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

Xi Luo,Jianqiao Liu and Yue Yu,*

1 College of Science,University of Shanghai for Science and Technology,Shanghai 200093,China

2 Department of Physics,Fudan University,Shanghai 200433,China

Abstract With the Becchi–Rouet–Stora–Tyutin (BRST) quantization of gauge theory,we solve the longstanding 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 λi which ensures the constraint: in the Hamiltonian formalism of the theory,λi is time-independent and commutes with the Hamiltonian while in the Lagrangian formalism,λi(t) becomes time-dependent and plays a role of gauge field.This implies that the redundant degrees of freedom of λi(t)are introduced and must be removed by the additional constraint,the gauge fixing condition (GFC) ∂tλi(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 λi(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 λi(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 Tc is then close to that in cuprates.

Keywords: slave particle,BRST,gauge theory,high Tc superconductivity,Hubbard model t–J model

1.Introductions

The Hubbard model,though it is simple,is in the central position for understanding strongly correlated electron systems [1,2].The single-band Hubbard model [3] was considered as the starting point to explain the high-Tcsuperconductivity (SC) [4].The strong Hubbard repulsion limit of the Hubbard model tends to the t–J model,which also was derived from a more realistic model for cuprates [5–7].Numerous subsequent studies on these two models were done either analytically or numerically.Many numerical simulation results are very impressive but they are basically subject to the computational resources and so are far from conclusive ones.Useful analytical approaches include the Gutzwiller approximation [8],mean field (MF) theories [9–17],and the gauge theory [18–24] based on the slave particle formalism[25–31].Analog to the slave particle models,a large class of models,e.g.spin-fermion models,were developed to study the strongly correlated systems such as cuprates based on the spin fluctuations [32–36].

Recently,the renormalized MF theory based on the Gutzwiller projection [14–17] has been generalized to that in the form of statistically-consistent Gutzwiller approximation[37],which was proved to be equivalent to the slave boson theory.The results in terms of the further subsequent generalization,i.e.a systematic diagrammatic expansion of the variational Gutzwiller-type wave function may be quantitatively compared with the experimental properties of cuprates [38].

We are not going to focus on the results obtained by these methods because they are too fruitful to be summarized.We will try to improve the slave boson method and fix some shortcomings of the theory.For example,the slave particle theory looks very powerful because it exactly maps a strongly correlated electron system to a weakly coupled slave particle one but things become difficult when dealing with local constraint conditions Ti=0 (see equation (3)),i.e.only one type of the single slave particle can occupy a lattice site i.The temporal component of the gauge field,λi(t) and the spatial components of the gauge field which are introduced to compensate for the gauge symmetry breaking by the MF approximation are not dynamic so the conventional perturbation theory is not applicable.In this paper,we try to solve these problems.

In terms of Dirac’s approach to solve the first-class constraint systems,a term -∑iλiTiwith λibeing the Lagrange multiplier is added to the Hamiltonian H.Since there are no temporal or spatial derivatives of λiin the Hamiltonian,[H,λi]=0 and then λiwill not evolve with time.In literature,λiwas simply relaxed to a time-dependent field λi(t) and as the temporal component of the gauge field.This introduces the redundant degrees of freedom because λishould be kept static,i.e.an additional constraint∂t λi(t)≡λ˙i(t)=0must be enforced.This point was missed before.Instead,in the MF approximation,a conventional approximationλi(t)=a constant with no spatial and temporal dependence,was taken.Althoughis zero,obviously this brings many unphysical degrees of freedom so that the MF theory after this approximation is not reliable or controllable.Many further improvements are proposed to deal with this issue but they do not bring conclusive results[18–21].Recent development in the statistical Gutzwiller approximation sheds light to systematically relieving the difficulties that original renormalization MF theory meets[37,38].In this paper,we make efforts to improve the slave particle theory by considering the additional constraint(t)=0instead ofλi=

When an electron operator is decomposed into slave particles,a gauge symmetry is induced and λi(t)behaves as a gauge potential in the temporal direction.To remove the redundant gauge degrees of freedom,one has to introduce a GFC while keeping the physical observables is gauge invariant.Simply settingλi(t)=is a GFC but it is not a good GFC because it violates the constraint Ti=0 and brings unphysical degrees of freedom.For a gauge theory with constraints,the GFC must be consistent with the constraints.

In this paper,we focus on the pairing states of the spinons and study the BCS-type MF states.In this case,the spatial components of the gauge field are not necessary to be introduced.We examine the spinion pairing gap of the Hubbard and t–J models.For the Hubbard model,it was found that the MF state of the slave bosons at half-filling in a small U is an s-wave SC state [41,42].This is obviously wrong because the Hubbard model in the weak coupling limit is a conventional metal.We show that after considering the dynamics of λi(t)induced by the constraint(t)=0,the SC state is not stable because integrating over λi(t) induces an unusual pairing instability of spinon’s Fermi surface and the spinon is gapped.This destroys the SC at half-filling.

Similarly,for the t–J model at the spinon pairing gap state or the SC state,integrating over λi(t) contributes an additional unusual term to the spinon pairing.This additional contribution does not destroy the MF SC gap but may substantially suppress it.Numerically,the gap will be smaller than at least a factor of one-fifth of the MF SC gap.Thus,one can expect the d-wave SC critical temperature Tc,whose MF value is of the order 1000 K,is substantially lowered and might be comparable with that of cuprates.

This paper is organized as follows: in section 2,weexplain why the additional constraint(t)=0is necessary for the slave particle theory.We relate this additional constraint to the BRST quantization of the gauge theory.In section 3,we apply our theory to the Hubbard model at halffilling and show the SC state in the small U is unstable.In section 4,we study the spinon gap suppression by λi(t)'s fluctuation in the t–J model and discuss the implication to Tcof curpates.In section 5,we conclude this work and schematically look forward to the prospect of the applications of our theory to the strongly correlated systems.

2.Constraint to the Lagrange multiplier and BRST quantization

For a strongly correlated electron many-body system,a conventional perturbation theory based on the Fermi liquid theory does not work.In order to turn the strong interacting electron model to an equivalent weak coupling theory,a powerful method called the slave boson/fermion theory is applied[25–28].For the electron operator ciσat a lattice site i,the local quantum space is{|0〉,|↑〉,|↓〉,|↑↓〉}.The completeness condition reads

and so on.The completeness condition (1) maps to a local constraint

2.1.Additional constraint

For a Hamilton system with constraints,we follow Dirac’s method to solve the constrained system and introduce a Lagrange multiplier λi.The Hamiltonian for the constrained problem is given by

In Schrödinger’s picture,H,λiand Tiare all time-independent.Notice that [Hλ,λi]=0 and then λidoes not evolve as time.

where∏Φiare the canonical conjugate fields of Φi.The Lagrangian (7) is exactly the same as (6).

2.2.Gauge symmetry

One way to remove the redundant gauge degrees of freedom is replacing LGIby the Lagrangian (6)

where

This is a correct gauge fixing Lagrangian of the Abelian gauge theory but equation (11) is not gauge invariant.In order to resolve this paradox,we recall the Faddeev–Popov quantization of the gauge theory.We insert 1 into the gauge invariant (9) to fix the redundant gauge degrees of freedom in terms of

and finally [43]

where N(ξ)is an unimportant infinity constant.The path integral(14) is gauge invariant.Comparing (14) and (11),they differ from a factordet()after dropping N(ξ).At the present case,this determinant does not contain any fields and is a constant.This means that (11) is equivalent to (14).Therefore,up to a constant determinant,(11) is gauge invariant.However,for a non-Abelian gauge theory,the determinant in general is dependent on the gauge field and can not be dropped.This is why Faddeev–Popov ghost fields are introduced.

2.3.BRST quantization

Integrating over the ghost fields,the path integral WBRSTrecovers the path integral (11).Therefore,the BRST quantization of the gauge theory is exactly equivalent to the conventional path integral quantization.Notice that this BRST quantization may also be applied to non-Abelian gauge theory such as the SU(2)gauge theory of the slave boson[20].In the quantization of the non-Abelian gauge theory,the determinant for the non-Abelian gauge theory will not be easily treated without introducing the Faddeev–Popov ghosts.

The benefits gained from the BRST quantization are that:

(1) Because the BRST symmetry is a global symmetry with respect to the fermionic constant ∈,one can define the conservation fermionic charge from Nother’s theorem

All the physical states which are ghost-free obey

This recovers Ti=0 and(t)=0because ξ is an arbitrary constant.From the gauge theory point of view,λi(t)=is also a GFC,which removes the redundant degrees of freedom but the constraint Ti=0 is relaxed to 〈Ti〉=0.This brings other unphysical degrees of freedom into the quantum state space.The gauge theory developed in [18–21] tried to solve this problem in a different way from ours.

(2) The nilpotency Q2=0 resembles the external differential operator d2=0 in the deRahm cohomology.The constraint (18) is called a BRST cocycle condition and all physical states form the BRST cohomology group which topologically classifies the strongly correlated systems.

(3) Introducing the ghost fields greatly simplifies the quantization of the non-Abelian gauge theory which we will not involve in here.For the Abelian gauge theory considered in this paper,the ghost fields are decoupled to the gauge field and can be integrated away.Therefore,we will use the path integral(11).For finite temperature T,if mapping t →iτ,the path integral turns to the partition function

3.The Hubbard model at half-filling

To be concrete,we take the repulsive Hubbard model on a square lattice at half-filling as an example.The model Hamiltonian is given by

As we have argued,to study the detailed properties the various phases,we need to do various MF approximations fluctuated by the spatial components of the gauge field which is not the task in this work.We only restrict on the fluctuation from λi(t)and examine the instability of the BCS MF states at half-filling for small U.The original Hubbard model at halffilling is metallic for small U,while the previous slave boson MF theory gave a SC phase[41,42].In this SC phase,charge and spin excitations are gapless and the slave bosons condense [41].Neglecting the boson fluctuation of the condensate,the effective Lagrangian reads

where g is a constant which is arbitrary according to the constraint Ti=0.The a-dependent part inreads

Integrating away ak(ω),the effective interacting Lagrangian between the spinons reads

The pairing Hamiltonian is then given by

4.t-J model

We are going to the large U limit.It was known that the SC MF theory of the t–J model has a SC Tc～1000 K.Let us see if λifluctuation can suppress it.The t–J model Hamiltonian on the square lattice is given by

The effective slave boson t–J Lagrangian then reads

Replacing Leffin equation (11) by,one has the path integral for the t–J model.If we ignore the condensed holon fluctuation,the effective low-lying Lagrangian in the SC MF state is given by

Fig.2.Numerical result ofof ξ.We choose the material data of cuprates [44],i.e.J～0.12 eV,χf～0.2-0.3,ρh～0.18-0.25,and μf～-0.05 eV.The four curves in the diagram,from the top to bottom,correspond to Jχf/4+tρh=0.10 eV,0.11 eV,0.12 eV,and 0.13 eV,respectively.

5.Conclusions and prospects

We have properly dealt with the local constraint conditions in the slave boson representation of the strongly correlated systems.We argued that as a gauge theory with Dirac’s firstclass constraint,taking the GFC that removes the redundant gauge degrees of freedom must be consistent with the constraint.The BRST quantization is a consistent method to do that although the final path integral for the Abelian gauge theory is decoupled to the ghost fields.We have applied our theory to the Hubbard model at half-filling and found that the ground state of the system in small U is indeed a conventional metal.We showed that the MF s-wave SC state obtained by the slave boson representation in a previous study is not stable against the gauge fluctuation of the gauge field λi(t).For the strong coupling system,we studied the t–J model.We focused on the gauge fluctuation to the d-wave SC gap and found that it was substantially suppressed to a factor of onefifth.For cuprates,this means that the MF SC Tcis lowered from 1000 to 200 K.As we have mentioned,the gauge fluctuation from the spatial components of the gauge field was not considered.It might further reduce Tcto be comparable to that of the cuprates materials.

Historically,the MF phase diagram of the t–J model was studied in the early days when the high TcSC was found in cuprates.The gauge fluctuation to various MF states was studied.However,as we see here,the GFC might not be introduced properly because the spatial components of the gauge field also play a role of the Lagrange multiplier to the constraint on the currents.Additional constraints are also needed.This may endow the gauge field with dynamics.Then,the gauge invariant physical quantities can be calculated with perturbation theory.For instance,one can calculate the renormalized pairing gap by Dyson’s equation using the perturbation theory

Acknowledgments

This work is in memory of Professor Zhong-Yuan Zhu for his discussions with YY in the possible application of the BRST quantization to strongly correlated systems thirty years ago.The authors thank Professor Qian Niu for his insightful comments and resultful discussions.We are grateful to Jianhui Dai and Long Liang for useful discussions.This work is supported by NNSF of China with No.12174067.

Appendix.Dispersion and renormalized pairing gap

The SC dispersion relation in the t–J model can be solved by equation (37)

This gives the dispersion relation.Defining the renormalized gap by

we have

The ground state energy is given by

where the … stands for the ξ-independent part.In the numerical calculation,we choose the parameters of cuprates,i.e.J～0.12 eV, χf～0.2-0.3, ρh～0.18-0.25,and Jχf/4+tρh～0.1-0.13 eV [44].The spinon chemical potential μfis determined by

where ρfis the spinon density and G is the Gibbs free energy.To the zeroth order of ξ,it reduces to