Topological phase transition of deformed ${\mathbb Z}_3$ toric code

This paper investigates the topological phase transitions of a deformed $\mathbb{Z}_3$ toric code by mapping its wavefunction norm to classical partition functions, revealing a rich phase diagram with three distinct phases separated by critical lines characterized by $\mathbb{Z}_3$ and $\mathbb{Z}_4$ parafermion conformal field theories, as well as isolated critical points exhibiting Hilbert space fragmentation and quantum many-body scars.

The Gist

Here is an explanation of the paper "Topological phase transition of deformed Z3 toric code," translated into simple, everyday language using creative analogies.

The Big Picture: A Quantum Puzzle Game

Imagine you have a giant, magical puzzle board made of tiny tiles. This isn't just any puzzle; it's a Quantum Toric Code. Think of it as a super-stable, invisible net that holds information together. In the world of quantum computers, this net is special because it can protect data from errors, much like a fortress protects a castle.

Usually, scientists study this net using simple "on/off" switches (like a light switch). But in this paper, the researchers decided to upgrade the game. Instead of just "on" and "off," they introduced three states (like a traffic light: Red, Yellow, Green). This is the $Z_3$ Toric Code.

The goal of the paper is to see what happens when you start "deforming" or stretching this quantum net. Does it stay strong? Does it break? Or does it turn into something completely new?

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The Setup: The "Cluster" and the "Measurement"

To build this quantum net, the researchers start with a Cluster State.

• The Analogy: Imagine a room full of people (the particles) holding hands in a specific pattern. They are all perfectly synchronized, holding a "dance pose."
• The Process: To turn this dance into the "Toric Code" (the protective net), you tell everyone to stop dancing and take a snapshot (a measurement).
• The Result: If everyone was dancing perfectly, the snapshot reveals a beautiful, closed-loop pattern. It's like looking at a photo of a dance and seeing that everyone formed perfect circles. These circles are the "loops" that protect the quantum information.

The Experiment: "Deforming" the Dance

The researchers asked: What if we don't let the dancers start in the perfect pose?

They introduced a deformation. Imagine telling the dancers to tilt their bodies slightly or change their rhythm before the snapshot is taken.

• The Parameter ($\beta$): Think of this as a "tilt knob."
  • Knob at 0: Perfect dance, perfect net (The Toric Code).
  • Knob turned up: The dancers are messy. The loops get distorted.

They tested two ways to turn this knob:

1. The $\beta_z$ Deformation (The "Weight" Knob): This changes how heavy or likely certain dance moves are. Some loops become "expensive" to make, so they disappear.
2. The $\beta_x$ Deformation (The "Confusion" Knob): This makes the dancers' poses overlap. It's hard to tell if a dancer is doing a "Red" move or a "Green" move. The boundaries between states get blurry.

The Discovery: Three Different Worlds

As they turned these knobs, they found the quantum net didn't just break; it transformed into three distinct "phases" (like water turning into ice, steam, or liquid).

1. The Topological Phase (The Fortress)

• What it is: The original, perfect net.
• The Analogy: A sturdy fortress with high walls. Inside, the "guards" (particles) are free to move around without getting stuck. The information is safe.
• Status: Deconfined. The particles can roam freely.

2. The Confined Phase (The Prison)

• What it is: The loops get too heavy (due to the $\beta_z$ knob).
• The Analogy: Imagine the dance floor is covered in sticky glue. The dancers can't move far without getting stuck. The loops shrink and snap.
• Status: Confined. The particles are trapped. They can't move freely, and the "fortress" walls collapse.

3. The Condensed Phase (The Blur)

• What it is: The loops get too blurry (due to the $\beta_x$ knob).
• The Analogy: The dancers are moving so fast and overlapping so much that you can't tell where one ends and another begins. The distinct loops dissolve into a fog.
• Status: Condensed. The particles lose their individual identity and merge into a single, messy state.

The Magic Moments: Critical Points

Between these three worlds, there are "borderlands" where the rules of physics change. The researchers found some very special spots on the map:

• The "Ice" Point (Square Ice): At a specific extreme tilt, the system turns into something called Square Ice.
  • The Analogy: Imagine a grid of arrows. In this state, every intersection must have exactly two arrows pointing in and two pointing out (like a traffic intersection with perfect flow).
  • The Surprise: This state has a hidden symmetry. It creates "Scar States"—special patterns that are so rigid they can't be disturbed. It's like a dance routine that, once started, cannot be interrupted, no matter what. This leads to "Hilbert Space Fragmentation," which is a fancy way of saying the universe splits into separate, non-communicating rooms.

The New Map: The Ashkin-Teller Model

For the $Z_2$ (two-state) version of this code, scientists already knew the map. But for the $Z_3$ (three-state) version, the map is much more complex.

• The Old Map (Z2): Had a simple symmetry. If you flipped the switch, the map looked the same.
• The New Map (Z3): The researchers discovered there is no simple flip symmetry. The map is lopsided and richer.
• The Tool: To draw this new map, they invented a new mathematical tool called the AT3 Model (a $Z_3$ version of the Ashkin-Teller model). Think of this as a new type of board game that combines two different games into one, allowing for more complex strategies and outcomes.

Why Does This Matter?

1. Better Quantum Computers: Understanding how these nets break and reform helps us build quantum computers that are less likely to crash.
2. New Physics: They found that the $Z_3$ code behaves differently than the $Z_2$ code. It's not just a bigger version; it's a different kind of physics.
3. The "Scar" Mystery: The discovery of these rigid "Scar States" at the Ice point is exciting. It suggests there are quantum systems that refuse to "thermalize" (settle down), which could be useful for storing information for a long time.

Summary in One Sentence

The researchers took a quantum safety net made of three-state particles, twisted it in different ways, and discovered it can turn into a prison, a fog, or a rigid ice-grid, revealing a much more complex and beautiful world of quantum phases than we saw with simple two-state systems.

Technical Summary

Here is a detailed technical summary of the paper "Topological phase transition of deformed Z3 toric code."

1. Problem Statement

The paper investigates the topological phase transitions of the deformed $Z_3$ toric code. While the phase diagram of the deformed $Z_2$ toric code (qubits) is well-understood and mapped to the Ashkin-Teller model, the behavior of higher-dimensional qudits (specifically qutrits, $N=3$) remains less explored. The authors aim to:

• Construct a deformed $Z_3$ toric code state via measurement-based quantum computation (MBQC) protocols starting from a $Z_3$ cluster state.
• Map the wavefunction norm of this deformed state to classical statistical mechanical models.
• Determine the phase diagram, identifying distinct topological phases (toric code, confined, and condensed) and the nature of the critical points separating them.
• Explore how the absence of specific dualities in $Z_3$ (compared to $Z_2$) alters the phase structure.

2. Methodology

The authors employ a multi-faceted approach combining quantum information theory, statistical mechanics, and tensor network methods:

• Construction via MBQC:

  • They start with a $Z_3$ cluster state defined on a Lieb lattice.
  • Deformation: The local qutrit basis states on the links are rotated (parameterized by $\theta$ or $\beta$) before performing forced projective measurements on vertex qutrits. This generates a deformed toric code state $|\tilde{\Psi}\rangle$.
  • Dualities: They utilize the electric-magnetic duality ($D_{e-m}$) to relate deformations in the $X$-basis ($\beta_x$) to those in the $Z$-basis ($\beta_z$).

• Loop-Gas and Net Configuration Framework:

  • The ground state is represented as a superposition of closed loop configurations.
  • The norm of the deformed wavefunction, $\langle \tilde{\Psi} | \tilde{\Psi} \rangle$, is analyzed by examining overlaps between loop configurations.
  • Single-Parameter Deformations:
    • For $\beta_z$ deformation, the norm maps to the partition function of the $Q=3$ Potts model.
    • For $\beta_x$ deformation, the analysis involves "discrepancy lines" (overlaps of different loop configurations), also mapping to a $Q=3$ Potts model but with different physical interpretations regarding anyon condensation.
  • Two-Parameter Deformation: The general case ($\beta_x, \beta_z$) maps to a novel classical model, the $Z_3$ Ashkin-Teller model (AT3), involving two coupled $Z_3$ spin variables per site.

• Tensor Network Analysis:

  • The authors use Projected Entangled Pair States (PEPS) to represent the wavefunction.
  • They employ the Variational Uniform Matrix Product State (VUMPS) method to numerically compute the fixed-point eigenstates of the transfer matrix in the thermodynamic limit.
  • Order parameters derived from the transfer matrix symmetries are used to distinguish phases.

• Criticality Analysis:

  • Central charges ($c$) are extracted by fitting the entanglement entropy ($S$) versus the correlation length ($\xi$) using the Calabrese-Cardy formula ($S \sim \frac{c}{6} \ln \xi$).

3. Key Contributions

• Generalization to $Z_3$: Successfully extends the deformed toric code framework from qubits ($Z_2$) to qutrits ($Z_3$), revealing a richer phase structure due to the lack of sign-change duality present in $Z_2$.
• Discovery of the AT3 Model: Identifies a new classical statistical model, the $Z_3$ Ashkin-Teller model (AT3), as the exact mapping for the norm of the general deformed $Z_3$ toric code.
• Identification of Critical Points: Precisely locates and characterizes critical lines and isolated points with distinct conformal field theories (CFTs):
  • $c = 4/5$ ($Z_3$ parafermion CFT).
  • $c = 8/5$ (merged critical line).
  • $c = 1$ ($Z_4$ parafermion CFT at isolated antiferromagnetic points).
• Hilbert Space Fragmentation: Demonstrates that at specific critical limits (square ice point), the system exhibits an emergent $U(1)$ 1-form symmetry, leading to Hilbert space fragmentation and the existence of quantum many-body scar states.

4. Results

Phase Diagram

The phase diagram in the $(\beta_x, \beta_z)$ plane (or variational parameters $t_x, t_z$) reveals three distinct phases:

1. Toric Code (TC) Phase: The topologically ordered phase where $e$-anyons are deconfined and not condensed. Corresponds to the disordered phase of the classical AT3 model.
2. $e$-Confined Phase: $e$-anyons are confined (loops are suppressed). Corresponds to the fully ordered ferromagnetic phase of the AT3 model.
3. $e$-Condensed Phase: $e$-anyons are condensed (discrepancy lines proliferate). Corresponds to the paramagnetic phase of the AT3 model.

Critical Lines and Points

• TC to Confined/Condensed Boundaries: Separated by critical lines governed by the $Z_3$ parafermion CFT with central charge $c = 4/5$. This corresponds to the phase transition of the $Q=3$ Potts model.
• Confined to Condensed Boundary: A merged critical line separating the confined and condensed phases exhibits a central charge of $c = 8/5$.
• Isolated Antiferromagnetic (AFM) Points: Located at $(t_x, t_z) = (-1, 0)$ and $(0, -1)$. These points correspond to the $T=0$ limit of the AFM $Q=3$ Potts model and are described by a $Z_4$ parafermion CFT with central charge $c = 1$.

Square Ice and Scar States

At the critical point where $\beta_z \to -\infty$ (or $\beta_x \to -\infty$ under duality), the system reduces to a square ice model (fully packed loops).

• Emergent Symmetry: The system acquires an emergent $U(1)$ 1-form symmetry.
• Hilbert Space Fragmentation: This symmetry fragments the Hilbert space into disconnected sectors.
• Quantum Many-Body Scars: The system hosts "scar" states (configurations with no flippable plaquettes) whose number grows exponentially with system size ($2^{L+2}-4$). These states violate the Eigenstate Thermalization Hypothesis (ETH).

Comparison with $Z_2$

Unlike the $Z_2$ case, the $Z_3$ model lacks a sign-change duality (arising from $\{X, Z\}=0$ in $Z_2$). Consequently, the $Z_3$ phase diagram is not symmetric about the axes $t_x=0$ or $t_z=0$, leading to a more complex and asymmetric phase structure.

5. Significance

• Theoretical Insight: The work provides a rigorous link between measurement-based quantum computation, topological order, and classical statistical mechanics for higher-dimensional qudits.
• New Universality Classes: The identification of the $c=8/5$ critical line and the $c=1$ isolated points in the context of deformed topological codes expands the known landscape of topological phase transitions.
• Quantum Simulation: The findings suggest that programmable quantum simulators can be used to explore complex topological phases and non-ergodic dynamics (scar states) by tuning measurement parameters.
• Foundation for Future Work: The introduction of the AT3 model opens avenues for studying $Z_N$ generalizations of topological codes and their connections to non-Abelian anyons and more complex classical models.

In summary, the paper establishes a comprehensive framework for understanding how local deformations in $Z_3$ topological codes drive transitions between topological, confined, and condensed phases, revealing rich critical phenomena and novel dynamical properties like Hilbert space fragmentation.