Symmetry Protected Bulk-Boundary Correspondence in Interacting Topological Insulators

This paper establishes a quantitative bulk-boundary correspondence in interacting topological insulators by demonstrating that a gauge-invariant many-body winding number, constructed from Pancharatnam geometric phases, uniquely predicts universal entanglement spectrum degeneracies that scale as $4^\nu$ with the winding number $\nu$.

The Gist

Imagine you have a long, magical necklace made of beads. In the world of physics, this necklace represents a material (like a wire or a crystal). Usually, to know if this necklace is "special" (topological), you have to look at the beads right at the very ends. If the ends are loose or wiggly, the necklace is special. If the ends are tight and normal, it's just a regular necklace.

This rule is called the Bulk-Boundary Correspondence. It's like saying: "The secret of the whole necklace is hidden in the knots at the ends."

However, there's a problem. In the real world, beads don't just sit there; they bump into each other, push, and pull (these are called interactions). When beads interact, the old rules break down. The "ends" might disappear, or the "knots" might get messy. Physicists have been struggling to find a new way to tell if the necklace is special when the beads are all jostling around.

This paper is like a new detective kit that solves this mystery. Here is how they did it, using simple analogies:

1. The Problem: The "Ghost" Ends

In a normal, non-interacting necklace, you can count the "ghost ends" (virtual edges) that appear if you cut the necklace in half in your mind.

• The Old Way: They used a "Berry Phase" (think of it as a compass needle that spins as you walk around the necklace). In a simple necklace, the compass points either North (0) or South (180 degrees). This tells you if the necklace is trivial or special.
• The Issue: What if the necklace is so complex that the compass spins 360 degrees, or 720 degrees? The old compass can't tell the difference between spinning once and spinning twice; it just says "North" again. Also, when the beads push each other (interactions), the compass gets confused and stops working.

2. The Solution: A New "Winding" Counter

The authors invented a new tool: a Many-Body Winding Invariant.

• The Analogy: Imagine you are walking around the necklace, but instead of just looking at a compass, you are holding a long piece of string attached to a fixed point on the ground. As you walk, you wrap the string around the necklace.
• The Magic: This new tool counts exactly how many times you wrapped the string.
  • 0 wraps = Normal necklace.
  • 1 wrap = Special necklace (Level 1).
  • 2 wraps = Super-special necklace (Level 2).
• Even when the beads are pushing and pulling (interactions), this string-counting method stays accurate. It doesn't get confused by the chaos.

3. The Proof: The "Entanglement Spectrum" (The Invisible Mirror)

How do we know the necklace is special without cutting it? The authors looked at the Entanglement Spectrum.

• The Analogy: Imagine you take a photo of the necklace, but you only look at the "shadow" it casts when you split the necklace in half mentally. This shadow is the "Entanglement Spectrum."
• The Discovery: They found a perfect pattern in the shadow!
  • If the necklace is normal (0 wraps), the shadow has 1 distinct line.
  • If it's Level 1 special (1 wrap), the shadow has 4 identical lines (a 4-fold degeneracy).
  • If it's Level 2 special (2 wraps), the shadow has 16 identical lines (a 16-fold degeneracy).
• The Rule: The number of identical lines in the shadow is always $4^\nu$ (where $\nu$ is the number of wraps).
• Why it matters: This proves that the "bulk" (the whole necklace) and the "boundary" (the shadow) are perfectly connected, even when the beads are interacting. It's like saying, "No matter how much the beads fight, the shadow always reveals the true number of knots."

4. The Guardian: Inversion Symmetry

What if you throw the necklace into a storm (disorder)? Usually, the magic disappears.

• The Finding: The authors discovered that if the necklace has a specific kind of balance called Inversion Symmetry (it looks the same if you flip it inside out), the magic survives the storm.
• Even if the beads are messy and the necklace is dirty, as long as it can be flipped and still look the same, the "string counter" and the "shadow pattern" remain perfect. This is a huge deal because it means we don't need perfect conditions to find these special materials.

5. The Big Picture: Beyond 1D

Finally, they showed this trick works even if the necklace is actually a 2D sheet or a 3D block (like a "Thouless Pump"). By treating time or a changing parameter as a "fake dimension," they could use their 1D string-counting trick to solve 2D and 3D puzzles.

Summary in One Sentence

The authors found a new, interaction-proof way to count the "knots" in a quantum material by looking at its "shadow," proving that even when particles push and pull each other, the deep connection between the inside of the material and its edges remains unbreakable, as long as the material has a specific type of symmetry.

Why should you care?
This gives scientists a reliable map to find new, exotic materials that could be used for super-fast, error-proof quantum computers. It tells us that even in a messy, interacting world, order and topology can still be found and measured.

Technical Summary

1. Problem Statement

The bulk-boundary correspondence (BBC) is a foundational principle in condensed matter physics, linking quantized bulk topological invariants to protected gapless boundary states. While this correspondence is well-established for non-interacting systems (e.g., the integer quantum Hall effect and topological band insulators), its extension to interacting many-body systems remains a significant challenge.

Key issues addressed in this work include:

• Breakdown of Single-Particle Descriptions: Interactions can invalidate single-particle band theories, rendering standard invariants (like the single-particle winding number or Zak phase) insufficient or ill-defined.
• Limitations of Many-Body Berry Phase (MBBP): While the MBBP generalizes the geometric phase to interacting systems, it is defined modulo $2\pi$. Consequently, it cannot distinguish between topological phases with winding numbers differing by even integers (e.g., $\nu=0$ vs. $\nu=2$).
• Lack of Quantitative BBC in Interacting Regimes: There is no universally applicable formulation that quantitatively links interacting bulk invariants to boundary signatures (such as entanglement spectrum degeneracies) beyond the non-interacting limit.
• Symmetry Protection: It is unclear which symmetries are minimal requirements to protect these topological features against disorder in the presence of interactions.

2. Methodology

The authors employ a combination of analytical theory and numerical exact diagonalization (ED) to study generalized Su–Schrieffer–Heeger (SSH) chains.

• Model System:

  • Base Model: A 1D SSH chain with chiral symmetry, described by intra-cell hopping ($v$) and inter-cell hopping ($w$).
  • Interactions: Density-density interactions ($U$ within unit cells, $V$ between unit cells) are introduced, preserving chiral and inversion symmetries.
  • Generalization: The model is extended to include long-range hopping ($t$) to support higher winding numbers ($\nu > 1$).
  • Disorder: Both generic onsite disorder (breaking all symmetries) and inversion-symmetric disorder are implemented to test robustness.

• Key Diagnostic Tools:

  1. Many-Body Berry Phase (MBBP, $\gamma$): Calculated via adiabatic flux insertion ($2\pi$) with twisted boundary conditions.
  2. Many-Body Winding Invariant ($\nu$): A novel gauge-invariant invariant constructed using Pancharatnam geometric phases. It tracks the relative phase of the ground state $|G(\phi)\rangle$ against a fixed reference state $|G(0)\rangle$ along the flux insertion path. This resolves the $2\pi$ ambiguity of the MBBP.
  3. Entanglement Spectrum (ES): Obtained by bipartitioning the periodic chain into two equal halves. The ES ($\xi_l = -\ln \lambda_l$) acts as a proxy for physical edge states, where characteristic degeneracies reflect topological order.
  4. Symmetry Analysis: The authors systematically break chiral symmetry while preserving inversion symmetry to identify the minimal protecting symmetry.

3. Key Contributions

A. Construction of a Gauge-Invariant Many-Body Winding Invariant

The authors introduce a many-body winding invariant $\nu$ based on Pancharatnam overlaps. Unlike the MBBP, which is defined modulo $2\pi$, this invariant:

• Is constructed by summing the arguments of triple overlaps: $\langle G(0)|G(\phi_j)\rangle \langle G(\phi_j)|G(\phi_{j+1})\rangle \langle G(\phi_{j+1})|G(0)\rangle$.
• Remains quantized to integer values ($\nu \in \mathbb{Z}$) even in the presence of strong interactions.
• Successfully distinguishes phases with winding numbers $\nu = 0, 1, 2$, which are indistinguishable by the standard MBBP.

B. Quantitative Bulk-Boundary Correspondence (BBC)

The paper establishes a precise quantitative relationship between the bulk invariant $\nu$ and the low-lying Entanglement Spectrum (ES) degeneracy:

• Scaling Law: The low-lying ES exhibits a universal degeneracy scaling of $4^\nu$.
  • $\nu = 0$ (Trivial): Non-degenerate ES.
  • $\nu = 1$ (Topological): Fourfold degeneracy ($4^1$).
  • $\nu = 2$ (Higher Winding): Sixteenfold degeneracy ($4^2$).
• This provides a concrete "many-body" formulation of BBC, linking the bulk invariant directly to the structure of the virtual edge modes created by the entanglement cut.

C. Identification of Inversion Symmetry as the Minimal Protector

A critical finding is the role of symmetry in protecting the interacting BBC:

• Chiral Symmetry is Not Essential: The authors demonstrate that the quantization of the invariant and the ES degeneracies persist even when chiral symmetry is broken, provided inversion symmetry is preserved.
• Disorder Robustness:
  • Under generic disorder (breaking inversion symmetry), the MBBP becomes a smooth crossover, and ES degeneracies are lifted.
  • Under inversion-symmetric disorder, the MBBP remains sharply quantized, and the ES degeneracies remain exact.
• This identifies inversion symmetry as the minimal symmetry required to stabilize interacting topological phases in 1D.

D. Extension to Synthetic Dimensions

The framework is shown to extend to higher-dimensional topology via synthetic parameters. By modulating onsite potentials in an SSH chain (realizing a Thouless pump), the authors map the 1D interacting system to an effective 2D Chern insulator. The many-body winding invariant correctly reproduces the Chern number ($C$), and the ES degeneracy scales as $4|C|$.

4. Key Results

1. Interaction-Induced Phase Shifts: Interactions ($U \neq V$) renormalize the effective hopping amplitudes, shifting the topological phase boundaries. However, the BBC (linking $\gamma$ or $\nu$ to ES degeneracy) remains intact.
2. Resolution of Higher Winding Numbers: The Pancharatnam-based invariant successfully resolves the topological structure of generalized SSH models with $\nu=2$, a regime where the standard Berry phase fails.
3. Universal Scaling: Numerical results confirm the $4^\nu$ scaling of ES degeneracy for $\nu=0, 1, 2$ in the interacting regime.
4. Symmetry Protection: The study proves that inversion symmetry alone is sufficient to protect the topological phase and the BBC against disorder, even in the absence of chiral symmetry.
5. Mean-Field Validation: A Hartree-Fock mean-field analysis qualitatively explains the interaction-induced shift of phase boundaries via renormalized hopping parameters, supporting the exact diagonalization results.

5. Significance

This work provides a unified framework for diagnosing interacting topological phases beyond band theory. Its significance lies in:

• Bridging the Gap: It connects geometric-phase invariants (bulk) with entanglement diagnostics (boundary) in a fully interacting many-body context.
• New Diagnostic Tool: The introduction of the Pancharatnam-based winding invariant offers a robust tool for identifying topological phases where single-particle concepts fail.
• Symmetry Insights: It redefines the understanding of symmetry protection in 1D topological insulators, highlighting inversion symmetry as a crucial, previously underappreciated protector in interacting systems.
• Experimental Relevance: By linking bulk invariants to entanglement spectrum features (which can be inferred from correlation matrices or measured in quantum simulators), the paper offers a pathway to experimentally identify interacting topological phases in cold atom systems or quantum simulators.

In summary, the paper establishes a rigorous, symmetry-protected bulk-boundary correspondence for interacting topological insulators, demonstrating that topological order persists and can be quantitatively characterized even in the presence of strong interactions and disorder, provided inversion symmetry is maintained.