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 \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}Z4 algebraic structures. Our findings will inspire researches seeking feasible and promising topological systems, which may lead to new practical applications such as solitronics.


. Bloch Hamiltonian
We consider the double-Peierls-chain model [1] which is composed of two identical SSH chains with interchain coupling (see inset of Fig. 1a in the main text). The Bloch Hamiltonian of this model is given by where H i = (2t 0 cos(k x a 0 /2), −∆ i sin(k x a 0 /2), 0) · σ, Here, t 0 is the hopping integral between nearest atoms along the chain in the undimerized phase, a 0 is the lattice constant of the dimerized phases, δ is the interchain coupling strength, σ = (σ x , σ y , σ z ) are the Pauli matrices, and ∆ i is the energy-valued Peierls dimerization displacement of the i-th chain (∆ i = 4αu (i) ). Here, u (i) is the atomic dimerization displacement for the i-th chain and α is the electron-phonon coupling coefficient. Thus, the strong and week hopping integrals in dimerized chains are given by t 0 + ∆ i /2 and t 0 − ∆ i /2, respectively.

S1.4. Notations for twelve topological chiral solitons
In the double-Peierls-chain model, the twelve topological chiral solitons shown in the order-parameter space of Fig. 1b in the main text are listed in Table S1. Table S1. Notations for twelve topological chiral solitons and their phase shifts in the order parameter space. For the soliton profiles, ∆i(x) at x = ±∞ are represented. ∆0 is the energy-valued dimerization displacement of groundstates.

S2. SYMMETRY ANALYSIS FOR TOPOLOGICAL CHIRAL SOLITONS
In this section, we explicitly prove transformation properties among topological chiral solitons in low energy effective theory using three classes of nonsymmorphic operators.

S2.1. Three classes of nonsymmorphic operators
The first one is the glide reflection operatorĜ y = {M y | − a0 4 } which flips the atoms in the upper and lower chains with respect to the horizontal line between two chains (or mirror reflection operation M y as shown in Fig. S1) followed by a fractional translation − a0 4 along the chain direction. The second class operator is the nonsymmorphic charge-conjugation operatorĈ (i) RL = G (i)Ĉ that gives the PA duality between RC-and LC-solitons, whereĜ (i) ≡ {E|(−1) i−1 a0 2 } (i) is a fractional translation operator along the i-th chain only. The third class operator is the nonsymmorphic charge-conjugation operatorĈ (i) AC that gives the PA dualities among AC-solitons. Their representations, cyclic properties, and roles are explicitly listed in Table S2.

S2.2. Transformations among topological chiral solitons
In this subsection, we explicitly prove the three relations among topological chiral solitons under the three classes of the nonsymmorphic operators using the low energy effective Hamiltonians. The results are summarized in Table S3.

Operator
Bloch Continuum Cyclic Rolê Table S2. Three types of nonsymmorphic symmetry operators in the Bloch basis and low energy continuum theory. Here, the representations of the charge conjugation operatorĈ in the Bloch basis and low energy continuum theory are given by σz 0 0 σz K and respectively.K is the complex conjugate operator.
Consider a soliton solution described by a wavefunction Ψ(x) and dimerization profiles [∆ 1 (x), ∆ 2 (x)] with an energy eigenvalue E that satisfy the following eigenvalue equation: Then, the transformed soliton under an operatorÔ is also a soliton solution which is described by Depending on the dimerization profiles [∆ 1 (x), ∆ 2 (x)], the transformed soliton corresponds to either RC-, LC-, or AC-soliton. For example, under the operatorĜ y , the transformed soliton has , and E = E. Thus, the solitons with the same chirality are physically equivalent and have the same energy eigenvalues. Symbolically, we find that where k is either R, L or A.

S2.3. PA duality between RC-and LC-solitons
Let us consider the PA-duality between RC-and LC-solitons via the nonsymmorphic charge-conjugation operatorĈ exchanging the energy spectra, leading to PA duality between RC-and LC-solitons. Moreover, the RC-and LC-soliton wavefunctions [Ψ RC (x) and Ψ LC (x)] also satisfy the PA duality: In the view point of quantum field theory, the wavefunctions can be promoted to the field creation and annihilation operators. Thus, Eq. (9) indicates the PA duality between an RC-and LC-soliton pair in the quantum level.

S2.4. PA duality between AC-solitons
Similarly, one can derive the PA duality between two AC-solitons shown in Fig. 2c of the main text. See the rows for the PA duality ofĈ Table S3.
Soliton relation Rolê    Table S3), which implies the AC-soliton has a particle-hole symmetric spectra ±E. For instance, the bonding and antibonding energy spectra of an AC-soliton (S A 1 ) are exchanged leading to the self PA duality as shown in Fig. 1f of the main text (dashed green arrow). In addition, the bonding and antibonding wavefunctions of the AC-soliton, Ψ B (x) and Ψ AB (x), satisfy the PA relation: Since (Ĉ where U is an 8 × 8 unitary matrix. The eigenvalue equation HΨ = EΨ for the soliton wavefunction Ψ = (U, V) T is given by where U and V are two-component spinor wavefunctions for the upper and lower chains, respectively. These equations are decoupled into two effective Hamiltonians for each chain. Using the slowly-varying soliton field approximation in space, the two effective Hamiltonians can be written as where we set v F = 1 for simplicity. Using these effective Hamiltonians, we obtain the energy eigenvalues of RC-and LCsolitons in a similar way of the Rice-Mele method [8]. For an RC-soliton, the primary and induced subsoliton energy spectra are given by in the leading order. For an LC-soliton, the primary and induced subsoliton energy spectra are given by E LC P = −E RC P and E LC I = −E RC I , which explicitly supports the PA duality between RC-and LC-solitons. In a similar way, we obtain bonding and antibonding subsoliton spectra of an AC-soliton considering the hybridized wavefunction U ± V . The spectra are given by E AC B = −E AC AB = −t 0 δ 0 , which supports the self PA duality of an AC-soliton.

S3.2. Fermion numbers of topological chiral solitons
For an effective Dirac Hamiltonian H eff (x) = −i∂ x σ x + m x (x)σ y + m z (x)σ z , the fermion number of a soliton is given ] are scalar fields that support a soliton [9]. Now, using the effective Hamiltonian for each chain, we calculate the fermion numbers of RC-and LC-solitons. If a soliton kink resides in chain 1, for instance, the fermion numbers of the primary and induced subsolitons are given by N RC/LC n is the fermion number from i-th chain and n-th band. Here, n = 1 and n = 2 indicate the filled lowest and next lowest bands, respectively. Because the effective Hamiltonian is reduced to 2 × 2 Hamiltonian comparing to the original 4 × 4 Hamiltonian, the normalization factor 1/2 is needed for the total fermion number conservation. Similarly, for an AC-soliton, the fermion numbers of bonding and antibonding subsoliton states are obtained as N B = N AB = −1/2.

S4. BERRY CURVATURE AND FRACTIONALIZED BERRY CURVATURE
We calculate topological charges of a soliton and its subsolitons using Berry curvature and fractionalized Berry curvatures under an adiabatic evolution, respectively [1,10,11]. The adiabatic evolution is generated by transporting a topological soliton slowly, which is represented by the time-dependent phase-space Hamiltonian H[k x , t] with time-varying displacement fields [∆ 1 (t), ∆ 2 (t)]. For example, transporting an RC-soliton S R 1 generates an adiabatic evolution of AA → BA. Then, the total topological charge of a soliton is given by Q = eC sol where C sol is the phase-space partial Chern number obtained under the corresponding adiabatic evolution. The phase-space partial Chern number for multibands from the initial time t i to the final time t f (which corresponds to a soliton) is defined as where the summation is done over the occupied bands and Ω n = ∂ kx u n | ∂ t u n − ∂ t u n | ∂ kx u n is the phase-space Berry curvature for n-th band. Here, |u n is a normalized eigenvector of n-th band for the Hamiltonian H[k x , t]. The Chern numbers for RC-and LC-solitons can be decomposed into fractionalized Chern numbers of primary and induced subsoliton states; C sol = C P + C I where Here, Ω P and Ω I are the fractionalized Berry curvatures for the primary and induced subsoliton states, which are given by n is the fractionalized Berry curvature for n-th band using an effective Hamiltonian of the i-th chain, which is given by Here, u In Fig. 3g-i of the main text, the numerically calculated (fractionalized) Berry curvatures are shown in the (k x , k y ) space, where the time t is replaced by the k y using the dimensional extension [1,10].  Figure S2c shows that the bonding and antibonding states of an AC-soliton equally reside in both chains due to the self PA duality. The relative phase between wavefunctions at upper and lower chains distinguishes bonding and antibonding states.

S5.2. Pseudospin vectors
Various PA dualities among the subsolitons can be easily visualized by the pseudospin vectors along each chain as discussed in the main text. The wavefunctions on the four atoms in the unit-cell are decomposed into two spinors (one spinor to each chain), and each spinor is represented by a pseudospin vector (d z ) along the i-th chain (Fig. 3j-l of the main text).
Thus, the pseudospin vector corresponds to the spinor wavefunction where θ and φ are polar and azimuthal angles of the pseudospin vector, respectively.

S5.3. PA duality between RC-and LC-solitons
We now explicitly prove that the PA duality between wavefunctions of RC-and LC-subsolitons using the nonsymmorphic charge-conjugation operatorsĈ (i) RL . Since the RC-and LC-solitons (S R 1 and S L 2 ) satisfy the PA duality of S L 2 =Ĉ (2) RL S R 1 , each fractionalized wavefunction of RC-and LC-solitons [Ψ RC P/I (x) and Ψ LC P/I (x)] also satisfies the PA duality: where the convention II for the Bloch wavefunction is used (see Table S4). Then, the direction vector of a pseudospin in chain 1 transforms as (d (1) x , d z ). Similarly, for chain 2, (d (2) x , d y , d z ) → (d (2) x , d y , −d (2) z ). Thus, the pseudospin vectors for S R 1 and S L 2 rotate oppositely in the xz plane, which is consistent with the numerical results shown in Fig. 3j, k of the main text.
where the convention II for the Bloch wavefunction is used (see Table S4). Then, the pseudospin vectors are transformed as (d (1) x , d y , d z ) → (−d (2) x , d (2) y , d z ) and (d (2) x , d (2) y , d z ) → (−d (1) x , d y , d z ), which explains the numerically obtained pseudospin profiles of the bonding and antibonding soliton states in Fig. 3l of the main text.

S5.5. Nonsymmorphic charge-conjugation operators in convention II
For the Bloch Hamiltonian and the symmetry analysis in Sec. S1 and S2, we used the convention I [12]. In convention I, the Bloch wavefunction has the exponential factor for each atom: 4 j=1 e ikRn C n,j e ikrj φ j (x − R n − r j ) ,

Operator
Continuum Rolê duality   Table S4. Representations of the nonsymmorphic charge-conjugation operators for the Bloch basis and low energy continuum theory in the convention II.
where φ j (x − R n − r j ) is the basis orbital of j-th atom in the n-th cell, R n is the position vector of the n-th cell's center, r j is the relative position vector of the j-th atom in the unit-cell, and C n,j is the coefficient. For the wavefunction and pseudospin analysis in Sec. S5, we used the convention II where the exponential factor for each atom is not included: ψ k (x) = N n=1 4 j=1 e ikRn C n,j φ j (x − R n − r j ) , whereC n,j = e ikrj C n,j . The representations for nonsymmorphic chiral operators in convention II are summarized in Table S4.