Particle-antiparticle duality and fractionalization of topological chiral solitons

Although a prototypical Su–Schrieffer–Heeger (SSH) soliton exhibits various important topological concepts including particle-antiparticle (PA) symmetry and fractional fermion charges, there have been only few advances in exploring such properties of topological solitons beyond the SSH model. Here, by considering a chirally extended double-Peierls-chain model, we demonstrate novel PA duality and fractional charge e/2 of topological chiral solitons even under the chiral symmetry breaking. This provides a counterexample to the belief that chiral symmetry is necessary for such PA relation and fractionalization of topological solitons in a time-reversal invariant topological system. Furthermore, we discover that topological chiral solitons are re-fractionalized into two subsolitons which also satisfy the PA duality. As a result, such dualities and fractionalizations support the topological Z4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Z}_4$$\end{document} algebraic structures. Our findings will inspire researches seeking feasible and promising topological systems, which may lead to new practical applications such as solitronics.

Topological solitons are ubiquitous in nature and have been widely investigated as exotic extended objects in the various systems 1 . In quantum topological physics, the famous Su-Schrieffer-Heeger (SSH) soliton manifests a variety of important concepts including Thouless topological charge pumping, fractional charge, particleantiparticle (PA) symmetry and spin-charge separation [2][3][4][5][6][7] . Such exotic properties and potential applications have stimulated many studies: conducting polymers 7 , diatomic chain model 8 , fractionalized domain wall excitations in 1D chain and wire 9,10 , acoustic experimental system 11 , realization of Zak phase and topological charge pumping in the cold atom system [12][13][14] , topological photonic crystal nanocavity lasers using SSH edge mode [15][16][17] , artificially engineered SSH lattices 18,19 , and topological quaternary operation using chiral soliton 20,21 . Among such exotic properties, PA symmetry and fractionalization are essential to understand not only the pair creation but also topological properties of topological solitons. Due to its PA symmetry, an SSH soliton is its own antisoliton (antiparticle) having a zero energy state with a fractional fermion number ± 1 2 in contrast to conventional fermion excitations in solids (electrons or holes) 2,4 . Though there were several attempts to extend SSH model (for example, considering more atoms in a unit cell, interactions beyond nearest neighbor, spin-orbit coupling, or nonsymmorphic symmetries) 8,9,[22][23][24][25][26][27][28] , there have been only few substantial studies seeking PA symmetry and fractionalization beyond the SSH model.
Here we consider a chirally extended SSH model, double-Peierls-chain model, realized in indium atomic chains on a silicon substrate 29,30 . The double-chain model possesses not only Z 4 symmetry and topological chiral solitons having new chiral degree of freedom 20 but also chiral switching between such solitons 21 . In this model, we find novel particle-antiparticle duality and the fractional charge e/2 of topological chiral solitons even in the presence of chiral symmetry breaking. We find the possible PA dualities and symmetries among topological chiral solitons using two classes of fundamental charge-conjugation operators combined with nonsymmorphic operators. The first class gives the distinguishable PA duality between right-chiral (RC-) and left-chiral (LC-) solitons, which allows that RC-and LC-solitons with the opposite quantum numbers can be pair created and pair annihilated; the second one provides the self PA duality to an achiral (AC-) soliton being its own antiparticle like Majorana fermions 31 . Furthermore, we demonstrate that each chiral soliton is re-fractionalized into two subsolitons, which we will refer to as "2nd fractionalization. " Through the 2nd fractionalization, RC-and LC-solitons are divided into primary and induced subsoliton states while AC-solitons are split into bonding and antibonding

Results and discussion
We investigate the PA duality and fractionalization of topological solitons in the double-Peierls-chain model 20 which has AI symmetry 32 (preserved time-reversal symmetry and broken chiral symmetry) and four degenerate groundstates in the order-parameter spaces (Fig. 1a). There are twelve topological chiral solitons that connect two of four groundstates and the interchain coupling induces dynamical chiral symmetry breaking leading to three types of topological chiral solitons: RC-, LC-, and AC-solitons (Fig. 1b).
RC-and LC-solitons are pair-created from a groundstate by adding a hole h while equally dividing that hole (Fig. 1c). However, the broken charge-conjugation symmetry makes RC-and LC-solitons distinguishable in various ways while the soliton and antisoliton in the SSH model are indistinguishable. The calculated spatiallyresolved local densities of states (LDOS's) of RC-and LC-solitons are located oppositely with respect to E = 0 and their pseudospin vectors rotate oppositely to each other as shown in Fig. 1e,g. These findings strongly imply the PA duality between RC-and LC-solitons. By the way, two subsoliton states appear in both RC-and LC-soliton spectra. This is the result of "2nd fractionalization, " which will be proved later.
On the other hand, two AC-solitons (an AC-soliton and an anti-AC-soliton) having a hole per soliton are generated through a pair production from a groundstate when two bonding electrons are removed (Fig. 1d). As shown in Fig. 1f, the calculated LDOS's of an AC-soliton and an anti-AC-soliton are both identical and symmetric each other (solid arrows) implying PA duality. Similar to the "2nd fractionalization" of RC-and LC-solitons, two AC-subsoliton states-bonding and antibonding states-appear in the gap of both an AC-soliton and an anti-AC-soliton (Fig. 1f). The calculated LDOS's of the bonding and antibonding states are located oppositely with respect to the E = 0 (dashed arrow) implying self PA duality. Moreover, the pseudospins of AC-solitons do not rotate (Fig. 1h) because an AC soliton does not have chirality 20 . Hence, we cannot distinguish an ACsoliton from its anti-AC-soliton with the same charges, energy spectra, and pseudospin vectors (Fig. 1d,f,h) like a pair of SSH soliton and antisoliton. These observations strongly suggest that each AC-soliton should show PA dualities with itself as well as with its anti-AC-soliton while a pair of RC-and LC-solitons exhibits PA duality.
To prove the various PA dualities among RC-, LC-, and AC-solitons, we perform a symmetry analysis in the framework of the low energy effective theory using nonsymmorphic operators. Since there is no chiral symmetry and the conventional charge-conjugation operator does not give any PA relation in this model, we find three classes of nonsymmorphic operators that give significant relations among solitons: a glide reflection operator ( Ĝ y ) and two nonsymmorphic charge-conjugation operators ( Ĉ (i) RL and Ĉ (i) AC ). See Section S2 for details in Supplementary information. www.nature.com/scientificreports/ We now discuss the main results of symmetry analysis. First, Ĝ y establishes equivalent relations among solitons having the same chirality ( Fig. 2a), which guarantees the same physical properties (energy spectra, soliton lengths, and topological charges) and Z 4 cyclic properties among solitons. Second, Ĉ (i) RL transforms RC-solitons to LC-solitons and vice versa (Fig. 2b), indicating that they are in PA duality. This operator assures that the quantum wavefunctions of RC-and LC-solitons are explicitly paired; � RC =Ĉ (i) RL � LC . Like the conventional charge-conjugation operator Ĉ , such PA duality supports that RC-and LC-solitons have opposite energy spectra and pseudospin vectors (Fig. 1e,g) as well as opposite topological charges (Fig. 3d,e). In this sense, two successive operations of Ĉ (i) RL bring a soliton to its original state due to (Ĉ (i) It is noteworthy that this PA duality confirms the experimentally measured RC-and LC-soliton spectra 20,21 . Third, we find that an ACsoliton and its anti-AC-soliton are transformed to each other under Ĉ (i) AC indicating PA duality (Fig. 2c). Together with the equivalent relation between AC-solitons (Fig. 2a), this PA duality of AC-solitons naturally leads to self PA duality (Fig. 2d), indicating that each AC-soliton becomes its own antisoliton similar to a Majorana fermion. Hence, within an AC-soliton, Ĉ (i) AC exchanges the bonding and antibonding subsoliton states as shown in Fig. 1f. While the SSH soliton is a self-charge-conjugate state ( S = * S , where S is the SSH soliton wavefunction 4 ), the AC-soliton is unexpectedly a pseudo-self-charge-conjugate state; AC = U * AC , where U is an unitary operator and AC is an AC-soliton wavefunction including both bonding and antibonding subsoliton states. We also find that the geometrical operations of three types of nonsymmorphic operators in the real space consistently prove the relations among solitons (Fig. 2e-g). Based on this symmetry analysis, we further confirm that dualities are well matched with the chiral characters of chiral solitons; PA dual partners have opposite chirality while self PA duality indicates achiral.
Beyond the fractionalization of the SSH soliton, we find that RC-and LC-solitons are fractionalized once more. Both RC-and LC-solitons are split into primary and induced subsolitons through "2nd fractionalization" as shown in the spectra (Fig. 3a,b). The primary subsoliton resides mainly in a chain having a kink (black triangles in Fig. 1c) while the induced one does in the other chain (see Fig. S2 in Supplementary information). If a primary subsoliton in one chain is generated, then an effective mass inversion simultaneously takes place in the other chain leading to an induced subsoliton 20 . "2nd fractionalization" is the process of dividing the one-half fermionsplit by "1st fractionalization" in the pair creation process (Fig. 1c)-into two fractional fermions possessed by two subsolitons (Fig. 3d,e). Naturally, the total fermion number of two subsolitons should be conserved with ± 1 2 .  Table S1 in Supplementary information). is obtained up to leading order, where t 0 and 0 are the hopping integral and dimerization displacement (see Section S3 for details in Supplementary information). To see the PA duality more physically, we consider the Fermi-level ( E = 0 ) and a neutralized system, which implies that the primary subsoliton state of RC-soliton (LC-soliton) is filled (empty). Then, the electrical charges for the primary and induced subsolitons of an RC-soliton are given by Q RC P = e 1 2 − x , Q RC I = ex . For an LC-soliton, Q LC P = −Q RC P , Q LC I = −Q RC I , which explicitly shows the opposite electrical charges of a PA pair. Analytically, the charge conservation Q RC/LC P + Q RC/LC I = ± e 2 is confirmed (Fig. 3d,e). Thus, the RC-and LC-solitons behave as a half electron and a half hole, respectively.
We confirm that the corresponding primary (induced) subsolitons for RC-and LC-solitons satisfy their own PA duality in their energy spectra. Figure 3a and b show the analytically calculated subsoliton spectra with respect to the interchain coupling strength δ . For an RC-soliton, with increasing δ , the energy state of the primary (induced) subsoliton decreases (increases) as shown in Fig. 3a. For an LC-soliton, the spectra of subsolitons show opposite behavior comparing with the RC-soliton (Fig. 3b) implying PA duality. Note that the level crossing between primary and induced subsolitons are observed for larger interchain coupling strength ( δ ≈ 0.2 ). However, when the interchain coupling strength is large, the low-energy effective Hamiltonian description no longer works and hence nature of the primary and induced subsolitons cannot be described within the lowenergy effective theory. Thus, the study about the effect of the strong interchain coupling on the topological solitons and their topological properties will be a future work.
To see the "2nd fractionalization" in the momentum space, we consider an adiabatic evolution from one to another groundstate, which corresponds to transporting a soliton very slowly. By taking the time-evolution as an extra momentum 34   www.nature.com/scientificreports/ information for details. The total Berry curvatures of RC-and LC-solitons are split into the fractionalized Berry curvatures of RC-and LC-subsolitons, respectively (Fig. 3g,h). Moreover, the fractionalized Berry curvatures of primary (induced) subsolitons for RC-and LC-solitons have opposite signs, which implies PA duality between subsolitons. Note that the fractionalized charges obtained by this fractionalized Berry curvature are consistent with those obtained by the Goldstone-Wilczek method. The quantum wavefunctions of each subsoliton state also manifest PA duality in real space as the pseudospin vectors of primary (induced) subsolitons for RC-and LC-solitons rotate oppositely (Fig. 3j,k). For an AC-soliton, two identical SSH solitons, which are created by "1st fractionalization" having a half fermion in both chains, are fractionalized into the hybridized bonding and antibonding subsoliton states due to the interchain coupling (Fig. 1f). Hence, the fermion numbers of bonding and antibonding subsoliton states of an AC-soliton are divided as N B = − 1 2 + n occ and N AB = − 1 2 + n occ , respectively. Likewise, due to the self PA duality, the electrical charges of such bonding and antibonding subsoliton states are opposite regardless of the interchain coupling when the system is neutralized; Q B = −Q AB = e/2 (Fig. 3f). This finding is consistent with the numerically obtained fractionalized Berry curvatures; the integrated values are π and −π for bonding and antibonding subsoliton states, respectively (Fig. 3i). The pseudospin vectors of the fractionalized bonding and antibonding AC-subsoliton states also respect to the self PA duality in the real space (Fig. 3l), so do the soliton wavefunctions (see Fig. S2c in Supplementary information).
For topological algebraic operation using solitons, the manipulation of pair creation and pair annihilation of solitons is essential. Even though the recent experiment showed the switching between LC-and RC-solitons 21 , the pair creation and pair annihilation have not been experimentally reported mainly due to poor temporal resolution of experiments. However, the robustness of topological solitons and their PA dualities strongly suggest that the pair creation and pair annihilation should be observable with suitable experimental methods. Here, as an example, we present the experimental evidence of the RC-and LC-soliton pair generation in the prototypical In/ Si(111) system using scanning tunneling microscopy (Fig. 4). Figure 4a shows the atomic model for pair created RC-and LC-solitons, where the upper chain has two separate kinks (indicated by closed and open triangles) while the lower chain does not have any kink. Figure 4b shows a scanning tunneling microscopy (STM) image of two chiral solitons corresponding to the same atomic model in Fig. 4a. To get more understanding, we take STM line profiles of both upper and lower chains from the STM image (Fig. 4c). In the upper line profile, each maximum in a unit cell is in phase with solid vertical lines at the left and right regions while each maximum is in phase with dashed vertical lines in the middle regions (or between two kinks). On the other hand, in the lower line profile, each maximum is always in phase with solid vertical lines. This observation strongly indicates that there are two soliton kinks on the upper chain and a single dimerized phase appears on the whole lower chain. This is consistent with the atomic model, showing the pair creation of S R 1 and S L 4 .

Conclusion
We have demonstrated that RC-and LC-solitons respect the PA duality and an AC-soliton does self PA duality using nonsymmorphic charge-conjugation operators. We found that "1st-fractionalized" solitons are "2ndfractionalized" into two subsoliton states leading to the emergence of a quartet of subsolitons from the vacuum in the process of pair creation. As a result, we extracted the important information that supports not only the topological pair creation but also topological Z 4 algebraic structures in the viewpoint of the topological operation. These theoretical concept and method can be easily generalized to other interesting topological systems such as 1D Rice-Mele 8 and 2D Haldane models 35 and the generalized model can be experimentally verified in various physical systems including atomic wires, cold atomic systems, and photonic crystals [36][37][38] . In the practical side, further study may focus on the fault-free manipulation of pair creation, pair annihilation, and topological operation among solitons in an appropriate time-scale. Such developments can in turn lead to developing topological quantum information technology and inspire researches seeking feasible and promising topological systems, which may lead to new practical applications such as solitronics.

Methods
To study various properties of topological solitons, we used the tight-binding as well as Bloch and low-energy effective Hamiltonians of the double-Peierls-chain model 20 Received: 27 July 2020; Accepted: 9 December 2020