Physics:Slave boson

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The slave boson method is a technique 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-valence cerium compound. The corresponding quantum states of these two states are the singlet [math]\displaystyle{ \vert f^0\rangle }[/math] state and the magnetic [math]\displaystyle{ \vert f^1:\sigma\rangle }[/math] state, where [math]\displaystyle{ \sigma=\uparrow,\ \downarrow }[/math] is the spin. The fermionic Hubbard operators that link these states are then

[math]\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 }[/math]

 

 

 

 

(1)

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

[math]\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 }[/math].

 

 

 

 

(2)

Together, these operators satisfy the graded Lie algebra

[math]\displaystyle{ [X_{ab},X_{cd}]_{\pm} = X_{ad}\delta_{bc}\pm X_{cb}\delta_{ad} }[/math]

 

 

 

 

(3)

where the [math]\displaystyle{ [A,B]_{\pm}=AB\pm BA }[/math] and the sign is chosen to be negative, unless both [math]\displaystyle{ A }[/math] and [math]\displaystyle{ B }[/math] are fermions, in which case 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" [math]\displaystyle{ \vert f^0\rangle= b^\dagger \vert 0 \rangle }[/math], whereas the magnetic configuration [math]\displaystyle{ \vert f^1:\sigma\rangle= f^\dagger_{\sigma} \vert 0 \rangle }[/math] is represented by an Abrikosov slave fermion. From these considerations, it is seen that the Hubbard operators can be written as

[math]\displaystyle{ X_{\sigma 0} = f^{\dagger}_\sigma b, \qquad X_{0\sigma}= b^\dagger f_{\sigma} }[/math]

 

 

 

 

(4)

and

[math]\displaystyle{ X_{00} = b^\dagger b , \qquad X_{\alpha\beta} = f^\dagger_{\alpha}f_{\beta} }[/math].

 

 

 

 

(5)

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

[math]\displaystyle{ Q = b^\dagger b + \sum_{\alpha = \uparrow, \downarrow} f^\dagger_{\alpha}f_{\alpha} }[/math].

 

 

 

 

(5)

In Hubbard's original approach, [math]\displaystyle{ Q=1 }[/math], 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 [math]\displaystyle{ N }[/math] component fermions, where the spin index [math]\displaystyle{ \alpha \in [1,N] }[/math] runs over [math]\displaystyle{ N }[/math] values. By allowing [math]\displaystyle{ N }[/math] to become large, while maintaining the ratio [math]\displaystyle{ Q/N }[/math], it is possible to develop a controlled large-[math]\displaystyle{ N }[/math] 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]

Bibliography

  1. Hubbard, John (1964). "Electron correlations in narrow energy bands. II. The degenerate band case". Proc. R. Soc. Lond. A (The Royal Society) 277 (1369): 237–259. doi:10.1098/rspa.1964.0019. Bibcode1964RSPSA.277..237H. 
  2. Piers Coleman (1984). "A New Approach to the Mixed Valence Problem". Phys. Rev. B (The American Physical Society) 29 (6): 3035–3044. doi:10.1103/PhysRevB.29.3035. Bibcode1984PhRvB..29.3035C. 
  3. N. Read and D. M. Newns (1983). "A new functional integral formalism for the degenerate Anderson model". Journal of Physics C: Solid State Physics 16 (29): L1055–L1060. doi:10.1088/0022-3719/16/29/007. 
  4. P. W. Anderson; G. Baskaran; Z. Zhou; T. Hsu (1987). "Resonating–valence-bond theory of phase transitions and superconductivity in La2CuO4-based compounds". Physical Review Letters (The American Physical Society) 58 (26): 2790–2793. doi:10.1103/PhysRevLett.58.2790. PMID 10034850. Bibcode1987PhRvL..58.2790A. 
  5. G. Kotliar and J. Liu (1988). "Superexchange mechanism and d-wave superconductivity". Physical Review B (The American Physical Society) 38 (7): 5142–5145. doi:10.1103/PhysRevB.38.5142. PMID 9946940. Bibcode1988PhRvB..38.5142K. 
  6. A. J. Millis; P.A. Lee (1986). "Large-orbital-degeneracy expansion for the lattice Anderson model". Physical Review B (The American Physical Society) 35 (7): 3394–3414. doi:10.1103/PhysRevB.35.3394. PMID 9941843.