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Constraint Quantization of Slave-Particle Theories

Christian Helm, Joachim Keller

Published Jan 1, 1996• 0 citations
OpenAlex

Abstract

We start from the Barnes-Coleman slave-particle description, where the Hubbard operators X are decomposed into a product of fermionic (fα) and bosonic (b) operators. The quantum mechanical constraint b † b+ ∑ α f † αfα = 1 is treated within the framework of Dirac’s method for the quantization of classical constrained systems. This leads to modified algebraic properties of the fundamental operators: bb † b = b, fαf † βfγ = δαβfγ and fαb † = 0. Thereby the algebra of the X-operators is preserved exactly on the operator level. Matrix representations of the above algebra are constructed and a resolvent-like perturbation theory for the single-impurity Anderson model is developed.

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We start from the Barnes-Coleman slave-particle description, where the Hubbard operators X are decomposed into a product of fermionic (fα) and bosonic (b) operators. The quantum mechanical constraint b † b+ ∑ α f † αfα = 1 is treated within the framework of Dirac’s method for the quantization of classical constrained systems. This leads to modified algebraic properties of the fundamental operators: bb † b = b, fαf † βfγ = δαβfγ and fαb † = 0. Thereby the algebra of the X-operators is preserved exactly on the operator level. Matrix representations of the above algebra are constructed and a resolvent-like perturbation theory for the single-impurity Anderson model is developed.
The quantization rules obtained by Le Guillou and Ragoucy in the slave particle theories of correlated systems have been applied to study the superconductivity in extended Hubbard and $t\ensuremath{-}J$ models. The quantization scheme has the virtue of treating the local constraint exactly. We compare our numerical results with those obtained in the mean-field approximation for usual slave boson technique. We also discuss the relation of the present approach with method of equation of motion for Hubbard operators. Where possible the obtained results are also compared with experimental data on high-temperature superconductors, as in these materials the essential physics is believed to depend on the presence of strong correlations.
We develop a supersymmetric representation of the Hubbard operator which unifies the slave boson and slave fermion representations into a single $U(1)\ifmmode\times\else\texttimes\fi{}SU(1|1)$ gauge theory, a group with larger symmetry than the product of two U(1)U(1)U(1) gauge groups. These representations of the Hubbard operator can be used to incorporate strong Hund's interactions in multielectron atoms as a constraint. We show how this method can be combined with the SP(N)\mathrm{SP}(N)SP(N) group to yield a locally supersymmetric large-NNN formulation of the $t\ensuremath{-}J$ model.
In this paper it is shown that the family of first-order Lagrangians for the t - J model and the corresponding correlation-generating functional previously found can be exactly mapped into the slave-fermion decoupled representation. Next, by means of the Faddeev-Jackiw symplectic method, a different family of Lagrangians is constructed and it is shown how the corresponding correlation-generating functional can be mapped into the slave-boson representation. Finally, in order to define the propagation of fermion modes we discuss two alternative ways to treat the fermionic sector in the path-integral formalism for the t - J model.
In this paper, we develop a new type of slave particle method which is similar to the slave rotor model except that the quantum rotor is substituted by a spin one slave particle. The spin-one slave particle itself can be represented in terms of Schwinger bosons/fermions. This approach is more conveniently applicable to the strongly correlated Hamiltonians with on-site Hubbard interaction and resolves the limitations of using the slave rotor model as well as the Anderson-Zou slave particle technique. For instance, the mean-field parameters of the slave spin method do not vanish above the Mott transition and this approach is smoothly connected to the non-interacting limit. As an example, we study the phase diagram of the Kane-Mele-Hubbard model using our current approach. In the absence of the spin-orbit interaction, the Mott transition occurs at Uc 3t1U_c ~ 3 t_1Uc​ 3t1​. Several aspects of the slave spin method, its gauge theory and various possible mean-field states associated with this approach have been discussed.
Abstract With the Becchi–Rouet–Stora–Tyutin (BRST) quantization of gauge theory, we solve the long-standing difficult problem of the local constraint conditions, i.e. the single occupation of a slave particle per site, in the slave particle theory. This difficulty is actually caused by inconsistently dealing with the local Lagrange multiplier λ i which ensures the constraint: in the Hamiltonian formalism of the theory, λ i is time-independent and commutes with the Hamiltonian while in the Lagrangian formalism, λ i (t) becomes time-dependent and plays a role of gauge field. This implies that the redundant degrees of freedom of λ i (t) are introduced and must be removed by the additional constraint, the gauge fixing condition (GFC) ∂ t λ i (t) = 0. In literature, this GFC was missed. We add this GFC and use the BRST quantization of gauge theory for Dirac’s first-class constraints in the slave particle theory. This GFC endows λ i (t) with dynamics and leads to important physical results. As an example, we study the Hubbard model at half-filling and find that the spinon is gapped in the weak U and the system is indeed a conventional metal, which resolves the paradox that the weak coupling state is a superconductor in the previous slave boson mean field (MF) theory. For the t–J model, we find that the dynamic effect of λ i (t) substantially suppresses the d-wave pairing gap and then the superconducting critical temperature may be lowered at least a factor of one-fifth of the MF value which is of the order of 1000 K. The renormalized T c is then close to that in cuprates.
In the framework of constrained systems, we give the classical Hamiltonian formulation of slave-particle models and their correct quantization. The electron-momentum distribution function in the t-J and Hubbard models is then studied in the framework of slave-particle approaches and within the decoupling scheme. We show that criticisms that have been addressed in this context coming from a violation of the sum rule for the physical electron are not valid. Due to the correct quantization rules for the slave particles, the sum rule for the physical electron is indeed obeyed, both exactly and within the decoupling scheme.