Renormalization Group Flows of Hamiltonians Using Tensor Networks.

Area of Science:

• Condensed Matter Physics
• Quantum Information Theory
• Statistical Mechanics

Background:

• Renormalization group (RG) methods are crucial for understanding critical phenomena in statistical physics.
• Tensor networks provide a powerful framework for representing and manipulating complex quantum states and classical partition functions.
• Traditional RG techniques like Kadanoff's spin blocking method face challenges, particularly with the area law of mutual information.

Purpose of the Study:

• To construct a renormalization group flow of Hamiltonians for two-dimensional classical partition functions using tensor networks.
• To develop a novel tensor network formalism that preserves positivity of tensors throughout the renormalization process.
• To provide a new perspective on the differences between tensor network approaches and classical RG methods.

Main Methods:

• Utilizing tensor networks to represent and evolve Hamiltonians under a renormalization group flow.
• Developing a formalism analogous to tensor network renormalization but ensuring positivity of tensors at each step.
• Introducing a change of the local basis at each decimation step to address limitations of previous methods.

Main Results:

• Achieved approximate fixed point tensor networks at criticality, analogous to tensor network renormalization.
• Demonstrated that the developed formalism preserves tensor positivity, allowing for Hamiltonian flow interpretation.
• Derived algebraic relations for fixed point tensors and calculated critical exponents.

Conclusions:

• The proposed tensor network approach offers a novel way to study renormalization group flows and critical phenomena.
• Preserving tensor positivity provides a physical interpretation in terms of Hamiltonian flows.
• The method successfully benchmarks on the Ising and six-vertex models, showing its efficacy.