Anisotropic SSH-Kitaev Model

• The anisotropic SSH-Kitaev model is a quantum lattice system that integrates dimerization (SSH) with bond-specific Kitaev interactions to explore topological phases.
• It employs tunable anisotropy in hopping, pairing, and exchange couplings, enabling controlled studies of zero modes, Majorana fermions, and fractionalized excitations.
• The model reveals non-equilibrium scaling laws and disorder-induced modifications that offer insights into defect production and critical dynamics.

The anisotropic SSH-Kitaev model designates a class of quantum lattice systems that merge the physics of dimerized (Su–Schrieffer–Heeger, SSH) chains or networks with bond- and/or site-dependent anisotropic Kitaev-type interactions. These models serve as archetypal settings for the realization and analysis of topologically non-trivial quantum phases, fractionalization of excitations (notably into emergent Majorana fermions and static fluxes), and unusual non-equilibrium quench dynamics. By tuning the pattern and magnitude of anisotropy—either in the hopping/pairing amplitudes or in the underlying exchange couplings—such models interpolate between the SSH regime (supporting solitonic and fermionic zero modes) and the Kitaev regime (harboring gapped/gapless spin liquids, non-abelian quasiparticles, and rich topological order), enabling a controlled paper of how topological, symmetry, and dynamical properties are shaped by microscopic details.

1. Lattice Structure and Hamiltonian Formulation

Anisotropic SSH-Kitaev models typically consider either one-dimensional (1D) chains or quasi-1D/2D networks in which the critical features are the alternation (“dimerization”) of bond strengths and the coexistence of Kitaev’s bond- or direction-dependent Ising-like couplings. A representative 1D minimal model, hybridizing the SSH chain and the Kitaev chain with site-dependent parameters, is characterized by the Hamiltonian: $$H = \sum_i \big[ t_1\, c_{B,i}^\dagger c_{A,i} + t_2\, c_{A,i+1}^\dagger c_{B,i} + \mathrm{h.c.} - \Delta_1\, (c_{B,i}^\dagger c_{A,i}^\dagger - c_{A,i}^\dagger c_{B,i}^\dagger) - \Delta_2\, (c_{A,i+1}^\dagger c_{B,i}^\dagger - c_{B,i}^\dagger c_{A,i+1}^\dagger) + \mathrm{h.c.} \big] - \mu \sum_{i,\alpha} c_{\alpha,i}^\dagger c_{\alpha,i}$$ where $t_{1,2}$, $\Delta_{1,2}$ control intra-/inter-cell hopping and pairing; their relative magnitude and possible anisotropy enable interpolation between pure SSH and pure Kitaev models or more general regimes (Griffith et al., 2019). Similar motifs are found in higher-dimensional generalizations—such as two-dimensional (2D) time-reversal symmetric Kitave-like models on the square lattice (Nakai et al., 2011), bond-alternated honeycomb systems, or anisotropic spin chains with bond-dependent Ising interactions (Gordon et al., 2021, Bhullar et al., 6 Jan 2025).

The essential feature is a breaking of uniformity and/or isotropy, encoded either as spatial alternation (SSH-like) or bond-selective anisotropy (Kitaev-like). This gives rise to non-uniform gap closings, distinct dynamical exponents, and a hierarchy of momentum-dependent dispersions.

2. Anisotropic Criticality and Non-equilibrium Scaling

Anisotropy profoundly modifies the criticality and excitation structure:

• In both 2D and 1D, suitably tuned anisotropy yields critical points at which the excitation gap vanishes with different exponents along distinct momentum directions: e.g., $\Delta_{\vec k} \sim k_{y}$ (linear) in one direction and $\sim k_x^2$ (quadratic) in another (Hikichi et al., 2010).
• This defines an “anisotropic critical point,” characterized by multiple dynamical exponents $(z, z')$ and a vector of correlation length exponents.

Upon driving such a system slowly across a quantum critical point (e.g., via a power-law quench of a control parameter), the density $n$ of generated defects and the residual energy $Q$ exhibit non-standard scaling: $$n \sim \tau^{-\frac{[m + (d-m)z/z'] \nu \alpha}{z\nu\alpha+1}};\quad Q \sim \tau^{-\frac{[(m+z)+(d-m)z/z'] \nu \alpha}{z\nu\alpha+1}}$$ where $m$ is the number of momentum directions scaling with $z$ (e.g., $m=1$ in Kitaev/SSH-Kitaev), $d$ the spatial dimension, $\nu$ the correlation length exponent, and $\alpha$ the ramp exponent (Hikichi et al., 2010).

For the 2D Kitaev case: $d=2$, $m=1$, $z=1$, $z'=2$, $\nu=1$, $\alpha=1$, yielding $n \sim \tau^{-3/4}$ and $Q \sim \tau^{-5/4}$. This nontrivial scaling generalizes to any anisotropic SSH-Kitaev system with equivalent dispersion near the gapless point; the precise exponents are determined by the structure of the underlying Hamiltonian.

3. Topological Phases, Edge Modes, and Bound States

Bond anisotropy and alternation lead to a wealth of phases, with distinct topological invariants and localized excitations:

• In the 1D anisotropic SSH-Kitaev model, the interplay between dimerization and p-wave pairing supports topological phase transitions, marked by changes in integer-valued winding numbers $W_1$, $W_2$ associated with chiral and particle–hole symmetries. The number and type of zero-energy states per edge—ranging from ordinary SSH soliton (fermionic) modes to isolated Majorana zero modes—are governed by these invariants (Griffith et al., 2019).
• Domain walls (kinks) that separate regions with different topological invariants host mid-gap soliton states; solitonic excitation at the chain center can hybridize and modify the edge Majorana character (Griffith et al., 2019).
• In higher-dimensional Kitaev systems with anisotropy, the presence or absence of gapless Majorana edge states is determined by a $\mathbb{Z}_2$ invariant (as in symmetry class DIII in 2D (Nakai et al., 2011)), with Kramers pairs of edge or vortex-bound Majorana states in the time-reversal symmetric case and single Majoranas in the BDI class of 1D chains.

4. Fractionalization, Anyon Dynamics, and Flux Constraints

Anisotropic SSH-Kitaev models exemplify spin (or fermion) fractionalization, leading to nontrivial emergent excitations:

• In the Kitaev limit, spins fractionalize into itinerant Majorana fermions and Z₂ gauge fluxes (static in the exactly solvable limit). The anisotropy tunes the low-energy excitations between gapless/gapped regimes, influencing the statistics and dynamics of the resulting anyons (Feng et al., 2022).
• In highly anisotropic limits with ordered (superlattice) flux configurations, the Majorana spectrum can be gapped by Brillouin zone folding effects, with the gap scaling inversely with the flux supercell size ($\Delta \sim 1/q$ for periodicity $q$) (Hashimoto et al., 2023).
• For models defined on the hypernonagon lattice or in three dimensions, elementary $Z_2$ flux excitations (loops) are subject to both local and global (surface/volume) parity constraints. Such global constraints force excitations to occur in pairs or closed loops, which modifies defect statistics and anyon dynamics in a manner not captured by purely local models (Kato et al., 2018).

The field-induced hybridization between composite $\epsilon$-type anyons and matter Majoranas can create new hybrid quasiparticles (e.g., a “$\psi$ fermion”) in the so-called “primordial fractionalized” regime (Feng et al., 2022).

5. Effects of Disorder

Disorder impacts non-equilibrium dynamics and the nature of low-energy excitations:

• In the SSH-Kitaev context, introducing disorder (e.g., random link variables in the Majorana representation) changes scaling laws for defects and residual energy during quenches. For linear quenches ending in the gapless phase: $n\sim\tau^{-1/2}$ and $Q\sim\tau^{-1}$; for quenches ending in a gapped phase: $Q\sim\tau^{-1/2}$. This is in contrast with the clean anisotropic scaling ($n\sim\tau^{-3/4}$) (Hikichi et al., 2010).
• The physical origin is a flattening of the low-energy density of states due to the randomness, and “smearing” of the anisotropic gap structure, leading to new universality classes for excitations and transport.

6. Realizations, Extensions, and Experimental Relevance

The physics of anisotropic SSH-Kitaev models has direct experimental ramifications:

• Bond-anisotropic exchange and dimerization are naturally realized in material candidates like $\alpha$-RuCl$_3$, YbOCl, RuI$_3$, and various 4$d$- and 5$d$-electron honeycomb compounds (Lampen-Kelley et al., 2018, Zhang et al., 2022, Ma et al., 22 Jul 2024). In these systems, anisotropy is encoded in the variation of exchange couplings $J_{\gamma}$, off-diagonal symmetry terms $\Gamma$, and longer-range interactions, and is detected via anisotropic susceptibility, torque magnetometry, and high-field measurements.
• Topological signatures—such as quantized thermal Hall signals—can depend critically on the anisotropic gap structure and field-induced topology (Yılmaz et al., 2022).
• Disorder and non-uniform modulation (experimental or intrinsic) provide further control knobs to tune the excitation spectrum and defect production.

Theoretical advances—such as the recognition that anisotropic Coulomb exchange (from quantum chemistry) can dominate over conventional superexchange mechanisms—give a microscopic framework for understanding and tuning the anisotropic interactions underpinning SSH-Kitaev physics (Bhattacharyya et al., 2023).

7. Summary Table: Scaling Laws and Topological Features

Model/Regime	Defect Scaling ($n$)	Residual Energy ($Q$)	Topological Invariant	Edge/Zero Modes
Clean anisotropic (2D)	$n\sim\tau^{-3/4}$	$Q \sim \tau^{-5/4}$	$\mathbb{Z}_2$, winds	Kramers/Majorana pairs
Disordered	$n\sim\tau^{-1/2}$	$Q \sim \tau^{-1}$ or $Q\sim \tau^{-1/2}$	Flattened (statistical)	Robust/fused under disorder
1D SSH-Kitaev hybrid	Varies	Varies	$\mathbb{Z}$	SSH soliton, Majoranas
Ordered flux supercell	Gap $\Delta\sim 1/q$	—	—	Modulated Majorana spectrum

Exponents, invariants, and edge/bulk states depend on anisotropy, disorder, and dimerization pattern; see references (Hikichi et al., 2010, Griffith et al., 2019, Feng et al., 2022).

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Anisotropic SSH-Kitaev models thus occupy a pivotal role in modern quantum condensed matter theory: they demonstrate how the conjunction of symmetry, dimerization, and bond/interaction anisotropy can be engineered to yield a hierarchy of phases, scaling behaviors, and emergent excitations—from fermionic solitons to non-abelian Majorana edge states—whose interplay is accessible both theoretically and experimentally.