Exact Diagonalization of Heisenberg SU(N) models

Area of Science:

• Quantum many-body physics
• High-energy physics
• Condensed matter theory

Background:

• The Heisenberg SU(N) model describes quantum interactions of N-color objects.
• Exact diagonalization of quantum systems is computationally intensive, limiting system size.
• Irreducible representations of SU(N) are crucial for understanding quantum symmetries.

Purpose of the Study:

• To develop a simplified method for analyzing the SU(N) Heisenberg model.
• To extend the capabilities of exact diagonalization techniques to larger system sizes.
• To investigate the quantum phases of matter in SU(N) models for various N.

Main Methods:

• Construction of an orthonormal basis using standard Young tableaux for each irreducible representation of SU(N).
• Exploiting the simplified matrix form of the Heisenberg SU(N) model in this basis.
• Extending exact diagonalization to larger N by leveraging the reduced complexity.

Main Results:

• An explicit and simple matrix form for the Heisenberg SU(N) model is achieved.
• Exact diagonalization is successfully extended to significantly larger values of N.
• Distinct quantum phases are identified: long-range color order (SU(5)), spontaneous dimerization (SU(8)), and quantum liquid behavior (SU(10)) on a square lattice.

Conclusions:

• The developed method provides a powerful tool for studying complex quantum systems with SU(N) symmetry.
• This approach significantly advances the feasibility of exact diagonalization for large N.
• New insights into the rich phase diagrams of SU(N) models are revealed, including novel quantum states of matter.