Topological Systems Show Distinct Quantum Phases

A thorough investigation into the extended Su-Schrieffer-Heeger model reveals a rich landscape of topological states and symmetry-breaking behaviours. Jing-Hua Niu and colleagues at Chongqing University detail how interactions fundamentally alter the model’s non-interacting phases, transforming symmetry-protected topological states into distinct interacting topological phases exhibiting telltale signs like entanglement-spectrum degeneracy and boundary modes. The study shows the emergence of charge-density-wave phases at strong attraction, alongside a new gapless symmetry-protected topological phase and a Luttinger liquid phase at intermediate attraction, characterised by unique edge states and charge modes. This collaboration between Chongqing University, University of Chicago, and University of California provides key insight into understanding correlated topological systems and their potential for quantum technologies.

Multiple topological phases emerge via extended entanglement-spectrum degeneracy in an interacting

Entanglement-spectrum degeneracies now extend beyond two-fold and four-fold patterns to encompass multiple interacting topological phases arising from a generalised Su-Schrieffer-Heeger model. Incorporating intracell, nearest-neighbour, and next-nearest-neighbour hopping alongside on-site interactions drove this expansion, a complexity previously hindering the observation of such diverse states. The original Su-Schrieffer-Heeger (SSH) model, a one-dimensional tight-binding model, describes electrons hopping between adjacent atoms in a lattice with alternating hopping amplitudes. This simple model already exhibits topological phases, characterised by the presence of zero-energy edge states protected by chiral symmetry. However, real materials invariably exhibit interactions between electrons, necessitating the extension of the SSH model to include these crucial correlations. The researchers specifically considered intracell hopping, representing electron movement within a unit cell, nearest-neighbour intercell hopping, and next-nearest-neighbour intercell hopping, increasing the model’s complexity and realism. The addition of an on-site inter-sublattice interaction, representing the energetic cost of an electron occupying a particular sublattice, further enriches the phase diagram.

These newly identified phases retain characteristic signatures, such as entanglement-spectrum degeneracy structures and boundary modes, even as free-fermion symmetry-protected topological states evolve under interaction. The entanglement spectrum, obtained by analysing the eigenvalues of the reduced density matrix, provides a powerful tool for characterising topological order. In symmetry-protected topological phases, the entanglement spectrum exhibits characteristic degeneracies, such as two-fold and four-fold patterns, reflecting the presence of protected edge states. The researchers found that these degeneracies are modified and extended in the interacting system, providing evidence for the emergence of new topological phases. Boundary modes, localized at the edges of the system, are another hallmark of topological phases and serve as a direct probe of the system’s topology.

The model reveals six distinct quantum phases, extending beyond the previously known trivial and symmetry-protected topological phases exhibiting two-fold and four-fold entanglement-spectrum degeneracies. Strong repulsive interactions induce symmetry-breaking, resulting in unequal sublattice densities, effectively creating a charge imbalance between the two sublattices. This symmetry breaking leads to the destruction of the topological protection and the emergence of a trivial phase. Conversely, strong attraction generates period-2 and period-4 charge-density-wave phases from particle clustering. These charge-density-wave phases arise from the tendency of electrons to localize and form a periodic modulation of the electron density. The period-2 phase exhibits a modulation with a wavelength of two lattice constants, while the period-4 phase exhibits a modulation with a wavelength of four lattice constants. At intermediate attraction, a gapless symmetry-protected topological phase and a Luttinger liquid phase emerge, distinguished by unique edge states and charge modes. The gapless symmetry-protected topological phase retains topological protection but exhibits gapless edge states, while the Luttinger liquid phase is a one-dimensional interacting electron system characterised by collective excitations known as spin-charge separation. These phases are further defined by central charges and critical exponents, detailing their behaviour and offering insights into their potential stability and response to external perturbations; however, these findings currently rely on theoretical modelling and do not yet demonstrate a clear pathway towards fabricating devices utilising these complex quantum states. The precise values of these parameters are crucial for understanding the universality class of the phase transition and the nature of the low-energy excitations.

Mapping entanglement and topological states guides future materials discovery

A potential route towards designing materials with tailored electronic properties is offered by identifying these new quantum phases within the Su-Schrieffer-Heeger model. Calculations meticulously mapped out phase transitions using entanglement, a measure of quantum connectedness, though the computational demands remain significant. The calculation of entanglement requires significant computational resources, particularly for larger systems. The researchers employed numerical techniques, such as density matrix renormalization group (DMRG), to overcome these challenges, but scaling the analysis to larger, more realistic systems therefore presents a challenge. The DMRG method is a powerful numerical technique for studying one-dimensional quantum systems, but its computational cost increases exponentially with the system size. Despite emerging within a deliberately simplified model, these quantum phases retain considerable importance.

These theoretical insights provide a key roadmap for materials scientists by identifying these distinct states, including novel Luttinger liquid and symmetry-protected topological phases. Unique entanglement patterns and edge currents offer specific, measurable signatures that can guide the search for real-world materials exhibiting tailored electronic behaviours, particularly relevant in the ongoing quest for more efficient conductors and semiconductors. The identification of these signatures is crucial for bridging the gap between theoretical predictions and experimental observations. A generalised Su-Schrieffer-Heeger model, a simplified representation of electron behaviour in materials, reveals a more complex arrangement of quantum phases than previously understood. Materials exhibiting strong electron correlations and structural features conducive to alternating hopping amplitudes are prime candidates for realising these topological phases. Examples include transition metal dichalcogenides and organic conductors. By incorporating interactions between electrons, the model demonstrates how symmetry-protected topological phases transform into distinct states with unique characteristics, and entanglement-spectrum degeneracy now serves as a key tool for identifying these phases, offering a new diagnostic for complex quantum systems and potentially accelerating materials discovery. Further research will focus on extending this model to two and three dimensions and exploring the effects of disorder and external fields on the stability of these topological phases.

The research identified a complex arrangement of quantum phases within a generalised Su-Schrieffer-Heeger model, revealing more states than previously known. This is important because it provides theoretical insights into the behaviour of electrons in materials and offers a roadmap for materials scientists seeking tailored electronic properties. Unique entanglement patterns and edge currents serve as measurable signatures to guide the search for materials exhibiting these phases, including potential candidates like transition metal dichalcogenides and organic conductors. The authors intend to extend this model to higher dimensions and investigate the impact of external factors on phase stability.