Order parameters and ground-state phase diagram of the interacting topological Su-Schrieffer-Heeger model with extended-range hoppings

This paper investigates the interacting Su-Schrieffer-Heeger model with extended-range hoppings, revealing a rich ground-state phase diagram comprising topological, superconducting-like, and charge-density-wave phases, and derives order parameters that demonstrate how interactions enable non-unidirectional hoppings distinct from the non-interacting case.

The Gist

Imagine a long, narrow hallway lined with pairs of lockers. In this hallway, we have tiny, invisible particles (let's call them "dancers") that can jump between these lockers. This setup is known in physics as the Su-Schrieffer-Heeger (SSH) model.

For years, scientists have studied how these dancers move when they are alone or when they only jump to the very next locker. They found that the dancers can form "topological" patterns—special arrangements that are robust and hard to break, kind of like a knot that stays tied even if you wiggle the rope.

However, this new paper asks a more complicated question: What happens if the dancers can jump further away (to lockers two or three steps down the hall) AND if they start interacting with each other (pushing or pulling on one another)?

Here is what the researchers discovered, explained simply:

1. The "Dance Floor" Rules

In the original version of this model, the dancers only jumped to the immediate neighbor, and they didn't really care about each other. The researchers added two new rules:

• Extended Hopping: The dancers can now jump further down the hall.
• Interactions: The dancers have feelings. Sometimes they hate being near each other (repulsion), and sometimes they love being near each other (attraction). Crucially, the "love" or "hate" between dancers in the same pair of lockers might be different from the "love" or "hate" between dancers in neighboring pairs.

2. A New Map of "States of Matter"

When the researchers turned up the volume on these interactions and long jumps, they didn't just find the old patterns. They discovered a rich "phase diagram" (a map of all possible states) containing 10 distinct phases.

Think of these phases as different ways the dancers can arrange themselves on the floor:

• The Topological Dancers: Some groups still form those special, knotted patterns (called winding numbers). Interestingly, the researchers found that even with the dancers pushing and pulling on each other, these special patterns didn't disappear; they just changed their dance moves.
• The Charge Density Waves (CDW): These are like a marching band where the dancers line up in a strict, repeating pattern (e.g., "two dancers here, two dancers there, empty, empty"). The paper found five different types of these marching bands. Two of these new types only appear because of the mix of long jumps and uneven interactions.
• The Phase Separation: In some extreme cases, the dancers get so attracted to each other that they all huddle together in one big pile, leaving the rest of the hallway empty.

3. The "Superconducting-Like" Surprise

The most exciting discovery is a Superconducting-Like (SC-like) phase.

• The Analogy: In real superconductors, electrons pair up (like dance partners) and move without friction. Here, the "dancers" (which are actually spinless fermions, a type of particle) also pair up.
• The Twist: Usually, 1D systems (like a single hallway) can't sustain perfect superconductivity because of quantum rules (the Mermin-Wagner theorem). However, this new phase shows quasi-long-range order.
• What this means: It's like a dance that is almost perfectly coordinated over a long distance. The partners stay in sync for a long time, but eventually, the rhythm drifts slightly. This happens because the dancers are using those "long jumps" and the specific imbalance in their interactions to create this unique pairing.

4. How They Knew What Was Happening (The "Order Parameters")

To figure out which phase the dancers were in, the scientists needed a way to "see" the pattern. In physics, this is called an Order Parameter (OP).

• The Old Way: In the simple, non-interacting version, the OP was like a unidirectional arrow. It only looked at jumps going one way (e.g., left to right).
• The New Discovery: When interactions are added, the dancers stop moving in just one direction. They start jumping back and forth in complex ways. The researchers had to invent new, more complex OPs. These new tools look at a "superposition" of all possible jump directions.
• The Metaphor: Imagine trying to describe a chaotic mosh pit. If you only look at people moving forward, you miss the whole picture. The new OPs look at the entire chaotic swirl of movement to correctly identify the phase.

5. The "Finite-Size" Glitch

The researchers used computer simulations to test this. They found that for some phases (specifically one they call "W1-like"), the results looked different when they simulated a small hallway versus a huge one.

• The Analogy: It's like trying to judge the weather by looking out a small window. In a small room, the air might feel stagnant, but in a large hall, there's a breeze. The "W1-like" phase is so sensitive to the size of the system that it's hard to pin down exactly what it is without a very large simulation. This highlights a limitation in their method: sometimes small models don't tell the whole story.

Summary

This paper is a deep dive into a quantum toy model. By adding long-range jumps and uneven interactions, the authors found that the system becomes much more complex than previously thought. They mapped out 10 different phases, including five new types of ordered patterns and a new "superconducting-like" state where particles pair up in a unique way. They also developed new mathematical tools (Order Parameters) to detect these phases, showing that interactions can actually enhance or modify topological features rather than just destroying them.

Technical Summary

Technical Summary: Order Parameters and Ground-State Phase Diagram of the Interacting Topological Su-Schrieffer-Heeger Model with Extended-Range Hoppings

Problem Statement
The Su-Schrieffer-Heeger (SSH) model is a paradigmatic system for studying topological insulators, characterized by a non-trivial topological invariant (winding number) and protected edge states. While the non-interacting SSH model and its extensions with long-range hoppings (Extended-SSH or ESSH) have been extensively studied, the interplay between electron-electron interactions and extended-range hoppings remains under-explored. A primary challenge in characterizing the ground-state phase diagram of the interacting ESSH model is the absence of well-established order parameters (OPs) that can distinguish between various topological and symmetry-broken phases, particularly when interactions induce complex many-body ground states that deviate from simple non-interacting limits.

Methodology
The authors employ a non-variational, subsystem-independent scheme proposed in Ref. [52] to construct order parameters for the interacting ESSH model. The methodology proceeds as follows:

1. Model Definition: The study focuses on the ESSH Hamiltonian including intra-cell ($U$) and inter-cell ($V$) density-density interactions, alongside nearest-neighbor ($t_a, t_b$) and next-nearest-neighbor ($t_c, t_d$) hoppings.
2. Order Parameter Construction: The authors analyze the dominant Fock states in the many-body ground state. By identifying the generic patterns of particle occupancy (0s and 1s) in these dominant states, they construct operators $\hat{O}$ that yield non-zero expectation values specifically for those phases.
3. Refinement for Interactions: Unlike the non-interacting case where dominant Fock states favor purely unidirectional hoppings, the interacting regime introduces imbalanced interactions that favor non-unidirectional hopping configurations. Consequently, the authors refine the OPs by summing over all possible combinations of local Hermitian conjugates of the hopping terms (e.g., $O'_{Wm} = \sum_{\kappa \in P(\{1,\dots,n\})} O_{n,\kappa}^{Wm}$).
4. Numerical Verification: The derived OPs are verified using:
   • Exact Diagonalization (ED): On small systems ($N=8$) to map out the phase diagram and identify dominant Fock states.
   • Density Matrix Renormalization Group (DMRG): On large systems ($N=90$ to $N=400$) to confirm the stability of phases and the behavior of OPs, entanglement entropy, and fidelity in the thermodynamic limit.

Key Contributions and Results
The paper uncovers a rich phase diagram containing ten distinct phases, including five topological phases, five charge-density-wave (CDW) phases, and two novel superconducting-like (SC-like) phases.

1. Refined Order Parameters for Topological Phases:
   The authors demonstrate that in the interacting regime, the ground state does not strictly favor unidirectional hoppings. They derive refined OPs ($O'_{Wm}$) that include superpositions of all hopping direction combinations. These OPs successfully capture the topological phases ($W_0, W_1, W_2, W_{-1}$) and show that interactions can enhance or preserve topological features rather than solely destroying them.

2. Identification of Novel CDW Phases:
   Five distinct CDW phases are identified:

   • CDW-1A: A topologically trivial Mott insulating phase (period 1).
   • CDW-2A and CDW-2B: Phases with period 2, arising from strong repulsive interactions.
   • CDW-4A and CDW-4B: Novel phases with period 4. These are direct consequences of the interplay between imbalanced interactions ($U \neq V$) and extended-range hoppings ($t_c$ or $t_d$). They do not appear in the nearest-neighbor interacting SSH model.

3. Discovery of a Superconducting-like (SC-like) Phase:
   A novel phase exhibiting quasi-long-range order is discovered in the regime of attractive interactions and specific extended hoppings.

   • Mechanism: The phase is characterized by the pairing of spinless fermions across sublattices, analogous to Cooper pairs. The dominant Fock states involve configurations where pairs of fermions hop between sites in a manner that preserves the total particle number but exhibits pairing correlations.
   • Order Parameter: An SC-like OP ($C_{SC}$) is constructed based on the correlation of pairing operators $\Delta^\dagger_j = c^\dagger_{j+1,A}c^\dagger_{j,B}$.
   • Properties: Consistent with the Mermin-Wagner theorem, the phase exhibits quasi-long-range order with algebraic decay of correlations, rather than true long-range order.

4. Phase Diagram and Finite-Size Effects:
   The study maps the ground-state phase diagram as a function of $U$ and $V$ for various hopping parameters.

   • W1-like Phase: A phase resembling the $W_1$ topological phase but with distinct properties (e.g., ground state degeneracy) is identified. The authors note that this phase suffers from strong finite-size effects, where the ground state degeneracy and fidelity fluctuations persist and even expand as system size increases, making the construction of a perfect OP challenging.
   • Phase Separation (PS): A phase separation region is identified where fermions cluster into high-density domains.

Significance
The paper claims significant contributions to the understanding of interacting topological systems:

• Beyond Symmetry Breaking: It provides a framework for characterizing phases in interacting topological systems where traditional symmetry-breaking order parameters are insufficient.
• Interplay of Interactions and Topology: It challenges the notion that interactions always destroy topological order, showing that specific interaction regimes can enhance topological features or stabilize new topological phases.
• New Physics from Extended Hoppings: The identification of CDW-4A/4B and SC-like phases highlights that extended-range hoppings, when combined with imbalanced interactions, can generate entirely new quantum phases not present in nearest-neighbor models.
• Methodological Insight: The work demonstrates the utility and limitations of constructing OPs from dominant Fock states. While successful for most phases, it exposes a limitation in handling phases with strong finite-size effects and ground state degeneracy (like the $W_1$-like phase), where small-system behavior may not qualitatively converge to the large-system limit.

The authors conclude that their derived OPs provide physical insights into the ground-state configurations and offer a complementary perspective to existing studies (such as those using string order parameters) for understanding the interacting ESSH model.