Hubbard-I

Official Resources

• Homepage: https://triqs.github.io/hubbardI/latest/
• Source Repository: https://github.com/TRIQS/hubbardI
• Documentation: https://triqs.github.io/hubbardI/latest/documentation.html
• License: Open-source (part of TRIQS ecosystem)

Overview

The Hubbard-I approximation is an impurity solver for dynamical mean-field theory (DMFT) that approximates the solution of the impurity model by neglecting hybridization with the bath and solving the purely atomic problem. The TRIQS-based Hubbard-I solver implements this approximation as part of the TRIQS (Toolbox for Research on Interacting Quantum Systems) ecosystem, providing a fast and simple impurity solver for DMFT calculations.

The Hubbard-I approximation was originally proposed by Hubbard in 1963 and is closely related to the moments expansion method. It works by solving the atomic Hamiltonian to obtain the local Green's function, which becomes diagonal with atomic energy levels. The solver is particularly useful as a fast starting point for DMFT calculations and for systems where hybridization is weak. It has been combined with hybridization expansion CT-QMC to create improved solvers that inherit advantages of both methods.

Scientific domain: DMFT, strongly correlated electron systems, impurity solvers
Target user community: Researchers in strongly correlated electron physics

Theoretical Methods

• Hubbard-I approximation (neglecting hybridization)
• Atomic problem solving with local interaction Hamiltonian
• DMFT self-consistency condition
• Hubbard operator representation
• Equation of motion decoupling
• Atomic Green's function computation
• Multi-orbital Anderson impurity model

Capabilities (CRITICAL)

• Fast impurity solver for DMFT
• Atomic Green's function G(iw) and self-energy Sigma(iw)
• Real-frequency Green's function G(w) and Sigma(w)
• Imaginary-time Green's function G(tau)
• Legendre expansion G(legendre)
• Density matrix computation
• Multi-orbital support
• Integration with TRIQS DMFT workflow
• DFT+DMFT example with VASP for Ce

Inputs & Outputs

Input formats:

• TRIQS Green's function objects (gf_struct)
• Local Hamiltonian h_int (Operator object)
• Inverse temperature beta
• Bath Green's function high-frequency behavior

Output data types:

• Impurity Green's function G(iw)
• Self-energy Sigma(iw)
• G(tau) imaginary-time Green's function
• G(w) real-frequency Green's function
• G(legendre) Legendre representation
• Density matrix

Interfaces & Ecosystem

• Programming language: C++ with Python (TRIQS framework)
• Part of TRIQS: Integrated with TRIQS DMFT tools
• DFT+DMFT: Interface with VASP for DFT+DMFT calculations
• Parallel computing: Limited (atomic problem is small)
• Related solvers: CT-QMC, ED (exact diagonalization) in TRIQS

Limitations & Known Constraints

• Neglects hybridization entirely (atomic limit)
• Less accurate than CT-QMC or ED for systems with strong hybridization
• Cannot capture Kondo physics or Fermi liquid behavior
• Limited to weakly hybridized systems or as starting approximation
• Multi-orbital atomic problem can still be large for many orbitals

Performance Characteristics

• Extremely fast (atomic problem, no Monte Carlo sampling)
• No sign problem
• Provides real-frequency results directly (no analytic continuation needed)
• Suitable for high-throughput DMFT screening
• Can be used as starting point for more expensive solvers

Comparison with Other Codes

• vs CT-QMC: Hubbard-I is much faster but less accurate; CT-QMC is exact but has sign problem
• vs ED (Exact Diagonalization): ED includes finite bath; Hubbard-I has no bath
• vs NRG: NRG is accurate for low-energy physics; Hubbard-I is simpler but less precise
• vs Improved Hubbard-I: Combined with CT-QMC for better accuracy while maintaining speed

Best Practices

• Use as starting point for DMFT iterations before switching to CT-QMC
• Appropriate for systems with weak hybridization (e.g., localized f-electrons)
• Combine with hybridization expansion CT-QMC for improved accuracy
• Use for DFT+DMFT with LDA+U starting point for f-electron systems
• Verify results against CT-QMC for systems with moderate hybridization

Verification & Sources

Primary sources:

1. TRIQS Hubbard-I documentation: https://triqs.github.io/hubbardI/latest/documentation.html
2. J. Hubbard, Proc. R. Soc. A 276, 238 (1963) - original Hubbard-I
3. Improved Hubbard-I: J. Phys.: Condens. Matter (2019)

Confidence: VERIFIED - TRIQS-based solver with documentation and GitHub repository