Characterizing topology at nonzero temperature: Topological invariants and indicators in the extended SSH model

This paper characterizes topology in the Su-Schrieffer-Heeger chain at nonzero temperature by comparing three complementary diagnostics for mixed Gaussian states: the ensemble geometric phase, local twist operators, and a generalized local chiral marker, ultimately proposing practical local indicators that remain robust in the thermodynamic limit.

The Gist

Imagine you are trying to identify a specific type of knot in a piece of rope. At absolute zero temperature (the coldest possible state), the rope is frozen solid. The knot is perfectly rigid, and you can easily see its shape and count its loops. In physics, this "knot" is called topology, and it describes how a material's electrons are arranged in a way that is robust and hard to change.

But what happens when you heat the rope up? It starts to wiggle, vibrate, and shake. The perfect knot gets fuzzy. The electrons are no longer in a single, neat state; they are in a "mixed" state, jiggling around due to heat. The big question this paper asks is: How do we still recognize the knot when the rope is shaking?

The authors, Julia Hannukainen and Nigel Cooper, tested three different ways to find this "knot" in a specific model called the SSH chain (think of it as a simple, one-dimensional necklace of atoms). Here is how they did it, explained simply:

1. The "Global Map" Method (The Ensemble Geometric Phase)

The Idea: Imagine trying to describe the knot by looking at the entire rope at once and measuring a single, giant angle that tells you how the rope twists from start to finish. This is the "Ensemble Geometric Phase."

The Problem: When the rope is cold, this angle is sharp and clear. But as you heat it up, the rope vibrates so much that the signal gets lost in the noise.

• The Analogy: It's like trying to hear a whisper in a crowded stadium. Even if the whisper (the topological phase) is still technically there, the volume of the crowd (the heat) is so loud that the whisper's volume drops to zero.
• The Result: The authors proved that for large systems, this "whisper" disappears completely. You can't measure it anymore because the signal vanishes as the system gets bigger. It's a great theory, but a bad tool for big, hot systems.

2. The "Local Detective" Method (Local Twist Operators)

The Idea: Instead of looking at the whole rope, let's just look at two specific spots: a spot inside a single link of the necklace, and a spot between two links.

• The Analogy: Imagine the necklace has two types of links: "tight" links (trivial phase) and "loose" links (topological phase).
  • In the Trivial Phase, the atoms prefer to hold hands tightly with their immediate neighbor (inside the unit cell).
  • In the Topological Phase, they prefer to hold hands with the neighbor in the next cell (across the boundary).
• The Trick: The authors created two tiny "detectors" (operators) that measure how strong the connection is in these two specific spots.
  • If the inside connection is stronger, it's a Trivial knot.
  • If the outside connection is stronger, it's a Topological knot.
• Why it's great: Even when the rope is shaking (hot), these local connections don't vanish. You only need to measure two tiny spots in the middle of the chain to know the whole story. It's like checking the tension on two specific springs to know if the whole machine is broken, without needing to see the whole machine.

3. The "Flattened Shadow" Method (Local Chiral Marker)

The Idea: This method is a bit more mathematical but very clever. When the rope is hot, the electrons are in a messy, mixed state. However, if the "messiness" isn't too bad (a condition they call a "purity gap"), you can mathematically "flatten" the messy state back into a clean, sharp shadow.

• The Analogy: Imagine a crumpled piece of paper with a drawing on it. If you shine a light on it, the shadow is distorted. But if you know the rules of the distortion, you can mathematically "iron out" the paper in your mind to see the original drawing clearly.
• The Result: Once they "ironed out" the heat, they could calculate a number (the Chiral Marker) that tells them exactly which knot they have. This works even if the system isn't perfectly uniform, making it very robust.

The Big Takeaway

The paper compares these three tools:

1. The Global Map: Fails for big, hot systems because the signal disappears.
2. The Local Detective: Works perfectly! It's simple, requires only local measurements (which is great for experiments with cold atoms), and can even tell you about more complex knots if you add a third detector.
3. The Flattened Shadow: Also works great and gives a precise mathematical number, but requires more complex calculations.

In summary: When things get hot and messy, you can't always look at the whole picture to find the topological "knot." Instead, you should look at the local connections between neighbors. By comparing how strong the "inside" bonds are versus the "outside" bonds, you can instantly tell if the material is topological or not, even in a warm, vibrating world. This opens the door for scientists to detect these exotic states in real-world experiments where absolute zero is impossible to reach.

Technical Summary

1. Problem Statement

Topological phases are traditionally defined for pure quantum states at zero temperature (ground states). However, real-world quantum systems often exist in mixed states due to finite temperatures or dissipation. Characterizing topology in these mixed Gaussian states is challenging because:

• Standard topological invariants (like the Zak phase or winding number) are defined for wavefunctions, not density matrices.
• Existing mixed-state generalizations, such as the Ensemble Geometric Phase (EGP), face a critical practical limitation: while the phase of the expectation value remains well-defined, the modulus of the expectation value decays exponentially with system size at any nonzero temperature. This makes the EGP impossible to measure in large systems.
• There is a need for scalable, experimentally accessible diagnostics that can distinguish between different topological sectors (including those with next-nearest-neighbor hopping) in thermal states, provided the system retains a purity gap (a gap in the spectrum of the effective single-particle operator $G$).

2. Methodology

The authors focus on the Su-Schrieffer-Heeger (SSH) model and its inversion-symmetric extension (including next-nearest-neighbor hopping). They analyze three complementary approaches to characterize topology in mixed states:

1. Ensemble Geometric Phase (EGP): A generalization of the Zak phase defined via the expectation value of a global many-body twist operator $T$.
2. Local Twist Operators: The authors decompose the global twist operator into local operators acting on specific bonds (intracell, intercell, and next-nearest-neighbor). They analyze the expectation values of these local operators to find signatures of the topological phase.
3. Local Chiral Marker: A real-space topological invariant defined using the single-particle correlation matrix. For mixed states with a purity gap, the correlation matrix is "band-flattened" to create an effective projector, allowing the calculation of a winding number-like invariant.

Key Theoretical Framework:

• The system is described by a thermal density matrix $\rho \sim e^{-\beta H}$.
• The authors utilize the purity gap framework: if the effective Hamiltonian $G = \beta H$ has a gap around zero eigenvalues, the state is adiabatically connected to a pure state.
• The correlation matrix $f(G)$ is flattened to a projector $P$ (Eq. 11) to define topological invariants for the mixed state.

3. Key Contributions

A. Analysis of the Ensemble Geometric Phase (EGP)

• Theoretical Proof: The authors analytically demonstrate that while the phase of the EGP converges to the Zak phase of the lowest purity band in the thermodynamic limit, the modulus $|\langle T \rangle|$ vanishes exponentially with system size $N$ at any $T > 0$.
• Implication: The EGP is theoretically sound but practically useless for large systems because the signal-to-noise ratio for the modulus becomes zero, rendering the phase ambiguous in experimental settings.

B. Introduction of Local Twist Operators

To overcome the limitations of the global EGP, the authors introduce local twist operators:

• Intracell ($T^{\text{intra}}_j$): Acts on sites within a unit cell.
• Intercell ($T^{\text{inter}}_j$): Acts on bonds between unit cells.
• Next-Nearest-Neighbor ($T^{\text{nnn}}_j$): Acts on bonds connecting next-nearest neighbors (for the extended model).
• Diagnostic Mechanism: Instead of measuring a global phase, the method relies on comparing the magnitudes of the expectation values of these local operators at the center of the chain.
  • In the trivial phase, the intracell expectation value is larger.
  • In the topological phase, the intercell expectation value is larger.
  • For the extended model, the relative magnitudes of intercell vs. next-nearest-neighbor operators distinguish between topological sectors with winding numbers $\nu = 1$ and $\nu = -1$.
• Scalability: This method requires only local measurements (two or three expectation values), making it robust against system size and suitable for cold-atom quantum gas microscopes.

C. Generalization of the Local Chiral Marker

• The authors extend the local chiral marker to mixed Gaussian states with a purity gap.
• By applying the marker to the band-flattened correlation matrix, they obtain a real-space invariant that coincides with the winding number in the zero-temperature limit.
• This marker successfully distinguishes all three topological sectors ($\nu = 0, \pm 1$) in the extended SSH model at finite temperature, provided the purity gap remains open.

4. Results

• Exponential Decay of EGP: Numerical and analytical results confirm that $|\langle T \rangle| \sim e^{-cN}$ for $T > 0$, validating the impracticality of the global twist operator for large systems.
• Robustness of Local Indicators:
  • The quantity $T = |\langle T^{\text{intra}}_c \rangle| - |\langle T^{\text{inter}}_c \rangle|$ (where $c$ is the center cell) serves as a clear indicator. $T > 0$ indicates the trivial phase, and $T < 0$ indicates the topological phase.
  • This distinction persists at nonzero temperatures, although thermal fluctuations broaden the transition region.
• Extended SSH Model:
  • The local twist operators successfully resolve the three phases: Trivial ($\nu=0$), Topological 1 ($\nu=1$), and Topological 2 ($\nu=-1$).
  • The EGP fails to distinguish $\nu = 1$ from $\nu = -1$ because it is quantized modulo $2\pi$ (both yield $\pi$).
  • The Local Chiral Marker correctly identifies all three sectors ($\nu = 0, 1, -1$).
• Temperature Dependence: As temperature increases, the purity gap decreases, reducing the numerical contrast of the markers. However, as long as the purity gap is non-zero, the topological classification remains valid.

5. Significance

• Experimental Feasibility: The paper provides a practical roadmap for detecting topology in thermal systems. The local twist operator method is particularly significant because it relies on local occupation measurements, which are directly accessible in current cold-atom quantum gas microscope experiments.
• Complementary Diagnostics: The work establishes that while the EGP offers a fundamental theoretical link to polarization, local indicators (twist operators) and real-space invariants (chiral markers) are superior for practical characterization in large, finite-temperature systems.
• Beyond Pure States: It solidifies the framework for classifying topology in mixed states based on the purity gap, demonstrating that topological order can survive thermalization as long as the system is not maximally mixed.
• Resolution of Degeneracy: The proposed methods successfully distinguish between topological phases that are indistinguishable by the Zak phase alone (e.g., $\nu = \pm 1$), offering a complete characterization of the extended SSH model's phase diagram at finite temperature.

In summary, the paper moves the field from theoretical definitions of mixed-state topology to experimentally viable protocols, proving that local measurements can robustly identify topological phases in thermal environments.