Topological order in a three-dimensional toric code at finite temperature

2008/04/30 by Claudio Castelnovo, Claudio Chamon · 4 citations
Mathematics · Physics and Astronomy · #Combinatorics #Entropy (arrow of time) #Gauge theory #Infinitesimal #Mathematical analysis #Mathematical physics #Mathematics #Order (exchange) #Physics #Physics of Superconductivity and Magnetism #Pure mathematics #Quantum #Quantum and electron transport phenomena #Quantum many-body systems #Quantum mechanics #Theoretical physics #Topological entropy #Topological entropy in physics #Topological order #Topological quantum number #Topology (electrical circuits) #Toric code #Zero temperature #cond-mat.stat-mech #cond-mat.str-el #quant-ph

paper · pdf · doi:10.1103/physrevb.78.155120

published as Phys. Rev. B 78, 155120 (2008) · (28 pages, 9 figures)

arxiv created 2008/10/21 · openalex publication_date 2008/10/21 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

Abstract

We study topological order in a toric code in three spatial dimensions or a 3+1D ℤ₂ gauge theory at finite temperature. We compute exactly the topological entropy of the system and show that it drops, for any infinitesimal temperature, to half its value at zero temperature. The remaining half of the entropy stays constant up to a critical temperature T_c, dropping to zero above T_c. These results show that topologically ordered phases exist at finite temperatures, and we give a simple interpretation of the order in terms of fluctuating strings and membranes and how thermally induced point defects affect these extended structures. Finally, we discuss the nature of the topological order at finite temperature and its quantum and classical aspects.

Citations

• TOPOLOGICAL ORDERS IN RIGID STATES
• Detecting Topological Order in a Ground State Wave Function
• Duality in Generalized Ising Models and Phase Transitions without Local Order Parameters
• Ground-state degeneracy of the fractional quantum Hall states in the presence of a random potential and on high-genus Riemann surfaces
• Duality in field theory and statistical systems
• Entanglement and topological entropy of the toric code at finite temperature
• Bipartite entanglement and entropic boundary law in lattice spin systems
• Topological quantum memory
• An introduction to lattice gauge theory and spin systems
• Fault-tolerant quantum computation by anyons
• Quantum orders and symmetric spin liquids

Cited by

• Quantized topological invariant of symmetry-projected Gibbs states
• Finite-temperature phase transition to a quantum spin liquid in a three-dimensional Kitaev model on a hyperhoneycomb lattice
• Scaling law for topologically ordered systems at finite temperature
• Localization of extended quantum objects