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arXiv:2212.13734 · Communications in Theoretical Physics

Solving local constraint condition problem in slave particle theory with the BRST quantization

Xi Luo, Jianqiao Liu, Yue Yu

Published Jul 13, 2023• 1 citation
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Abstract

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.

DOI
10.1088/1572-9494/ace763

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arXiv:hep-th/9412137•Cambridge University Press eBooks•Jan 5, 1995•11 citations
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.
Abstract We aim to clarify the confusion and inconsistency in our recent works (Luo et al 2023 Commun. Theor. Phys. 75 095702; Liang et al 2024 Phys. Rev. B 110 075125), and to address the incompleteness therein. In order to avoid the ill-defined nature of the free propagator of the gauge field in the ordered states of the t – J model, we adopted a gauge fixing that was not of the Becchi–Rouet–Stora–Tyutin (BRST) exact form in our previous work (Liang et al 2024 Phys. Rev. B 110 075125). This led to the situation where Dirac’s second-class constraints, namely, the slave particle number constraint and the Ioffe–Larkin current constraint, were not rigorously obeyed. Here we show that a consistent gauge fixing condition that enforces the exact constraints is BRST-exact in our theory. An example is the Lorenz gauge. On the other hand, we prove that although the free propagator of the gauge field in the Lorenz gauge is ill-defined, the full propagator is still well-defined. This implies that the strongly correlated t – J model can be exactly mapped to a perturbatively controllable theory within the slave particle representation.
In the slave particle representation with U(1)U(1)U(1) gauge symmetry, local constraints on physical states characterized by various mean field solutions belong to Dirac's second-class ones. Although constrained systems are extensively investigated, realistic methods to solve the gauge theory problem with second-class constraints are yet to be developed. We formulate a Becchi-Rouet-Stora-Tyutin (BRST) quantization theory, called consistent U(1)U(1)U(1) gauge theory, that is consistent with both first- and second-class local constraints for strongly correlated condensed matter systems. In our consistent U(1)U(1)U(1) gauge theory, the redundant gauge degrees of freedom are removed by proper gauge fixing conditions while the constraints are exactly retained and the gauge invariance is guaranteed by the BRST symmetry. Furthermore, the gauge fixing conditions endow the gauge field with dynamics. This turns the strongly correlated electron model into a weakly coupled slave boson model, so most of the system's physical properties can be calculated by the conventional quantum many-body perturbation method. We focus on the property of the strange metal phase in the $t\text{\ensuremath{-}}J$ model. The electron momentum distribution and the spectral function are calculated, and the non-Fermi-liquid behavior agrees with the angle-resolved photoemission spectroscopy measurements for cuprate materials. We also study the electromagnetic responses of the strange metal state. The observed non-Fermi-liquid anomalies are captured by our calculations. Especially, we find that the Hall resistivity decreases as temperature increases, and the sign of the Hall resistivity varies from negative to positive when the dopant concentration varies from optimal doping to underdoping in the strange metal regime.
Slave bosons are commonly introduced in order to implement an infinite Hubbard U by means of a local constraint. The usual starting point for investigations within this scheme is a mean-field theory in which the constraint is taken to be global. This approximate treatment of the constraint is studied in the context of a two-band Hubbard model, and it is shown that (i) the ground state has a significant number of doubly occupied sites, despite the infinite on-site repulsion in the original model, and (ii) there is an unphysical tendency for pairing. However, it is found that if the local constraint is retained for the insulator at half filling, then mean-field theory gives the correct result that the double occupancy is zero.
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.
Slave-particle method is a powerful tool to tackle the correlation effect in quantum many-body physics. Although it has been successfully used to comprehend various intriguing problems, such as Mott metal–insulator transition and Kondo effect, there is still no convincing theory so far on the availability and limitation of this method. The abuse of slave-particle method may lead to wrong physics. As the simplest slave-particle method, Z 2 slave spin, which is widely applied to many strongly correlated problems, is highly accessible and researchable. In this work, we will uncover the nature of the Z 2 slave-spin method by studying a two-site Hubbard model. After exploring aspects of properties of this toy model, we make a comparative analysis of the results obtained by three methods: (i) slave-spin method on mean-field level, (ii) slave-spin method with gauge constraint, and (iii) the exact solution as a benchmark. We find that, protected by the particle–hole symmetry, the slave-spin mean-field method can recover the static properties of ground state exactly at half filling. Furthermore, in the parameter space where both U and T are small enough, the slave-spin mean-field method is also reliable in calculating the dynamic and thermal dynamic properties. However, when U or T is considerably large, the mean-field approximation gives ill-defined behaviors, which result from the unphysical states in the enlarged Hilbert space. These findings lead to our conclusion that the accuracy of slave particle can be guaranteed if we can exclude all unphysical states by enforcing gauge constraints. Our work demonstrates the promising prospect of slave-particle method in studying complex strongly correlated models with specific symmetry or in certain parameter space.
The relationship between the Dirac and reduced phase space quantizations is investigated for spin models belonging to the class of Hamiltonian systems having no gauge conditions. It is traced out that the two quantization methods may give similar, or essentially different physical results, and, moreover, it is shown that there is a class of constrained systems, which can be quantized only by the Dirac method. A possible interpretation of the gauge degrees of freedom is given. Introduction There are two main methods to quantize the Hamiltonian systems with first class constraints: the Dirac quantization [1] and the reduced phase space quantization [2], whereas two other methods, the path integral method [3, 2] and the BRST quantization [4] being the most popular method for the covariant quantization of gauge-invariant systems, are based on and proceed from them [2, 5]. The basic idea of the Dirac method consists in imposing quantum mechanically the first class constraints as operator conditions on the states for singling out the physical ones [1]. The reduced phase space quantization first identifies the physical degrees of freedom at the classical level by the factorization of the constraint surface with respect to the action of the gauge group, generated by the constraints. Then the resulting Hamiltonian system is quantized as a usual unconstrained system [2]. Naturally, the problem of the relationship of these two methods arises. It was discussed in different contexts in literature [6], and there is an opinion that the differences between the two quantization methods can be traced out to a choice of factor ordering in the construction of various physical operators.