[Paper Review] Thermal States of Anyonic Systems

S. Iblisdir, David Pérez-Garcı́a+2 more|arXiv (Cornell University)|Dec 30, 2008

Quantum many-body systems12 references4 citations

TL;DR

This paper investigates the thermal stability of topological order in two-dimensional anyonic systems using the topological mutual information $I_{\mathrm{topo}}$ as a thermal order parameter. It establishes a scaling law linking system size and temperature, showing that $I_{\mathrm{topo}}$ depends on the Kullback-Leibler divergence between thermal and quantum dimension-induced topological charge distributions, with non-Abelian models like $D(S_3)$ exhibiting similar behavior to the toric code.

ABSTRACT

A study of the thermal properties of two-dimensional topological lattice models is presented. This work is relevant to assess the usefulness of these systems as a quantum memory. For our purposes, we use the topological mutual information $I_{\mathrm{topo}}$ as a "topological order parameter". For Abelian models, we show how $I_{\mathrm{topo}}$ depends on the thermal topological charge probability distribution. More generally, we present a conjecture that $I_{\mathrm{topo}}$ can (asymptotically) be written as a Kullback-Leitner distance between this probability distribution and that induced by the quantum dimensions of the model at hand. We also explain why $I_{\mathrm{topo}}$ is more suitable for our purposes than the more familiar entanglement entropy $S_{\mathrm{topo}}$. A scaling law, encoding the interplay of volume and temperature effects, as well as different limit procedures, are derived in detail. A non-Abelian model is next analysed and similar results are found. Finally, we also consider, in the case of a one-plaquette toric code, an environment model giving rise to a simulation of thermal effects in time.

Motivation & Objective

• To assess the viability of topological lattice models as quantum memories under thermal noise.
• To establish $I_{\mathrm{topo}}$ as a superior order parameter over entanglement entropy $S_{\mathrm{topo}}$ for detecting topological order at finite temperature.
• To derive a scaling law that quantifies the interplay between system size, temperature, and topological order persistence.
• To extend the analysis to non-Abelian models, specifically $D(S_3)$, and confirm universal scaling behavior.
• To simulate thermal effects in a one-plaquette toric code using an open quantum system model.

Proposed method

• Uses the topological mutual information $I_{\mathrm{topo}}$ as a thermal order parameter, derived from the generalized topological entropy of Kitaev-Preskill and Levin-Wen.
• Relies on the Kullback-Leibler divergence (Kullback-Leibler distance) to relate $I_{\mathrm{topo}}$ to the thermal distribution of topological charges and the quantum dimensions of anyons.
• Applies the formalism to Abelian (toric code) and non-Abelian ($D(S_3)$) anyonic models via exact diagonalization and group-theoretic decomposition of the ground state.
• Derives a Schmidt decomposition of the ground state using quotient groups $\mathbb{G}_{AB}/\mathbb{G}_{\mathrm{d}}$ to compute $S(\rho_A)$ and hence $I_{\mathrm{topo}}$.
• Uses a one-plaquette toric code model coupled to a thermal environment to simulate thermalization dynamics and $I_{\mathrm{topo}}$ evolution in time.
• Employs group-theoretic lemmas to prove bi-orthogonality of basis states, ensuring the validity of the Schmidt decomposition for entropy computation.

Experimental results

Research questions

• RQ1 How does the topological mutual information $I_{\mathrm{topo}}$ behave as a function of temperature and system size in topological lattice models?
• RQ2 Can $I_{\mathrm{topo}}$ be asymptotically expressed as a Kullback-Leibler divergence between thermal and anyonic quantum dimension distributions?
• RQ3 Why is $I_{\mathrm{topo}}$ a more suitable order parameter than $S_{\mathrm{topo}}$ for detecting topological order at finite temperature?
• RQ4 Does the scaling behavior of $I_{\mathrm{topo}}$ under temperature and size variation hold for non-Abelian models like $D(S_3)$?
• RQ5 Can thermal effects in a small system (one-plaquette toric code) be effectively simulated using an open quantum system approach?

Key findings

• For the toric code, $I_{\mathrm{topo}}$ is non-zero only for finite systems at finite temperature, and it remains constant over a temperature regime before decaying to zero.
• The scaling law shows that to maintain topological order when increasing system size, temperature must be reduced proportionally to compensate for volume effects.
• The asymptotic form of $I_{\mathrm{topo}}$ is given by the Kullback-Leibler divergence between the thermal topological charge distribution and the distribution defined by quantum dimensions.
• For the non-Abelian $D(S_3)$ model, $I_{\mathrm{topo}}$ exhibits the same qualitative and quantitative dependence on size and temperature as in the toric code, confirming universality of the scaling.
• The ground state of the $D(S_3)$ model admits a Schmidt decomposition with $S(\rho_A) = \log_2 |S_3| (N_{\partial A} - 1)$, where $N_{\partial A}$ is the number of boundary vertices.
• The simulation of thermal effects in a one-plaquette toric code is feasible using a master equation approach, enabling time-resolved study of $I_{\mathrm{topo}}$ decay.