Entanglement Renormalization of Thermofield Double States

Area of Science:

• Quantum Information Theory
• Condensed Matter Physics
• Statistical Mechanics

Background:

• Entanglement renormalization is a powerful technique for coarse-graining quantum states in real space.
• The multiscale entanglement renormalization ansatz (MERA) is a prominent example, effective for ground states.
• Extending these methods to finite-temperature states is crucial for understanding thermal quantum systems.

Purpose of the Study:

• To develop an entanglement renormalization scheme applicable to finite-temperature (Gibbs) states.
• To investigate the properties of quantum systems at finite temperatures using this new scheme.
• To demonstrate the loss of topological order and analyze critical thermal states.

Main Methods:

• Applied the multiscale entanglement renormalization ansatz (MERA) to the canonical purification of Gibbs states, known as the thermofield double state.
• Derived an analytically exact renormalization circuit for a finite-temperature two-dimensional toric code.
• Applied the scheme to one-dimensional free boson models at finite temperatures and performed numerical analysis of perturbations.

Main Results:

• An entanglement renormalization scheme for finite-temperature states was successfully obtained.
• The renormalization circuit for the 2D toric code explicitly demonstrated its lack of topological order at finite temperatures.
• The thermofield double of critical thermal states in 1D free boson models was shown to be described by Lifshitz theory.

Conclusions:

• The developed scheme provides a new tool for coarse-graining finite-temperature quantum states.
• The study explicitly shows the disappearance of topological order at finite temperatures in specific models.
• The connection between thermal states and Lifshitz theories is established, offering insights into critical phenomena.