Slave boson

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The slave boson method is a way to study systems where electrons interact strongly and move in ways that are heavily restricted by their surroundings. It focuses on how an ion can fluctuate between just a few charge states (valence configurations).

In the 1960s, John Hubbard introduced operators that describe adding or removing an electron within a limited set of valence states. For example, a cerium ion might flip between Ce4+ (no 4f electron) and Ce3+ (one 4f electron). The two local states can be written as |f0> and |f1; σ>, where σ represents spin up or down.

The algebra of these Hubbard operators is unusual and not the same as ordinary fermions, which makes standard diagrammatic methods hard to apply. In 1983, Piers Coleman proposed the slave boson approach to handle this. In this framework, the empty configuration |f0> is represented by a spinless boson b† acting on a vacuum, while the singly occupied state |f1; σ> is represented by a fermion fσ† acting on the vacuum.

Using these representations, the Hubbard operators that connect the two states can be expressed in terms of the slave boson and slave fermion operators. This rewrite preserves the necessary algebra and allows the use of field-theoretic techniques.

There is a conserved quantity Q that counts the number of auxiliary particles; originally Q = 1, but the idea can be generalized to larger values to explore more complex representations. The approach can also be extended to N-component fermions (with spin index α = 1,…,N), and by taking N large, one can perform a controlled large-N expansion.

The slave boson method has become a widely used tool for studying strongly correlated electron systems. It has helped advance ideas like resonating valence bond theory for high-temperature superconductivity and the understanding of heavy fermion materials.