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.
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.
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 slave particle representation with 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) gauge theory, that is consistent with both first- and second-class local constraints for strongly correlated condensed matter systems. In our consistent 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.
The flux phase solutions in the t-J model, the gauge theories with Chern-Simons term, promising new exciting physics, are so far obtained only within slave-boson (or fermion) approaches, where local gauge invariance is not taken into account exactly. We inspect the possibility to do this, mainly, in the example of the t-J model. The analysis of different parametrizations in the functional representation shows that (i) calculations which take into account constraints for the number of on-site-available states are describing quasiparticles in terms of the wrong local statistics, and contain gauge non-invariant objects; (ii) application of the projection technique in the slave-boson(fermion) representation reproduces the correct statistics, and is exactly equivalent to the conventional diagram technique for Hubbard operators. In the language of slave bosons this means that the Fourier transformation leading to Bloch states should be made only after accounting constraints in the measure of functional integral.
We extend Abelian and non-Abelian bosonization formulas found in a previous paper, in combination with coherent-state methods, to present a systematic derivation of the slave-fermion, slave-boson, and semion representations of the two-dimensional t-J model in path-integral form. The slave-fermion and slave-boson representations are shown to arise from two different gauge fixing constraints introduced in the path-integral representation of the bosonized t-J model. For each representation we discuss the approximations leading to a mean-field theory. In the mean-field theory based on the semion representation, the holons are shown to be ``Dirac semions.''
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.
In a recent Letter, Nayak argued that slave partices are always confined in gauge theories of strongly-correlated electron systems like the t-J model. The argument mostly relies on Elitzur's theorem and the fact that the effective gauge theory under discussion is at infinite coupling. On the other hand, in the previous papers we studied dynamics of the gauge theory of the t-J model showing that the slave particles are in the Coulomb phase below certain critical temperature T{CSS} that depends on hole doping. We comment on the discrepancy between the results in two papers.