Slave boson

The Slave Boson method is a method for dealing with models of strongly correlated systems, providing a method to second-quantize valence fluctuations within a restrictive manifold of states. In the 1960s the physicist John Hubbard introduced an operator, now named the "Hubbard Operator"[1] to describe the creation of an electron within a restrictive manifold of valence configurations. Consider for example, a rare earth or actinide ion in which strong Coulomb interactions restrict the charge fluctuations to two valence states, such as the Ce⁴⁺(4f⁰) and Ce³⁺ (4f¹) configurations of a mixed valent Cerium compound. The corresponding quantum states of these two states are the singlet ${\displaystyle \vert f^{0}\rangle }$ state and the magnetic ${\displaystyle \vert f^{1}:\sigma \rangle }$ state, where ${\displaystyle \sigma =\uparrow ,\ \downarrow }$ is the spin. The fermionic Hubbard operators that link these states are then

${\displaystyle X_{\sigma 0}=\vert f^{1}:\sigma \rangle \langle f^{0}\vert ,\qquad X_{0\sigma }=\vert f^{0}\rangle \langle f^{1}:\sigma \vert }$

(1)

The algebra of operators is closed by introducing the two bosonic operators

${\displaystyle X_{00}=\vert f^{0}\rangle \langle f^{0}\vert ,\qquad X_{\alpha \beta }=\vert f^{1}:\alpha \rangle \langle f^{1}:\beta \vert }$.

(2)

Together, these operators satisfy the graded Lie algebra

${\displaystyle [X_{ab},X_{cd}]_{\pm }=X_{ad}\delta _{bc}\pm X_{cb}\delta _{ad}}$

(3)

where the ${\displaystyle [A,B]_{\pm }=AB\pm BA}$ and the sign is chosen to be negative, unless both A and B are fermions, when it is positive. The Hubbard operators are the generators of the super group SU(2|1). This non-canonical algebra means that these operators do not satisfy a Wick's theorem, which prevents a conventional diagrammatic or field theoretic treatment.

In 1983 Piers Coleman introduced the Slave Boson formulation of the Hubbard operators,[2] which enabled valence fluctuations to be treated within a field-theoretic approach.[3] In this approach, the spinless configuration of the ion is represented by a spinless "slave boson" ${\displaystyle \vert f^{0}\rangle =b^{\dagger }\vert 0\rangle }$, whereas the magnetic configuration ${\displaystyle \vert f^{1}:\sigma \rangle =f_{\sigma }^{\dagger }\vert 0\rangle }$ is represented by an Abrikosov slave fermion. From these considerations, it is seen that the Hubbard operators can be written as

${\displaystyle X_{\sigma 0}=f_{\sigma }^{\dagger }b,\qquad X_{0\sigma }=b^{\dagger }f_{\sigma }}$

(4)

and

${\displaystyle X_{00}=b^{\dagger }b,\qquad X_{\alpha \beta }=f_{\alpha }^{\dagger }f_{\beta }}$.

(5)

This factorization of the Hubbard operators faithfully preserves the graded Lie algebra. Moreover, the Hubbard operators so written commute with the conserved quantity

${\displaystyle Q=b^{\dagger }b+\sum _{\alpha =\uparrow ,\downarrow }f_{\alpha }^{\dagger }f_{\alpha }}$.

(5)

In Hubbard's original approach, Q=1, but by generalizing this quantity to larger values, higher irreducible representations of SU(2|1) are generated. The slave boson representation can be extended from two component to N component fermions, where the spin index ${\displaystyle \alpha \in [1,N]}$ runs over N values. By allowing N to become large, while maintaining the ratio Q/N, it is possible to develop a controlled large N expansion.

The slave boson approach has since been widely applied to strongly correlated electron systems, and has proven useful in developing the resonating valence bond theory(RVB) of high temperature superconductivity[4][5] and the understanding of heavy fermion compounds.[6]