framework the results are computed within a
wave-guide theory. The hopping dimerization has a strong effect on current
enhancement which we establish critically by analyzing current under different
input conditions. The specific roles of disorder, Fermi energy, ring size and
junction configurations on $I_{c}$ are thoroughly investigated. The results
are valid for a broad range of physical parameters which proves the robustness
of our work. The present analysis may provide a new insight on localization
behavior in disordered lattices in presence of finite correlation among the
hopping integrals. Before an end, we would like to point out that, instead of
AAH modulation if we consider random site energies, an enhancement of circular
might also be expected in the presence of hopping dimerization. However, the
response is relatively weak compared to the AAH case.

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