Beyond Topological Invariants: Order Parameters from Dominant Fock-state Patterns

This paper introduces a general framework for constructing real-space order parameters from dominant Fock-state patterns that reveal hidden sub-structures within topological phases, offering a more refined classification and robust diagnostic tool for both ordered and disordered quantum many-body systems.

The Gist

Imagine you are trying to identify different types of crowds at a massive festival. Traditionally, physicists have used a "topological map" (like a winding number) to tell these crowds apart. This map counts how many times the crowd loops around a central point. If the loop count is different, it's a different crowd. If the loop count is the same, the map says, "These are the same crowd."

However, the authors of this paper argue that the old map isn't detailed enough. They propose a new method: instead of just counting loops, they look at the dominant patterns of the people standing in the crowd. They call these patterns "Fock states" (which is just a fancy way of saying "specific arrangements of particles").

Here is the breakdown of their discovery using simple analogies:

1. The Problem: The "Loop Counter" is Blurry

In the world of quantum physics, there are models (like the Su-Schrieffer-Heeger or SSH model) that describe how particles hop between sites, like people moving between houses on a street.

• The Old Way: Scientists used a "winding number" to classify these states. It's like saying, "If everyone walks in a circle once, it's a Type A crowd. If they walk in a circle twice, it's a Type B crowd."
• The Issue: The authors found that sometimes, two crowds have the exact same winding number (they both walk in a circle once), but they are actually fundamentally different. The old map couldn't tell them apart. It was like trying to distinguish between a jazz band and a rock band just by counting how many times they played a C-note.

2. The Solution: Looking at the "Dominant Patterns"

The authors suggest a new strategy: Look at the most common arrangements.

• Imagine a crowd of people. Even though everyone is moving, there are certain "dominant patterns" of who is standing next to whom that happen most often.
• The authors say: "Let's ignore the rare, chaotic arrangements and focus only on the most frequent, dominant patterns."
• By analyzing these dominant patterns, they can build a new "Order Parameter" (a measuring tool). Think of this as a specialized detector that only "beeps" loudly if it sees a specific, dominant pattern.

3. The Discovery: Hidden Sub-Structures

When they applied this new detector to the SSH model, they found something surprising:

• The Split: Every "winding number" phase (every loop count) actually splits into two distinct sub-phases.
• The Analogy: Imagine the "Type A" crowd (one loop). The old map said it was all one group. The new map reveals that inside that group, there are actually two distinct tribes: the "Electron-like" tribe and the "Hole-like" tribe. They look similar from a distance (same loop count), but their internal seating arrangements are different.
• The "Depth" Meter: The new tool doesn't just say "You are in Phase A." It also tells you how deep you are in that phase. If the signal is strong, you are in the heart of the phase. If it's weak, you are near the edge or transitioning.

4. Testing the Tool: Disorder and Chaos

The authors tested their new tool in two difficult scenarios:

• The Messy Room (Disordered Systems): They added "disorder" (randomness) to the model, like throwing random obstacles into the street. The old tools struggled here, but their new pattern-based detector remained robust. It could still identify the phases even when the system was messy.
• The BKT Transition: They also tested it on a different model (the XXZ spin model) involving a very tricky type of phase transition called Berezinskii–Kosterlitz–Thouless (BKT). This is notoriously hard to spot in small systems. Their new tool successfully identified this transition, acting like a precise diagnostic tool for a condition that was previously hard to diagnose.

5. How It Works (The "Recipe")

The paper doesn't just guess these patterns; they have a recipe:

1. Find the Ground State: Look at the most stable, lowest-energy state of the system.
2. Identify the Heavyweights: Find the specific arrangements (Fock states) that appear most frequently (have the highest "weight").
3. Spot the Pattern: Look for a repeating rule in these heavyweights (e.g., "In this phase, Site A is always empty while Site B is always full").
4. Build the Detector: Create a mathematical operator (a measuring tool) that specifically looks for that rule. If the rule is present, the tool gives a big number. If not, it gives zero.

Summary

In short, the authors moved beyond the "big picture" topological maps (which count loops) to a "microscopic" view that counts specific, dominant arrangements of particles. This new approach revealed that the quantum world is more nuanced than previously thought, splitting known phases into hidden sub-groups and providing a sharper, more robust way to detect phase transitions, even in messy, disordered environments.

Technical Summary

Technical Summary: Beyond Topological Invariants: Order Parameters from Dominant Fock-state Patterns

Problem Statement
The construction of appropriate order parameters (OPs) is a fundamental challenge in many-body physics. While topological phases are traditionally characterized by non-local invariants (e.g., winding numbers), these invariants often fail to provide a refined classification of phases, particularly in systems with long-range hoppings or disorder. Existing methods for constructing OPs, such as variational approaches detecting quasi-degenerate states or analyzing reduced density matrix spectra, rely heavily on physical intuition or require the selection of suitable subsystems, which can be ambiguous in systems with long-range couplings. Furthermore, standard topological invariants may not distinguish between distinct phases sharing the same topological index, nor do they quantify the "depth" of a phase or robustly characterize transitions in disordered systems.

Methodology
The authors propose a general scheme for constructing order parameters by extracting generic patterns from the dominant Fock states of many-body ground states (GS). The core idea is to identify the Fock states $|n_1 n_2 \dots n_L\rangle$ that possess the largest weights $|\xi_{n_1 \dots n_L}|^2$ in the GS expansion $|GS\rangle = \sum \xi |n_1 \dots n_L\rangle$. By analyzing the occupancy patterns of these dominant states, the authors infer the form of an operator $\hat{O}$ such that $\langle GS | \hat{O} | GS \rangle$ yields a non-zero value within its specific phase and vanishes (or approaches zero) in other phases.

The scheme involves:

1. Dominant State Analysis: Identifying the specific configurations of site occupancies (0s and 1s for fermions, spin states for spins) that dominate the GS wavefunction.
2. Operator Construction: Formulating $\hat{O}$ as a product of creation and annihilation operators (for fermions) or spin operators (for spins) that selectively act on these dominant configurations.
3. Generalization: Extending the operator to include $n$ hopping terms or spin correlations, where $n$ is optimized (typically $n \approx N/2$) to maximize the distinction between phases.

Key Contributions and Results

• Refined Classification in the Extended Su-Schrieffer-Heeger (ESSH) Model:
  The authors apply their scheme to the ESSH model, which includes hopping beyond nearest-neighbors ($t_a, t_b, t_c, t_d$). They demonstrate that the standard winding number $W$ is insufficient to distinguish all phases. Specifically, they identify that each winding number sector $W_m$ splits into two distinct phases, denoted as $W_m^e$ (electron-like) and $W_m^p$ (hole-like).

  • Discovery of Hidden Sub-structure: Using fidelity analysis, they show that ground states from $W_m^e$ and $W_m^p$ have zero overlap (fidelity) in the thermodynamic limit, confirming they are distinct phases despite sharing the same winding number.
  • Real-Space Patterns: Analysis of dominant Fock states reveals that $W_m^e$ and $W_m^p$ phases are characterized by specific real-space pairings of 0s and 1s corresponding to specific hopping terms (e.g., $t_d$ hopping for $W_2$).
  • New Order Parameters: The authors construct explicit OPs ($O_{W_m}$) based on these patterns. These operators not only distinguish between $W_m^e$ and $W_m^p$ via their sign but also quantify the "depth" of the phase via their magnitude, a feature absent in the winding number.

• Robustness in Disordered Systems:
  The scheme is applied to a disordered SSH model with site-dependent hopping parameters. The constructed OPs successfully identify phase boundaries consistent with previous studies using non-commutative winding numbers but provide additional information regarding the phase depth. The OPs remain robust, attaining significant finite values in their respective phases and vanishing elsewhere, even in the presence of disorder.

• Application to the Interacting Spin-1/2 XXZ Model:
  The framework is extended to the interacting XXZ model, which exhibits ferromagnetic (FM), XY, and anti-ferromagnetic (AFM) phases, including a Berezinskii–Kosterlitz–Thouless (BKT) transition at $\Delta=1$.

  • Diagnosing BKT Transition: The authors derive a class of OPs ($O_{FM,n}, O_{AFM,n}, O_{XY,n}$) involving products of spin operators. These OPs correctly identify the phase boundaries, including the difficult-to-detect BKT transition in finite-size systems, by attaining maximum values in their respective phases and vanishing in others.

Significance and Claims
The paper claims to offer a broadly applicable, practical tool for uncovering phase diagrams in diverse interacting and non-interacting quantum many-body systems. The significance of the work lies in:

1. Moving Beyond Topological Invariants: Providing a real-space classification that reveals hidden sub-structures (like the $e/p$ splitting) that topological invariants miss.
2. Physical Intuition: The method directly links the order parameter to the physical configuration of dominant Fock states, offering immediate physical insight into the electronic or spin configurations of a phase.
3. Versatility and Robustness: The scheme does not rely on symmetry breaking (unlike Landau theory) and avoids the ambiguity of choosing subsystems required by reduced density matrix methods. It remains effective in disordered settings and for finite-size diagnostics of critical transitions like BKT.
4. Quantitative Depth: Unlike binary topological invariants, the magnitude of the proposed OPs provides a quantitative measure of how "deep" a system is within a specific phase.

The authors note that while the OPs successfully distinguish the $W_m^e$ and $W_m^p$ phases, the full physical interpretation of this distinction remains a subject for future work. The framework is presented as a general diagnostic tool rather than a specific proposal for new experimental platforms, though it aligns with experimental interests in long-range coupled systems.