Gapless Symmetry-Protected Topological States in Measurement-Only Circuits

This paper investigates gapless symmetry-protected topological (gSPT) states in measurement-only quantum circuits, utilizing large-scale simulations and a Majorana loop model mapping to uncover a symmetry-enriched percolation critical point in Ising cluster circuits and demonstrate a robust gSPT phase in a $\mathbb Z_4$ model.

The Gist

Imagine you are trying to build a complex structure, like a castle, but instead of using bricks and mortar, you are using quantum bits (qubits). Usually, to build a stable castle, you need to follow strict rules of construction (like gravity and physics). But in the quantum world, there's a new, weird way to build: Measurement-Only Circuits.

Think of this like a game where you don't build the castle by placing blocks. Instead, you constantly peek at the blocks to see what they are. Every time you peek (measure), you force the block to "decide" what it is, but you do it in a way that creates a tug-of-war. Some peeks say "be a wall," others say "be a window." This constant, conflicting peeking creates a unique, living state of matter that doesn't exist in the real world.

This paper explores what happens when you play this "peeking game" with a specific goal: creating Gapless Symmetry-Protected Topological (gSPT) states.

Here is the breakdown of their discovery using simple analogies:

1. The Problem: The "Gap" in the Road

In the quantum world, most stable states are like a car driving on a smooth highway with a "gap" (a safety buffer) between the car and the edge of the cliff. If you hit a bump, the car stays on the road.

• Gapless states are like driving on a tightrope. There is no safety buffer. The system is "critical," meaning it's on the edge of chaos. Usually, if you have a tightrope, you can't have a special "edge" feature (like a guardrail) without the whole thing collapsing.
• The Goal: The researchers wanted to see if you could have a tightrope (gapless) that still has a guardrail (topological edge state) that protects it, even though it's supposed to be unstable.

2. The Experiment: The "Peeking" Game

The team simulated two different types of "peeking games" (circuits) on a computer.

Game A: The Ising Cluster (The "Symmetry-Enriched Percolation")

Imagine a line of people holding hands.

• The Rules: Sometimes you ask, "Are you holding hands?" (Measurement A). Sometimes, "Are you holding hands with your neighbor?" (Measurement B).
• The Result: They found a special point where the people are in a constant state of flux (critical), but the people at the very ends of the line are still holding hands in a special, unbreakable way.
• The Discovery: They found a new type of "percolation" (like water soaking through a sponge). Usually, water soaking through a sponge is boring and random. But here, the water soaking through has a secret code (symmetry) that protects the edges. They call this "Symmetry-Enriched Percolation." It's like a sponge that, while soaking, secretly builds a fortress at its edges.

Game B: The Z4 Circuit (The "Steady-State gSPT")

This is a more complex game with two types of people (let's call them Team Alpha and Team Beta) standing in pairs.

• The Rules: You peek at them in different combinations.
• The Result: They found a huge region in the game where the system is in a steady state of chaos (critical), yet it still has robust edge modes.
• The Analogy: Imagine a crowd of people dancing wildly in the middle of a room (chaos/criticality), but the two people standing at the very doors are locked in a perfect, silent dance that never breaks, no matter how wild the crowd gets. Even if you try to mess with the crowd, those two dancers at the door stay protected. This is the gSPT phase.

3. The Secret Weapon: The "Majorana Loop" Map

How did they understand this? They used a mathematical trick called the Majorana Loop Model.

• The Analogy: Imagine trying to understand a tangled ball of yarn. It looks like a mess. But if you look at it from a different angle, you realize it's actually two separate balls of yarn that are just sitting next to each other.
• The researchers mapped their complex quantum circuit onto a model of loops of "Majorana fermions" (a type of quantum particle).
• They realized that in their "gSPT" phase, the system is essentially two copies of a percolation game happening side-by-side. One copy is the "chaos" in the middle, and the other copy is the "protection" at the edges. Because they are linked in a specific way, the chaos in the middle actually protects the edges, rather than destroying them.

4. Why Does This Matter?

• New Physics: We thought "critical" (chaotic) states and "topological" (protected) states were enemies. This paper shows they can be best friends. You can have a system that is constantly changing and critical, yet still has a protected edge.
• Quantum Computers: Building quantum computers is hard because they are fragile. If you can create states that are naturally protected at the edges (even when they are critical), it might help us build more stable quantum memory or processors.
• The "Measurement" Revolution: This proves that you don't need to control every single particle perfectly. You can just "peek" at them in the right way to create exotic, useful states of matter.

Summary

The authors discovered a new way to build quantum matter using only "peeking" (measurements). They found that even in a state of total quantum chaos (criticality), you can have a "guardrail" at the edges that protects the system. They called this Gapless Symmetry-Protected Topological (gSPT) states.

Think of it as a stormy sea (the critical bulk) where the waves are crashing everywhere, but the lighthouse at the edge (the topological mode) remains perfectly still and unbreakable, thanks to a hidden symmetry that links the storm to the light. This is a brand new kind of quantum phase that could be the key to future quantum technologies.

Technical Summary

1. Problem Statement

The paper addresses two primary gaps in the current understanding of quantum many-body physics:

1. Non-Equilibrium Topology: While measurement-only quantum circuits (involving non-commuting projective measurements) have been shown to host distinct quantum phases and entanglement transitions, the existence of Gapless Symmetry-Protected Topological (gSPT) states in these non-equilibrium settings remains largely unexplored.
2. Critical Edge Modes: In equilibrium, gSPT phases are characterized by robust topological edge modes coexisting with critical bulk fluctuations (gapless excitations). It is an open question whether such "symmetry-enriched" criticality can be realized and stabilized in measurement-only circuits, where energy conservation does not apply and dynamics are driven by stochastic measurements.

2. Methodology

The authors employ a combination of large-scale numerical simulations and analytical mapping to investigate two families of 1+1D measurement-only circuits:

• Numerical Simulations:

  • Models: They simulate the Ising cluster circuit (Z2 symmetric) and a Z4 symmetric circuit (intrinsic gSPT model).
  • Protocol: Starting from an initial state (either a product state or a maximally mixed state for purification dynamics), the system evolves via random projective measurements applied with specific probabilities ($p_x, p_{zz}, p_{zxz}$, etc.) over $L$ time steps.
  • Observables: They calculate:
    • Half-chain entanglement entropy ($S_{Half}$): To detect criticality and effective central charges ($c_{eff}$).
    • Generalized topological entanglement entropy ($S_{topo}$): To distinguish topological phases from trivial ones.
    • String Operators: $O_{SPT}$ and $O_{PM}$ to detect topological order and symmetry flux.
    • Edge Magnetization ($M_b$): To identify topological edge modes under open boundary conditions (OBC).
    • Purification Dynamics: To analyze the residue entropy ($S(t \to \infty)$) starting from a maximally mixed state, which serves as a signature of topological edge states.

• Analytical Framework:

  • Majorana Loop Model: The authors map the spin-based measurement circuits to Majorana fermion loop models via the Jordan-Wigner transformation. This provides a unified theoretical framework to understand the pairing configurations of Majorana modes and the resulting phase transitions.

3. Key Contributions and Results

A. Symmetry-Enriched Percolation in Ising Cluster Circuits

• Discovery: The authors identify a new universality class termed "symmetry-enriched percolation" at the transition between the Spontaneous Symmetry Breaking (SSB) and Symmetry-Protected Topological (SPT) phases.
• Critical Properties:
  • The transition occurs at $p_{zz} = p_{zxz} = 1/2$ (with $p_x=0$).
  • It exhibits the critical exponent $\nu = 4/3$ and an effective central charge $c_{eff} \approx \frac{\sqrt{3}\ln 2}{4\pi}$, characteristic of bond percolation.
  • Topological Distinction: Unlike the standard SSB-Paramagnetic (PM) transition (which is also percolation-like), the SSB-SPT transition hosts robust topological edge modes.
    • Evidence: Finite edge magnetization ($M_b$) as boundary fields vanish, and a non-zero residue entropy ($S=1$) in purification dynamics.
    • String operators: The SPT string operator dominates at this critical point, carrying a non-trivial symmetry flux odd under time-reversal, distinguishing it from the PM string operator.
• Significance: This is the first example of a symmetry-enriched non-unitary Conformal Field Theory (CFT), where topological edge states survive at a critical point.

B. Steady-State gSPT Phase in Z4 Circuits

• Model: A Z4-symmetric circuit inspired by the equilibrium intrinsic gSPT model, involving two species of spins ($\sigma, \tau$) per unit cell.
• Phase Diagram:
  • Phase I (gSPT): A robust gapless SPT phase exists over a significant region of the parameter space. It features:
    • Topological edge states (residue entropy $S=1$).
    • Critical bulk fluctuations (power-law string correlations with exponent $\Delta \approx 4/3$).
    • $c_{eff} \approx 2 \times \frac{\sqrt{3}\ln 2}{4\pi}$, indicating the criticality is equivalent to two copies of percolation.
  • Phase II (Trivial Critical): A gapless phase without topological edge states (residue entropy vanishes).
  • Phase III & IV: Symmetry-breaking (SSB) phases.
• Transitions:
  • gSPT $\to$ SSB: Identified as a Berezinskii–Kosterlitz–Thouless (BKT) transition.
  • gSPT $\to$ Trivial Critical: Identified as a percolation transition.
• Robustness: The gSPT phase persists under symmetry-preserving perturbations, demonstrating that topology can be protected even in non-equilibrium steady states.

C. Theoretical Unification via Majorana Loop Models

• The authors map the circuits to Majorana loop models to explain the mechanisms:
  • Ising Cluster: The SSB-SPT critical point corresponds to two decoupled Majorana chains where specific boundary conditions leave two unpaired Majorana modes (edge states) at the ends.
  • Z4 Circuit: The gSPT phase is understood as two decoupled Majorana chains (one critical, one dimerized). The topological edge states arise from the dimerized chain, while the gapless fluctuations come from the critical chain.
  • Perturbations: The paper explains how specific measurements (e.g., single-site $\tau^x$) act as relevant perturbations that gap out the critical modes, destroying the gSPT phase and leading to SSB.

4. Significance

• New Phase of Matter: The work establishes the existence of steady-state gSPT phases in non-equilibrium settings, expanding the landscape of quantum phases beyond equilibrium ground states.
• New Universality Class: The identification of "symmetry-enriched percolation" provides the first concrete example of a non-unitary CFT enriched by symmetry, where topological edge modes coexist with critical bulk fluctuations.
• Platform for Quantum Simulation: The results highlight measurement-only circuits as a powerful platform for realizing exotic quantum critical states that are difficult to achieve in solid-state materials.
• Theoretical Framework: The mapping to Majorana loop models offers a unified and intuitive theoretical tool for analyzing complex measurement-induced phase transitions and topological phenomena in 1+1D systems.

In summary, this paper bridges the gap between topological order and quantum criticality in non-equilibrium systems, demonstrating that measurement-induced dynamics can stabilize robust topological edge modes even within gapless, critical steady states.