Solving local constraint conditions in slave particle theory

TL;DR

This paper applies BRST quantization to slave particle theory, resolving the local constraint issue by properly incorporating gauge fixing, leading to more accurate physical predictions for models like the Hubbard and t-J models.

Contribution

It introduces a gauge fixing condition in slave particle theory and uses BRST quantization to correctly handle local constraints, improving the understanding of strongly correlated systems.

Findings

The spinon is gapped in the weak U regime, indicating a conventional metal.

Dynamic effects of the Lagrange multiplier suppress the d-wave pairing gap.

The superconducting T_c is reduced to values close to experimental cuprates.

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 $\lambda_i$ which ensures the constraint: In the Hamiltonian formalism of the theory, $\lambda_i$ is time-independent and commutes with the Hamiltonian while in the Lagrangian formalism, $\lambda_i(t)$ becomes time-dependent and plays a role of gauge field. This implies that the redundant degrees of freedom of $\lambda_i(t)$ are introduced and must be removed by the additional constraint, the gauge fixing condition $\partial_t \lambda_i(t)=0$. In literature, this gauge fixing condition was missed. We add this gauge fixing condition and use the BRST quantization of gauge theory for Dirac's first-class constraints in the slave particle theory. This gauge fixing condition endows $\lambda_i(t)$ 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 theory. For the $t$-$J$ model, we find that the dynamic effect of $\lambda_i(t)$ 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 mean field value which is of the order of 1000 K. The renormalized $T_c$ is then close to that in cuprates.