Introduction

The phenomenon of localization has been a vibrant area of research in condensed matter physics ever since its prediction by Anderson1. Over the years, the interest in this topic has grown exponentially with the exploration of various fascinating systems across different branches of physics2,3,4,5,6,7,8. Anderson’s seminal work demonstrates a metal to insulator transition in a three-dimensional system with uncorrelated (random) disorder, where all the energy eigenstates become completely localized beyond a critical disorder strength1,9. On the contrary, for the lower-dimensional cases, all the energy eigenstates get completely localized regardless of the strength of the disorder1. Therefore, such an uncorrelated disordered system, in lower-dimensional cases, is considered relatively trivial due to the absence of a finite critical disorder strength. By imposing constraints on the site energies, one can unveil captivating dynamics and explore more intriguing phenomena within correlated disordered systems10,11,12,13,14.

To date, a wide range of correlated systems have been employed across various fields, and among them, the Aubry–André–Harper (AAH) model15,16 stands out as the most prevalent and adaptable example. In the nearest-neighbor tight-binding (TB) framework, the 1D AAH model with an incommensurate potential demonstrates a distinct transition between localization and delocalization. Below a critical point, all eigenstates are found to be delocalized, while beyond that critical point, they become completely localized15,16,17. Recent advancements in the field have introduced several generalizations of this model. These include exponential short-range hopping17, flatband networks18, higher dimensions19, power-law hopping20, flux-dependent hopping21, and non-equilibrium generalized AAH model22, etc. The studies have revealed that beyond the nearest-neighbor TB framework, there is typically a single-particle mobility edge (SPME), which represents a critical energy that differentiates localized states from extended states within the system23. AAH systems have also been experimentally realized using cold atoms and optical wave-guides24,25.

Based on current understanding, it has been firmly established that following a localization transition, all states continue to exhibit localization indefinitely as the disorder strength increases. However, recent studies have revealed that under certain constraints or conditions imposed on the system, this characteristic of indefinite localization may alter. In more recent findings, an intriguing occurrence of reentrant localization has been discovered in 1D quasi-periodic disordered systems26,27,28,29,30,31,32,33,34,35, which can be attributed to the interplay between hopping dimerization and the presence of staggered AAH disorder. In this localization phenomenon, as the strength of the staggered potential increases, certain states that were previously localized undergo a transition and regain their extended character. Up to this point, reentrant localization has paved the way for obtaining a comprehensive understanding of Anderson localization theory. As a consequence, it has generated significant interest from both the theoretical and experimental domains. Building upon this line of research, we introduce a methodology to observe the reentrant behavior in systems that go beyond the nearest-neighbor hopping scenario.

Specifically, we focus on a DSH system and investigate its response when exposed to a transverse electric field. Interestingly, the application of a transverse electric field has the intriguing capability to introduce AAH disorder as well as the staggered scenario into the helical system36,37. This makes the system highly desirable from an experimental standpoint. In the context of the geometrical structure of a helical system, the nature of hopping interactions can vary, being either short-range or long-range. However, recent research31 suggests that long-range hopping (LRH) can weaken the competition between dimerized hopping and the staggered potential, leading to the absence of reentrant behavior. Considering this insight, we focus solely on short-range hopping (SRH) interactions in our current work. By exclusively examining short-range hopping, we are able to observe and validate the presence of multiple localization phenomena using various numerical techniques. These techniques include analyzing the eigenvalue spectrum, inverse participation ratio, and local probability amplitudes, among others. Through these investigations, we gain valuable insights into the behavior and characteristics exhibited by the system.

The key aspects of the present work are: (1) establishment of multiple reentrant behavior, (2) appearance of truly extend states in the first reentrant region in the thermodynamic limit, (3) observation of quasi-extended states in the second reentrant region, and (4) implementation of reentrant phenomenon in realistic biologic systems simply by applying an electric field.

The rest of the paper is organized as follows. In “System and theoretical framework”, we describe the helical geometry, tight-binding Hamiltonian in presence of transverse electric field, and theoretical formulae for the quantities required to study the localization phenomenon. The numerical results and our analysis are presented in “Results and discussion”. Finally, in “Conclusion”, we conclude our findings.

Figure 1
figure 1

Schematic of a right-handed double-stranded helical geometry in presence of an external electric field of strength \(E_g\). The blue balls represent the sites in strand-I, and the red balls represent the sites in strand-II. R is the radius of the helix and \(\Delta h\) is the stacking distance between adjacent sites. \(\phi = n\Delta \phi \), where \(\Delta \phi \) is the twisting angle between the neighboring sites and n is the site index in each strand. The alternating black (dotted black) and blue (dotted blue) lines indicate dimerization of the adjacent hoppings in strand-I (strand-II).

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System and theoretical framework

Figure 1 depicts the schematic diagram of a right-handed double-stranded helical geometry. The alternating bonds, characterized by two different hopping strengths, are denoted by the black (dotted black) and blue (dotted blue) lines in strand-I (strand-II). An electric field \(E_g\) is applied perpendicular to the axis of the helix. The nature of electron hopping beyond usual nearest-neighbor sites is determined by two important helical parameters: the stacking distance \(\Delta h\) and the twisting angle \(\Delta \phi \). When \(\Delta h\) is sufficiently small, indicating densely packed atoms, long-range hopping becomes significant as electrons can effectively hop over larger distances. In contrast, when \(\Delta h\) is considerably large, with atoms separated by greater distances, electron motion is restricted to shorter distances, resulting in a short-range hopping helix. In practice, two prominent examples that fall into the short-range hopping and long-range hopping groups are DNA and protein molecules, respectively38. However, in this study, our focus is solely on SRH systems, as mentioned previously.

The tight-binding Hamiltonian for the DSH system in the presence of an external electric field is expressed as

$$\begin{aligned} H= & {} \sum _{j=I,II} \left[ \sum _{\begin{array}{c} n=1\\ \end{array}}^N \epsilon _{j,n} c_{j,n}^\dagger c_{j,n} \right. \nonumber \\{} & {} + \left. t_1\sum _{n=1,3,5,\ldots }^N\left( c_{j,n}^\dagger c_{j,n+1} +h.c.\right) \right. \nonumber \\{} & {} + \left. t_2\sum _{n=2,4,6,\ldots }^N\left( c_{j,n}^\dagger c_{j,n+1} +h.c.\right) \right. \nonumber \\{} & {} + \left. \sum _{n=1}^{N} \sum _{\begin{array}{c} m=1\\ m-n\ne \pm 1 \end{array}}^{N} t_{j,(n,m)}\left( c_{j,n}^\dagger c_{j,m} + h.c. \right) \right] \nonumber \\{} & {} +\sum _n t_3\left( c_{I,n}^\dagger c_{II,n} + h.c.\right) . \end{aligned}$$
(1)

Here \(j(=I,II)\) represents the strand index, \(c_{j,n}^\dagger \) and \(c_{j,n}\) are the usual fermionic creation and annihilation operators at the nth site of strand-j, respectively.

In the first term of Eq. (1), \(\epsilon _{j,n}\) represents the site energy at site n of strand-j. When an electric field is applied perpendicular to the helix axis, the site energies are modified following the relations36,39,40

$$\begin{aligned} \epsilon _{I,n} = -\epsilon _{II,n} = eV_g \cos {(n\Delta \phi - \beta )}, \end{aligned}$$
(2)

where e is the electronic charge and \(V_g\) corresponds to the gate voltage associated with the applied electric field. The relationship between the gate voltage and the electric field can be expressed as \(V_g = E_g R\). The reversal in sign observed between the two strands can be attributed to the combined effects of the perpendicular electric field and the helix conformation of the strands36. The phase factor \(\beta \) represents the angle between the positive x-axis and the incident electric field. This phase factor can be adjusted or modified by changing the direction of the electric field. Equation (2) illustrates that the presence of a perpendicular electric field leads to a harmonic modulation of the site energies along the helical strands. Interestingly, such a modulation is identical to the well-known AAH model15,16. The factor \(eV_g\) is analogous to the AAH modulation strength W, \(\Delta \phi \) can be identified with the term \(2\pi b\) (b an irrational number) and the phase \(\beta \) with the Aubry phase \(\phi _\nu \) in the AAH model. By selectively choosing the term \(\Delta \phi \), it becomes possible to achieve a deterministic disordered double-stranded helical system, where the site energies exhibit a correlated pattern resembling the AAH model. This correlation is realized when the DSH system is subjected to the electric field \(E_g\).

The second and third terms in Eq. (1) represent the nearest-neighbor hopping terms in the Hamiltonian. The parameters \(t_1\) and \(t_2\) indicate that the NNH integrals in the DSH are dimerized.

The fourth term in Eq. (1) is associated with the electron hopping beyond the conventional NNH. \(t_{j,(n,m)}\) is the hopping integral between the sites n and m in strand-j and reads as36,41

$$\begin{aligned} t_{j,(n,m)} = \left( \frac{t_1 + t_2}{2}\right) \mathrm{{e}}^{-\left( l_{j,(n,m)} - l_1\right) /l_c}, \end{aligned}$$
(3)

where \(l_{j,(n,m)}\) is the Euclidean distance between sites n and m. With \(n-m=k\), it is expressed as

$$\begin{aligned} l_{j,(n,m)} = \sqrt{\left[ 2R\sin {\left( \frac{k\Delta \phi }{2}\right) }\right] ^2 + \left[ k\Delta h\right] ^2}, \end{aligned}$$
(4)

and \(l_1\) represents the distance between neighboring sites in both strands, and its value can be calculated using Eq. (4) when \(k=1\). On the other hand, \(l_c\) denotes the decay exponent. In Eq. (3), the first term within the parentheses, \(\left( t_1 + t_2\right) /2\), accounts for an average over a unit cell. This average is utilized in the computation of hopping integrals beyond the nearest-neighbor interactions.

The final term in Eq. (1) corresponds to the inter-strand coupling, which describes the interaction between the two strands and \(t_3\) represents the inter-strand hopping integral.

The inverse participation ratio (IPR) serves as a valuable tool for detecting the transition from a localized state to a delocalized state. This measure allows us to quantify and analyze the spatial distribution of a wavefunction or probability density, providing insights into whether the state is confined to a specific region or spread out across multiple locations. By observing changes in the IPR, we can effectively identify and track the transition as the wavefunction evolves from a localized state to a more delocalized one or vice-versa. For the nth eigenstate, IPR is defined as42,43

$$\begin{aligned} \text {IPR}_n = \frac{ \sum _i |\psi ^n_i|^4}{ \sum _i |\psi ^n_i|^2} \end{aligned}$$
(5)

In the case of a highly extended state, in the thermodynamic limit, the IPR tends to zero. On the other hand, for a strongly localized state, the IPR approximately approaches to unity.

A complementary tool to characterize the localization transition is the normalized participation ratio (NPR), which for the nth normalized eigenstate is defined as42,43

$$\begin{aligned} \text {NPR}_n = \left( 2N\sum _i |\psi _n^i |^4\right) ^{-1}, \end{aligned}$$
(6)

where 2N is the total number of sites present in the DSH system. In the case of a highly extended state, the NPR tends to unity in the thermodynamic limit. Conversely, for a strongly localized state, the NPR approaches to zero.

The earlier defined IPR\(_n\) and NPR\(_n\) can be modified to characterize the parameter space region where localized and delocalized states coexist. One defines their average over a subset of states \(N_L\) as follows43

$$\begin{aligned} \langle \text {IPR} \rangle = \sum _n^{N_L} \frac{\text {IPR}_n}{N_L}, \langle \text {NPR} \rangle = \sum _n^{N_L} \frac{\text {NPR}_n}{N_L}. \end{aligned}$$
(7)

When all \(N_L\) states are localized, \(\langle \text {IPR} \rangle \) tends to unity, while \(\langle \text {NPR} \rangle \) tends to zero when all \(N_L\) states are delocalized. However, in the regime where both \(\langle \text {IPR} \rangle \) and \(\langle \text {NPR} \rangle \) remain finite, the Hamiltonian’s spectrum features an intermediate phase with coexisting spatially extended and localized eigenstates, along with the presence of SPME.

Results and discussion

Let us mention the common parameter values before presenting the numerical results. To implement the short-range hopping in DSH system, we consider physical parameters analogous to those found in the real biological system44. DNA has been proposed as an ideal and established example of short-range hopping system by various research groups. The structural parameters for the said geometry are as follows: the radius is considered as \(R=8\,\)Å, the stacking distance as \(\Delta h=4.3\,\)Å, the twisting angle \(\Delta \phi =\pi \left( \sqrt{5}-1\right) /4\), and the decay exponent \(l_c=0.8\,\)Å. From the relation \(\Delta \phi = 2\pi b\), we can determine the value of b for the SRH case, which is incommensurate. All the energies are measured in units of eV. The number of sites in each of the strands is taken as \(N=500\). Unless stated otherwise, we set the dimerized hopping integrals as \(t_1=0.5\), \(t_2=2.2\), and the inter-strand coupling \(t_3=1\).

Figure 2
figure 2

Density plot. (a) The energy spectrum vs gate voltage \(V_g\) with \(t_1 = 0.5\), \(t_2 = 2.2\), and \(t_3=1\). (b) The energy index vs gate voltage \(V_g\). (c) A magnified version of Fig. 2b to provide enhanced clarity. The color map shows IPR values of different energy eigenstates.

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To study the localization transition, the energy spectrum corresponding to the Hamiltonian in Eq. (1) is plotted as a function of gate voltage \(V_g\) (in units of Volts) as shown in Fig. 2a. Each energy point in the plot is color-coded based on its corresponding IPR value, which is computed according to Eq. (5). To capture the localization transition, our colorbar uses purple for the lowest 10\(\%\) of the maximum IPR value, highlighting the extended states and a gradient of increasing gray shades for the rest, reflecting higher degree of localization. In Fig. 2a, the purple color extends throughout the entire region below \(V_g\sim 1\), indicating that the IPR values in this region are significantly below 0.1. This observation strongly suggests that the states within this region exhibit extended behavior. Above \(V_g\sim 1\), a mixed phase emerges where some states begin to localize while others remain extended, resulting in a combination of both localized and extended states. This mixed phase persists until approximately \(V_g\sim 2\). However, beyond this critical value, all the states undergo localization, indicating a complete transition to a fully localized state. Around \(V_g\sim 2.6\), an intriguing phenomenon occurs as a small number of states around zero energy undergo a reentrant localization, indicated by a narrow purple patch within the predominantly localized region. This region exhibits a transient delocalization, where a few states regain their extended nature in contrast to the surrounding localized states. Upon crossing the reentrant zone, all the states return to a localized state. However, as we approach \(V_g\sim 4\), a noteworthy phenomenon occurs. Several small purple spots emerge, indicating the presence of a second reentrant region. Within this region, a few states exhibit a transient delocalization before ultimately undergoing localization once again. A better visibility of the situation can be obtained by examining the individual eigenstates’ IPR, as depicted in Fig. 2b. This plot provides a comprehensive view of the localization behavior and allows for a more detailed analysis of the reentrant regions and the transition between extended and localized states. The presence of the first reentrant region, spanning from approximately \(V_g\sim 2.5\) to 2.9, is clearly evident in the plot. Within this range, a significant number of eigenstates exhibit delocalization, marked by a distinct decrease in their IPRs. Similarly, the occurrence of the second reentrant region around \(V_g\sim 3.9\) to 4.1 is also observed, with a noticeable deviation from the localized behavior as indicated by a cluster of eigenstates displaying lower IPR values. For a more enhanced visualization, a magnified section of Fig. 2b is illustrated in Fig. 2c, providing a clear depiction of the aforementioned description. In Fig. 2c, we observe the presence of two horizontal lines highlighted in purple color immediately following the first reentrant localization. To assess the potential occurrence of another reentrant localization, we thoroughly analyze the IPR values and the \(V_g\)-window associated with these two lines. Upon investigation, it becomes evident that the IPR values associated with these horizontal lines are approximately 0.09, indicating the presence of quasi-extended states. However, it should be noted that these horizontal lines appear before the completion of the first reentrant localization. Consequently, we can conclude that these two lines do not represent another instance of reentrant behavior.

To gain insight into the mixed phase zone, we compute the average IPR and NPR over a subset of states \(N_L\) from the spectrum of Fig. 2,

Figure 3
figure 3

\(\langle \text {IPR} \rangle \) and \(\langle \text {NPR} \rangle \) as a function of \(V_g\) for a subset of states ranging from 400 to 600 of Fig. 2b. The shaded regions indicate the critical zones. All the system parameters remain the same as described in Fig. 2.

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as defined in Eq. (7). The quantities \(\langle \text {IPR} \rangle \) and \(\langle \text {NPR} \rangle \) are plotted as a function of \(V_g\) in Fig. 3. In this analysis, \(N_L\) is considered to be the subset of eigenstates with indices ranging from 400 to 600, taken from Fig. 2b. All the system parameters remain unchanged as described earlier. In Fig. 3, both \(\langle \text {IPR} \rangle \) and \(\langle \text {NPR} \rangle \) exhibit finite values within the range of \(1.1< V_g < 1.7\), indicating the presence of a critical region where a mixture of extended and localized states coexist. For \(V_g > 1.7\), the system undergoes a transition into a fully localized state, where all states become localized. Moreover, in the approximate window of \(2.5< V_g < 2.9\), a dip in the \(\langle \text {IPR} \rangle \) value is observed, accompanied by a bump in \(\langle \text {NPR} \rangle \). This specific region corresponds to the occurrence of the first reentrant region. Within the chosen subset of states, the system hosts two SPMEs. Considering the limited number of extended states in the second reentrant region, detection of the transition becomes challenging within the same plot. Nevertheless, it is important to note that when considering the entire spectrum, the system reveals the presence of three distinct SPMEs.

To explore the extension of states within the reentrant regions, we analyze the local probability amplitudes of different states at varying gate voltages. This analysis provides insights into the robustness of state extension or localization within the system as the gate voltage, \(V_g\) changes. The results are presented in Fig. 4. Firstly, we calculate the local probability amplitude \(|\psi _n^i|^2\) for the state \(n=500\) under zero-field condition, as illustrated in Fig. 4a. In this disorder-free case, as expected, the local probability amplitudes \(|\psi _n^i |^2\) for all sites exhibit extended behavior. This is evident from the smooth sinusoidal curve and the relatively lower values of probability amplitudes throughout the system. Next, we examine the case where \(V_g=1\) and focus on the state \(n=500\), with the corresponding result depicted in Fig. 4b. Notably, the envelope of the local probability distribution maintains the characteristics observed in the disorder-free scenario. Consequently, the state remains within the extended region. Subsequently, we raise the gate voltage to \(V_g=2.3\) and examine the state \(n=249\). As depicted in Fig. 4c, the probability amplitudes for all sites, except for site index \(i=500\), become vanishingly small. Notably, at this specific site, the probability amplitude assumes a relatively large value of approximately 0.8. This observation indicates that the chosen \(V_g\) value indeed induces a fully localized state within the system. In Fig. 4d, we examine the case where \(V_g\) is fixed at 2.65, corresponding to the first reentrant localization region. We consider the state \(n=436\), and observe that the probability amplitudes range from 0 to 0.02, indicating relatively low values. Therefore, it is evident that within the first reentrant region, the considered state regains its extended nature. Upon further increasing \(V_g\) to 3.5 and examining the state \(n=249\), it is evident from Fig. 4e that the system transitions into a fully localized phase once again. To investigate the second reentrant localization, we examine the case where \(V_g=4\) and focus on the state \(n=332\). Interestingly, we observe two broad peaks in the distribution of \(|\psi _n^i |^2\), as shown in Fig. 4f. The values of the probability amplitude for these peaks are relatively low. Upon closer inspection, we find that these peaks are spread over a span of approximately 40–50 sites, as demonstrated in the two insets of Fig. 4f. Consequently, this region exhibits a quasi-extended behavior. In Fig. 4g, we set the gate voltage to \(V_g=5\) and examine the state \(n=249\). Notably, the probability amplitude is localized predominantly at site \(n=500\) with a value of approximately 0.8, while the amplitudes at all other sites are vanishingly small. This observation confirms the presence of a fully localized state within the system.

Figure 4
figure 4

Local probability amplitude \(|\psi _n^i|^2\) vs site index i for the nth eigen state. (a) \(V_g=0\) and \(n=500\), (b) \(V_g=1\) and \(n=500\), (c) \(V_g=2.3\) and \(n=249\), (d) \(V_g=2.65\) and \(n=436\), (e) \(V_g=3.5\) and \(n=249\), (f) \(V_g=4\) and \(n=332\), and (g) \(V_g=5\) and \(n=249\).

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To address and account for any potential finite size effects, we examine the relationship between the minimum IPR value and the system size in the two reentrant regions as shown in Fig. 5. The minimum IPR value for a given system size is determined by identifying the lowest IPR among all states,

Figure 5
figure 5

Minimimum IPR\(_n\) as a function of 1/2N in the two reentrant regions, namely, at (a) \(V_g=2.65\) and (b) \(V_g=4\).

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achieved at a specific value of \(V_g\). We plot IPR\(_n\) as a function of the inverse of the system size 1/2N in the first reentrant region, namely at \(V_g=2.65\) as shown in Fig. 5a. As the system size increases, the IPR\(_n\) value decreases following a scaling behavior of \(\mathcal {O}(1/L)\), where L represents the system size. Consequently, in the thermodynamic limit, the states within the first reentrant region display a tendency towards a true extended nature. In contrast, the results shown in Fig. 5b for the second reentrant region do not exhibit a scaling behavior similar to the first reentrant region. Instead, the IPR\(_n\) value decreases with increasing system size in a step-like fashion. In the limit of large system sizes, it converges to a finite value of approximately 0.055. Considering the lower values of IPR and its behavior with respect to system size, it becomes evident that the states within the second reentrant region do not exhibit a genuine extended nature in the thermodynamic limit. Instead, these states can be characterized as quasi-extended, as observed in Fig. 4f, where they demonstrate a partial spreading throughout the system.

Finally, we examine the parameter space between the gate voltage \(V_g\) and the hopping integrals in terms of average IPR \((\langle \text {IPR} \rangle )\) to identify the regions where the phenomenon of reentrant localization emerges. Here, our focus is solely on the first reentrant region, and we do not investigate the second reentrant region due to the aforementioned reasons. In Fig. 6a, we plot the color-coded \(\langle \text {IPR} \rangle \) as functions of \(V_g\) and \(t_2/t_1\). All other physical parameters are the same as described in Fig. 2. We calculate \(\langle \text {IPR} \rangle \) using the

Figure 6
figure 6

Density plot. The phase diagram in (a) \(t_2/t_1\) and \(V_g\) space, (b) \(t_3/t_1\) and \(V_g\) spce. The color map shows \(\langle \text {IPR} \rangle \) value associated with the subset of states mentioned in Fig. 3. The physical parameters remain the same as in Fig. 2.

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same method as described in Fig. 3. The maximum IPR value in the corresponding color bar is 0.7, and the extended nature is attributed to the range from 0 to 0.07, which corresponds to \(10\%\) of the maximum IPR value. Based on the plot, we observe that the reentrant region emerges for values of \(t_2/t_1\) ranging approximately from 2.9 to 4.5 and for \(V_g\) within the range of 1.3 to 2.8. By adjusting the inter-strand coupling \(t_3\), it is also feasible to alter the extent of the reentrant region. To visualize this, we plot the color-coded \(\langle \text {IPR} \rangle \) as functions of \(V_g\) and \(t_3/t_1\) in Fig. 6b. We observe an approximate reentrant window occurring for \(t_3/t_1\) values ranging from 2.2 to 4 and for \(V_g\) within the range of 1 to 2.6. It is important to note that the parameter space considered is based on the selected subset of eigenstates, as mentioned earlier. The specific values might slightly vary if we were to consider the entire spectrum.

To explore the influence of the decay constant on reentrant localization phenomena, we present a color-coded plot illustrating the average IPR in relation to both \(V_g\) and decay constant \(l_c\) as shown in Fig. 7. All the relevant parameters remain the same as Fig. 2, and the computation of IPR follows Eq. (5) as previously explained.

Figure 7
figure 7

\(V_g\)–\(l_c\) phase diagram of IPR. The color map shows \(\langle \text {IPR} \rangle \) value associated with the subset of states mentioned in Fig. 3. All the other relevant parameters remain the same as taken in Fig. 2.

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The decay constant \(l_c\) is systematically varied from 0.5 to 8.5, leading to a gradual transition from a short-range hopping regime to a long-range hopping scenario. Form the phase diagram, we find that starting from \(l_c =0.5\), the reentrant signature persists up to \(l_c \sim 7.5\) beyond which the system makes a direct transition from delocalized to localized states for disorder strength \(V_g \sim 3.2\). The presence of reentrant behavior for wide range of \(l_c\) value, underscores the robustness of the reentrant signature, indicating its persistence regardless of the nature of the interaction, be it short-range or long-range. For low values of decay constant \(l_c\) (SRH) within the range of 0.5 to 4, the \(V_g\) window associated with the reentrant region remains nearly constant and it expands as \(l_c\) increases. Beyond \(l_c=7.5\), the reentrant region disappears.

Conclusion

We have focused on the localization behavior of a DSH system under the influence of a transverse electric field. Each strand of the DSH system is assumed to possess dimerized hopping. The introduction of a transverse electric field gives rise to the emergence of correlated disorder within the system, accompanied by a strand-wise staggered arrangement of site energies. Notably, we have observed a distinctive multiple reentrant behavior, specifically characterized by two instances of reentrant localizations. This observation has been made by examining the behavior of the IPR of individual eigenstates within the single-particle spectrum. Each localization transition is accompanied by an SPME, and our system exhibits a total of three SPMEs, two of which are associated with the two reentrant regions. By examining the local probability amplitude and scaling behavior of IPR, we have found that the states corresponding to the first reentrant region demonstrate genuine extended characteristics in the thermodynamic limit. However, states within the second reentrant region display a quasi-extended nature. Our investigation reveals that the reentrant region can be influenced and adjusted by modulating both the gate voltage and hopping integrals.

Considering the ongoing progress in experimental feasibility to achieve hopping dimerization45,46,47,48,49,50 and the potential for inducing correlated disorder (AAH) through a transverse electric field, our proposal is highly compelling and is expected to generate significant interest within the research community. The incorporation of these factors in our study presents a valuable opportunity to observe and study the reentrant localization behavior using different kinds of other helical systems as well.