[Paper Review] Finite temperature topological order in 2D topological color codes

Mehdi Kargarian|ArXiv.org|Apr 28, 2009

Neural dynamics and brain function6 references3 citations

TL;DR

This paper investigates finite-temperature topological order in two-dimensional topological color codes using topological entanglement entropy as a diagnostic. It analytically computes the temperature-dependent topological entropy in the hard-core limit, showing that topological order vanishes upon thermalization of one string-net type in finite systems and is fragile even at low temperatures in the thermodynamic limit, with non-commuting zero-temperature and thermodynamic limits tied to region topology.

ABSTRACT

In this work the topological order at finite temperature in two-dimensional color code is studied. The topological entropy is used to measure the behavior of the topological order. Topological order in color code arises from the colored string-net structures. By imposing the hard constrained limit the exact solution of the entanglement entropy becomes possible. For finite size systems, by raising the temperature, one type of string-net structure is thermalized and the associative topological entropy vanishes. In the thermodynamic limit the underlying topological order is fragile even at very low temperatures. Taking first the thermodynamic limit and then the zero-temperature limit and vice versa does not commute, and their difference is related only to the topology of regions. The contribution of the colors and symmetry of the model in the topological entropy is also discussed. It is shown how the gauge symmetry of the color code underlies the topological entropy.

Motivation & Objective

• To understand the stability of topological order in 2D topological color codes at finite temperature.
• To investigate how temperature affects the topological entanglement entropy, a key indicator of topological order.
• To examine the role of gauge symmetry and color structure in determining topological entropy.
• To clarify the non-commutativity of thermodynamic and zero-temperature limits in the context of topological order.

Proposed method

• The study employs the topological entanglement entropy as a probe of topological order, derived from the von Neumann entropy of a subsystem.
• It uses the hard-core limit to enable exact computation of entanglement entropy, simplifying the many-body Hamiltonian.
• The analysis involves partitioning the system into regions and computing the entropy contributions from different string-net configurations.
• The method accounts for the Z₂×Z₂ gauge symmetry of the color code, which underlies the string-net structure and topological order.
• The topological entropy is computed via a sum over four distinct configurations of the system, weighted by Boltzmann factors depending on energy differences.
• The final expression for finite-temperature topological entropy is derived by subtracting the zero-temperature value from the thermalized entropy, with explicit dependence on color-specific energy terms and partition functions.

Experimental results

Research questions

• RQ1 How does finite temperature affect the topological entanglement entropy in 2D topological color codes?
• RQ2 What is the role of the Z₂×Z₂ gauge symmetry in sustaining or breaking topological order at finite temperature?
• RQ3 Why does the topological order vanish upon thermalization of a single string-net structure in finite systems?
• RQ4 How do the zero-temperature and thermodynamic limits fail to commute, and what does this imply about topological order?
• RQ5 What is the quantitative relationship between the topological entropy and the color structure (red, blue, green) in the model?

Key findings

• The topological entanglement entropy at zero temperature is exactly S_cc(0) = 4 ln 2, confirming the presence of topological order in the ground state.
• At finite temperature, the topological entropy vanishes when one type of string-net structure is thermally excited, indicating loss of topological order.
• In the thermodynamic limit, topological order is fragile and destroyed even at arbitrarily low temperatures, indicating no finite-temperature topological phase.
• The non-commutativity of the zero-temperature and thermodynamic limits is shown to depend only on the topology of the region, not on system details.
• The gauge symmetry Z₂×Z₂ of the color code is directly linked to the structure of the topological entropy, with each color contributing to the entropy through distinct energy terms.
• The finite-temperature topological entropy is expressed as a sum over four configurations, each weighted by exponential factors involving color-specific energy differences and partition functions.