Topological properties of gapless phases in an interacting spinful wire

This paper identifies and characterizes two distinct topologically non-trivial gapless phases—the topological Luther-Emery liquid and the topological Mott insulator—at the boundaries of interacting spinful wires, demonstrating that despite lacking a mean-field description or local order parameter, these states can be adiabatically connected to a non-interacting topological metal with winding number $\nu=2$.

The Gist

Imagine a long, one-dimensional wire made of atoms, like a string of beads. In the world of quantum physics, the electrons moving through this wire are like tiny, mischievous dancers. Usually, when we talk about "topological" materials (the kind that might power future quantum computers), we think of them as being very rigid and "gapped." Think of a gapped material as a dance floor where the dancers are stuck in a specific formation; they can't move freely, but they have a special, protected "edge" where a few dancers can wiggle around safely.

This paper explores a much weirder scenario: What happens when the dancers are allowed to move freely (a "gapless" or metallic state), but the system still has those special, protected edge dancers?

Here is the story of their discovery, broken down into simple concepts:

1. The Magic Trick: Separating the Dancers

In these special wires, the electrons have a superpower called Spin-Charge Separation.

• Charge is the electron's "weight" or electricity.
• Spin is its "magnetic personality" (like a tiny arrow pointing up or down).

In normal materials, these two move together like a couple holding hands. But in this wire, they split up! The "Charge" dancers run one way, and the "Spin" dancers run another. Sometimes, one group gets stuck (gapped), while the other keeps dancing freely (gapless).

2. The Two Special "In-Between" States

The researchers found two unique states that exist right on the boundary between two different types of insulating materials. Think of these as the "twilight zones" between two distinct neighborhoods.

A. The Topological Luther-Emery Liquid (The "Spin-Edge" Metal)

Imagine a highway where the electricity (charge) flows freely like traffic, but the magnetic arrows (spin) are frozen in place.

• The Weird Part: Even though the electricity is flowing, the edges of this wire have a special guest. You can add a single electron to the edge without paying any extra energy cost.
• The Metaphor: It's like a VIP lounge at a busy airport. The main terminal is chaotic and full of people, but at the edge, there's a special seat where a guest can sit with fractional spin (imagine a person who is only "half" a person magnetically). This edge state is protected; you can't easily kick them out unless you break the rules of the game (symmetry).

B. The Topological Mott Insulator (The "Charge-Edge" Insulator)

Now, flip the script. Imagine a highway where the magnetic arrows are dancing freely, but the electricity is stuck in traffic (gapped).

• The Weird Part: Even though electricity can't flow through the middle, the edges have a special guest again.
• The Metaphor: This time, the VIP at the edge carries fractional charge (imagine a guest who is only "half" a person electrically). You can add or remove half an electron's worth of charge at the edge without breaking the system.

3. The "Bridge" to the Simple World

Usually, interacting systems (where electrons talk to each other constantly) are a nightmare to solve. They don't have simple "mean-field" descriptions (like a simple map). They are messy and complex.

However, the authors discovered a magic bridge. They showed that you can slowly, smoothly deform these complex, interacting "twilight zones" into a simple, non-interacting metal.

• The Analogy: Imagine a complex, tangled knot of yarn (the interacting phase). The researchers found a way to slowly untangle it without cutting the string, until it becomes a simple, straight line (the non-interacting metal).
• The Result: This proves that these messy, interacting states are actually "cousins" to simple, well-understood topological metals. They belong to the same family.

4. Why Does This Matter?

• Robustness: These edge states are "protected." Just like a topological insulator, you can poke and prod the system, and as long as you don't break the fundamental symmetry (like the rhythm of the dance), these special edge guests stay put.
• New Physics: It shows that "topology" (the study of shapes and connections) isn't just for rigid, frozen materials. It exists in flowing, liquid-like metals too.
• Fractionalization: The idea that you can have "half an electron" or "quarter of a spin" at the edge is a hallmark of exotic quantum matter. This paper confirms that these fractional particles can exist in gapless, conducting wires, not just in insulators.

Summary

The paper is about finding protected VIPs at the edges of a wire where the electrons are running wild. They found two types of VIPs: one that carries a fractional spin and one that carries a fractional charge. Surprisingly, even though the electrons are interacting and messy, these states can be smoothly transformed into a simple, clean metal, proving that the "topology" is a fundamental property that survives even in the chaos of strong interactions.

Technical Summary

1. Problem Statement

The paper addresses a fundamental gap in the classification of topological phases of matter. While the mathematical classification of non-interacting topological phases (both gapped and gapless) is well-established, the classification of interacting systems remains incomplete, particularly for gapless states.

• Context: Topological properties are usually discussed in the context of fully gapped systems (e.g., topological insulators) where edge modes are protected by symmetry.
• Challenge: It is unclear whether protected edge modes exist in interacting gapless systems, especially when there is no local order parameter or mean-field description to characterize the phase.
• Specific Focus: The authors investigate one-dimensional interacting spinful wires exhibiting spin-charge separation. They focus on the gapless boundaries between different $Z_2$ symmetry-breaking gapped phases to determine if these critical lines host topologically non-trivial states.

2. Methodology

The authors employ a combination of bosonization techniques, analysis of edge solitons, and adiabatic continuity arguments.

• Model System: They utilize an effective bosonic Hamiltonian describing 1D spinful electrons with spin-charge separation:
  $$H = H_{LL} + V$$
  Where $H_{LL}$ is the Luttinger liquid part (quadratic in charge $\phi_c$ and spin $\phi_s$ fields) and $V$ contains cosine terms representing dynamically generated gaps:
  $$V = \frac{g_c}{(\pi a)^2} \cos[\sqrt{8\pi}\phi_c] + \frac{g_s}{(\pi a)^2} \cos[\sqrt{8\pi}\phi_s]$$
  The signs of coupling constants $g_c$ and $g_s$ determine four distinct gapped phases (CDW, SDW, SSH+, SSH-), which spontaneously break $Z_2$ symmetry.

• Topological Characterization:

  • Edge Solitons: Instead of relying on local order parameters, the authors analyze the degeneracy of edge solitons in a finite system. They examine how bosonic fields interpolate between bulk values and boundary conditions.
  • Fractionalization: They identify topological phases by the presence of edge states carrying fractional quantum numbers (fractional spin or fractional charge).
  • Adiabatic Continuity: To bridge the gap between interacting and non-interacting descriptions, they introduce a competing single-particle gap term ($\delta t$) and demonstrate that the interacting gapless phases can be adiabatically deformed into a known non-interacting topological metal.

• Non-Interacting Reference: They establish a baseline using two decoupled Su-Schrieffer-Heeger (SSH) chains. A transition between a phase with winding number $\nu=2$ (both chains topological) and $\nu=1$ (one chain trivial) creates a gapless "topological metal" with robust edge modes.

3. Key Contributions and Results

The study identifies two distinct topologically non-trivial gapless phases occurring at the boundaries between gapped symmetry-broken phases:

A. Topological Luther-Emery Liquid

• Location: Occurs at the boundary between the Spin Density Wave (SDW) and SSH- phases ($g_c = 0$, $g_s > 0$).
• Properties:
  • Gapless Sector: Charge excitations are gapless (metallic); spin sector is gapped.
  • Edge Modes: Features edge states carrying fractional spin ($s = \pm 1/4$).
  • Tunneling: Two electrons can tunnel into the bulk without energy cost, but they must form a triplet state (unlike the singlet tunneling in trivial Luther-Emery liquids).
  • Symmetry: Protected by a $Z_4$ symmetry associated with real-space translations by half a lattice constant ($a/2$). This symmetry is broken to $Z_2$ when a charge gap opens.

B. Topological Mott Insulator

• Location: Occurs at the boundary between the Charge Density Wave (CDW) and SSH- phases ($g_s = 0$, $g_c \neq 0$).
• Properties:
  • Gapless Sector: Spin excitations are gapless; charge sector is gapped (insulating).
  • Edge Modes: Features edge states carrying fractional charge ($Q = \pm e/2$). The charge is localized at the edge, while spin is delocalized.
  • Symmetry: Similarly protected by $Z_4$ translation symmetry, reducing to $Z_2$ upon opening a spin gap.

C. Adiabatic Connection to Non-Interacting Metals

A major result is the demonstration that these strongly correlated interacting phases are adiabatically connected to a non-interacting topological metal.

• By adding a single-particle term ($V_{\nu=2} \propto \cos[\sqrt{2\pi}\phi_c]\cos[\sqrt{2\pi}\phi_s]$) that competes with the interaction-induced gaps, the authors construct a path in parameter space.
• At a specific critical point ($g_c = g_s = \delta t/4$), the interacting model maps exactly to a non-interacting system of two decoupled SSH chains with total winding number $\nu=1$ (one chain topological, one trivial).
• This proves that the topological nature of the interacting gapless phases is robust and can be understood via their connection to single-particle topology, despite the absence of a mean-field description.

4. Significance and Implications

1. Extension of Topological Classification: The work extends the concept of symmetry-protected topological (SPT) order to gapless interacting systems. It confirms that "protected edge modes" are not exclusive to gapped systems.
2. New Phases of Matter: It rigorously defines and characterizes the Topological Luther-Emery Liquid and Topological Mott Insulator, distinguishing them from their trivial counterparts by their fractionalized edge excitations and specific tunneling selection rules (triplet vs. singlet).
3. Bridging Interacting and Non-Interacting Physics: The demonstration of adiabatic continuity provides a powerful tool for understanding complex interacting systems. It suggests that certain interacting gapless phases can be effectively described by non-interacting topological metals, offering a new perspective on "topological metals" in the presence of strong correlations.
4. Symmetry Protection: The identification of $Z_4$ translation symmetry as the protector of these phases offers a concrete mechanism for stability in 1D interacting wires, relevant for experimental realizations in quantum wires and cold atom systems.

In summary, the paper establishes that gapless boundaries in interacting 1D spinful systems can host robust topological order characterized by fractionalized edge modes, and these phases are fundamentally linked to non-interacting topological metals through adiabatic deformation.