Evaluation of matrix elements in partially projected wave functions

Evaluation of matrix elements in partially projected wave functions

Noboru Fukushima,1 Bernhard Edegger,1 V. N. Muthukumar,2 and Claudius Gros1

1Department of Physics, University of the Saarland, 66041 Saarbrücken, Germany2Department of Physics, City College at the City University of New York, New York, New York 10031, USA

�Received 24 February 2005; published 7 October 2005�

We generalize the Gutzwiller approximation scheme to the calculation of nontrivial matrix elements betweenthe ground state and excited states. In our scheme, the normalization of the Gutzwiller wave function relativeto a partially projected wave function with a single nonprojected site �the reservoir site� plays a key role. Forthe Gutzwiller projected Fermi sea, we evaluate the relative normalization both analytically and by variationalMonte Carlo �VMC�. We also report VMC results for projected superconducting states that show noveloscillations in the hole density near the reservoir site.

DOI: 10.1103/PhysRevB.72.144505 PACS number�s�: 75.30.Gw, 75.10.Jm, 78.30.�j

I. INTRODUCTION

This paper concerns the calculation of matrix elementsusing projected wave functions of the form, ��= P��0�.Here, P=�i�1−ni↑ni↓� is a projection operator which ex-cludes double occupancies at sites i, and ��0�, a trial wavefunction. Projected wave functions of this form were origi-nally proposed by Gutzwiller to study electronic systemswith repulsive on-site interactions.1 The choice of ��0� de-pends on the problem under consideration. For instance, aprojected Fermi liquid state,

P��FS� = P �k�kF

ck↑† ck↓

† �0� , �1�

was used successfully in the description of liquid 3He as analmost localized Fermi liquid.2,3 Soon after the discovery ofhigh temperature superconductivity in the cuprates, projectedBCS wave functions were proposed as possible ground statesof the so-called t-J model.4,5 Early results from variationalMonte Carlo �VMC� studies as well as a renormalized meanfield theory based on Gutzwiller approximation showed thata projected d-wave BCS state,

PNP��BCS� = PNP�k

�uk + vkck↑† c−k↓

† ��0� , �2�

reproduces many features seen in the phase diagram of thehigh temperature superconductors.6–8 The projection opera-tor PN which fixes the particle number N in �2�, is usefulwhen considering the phase diagram near half filling.6 With-out PN in �2�, one would need to consider the effects ofparticle number fluctuations, which become singular nearhalf-filling.9–11

Detailed VMC studies have been carried out recently us-ing projected d-wave BCS states as variational wave func-tions for the two-dimensional Hubbard model,12 after a suit-able canonical transformation.3 Similar wave functions havebeen proposed in the literature for cobaltate superconductorsas well as organic superconductors.13,14 The Gutzwiller ap-proximation has been extended to finite temperatures15

within the slave-boson approach.16 The latter is equivalent tothe Gutzwiller approach at the mean field level, but leads to

inconsistencies in its orignal formulation when fluctuationsare taken into acount.17 Further developments include appli-cation to the attractive Hubbard model,18 extension to multi-band Hubbard models,19 and the formulation of a time-dependent Gutzwiller approximation allowing for theevaluation of the dynamical conductivity.20

In this paper, we are interested in extending Gutzwiller’sscheme to construct normalized single particle excitationsand to calculate relevant matrix elements and nontrivial nor-malization factors. These calculations are relevant, for in-stance, in the calculation of matrix elements in spectroscopicprocesses.21,22

In his original paper, Gutzwiller proposed that in calculat-ing expectation values of operators with projected wavefunctions, the effects of projection on the state ��0� could beapproximated by a classical statistical weight factor, whichmultiplies the quantum result.23 Thus, for example,

��O��

� g��0�O��0��0��0�

, �3�

where O is any operator, and g, a statistical weight factor.The basic idea is that the projection operator P reduces thenumber of allowed states in the Hilbert space, and invoking asimple approximation, such a reduction can be taken intoaccount through combinatorial factors. For example, expec-tation values of the kinetic energy operator ci

†cj +cj†ci and the

superexchange interaction between sites i and j, S� i ·S� j in theprojected subspace of states are renormalized by theGutzwiller factors,

gt =1 − n

1 − n/2, gs =

1

�1 − n/2�2 , �4�

where n is the density of electrons. In deriving these renor-malization factors, one considers the number of states that

contribute to ��O�� and to ��0�O��0�, respectively. Theratio of these two contributions is identified as the renormal-ization factor.

It is clear that this approach can be generalized to evaluate

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matrix elements of an operator O between different projectedstates. However, as we will see in this paper, many of thematrix elements that are of interest are reduced to the calcu-lation of matrix elements between partially projected wavefunctions of the form

��l�� = Pl��0�, Pl� = �i�l

�1 − ni↑ni↓� . �5�

The wave function ��l�� describes a state where double oc-cupancies are projected out on all sites except the site l,which we call the reservoir site. The reason for the appear-ance of reservoir sites is not far to seek. Consider, for ex-ample, the operator Pcl↑. Clearly, it can be rewritten as cl↑Pl�.Since calculation of matrix elements involving excited statesinvolve the commutation of projection operators withcreation/destruction operators, partially projected states ariseinevitably within the Gutzwiller scheme.

In this paper, we present a method to calculate matrixelements between a partially projected Fermi sea, i.e., a pro-jected Fermi sea with a reservoir site at l, as in �5�. We willshow that this problem has to be solved if we were to con-struct normalized particle/hole excitations of the �fully� pro-jected Fermi sea. The same problem arises when calculatingmatrix elements for particle/hole tunneling into the projectedFermi sea. We develop an analytical approximation to solvethis problem, and use it to calculate various matrix elements.We use VMC to test the validity of the approximation andfind that our analytical results for the partially projectedFermi sea are in good agreement with the results from VMC.

The outline of the paper is as follows: In Sec. II, wepresent results for the occupancy of the reservoir site. We usethese results in Sec. III, where we show how normalizedsingle particle excitations can be constructed from the pro-jected Fermi sea. In Sec. IV, we calculate the matrix ele-ments for particle/hole tunneling into the projected Fermisea. VMC results for density oscillations in the vicinity ofthe reservoir site for both projected Fermi sea and BCS statesare presented in Sec. V. The final section contains a summaryand discussion of results.

II. OCCUPANCY OF THE RESERVOIR SITE

Consider a partially projected wave function,

��l�� = Pl��0�, Pl� = �i�l

�1 − ni↑ni↓� . �6�

Double occupancy is projected out on all sites except the sitel, called the reservoir site. Unless specified otherwise, wetake ��0� to mean the Fermi sea. For the calculation of singleparticle excitations and matrix elements, we need expecta-tion values such as

��l��O��l��l��l��

= g��0�O��0��0��0�

, �7�

that generalize the Gutzwiller renormalization scheme �3� topartially projected wave functions. Note that the reservoirsite does not enter into the unprojected wave function ��0�.

This is in contrast to the impurity problem �which we do notconsider here�, where an impurity site breaks the transla-tional invariance of the unprojected wave function.

A. Gutzwiller approximation

In order to evaluate the generalized renormalization pa-rameters g� in �7�, we obviously need the normalization��l� ��l��. We define

X =��0�PP��0�

��0�Pl�Pl��0�=

��l��l��

, �8�

the norm of the fully projected state relative to the state withone reservoir site. Invoking the Gutzwiller approximation,we estimate this ratio by considering the relative sizes of theHilbert spaces,

X �

L!

N↑!N↓!Nh!

L!

N↑!N↓!Nh!+

�L − 1�!�N↑ − 1�!�N↓ − 1�!�Nh + 1�!

, �9�

where L=N↑+N↓+Nh is the number of lattice sites, N↑, N↓,and Nh, the number of up spins, down spins, and empty sites,respectively. The first term in the denominator of �9� repre-sents the number of states with the reservoir site being emptyor singly occupied; the second term represents the state withthe reservoir site being doubly occupied.

Equation �9� can be simplified in the thermodynamiclimit. We get

X =1 − n

�1 − n↑��1 − n↓�, �10�

where the particle densities, n�=N� /L ��= ↑ , ↓ � and n=n↑+n↓. The above argument can be extended to the case of twounprojected sites in an otherwise projected Fermi sea. Wethen get,

��0�PP��0��0�Plm� Plm� ��0�

= X2, �11�

where Plm=�i�l,m�1−ni,↑ni,↓�. We note for later use that

1 − X

X=

n↑n↓�1 − n�

. �12�

B. Exact relations

Assuming translation invariance for the unprojectedwavefunction ��0�, it is possible to derive the following ex-act expressions:

��1 − nl↑��1 − nl↓��l�= X�1 − n� , �13�

�nl��1 − nl−��l�= Xn�, �14�

�d��l� �nl↑nl↓��l�

= 1 − X , �15�

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for the occupancy of the reservoir site, where

�¯��l� ��l�� ¯ ��l��/��l��l�� .

The proof is straightforward. Consider for instance, the prob-ability �13� of finding the reservoir site empty. Since,

��0�P�1 − nl�P��0� = ��0�Pl��1 − nl↑��1 − nl↓�Pl��0��16�

we have,

��1 − nl↑��1 − nl↓��l�=

��1 − nl��

��l��l��

= �1 − n�X .

Equations �14� and �15� can be proved analogously.

C. VMC results for projected Fermi sea and BCS states

In Fig. 1, we compare �10� with VMC results for �d��l�=1−X. We find that the results from the generalizedGutzwiller approximation are in excellent qualitative agree-ment with the VMC results for a partially projected Fermisea. We also used VMC to obtain the same quantity usingprojected s /d-wave BCS states24 as variational states in thesimulation. The results for �d��l�

in BCS states are shown inFig. 2. In contrast to the projected Fermi sea, a clear devia-tion from the Gutzwiller approximation is seen. This under-scores the importance of pairing correlations in the un-projected wave function that are not completely taken intoaccount by the Gutzwiller approximation scheme. These dif-ferences between Fermi sea and BCS states are discussed inmore detail in Sec. V, where we consider density oscillationsin the vicinity of the reservoir site.

In the following we discuss some details of the VMCcalculations with one unprojected �reservoir� site l. As men-tioned earlier, single occupancy is enforced �by projection�on all other sites. Simulations are performed on a finitesquare lattice spanned by two vectors �Lx ,Ly� and �−Ly ,Lx�with periodic boundary conditions.25 The number of sites,L=Lx

2+Ly2. The numbers of up- and down-electrons are cho-

sen to be equal, N↑=N↓. The simulation for the local quantity

�d��l�= �nl↑nl↓��l�

has a larger statistical error than results formacroscopic quantities in uniform systems because the sum-mation over site indices yields effectively L times more sta-tistics for the latter. In order to overcome this problem, weupdate the reservoir site more often than the projected sites.Accordingly, the transition probability needs an extra weight-ing factor to keep the local balance. With this procedure, wecan improve the statistical accuracy by about one order ofmagnitude. In addition, we carry out measurements after ev-ery update. Usually, in VMC simulations, measurements areperformed every O�L� updates to obtain independent samplessince similar states return similar sampled data. However, inthe case of nl↑nl↓, a measurement returns only 0 or 1; viz., thesampled data can be different even when the states are simi-lar. Given this, the measurement after every update seemsmore reasonable as it reduces statistical errors. Furthermore,we have restricted updates to the transfer of a single electronto an unoccupied site, and excluded updates via the exchangeof two electrons. The calculation of the transition probabili-ties for the former update consumes time of O�N��, whereasthe time taken for the latter update is O�N�

2�. As the systemsize increases, this restriction achieves efficiency. We havecollected statistics from up to 60 independent runs over twodays, and the total number of updates amounts to 108–109.

For superconducting states, one can perform the VMCsimulation either with fixed particle number PNP��BCS�, orwith a fixed phase P��BCS�.10,25 For the latter choice, particlenumber fluctuation hinders the variational wave functionfrom reaching half filling unless the chemical potential �goes to infinity. On the other hand, the wave function can beoptimized by varying the gap �k even at half filling, if wechoose to fix the particle number. It is important to note thatsimulations with fixed particle number are done not with themost probable N of P��BCS�, but that of ��BCS�. This isbecause P decreases the average particle number.10 Through-out this paper we choose to fix the particle number whileworking with projected BCS states.

Let us define akvk /uk. For the d-wave BCS state, ak=0=0 in the thermodynamic limit. However, if one choosesak=0=0 in the finite system, the k=0 state is unoccupied,although it is the lowest energy state. One can also choose alarge value for ak=0. Usually, the difference between these

FIG. 1. �Color online� Double occupancy of the reservoir site,�n↑n↓��l�

=1−X, as a function of doping, for the partially projectedFermi sea. See �6� and �1�. Note the good agreement between theGutzwiller result �solid line�, Eqs. �10� and �15� and the VMC re-sults for the projected Fermi sea �open circles�. Statistical errors andfinite-size corrections are estimated to be smaller than the symbols.

FIG. 2. �Color online� Double occupancy at the reservoir site�n↑n↓��l�

=1−X, as a function of doping, for the partially projectedBCS wave function, see �6� and �2�. The parameterization followsRef. 25. Statistical errors and finite-size corrections are estimated tobe smaller than the symbols.

EVALUATION OF MATRIX ELEMENTS IN PARTIALLY… PHYSICAL REVIEW B 72, 144505 �2005�

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choices is O�1/N�. We expect an N electron system withak=0=0 to be similar to an N+2 electron system with largeak=0, because the two electrons that are no longer in the k=0 state can occupy other available states. However, thisargument fails at half filling for projected states, becausethere are no available states left for the two extra electrons.So it should not be surprising that X depends strongly onthese choices close to half filling; the ak=0=0 definition giveslarger X than the other does as shown in Fig. 3. At the otherfillings, our results show only O�1/N� of difference betweenthese choices. Except for Fig. 3, where we show both cases,all other results in this paper are obtained for a choice oflarge ak=0, i.e., we take ak=0 larger than any other ak.

The system size dependence is quite small except in thevicinity of half filling. In fact, it is qualitatively consistentwith the Gutzwiller approximation; size dependence entersonly as �Nh+1� /L in Eq. �9� and is negligible for large Nh. InFig. 3, we show the dependence of �d� on the system size. Asshown in Fig. 3, �d� approaches unity for the projected Fermisea. For the projected d-wave BCS state, we speculate thatthe value of �d� goes to unity too, because it does not satu-rate, but increases more rapidly as 1/L decreases.

III. SINGLE PARTICLE EXCITATIONS OF THEPROJECTED FERMI SEA

We consider the particle excitation

��k�+ � = Pck�

† ��0� , �17�

and the hole excitation

��k�− � = Pck��0� . �18�

Any calculation involving ��k�± � needs the respective norms,

Nk�± = ��k�

± ��k�± �. We now calculate these norms within the

generalized Gutzwiller approximation.

A. Particle excitation

For the particle excitation, we get,

Nk�+

NG= 1 − n + gt�n� − nk�

0 � = gt�1 − nk�0 � , �19�

where gt= �1−n� / �1−n��, NG= �� , and nk�0 = �ck�

† ck��0is the momentum distribution function in the unprojectedsite.

Equation �19� has appeared frequently in the literature.Here, we repeat its derivation to facilitate a comparison withthe analogous problem for hole excitations. The norm��k�

+ ��k�+ � is given by

Nk�+ = ��0�ck�PPck�

† ��0�

=1

Ll,m

eik�l−m��0�Pl��1 − nl−��cl�cm�† �1 − nm−��Pm� ��0�

=1

L

l

��0�Pl��1 − nl��1 − nl−��Pl��0�

+1

Ll�m

eik�l−m��0�Pcl�cm�† P��0�

= NG

��1 − n��

−NG

Ll�m

eik�l−m� ��cm�† cl��

��,

�20�

where we have used �16� for the diagonal contribution in thelast step. Invoking the Gutzwiller approximation for the off-diagonal term, Eq. �19� follows directly from Eq. �20�.

B. Hole excitation

The normalization of the hole excitation can be doneanalogously. We get,

Nk�−

NG=

1

NGLl,m

eik�l−m��0�Pl�cl�† cm�Pm� ��0�

=1

X�Xn� + �1 − X��

+1

NGLl�m

eik�l−m��0�Plm� cl�† cm�Plm� ��0� ,

where Plm=�i�l,m�1−ni,↑ni,↓�. The last term in the aboveequation corresponds to a hopping process between two res-ervoir sites. The generalized Gutzwiller approximation as-sumes that the matrix elements are proportional to the squareroots of the corresponding densities �13�–�15�.

Invoking the Gutzwiller approximation and using �11�, weget,

FIG. 3. �Color online� The size dependence of the double occu-pancy at the reservoir site, �15�, at half filling. The VMC result forthe Gutzwiller state �upper curve� seems to converge nicely to unityin the thermodynamic limit, in agreement with Eqs. �10� and �12�.The result for the projected d-wave show a pronounced dependenceon the occupancy of the k=0 state. See text for details.


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Nk�−

NG=

��k�− ��k�

− ��

= n� +1 − X

X

+nk�

0 − n�

X2n��1 − n�� X�1 − n� Xn� + Xn−�

1 − X�2,

�21�

for the normalization of the hole excitation relative to thenorm of the Gutzwiller wave function.

The general expression �21� for the hole normalizationcan be simplified upon using the Gutzwiller result �12� forthe relative norm X. We then get,

nk�0 − n�

n��1 − n�� 1 − n n� + n−�

�1 − X�/X�2

= �nk�0 − n��

��1 − n� + n−��2

�1 − n��1 − n�

= �nk�0 − n��

1 − n−�

�1 − n�,

for the last term in �21�. Finally, we get the simple result,

Nk,�−

NG= nk�

0 1 − n−�

�1 − n�=

nk�0

gt. �22�

It is interesting to compare this result for the normalizationof the hole excitation with the corresponding expression �19�for the particle excitation. The vanishing of the latter at halffilling could have been expected. But the divergence of Nk�

−

as n→1 is surprising. We will return to this point in the nextsection.

C. Consistency check

The norm Nk�+ has to vanish whenever ��k�

+ �= Pck�† ��0�

vanishes. For the Fermi sea this is the case when k�kF, i.e.,when nk�

0 =1. This physical condition is obviously fulfilledby �19�. Similarly, we expect Nk�

− to vanish for nk�0 =0, which

is satisfied by �22�. Thus, the Gutzwiller result �10� obeys thenormalization condition for the hole excitation and thetheory is consistent.

IV. TUNNELING MATRIX ELEMENTS

We now consider the tunneling of electrons and holes intoa projected wave function. Single particle tunneling into a

projected superconducting state has been considered recentlyby Anderson and Ong9 and Randeria et al.22 Here, we restrictourselves to the projected Fermi liquid state and evaluate thetunneling matrix elements by retaining systematically, allterms arising from the commutation of the electron creationand destruction operators with the projection operator P, asoutlined in Sec. III.

A. Particle tunneling

Consider first, the matrix element

Mk�+ =

��k�+ �ck�

† ��2

Nk�+ NG

. �23�

The numerator may be calculated easily by using the resultof �19�,

��0�ck�Pck�† P��0�

NG=

��0�ck�PPck�† ��0�

NG= gt�1 − nk�

0 � .

From the above expression we find that the particle tun-neling matrix element takes the form,

Mk�+ =

gt2�1 − nk�

0 �2

gt�1 − nk�0 �

= gt�1 − nk�0 � . �24�

It vanishes at half filling n→1, implying that the addition ofelectrons is not possible exactly at half filling because of therestriction in the Hilbert space.

B. Hole tunneling

Next we evaluate the matrix element

Mk�− =

��k�− �ck��2

Nk�− NG

, �25�

corresponding to the tunneling of holes into the projectedstate. Naively, we might expect this process to be allowed athalf filling, since the removal of electrons is not forbidden bythe projection operator. Consider now the matrix element inthe numerator of �25�. We follow the same procedure used toevaluate the norm of the hole wave function in Sec. III B anduse �12� and �14� and find,

��0�ck�† Pck�P��0�

NG=

1

NGLl,m

eik�l−m��0�Pl�cl�† cm�P��0� =

Xn�

X+

1

NGLl�m

eik�l−m��0�Pl�cl�† cm�Pl��0�

=Xn�

X+ �nk�

0 − n�� Xn−�

1 − X + X�1 − n� Xn�� 1 − n n�

X�1 − n��n�

= n� + �nk�0 − n��

n−�n� + �1 − n�n�

�1 − n��n�

= nk�0 . �26�

Using this expression together with the norm �21� of the hole excitation, we obtain the hole tunneling matrix element �25�,


144505-5

Mk�− =

nk�0 nk�

0

nk�0 /gt

= gtnk�0 , �27�

a surprising result, in that it vanishes at half filling �n↑=n↓=1/2� too.

The vanishing of the hole tunneling matrix element at halffilling is clearly related to the divergence of the norm of thehole excitation. This, in turn, is related to the fact that X→0, as n→1 �cf. Eq. �10��. The vanishing hole tunnelingmatrix element can then be understood as follows. When thereservoir site is doubly occupied, a single hole in the other-wise projected Fermi sea can be found in any of the latticesites. Consequently, when double occupancy of the reservoirsite occurs with probability 1, as it does at half filling, an“orthogonality catastrophe” occurs leading to zero overlapfor the tunneling matrix element. Note that the result �27�hinges on the exact functional dependence �12� of �1−X� /X on the particle densities n�. On the other hand, theparticle tunneling matrix element Mk�

+ is not affected by thefunctional form of the relative normalization factor X. If Xwere to vanish more slowly than �1−n� at half filling, then

from �21� and �26�, one could conclude that the hole tunnel-ing matrix element Mk�

− does not vanish as n→1, possiblyleading to an asymmetry between particle and hole tunnel-ing. Our analytical results preclude this possibility for theprojected Fermi sea. But we are unable to provide a definiteanswer for the projected superconducting states, in view ofthe discrepancy between the Gutzwiller approximation andthe VMC results �Fig. 2�. To understand this discrepancy, westudy density oscillations in the vicinity of the reservoir siteusing VMC.

V. DENSITY OSCILLATIONS NEAR THE RESERVOIRSITE

To clarify the limitations of the Gutzwiller approximationfor projected superconducting states, we use VMC to calcu-late the hole density in the vicinity of the reservoir site. Wefind that the density oscillations seen are very different forthe projected Fermi sea and the BCS states.

In Fig. 4, VMC results for the hole density

FIG. 4. �Top row� VMC results for the hole density nh�m�= ��1−nm��l��white/black correspond to high/low values of nh�m�� in the

partially projected state ��l��, for sites m other than the reservoir site l �marked by the cross�. Left: Fermi sea. Middle: s-wave state. Right:d-wave state. �Second row� Exact results for dh�m�, see Eq. �28�, in the unprojected state �color coding: white/black correspond to high/lowvalues�, for sites m other than the doubly occupied site l �marked by the cross�. Left: Fermi sea. Middle: s-wave state. Right: d-wave state.�All� The lattice has 372+1=1370 sites and n=1. For the BCS states, �=1.


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nh�m� = �1 − nm��l�,

are presented in the first row for the partially projected state��l��. The sites m are distinct from the reservoir site l�marked by a cross in the figure�. All results shown corre-spond to half filling, viz., n↑=n↓=0.5. We choose �=1 forthe BCS states.24 The vectors of periodic boundary condi-

tions are L�1= �Lx ,Ly� and L�2= �−Ly ,Lx�, respectively, withLx=37, Ly =1; including the reservoir site, L=Lx

2+Lx2=1370

sites. In the figure, white/black correspond to high/low val-ues of nh�m�, which is scaled by a logarithmic gray scalevarying in the range −8.5� log nh�m��−6. Thus, the samegray represents the same value in all three cases shown.

For the Fermi sea, we see that the hole is distributed moreuniformly than the other cases even though the diagonal di-rection has a larger probability of being occupied by a hole.The s-wave shows a checker-board pattern. The d-wave has aquasi checkerboard pattern where only one of four sites isblack, and the hole tends to be near the reservoir site. TheVMC results for the projected BCS wave functions are strik-ingly different in that the hole density is not uniform. On theother hand, the Gutzwiller approximation would be exact, ifall states in the Hilbert space contribute equally to the wavefunction. That would correspond to a uniform density ofholes. Clearly, the Gutzwiller approximation has to be ex-tended to treat projected superconducting wave functions.This is in agreement with our previous considerations, wherewe found that the functional form of X �Eq. �10�, derivedusing Gutzwiller approximation� agrees with the VMC cal-culations only for the projected Fermi sea, but not for BCSstates �see Figs. 1 and 2�.

To further investigate the effects of Gutzwiller projection,we also plot �second row of Fig. 4� the correlation function

dh�0��m� = �nl↑nl↓�1 − nm↑��1 − nm↓��

− �nl↑nl↓��1 − nm↑��1 − nm↓�� 28�

in systems without the Gutzwiller projection. This correlationfunction between a hole at site m and a doubly occupied siteat l corresponds to the quantity nh�m� for the partially pro-jected wave function close to half filling. This is because, inthe latter case, the unprojected site is doubly occupied. Notethat translation invariance implies that the second term in�28� does not depend on the site indices l and m, and is aconstant factor. Then, using Wick’s theorem, the correlationfunction dh

�0��m� is reduced to a function of �ci↑† cj↑� and

�ci↑cj↓�. The quantity dh�0��m� can be evaluated exactly, per-

forming a Fourier transform �we use the same system sizeand boundary conditions�. The logarithm of the correlationfunction is scaled in the second row of Fig. 4 by the grayscale varying in the range �−22,−4�. Both �ci↑

† cj↑� and�ci↑cj↓� show Friedel oscillations.

For the Fermi sea, only �ci↑† cj↑� is finite. The nesting of the

Fermi surface by Q= �� ,�� then leads to the checkerboardpattern for the hole density observed in Fig. 4 �second row�.

For the s-wave, the Friedel oscillations of �ci↑† cj↑� are

similar to that of the Fermi sea while the oscillation of�ci↑cj↓� is phase shifted by � /2. Summing both contributions

to dh�0��m� the oscillations are smeared out. In contrast, for the

d-wave both �ci↑† cj↑� and �ci↑cj↓� oscillate in phase, leading to

the oscillation observed.Let us compare these results with those obtained after

projection. For the Fermi sea, we see clearly that the densityoscillations are suppressed by projection. This is likely be-cause projection reduces the discontinuity at the Fermi level,thereby suppressing the nesting by Q and the correspondingFriedel oscillations.

The emergence of the checkerboard pattern in the pro-jected s-wave suggests that Gutzwiller projection affects�ci↑cj↓� stronger than �ci↑

† cj↑�. With only one contribution, theFriedel oscillations are no longer smeared out and are ob-served.

Projection changes the pattern qualitatively for thed-wave too. The observed pattern resembles approximatelythe function, �sin2�x� /2�sin2�y� /2� �see Fig. 4, top row�,with m= �x ,y�. This indicates that the nodal points at�±� /2 , ±� /2� contribute dominantly after projection. Fur-thermore, in this case, the hole tends to stay near the reser-voir site. It means that only a part of the Hilbert space has alarge weight, leading to a deviation from the Gutzwiller ap-proximation. We believe this effect cannot be capturedwithin the Gutzwiller approximation without invoking off-site correlations.26,27

VI. DISCUSSION

In this paper, we extended the Gutzwiller approximationscheme to construct normalized excitations and matrix ele-ments for the projected Fermi sea. In typical calculations,one needs to determine matrix elements between partiallyprojected Gutzwiller projected states, where double occupan-cies are projected out at all but one site l �called the “reser-voir” site�. The occupancy of the reservoir site, nl turns outto be an important quantity in the calculation of matrix ele-ments. Since the wave function projects out double occupan-cies on all sites m� l, it follows that the occupancy nm�1.But, nl= �0,1 ,2�. Therefore, our results for nl are nontrivialin that the Gutzwiller approximation is extended to calculatethe occupancy at an unprojected site.

We presented an analytical method to calculate such ma-trix elements and showed that the approximations are ingood agreement with results from variational Monte Carlo�VMC� for the Fermi sea. Our results were obtained consid-ering combinatiorial arguments for projected wave functions.As shown by Gebhard, combinatorial arguments become ex-act in the limit of infinite spatial dimensions, for translationalinvariant wave functions.28 We then used our results to con-struct normalized single particle excitations of the fully pro-jected Fermi sea, and to calculate matrix elements for tun-neling into the projected Fermi sea.

Single particle tunneling in projected BCS wave functionshas been discussed recently by Anderson and Ong,9 and Ran-deria et al.22 In our calculations for tunneling into the pro-jected Fermi sea, we find that the matrix elements for bothparticle and hole tunneling vanish as n→1 �half filling�.Within our scheme, the result follows from the behavior of


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the charge density in the vicinity of the reservoir site. Thetunneling matrix elements Mk�

± correspond for k→kF±within Fermi liquid theory to the quasiparticle renormaliza-tion factors Z± for particle and hole excitations, respectively.For the projected Fermi sea, we find Eqs. �24� and �27�, thatZ+Z− for all fillings, reflecting particle hole symmetry.However, the analytical results, Eqs. �24� and �27�, hinge onthe expression for single occupancy of the reservoir site, Eq.�10�. As can be seen in Fig. 2, the analytical result does notagree with numerical calculations done for projected BCSwave functions. This discrepancy underscores the impor-tance of pairing correlations in the unprojected wave func-tions that are not taken into account within the Gutzwillerapproximation scheme.

There are two ways by which electron correlations arisein the Gutzwiller scheme: one is through the mean field ortrial wave function ��0�, and the other via the projection onthe subspace of no double occupancy, ��= P��0�. The lattereffect, which results in the reduction in the size of the Hilbertspace, can be described by combinatorial arguments, leadingto �10�. As seen in Fig. 1, the analytical and VMC results arein good agreement for the case of the projected Fermi sea.We can trace this agreement back to the fact that the Fermisea does not contain any additional explicit correlations.

Consider instead ��BCS�, which contains additional, mo-lecular field correlations in the unprojected wave function.Here, we may expect deviations for quantities like the rela-tive normalization X from the combinatorial result �10�. In-deed, the VMC data presented in Fig. 2 confirms this expec-tation. For instance, the data show a qualitatively differentdependence of X on doping, for the s-wave BCS states. TheVMC data indicates a possibly different limiting behavior forX in the limit n→1, as indicated by the analysis of the data

as a function of inverse cluster-size, presented in Fig. 3.For the s-wave BCS state, we observe a dramatic en-

hancement in the double occupancy at the reservoir site forlow doping, which we understand as a consequence of en-hanced on-site pairing, relative to the Fermi liquid state. Onthe other hand, the double occupancy of the reservoir site isreduced for the d-wave, since the d-wave state suppresseson-site pairing fluctuations. The quantitative behavior of thenormalization ratio X as a function of doping, for projectedsuperconducting states is thus a subtle problem which wehope to solve in the future.

We also studied the hole density near the reservoir site forprojected superconducting wave functions at half filling us-ing VMC. The results are shown in the top row of Fig. 4. Forthe projected Fermi sea, we find that the hole density is uni-form. However, for the superconducting states, we find thatprojection induces oscillations in the hole density near thereservoir site. For the projected d-wave state, we find that thehole density is mostly near the reservoir site. We believe thatthe Gutzwiller approximation needs to be extended to treatpairing correlations in the superconducting wave functions tounderstand these results fully. This issue, along with thestudy of systems away from half filling and their possiblerelevance to the checker-board pattern observed in scanningtunneling microscopy of the high temperature superconduct-ors is left to future research.

ACKNOWLEDGMENTS

We thank P. W. Anderson, N. P. Ong, and H. Yokoyamafor several discussions. N.F. is supported by the DeutscheForschungsgemeinschaft. V.N.M. acknowledges partial fi-nancial support from The City University of New York, PSC-CUNY Research Award Program.

1 M. C. Gutzwiller, Phys. Rev. Lett. 10, 159 �1963�.2 K. Seiler, C. Gros, T. M. Rice, K. Ueda, and D. Vollhardt, J. Low

Temp. Phys. 64, 195 �1986�.3 C. Gros, R. Joynt, and T. M. Rice, Phys. Rev. B 36, 381 �1987�.4 P. W. Anderson, Science 235, 1196 �1987�.5 G. Baskaran, Z. Zou, and P. W. Anderson, Solid State Commun.

63, 973 �1987�.6 Claudius Gros, Phys. Rev. B 38, R931 �1988�.7 F. C. Zhang, C. Gros, T. M. Rice, and H. Shiba, Supercond. Sci.

Technol. 1, 36 �1988�; also available as cond-mat/0311604.8 P. W. Anderson, P. A. Lee, M. Randeria, T. M. Rice, N. Trivedi,

and F. C. Zhang, J. Phys.: Condens. Matter 16, R755 �2004�.9 P. W. Anderson �private communication�; P. W. Anderson and N.

P. Ong, cond-mat/0405518 �unpublished�.10 H. Yokoyama and H. Shiba, J. Phys. Soc. Jpn. 57, 2482 �1988�.11 Bernhard Edegger, Noboru Fukushima, Claudius Gros, and V. N.

Muthukumar, cond-mat/0506120 �unpublished�.12 A. Paramekanti, M. Randeria, and N. Trivedi, Phys. Rev. Lett.

87, 217002 �2001�; M. Randeria, A. Paramekanti, and N.Trivedi, Phys. Rev. B 69, 144509 �2004�; 70, 054504 �2004�.

13 Masao Ogata, J. Phys. Soc. Jpn. 72, 1839 �2003�; also availableas cond-mat/0304405.

14 Q.-H. Wang, D.-H. Lee, and P. A. Lee, Phys. Rev. B 69, 092504�2004�.

15 M. Lavagna, Phys. Rev. B 41, 142 �1990�.16 G. Kotliar and A. E. Ruckenstein, Phys. Rev. Lett. 57, 1362

�1986�.17 E. Arrigoni and G. C. Strinati, Phys. Rev. Lett. 71, 3178 �1993�.18 M. Bak and R. Micnas, J. Phys.: Condens. Matter 10, 9029

�1998�; B. R. Bulka and S. Robaszkiewicz, Phys. Rev. B 54,13138 �1996�.

19 J. Bünemann, W. Weber, and F. Gebhard, Phys. Rev. B 57, 6896�1998�.

20 G. Seibold and J. Lorenzana, Phys. Rev. Lett. 86, 2605 �2001�.21 R. B. Laughlin, cond-mat/0209269 �unpublished�; Bogdan A.

Bernevig, George Chapline, Robert B. Laughlin, Zaira Nazario,and David I. Santiago, cond-mat/0312573 �unpublished�.

22 M. Randeria, R. Sensarma, N. Trivedi, and F.-C. Zhang, cond-mat/0412096 �unpublished�.

23 D. Vollhardt, Rev. Mod. Phys. 56, 99 �1984�.24 The BCS states are defined by �2�, �vk�2=1/2�1−k /Ek�, and

ukvk*=�k / �2Ek�, where k=−2�cos kx+cos ky�−� and

Ek= ��k�2+k2 �s-wave: �k=�; d-wave: �k=��cos kx−cos ky��.

25 C. Gros, Ann. Phys. �N.Y.� 189, 53 �1989�.


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26 An extension of the Gutzwiller approximation, which includesoff-site correlations, is presented by a recent work of M. Ogataand A. Himeda, J. Phys. Soc. Jpn. 72, 374 �2003�.

27 However, we have indications that the hole is not bound to the

reservoir site, since the density correlation seems to decay alge-braically and because they decrease in magnitude with increas-ing lattice size.

28 F. Gebhard, Phys. Rev. B 41, 9452 �1990�.


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Evaluation of matrix elements in partially projected wave functions