🔬 mesoscale physics

Topologically nontrivial multicritical points

This paper investigates topologically nontrivial multicritical points in one-dimensional chains with third-nearest-neighbor couplings, characterizing their stable localized edge modes and quadratic dispersion through topological invariants and discriminant analysis while revealing a gapless Anderson-localized phase under strong disorder.

Original authors: Ranjith R Kumar, Pasquale Marra

Published 2026-07-08

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Ranjith R Kumar, Pasquale Marra

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a long, one-dimensional chain of atoms, like a string of beads. In the world of quantum physics, these chains can act like special "insulators" or "superconductors." Usually, these materials have a clear rule: if the inside of the chain is "safe" (has an energy gap), the ends might have special, protected particles called edge modes. If the inside becomes "unsafe" or "gapless" (like a critical point where the energy gap closes), the old rules say those special particles at the ends should disappear and spread out into the middle of the chain.

This paper explores a fascinating exception to that rule. The authors found a way to create a "multicritical point"—a very specific, rare spot in the system's settings where different phases of matter meet. At this specific spot, even though the inside of the chain is "gapless" (usually a sign of chaos), the special particles at the ends stay put. They don't disappear; they remain locked to the edges, protected by the topology of the system.

Here is a breakdown of their discovery using simple analogies:

1. The Map of the Chain (The Phase Diagram)

Think of the system's settings (how strongly the atoms interact with their neighbors) as a map.

Gapped Phases: These are like solid, stable islands on the map. Depending on the settings, you can have islands with 1, 2, or 3 "guardians" (edge modes) at each end of the chain.
Critical Lines: These are the rivers separating the islands. Usually, when you cross a river, the guardians vanish because the water (the gap) is too high.
Multicritical Points: These are rare intersections where two or more rivers meet. In most previous studies, these intersections were "trivial"—meaning if you stood there, the guardians would dissolve into the water.

2. The Big Discovery: The "Non-Trivial" Intersection

The authors found a specific region on their map where every single island and every river is special (non-trivial). There are no "boring" or "trivial" islands where the guardians don't exist.

The Result: At the intersection of these special rivers (the multicritical points), the guardians do not dissolve. Instead, they stay locked at the ends of the chain, even though the middle of the chain is gapless.
The Analogy: Imagine a busy intersection in a city where traffic usually stops and cars scatter. But in this specific city, the traffic lights are rigged so that even at the busiest intersection, two specific cars are magically forced to stay parked at the curb, no matter how chaotic the intersection gets.

3. How They Proved It (The "Discriminant" Test)

To prove these points were truly special and not just a fluke, the authors used a mathematical tool called a discriminant.

Think of the system's behavior as a polynomial equation (a math puzzle with roots).
In "trivial" intersections (where guardians disappear), the solution to this puzzle stays positive on both sides of the intersection.
In their "non-trivial" discovery, the solution flips sign (goes from positive to negative) right at the intersection. This sign flip is the unique fingerprint that proves the intersection is topologically protected.

4. The "Kinetic Inversion" Mechanism

Why does this happen? The authors explain it using a concept called kinetic inversion.

Imagine the energy of the particles as a ball rolling on a hill. Usually, the shape of the hill determines if the ball rolls away or stays put.
At these special points, the "hill" changes shape in a very specific way (it becomes quadratic, like a bowl rather than a slope). This shape change acts like a trap, forcing the edge particles to stay localized even when the rest of the system is unstable.

5. What Happens with Disorder (Messy Conditions)

Real-world systems are never perfect; they have "disorder" (randomness or impurities).

Weak Disorder: If you add a little bit of mess to the chain, these special edge guardians remain safe. They are robust.
Strong Disorder: If the mess gets too strong, the special intersection points eventually vanish. However, something surprising happens: the whole chain turns into a gapless, disordered phase that is still topologically non-trivial. It's like a chaotic storm where, despite the chaos, two guardians still manage to stay locked at the ends of the chain.

Summary

The paper claims that by carefully tuning a quantum chain with long-range connections, you can create a "sweet spot" (a multicritical point) where the usual rules of physics are bent. At this spot, the system is unstable in the middle, yet the edges remain perfectly protected. This happens because the system is forced to choose between different "special" states, leaving no room for a "boring" state where the protection would fail.

The authors suggest this could be useful for topological quantum computing, where these stable edge modes could carry quantum information without losing it (decoherence) during phase transitions. They note this could be built using superconducting circuits, cold atoms, or quantum walks.

Technical Summary: Topologically Nontrivial Multicritical Points

Problem Statement
The interplay between topology and quantum criticality in one-dimensional (1D) systems has recently revealed that topologically distinct critical phases can host localized edge modes even when the bulk gap vanishes. Conventionally, the bulk-boundary correspondence dictates that at a quantum critical point (where the bulk gap closes), edge modes delocalize into the bulk, rendering the point topologically trivial. While previous studies identified that critical points separating nontrivial gapped phases can host edge modes, it was generally understood that at multicritical points—where distinct critical lines intersect—the localization length of edge modes diverges, causing them to delocalize and the point to become topologically trivial.

This work addresses the question of whether multicritical points can remain topologically nontrivial under specific constraints. Specifically, the authors investigate a regime where the parameter space hosts only topologically nontrivial phases (both gapped and critical), excluding any trivial ( $w=0$ ) phases. They aim to determine if, in such a "fully nontrivial" parameter space, localized edge modes can persist at the intersection of critical lines (multicritical points).

Methodology
The authors study a generalized 1D topological chain model, encompassing both the Su–Schrieffer–Heeger (SSH) model (topological insulator) and the Kitaev chain (topological superconductor), extended to include hopping and pairing terms up to the third nearest neighbors. The system is described by a two-band Bloch Hamiltonian $H(\Gamma, k) = \chi(\Gamma, k) \cdot \sigma$ , where the components $\chi_x$ and $\chi_y$ are trigonometric polynomials involving parameters $\Gamma_0, \Gamma_1, \Gamma_2, \Gamma_3$ (representing onsite potentials and hopping/pairing amplitudes).

The analysis employs several complementary approaches:

Topological Invariants: For gapped phases, the winding number $w$ is calculated via the curvature function of the bulk Bloch states. For critical (gapless) phases, where the conventional winding number is ill-defined, the authors utilize an alternative invariant $w_c$ derived from the zeros and poles of a complex function $f(\zeta)$ associated with the Hamiltonian (where $\zeta = e^{ik}$ ). $w_c$ is defined as the number of zeros of $f(\zeta)$ inside the unit circle.
Dispersion Analysis: The nature of the gap closing at critical and multicritical points is analyzed by calculating the dynamical critical exponent $z$ from the energy dispersion $E \propto k^z$ .
Complex Plane Analysis: The behavior of the roots (zeros) of the cubic polynomial $f(\zeta)$ is tracked as parameters vary. The authors analyze the discriminant ( $\Delta$ ) of this polynomial to distinguish between real and complex conjugate roots and to identify degeneracies.
Numerical Simulations: Under open boundary conditions, the authors compute eigenvalue distributions, probability densities of wavefunctions, and localization lengths. They also study the effects of disorder by introducing random variations in the hopping parameters and calculating the Inverse Participation Ratio (IPR) and finite-size scaling of eigenvalues.

Key Contributions and Results

Identification of Topologically Nontrivial Multicritical Points: The authors demonstrate that by fixing parameters such that the trivial gapped phase ( $w=0$ ) is eliminated (specifically when $\Gamma_0 \leq \Gamma_2/3$ ), the resulting phase diagram contains only nontrivial gapped phases ( $w=1, 2, 3$ ) and nontrivial critical phases. In this regime, the intersection of critical lines (multicritical points $M_1$ and $M_2$ ) hosts stable, localized edge modes. This contradicts the conventional expectation that edge modes delocalize at multicritical points.
Characterization via Complex Zeros and Discriminants:
- At the multicritical points, the complex function $f(\zeta)$ possesses one zero inside the unit circle and two degenerate zeros on the unit circle. This yields a topological invariant $w_{mc} = 1$ , corresponding to one localized edge mode at each end of the chain.
- The authors introduce the discriminant $\Delta$ of the cubic polynomial as a unique identifier for these points. At the topological multicritical points, $\Delta = 0$ , and the sign of $\Delta$ flips as one crosses the point along high-symmetry critical lines (transitioning between $w_c=2$ and $w_c=1$ phases). This sign-flip behavior is absent in trivial multicritical points (where a $w=0$ phase exists), where $\Delta$ remains positive on both sides.
Quadratic Dispersion and Kinetic Inversion: The multicritical points are characterized by a quadratic dispersion ( $z=2$ ) at the gap-closing momenta, distinct from the linear ( $z=1$ ) dispersion of the surrounding critical phases. The authors attribute the emergence of topological protection at these points to a "kinetic inversion" mechanism in higher-order momentum terms of the effective Dirac equation. Specifically, the coefficient of the quadratic term ( $\alpha_1$ ) acts as a mass term; its sign determines whether the multicritical point is trivial ( $\alpha_1 < 0$ ) or nontrivial ( $\alpha_1 > 0$ ).
Robustness and Disorder Effects:
- Weak Disorder: The topological multicritical points are robust against weak disorder, though their positions in the parameter space shift.
- Strong Disorder: At strong disorder strengths, the gapped phases and critical lines vanish, leading to a gapless phase. However, this phase is identified as a topologically nontrivial Anderson-localized phase. It is characterized by the presence of two zero-energy modes localized at the chain ends, confirmed by non-zero IPR values and exponential finite-size scaling of the edge mode energies.

Significance
The paper claims that the existence of topologically nontrivial multicritical points is contingent upon the parameter space being entirely nontrivial (i.e., devoid of $w=0$ phases). This finding revises the understanding of multicritical behavior, showing that edge modes do not necessarily delocalize at these points if the system is constrained to a regime where it cannot be perturbed into a trivial state.

The work provides a rigorous characterization of these points using the discriminant of the associated polynomial, offering a tool to distinguish topological from trivial multicriticalities. Furthermore, the discovery of a gapless, topologically nontrivial Anderson-localized phase at strong disorder suggests new avenues for topological protection in disordered systems. The authors suggest that these unique features, particularly the persistence of localized edge modes at multicritical points without delocalization, could be relevant for topological quantum computing, where protected edge modes carry quantum information. The proposed models can be realized in various physical platforms, including superconducting circuits, ultracold atoms, and quantum walks, which support long-range interactions.