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Communications Physics

Gapless topological behaviors in a long-range quantum spin chain

Sheng Yang, Hai‐Qing Lin, Xue-Jia Yu

Published Jan 18, 2025• 21 citations
Publisher

Abstract

Topology is a fascinating phenomena in condensed-matter physics typically associated with a bulk gap. However, recent research shifts focus to quantum critical points or phases that exhibit nontrivial topological properties. Here we explore a cluster-Ising chain with long-range antiferromagnetic interactions that decay as a power law with distance. Using large-scale density matrix renormalization group simulations, we demonstrate that the nontrivial topology at the critical point remains stable against long-range interactions, resulting in a topologically nontrivial critical line. Moreover, even within the gapped region, the interplay between topology and long-range interaction can give rise to a topological phase featuring algebraically decaying correlations and edge modes, similar to gapless topological phases. We refer to this phase as the algebraic topological phase, which exhibits nontrivial gapless topological behaviors and arises solely from long-range interactions without short-range counterparts. The findings pave the way for more studies on topological states in long-range systems. Long-range interactions play a crucial role in condensed-matter systems and can lead to range of emerging and exotic physical phenomenon. Here using simulations, the authors investigate the non-trivial topological properties that can arise in a cluster Ising chain with long-range antiferromagnetic interactions, finding evidence for what they classify as an algebraic symmetry-protected topological phase.

DOI
10.1038/s42005-025-01947-z

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Topology is a fascinating phenomena in condensed-matter physics typically associated with a bulk gap. However, recent research shifts focus to quantum critical points or phases that exhibit nontrivial topological properties. Here we explore a cluster-Ising chain with long-range antiferromagnetic interactions that decay as a power law with distance. Using large-scale density matrix renormalization group simulations, we demonstrate that the nontrivial topology at the critical point remains stable against long-range interactions, resulting in a topologically nontrivial critical line. Moreover, even within the gapped region, the interplay between topology and long-range interaction can give rise to a topological phase featuring algebraically decaying correlations and edge modes, similar to gapless topological phases. We refer to this phase as the algebraic topological phase, which exhibits nontrivial gapless topological behaviors and arises solely from long-range interactions without short-range counterparts. The findings pave the way for more studies on topological states in long-range systems. Long-range interactions play a crucial role in condensed-matter systems and can lead to range of emerging and exotic physical phenomenon. Here using simulations, the authors investigate the non-trivial topological properties that can arise in a cluster Ising chain with long-range antiferromagnetic interactions, finding evidence for what they classify as an algebraic symmetry-protected topological phase.
Topology in condensed matter physics is typically associated with a bulk energy gap. However, recent research has shifted focus to topological phases without a bulk energy gap, exhibiting nontrivial gapless topological behaviors. In this letter, we explore a cluster Ising chain with long-range antiferromagnetic interactions that decay as a power law with the distance. Using complementary numerical and analytical techniques, we demonstrate that long-range interactions can unambiguously induce an algebraic topological phase and a topological Gaussian universality, both of which exhibit nontrivial gapless topological behaviors. Our study not only provides a platform to investigate the fundamental physics of quantum many-body systems but also offers a novel route toward searching for gapless topological phases in realistic quantum simulators.
Long-range interactions can fundamentally alter properties in gapped topological phases such as emergent massive edge modes. However, recent research has shifted attention to topological nontrivial critical points or phases, and it is natural to explore how long-range interactions influence them. In this work, we investigate the topological behavior at the quantum critical point of extended Kitaev chains with long-range interactions, which can be derived from the critical Ising model via the Jordan-Wigner transformation in the short-range limit. Specifically, we analytically find that the critical edge modes at the critical point remain stable against long-range interactions. More importantly, we observe that these critical edge modes remain massless even when long- range interactions become very strong. As a by-product, we numerically find that the critical behavior of the long-range model belongs to the free-Majorana-fermion universality class, which is entirely different from the long-range universality class in the usual long-range spin models. Our work could shed new light on the interplay between long-range interactions (frustrated) and the gapless topological phases of matter.
ABSTRACT The role of long‐range interactions in topological quantum systems is reviewed with emphasis on the long‐range ‐wave superconductor and its variants. Basic concepts of the lattice model with couplings decaying by power law of distance are introduced to frame the underlying theoretical formalism. This review centers on properties of the long‐range phase that finds no counterpart in the short‐range limit. The breakdown of conformal symmetry in critical regimes and the appearance of massive edge modes differing from conventional Majorana zero modes are discussed. Relevant features including the fractional topological invariant, violations of the area law, and anomalous transport characteristics are also described. Theoretical studies of realistic condensed‐matter realizations such as helical Shiba chains, hybrid nanowires, and other platforms are covered to assess their feasibility under various experimental constraints. Finally, experimental efforts are surveyed where implementations in solid‐state devices and other systems that can simulate effective long‐range topological Hamiltonians are both highlighted.
The phase transition between gapped topological phases represents a class of unconventional criticality beyond the Landau paradigm. However, recent research has shifted attention to topological phases without a bulk gap, where the phase transitions between them are still elusive. In this work, based on large-scale density-matrix renormalization-group techniques, we investigate the critical behaviors of the extended quantum XXZXXZXXZ model obtained by the Kennedy-Tasaki transformation. Using fidelity susceptibility as a diagnostic, we obtain a complete phase diagram, which includes both topological nontrivial and trivial gapless phases. Furthermore, as the XXZXXZXXZ-type anisotropy parameter $\mathrm{\ensuremath{\Delta}}$ varies, both the critical points hc{h}_{c}hc​ and correlation length exponent $\ensuremath{\nu}$ remain the same as in the $\mathrm{\ensuremath{\Delta}}=0$ case, characterized by c=3/2c=3/2c=3/2 (Ising plus free boson) conformal field theory. Our results indicate that fidelity susceptibility can effectively detect and reveal a stable unconventional critical line between the topologically distinct gapless phases for general $\mathrm{\ensuremath{\Delta}}$. This work serves as a valuable reference for further research on phase transitions within the gapless topological phase of matter.
We introduce topological invariants for gapless systems and study the associated boundary phenomena.More generally, the symmetry properties of the low-energy conformal field theory (CFT) provide discrete invariants establishing the notion of symmetry-enriched quantum criticality.The charges of nonlocal scaling operators, or more generally, of symmetry defects, are topological and imply the presence of localized edge modes.We primarily focus on the 1 þ 1d case where the edge has a topological degeneracy, whose finite-size splitting can be exponential or algebraic in system size depending on the involvement of additional gapped sectors.An example of the exponential case is given by tuning the spin-1 Heisenberg chain to a symmetry-breaking Ising phase.An example of the algebraic case arises between the gapped Ising and cluster phases: This symmetry-enriched Ising CFT has an edge mode with finite-size splitting scaling as 1=L 14 .In addition to such new cases, our formalism unifies various examples previously studied in the literature.Similar to gapped symmetry-protected topological phases, a given CFT can split into several distinct symmetry-enriched CFTs.This raises the question of classification, to which we give a partial answer-including a complete characterization of symmetry-enriched 1 þ 1d Ising CFTs.Nontrivial topological invariants can also be constructed in higher dimensions, which we illustrate for a symmetry-enriched 2 þ 1d CFT without gapped sectors.
Recently, the intriguing interplay between topology and quantum criticality has been unveiled in one-dimensional topological chains with extended nearest-neighbor couplings. In these systems, topologically distinct critical phases emerge with localized edge modes despite the vanishing bulk gap. In this work, we study the topological multicritical points at which distinct gapped and critical phases intersect. Specifically, we consider a topological chain with coupling up to the third nearest neighbors, which shows stable localized edge modes at the multicritical points. These points possess only nontrivial gapped and critical phases around them and are also characterized by the quadratic dispersion around the gap-closing points. We characterize the topological multicritical points in terms of the topological invariant obtained from the zeros of the complex function associated with the Hamiltonian. Further, we analyze the nature of zeros in the vicinity of the multicritical points by calculating the discriminants of the associated polynomial. The discriminant uniquely identifies the topological multicritical points and distinguishes them from the trivial ones. Moreover, we identify the underlying physical mechanism in terms of kinetic inversion in higher-order terms. We finally study the robustness of the zero-energy modes at the multicritical points at weak disorder strengths, and reveal the presence of a topologically nontrivial gapless Anderson-localized phase at strong disorder strengths.