Slave Particle Studies of the Electron Momentum Distribution in the Low Dimensional $t-J$ M

a r X i v :c o n d -m a t /9304011v 1 8 A p r 1993Slave Particle Studies of the Electron Momentum Distribution in the Low Dimensional t ?J Model Shiping Feng ?,1J.B.Wu,2Z.B.Su,1,2and L.Yu 1,21International Centre for Theoretical Physics,P.O.Box 586,34100Trieste,Italy 2Institute of Theoretical Physics,Chinese Academy of Sciences,Beijing 100080,China The electron momentum distribution function in the t ?J model is studied in the framework of slave particle approach.Within the decoupling scheme used in the gauge ?eld and related theories,we treat formally phase and amplitude ?uctuations as well as constraints without further approximations.Our result indicates that the

electron Fermi surface observed in the high-resolution angle-resolved photoemission and inverse photoemission experiments cannot be explained within this framework,and the sum rule for the physical electron is not obeyed.A correct scaling behavior of the electron momentum distribution function near k ～k F and k ～3k F in one dimension can be reproduced by considering the nonlocal string ?elds [Z.Y.Weng et al.,Phys.Rev.B45,7850(1992)],but the overall momentum distribution is still not correct,at least at the mean ?eld level.

PACS numbers:71.45.-d,75.10.Jm

I.INTRODUCTION

The t?J model is one of the simplest models containing the essence of strong correlations and its implications for oxide superconductivity[1,2]still remain an outstanding problem. The t?J model was originally introduced as an e?ective Hamiltonian of the Hubbard model in the strong coupling regime,where the on-site Coulomb repulsion U is very large as compared with the electron hopping energy t,and therefore the electrons become strongly correlated to avoid the double occupancy.In this case,the electron’s Hilbert space is severely restricted due to this constraint σC?iσC iσ≤1.Anderson[1]and later Zhang and Rice[2]

have argued strongly that the basic physics of oxide superconductors can be described by the t?J model.

The normal state properties of oxide superconductors exhibit a number of anomalous properties in the sense that they do not?t in the conventional Fermi liquid theory[3,4]. Some properties can be interpreted only in terms of a doped Mott insulator[3,4].A central question in the theory of these strongly correlated systems concerns the nature of the elec-tron Fermi surface(EFS)[4].The high-resolution angle-resolved photoemission and inverse photoemission experiments[5]demonstrate the existence of a large EFS,with an area con-sistent with the band structure calculations.Since the band theory is consistent with the Luttinger theorem[6],this means that the EFS area contains1-δelectron per site,where δis the hole doping concentration.Although the topology of EFS is in general agreement with one-electron-band calculations,the Fermi velocity is quite di?erent.This indicates that electron correlations renormalize considerably the results obtained in the framework of a single-particle treatment.It has recently been shown by small cluster diagonalization that in a two-dimensional(2D)square lattice the EFS within the t?J model is consistent with Luttinger’s theorem[7].Monte Carlo simulations for a nearly half-?lled2D Hubbard model also support this result[8].Moreover,the electron momentum distribution function of a2D t?J model was studied[9]by using the Luttinger-Jastrow-Gutzwiller variational

Fermi edge.For one-dimensional(1D)large U limit Hubbard model which is equivalent to the t?J model,Ogata and Shiba[10]obtained the electron momentum distribution func-tion by using Lieb-Wu’s exact wave function[11].Their result also shows the existence of EFS as well as the singular behavior at k～k F and k～3k F in the momentum distribution function.Furthermore,Yokoyama and Ogata[12]studied the1D t?J model by using exact diagonalization of small systems and found the power-law singularity appearing at k F in the momentum distribution function.

So far,strong correlation e?ects can be properly taken into account only by numerical methods[7–10,12],such as variational Monte-Carlo technique[13],exact cluster diagonaliza-tion[14],and various realizations of quantum Monte-Carlo method[15].Apart from these numerical techniques,an analytical approach to the t?J model receiving a great deal of attention is the slave particle theory[4,16],where the electron operator C iσis presented as C iσ=a?i f iσwith a?i as the slave boson and f iσas the fermion,or vice versa.This way the non-holonomic constraint

C?iσC iσ≤1(1)

σ

is converted into a holonomic one

a?i a i+ σf?iσf iσ=1,(2) which means a given site cannot be occupied by more than one particle.A new gauge degree of freedom must be introduced to incorporate the constraint,which means that the slave particle representation should be invariant under a local gauge transformation a i→a i e iθ(r i,t),f iσ→f iσe iθ(r i,t),and all physical quantities should be invariant with respect to this transformation.We call such slave particle approach as”conventional”.The advantage of this formalism is that the charge and spin degrees of freedom of electron may be separated at the mean?eld level,where the elementary charge and spin excitations are called holons

?uids of spinons and holons represent the same set of electrons,and they must,on average,?ow together.The decoupling of charge and spin degrees of freedom in the large U limit Hubbard model is undoubtedly correct in1D[10],where the charge degrees of freedom of the ground state are expressed as a Slater determinant of spinless fermions,while its spin degrees of freedom are equivalent to the1D S=1

following Ref.20,to show that the scaling behavior of the electron momentum distribution function near k～k F and k～3k F in1D can be reproduced by considering the nonlocal string?elds in the mean?eld approximation,but there is still no EFS,and the sum rule for the electron number is still violated.If Luttinger theorem is obeyed,the Fermi volume is invariant under interactions.Thus our mean?eld result seems to indicate that the correct scaling behavior of the electron momentum distribution does not guarantee the existence of EFS(in the sense of global distribution)in the same theoretical framework.Section IV is devoted to a summary and discussions on related problems.

II.FORMAL STUDY OF ELECTRON MOMENTUM DISTRIBUTION

The slave particle theory can be the slave boson or the slave fermion theories according to statistics assigned to spinons and holons.The slave boson formulation[21]is one of the popular methods of treating the t?J model.This method,however,does not give a good energy of the ground state[22].For example,in the half-?lled case,where the t?J model reduces to the antiferromagnetic Heisenberg model,the lowest energy state obtained by this method fails to show the expected long-range N′e el order,and the energy is considerably higher than numerical estimates[13].Also,it does not satisfy the Marshall sign rule[23].However,this theory provides a spinon Fermi surface[17]even in the mean ?eld approximation.Alternatively,the slave fermion approch naturally gives an ordered N′e el state at half-?lling[24],which obeys the Marshall[23]sign rule.The ground state energy obtained in this case is much better than the slave boson case.At the mean?eld level,they are quite di?erent although in princple they should be equivalent to each other. The di?erences may be reduced by going beyond the mean?eld[25]approximation.In the following,we use both approaches to discuss the electron momentum distribution function and the sum rule obeyed by it,and a direct comparison of the obtained results is made.

First consider the slave fermion representation in which the electron operator can be

with the constraint σb?iσb iσ+e?i e i=1.In this case,the Lagrangian L sf and the partition function Z sf of the t?J model in the imaginary timeτcan be written as

L sf= iσb?iσ?τb iσ+ i e?i?τe i+H+ iλi(e?i e i+ σb?iσb iσ?1),(3) H=?t σe i e?j b?iσb jσ+J

DeDe?DbDb?Dλe?i e i e? dτL sf(τ),(6)

Z sf

1?δ= σ=1

Dσ(R i?R j,τ?τ′)=?

=?

1

Z sf

DeDe?DbDb?Dλe?i(τ)e j(τ′)b iσ(τ)b?jσ(τ′)e? dτL sf(τ).(10)

The spinon-holon scattering contained in eq.(10)is a four particle process,therefore the spinon and holon are strongly coupled.At the mean?eld level,the holons and spinons are separated completely.However,in many theoretical frameworks,such as the usual gauge theories discussed by many authors[17,26],where the vertex corrections are ignored,but the RPA bubbles are included,the spinons and holons are still strongly coupled by phase ?uctuations(gauge?elds)[17],amplitude?uctuations[25]and other e?ects.Nevertheless, in all these cases the electron Green’s function Gσ(k,iw n)can be presented as a convolution of the fermion Green’s function g(k,iw n)and boson Green’s function Dσ(k,iw n)

Gσ(k,iw n)=1

β w m g(q,iw m)Dσ(q+k,iw m+iw n).(11)

The coupling of the gauge?eld to these particles can be strong and a partial resummation of digrams has been carried out[17],but the vertex corrections were neglected,being the essence of the decoupling approximation.This is an important but the only approximation apart from the sum rules for slave particles(6)-(7)in our formal study.In what follows, one will?nd that many di?culties might appear due to this approximation.The fermion and boson Green’s functions can be expressed as frequency integrals of fermion and boson spectral functions as

g(k,iw n)= ∞?∞dw iw n?w,(12)

respectively[27].Substituting eqs.(12)and(13)into eq.(11),we obtain the electron Green’s function by summing over the Matsubara frequency iw m,

Gσ(k,iw n)=

1

2π ∞

?∞

dw′′

iw n+w′?w′′

,(14)

where n F(w′)and n B(w′′)are the Fermi and Bose distribution functions,respectively.The spectral functions A e(q,w′)and A bσ(k,w′′)obey the sum rules coming from the commutation relations,

∞

?∞

dw

2π

A bσ(k,w)=2.(16) The electron’s Hilbert space has been severely restricted,but the fermion and boson them-selves are not restricted.By an analytic continuation iw n→w+iηin the electron Green’s function(14),the electron spectral function A cσ(k,w)=?2ImG cσ(k,w)can be obtained as

A cσ(k,w)=

1

2π

A e(q,w′)A bσ(q+k,w+w′)[n F(w′)+n B(w+w′)].(17) Therefore,the electron spectral function A cσ(k,w)obeys the sum rule

σ ∞

?∞

dw

2π

n B(w)A bσ(k,w),(19)

This is because n B(w)A bσ(k,w)and n F(w)A e(k,w)can be interpreted as the probability functions of state k with energy w for boson and fermion,respectevely.A similar interpretion is also valid for the electron spectral function A cσ(k,w).Thus the number of electrons in state k is obtained by summing over all energies w,weighted by the electron spectral function

n c(k)= σ ∞?∞dw N qσn bσ(k+q)n e(q).(21) In eq.(21),the?rst term of the righthand side1-δis independent of k,and therfore it

is true for all k states of the entire Brillouin zone.The value of the second term of the righthand side is of the order ofδ,and hence it is not enough to restore the EFS,i.e., the distribution outside the should-be EFS is still of the order1.Fig.1shows the mean ?eld electron momentum distribution n c(k)for dopingδ=0.125in1D.Beyond the mean ?eld approximation,but still within the decoupling scheme(11),there are no important corrections for the global features of the electron momentum distribution and EFS,but the Fermi velocity will be modi?ed.This is because the essential global features of the electron momentum distribution are dominated by the?rst term of the righthand side,1?δin eq.

(21),and the second term of the righthand side in eq.(21),which is of the order ofδ,is not enough to cancel out1?δat each k outside the k F state,i.e.,k F0, and?π0,and?k F

1

Alternatively,in the slave boson representation,the electron operator can be expressed as

C iσ=f iσb ?i ,where f iσis fermion and b ?i is slave boson,with the contraint

σf ?iσf iσ+b ?i b i =1.In this case,the Lagrangian L sb and the partition function Z sb of the t ?J model may be written as

L sb = iσf ?iσ?τf iσ+ i b ?i ?τb i

+H + i λi (b ?i b i

+ σf ?iσf iσ?1),(23)H =?t

σ

b i b ?j f ?iσf jσ?μ iσf ?iσf iσ+J Z sb

DbDb ?DfDf ?Dλb ?i b i e ? dτL sb (τ),(26)1?δ= σ=1

2π

A cσ(k,w )=1+δ,(28)

n c (k )=1?δ+

1

In comparison with eq.(21),the second term of the righthand side of eq.(29)changes sign as an essential di?erence of the electron momentum distribution function between the slave boson and slave fermion representations.However,the value of the second term of the righthand side of eq.(29)is also of the order ofδ,and it is also not enough to restore the EFS.Fig.2shows the mean?eld electron momentum distribution n c(k)for dopingδ=0.125 in1D.Beyond the mean?eld approximation,but still within the decoupling scheme,there are no important corrections for the global features of the electron momentum distribution. The reason is almost the same as in the conventional slave fermion approach.The essential global features of the electron momentum distribution are dominated by the?rst term of the righthand side1?δin eq.(29).Since n fσ(k+q)≥0and n b(q)≥0,and a plus sign appears between the?rst and the second terms of the righthand side in eq.(29),it is impossible to remove those1?δstates beyond the Fermi points.The best situation is that an amount of order ofδis added into each k′state within the Fermi points.The above discussion is also valid for the2D case.Therefore,there is no EFS within the decoupling scheme in the conventional slave boson approach.In this case,the sum rule of the electron number is also violated as in the slave fermion case.The di?erence between these two approaches is that the electron number is more than the expected value in the slave boson representation,but less than it in the slave fermion representation.

These results indicate that there is no real EFS for the electron momentum distribution function n c(k)near k～k F within the decoupling scheme in the conventional slave particle approach.The sum rule of the physical electron number is violated.The scaling behavior of the electron momentum distribution near k F is also not correctly described by this scheme. The theory can be applied to both1D and2D systems.It was proposed[17,26]that the low energy physics of the t?J model can be described by a theory of fermions and bosons coupled by a gauge?eld.Our results also indicate that the gauge?elds(phase?uctuations)discussed by many authors[17,26,28]are not strong enough to restore the EFS under the decoupling

experiments[5]that oxide superconducting materials exhibit an EFS,and the EFS obeys Luttinger theorem[6],which means the Fermi volume is invariant under interaction.In this sense the EFS of strong correlated systems should also be described by an adequate theory even in the mean?eld approximation.In fact,the rearrangements of spin con?guations in the electron hopping process[10],which are nonlocal e?ects and beyond the conventional slave particle approach,play an essential role to restore EFS for a system of decoupled charge and spin degrees of freedom.Thus the electron is not a composite of holon and spinon only, and it should include other?elds which describe the nonlocal e?ects.In the next Section, we will see that the scaling behavior of n c(k)near k～k F and k～3k F can be obtained in1D by considering some nonlocal e?ects.Before going to the next Section,we would like to emphasize that the”Fermi Surface”obtained previously[29]from the slave boson theory is a spinon Fermi surface,not the real EFS.In order to interpret the high-resolution angle-resolved photoemission and inverse photoemission experiments[5],the spinon Fermi surface is not enough because experiments have shown the EFS for real electrons.

III.THE SCALING BEHA VIOR OF THE ELECTRON MOMENTUM

DISTRIBUTION

FUNCTION AND THE EFFECTS OF NONLOCAL STRING FIELD IN1D

Interacting1D electron systems generally behave like Luttinger liquids[18]in which the correlation functions have power-law decay with exponents which depend on the interaction strength.For the1D Hubbard model,an exact solution was explicitly obtained by Lieb and Wu[11].In the limit of J→0,the t?J model is equivalent to the large U limit Hubbard model[1,2,10].Thus the1D t?J model provides a good test for various approaches.The asymptotic forms of some correlation functions as well as the single electron Green’s function have been obtained by many authors using di?erent approximations[20,30].In particular, Ogata and Shiba[10]obtained the electron momentum distribution function n c(k)by using

and the correct scaling behavior of n c(k)at k～k F and k～3k F.It is well established that the Landau Fermi liquid theory breaks down in1D,namely,(1)there is no?nite jump of the momentum distribution at the Fermi surface;(2)there is no quasi-particle propagation and(3)the spin and charge are separated.On the other hand,there is still a well de?ned Fermi surface at k F,as one would expect from the Luttinger theorem.It is remarkable that the exact solution of the1D Hubbard model demonstrates explicitly these two aspects at the same time.In this sense it is important to check whether both aspects remain in any appropriate approximate treatment of the1D model.To our knowledge,such global features of the electron momentum distribution function even in1D were obtained only by using numerical techniqes.In this Section,we try to study analytically this problem.A CP1 boson-soliton approach including the e?ects of the nonlocal string?eld to study the large U Hubbard model was recently developed by Weng et al.[20].They have shown that the electron is a composite particle of holon,spinon,together with a nonlocal string?eld.We will draw heavily on their results,but we try to make the presentation self-contained.The correct scaling form of the electron momentum distribution function can be obtained even in the mean?eld approximation if one considers the nonlocal string?eld.

For convenience,we begin with the t?J model,but consider the limit J→0+which is a?xed point di?erent from J=0.In this case,the t?J Hamiltonian may be written as

H=?t σC?iσC jσ+h.c.(31) As in the CP1boson-soliton approach[20],the electron operator C iσcan be expressed in

the slave fermion representation including the e?ects of nonlocal string?elds as

C iσ=e?i e?iπ

2 l

that b iσannihilates the spinσand is a nonlocal phase shift G i=e iπ

2X j h

j

,(34)

and the Hamiltonian may be rewritten as

H=?t iσ[b?iσb i+1σh?i+1h i+h.c]?μ i h?i h i

+ iλi(h?i h i+ σb?iσb iσ?1).(35) Following Weng et al.[20],we can obtain the asymptotic singular behavior of the electron momentum distribution by a similar calculation,and the result is in agreement with theirs.

factor will contribute an additional power-law decay in the asymptotic single electron Green’s function,and one may negelect it for simplicity if only interested in the leading contribution. The main e?ect due to the nonlocal string is already present.In the MFA,our situation is very similar.Eq.(32)holds exactly in the N′e el limit of the spin con?guration,and therefore the quantum many-body e?ects are neglected.This does not a?ect the leading long-wavelength behavior.In this paper,however,we are mainly interested in the global features of the electron momentum distribution.If Luttinger theorem[6]is obeyed,the Fermi volume is invariant under interaction and a strongly interacting system should also show a large EFS even in the mean?eld approximation.Thus a mean?eld treatment is a useful test for the present approach.

The mean?eld approximation to the Hamiltoian(35)amounts to treatingλi as a constant, independent of position and to decoupling the spinon-holon interaction in a Hartree-like approximation by introducing the order paramerters

χ= σ,(36)

φ=,(37)

where we have considered two sublattices A and B with i∈A,i+1∈B.The self-consistent equations aboutλ,χ,φ,andμcan be obtained by minimizing the free energy.

Under the same approximation as the one used to obtain the asymptotic single electron Green’s function(i.e.,G i factor in eq.(32)has been dropped),the single particle electron Green’s function G cσ(k,iw n)in the present mean?eld approxmation can be obtained as

G cσ(k,iw n)=1

2

(1+δ)

)

2

(1+δ)

?w q

,(38)

where

w k=λ?tφγk,εk=λ?μ?tχγk,γk=2cos(k).(39)

A cσ(k,w)=

π

2

(1+δ)

)]δ(w+εq?k±π

2π

A cσ(k,w)=1+δ,(41) n c(k)=1?δ?

1

2

(1+δ)

),(42) 1

2

(1+δ)

due to the presence of nonlocal string?elds.Fig.3shows the electron momentum distribu-tion n c(k)in the mean?eld approximation,including the nonlocal string?elds for doping δ=0.125in1D.The singular behavior of the electron momentum distribution function at k F and3k F is in qualitative agreement with the numerical results[10],but the sum rule for the total electron number is still violated and EFS still does not exist in the present framework under the above approximations.Beyond these approximations,the situation is not clear yet.If Luttinger theorem[6]is obeyed,then there are no important corrections to the global features of the electron momentum distribution function and no EFS beyond the mean?eld approximation in the same theoretical framework.Since the rearrangements of quantum spin con?gurations in the hopping process play an essential role to restore the EFS[10],then eq.(32)which holds exactly[20]in the N′e el limit of the spin con?guration, is perhaps not enough to describe all of these quantum spin con?gurations.Thus our feeling is that in this theoretical framework,the correct singular behavior of electron momentum distribution may be obtained,but it does not guarantee the existence of EFS.We also note that only the asymptotic forms were discussed in the previous works by many authors[30], while the global features of the momentum distribution function have not been considered by their theoretical methods.

IV.SUMMARY AND DISCUSSIONS

We have discussed the electron momentum distribution function and the sum rule of the electron number in the t?J model by using the slave particle approach.Under the decoupling scheme(11)used in the gauge?eld[17]and related theories,we have formally proved that EFS can not be restored and the sum rule of electron number is violated in the framework of conventional slave particle approach.For1D t?J model,the correct singularity forms can be reproduced by considering the nonlocal string?elds in the special mean?eld approximation,but there is still no EFS and the sum rule is still not satis?ed.

In the present slave particle approach to study t?J model,we have pushed out the upper Hubbard band.This means we have neglected the doubly occupied sites in the original slave particle approach[21]:C iσ=a?i f iσ+σd i f?i?σ,with a constraint a?i a i+ σf?iσf iσ+d?i d i=1,

where a?i is the slave boson,f iσis fermion,and d i is boson which describes the doubly occupied sites,or vice versa.In fact in the original slave particle approach the electron spectral function obeys the sum rule of conventional electron,i.e., σ ∞?∞dw

vertex corrections are neglected,i.e.,the decoupling scheme is adopted,the di?culty will remain.On the other hand,it is di?culty to introduce vertex corrections in the present form of the gauge?eld theory,because the infrared divergence has not yet been properly handled. Another possibility to avoid this di?culty is that one should make a new interpretation of the constraint(2)and the physical meaning of spinons and holons.In fact the constraint (2)is an operator identity,and one replaces the constraint(2)by eqs.(6)and(7)in the slave fermion approach,or eqs.(26)and(27)in the slave boson case.It is not clear that this is a correct way of imposing the constraint,i.e.,the charge is represented by a fermion, while the spin is represented by a boson in the slave fermion approach and vice versa in the slave boson version.The crucial point is to implement the local constraint rigorously[34]. Finally,according to the implications of numerical solutions[10],the electron is perhaps not a composite of holon and spinon only,but it rather should be a composite of holon, spinon together with something else.This is because the Bethe-ansatz Lieb-Wu’s exact wave function[10]for1D Hubbard model at large U limit may be written as

Ψ(x1,···,x N)=(?1)Q det[exp(ik i x Q

i

)]Φ(y1,···,y M),(44)

where the determinant depends only on the coordinates of particles(x Q

1<···

N

)but

not on their spins.Thus it is the same as the Slater determinant of spinless fermions with momenta k j’s.The spin wave functionφ(y1,···,y M)is the same as the Bethe’s exact solution of1D Heisenberg spin system.In the second quantization representation,a spinless fermion operator may be responsible for the Slater determinant of spinless fermions,but a simple fermion or boson operator may be not enough to describe the spin wave function.Perhaps it should be a fermion or boson operator together with something else,such as the Jordan-Wigner’s form[35]in1D,or a map of the three spin operators of the SU(2)algebra onto a couple of canonical boson operators,like the Holstein-Primako?form[36],are responsible for the spin wave function.We believe that a successful theoretical framework must include

To summarize,we are facing a dilemma:if we would like to split one electron into two particles,one keeping track of the charge,the other keeping track of the spin,and impose corresponding sum rules on them[eqs.(6)-(7)or eqs.(26)-(27)]and then use the decoupling scheme for their expectation values,the resulting electron distribution does not satisfy the sum rule and does not show an EFS.In this sense,we have proved in this paper a No-Go theorem.The alternative would be to give up the attractively simple interpretation of spinons and holons and to look for more complicated charge and spin collective excitations.

Finally we note that the present slave particle approach is di?erent from those?rst proposed by Barnes[37]and rediscovered and extended by Coleman[38],Read and Newns [39],and Kotliar and Ruckenstein[40]in their works on the mixed valence problem and the heavy fermion systems.In their formulation,the spin and charge degrees of freedom are not decoupled,where the auxiliary bosons keep track only of the environment by measuring the occupation numbers in each of the possible states for electron hopping.Their theory describes the properties of a Fermi liquid,and is another story.

ACKNOWLEDGMENTS

The authors would like to thank Dr.Z.Y.Weng for helpful discussions.S.P.Feng and Z.B.Su would like to thank the ICTP for the hospitality.

REFERENCES

?On leave from the Department of Physics,Beijing Normal University,Beijing100875, China.

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九年级课文翻译完整版

九年级课文翻译 Document serial number【NL89WT-NY98YT-NC8CB-NNUUT-NUT108】

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九年级英语u n i t课文 翻译 Document serial number【KK89K-LLS98YT-SS8CB-SSUT-SST108】

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