How can I calculate spin Hall conductivity for a 4*4 Hamiltonian?

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SUMMARY

This discussion focuses on calculating spin Hall conductivity for a 4 × 4 Hamiltonian using Dirac matrices, specifically the gamma matrices associated with the Dirac equation in quantum field theory (QFT). The user references the connection between these matrices and the 2x2 Pauli matrices, which also serve as generators of the Clifford algebra. A key reference for this topic is the paper with DOI: 10.1103/PhysRevLett.94.226601, which provides foundational insights into the subject matter.

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  • Understanding of 4 × 4 Hamiltonians in quantum mechanics
  • Familiarity with Dirac matrices and their role in QFT
  • Knowledge of spin Hall conductivity and its significance
  • Basic concepts of Clifford algebra and its applications
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  • Research the properties and applications of Dirac matrices in quantum mechanics
  • Study the derivation of spin Hall conductivity formulas in condensed matter physics
  • Explore the implications of the paper with DOI: 10.1103/PhysRevLett.94.226601
  • Learn about the relationship between Clifford algebra and quantum spin systems
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Physicists, quantum mechanics researchers, and graduate students studying condensed matter physics or quantum field theory who are interested in advanced topics like spin Hall conductivity and Hamiltonian systems.

Mohammad-gl
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Spin Hall conductivity formula
I'm trying to calculate spin Hall conductivity for a 4 × 4 Hamiltonian and I found the attached formula for spin Hall conductivity. What are Dirac matrices for 4× 4 Hamiltonian?
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I have not checked it but from context I assume these are the Dirac matrices that you are familiar with from the Dirac equation in QFT, i.e. the gamma matrices that generate the Clifford algebra (as the 2x2 Pauli matrices that you are used to when talking about spin are also generators of the Clifford algebra, hence the connection)
 
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Source?
 
pines-demon said:
Source?
This is the reference.
DOI: 10.1103/PhysRevLett.94.226601
 
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phyro said:
I have not checked it but from context I assume these are the Dirac matrices that you are familiar with from the Dirac equation in QFT, i.e. the gamma matrices that generate the Clifford algebra (as the 2x2 Pauli matrices that you are used to when talking about spin are also generators of the Clifford algebra, hence the connection)
Thank you. I want to know metrices form.
 

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